{"id": 1, "thm_name": "Binius.BinaryBasefold.fiberwise_dist_lt_imp_dist_lt_unique_decoding_radius", "thm_stmt": "theorem fiberwise_dist_lt_imp_dist_lt_unique_decoding_radius (i : Fin ℓ) (steps : ℕ)\n    [NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)\n    (f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩)\n  (h_fw_dist_lt : fiberwiseClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)\n    (i := i) (steps := steps) (h_i_add_steps := h_i_add_steps) (f := f)) :\n  hammingClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩ f", "lean_root": "ArkLib", "rel_path": "ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean", "imports": ["import ArkLib.Data.MvPolynomial.Multilinear", "import ArkLib.Data.CodingTheory.Basic", "import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.CodingTheory.ReedSolomon", "import ArkLib.Data.Vector.Basic", "import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound", "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Submodule.map", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Module.Basis", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Disjoint", "module": "Mathlib.Order.Disjoint"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "LinearEquiv", "module": "Mathlib.Algebra.Module.Equiv.Defs"}, {"name": "LinearEquiv.ofBijective", "module": "Mathlib.Algebra.Module.Submodule.Equiv"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap.codRestrict", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "LinearMap.ker", "module": "Mathlib.Algebra.Module.Submodule.Ker"}, {"name": "Module.Basis.span", "module": "Mathlib.LinearAlgebra.Basis.Basic"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.subtype", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "context", "module": "Examples.FrankingProtocol"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.degreeLT", "module": "Mathlib.RingTheory.Polynomial.Basic"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}, {"name": "hammingDist", "module": "Mathlib.InformationTheory.Hamming"}, {"name": "Finset.max", "module": "Mathlib.Data.Finset.Max"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Module.finrank", "module": "Mathlib.LinearAlgebra.Dimension.Finrank"}, {"name": "ENat", "module": "Mathlib.Data.ENat.Defs"}, {"name": "ENat.toNat", "module": "Mathlib.Data.ENat.Basic"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Finset.biUnion", "module": "Mathlib.Data.Finset.Union"}, {"name": "Finset.filter", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "Function.onFun", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "SetLike", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "WithTop", "module": "Mathlib.Order.TypeTags"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.equivFunOnFinite", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Fin.cast", "module": "Init.Data.Fin.Basic"}, {"name": "Nat.succ", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "hammingDist", "content": "notation \"Δ₀(\" u \", \" v \")\" => hammingDist u v"}, {"name": "distFromCode", "content": "notation \"Δ₀(\" u \", \" C \")\" => distFromCode u C"}, {"name": "scoped macro_rules", "content": "scoped macro_rules\n  | `(ρ $t:term) => `(LinearCode.rate $t)"}, {"name": "sDomain", "content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n  let W_i_norm := normalizedW 𝔽q β i\n  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n    AdditiveNTT.normalizedW_is_additive 𝔽q β i\n  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)\n    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "polyEvalLinearMap", "content": "noncomputable def polyEvalLinearMap {L 𝔽q : Type*} [Field L] [Field 𝔽q] [Algebra 𝔽q L]\n  (p : L[X]) (hp_add : IsLinearMap 𝔽q (fun x : L => p.eval x)) : L →ₗ[𝔽q] L :=\n{\n  toFun    := fun x => p.eval x,\n  map_add' := hp_add.map_add,\n  map_smul' := hp_add.map_smul\n}"}, {"name": "binaryFinMapToNat", "content": "def binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary : ∀ j: Fin n, m j ≤ 1) : Fin (2^n) :="}, {"name": "sDomain_basis", "content": "noncomputable def sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    Basis (Fin (ℓ + R_rate - i)) 𝔽q (\n      sDomain 𝔽q β h_ℓ_add_R_rate i) :="}, {"name": "sBasis", "content": "def sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=\n  fun k => β ⟨i + k.val, by admit /- proof elided -/\n  ⟩"}, {"name": "iteratedQuotientMap", "content": "noncomputable def iteratedQuotientMap (i : Fin ℓ) (k : ℕ)\n    (h_bound : i.val + k ≤ ℓ) (x : (sDomain 𝔽q β\n    h_ℓ_add_R_rate) ⟨i, by omega⟩) :\n    (sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i.val + k, by omega⟩ :="}, {"name": "intermediateNormVpoly", "content": "noncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)"}, {"name": "qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "code", "content": "def code (deg : ℕ) [Semiring F]: Submodule F (ι → F) :=\n  (Polynomial.degreeLT F deg).map (evalOnPoints domain)"}, {"name": "evalOnPoints", "content": "def evalOnPoints [Semiring F] : F[X] →ₗ[F] (ι → F) where\n  toFun := fun p => fun x => p.eval (domain x)\n  map_add' := fun x y => by admit /- proof elided -/"}, {"name": "distFromCode", "content": "noncomputable def distFromCode (u : n → R) (C : Set (n → R)) : ℕ∞ :=\n  sInf {d | ∃ v ∈ C, hammingDist u v ≤ d}"}, {"name": "toFinset", "content": "def toFinset (domain : ι ↪ F) (deg : ℕ) : Finset (ι → F) :=\n  (RScodeSet domain deg).toFinset"}, {"name": "RScodeSet", "content": "abbrev RScodeSet (domain : ι ↪ F) (deg : ℕ) : Set (ι → F) := (ReedSolomon.code domain deg).carrier"}, {"name": "minDist", "content": "noncomputable def minDist (C : Set (n → R)) : ℕ :=\n  sInf {d | ∃ u ∈ C, ∃ v ∈ C, u ≠ v ∧ hammingDist u v = d}"}, {"name": "rate", "content": "noncomputable def rate [Semiring F] (LC : LinearCode ι F) : ℚ≥0 :=\n  (dim LC : ℚ≥0) / length LC"}, {"name": "dim", "content": "noncomputable def dim [Semiring F] (LC : LinearCode ι F) : ℕ :=\n  Module.finrank F LC"}, {"name": "LinearCode.{u,", "content": "abbrev LinearCode.{u, v} (ι : Type u) [Fintype ι] (F : Type v) [Semiring F] : Type (max u v) :=\n  Submodule F (ι → F)"}, {"name": "length", "content": "def length [Semiring F] (_ : LinearCode ι F) : ℕ := Fintype.card ι"}, {"name": "symm", "content": "def symm (eqv : Equiv pSpec pSpec') : Equiv pSpec' pSpec where\n  round_eq := eqv.round_eq.symm\n  dir_eq := fun i => by admit /- proof elided -/"}, {"name": "Equiv", "content": "@[ext]\nstructure Equiv {m n : ℕ} (pSpec : ProtocolSpec m) (pSpec' : ProtocolSpec n) where\n  round_eq : m = n\n  dir_eq : ∀ i, pSpec.dir i = pSpec'.dir (Fin.cast round_eq i)\n  typeEquiv : ∀ i, pSpec.«Type» i ≃ pSpec'.«Type» (Fin.cast round_eq i)"}, {"name": "ProtocolSpec", "content": "@[ext]\nstructure ProtocolSpec (n : ℕ) where\n   \n  dir : Fin n → Direction\n   \n  «Type» : Fin n → Type\nderiving Inhabited"}, {"name": "Direction", "content": "inductive Direction where\n  | P_to_V  \n  | V_to_P \nderiving DecidableEq, Inhabited, Repr"}, {"name": "qCompositionChain", "content": "noncomputable def qCompositionChain (i : Fin r) : L[X] :=\n  match i with\n  | ⟨0, _⟩ => X\n  | ⟨k + 1, h_k_add_1⟩ => (qMap 𝔽q β ⟨k, by admit /- proof elided -/\n  ⟩).comp (qCompositionChain ⟨k, by admit /- proof elided -/\n  ⟩)"}, {"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}, {"name": "trans", "content": "def trans (eqv : Equiv pSpec pSpec') (eqv' : Equiv pSpec' pSpec'') : Equiv pSpec pSpec'' where\n  round_eq := eqv.round_eq.trans eqv'.round_eq\n  dir_eq := fun i => by admit /- proof elided -/"}, {"name": "minDist", "content": "notation \"Δ\" IC => minDist IC"}, {"name": "distFromCode", "content": "notation \"Δ₀(\" u \", \" C \")\" => distFromCode u C"}], "lib_lemmas": [{"name": "Fin.is_le", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.lt_of_add_right_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.lt_of_le_of_lt", "module": "Init.Prelude"}, {"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Module.Basis.repr_linearCombination", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Module.Basis.repr_symm_apply", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Nat.add_zero", "module": "Init.Core"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}, {"name": "tsub_zero", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Decidable.not_not", "module": "Init.PropLemmas"}, {"name": "Fin.eq_of_val_eq", "module": "Init.Prelude"}, {"name": "Fin.is_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fintype.card_fin", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Fintype.card_setUniv", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Set.card_image_of_injective", "module": "Mathlib.Data.Set.Finite.Basic"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "left_eq_ite_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "right_eq_ite_iff", "module": "Init.PropLemmas"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_lt_iff_neg_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "add_tsub_cancel_right", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Nat.le_of_not_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_lt_sub_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_sub", "module": "Init.Data.Nat.Basic"}, {"name": "dite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "Finset.disjoint_iff_inter_eq_empty", "module": "Mathlib.Data.Finset.Lattice.Lemmas"}, {"name": "Finset.mem_of_mem_inter_left", "module": "Mathlib.Data.Finset.Lattice.Basic"}, {"name": "Finset.mem_of_mem_inter_right", "module": "Mathlib.Data.Finset.Lattice.Basic"}, {"name": "Finset.nonempty_of_ne_empty", "module": "Mathlib.Data.Finset.Empty"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_toFinset", "module": "Mathlib.Data.Fintype.Sets"}, {"name": "ENat.coe_le_coe", "module": "Mathlib.Data.ENat.Basic"}, {"name": "ENat.coe_lt_coe", "module": "Mathlib.Data.ENat.Basic"}, {"name": "ENat.coe_mul", "module": "Mathlib.Data.ENat.Basic"}, {"name": "ENat.coe_toNat", "module": "Mathlib.Data.ENat.Basic"}, {"name": "Finset.card_biUnion", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.card_le_card", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.coe_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.mem_biUnion", "module": "Mathlib.Data.Finset.Union"}, {"name": "Finset.mem_filter", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Finset.mem_image", "module": "Mathlib.Data.Finset.Image"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_const", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Fintype.card_ofFinset", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.card_coe_set_eq", "module": "Mathlib.Data.Set.Card"}, {"name": "Nat.card_eq_fintype_card", "module": "Mathlib.SetTheory.Cardinal.Finite"}, {"name": "Nat.le_of_lt_succ", "module": "Init.Prelude"}, {"name": "Nat.lt_add_of_pos_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.lt_of_le_pred", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.pos_of_neZero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.pow_lt_pow_right", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.sInf_mem", "module": "Mathlib.Data.Nat.Lattice"}, {"name": "Nat.sub_mul", "module": "Init.Data.Nat.Basic"}, {"name": "Set.image_nonempty", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_univ", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.nonempty_iff_ne_empty'", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.toFinset_card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Set.toFinset_range", "module": "Mathlib.Data.Fintype.Sets"}, {"name": "Set.toFinset_setOf", "module": "Mathlib.Data.Fintype.Sets"}, {"name": "Set.univ_nonempty", "module": "Mathlib.Data.Set.Basic"}, {"name": "SetLike.mem_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "Submodule.coe_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.nonempty", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Subtype.coe_eta", "module": "Mathlib.Data.Subtype"}, {"name": "WithTop.coe_lt_coe", "module": "Mathlib.Order.WithBot"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "le_of_eq_of_le", "module": "Init.Core"}, {"name": "le_refl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_le_mul_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "nonempty_subtype", "module": "Mathlib.Logic.Nonempty"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_pos", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "sInf_le", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "smul_eq_mul", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "lt_add_of_pos_right_of_le", "content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"}, {"name": "eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "getBit_of_lt_two_pow", "content": "lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :\n  getBit k a = if k < n then getBit k a else 0"}, {"name": "getBit_eq_testBit", "content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"}, {"name": "getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "getBit_eq_zero_or_one", "content": "lemma getBit_eq_zero_or_one {k n : Nat} :\n  getBit k n = 0 ∨ getBit k n = 1"}, {"name": "getSDomainBasisCoeff_of_iteratedQuotientMap", "content": "omit [DecidableEq 𝔽q] hF₂ in\nlemma getSDomainBasisCoeff_of_iteratedQuotientMap\n    [NeZero R_rate] (i : Fin ℓ) (k : ℕ)\n    (h_bound : i.val + k ≤ ℓ) (x : (sDomain 𝔽q β\n    h_ℓ_add_R_rate) ⟨i, by omega⟩) :\n    let y"}, {"name": "base_intermediateNormVpoly", "content": "theorem base_intermediateNormVpoly\n  (k : Fin (ℓ + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨0, by\n    by_contra ht\n    simp only [not_lt, nonpos_iff_eq_zero] at ht\n    contradiction\n  ⟩ ⟨k, by simp only [tsub_zero]; omega⟩ =\n  normalizedW 𝔽q β ⟨k, by omega⟩"}, {"name": "normalizedW_eq_qMap_composition", "content": "lemma normalizedW_eq_qMap_composition (ℓ R_rate : ℕ) (i : Fin r) :\n    normalizedW 𝔽q β i = qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i"}, {"name": "qMap_comp_normalizedW", "content": "lemma qMap_comp_normalizedW (i : Fin r) (h_i_add_1 : i + 1 < r) :\n  (qMap 𝔽q β i).comp (normalizedW 𝔽q β i) = normalizedW 𝔽q β (i + 1)"}, {"name": "qCompositionChain_eq_foldl", "content": "lemma qCompositionChain_eq_foldl (i : Fin r) :\n  qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i =\n  Fin.foldl (n:=i) (fun acc j =>\n    (qMap 𝔽q β ⟨j, by omega⟩).comp acc) (X)"}, {"name": "getSDomainBasisCoeff_of_sum_repr", "content": "omit [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 in\nlemma getSDomainBasisCoeff_of_sum_repr [NeZero R_rate] (i : Fin (ℓ + 1))\n    (x : (sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i, by omega⟩)\n    (x_coeffs : Fin (ℓ + R_rate - i) → 𝔽q)\n    (hx : x = ∑ j_x, (x_coeffs j_x) • (sDomain_basis 𝔽q β\n      h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (h_i := by\n        simp only; apply Nat.lt_add_of_pos_right_of_le; omega) j_x).val) :\n    ∀ (j: Fin (ℓ + R_rate - i)), ((sDomain_basis 𝔽q β\n      h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (h_i := by\n        simp only; apply Nat.lt_add_of_pos_right_of_le; omega)).repr x) j = x_coeffs j"}, {"name": "get_sDomain_basis", "content": "omit [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 in\nlemma get_sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    ∀ (k : Fin (ℓ + R_rate - i)),\n    (sDomain_basis 𝔽q β h_ℓ_add_R_rate\n      i (by omega)) k = eval (β ⟨i + k.val, by omega⟩) (normalizedW 𝔽q β i)"}, {"name": "intermediateNormVpoly_comp", "content": "omit [DecidableEq L] [DecidableEq 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in\ntheorem intermediateNormVpoly_comp (i : Fin ℓ) (k : Fin (ℓ - i + 1))\n  (l : Fin (ℓ - (i.val + k.val) + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k + l, by\n      simp only; omega⟩) =\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i+k, by omega⟩) (k:=⟨l, by\n      simp only; omega⟩)).comp (\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k, by\n      simp only; omega⟩)\n  )"}, {"name": "intermediateNormVpoly_eval_is_linear_map", "content": "omit [DecidableEq L] [DecidableEq 𝔽q] hF₂ hβ_lin_indep h_β₀_eq_1 in\nlemma intermediateNormVpoly_eval_is_linear_map (i : Fin (ℓ + 1)) (k : Fin (ℓ - i + 1)) :\n  IsLinearMap 𝔽q (fun x : L =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i k).eval x)"}, {"name": "qMap_is_linear_map", "content": "theorem qMap_is_linear_map (i : Fin r) :\n  IsLinearMap 𝔽q (f:=fun inner_p ↦ (qMap 𝔽q β i).comp inner_p)"}, {"name": "𝔽q_element_eq_zero_or_eq_one", "content": "omit h_Fq_char_prime in\nlemma 𝔽q_element_eq_zero_or_eq_one : ∀ c: 𝔽q, c = 0 ∨ c = 1"}, {"name": "getBit_of_binaryFinMapToNat", "content": "lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :\n    ∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val\n      = if h_k: k < n then m ⟨k, by omega⟩ else 0"}, {"name": "and_two_pow_eq_zero_of_getBit_0", "content": "lemma and_two_pow_eq_zero_of_getBit_0 {n i : ℕ} (h_getBit : getBit i n = 0)\n    : n &&& (2 ^ i) = 0"}, {"name": "and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "getBit_of_xor", "content": "lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m"}, {"name": "getBit_zero_eq_zero", "content": "lemma getBit_zero_eq_zero {k : Nat} : getBit k 0 = 0"}, {"name": "sum_of_and_eq_zero_is_xor", "content": "lemma sum_of_and_eq_zero_is_xor {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ^^^ m"}, {"name": "sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||"}, {"name": "xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}, {"name": "distFromCode_eq_top_iff_empty", "content": "theorem distFromCode_eq_top_iff_empty (u : n → R) (C : Set (n → R)) : Δ₀(u, C) = ⊤ ↔ C = ∅"}, {"name": "sub_add_eq_sub_sub_rev", "content": "theorem sub_add_eq_sub_sub_rev (a b c : Nat) (h1 : c ≤ b) (h2 : b ≤ a) :\n  a - b + c = a - (b - c)"}], "used_local_defs": [{"name": "Binius.BinaryBasefold.OracleFunction", "content": "abbrev OracleFunction (i : Fin (ℓ + 1)) : Type _ := sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ → L"}, {"name": "Binius.BinaryBasefold.fiber_coeff", "content": "noncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber", "content": "noncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/"}, {"name": "Binius.BinaryBasefold.pointToIterateQuotientIndex", "content": "def pointToIterateQuotientIndex (i : Fin (ℓ + 1)) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩)) : Fin (2 ^ steps) :="}, {"name": "Binius.BinaryBasefold.BBF_Code", "content": "def BBF_Code (i : Fin (ℓ + 1)) : Submodule L ((sDomain 𝔽q β h_ℓ_add_R_rate)\n    ⟨i, by admit /- proof elided -/\n        ⟩ → L) :=\n  let domain : (sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i, by admit /- proof elided -/\n  ⟩ ↪ L :=\n    ⟨fun x => x.val, fun x y h => by admit /- proof elided -/\n    ⟩\n  ReedSolomon.code (domain := domain) (deg := 2^(ℓ - i.val))"}, {"name": "Binius.BinaryBasefold.BBF_CodeDistance", "content": "def BBF_CodeDistance (ℓ 𝓡 : ℕ) (i : Fin (ℓ + 1)) : ℕ :=\n  2^(ℓ + 𝓡 - i.val) - 2^(ℓ - i.val) + 1"}, {"name": "Binius.BinaryBasefold.fiberwiseDisagreementSet", "content": "def fiberwiseDisagreementSet (i : Fin ℓ) (steps : ℕ) [NeZero steps]\n    (h_i_add_steps : i.val + steps ≤ ℓ) (f g : OracleFunction 𝔽q β (h_ℓ_add_R_rate :=\n      h_ℓ_add_R_rate) ⟨i, by admit /- proof elided -/\n      ⟩) :\n  Set ((sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i.val + steps, by admit /- proof elided -/\n  ⟩) :=\n  \n    \n  {y | ∃ x, iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate (i := i)\n    (k := steps) (h_bound := by admit /- proof elided -/\n    ) x = y ∧ f x ≠ g x}"}, {"name": "Binius.BinaryBasefold.fiberwiseDistance", "content": "def fiberwiseDistance (i : Fin ℓ) (steps : ℕ) [NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)\n  (f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i.val, by admit /- proof elided -/\n  ⟩) : ℕ :=\n  \n  \n  let C_i := BBF_Code 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i.val, by admit /- proof elided -/\n  ⟩\n  let disagreement_sizes := (fun (g : C_i) =>\n    (fiberwiseDisagreementSet 𝔽q β i steps h_i_add_steps f g).ncard) '' Set.univ\n  sInf disagreement_sizes"}, {"name": "Binius.BinaryBasefold.fiberwiseClose", "content": "def fiberwiseClose (i : Fin ℓ) (steps : ℕ) [NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)\n    (f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)\n      ⟨i, by admit /- proof elided -/\n      ⟩) : Prop :=\n  2 * fiberwiseDistance 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) (i := i) steps\n    (h_i_add_steps := h_i_add_steps) (f := f) < (BBF_CodeDistance ℓ 𝓡 ⟨i + steps, by admit /- proof elided -/\n    ⟩ : ℕ∞)"}, {"name": "Binius.BinaryBasefold.hammingClose", "content": "def hammingClose (i : Fin (ℓ + 1)) (f : OracleFunction 𝔽q β\n    (h_ℓ_add_R_rate := h_ℓ_add_R_rate) i) : Prop :=\n  2 * Code.distFromCode (u := f)\n    (C := BBF_Code 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) i) <\n    (BBF_CodeDistance ℓ 𝓡 i : ℕ∞)"}], "used_local_lemmas": [{"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R", "content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n    : i.val + steps < ℓ + 𝓡"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff", "content": "lemma qMap_total_fiber_repr_coeff (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩))\n    (k : Fin (2 ^ steps)) :\n    let x := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y) k\n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)\n      (h_i := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps)\n    let y_coeffs := basis_y.repr y\n    ∀ j, -- j refers to bit index of the fiber point x\n      ((sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (by simp only; omega)).repr x) j\n      = fiber_coeff (i := i) (steps := steps) (j := j) (elementIdx := k)\n        (y_coeffs := y_coeffs)"}, {"name": "Binius.BinaryBasefold.generates_quotient_point_if_is_fiber_of_y", "content": "theorem generates_quotient_point_if_is_fiber_of_y\n    (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩))\n    (hx_is_fiber : ∃ (k : Fin (2 ^ steps)), x = qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := steps) (h_i_add_steps := by\n        simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y) k) :\n    y = iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i (k := steps) (h_bound := h_i_add_steps) x"}, {"name": "Binius.BinaryBasefold.is_fiber_iff_generates_quotient_point", "content": "theorem is_fiber_iff_generates_quotient_point (i : Fin ℓ) (steps : ℕ)\n    (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :\n    let qMapFiber := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y)\n    let k := pointToIterateQuotientIndex (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := h_i_add_steps) (x := x)\n    y = iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i (k := steps) (h_bound := h_i_add_steps) x ↔\n    qMapFiber k = x"}, {"name": "Binius.BinaryBasefold.card_qMap_total_fiber", "content": "omit [CharP L 2] [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 [NeZero ℓ] in\ntheorem card_qMap_total_fiber (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :\n    Fintype.card (Set.image (qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps)\n      (y := y)) Set.univ) = 2 ^ steps"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber_disjoint", "content": "theorem qMap_total_fiber_disjoint\n  (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i + steps ≤ ℓ)\n  {y₁ y₂ : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i.val + steps, by omega⟩}\n  (hy_ne : y₁ ≠ y₂) :\n  Disjoint\n    ((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) y₁ '' Set.univ).toFinset)\n    ((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) y₂ '' Set.univ).toFinset)"}], "local_ctx": "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch\n\nimport ArkLib.Data.CodingTheory.ReedSolomon\n\nimport ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT\n\nimport ArkLib.Data.MvPolynomial.Multilinear\n\nimport ArkLib.Data.Vector.Basic\n\nimport ArkLib.ProofSystem.Sumcheck.Spec.SingleRound\n\nnamespace Binius.BinaryBasefold\n\nopen OracleSpec OracleComp ProtocolSpec Finset AdditiveNTT Polynomial MvPolynomial\n  Binius.BinaryBasefold\n\nopen scoped NNReal\n\nopen ReedSolomon Code BerlekampWelch\n\nopen Finset AdditiveNTT Polynomial MvPolynomial Nat Matrix\n\nsection Preliminaries\n\nvariable {L : Type} [CommRing L] (ℓ : ℕ) [NeZero ℓ]\n\nvariable (𝓑 : Fin 2 ↪ L)\n\nend Preliminaries\n\nnoncomputable section       -- expands with 𝔽q in front\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type} [Field L] [Fintype L] [DecidableEq L] [CharP L 2]\n\nvariable (𝔽q : Type) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ 𝓡 ϑ : ℕ} (γ_repetitions : ℕ) [NeZero ℓ] [NeZero 𝓡] [NeZero ϑ] -- Should we allow ℓ = 0?\n\nvariable {h_ℓ_add_R_rate : ℓ + 𝓡 < r} -- ℓ ∈ {1, ..., r-1}\n\nvariable {𝓑 : Fin 2 ↪ L}\n\nsection Essentials\n\nabbrev OracleFunction (i : Fin (ℓ + 1)) : Type _ := sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ → L\n\nnoncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩\n\nnoncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/\n\ndef pointToIterateQuotientIndex (i : Fin (ℓ + 1)) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩)) : Fin (2 ^ steps) :=\n\nend Essentials\n\nsection SoundnessTools\n\ndef BBF_Code (i : Fin (ℓ + 1)) : Submodule L ((sDomain 𝔽q β h_ℓ_add_R_rate)\n    ⟨i, by admit /- proof elided -/\n        ⟩ → L) :=\n  let domain : (sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i, by admit /- proof elided -/\n  ⟩ ↪ L :=\n    ⟨fun x => x.val, fun x y h => by admit /- proof elided -/\n    ⟩\n  ReedSolomon.code (domain := domain) (deg := 2^(ℓ - i.val))\n\ndef BBF_CodeDistance (ℓ 𝓡 : ℕ) (i : Fin (ℓ + 1)) : ℕ :=\n  2^(ℓ + 𝓡 - i.val) - 2^(ℓ - i.val) + 1\n\ndef fiberwiseDisagreementSet (i : Fin ℓ) (steps : ℕ) [NeZero steps]\n    (h_i_add_steps : i.val + steps ≤ ℓ) (f g : OracleFunction 𝔽q β (h_ℓ_add_R_rate :=\n      h_ℓ_add_R_rate) ⟨i, by admit /- proof elided -/\n      ⟩) :\n  Set ((sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i.val + steps, by admit /- proof elided -/\n  ⟩) :=\n  \n    \n  {y | ∃ x, iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate (i := i)\n    (k := steps) (h_bound := by admit /- proof elided -/\n    ) x = y ∧ f x ≠ g x}\n\ndef fiberwiseDistance (i : Fin ℓ) (steps : ℕ) [NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)\n  (f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i.val, by admit /- proof elided -/\n  ⟩) : ℕ :=\n  \n  \n  let C_i := BBF_Code 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i.val, by admit /- proof elided -/\n  ⟩\n  let disagreement_sizes := (fun (g : C_i) =>\n    (fiberwiseDisagreementSet 𝔽q β i steps h_i_add_steps f g).ncard) '' Set.univ\n  sInf disagreement_sizes\n\ndef fiberwiseClose (i : Fin ℓ) (steps : ℕ) [NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)\n    (f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)\n      ⟨i, by admit /- proof elided -/\n      ⟩) : Prop :=\n  2 * fiberwiseDistance 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) (i := i) steps\n    (h_i_add_steps := h_i_add_steps) (f := f) < (BBF_CodeDistance ℓ 𝓡 ⟨i + steps, by admit /- proof elided -/\n    ⟩ : ℕ∞)\n\ndef hammingClose (i : Fin (ℓ + 1)) (f : OracleFunction 𝔽q β\n    (h_ℓ_add_R_rate := h_ℓ_add_R_rate) i) : Prop :=\n  2 * Code.distFromCode (u := f)\n    (C := BBF_Code 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) i) <\n    (BBF_CodeDistance ℓ 𝓡 i : ℕ∞)", "target_theorem": "theorem fiberwise_dist_lt_imp_dist_lt_unique_decoding_radius (i : Fin ℓ) (steps : ℕ)\n    [NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)\n    (f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩)\n  (h_fw_dist_lt : fiberwiseClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)\n    (i := i) (steps := steps) (h_i_add_steps := h_i_add_steps) (f := f)) :\n  hammingClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩ f :=", "ground_truth_proof": ":= by\n  unfold fiberwiseClose at h_fw_dist_lt\n  unfold hammingClose\n  -- 2 * Δ₀(f, ↑(BBF_Code 𝔽q β ⟨↑i, ⋯⟩)) < ↑(BBF_CodeDistance ℓ 𝓡 ⟨↑i, ⋯⟩)\n  let d_fw := fiberwiseDistance 𝔽q β (i := i) steps h_i_add_steps f\n  let C_i := (BBF_Code 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩)\n  let d_H := Code.distFromCode f C_i\n  let d_i := BBF_CodeDistance ℓ 𝓡 (⟨i, by omega⟩)\n  let d_i_plus_steps := BBF_CodeDistance ℓ 𝓡 ⟨i.val + steps, by omega⟩\n\n  have h_d_i_gt_0 : d_i > 0 := by\n    dsimp [d_i, BBF_CodeDistance] -- ⊢ 2 ^ (ℓ + 𝓡 - ↑i) - 2 ^ (ℓ - ↑i) + 1 > 0\n    have h_exp_lt : ℓ - i.val < ℓ + 𝓡 - i.val := by\n      exact Nat.sub_lt_sub_right (a := ℓ) (b := ℓ + 𝓡) (c := i.val) (by omega) (by\n        apply Nat.lt_add_of_pos_right; exact pos_of_neZero 𝓡)\n    have h_pow_lt : 2 ^ (ℓ - i.val) < 2 ^ (ℓ + 𝓡 - i.val) := by\n      exact Nat.pow_lt_pow_right (by norm_num) h_exp_lt\n    omega\n\n  have h_C_i_nonempty : Nonempty C_i := by\n    simp only [nonempty_subtype, C_i]\n    exact Submodule.nonempty (BBF_Code 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i.val, by omega⟩)\n\n  -- 1. Relate Hamming distance `d_H` to fiber-wise distance `d_fw`.\n  obtain ⟨g', h_g'_mem, h_g'_min_card⟩ : ∃ g' ∈ C_i, d_fw\n    = (fiberwiseDisagreementSet 𝔽q β i steps h_i_add_steps f g').ncard := by\n    -- Let `S` be the set of all possible fiber-wise disagreement sizes.\n    let S := (fun (g : C_i) => (fiberwiseDisagreementSet 𝔽q β i steps h_i_add_steps\n      f g).ncard) '' Set.univ\n    -- The code `C_i` (a submodule) is non-empty, so `S` is also non-empty.\n    have hS_nonempty : S.Nonempty := by\n      refine Set.image_nonempty.mpr ?_\n\n      exact Set.univ_nonempty\n    -- For a non-empty set of natural numbers, `sInf` is an element of the set.\n    have h_sInf_mem : sInf S ∈ S := Nat.sInf_mem hS_nonempty\n    -- By definition, `d_fw = sInf S`.\n    unfold d_fw at h_sInf_mem\n    -- Since `sInf S` is in the image set `S`, there must be an element `g_subtype` in the domain\n    -- (`C_i`) that maps to it. This `g_subtype` is the codeword we're looking for.\n    rw [Set.mem_image] at h_sInf_mem\n    rcases h_sInf_mem with ⟨g_subtype, _, h_eq⟩\n    -- Extract the codeword and its membership proof.\n    exact ⟨g_subtype.val, g_subtype.property, by exact id (Eq.symm h_eq)⟩\n\n  -- The Hamming distance to any codeword `g'` is bounded by `d_fw * 2 ^ steps`.\n  have h_dist_le_fw_dist_times_fiber_size : (hammingDist f g' : ℕ∞) ≤ d_fw * 2 ^ steps := by\n    -- This proves `dist f g' ≤ (fiberwiseDisagreementSet ... f g').ncard * 2 ^ steps`\n    -- and lifts to ℕ∞. We prove the `Nat` version `hammingDist f g' ≤ ...`,\n    -- which is equivalent.\n    change (Δ₀(f, g') : ℕ∞) ≤ ↑d_fw * ((2 ^ steps : ℕ) : ℕ∞)\n    rw [←ENat.coe_mul, ENat.coe_le_coe, h_g'_min_card]\n    -- Let ΔH be the finset of actually bad x points where f and g' disagree.\n    set ΔH := Finset.filter (fun x => f x ≠ g' x) Finset.univ\n    have h_dist_eq_card : hammingDist f g' = ΔH.card := by\n      simp only [hammingDist, ne_eq, ΔH]\n    rw [h_dist_eq_card]\n    -- Y_bad is the set of quotient points y that THERE EXISTS a bad fiber point x\n    set Y_bad := fiberwiseDisagreementSet 𝔽q β i steps h_i_add_steps f g'\n    simp only at * -- simplify domain indices everywhere\n\n    -- ⊢ #ΔH ≤ Y_bad.ncard * 2 ^ steps\n\n    have hFinType_Y_bad : Fintype Y_bad := by exact Fintype.ofFinite ↑Y_bad\n    -- Every point of disagreement `x` must belong to a fiber over some `y` in `Y_bad`,\n    -- BY DEFINITION of `Y_bad`. Therefore, `ΔH` is a subset of the union of the fibers\n    -- of `Y_bad`\n    have h_ΔH_subset_bad_fiber_points : ΔH ⊆ Finset.biUnion Y_bad.toFinset\n        (t := fun y => ((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n          (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y)) ''\n          (Finset.univ : Finset (Fin ((2:ℕ)^steps)))).toFinset) := by\n      -- ⊢ If any x ∈ ΔH, then x ∈ Union(qMap_total_fiber(y), ∀ y ∈ Y_bad)\n      intro x hx_in_ΔH; -- ⊢ x ∈ Union(qMap_total_fiber(y), ∀ y ∈ Y_bad)\n      simp only [ΔH, Finset.mem_filter] at hx_in_ΔH\n      -- Now we actually apply iterated qMap into x to get y_of_x,\n      -- then x ∈ qMap_total_fiber(y_of_x) by definition\n      let y_of_x := iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i steps h_i_add_steps x\n      apply Finset.mem_biUnion.mpr; use y_of_x\n      -- ⊢ y_of_x ∈ Y_bad.toFinset ∧ x ∈ qMap_total_fiber(y_of_x)\n      have h_elemenet_Y_bad :  y_of_x ∈ Y_bad.toFinset := by\n        -- ⊢ y ∈ Y_bad.toFinset\n        simp only [fiberwiseDisagreementSet, iteratedQuotientMap, ne_eq, Subtype.exists,\n          Set.toFinset_setOf, mem_filter, mem_univ, true_and, Y_bad]\n        -- one bad fiber point of y_of_x is x itself\n        let X := x.val\n        have h_X_in_source : X ∈ sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) := by\n          exact Submodule.coe_mem x\n        use X\n        use h_X_in_source\n        -- ⊢ Ŵ_steps⁽ⁱ⁾(X) = y (iterated quotient map) ∧ ¬f ⟨X, ⋯⟩ = g' ⟨X, ⋯⟩\n        have h_forward_iterated_qmap : Polynomial.eval X\n            (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩\n              ⟨steps, by simp only; omega⟩) = y_of_x := by\n          simp only [iteratedQuotientMap, X, y_of_x];\n        have h_eval_diff : f ⟨X, by omega⟩ ≠ g' ⟨X, by omega⟩ := by\n          unfold X\n          simp only [Subtype.coe_eta, ne_eq, hx_in_ΔH, not_false_eq_true]\n        simp only [h_forward_iterated_qmap, Subtype.coe_eta, h_eval_diff,\n          not_false_eq_true, and_self]\n      simp only [h_elemenet_Y_bad, true_and]\n\n      set qMapFiber := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n        (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y_of_x)\n      simp only [coe_univ, Set.image_univ, Set.toFinset_range, mem_image, mem_univ, true_and]\n      use (pointToIterateQuotientIndex (i := ⟨i, by omega⟩) (steps := steps)\n        (h_i_add_steps := by omega) (x := x))\n      have h_res := is_fiber_iff_generates_quotient_point 𝔽q β i steps (by omega)\n        (x := x) (y := y_of_x).mp (by rfl)\n      exact h_res\n    -- ⊢ #ΔH ≤ Y_bad.ncard * 2 ^ steps\n    -- The cardinality of a subset is at most the cardinality of the superset.\n    apply (Finset.card_le_card h_ΔH_subset_bad_fiber_points).trans\n    -- The cardinality of a disjoint union is the sum of cardinalities.\n    rw [Finset.card_biUnion]\n    · -- The size of the sum is the number of bad fibers (`Y_bad.ncard`) times\n      -- the size of each fiber (`2 ^ steps`).\n      simp only [Set.toFinset_card]\n      have h_card_fiber_per_quotient_point := card_qMap_total_fiber 𝔽q β\n        (h_ℓ_add_R_rate := h_ℓ_add_R_rate) i steps h_i_add_steps\n      simp only [Set.image_univ, Fintype.card_ofFinset,\n        Subtype.forall] at h_card_fiber_per_quotient_point\n      have h_card_fiber_of_each_y : ∀ y ∈ Y_bad.toFinset,\n          Fintype.card ((qMap_total_fiber 𝔽q β (i := ⟨↑i, by omega⟩) (steps := steps)\n            (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y)) ''\n            ↑(Finset.univ : Finset (Fin ((2:ℕ)^steps)))) = 2 ^ steps := by\n        intro y hy_in_Y_bad\n        have hy_card_fiber_of_y := h_card_fiber_per_quotient_point (a := y) (b := by\n          exact Submodule.coe_mem y)\n        simp only [coe_univ, Set.image_univ, Fintype.card_ofFinset, hy_card_fiber_of_y]\n      rw [Finset.sum_congr rfl h_card_fiber_of_each_y]\n      -- ⊢ ∑ x ∈ Y_bad.toFinset, 2 ^ steps ≤ Y_bad.encard.toNat * 2 ^ steps\n      simp only [sum_const, Set.toFinset_card, smul_eq_mul, ofNat_pos, pow_pos,\n        _root_.mul_le_mul_right, ge_iff_le]\n      conv_rhs => rw [←_root_.Nat.card_coe_set_eq] -- convert .ncard back to .card\n      -- ⊢ Fintype.card ↑Y_bad ≤ Nat.card ↑Y_bad\n      simp only [card_eq_fintype_card, le_refl]\n    · -- Prove that the fibers for distinct quotient points y₁, y₂ are disjoint.\n      intro y₁ hy₁ y₂ hy₂ hy_ne\n      have h_disjoint := qMap_total_fiber_disjoint (i := ⟨↑i, by omega⟩) (steps := steps)\n        (h_i_add_steps := by omega) (y₁ := y₁) (y₂ := y₂) (hy_ne := hy_ne)\n      simp only [Function.onFun, coe_univ]\n      exact h_disjoint\n\n  -- The minimum distance `d_H` is bounded by the distance to this specific `g'`.\n  have h_dist_bridge : d_H ≤ d_fw * 2 ^ steps := by\n    -- exact h_dist_le_fw_dist_times_fiber_size\n    apply le_trans (a := d_H) (c := d_fw * 2 ^ steps) (b := hammingDist f g')\n    · -- ⊢ d_H ≤ ↑Δ₀(f, g')\n      simp only [distFromCode, SetLike.mem_coe, hammingDist, ne_eq, d_H];\n      -- ⊢ Δ₀(f, C_i) ≤ ↑Δ₀(f, g')\n      -- ⊢ sInf {d | ∃ v ∈ C_i, ↑(#{i | f i ≠ v i}) ≤ d} ≤ ↑(#{i | f i ≠ g' i})\n      apply sInf_le\n      use g'\n    · exact h_dist_le_fw_dist_times_fiber_size\n\n  -- 2. Use the premise : `2 * d_fw < d_{i+steps}`.\n  -- As a `Nat` inequality, this is equivalent to `2 * d_fw ≤ d_{i+steps} - 1`.\n  have h_fw_bound : 2 * d_fw ≤ d_i_plus_steps - 1 := by\n    -- Convert the ENat inequality to a Nat inequality using `a < b ↔ a + 1 ≤ b`.\n    exact Nat.le_of_lt_succ (WithTop.coe_lt_coe.1 h_fw_dist_lt)\n\n  -- 3. The Algebraic Identity.\n  -- The core of the proof is the identity : `(d_{i+steps} - 1) * 2 ^ steps = d_i - 1`.\n  have h_algebraic_identity : (d_i_plus_steps - 1) * 2 ^ steps = d_i - 1 := by\n    dsimp [d_i, d_i_plus_steps, BBF_CodeDistance]\n    rw [Nat.sub_mul, ←Nat.pow_add, ←Nat.pow_add];\n    have h1 : ℓ + 𝓡 - (↑i + steps) + steps = ℓ + 𝓡 - i := by\n      rw [Nat.sub_add_eq_sub_sub_rev (h1 := by omega) (h2 := by omega),\n        Nat.add_sub_cancel (n := i) (m := steps)]\n    have h2 : (ℓ - (↑i + steps) + steps) = ℓ - i := by\n      rw [Nat.sub_add_eq_sub_sub_rev (h1 := by omega) (h2 := by omega),\n        Nat.add_sub_cancel (n := i) (m := steps)]\n    rw [h1, h2]\n\n  -- 4. Conclusion : Chain the inequalities to prove `2 * d_H < d_i`.\n  -- We know `d_H` is finite, since `C_i` is nonempty.\n  have h_dH_ne_top : d_H ≠ ⊤ := by\n    simp only [ne_eq, d_H]\n    rw [Code.distFromCode_eq_top_iff_empty f C_i]\n    exact Set.nonempty_iff_ne_empty'.mp h_C_i_nonempty\n\n  -- We can now work with the `Nat` value of `d_H`.\n  let d_H_nat := ENat.toNat d_H\n  have h_dH_eq : d_H = d_H_nat := (ENat.coe_toNat h_dH_ne_top).symm\n\n  -- The calculation is now done entirely in `Nat`.\n  have h_final_inequality : 2 * d_H_nat ≤ d_i - 1 := by\n    have h_bridge_nat : d_H_nat ≤ d_fw * 2 ^ steps := by\n        rw [←ENat.coe_le_coe]\n        exact le_of_eq_of_le (id (Eq.symm h_dH_eq)) h_dist_bridge\n    calc 2 * d_H_nat\n      _ ≤ 2 * (d_fw * 2 ^ steps) := by gcongr\n      _ = (2 * d_fw) * 2 ^ steps := by rw [mul_assoc]\n      _ ≤ (d_i_plus_steps - 1) * 2 ^ steps := by gcongr;\n      _ = d_i - 1 := h_algebraic_identity\n\n  simp only [d_H, d_H_nat] at h_dH_eq\n  -- This final line is equivalent to the goal statement.\n  rw [h_dH_eq]\n  -- ⊢ 2 * ↑Δ₀(f, C_i).toNat < ↑(BBF_CodeDistance ℓ 𝓡 ⟨↑i, ⋯⟩)\n  change ((2 : ℕ) : ℕ∞) * ↑Δ₀(f, C_i).toNat < ↑(BBF_CodeDistance ℓ 𝓡 ⟨↑i, by omega⟩)\n  rw [←ENat.coe_mul, ENat.coe_lt_coe]\n  apply Nat.lt_of_le_pred (n := 2 * Δ₀(f, C_i).toNat) (m := d_i) (h := h_d_i_gt_0)\n    (h_final_inequality)", "nesting_depth": 7, "transitive_dep_count": 232, "subset_aristotle": false, "category": "Applied verif."}
{"id": 2, "thm_name": "ConcreteBinaryTower.minPoly_of_powerBasisSucc_generator", "thm_stmt": "@[simp]\ntheorem minPoly_of_powerBasisSucc_generator (k : ℕ) :\n  (minpoly (ConcreteBTField k) (powerBasisSucc k).gen) = X^2 + (Z k) • X + 1", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "AddCommGroup", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.negSucc", "module": "Init.Data.Int.Basic"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "NNRat", "module": "Mathlib.Data.Rat.Init"}, {"name": "NNRat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "Rat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.toAlgebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "invFun", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "left_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "right_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "absurd", "module": "Init.Prelude"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "instFintypeProd", "module": "Mathlib.Data.Fintype.Prod"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "CommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Module.Basis", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Algebra.algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Module.Basis.mk", "module": "Mathlib.LinearAlgebra.Basis.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "PowerBasis", "module": "Mathlib.RingTheory.PowerBasis"}, {"name": "gen", "module": "VCVio.CryptoFoundations.FiatShamir"}, {"name": "minpoly", "module": "Mathlib.FieldTheory.Minpoly.Basic"}, {"name": "IsUnit", "module": "Mathlib.Algebra.Group.Units.Defs"}, {"name": "Polynomial.leadingCoeff", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}, {"name": "Polynomial.aeval", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "MonoidHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "OneHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}, {"name": "AlgebraTower.toIsScalarTower", "content": "@[simp]\ninstance AlgebraTower.toIsScalarTower (a : AlgebraTower C) {i j k : ι}\n    (h1 : i ≤ j) (h2 : j ≤ k) :\n    letI : Algebra (C i) (C j) :="}, {"name": "AlgebraTower.toAlgebra", "content": "@[simp]\ndef AlgebraTower.toAlgebra {i j : ι} (h : i ≤ j) : Algebra (A i) (A j) :=\n  (AlgebraTower.algebraMap (i:=i) (j:=j) (h:=h)).toAlgebra"}], "lib_lemmas": [{"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}, {"name": "BitVec.extractLsb_ofNat", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.zero_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.zero_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.ne_zero_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.one_lt_two_pow_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_eq_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.one_mod_two_pow_eq_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.one_mod_two_pow", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.zero_lt_two", "module": "Init.Data.Nat.Basic"}, {"name": "pow_pos", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "BitVec.zero_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "pow_two", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "BitVec.ofNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.toNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Ne.dite_eq_left_iff", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.left_eq_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingHom.comp_id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "BitVec.xor_assoc", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_self", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Polynomial.C_mul'", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.aeval_def", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.eval₂_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval₂_X_pow", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval₂_add", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval₂_one", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval₂_smul", "module": "Mathlib.Algebra.Polynomial.Eval.SMul"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "MonoidHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "Nat.sub_one_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "OneHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "RingHom.coe_mk", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_cast", "module": "Init.PropLemmas"}, {"name": "eqRec_eq_cast", "module": "Batteries.Logic"}, {"name": "Polynomial.aeval_C", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.aeval_X", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.degree_eq_iff_natDegree_eq", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.degree_ne_bot", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eq_C_of_natDegree_eq_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.eq_X_add_C_of_degree_eq_one", "module": "Mathlib.Algebra.Polynomial.Degree.SmallDegree"}, {"name": "Polynomial.leadingCoeff_eq_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.natDegree_lt_iff_degree_lt", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "map_add", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_eq_zero", "module": "Mathlib.Algebra.GroupWithZero.Units.Lemmas"}, {"name": "map_smul", "module": "Mathlib.GroupTheory.GroupAction.Hom"}, {"name": "map_zero", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "minpoly.unique'", "module": "Mathlib.FieldTheory.Minpoly.Basic"}, {"name": "ne_zero_of_eq_one", "module": "Mathlib.Algebra.NeZero"}, {"name": "not_true_eq_false", "module": "Init.SimpLemmas"}, {"name": "or_false", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}, {"name": "zero_lt_pow_n", "content": "theorem zero_lt_pow_n (m : ℕ) (n : ℕ) (h_m : m > 0): 0 < m^n"}, {"name": "is_unit_iff_deg_0", "content": "theorem is_unit_iff_deg_0 {R : Type*} [Field R] {p : R[X]} : p.degree = 0 ↔ IsUnit p"}, {"name": "definingPoly_is_monic", "content": "lemma definingPoly_is_monic {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (definingPoly s).Monic"}, {"name": "degree_s_smul_X_add_1", "content": "lemma degree_s_smul_X_add_1 {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (C s * (X : (F)[X]) + 1).degree = 1"}, {"name": "degree_s_smul_X", "content": "lemma degree_s_smul_X {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (C s * (X : (F)[X])).degree = 1"}, {"name": "degree_definingPoly", "content": "lemma degree_definingPoly {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (definingPoly s).degree = 2"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.add", "content": "def add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y"}, {"name": "ConcreteBinaryTower.neg", "content": "def neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.ConcreteBTFAddCommGroupProps", "content": "structure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel"}, {"name": "ConcreteBinaryTower.mkAddCommGroupInstance", "content": "def mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}"}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.equivProd", "content": "def equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)"}, {"name": "ConcreteBinaryTower.concrete_mul", "content": "def concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.concrete_inv", "content": "def concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res"}, {"name": "ConcreteBinaryTower.natCast", "content": "def natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.natCast_zero", "content": "def natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :="}, {"name": "ConcreteBinaryTower.natCast_succ", "content": "def natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :="}, {"name": "ConcreteBinaryTower.intCast", "content": "def intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.intCast_ofNat", "content": "def intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :="}, {"name": "ConcreteBinaryTower.intCast_negSucc", "content": "def intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :="}, {"name": "ConcreteBinaryTower.ConcreteBTFRingProps", "content": "structure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c"}, {"name": "ConcreteBinaryTower.ConcreteBTFDivisionRingProps", "content": "structure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldProps", "content": "structure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a"}, {"name": "ConcreteBinaryTower.mkRingInstance", "content": "def mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n"}, {"name": "ConcreteBinaryTower.mkDivisionRingInstance", "content": "def mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)"}, {"name": "ConcreteBinaryTower.mkFieldInstance", "content": "def mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm"}, {"name": "ConcreteBinaryTower.ConcreteBTFStepResult", "content": "structure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))"}, {"name": "ConcreteBinaryTower.liftBTFieldProps", "content": "def liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.liftConcreteBTField", "content": "def liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.concreteCanonicalEmbedding", "content": "def concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :="}, {"name": "ConcreteBinaryTower.instAlgebraLiftConcreteBTField", "content": "instance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))"}, {"name": "ConcreteBinaryTower.getBTFResult", "content": "def getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.canonicalAlgMap", "content": "def canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap", "content": "def concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :="}, {"name": "ConcreteBinaryTower.instAlgebraTowerConcreteBTF", "content": "instance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldAlgebra", "content": "def ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le"}, {"name": "ConcreteBinaryTower.join_via_add_smul", "content": "def join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.basisSucc", "content": "def basisSucc (k : ℕ) : Basis (Fin 2) (ConcreteBTField k) (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.powerBasisSucc", "content": "def powerBasisSucc (k : ℕ) :\n    PowerBasis (ConcreteBTField k) (ConcreteBTField (k + 1)) :="}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq", "content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n  ConcreteBTField k = ConcreteBTField m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq_zero", "content": "theorem BitVec.dcast_bitvec_eq_zero {l r : ℕ} (h_width_eq : l = r) :\n  dcast (h_width_eq) 0#(l) = 0#(r)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.sum_fromNat_eq_from_xor_Nat", "content": "theorem sum_fromNat_eq_from_xor_Nat {k : ℕ} (x y : Nat) :\n  fromNat (k:=k) (x ^^^ y) = fromNat (k:=k) x + fromNat (k:=k) y"}, {"name": "ConcreteBinaryTower.add_self_cancel", "content": "lemma add_self_cancel {k : ℕ} (a : ConcreteBTField k) : a + a = 0"}, {"name": "ConcreteBinaryTower.add_assoc", "content": "lemma add_assoc {k : ℕ} : ∀ (a b c : ConcreteBTField k), a + b + c = a + (b + c)"}, {"name": "ConcreteBinaryTower.zero_add", "content": "lemma zero_add {k : ℕ} (a : ConcreteBTField k) : 0 + a = a"}, {"name": "ConcreteBinaryTower.add_zero", "content": "lemma add_zero {k : ℕ} (a : ConcreteBTField k) : a + 0 = a"}, {"name": "ConcreteBinaryTower.cast_join", "content": "lemma cast_join {k n : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) (heq : k = n) :\n  join (k:=k) h_pos hi lo = cast (by rw [heq])\n    (join (k:=n) (by omega) (cast (by subst heq; rfl) hi) (lo:=cast (by subst heq; rfl) lo))"}, {"name": "ConcreteBinaryTower.zero_is_0", "content": "lemma zero_is_0 {k : ℕ} : (zero (k:=k)) = (0 : ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.one_is_1", "content": "lemma one_is_1 {k : ℕ} : (one (k:=k)) = 1"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_join_iff", "content": "theorem join_eq_join_iff {k : ℕ} (h_pos : k > 0) (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1)) :\n  《 hi₀, lo₀ 》 = 《 hi₁, lo₁ 》 ↔ (hi₀ = hi₁ ∧ lo₀ = lo₁)"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_of_split", "content": "theorem join_of_split {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_split_eq : split h_pos x = (hi_btf, lo_btf)) :\n    x = 《 hi_btf, lo_btf 》"}, {"name": "ConcreteBinaryTower.split_of_join", "content": "theorem split_of_join {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_join : x = 《hi_btf, lo_btf》) :\n    (hi_btf, lo_btf) = split h_pos x"}, {"name": "ConcreteBinaryTower.split_join_eq_split", "content": "lemma split_join_eq_split {k : ℕ} (h_pos : k > 0)\n    (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    split h_pos (《 hi_btf, lo_btf 》) = (hi_btf, lo_btf)"}, {"name": "ConcreteBinaryTower.eq_iff_split_eq", "content": "theorem eq_iff_split_eq {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k) :\n  x₀ = x₁ ↔ (split h_pos x₀ = split h_pos x₁)"}, {"name": "ConcreteBinaryTower.split_sum_eq_sum_split", "content": "theorem split_sum_eq_sum_split {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k)\n  (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1))\n  (h_split_x₀ : split h_pos x₀ = (hi₀, lo₀))\n  (h_split_x₁ : split h_pos x₁ = (hi₁, lo₁)) :\n  split h_pos (x₀ + x₁) = (hi₀ + hi₁, lo₀ + lo₁)"}, {"name": "ConcreteBinaryTower.join_add_join", "content": "theorem join_add_join {k : ℕ} (h_pos : k > 0) (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1)) :\n  《 hi₀, lo₀ 》 + 《 hi₁, lo₁ 》 = 《 hi₀ + hi₁, lo₀ + lo₁ 》"}, {"name": "ConcreteBinaryTower.split_zero", "content": "theorem split_zero {k : ℕ} (h_pos : k > 0) : split h_pos zero = (zero, zero)"}, {"name": "ConcreteBinaryTower.split_Z", "content": "theorem split_Z {k : ℕ} (h_pos : k > 0) :\n    split h_pos (Z k) = (one (k:=k - 1), zero (k:=k - 1))"}, {"name": "ConcreteBinaryTower.one_bitvec_toNat", "content": "lemma one_bitvec_toNat {width : ℕ} (h_width : width > 0) : (1#width).toNat = 1"}, {"name": "ConcreteBinaryTower.one_bitvec_shiftRight", "content": "lemma one_bitvec_shiftRight {d : ℕ} (h_d : d > 0) : 1 >>> d = 0"}, {"name": "ConcreteBinaryTower.split_one", "content": "lemma split_one {k : ℕ} (h_k : k > 0) :\n    split h_k (one (k:=k)) = (zero (k:=k - 1), one (k:=k - 1))"}, {"name": "ConcreteBinaryTower.join_zero_zero", "content": "lemma join_zero_zero {k : ℕ} (h_k : k > 0) :\n  《 zero (k:=k - 1), zero (k:=k - 1) 》 = zero (k:=k)"}, {"name": "ConcreteBinaryTower.join_zero_one", "content": "theorem join_zero_one {k : ℕ} (h_k : k > 0) :\n    《 zero (k:=k - 1), one (k:=k - 1) 》 = one (k:=k)"}, {"name": "ConcreteBinaryTower.Z_square_eq", "content": "lemma Z_square_eq (k : ℕ) (prevBTFieldProps : ConcreteBTFieldProps (k := k))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI : Field (ConcreteBTField (k + 1)) := mkFieldInstance curBTFieldProps\n  (Z (k + 1)) ^ 2 = 《 Z (k), 1 》"}, {"name": "ConcreteBinaryTower.Z_square_mul_form", "content": "lemma Z_square_mul_form\n  (k : ℕ)\n  (prev : ConcreteBTFStepResult (k := k)) :\n  letI : Field (ConcreteBTField k) := mkFieldInstance (prev.toConcreteBTFieldProps)\n  letI : Field (ConcreteBTField (k + 1)) := mkFieldInstance (k:=k+1)\n    (props:=liftBTFieldProps (k:=k) (prevBTFResult:=prev))\n  letI : Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n    instAlgebraLiftConcreteBTField k prev\n  Z (k + 1) ^ 2\n    = Z (k + 1)\n      * (algebraMap (ConcreteBTField k) (ConcreteBTField (k + 1))) (Z k)\n      + 1"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_eq_of_dest_eq", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_eq_of_dest_eq (k m n : ℕ) (h_eq : m = n) :\n    (ConcreteBTField k →+* ConcreteBTField m)\n    = (ConcreteBTField k →+* ConcreteBTField n)"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_cast_dest_apply", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_cast_dest_apply (k m n : ℕ) (h_eq : m = n)\n  (f : ConcreteBTField k →+* ConcreteBTField m) (x : ConcreteBTField k) :\n    (cast (ConcreteBTField.RingHom_eq_of_dest_eq (k:=k) (m:=m) (n:=n) h_eq) f) x\n    = cast (by apply cast_ConcreteBTField_eq (h_eq:=h_eq)) (f x)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_id", "content": "lemma concreteTowerAlgebraMap_id (k : ℕ) :\n    concreteTowerAlgebraMap (h_le:=by omega) = RingHom.id (ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_succ_1", "content": "lemma concreteTowerAlgebraMap_succ_1 (k : ℕ) :\n  concreteTowerAlgebraMap (l:=k) (r:=k + 1) (h_le:=by omega) = canonicalAlgMap k"}, {"name": "ConcreteBinaryTower.split_algebraMap_eq_zero_x", "content": "lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x) = (0, x)"}, {"name": "ConcreteBinaryTower.algebraMap_succ_eq_zero_x", "content": "lemma algebraMap_succ_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x = 《 0, x 》"}, {"name": "ConcreteBinaryTower.split_smul_Z_eq_zero_x", "content": "lemma split_smul_Z_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (x • Z k) = (x, 0)"}, {"name": "ConcreteBinaryTower.smul_Z_eq_zero_x", "content": "lemma smul_Z_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  x • Z k = 《 x, 0 》"}, {"name": "ConcreteBinaryTower.aeval_definingPoly_at_Z_succ", "content": "lemma aeval_definingPoly_at_Z_succ (k : ℕ) :\n  (aeval (Z (k + 1))) (definingPoly (s:=Z (k))) = 0"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y\n\ndef neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\nstructure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel\n\ndef mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\ndef equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)\n\ndef concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/\n\ndef concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res\n\nsection FieldLemmasOfLevel0\n\nend FieldLemmasOfLevel0\n\nsection NumericCasting\n\ndef natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :=\n\ndef natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :=\n\ndef intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :=\n\ndef intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :=\n\nend NumericCasting\n\nstructure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c\n\nstructure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one\n\nstructure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a\n\ndef mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n\n\ndef mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)\n\ndef mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm\n\nstructure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))\n\nend FieldOperationsAndInstances\n\nsection BTFieldPropsOneLevelLiftingLemmas\n\nvariable {k : ℕ} {h_k : k > 0}\n\nend BTFieldPropsOneLevelLiftingLemmas\n\nsection TowerFieldsConstruction\n\ndef liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/\n\ndef liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :=\n\ndef concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :=\n\ninstance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))\n\ndef getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/\n\nend TowerFieldsConstruction\n\nsection ConcreteBTFieldAlgebraConstruction\n\ndef canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))\n\ndef concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :=\n\ninstance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/\n\ndef ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le\n\ndef join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :=\n\nend ConcreteBTFieldAlgebraConstruction\n\nnoncomputable section ConcreteMultilinearBasis\n\nopen Module\n\ndef basisSucc (k : ℕ) : Basis (Fin 2) (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n\ndef powerBasisSucc (k : ℕ) :\n    PowerBasis (ConcreteBTField k) (ConcreteBTField (k + 1)) :=", "target_theorem": "@[simp]\ntheorem minPoly_of_powerBasisSucc_generator (k : ℕ) :\n  (minpoly (ConcreteBTField k) (powerBasisSucc k).gen) = X^2 + (Z k) • X + 1 :=", "ground_truth_proof": ":= by\n  unfold powerBasisSucc\n  simp only\n  rw [←C_mul']\n  letI: Fintype (ConcreteBTField k) := (getBTFResult k).instFintype\n  refine Eq.symm (minpoly.unique' (ConcreteBTField k) (Z (k + 1)) ?_ ?_ ?_)\n  · exact (definingPoly_is_monic (s:=Z (k)))\n  · exact aeval_definingPoly_at_Z_succ k\n  · intro q h_degQ_lt_deg_minPoly\n    -- h_degQ_lt_deg_minPoly : q.degree < (X ^ 2 + Z k • X + 1).degree\n    -- ⊢ q = 0 ∨ (aeval (Z (k + 1))) q ≠ 0\n    have h_degree_definingPoly : (definingPoly (s:=Z (k))).degree = 2 := by\n      exact degree_definingPoly (s:=Z (k))\n    rw [←definingPoly, h_degree_definingPoly] at h_degQ_lt_deg_minPoly\n    if h_q_is_zero : q = 0 then\n      rw [h_q_is_zero]\n      simp only [map_zero, ne_eq, not_true_eq_false, or_false]\n    else\n      -- reason stuff related to IsUnit here\n      have h_q_is_not_zero : q ≠ 0 := by omega\n      simp only [h_q_is_zero, ne_eq, false_or]\n      -- ⊢ ¬(aeval (Z (k + 1))) q = 0\n      have h_deg_q_ne_bot : q.degree ≠ ⊥ := by\n        exact degree_ne_bot.mpr h_q_is_zero\n      have q_natDegree_lt_2 : q.natDegree < 2 := by\n        exact (natDegree_lt_iff_degree_lt h_q_is_zero).mpr h_degQ_lt_deg_minPoly\n      -- do case analysis on q.degree\n      interval_cases hqNatDeg : q.natDegree\n      · simp only [ne_eq]\n        have h_q_is_c : ∃ r : ConcreteBTField k, q = C r := by\n          use q.coeff 0\n          exact Polynomial.eq_C_of_natDegree_eq_zero hqNatDeg\n        let hx := h_q_is_c.choose_spec\n        set x := h_q_is_c.choose\n        simp only [hx, aeval_C, map_eq_zero, ne_eq]\n        -- ⊢ ¬x = 0\n        by_contra h_x_eq_0\n        simp only [h_x_eq_0, map_zero] at hx -- hx : q = 0, h_q_is_not_zero : q ≠ 0\n        contradiction\n      · have h_q_natDeg_ne_0 : q.natDegree ≠ 0 := by exact ne_zero_of_eq_one hqNatDeg\n        have h_q_deg_ne_0 : q.degree ≠ 0 := by\n          by_contra h_q_deg_is_0\n          have h_q_natDeg_is_0 : q.natDegree = 0 := by exact\n            (degree_eq_iff_natDegree_eq h_q_is_zero).mp h_q_deg_is_0\n          contradiction\n        have h_natDeg_q_is_1 : q.natDegree = 1 := by exact hqNatDeg\n        have h_deg_q_is_1 : q.degree = 1 := by\n          apply (degree_eq_iff_natDegree_eq h_q_is_zero).mpr\n          exact hqNatDeg\n        have h_q_is_not_unit : ¬IsUnit q := by\n          by_contra h_q_is_unit\n          rw [←is_unit_iff_deg_0] at h_q_is_unit\n          contradiction\n        let c := q.coeff 1\n        let r := q.coeff 0\n        have hc : c = q.leadingCoeff := by\n          rw [Polynomial.leadingCoeff]\n          exact congrArg q.toFinsupp.2 (id (Eq.symm hqNatDeg))\n        have hc_ne_zero : c ≠ 0 := by\n          rw [hc]\n          by_contra h_c_eq_zero\n          simp only [leadingCoeff_eq_zero] at h_c_eq_zero -- h_c_eq_zero : q = 0\n          contradiction\n        have hq_form : q = c • X + C r := by\n          rw [Polynomial.eq_X_add_C_of_degree_eq_one (p:=q) (h:=by exact h_deg_q_is_1)]\n          congr\n          rw [hc]\n          exact C_mul' q.leadingCoeff X\n        -- ⊢ ¬(aeval (Z (k + 1))) q = 0\n        simp only [hq_form, map_add, map_smul, aeval_X, aeval_C, ne_eq]\n        -- ⊢ ¬Z k • Z (k + 1) + (algebraMap (ConcreteBTField k) (ConcreteBTField (k + 1))) x = 0\n        have h_split_smul := split_smul_Z_eq_zero_x (k:=k+1) (h_pos:=by omega) (x:=c)\n        rw [smul_Z_eq_zero_x (k:=k+1) (h_pos:=by omega) (x:=c)]\n        have h_alg_map_x := algebraMap_succ_eq_zero_x (k:=k+1) (h_pos:=by omega) (x:=r)\n        simp only [Nat.add_one_sub_one] at h_alg_map_x\n        rw [h_alg_map_x, join_add_join]\n        simp only [Nat.add_one_sub_one, _root_.add_zero, _root_.zero_add,\n          ne_eq]\n        -- ⊢ ¬join ⋯ c x = 0\n        by_contra h_join_eq_zero\n        conv_rhs at h_join_eq_zero =>\n          rw [←zero_is_0];\n          rw! [←join_zero_zero (k:=k+1) (h_k:=by omega)]\n        rw [join_eq_join_iff] at h_join_eq_zero\n        have h_c_eq_zero := h_join_eq_zero.1\n        contradiction", "nesting_depth": 16, "transitive_dep_count": 324, "subset_aristotle": false, "category": "Applied verif."}
{"id": 3, "thm_name": "AdditiveNTT.evaluation_poly_split_identity", "thm_stmt": "theorem evaluation_poly_split_identity (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) :\n  let P_i: L[X] := intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ coeffs\n  let P_even_i_plus_1: L[X] := evenRefinement 𝔽q β h_ℓ_add_R_rate i coeffs\n  let P_odd_i_plus_1: L[X] := oddRefinement 𝔽q β h_ℓ_add_R_rate i coeffs\n  let q_i: L[X] := qMap 𝔽q β ⟨i, by omega⟩\n  P_i = (P_even_i_plus_1.comp q_i) + X * (P_odd_i_plus_1.comp q_i)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean", "imports": ["import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import Mathlib.Data.Finsupp.Defs", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Tactic", "import ArkLib.Data.Nat.Bitwise", "import Mathlib.LinearAlgebra.LinearIndependent.Defs"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "context", "module": "Examples.FrankingProtocol"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "AddCommMonoid", "module": "Mathlib.Algebra.Group.Defs"}], "used_repo_defs": [{"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "g", "content": "def g (n : ℕ) (c : ℕ) (x : ℕ) := (x * x + c) % n"}], "lib_lemmas": [{"name": "Polynomial.comp_assoc", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "Fin.coe_ofNat_eq_mod", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.foldl_succ", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Polynomial.X_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_succ", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mul_add_mod_self_right", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Polynomial.pow_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.prod_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Finset.prod_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pow_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.is_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.sum_univ_eq_sum_range", "module": "Mathlib.Data.Fintype.BigOperators"}, {"name": "Finset.mul_sum", "module": "Mathlib.Algebra.BigOperators.Ring.Finset"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "Polynomial.C_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.mul_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.sum_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "map_eq_zero", "module": "Mathlib.Algebra.GroupWithZero.Units.Lemmas"}, {"name": "mul_eq_mul_left_iff", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "or_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "getBit_eq_succ_getBit_of_mul_two_add_one", "content": "lemma getBit_eq_succ_getBit_of_mul_two_add_one {n k : ℕ} : getBit (k+1) (2*n + 1) = getBit k n"}, {"name": "getBit_of_shiftRight", "content": "lemma getBit_of_shiftRight {n p : ℕ}:\n  ∀ k, getBit k (n >>> p) = getBit (k+p) n"}, {"name": "mul_two_add_bit_lt_two_pow", "content": "theorem mul_two_add_bit_lt_two_pow (a b c : ℕ) (i : Fin 2)\n    (h_a : a < 2 ^ b) (h_b : b < c) :\n    a * 2 + i.val < 2^c"}, {"name": "lt_two_pow_of_lt_two_pow_exp_le", "content": "lemma lt_two_pow_of_lt_two_pow_exp_le (x i j: ℕ)\n    (h_x_lt_2_pow_i: x < 2^i) (h_i_le_j: i ≤ j): x < 2^j"}, {"name": "getBit_zero_of_two_mul", "content": "lemma getBit_zero_of_two_mul {n : ℕ} : getBit 0 (2*n) = 0"}, {"name": "getBit_eq_succ_getBit_of_mul_two", "content": "lemma getBit_eq_succ_getBit_of_mul_two {n k : ℕ} : getBit (k+1) (2*n) = getBit k n"}, {"name": "Fin.sum_univ_odd_even", "content": "theorem Fin.sum_univ_odd_even {n : ℕ} {M : Type*} [AddCommMonoid M] (f : ℕ → M) :\n    (∑ i : Fin (2 ^ n), f (2 * i)) + (∑ i : Fin (2 ^ n), f (2 * i + 1))\n    = ∑ i: Fin (2 ^ (n+1)), f i"}], "used_local_defs": [{"name": "AdditiveNTT.qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "AdditiveNTT.intermediateNormVpoly", "content": "noncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)"}, {"name": "AdditiveNTT.intermediateNovelBasisX", "content": "noncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))"}, {"name": "AdditiveNTT.intermediateEvaluationPoly", "content": "noncomputable def intermediateEvaluationPoly (i : Fin (ℓ + 1))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i))), C (coeffs ⟨j, by admit /- proof elided -/\n  ⟩) *\n    (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate i ⟨j, by admit /- proof elided -/\n    ⟩)"}, {"name": "AdditiveNTT.evenRefinement", "content": "noncomputable def evenRefinement (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i-1))), C (coeffs ⟨j*2, by admit /- proof elided -/\n  ⟩) * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by admit /- proof elided -/\n  ⟩ ⟨j, hj⟩)"}, {"name": "AdditiveNTT.oddRefinement", "content": "noncomputable def oddRefinement (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i-1))), C (coeffs ⟨j*2+1, by admit /- proof elided -/\n  ⟩) * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by admit /- proof elided -/\n  ⟩ ⟨j, hj⟩)"}], "used_local_lemmas": [{"name": "AdditiveNTT.Polynomial.foldl_comp", "content": "omit [Fintype L] [DecidableEq L] in\ntheorem Polynomial.foldl_comp (n : ℕ) (f : Fin n → L[X]) : ∀ initInner initOuter: L[X],\n    Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter.comp initInner)\n    = (Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter)).comp initInner"}, {"name": "AdditiveNTT.intermediateNormVpoly_comp_qmap", "content": "theorem intermediateNormVpoly_comp_qmap (i : Fin (ℓ))\n    (k : Fin (ℓ - i - 1)) : -- corresponds to intermediateNormVpoly_comp"}, {"name": "AdditiveNTT.intermediateNormVpoly_comp_qmap_helper", "content": "theorem intermediateNormVpoly_comp_qmap_helper (i : Fin (ℓ))\n    (k : Fin (ℓ - (↑i + 1))) :\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate\n      ⟨↑i + 1, by omega⟩ (k:=⟨k, by simp only; omega⟩)).comp (qMap 𝔽q β ⟨↑i, by omega⟩) =\n    intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate\n      ⟨↑i, by omega⟩ ⟨k + 1, by simp only; omega⟩"}, {"name": "AdditiveNTT.even_index_intermediate_novel_basis_decomposition", "content": "lemma even_index_intermediate_novel_basis_decomposition (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :\n  intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2, by\n    apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)\n  ⟩  = (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by omega⟩ ⟨j, by\n    apply lt_two_pow_of_lt_two_pow_exp_le j (ℓ-i-1) (ℓ-(i+1)) (by omega) (by omega)\n  ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩)"}, {"name": "AdditiveNTT.odd_index_intermediate_novel_basis_decomposition", "content": "lemma odd_index_intermediate_novel_basis_decomposition\n    (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :\n    intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2 + 1, by\n      apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)\n    ⟩  = X * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by omega⟩ ⟨j, by\n      apply lt_two_pow_of_lt_two_pow_exp_le j (ℓ-i-1) (ℓ-(i+1)) (by omega) (by omega)\n    ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩)"}], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis\n\nimport Mathlib.Tactic\n\nimport Mathlib.Data.Finsupp.Defs\n\nimport Mathlib.LinearAlgebra.LinearIndependent.Defs\n\nopen Polynomial AdditiveNTT Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\nsection IntermediateStructures\n\nnoncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))\n\nnoncomputable section DomainBijection\n\nend DomainBijection\n\nnoncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)\n\nnoncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))\n\nnoncomputable def intermediateEvaluationPoly (i : Fin (ℓ + 1))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i))), C (coeffs ⟨j, by admit /- proof elided -/\n  ⟩) *\n    (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate i ⟨j, by admit /- proof elided -/\n    ⟩)\n\nnoncomputable def evenRefinement (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i-1))), C (coeffs ⟨j*2, by admit /- proof elided -/\n  ⟩) * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by admit /- proof elided -/\n  ⟩ ⟨j, hj⟩)\n\nnoncomputable def oddRefinement (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i-1))), C (coeffs ⟨j*2+1, by admit /- proof elided -/\n  ⟩) * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by admit /- proof elided -/\n  ⟩ ⟨j, hj⟩)", "target_theorem": "theorem evaluation_poly_split_identity (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) :\n  let P_i: L[X] :=", "ground_truth_proof": ":= intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ coeffs\n  let P_even_i_plus_1: L[X] := evenRefinement 𝔽q β h_ℓ_add_R_rate i coeffs\n  let P_odd_i_plus_1: L[X] := oddRefinement 𝔽q β h_ℓ_add_R_rate i coeffs\n  let q_i: L[X] := qMap 𝔽q β ⟨i, by omega⟩\n  P_i = (P_even_i_plus_1.comp q_i) + X * (P_odd_i_plus_1.comp q_i) := by\n\n  simp only [intermediateEvaluationPoly, Fin.eta]\n  simp only [evenRefinement, Fin.eta, sum_comp, mul_comp, C_comp, oddRefinement]\n\n  set leftEvenTerm := ∑ ⟨j, hj⟩ : Fin (2 ^ (ℓ - ↑i - 1)), C (coeffs ⟨j * 2, by\n    exact mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)\n  ⟩) * intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩ ⟨j * 2, by\n    exact mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)\n  ⟩\n  set leftOddTerm := ∑ ⟨j, hj⟩ : Fin (2 ^ (ℓ - ↑i - 1)), C (coeffs ⟨j * 2 + 1, by\n    apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)\n  ⟩) * intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩ ⟨j * 2 + 1, by\n    exact mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)\n  ⟩\n\n  have h_split_P_i: ∑ ⟨j, hj⟩ : Fin (2 ^ (ℓ - ↑i)), C (coeffs ⟨j, by\n    apply lt_two_pow_of_lt_two_pow_exp_le j (ℓ-i) (ℓ-i) (by omega) (by omega)\n  ⟩) * intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩ ⟨j, by omega⟩ =\n  leftEvenTerm + leftOddTerm\n  := by\n    unfold leftEvenTerm leftOddTerm\n    simp only [Fin.eta]\n\n    -- ⊢ ∑ k ∈ Fin (2 ^ (ℓ - ↑i)), C (coeffsₖ) * Xₖ⁽ⁱ⁾(X) = -- just pure even odd split\n    -- ∑ k ∈ Fin (2 ^ (ℓ - ↑i - 1)), C (coeffs₂ₖ) * X₂ₖ⁽ⁱ⁾(X) +\n    -- ∑ k ∈ Fin (2 ^ (ℓ - ↑i - 1)), C (coeffs₂ₖ+1) * X₂ₖ+1⁽ⁱ⁾(X)\n\n    set f1 := fun x: ℕ => -- => use a single function to represent the sum\n      if hx: x < 2 ^ (ℓ - ↑i) then\n        C (coeffs ⟨x, hx⟩) *\n          intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩ ⟨x, by omega⟩\n      else 0\n\n    have h_x: ∀ x: Fin (2 ^ (ℓ - ↑i)), f1 x.val =\n      C (coeffs ⟨x.val, by omega⟩) *\n        intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩\n          ⟨x.val, by simp only; omega⟩ := by\n      intro x\n      unfold f1\n      simp only [Fin.is_lt, ↓reduceDIte, Fin.eta]\n\n    conv_lhs =>\n      enter [2, x]\n      rw [←h_x x]\n\n    have h_x_2: ∀ x: Fin (2 ^ (ℓ - ↑i - 1)), f1 (x*2) =\n      C (coeffs ⟨x.val * 2, by\n        calc _ < 2 ^ (ℓ - i - 1) * 2 := by omega\n          _ = 2 ^ (ℓ - i) := Nat.two_pow_pred_mul_two (w:=ℓ - i) (h:=by omega)\n      ⟩) *\n        intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩ ⟨x.val * 2, by\n          exact mul_two_add_bit_lt_two_pow x.val (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)\n        ⟩ := by\n      intro x\n      unfold f1\n      simp only\n      have h_x_lt_2_pow_i_minus_1 :=\n        mul_two_add_bit_lt_two_pow x.val (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)\n      simp at h_x_lt_2_pow_i_minus_1\n      simp only [h_x_lt_2_pow_i_minus_1, ↓reduceDIte]\n\n    conv_rhs =>\n      enter [1, 2, x]\n      rw [←h_x_2 x]\n\n    have h_x_3: ∀ x: Fin (2 ^ (ℓ - ↑i - 1)), f1 (x*2+1) =\n      C (coeffs ⟨x.val * 2 + 1, by\n        calc _ < 2 ^ (ℓ - i - 1) * 2 := by omega\n          _ = 2 ^ (ℓ - i) := Nat.two_pow_pred_mul_two (w:=ℓ - i) (h:=by omega)\n      ⟩) *\n        intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩ ⟨x.val * 2 + 1, by\n          exact mul_two_add_bit_lt_two_pow x.val (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)\n        ⟩ := by\n      intro x\n      unfold f1\n      simp only\n      have h_x_lt_2_pow_i_minus_1 := mul_two_add_bit_lt_two_pow x.val\n        (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)\n      simp only [h_x_lt_2_pow_i_minus_1, ↓reduceDIte]\n\n    conv_rhs =>\n      enter [2, 2, x]\n      rw [←h_x_3 x]\n\n    -- ⊢ ∑ x, f1 ↑x = ∑ x, f1 (↑x * 2) + ∑ x, f1 (↑x * 2 + 1)\n\n    have h_1: ∑ i ∈ Finset.range (2 ^ (ℓ - ↑i)), f1 i\n      = ∑ i ∈ Finset.range (2 ^ (ℓ - ↑i - 1 + 1)), f1 i := by\n      congr\n      omega\n\n    have res := Fin.sum_univ_odd_even (f:=f1) (n:=(ℓ - ↑i - 1))\n    conv_rhs at res =>\n      rw [Fin.sum_univ_eq_sum_range]\n      rw [←h_1]\n      rw [←Fin.sum_univ_eq_sum_range]\n\n    rw [←res]\n    congr\n    · funext i\n      rw [mul_comm]\n    · funext i\n      rw [mul_comm]\n\n  conv_lhs => rw [h_split_P_i]\n\n  set rightEvenTerm := ∑ ⟨j, hj⟩ : Fin (2 ^ (ℓ - ↑i - 1)),\n      C (coeffs ⟨j * 2, by\n        calc _ < 2 ^ (ℓ - i - 1) * 2 := by omega\n          _ = 2 ^ (ℓ - i) := Nat.two_pow_pred_mul_two (w:=ℓ - i) (h:=by omega)\n      ⟩) *\n        (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i + 1, by omega⟩ ⟨j, by\n          apply lt_two_pow_of_lt_two_pow_exp_le (x:=j)\n            (i := ℓ-↑i-1) (j:=ℓ-↑i-1) (by omega) (by omega)\n        ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩)\n\n  set rightOddTerm :=\n    X *\n      ∑ ⟨j, hj⟩ : Fin (2 ^ (ℓ - ↑i - 1)),\n        C (coeffs ⟨j * 2 + 1, by\n          calc _ < 2 ^ (ℓ - i - 1) * 2 := by omega\n            _ = 2 ^ (ℓ - i) := Nat.two_pow_pred_mul_two (w:=ℓ - i) (h:=by omega)\n        ⟩) *\n          (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i + 1, by omega⟩ ⟨j, by\n            apply lt_two_pow_of_lt_two_pow_exp_le (x:=j)\n              (i := ℓ-↑i-1) (j:=ℓ-↑i-1) (by omega) (by omega)\n          ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩)\n\n  conv_rhs => change rightEvenTerm + rightOddTerm\n\n  have h_right_even_term: leftEvenTerm = rightEvenTerm := by\n    unfold rightEvenTerm leftEvenTerm\n    apply Finset.sum_congr rfl\n    intro j hj\n    simp only [Fin.eta, mul_eq_mul_left_iff, map_eq_zero]\n    --  X₂ⱼ⁽ⁱ⁾ = Xⱼ⁽ⁱ⁺¹⁾(q⁽ⁱ⁾(X)) ∨ a₂ⱼ = 0\n    by_cases h_a_j_eq_0: coeffs ⟨j * 2, by\n      calc _ < 2 ^ (ℓ - i - 1) * 2 := by omega\n        _ = 2 ^ (ℓ - i) := Nat.two_pow_pred_mul_two (w:=ℓ - i) (h:=by omega)\n    ⟩ = 0\n    · simp only [h_a_j_eq_0, or_true]\n    · simp only [h_a_j_eq_0, or_false]\n      --  X₂ⱼ⁽ⁱ⁾ = Xⱼ⁽ⁱ⁺¹⁾(q⁽ⁱ⁾(X))\n\n      exact even_index_intermediate_novel_basis_decomposition\n        𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) j\n\n  have h_right_odd_term: rightOddTerm = leftOddTerm := by\n    unfold rightOddTerm leftOddTerm\n    simp only [Fin.eta]\n    conv_rhs =>\n      simp only [Fin.is_lt, odd_index_intermediate_novel_basis_decomposition, Fin.eta]\n      enter [2, x];\n      rw [mul_comm (a:=X)]\n\n    rw [Finset.mul_sum]\n    congr\n    funext x\n    ring_nf -- just associativity and commutativity of multiplication in L[X]\n\n  rw [h_right_even_term, h_right_odd_term]", "nesting_depth": 7, "transitive_dep_count": 78, "subset_aristotle": false, "category": "Applied verif."}
{"id": 4, "thm_name": "Nat.getBit_repr", "thm_stmt": "theorem getBit_repr {ℓ : Nat} : ∀ j, j < 2^ℓ →\n  j = ∑ k ∈ Finset.Icc 0 (ℓ-1), (getBit k j) * 2^k", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/Nat/Bitwise.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Data.Nat.Bitwise", "import Mathlib.Data.Finsupp.Basic", "import Mathlib.Algebra.Order.Ring.Star", "import Mathlib.Data.Nat.Digits.Defs", "import Mathlib.Algebra.BigOperators.Fin"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "And", "module": "Init.Prelude"}, {"name": "AddCommMonoid", "module": "Mathlib.Algebra.Group.Defs"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.shiftRight_add", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Finset.Icc_self", "module": "Mathlib.Order.Interval.Finset.Basic"}, {"name": "Finset.mem_Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "Finset.sum_bij'", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_mul", "module": "Mathlib.Algebra.BigOperators.Ring.Finset"}, {"name": "Finset.sum_singleton", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pow_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "tsub_self", "module": "Mathlib.Algebra.Order.Sub.Basic"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_tsub", "module": "Mathlib.Algebra.Order.Sub.Basic"}], "repo_lemmas": [{"name": "sum_Icc_split", "content": "theorem sum_Icc_split {α : Type*} [AddCommMonoid α] (f : ℕ → α) (a b c : ℕ)\n    (h₁ : a ≤ b) (h₂ : b ≤ c):\n    ∑ i ∈ Finset.Icc a c, f i = ∑ i ∈ Finset.Icc a b, f i + ∑ i ∈ Finset.Icc (b+1) c, f i"}], "used_local_defs": [{"name": "Nat.getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}], "used_local_lemmas": [{"name": "Nat.getBit_of_shiftRight", "content": "lemma getBit_of_shiftRight {n p : ℕ}:\n  ∀ k, getBit k (n >>> p) = getBit (k+p) n"}], "local_ctx": "import ArkLib.Data.Fin.BigOperators\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nimport Mathlib.Algebra.Order.Ring.Star\n\nimport Mathlib.Data.Nat.Bitwise\n\nimport Mathlib.Data.Nat.Digits.Defs\n\nimport Mathlib.Data.Finsupp.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Fin\n\nnamespace Nat\n\ndef getBit (k n : Nat) : Nat := (n >>> k) &&& 1", "target_theorem": "theorem getBit_repr {ℓ : Nat} : ∀ j, j < 2^ℓ →\n  j = ∑ k ∈ Finset.Icc 0 (ℓ-1), (getBit k j) * 2^k :=", "ground_truth_proof": ":= by\n  induction ℓ with\n  | zero =>\n    -- Base case : ℓ = 0\n    intro j h_j\n    have h_j_zero : j = 0 := by exact Nat.lt_one_iff.mp h_j\n    subst h_j_zero\n    simp only [zero_tsub, Finset.Icc_self, Finset.sum_singleton, pow_zero, mul_one]\n    unfold getBit\n    rw [Nat.shiftRight_zero, Nat.and_one_is_mod]\n  | succ ℓ₁ ih =>\n    by_cases h_ℓ₁ : ℓ₁ = 0\n    · simp only [h_ℓ₁, zero_add, pow_one, tsub_self, Finset.Icc_self, Finset.sum_singleton,\n      pow_zero, mul_one];\n      intro j hj\n      interval_cases j\n      · simp only [getBit, Nat.shiftRight_zero, Nat.and_one_is_mod, Nat.zero_mod]\n      · simp only [getBit, Nat.shiftRight_zero, Nat.and_one_is_mod]\n    · push_neg at h_ℓ₁\n      set ℓ := ℓ₁ + 1\n      have h_ℓ_eq : ℓ = ℓ₁ + 1 := by rfl\n      intro j h_j\n      -- Inductive step : assume theorem holds for ℓ₁ = ℓ - 1\n      -- => show j = ∑ k ∈ Finset.range (ℓ + 1), (getBit k j) * 2^k\n      -- Split j into lowBits (b) and higher getLowBits (m) &\n      -- reason inductively from the predicate of (m, ℓ₁)\n      set b := getBit 0 j -- Least significant getBit : j % 2\n      set m := j >>> 1 -- Higher getLowBits : j / 2\n      have h_b_eq : b = getBit 0 j := by rfl\n      have h_m_eq : m = j >>> 1 := by rfl\n      have h_getBit_shift : ∀ k, getBit (k+1) j = getBit k m := by\n        intro k\n        rw [h_m_eq]\n        exact (getBit_of_shiftRight (n := j) (p := 1) k).symm\n      have h_j_eq : j = b + 2 * m := by\n        calc\n          _ = 2 * m + b := by\n            have h_m_eq : m = j/2 := by rfl\n            have h_b_eq : b = j%2 := by\n              rw [h_b_eq]; unfold getBit; rw [Nat.shiftRight_zero]; rw [Nat.and_one_is_mod];\n            rw [h_m_eq, h_b_eq];\n            rw [Nat.div_add_mod (m := j) (n := 2)]; -- n * (m / n) + m % n = m := by\n          _ = b + 2 * m := by omega;\n      have h_m : m < 2^ℓ₁ := by\n        by_contra h_m_ge_2_pow_ℓ\n        push_neg at h_m_ge_2_pow_ℓ\n        have h_j_ge : j ≥ 2^ℓ := by\n          calc _ = 2 * m + b := by rw [h_j_eq]; omega\n            _ ≥ 2 * (2^ℓ₁) + b := by omega\n            _ = 2^ℓ + b := by rw [h_ℓ_eq]; omega;\n            _ ≥ 2^ℓ := by omega;\n        exact Nat.not_lt_of_ge h_j_ge h_j -- contradiction\n      have h_m_repr := ih (j := m) h_m\n      have getBit_shift : ∀ k, getBit (k + 1) j = getBit k m := by\n        intro k\n        rw [h_m_eq]\n        exact (getBit_of_shiftRight (n := j) (p := 1) k).symm\n      -- ⊢ j = ∑ k ∈ Finset.range ℓ, getBit k j * 2 ^ k\n      have h_sum : ∑ k ∈ Finset.Icc 0 (ℓ-1), getBit k j * 2 ^ k\n        = (∑ k ∈ Finset.Icc 0 0, getBit k j * 2 ^ k)\n        + (∑ k ∈ Finset.Icc 1 (ℓ-1), getBit k j * 2 ^ k) := by\n        apply sum_Icc_split\n        omega\n        omega\n      rw [h_sum]\n      rw [h_j_eq]\n      rw [Finset.Icc_self, Finset.sum_singleton, pow_zero, mul_one]\n\n      have h_sum_2 : ∑ k ∈ Finset.Icc 1 (ℓ-1), getBit k (b + 2 * m) * 2 ^ k\n        = ∑ k ∈ Finset.Icc 0 (ℓ₁-1), getBit k (m) * 2 ^ (k+1) := by\n        apply Finset.sum_bij' (fun i _ => i - 1) (fun i _ => i + 1)\n        · -- left inverse\n          intro i hi\n          simp only [Finset.mem_Icc] at hi ⊢\n          exact Nat.sub_add_cancel hi.1\n        · -- right inverse\n          intro i hi\n          norm_num\n        · -- function value match\n          intro i hi\n          rw [←h_j_eq]\n          rw [getBit_of_shiftRight]\n          have ⟨left_bound, right_bound⟩ := Finset.mem_Icc.mp hi\n          rw [Nat.sub_add_cancel left_bound]\n        · -- left membership preservation\n          intro i hi -- hi : i ∈ Finset.Icc 1 (ℓ - 1)\n          rw [Finset.mem_Icc]\n          have ⟨left_bound, right_bound⟩ := Finset.mem_Icc.mp hi\n          -- ⊢ 0 ≤ i - 1 ∧ i - 1 ≤ ℓ₁ - 1\n          apply And.intro\n          · exact Nat.pred_le_pred left_bound\n          · exact Nat.pred_le_pred right_bound\n        · -- right membership preservation\n          intro j hj\n          rw [Finset.mem_Icc]\n          have ⟨left_bound, right_bound⟩ := Finset.mem_Icc.mp hj -- (0 ≤ j ∧ j ≤ ℓ₁ - 1)\n          -- ⊢ 1 ≤ j + 1 ∧ j + 1 ≤ ℓ - 1\n          apply And.intro\n          · exact Nat.le_add_of_sub_le left_bound\n          · rw [h_ℓ_eq]; rw [Nat.add_sub_cancel]; -- ⊢ j + 1 ≤ ℓ₁\n            have h_j_add_1_le_ℓ₁ : j + 1 ≤ ℓ₁ := by\n              calc j + 1 ≤ (ℓ₁ - 1) + 1 := by apply Nat.add_le_add_right; exact right_bound;\n              _ = ℓ₁ := by rw [Nat.sub_add_cancel]; omega;\n            exact h_j_add_1_le_ℓ₁\n      rw [h_sum_2]\n\n      have h_sum_3 : ∑ k ∈ Finset.Icc 0 (ℓ₁-1), getBit k (m) * 2 ^ (k+1)\n        = 2 * ∑ k ∈ Finset.Icc 0 (ℓ₁-1), getBit k (m) * 2 ^ k := by\n        calc\n          _ = ∑ k ∈ Finset.Icc 0 (ℓ₁-1), ((getBit k (m) * 2^k) * 2) := by\n            apply Finset.sum_congr rfl (fun k hk => by\n              rw [Finset.mem_Icc] at hk -- hk : 0 ≤ k ∧ k ≤ ℓ₁ - 1\n              have h_res : getBit k (m) * 2 ^ (k+1) = getBit k (m) * 2 ^ k * 2 := by\n                rw [Nat.pow_succ, ←mul_assoc]\n              exact h_res\n            )\n        _ = (∑ k ∈ Finset.Icc 0 (ℓ₁-1), getBit k (m) * 2 ^ k) * 2 := by rw [Finset.sum_mul]\n        _ = 2 * ∑ k ∈ Finset.Icc 0 (ℓ₁-1), getBit k (m) * 2 ^ k := by rw [mul_comm]\n      rw [h_sum_3]\n      rw [←h_m_repr]\n      conv =>\n        rhs\n        rw [←h_j_eq]", "nesting_depth": 2, "transitive_dep_count": 24, "subset_aristotle": true, "category": "Applied verif."}
{"id": 5, "thm_name": "Nat.getBit_of_binaryFinMapToNat", "thm_stmt": "lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :\n    ∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val\n      = if h_k: k < n then m ⟨k, by omega⟩ else 0", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/Nat/Bitwise.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Data.Nat.Bitwise", "import Mathlib.Data.Finsupp.Basic", "import Mathlib.Algebra.Order.Ring.Star", "import Mathlib.Data.Nat.Digits.Defs", "import Mathlib.Algebra.BigOperators.Fin"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.binaryRec", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bit", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bodd", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.boddDiv2", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.div2", "module": "Mathlib.Data.Nat.Bits"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "Nat.mod_two_bne_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.one_and_eq_mod_two", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "Bool.not_eq_true", "module": "Init.SimpLemmas"}, {"name": "Nat.pow_le_pow_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.testBit_eq_false_of_lt", "module": "Mathlib.Data.Nat.Bitwise"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Nat.eq_of_testBit_eq", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "beq_eq_beq", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.and_assoc", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_comm", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_and_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_and", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_eq_zero_iff_lt", "module": "Init.Data.Nat.Div.Lemmas"}, {"name": "Nat.pow_lt_pow_right", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftLeft_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_add", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_mod_two_eq_zero", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.add_mul_div_left", "module": "Init.Data.Nat.Div.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.div_add_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mul_add_mod_self_right", "module": "Init.Data.Nat.Div.Basic"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.or_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_or", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_xor", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Bool.toNat_lt", "module": "Init.Data.Bool"}, {"name": "Nat.bit_decomp", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.bit_val", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.mul_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.right_distrib", "module": "Init.Data.Nat.Basic"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.pow_zero", "module": "Init.Data.Nat.Basic"}, {"name": "NeZero.ne'", "module": "Init.Data.NeZero"}, {"name": "lt_add_iff_pos_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "lt_self_iff_false", "module": "Mathlib.Order.Basic"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}, {"name": "sub_one_lt", "module": "Mathlib.Algebra.Order.Ring.Unbundled.Basic"}, {"name": "zero_lt_one", "module": "Mathlib.Algebra.Order.ZeroLEOne"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Nat.getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "Nat.binaryFinMapToNat", "content": "def binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary : ∀ j: Fin n, m j ≤ 1) : Fin (2^n) :="}], "used_local_lemmas": [{"name": "Nat.getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "Nat.getBit_eq_testBit", "content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"}, {"name": "Nat.getBit_zero_eq_zero", "content": "lemma getBit_zero_eq_zero {k : Nat} : getBit k 0 = 0"}, {"name": "Nat.eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "Nat.shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "Nat.and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "Nat.getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "Nat.and_two_pow_eq_zero_of_getBit_0", "content": "lemma and_two_pow_eq_zero_of_getBit_0 {n i : ℕ} (h_getBit : getBit i n = 0)\n    : n &&& (2 ^ i) = 0"}, {"name": "Nat.div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "Nat.and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "Nat.xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}, {"name": "Nat.or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||| m = (n1 ||| m1) * 2 + (bn ||| bm)"}, {"name": "Nat.sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "Nat.sum_of_and_eq_zero_is_xor", "content": "lemma sum_of_and_eq_zero_is_xor {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ^^^ m"}, {"name": "Nat.getBit_of_xor", "content": "lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m"}, {"name": "Nat.getBit_of_lt_two_pow", "content": "lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :\n  getBit k a = if k < n then getBit k a else 0"}], "local_ctx": "import ArkLib.Data.Fin.BigOperators\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nimport Mathlib.Algebra.Order.Ring.Star\n\nimport Mathlib.Data.Nat.Bitwise\n\nimport Mathlib.Data.Nat.Digits.Defs\n\nimport Mathlib.Data.Finsupp.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Fin\n\nnamespace Nat\n\ndef getBit (k n : Nat) : Nat := (n >>> k) &&& 1\n\ndef binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary : ∀ j: Fin n, m j ≤ 1) : Fin (2^n) :=", "target_theorem": "lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :\n    ∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val\n      = if h_k: k < n then m ⟨k, by omega⟩ else 0 :=", "ground_truth_proof": ":= by\n  -- We prove this by induction on `n`.\n  induction n with\n  | zero =>\n    intro k;\n    simp only [Nat.pow_zero, Fin.val_eq_zero, not_lt_zero', ↓reduceDIte]\n    exact getBit_zero_eq_zero\n  | succ n ih =>\n    -- Inductive step: Assume the property holds for `n`, prove it for `n+1`.\n    have h_lt: 2^n - 1 < 2^n := by\n      refine sub_one_lt ?_\n      exact Ne.symm (NeZero.ne' (2 ^ n))\n    intro k\n    dsimp [binaryFinMapToNat]\n    -- ⊢ (↑k).getBit (∑ j, 2 ^ ↑j * m j) = m k\n    rw [Fin.sum_univ_castSucc] -- split the msb\n    set prevSum := ∑ i: Fin n, (2 ^ i.castSucc.val) * (m i.castSucc)\n    let mPrev := fun i: Fin n => m i.castSucc\n    have h_getBit_prevSum := ih (m:=mPrev) (h_binary:=by exact fun j ↦ h_binary j.castSucc)\n    have h_prevSum_eq: prevSum = binaryFinMapToNat mPrev\n      (by exact fun j ↦ h_binary j.castSucc) := by rfl\n    set msbTerm := 2 ^ ((Fin.last n).val) * m (Fin.last n)\n    -- ⊢ (↑k).getBit (prevSum + msbTerm) = m k\n    have h_m_at_last: m ⟨n, by omega⟩ ≤ 1 := by exact h_binary (Fin.last n)\n    have h_sum_eq_xor: prevSum + msbTerm = prevSum ^^^ msbTerm := by\n      rw [sum_of_and_eq_zero_is_xor]\n      unfold msbTerm\n      interval_cases h_m_last_val: m ⟨n, by omega⟩\n      · simp only [Fin.last, h_m_last_val, mul_zero, Nat.and_zero]\n      · simp only [Fin.last, h_m_last_val, mul_one]\n        apply and_two_pow_eq_zero_of_getBit_0\n        have h_getBit_prevSum_at_n := getBit_of_lt_two_pow (k:=n) (n:=n) (a:=⟨prevSum, by omega⟩)\n        simp only [lt_self_iff_false, ↓reduceIte] at h_getBit_prevSum_at_n\n        rw [h_getBit_prevSum_at_n]\n    rw [h_sum_eq_xor, getBit_of_xor]\n    if h_k_eq: k = n then\n      rw [h_k_eq]\n      simp only [lt_add_iff_pos_right, zero_lt_one, ↓reduceDIte]\n      rw [h_prevSum_eq]\n      rw [getBit_of_lt_two_pow]\n      simp only [lt_self_iff_false, ↓reduceIte, zero_xor]\n      unfold msbTerm\n      -- ⊢ n.getBit (2 ^ ↑(Fin.last n) * m (Fin.last n)) = m ⟨n, ⋯⟩\n      interval_cases h_m_last_val: m ⟨n, by omega⟩\n      · -- ⊢ n.getBit (2 ^ ↑(Fin.last n) * m (Fin.last n)) = 0\n        rw [Fin.val_last, Fin.last]\n        rw [h_m_last_val, mul_zero]\n        exact getBit_zero_eq_zero\n      · -- ⊢ n.getBit (2 ^ ↑(Fin.last n) * m (Fin.last n)) = 1\n        simp only [Fin.last]\n        rw [h_m_last_val, mul_one]\n        rw [Nat.getBit_two_pow]\n        simp only [BEq.rfl, ↓reduceIte]\n    else\n      have hBitLhs := h_getBit_prevSum (k:=k)\n      simp only at hBitLhs\n      rw [h_prevSum_eq.symm] at hBitLhs\n      rw [hBitLhs]\n      if h_k_lt_n: k < n then\n        have h_k_lt_n_add_1: k < n + 1 := by omega\n        simp only [h_k_lt_n_add_1, ↓reduceDIte]\n        push_neg at h_k_eq\n        simp only [h_k_lt_n, ↓reduceDIte]\n        unfold msbTerm\n        interval_cases h_m_last_val: m ⟨n, by omega⟩\n        · simp only [Fin.last, h_m_last_val, mul_zero]\n          rw [Nat.getBit_zero_eq_zero, Nat.xor_zero]\n          rfl\n        · simp only [Fin.last, h_m_last_val, mul_one]\n          rw [Nat.getBit_two_pow]\n          simp only [beq_iff_eq]\n          simp only [h_k_eq.symm, ↓reduceIte, xor_zero]\n          rfl\n      else\n        have h_k_not_lt_n_add_1: ¬(k < n + 1) := by omega\n        have h_k_not_lt_n: ¬(k < n) := by omega\n        simp only [h_k_not_lt_n_add_1, h_k_not_lt_n, ↓reduceDIte, Nat.zero_xor]\n        unfold msbTerm\n        interval_cases h_m_last_val: m ⟨n, by omega⟩\n        · simp only [Fin.last, h_m_last_val, mul_zero]\n          exact getBit_zero_eq_zero\n        · simp only [Fin.last, h_m_last_val, mul_one]\n          rw [Nat.getBit_two_pow]\n          simp only [beq_iff_eq]\n          simp only [ite_eq_right_iff, one_ne_zero, imp_false, ne_eq]\n          omega", "nesting_depth": 4, "transitive_dep_count": 104, "subset_aristotle": true, "category": "Applied verif."}
{"id": 6, "thm_name": "ConcreteBinaryTower.towerEquiv_commutes_left_diff", "thm_stmt": "lemma towerEquiv_commutes_left_diff (i d : ℕ) : ∀ r : ConcreteBTField i,\n  (AlgebraTower.algebraMap i (i+d) (by omega)) ((towerEquiv i).ringEquiv r) =\n  (towerEquiv (i+d)).ringEquiv ((AlgebraTower.algebraMap i (i+d) (by omega)) r)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "AddCommGroup", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.negSucc", "module": "Init.Data.Int.Basic"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "NNRat", "module": "Mathlib.Data.Rat.Init"}, {"name": "NNRat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "Rat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.toAlgebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "invFun", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "left_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "right_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "absurd", "module": "Init.Prelude"}, {"name": "instFintypeProd", "module": "Mathlib.Data.Fintype.Prod"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "CommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Module.Basis", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Algebra.algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Module.Basis.mk", "module": "Mathlib.LinearAlgebra.Basis.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "PowerBasis", "module": "Mathlib.RingTheory.PowerBasis"}, {"name": "gen", "module": "VCVio.CryptoFoundations.FiatShamir"}, {"name": "RingEquiv", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Equiv.cast", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "AdjoinRoot.powerBasis", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Field.toCommRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}, {"name": "MonoidHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "OneHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "EquivLike", "module": "Mathlib.Data.FunLike.Equiv"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}, {"name": "join_via_add_smul", "content": "def join_via_add_smul {k : ℕ} (h_pos : k > 0) (hi_btf lo_btf : BTField (k - 1)) :\n    BTField k :="}, {"name": "binaryAlgebraTower", "content": "def binaryAlgebraTower {l r : ℕ} (h_le : l ≤ r) : Algebra (BTField l) (BTField r) :="}, {"name": "AlgebraTower.toAlgebra", "content": "@[simp]\ndef AlgebraTower.toAlgebra {i j : ι} (h : i ≤ j) : Algebra (A i) (A j) :=\n  (AlgebraTower.algebraMap (i:=i) (j:=j) (h:=h)).toAlgebra"}, {"name": "Z", "content": "@[simp]\ndef Z (k : ℕ) : BTField k := (BinaryTowerAux k).snd.specialElement "}, {"name": "AlgebraTower.toIsScalarTower", "content": "@[simp]\ninstance AlgebraTower.toIsScalarTower (a : AlgebraTower C) {i j k : ι}\n    (h1 : i ≤ j) (h2 : j ≤ k) :\n    letI : Algebra (C i) (C j) :="}, {"name": "split", "content": "def split (k : ℕ) (h_k : k > 0) (x : BTField k) : BTField (k-1) × BTField (k-1) :="}, {"name": "powerBasisSucc", "content": "def powerBasisSucc (k : ℕ) :\n    PowerBasis (BTField k) (BTField (k+1)) :="}, {"name": "BTField_succ_alg_equiv_adjoinRoot", "content": "def BTField_succ_alg_equiv_adjoinRoot (k : ℕ) :\n  AdjoinRoot (poly k) ≃ₐ[BTField k] BTField (k + 1) :="}, {"name": "poly", "content": "@[simp]\ndef poly (k : ℕ) : Polynomial (BTField k) := definingPoly (s:=(Z k))"}, {"name": "CommRing", "content": "@[simp]\ninstance CommRing (k : ℕ) : CommRing (BTField k) := Field.toCommRing"}, {"name": "towerAlgebraMap", "content": "def towerAlgebraMap (l r : ℕ) (h_le : l ≤ r) : BTField l →+* BTField r :="}, {"name": "canonicalEmbedding", "content": "def canonicalEmbedding (k : ℕ) : BTField k →+* BTField (k+1) :=\n  AdjoinRoot.of (poly k)"}, {"name": "(priority", "content": "instance (priority := 1000) algebra_adjacent_tower (l : ℕ) :\n  Algebra (BTField l) (BTField (l+1)) :="}, {"name": "polyMonic", "content": "instance polyMonic (n : ℕ) : Monic (poly n) := definingPoly_is_monic (Z n)"}], "lib_lemmas": [{"name": "Ne.dite_eq_left_iff", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.left_eq_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingHom.comp_id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Function.comp_apply", "module": "Init.Core"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "RingHom.coe_comp", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}, {"name": "MonoidHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "Nat.sub_one_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "OneHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "RingHom.coe_mk", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_cast", "module": "Init.PropLemmas"}, {"name": "eqRec_eq_cast", "module": "Batteries.Logic"}, {"name": "BitVec.extractLsb_ofNat", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.zero_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.zero_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Equiv.toFun_as_coe", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "EquivLike.coe_coe", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Nat.add_eq_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingEquiv.toEquiv_eq_coe", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "Nat.add_zero", "module": "Init.Core"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}, {"name": "join_via_add_smul_zero", "content": "lemma join_via_add_smul_zero {k : ℕ} (h_pos : k > 0) :\n  ⋘ 0, 0 ⋙ = 0"}, {"name": "algebraMap_eq_zero_x", "content": "lemma algebraMap_eq_zero_x {i j : ℕ} (h_le : i < j) (x : BTField i) :\n    letI instAlgebra"}, {"name": "algebraMap_succ_eq_zero_x", "content": "lemma algebraMap_succ_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : BTField (k - 1)) :\n  letI instAlgebra"}, {"name": "split_algebraMap_eq_zero_x", "content": "lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : BTField (k - 1)) :\n  letI instAlgebra"}, {"name": "towerAlgebraMap_succ_1", "content": "lemma towerAlgebraMap_succ_1 (k : ℕ) :\n  towerAlgebraMap (l:=k) (r:=k+1) (h_le:=by omega) = canonicalEmbedding k"}, {"name": "towerAlgebraMap_id", "content": "lemma towerAlgebraMap_id (k : ℕ) : towerAlgebraMap (h_le:=by omega) = RingHom.id (BTField k)"}, {"name": "eq_join_via_add_smul_eq_iff_split", "content": "theorem eq_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)\n    (x : BTField k) (hi_btf lo_btf : BTField (k - 1)) :\n    x = ⋘ hi_btf, lo_btf ⋙ ↔\n  split (k:=k) (h_k:=h_pos) x = (hi_btf, lo_btf)"}, {"name": "unique_linear_decomposition_succ", "content": "theorem unique_linear_decomposition_succ (k : ℕ) :\n  ∀ (x : BTField (k+1)), ∃! (p : BTField k × BTField k),\n    x = ⋘ p.1, p.2 ⋙"}, {"name": "algebraMap_adjacent_tower_succ_eq_Adjoin_of", "content": "lemma algebraMap_adjacent_tower_succ_eq_Adjoin_of (k : ℕ) :\n  (algebraMap (BTField k) (BTField (k + 1))) = of (poly k)"}, {"name": "algebraMap_adjacent_tower_def", "content": "lemma algebraMap_adjacent_tower_def (l : ℕ) :\n  (algebraMap (BTField l) (BTField (l + 1))) = canonicalEmbedding l"}, {"name": "binaryTowerAlgebra_def", "content": "lemma binaryTowerAlgebra_def (l r : ℕ) (h_le : l ≤ r) :\n    @binaryAlgebraTower (l:=l) (r:=r) (h_le:=h_le)\n    = (towerAlgebraMap l r h_le).toAlgebra"}, {"name": "poly_natDegree_eq_2", "content": "lemma poly_natDegree_eq_2 (k : ℕ) : (poly (k:=k)).natDegree = 2"}, {"name": "BTField.cast_BTField_eq", "content": "lemma BTField.cast_BTField_eq (k m : ℕ) (h_eq : k = m) :\n  BTField k = BTField m"}, {"name": "BTField.RingHom_cast_dest_apply", "content": "@[simp]\ntheorem BTField.RingHom_cast_dest_apply (k m n : ℕ) (h_eq : m = n)\n  (f : BTField k →+* BTField m) (x : BTField k) :\n    (cast (BTField.RingHom_eq_of_dest_eq (k:=k) (m:=m) (n:=n) h_eq) f) x\n    = cast (by apply cast_BTField_eq (h_eq:=h_eq)) (f x)"}, {"name": "BTField.RingHom_eq_of_dest_eq", "content": "@[simp]\ntheorem BTField.RingHom_eq_of_dest_eq (k m n : ℕ) (h_eq : m = n) :\n  (BTField k →+* BTField m) = (BTField k →+* BTField n)"}, {"name": "towerAlgebraMap_succ", "content": "lemma towerAlgebraMap_succ (l r : ℕ) (h_le : l ≤ r) :\n  towerAlgebraMap (l:=l) (r:=r+1) (h_le:=by omega) =\n  (towerAlgebraMap (l:=r) (r:=r+1) (h_le:=by omega)).comp\n  (towerAlgebraMap (l:=l) (r:=r) (h_le:=by omega))"}, {"name": "join_eq_join_iff", "content": "lemma join_eq_join_iff (k : ℕ) (h_pos : k > 0) (hi₁ hi₀ lo₁ lo₀ : BTField (k - 1)) :\n    ⋘ hi₁, lo₁ ⋙ = ⋘ hi₀, lo₀ ⋙ ↔\n  hi₁ = hi₀ ∧ lo₁ = lo₀"}, {"name": "split_join_via_add_smul_eq_iff_split", "content": "lemma split_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)\n    (hi_btf lo_btf : BTField (k - 1)) :\n    split (k:=k) (h_k:=h_pos) (⋘ hi_btf, lo_btf ⋙) = (hi_btf, lo_btf)"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.add", "content": "def add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y"}, {"name": "ConcreteBinaryTower.neg", "content": "def neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.ConcreteBTFAddCommGroupProps", "content": "structure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel"}, {"name": "ConcreteBinaryTower.mkAddCommGroupInstance", "content": "def mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}"}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.equivProd", "content": "def equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)"}, {"name": "ConcreteBinaryTower.concrete_mul", "content": "def concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.concrete_inv", "content": "def concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res"}, {"name": "ConcreteBinaryTower.natCast", "content": "def natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.natCast_zero", "content": "def natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :="}, {"name": "ConcreteBinaryTower.natCast_succ", "content": "def natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :="}, {"name": "ConcreteBinaryTower.intCast", "content": "def intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.intCast_ofNat", "content": "def intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :="}, {"name": "ConcreteBinaryTower.intCast_negSucc", "content": "def intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :="}, {"name": "ConcreteBinaryTower.ConcreteBTFRingProps", "content": "structure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c"}, {"name": "ConcreteBinaryTower.ConcreteBTFDivisionRingProps", "content": "structure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldProps", "content": "structure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a"}, {"name": "ConcreteBinaryTower.mkRingInstance", "content": "def mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n"}, {"name": "ConcreteBinaryTower.mkDivisionRingInstance", "content": "def mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)"}, {"name": "ConcreteBinaryTower.mkFieldInstance", "content": "def mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm"}, {"name": "ConcreteBinaryTower.ConcreteBTFStepResult", "content": "structure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))"}, {"name": "ConcreteBinaryTower.liftBTFieldProps", "content": "def liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.liftConcreteBTField", "content": "def liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.concreteCanonicalEmbedding", "content": "def concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :="}, {"name": "ConcreteBinaryTower.instAlgebraLiftConcreteBTField", "content": "instance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))"}, {"name": "ConcreteBinaryTower.getBTFResult", "content": "def getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.canonicalAlgMap", "content": "def canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap", "content": "def concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :="}, {"name": "ConcreteBinaryTower.instAlgebraTowerConcreteBTF", "content": "instance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldAlgebra", "content": "def ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le"}, {"name": "ConcreteBinaryTower.join_via_add_smul", "content": "def join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.basisSucc", "content": "def basisSucc (k : ℕ) : Basis (Fin 2) (ConcreteBTField k) (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.powerBasisSucc", "content": "def powerBasisSucc (k : ℕ) :\n    PowerBasis (ConcreteBTField k) (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.towerEquiv_zero", "content": "noncomputable def towerEquiv_zero : RingEquiv (R:=GF(2)) (S:=ConcreteBTField 0) :=  {\n  toFun := fun x => if x = 0 then 0 else 1,\n  invFun := fun x => if x = 0 then 0 else 1,\n  left_inv := fun x => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.towerRingEquiv0", "content": "noncomputable def towerRingEquiv0 : BTField 0 ≃+* ConcreteBTField 0 :="}, {"name": "ConcreteBinaryTower.towerRingEquivFromConcrete0", "content": "noncomputable def towerRingEquivFromConcrete0 : ConcreteBTField 0 ≃+* BTField 0 :="}, {"name": "ConcreteBinaryTower.towerRingHomForwardMap", "content": "noncomputable def towerRingHomForwardMap (k : ℕ) : ConcreteBTField k → BTField k :="}, {"name": "ConcreteBinaryTower.towerRingHomBackwardMap", "content": "noncomputable def towerRingHomBackwardMap (k : ℕ) : BTField k → ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.TowerEquivResult", "content": "structure TowerEquivResult (k : ℕ) where\n  ringEquiv : ConcreteBTField k ≃+* BTField k\n  ringEquivForwardMapEq : ringEquiv = towerRingHomForwardMap k"}, {"name": "ConcreteBinaryTower.towerEquiv", "content": "noncomputable def towerEquiv (n : ℕ) : TowerEquivResult n :="}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq", "content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n  ConcreteBTField k = ConcreteBTField m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq_zero", "content": "theorem BitVec.dcast_bitvec_eq_zero {l r : ℕ} (h_width_eq : l = r) :\n  dcast (h_width_eq) 0#(l) = 0#(r)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.cast_join", "content": "lemma cast_join {k n : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) (heq : k = n) :\n  join (k:=k) h_pos hi lo = cast (by rw [heq])\n    (join (k:=n) (by omega) (cast (by subst heq; rfl) hi) (lo:=cast (by subst heq; rfl) lo))"}, {"name": "ConcreteBinaryTower.zero_is_0", "content": "lemma zero_is_0 {k : ℕ} : (zero (k:=k)) = (0 : ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_of_split", "content": "theorem join_of_split {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_split_eq : split h_pos x = (hi_btf, lo_btf)) :\n    x = 《 hi_btf, lo_btf 》"}, {"name": "ConcreteBinaryTower.split_of_join", "content": "theorem split_of_join {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_join : x = 《hi_btf, lo_btf》) :\n    (hi_btf, lo_btf) = split h_pos x"}, {"name": "ConcreteBinaryTower.split_zero", "content": "theorem split_zero {k : ℕ} (h_pos : k > 0) : split h_pos zero = (zero, zero)"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_eq_of_dest_eq", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_eq_of_dest_eq (k m n : ℕ) (h_eq : m = n) :\n    (ConcreteBTField k →+* ConcreteBTField m)\n    = (ConcreteBTField k →+* ConcreteBTField n)"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_cast_dest_apply", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_cast_dest_apply (k m n : ℕ) (h_eq : m = n)\n  (f : ConcreteBTField k →+* ConcreteBTField m) (x : ConcreteBTField k) :\n    (cast (ConcreteBTField.RingHom_eq_of_dest_eq (k:=k) (m:=m) (n:=n) h_eq) f) x\n    = cast (by apply cast_ConcreteBTField_eq (h_eq:=h_eq)) (f x)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_id", "content": "lemma concreteTowerAlgebraMap_id (k : ℕ) :\n    concreteTowerAlgebraMap (h_le:=by omega) = RingHom.id (ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_succ_1", "content": "lemma concreteTowerAlgebraMap_succ_1 (k : ℕ) :\n  concreteTowerAlgebraMap (l:=k) (r:=k + 1) (h_le:=by omega) = canonicalAlgMap k"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_succ", "content": "lemma concreteTowerAlgebraMap_succ (l r : ℕ) (h_le : l ≤ r) :\n  concreteTowerAlgebraMap (l:=l) (r:=r + 1) (h_le:=by omega) =\n  (concreteTowerAlgebraMap (l:=r) (r:=r + 1) (h_le:=by omega)).comp\n  (concreteTowerAlgebraMap (l:=l) (r:=r) (h_le:=by omega))"}, {"name": "ConcreteBinaryTower.split_algebraMap_eq_zero_x", "content": "lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x) = (0, x)"}, {"name": "ConcreteBinaryTower.algebraMap_succ_eq_zero_x", "content": "lemma algebraMap_succ_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x = 《 0, x 》"}, {"name": "ConcreteBinaryTower.algebraMap_eq_zero_x", "content": "lemma algebraMap_eq_zero_x {i j : ℕ} (h_le : i < j) (x : ConcreteBTField i) :\n    letI instAlgebra := ConcreteBTFieldAlgebra (l:=i) (r:=j) (h_le:=by omega)\n    letI instAlgebraPred := ConcreteBTFieldAlgebra (l:=i) (r:=j-1) (h_le:=by omega)\n    algebraMap (ConcreteBTField i) (ConcreteBTField j) x\n      = 《 0, algebraMap (ConcreteBTField i) (ConcreteBTField (j-1)) x 》"}, {"name": "ConcreteBinaryTower.towerRingHomForwardMap_zero", "content": "lemma towerRingHomForwardMap_zero {k : ℕ} :\n  (towerRingHomForwardMap k) 0 = 0"}, {"name": "ConcreteBinaryTower.towerRingHomForwardMap_split_eq", "content": "lemma towerRingHomForwardMap_split_eq (k : ℕ) (h_pos : k > 0) (x : ConcreteBTField k) :\n  let p := split (k:=k) (h:=h_pos) x\n  towerRingHomForwardMap (k:=k) (x) =\n    BinaryTower.join_via_add_smul (k:=k) (h_pos:=h_pos)\n      (hi_btf := towerRingHomForwardMap (k:=k-1) (p.1))\n      (lo_btf := towerRingHomForwardMap (k:=k-1) (p.2))"}, {"name": "ConcreteBinaryTower.towerRingHomForwardMap_join", "content": "lemma towerRingHomForwardMap_join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) :\n  towerRingHomForwardMap (k:=k) (《 hi, lo 》) =\n    BinaryTower.join_via_add_smul (k:=k) (h_pos:=by omega)\n      (hi_btf := towerRingHomForwardMap (k:=k-1) hi)\n      (lo_btf := towerRingHomForwardMap (k:=k-1) lo)"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y\n\ndef neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\nstructure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel\n\ndef mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\ndef equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)\n\ndef concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/\n\ndef concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res\n\nsection FieldLemmasOfLevel0\n\nend FieldLemmasOfLevel0\n\nsection NumericCasting\n\ndef natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :=\n\ndef natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :=\n\ndef intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :=\n\ndef intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :=\n\nend NumericCasting\n\nstructure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c\n\nstructure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one\n\nstructure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a\n\ndef mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n\n\ndef mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)\n\ndef mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm\n\nstructure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))\n\nend FieldOperationsAndInstances\n\nsection BTFieldPropsOneLevelLiftingLemmas\n\nvariable {k : ℕ} {h_k : k > 0}\n\nend BTFieldPropsOneLevelLiftingLemmas\n\nsection TowerFieldsConstruction\n\ndef liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/\n\ndef liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :=\n\ndef concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :=\n\ninstance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))\n\ndef getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/\n\nend TowerFieldsConstruction\n\nsection ConcreteBTFieldAlgebraConstruction\n\ndef canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))\n\ndef concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :=\n\ninstance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/\n\ndef ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le\n\ndef join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :=\n\nend ConcreteBTFieldAlgebraConstruction\n\nnoncomputable section ConcreteMultilinearBasis\n\nopen Module\n\ndef basisSucc (k : ℕ) : Basis (Fin 2) (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n\ndef powerBasisSucc (k : ℕ) :\n    PowerBasis (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n\nend ConcreteMultilinearBasis\n\nsection TowerEquivalence\n\nopen BinaryTower\n\nnoncomputable def towerEquiv_zero : RingEquiv (R:=GF(2)) (S:=ConcreteBTField 0) :=  {\n  toFun := fun x => if x = 0 then 0 else 1,\n  invFun := fun x => if x = 0 then 0 else 1,\n  left_inv := fun x => by admit /- proof elided -/\n\nnoncomputable def towerRingEquiv0 : BTField 0 ≃+* ConcreteBTField 0 :=\n\nnoncomputable def towerRingEquivFromConcrete0 : ConcreteBTField 0 ≃+* BTField 0 :=\n\nnoncomputable def towerRingHomForwardMap (k : ℕ) : ConcreteBTField k → BTField k :=\n\nnoncomputable def towerRingHomBackwardMap (k : ℕ) : BTField k → ConcreteBTField k :=\n\nstructure TowerEquivResult (k : ℕ) where\n  ringEquiv : ConcreteBTField k ≃+* BTField k\n  ringEquivForwardMapEq : ringEquiv = towerRingHomForwardMap k\n\nnoncomputable def towerEquiv (n : ℕ) : TowerEquivResult n :=", "target_theorem": "lemma towerEquiv_commutes_left_diff (i d : ℕ) : ∀ r : ConcreteBTField i,\n  (AlgebraTower.algebraMap i (i+d) (by omega)) ((towerEquiv i).ringEquiv r) =\n  (towerEquiv (i+d)).ringEquiv ((AlgebraTower.algebraMap i (i+d) (by omega)) r) :=", "ground_truth_proof": ":= by\n  -- If d = 0, then this is trivial\n  -- For d > 0 : let j = i+d\n    -- lhs of goal : right => 《 0, ringMap x 》 => up => 《 algMap 0 = 0, algMap (ringMap x) 》\n    -- rhs of goal : up => 《 0, algMap x 》 => right => 《 ringMap 0 = 0, ringMap (algMap x) 》\n    -- where both `algMap (ringMap x)` and `ringMap (algMap x)` are in `BTField (j-1)`\n  -- => Strategy : For each i => do induction upwards on d\n  change ∀ r : ConcreteBTField i,\n    (BinaryTower.towerAlgebraMap (l:=i) (r:=i+d) (h_le:=by omega)) ((towerEquiv i).ringEquiv r) =\n    (towerEquiv (i+d)).ringEquiv ((concreteTowerAlgebraMap i (i+d) (by omega)) r)\n  induction d using Nat.rec with\n  | zero =>\n    intro r\n    simp only [Nat.add_zero]\n    rw [BinaryTower.towerAlgebraMap_id, concreteTowerAlgebraMap_id]\n    rfl\n  | succ d' ih =>\n    intro r\n    letI instAbstractAlgebra : Algebra (BTField i) (BTField (i + d' + 1)) :=\n      binaryAlgebraTower (by omega)\n    let : Algebra (ConcreteBTField i) (ConcreteBTField (i + d')) :=\n      ConcreteBTFieldAlgebra (l:=i) (r:=i+d') (h_le:=by omega)\n    letI instConcreteAlgebra : Algebra (ConcreteBTField i) (ConcreteBTField (i + d' + 1)) :=\n      ConcreteBTFieldAlgebra (l:=i) (r:=i+d'+1) (h_le:=by omega)\n    change (algebraMap (R:=BTField i) (A:=BTField (i + d' + 1))) ((towerEquiv i).ringEquiv r) =\n      (towerEquiv (i + d' + 1)).ringEquiv ((algebraMap (R:=ConcreteBTField i)\n      (A:=ConcreteBTField (i + d' + 1))) r)\n    have h_concrete_algMap_eq_zero_x := algebraMap_eq_zero_x (i:=i) (j:=i+d'+1) (h_le:=by omega) r\n    simp only [Nat.add_one_sub_one] at h_concrete_algMap_eq_zero_x\n    rw [algebraMap, Algebra.algebraMap] at h_concrete_algMap_eq_zero_x\n    have h_abstract_algMap_eq_zero_x := BinaryTower.algebraMap_eq_zero_x (i:=i) (j:=i+d'+1)\n      (h_le:=by omega) ((towerEquiv i).ringEquiv r)\n    simp only [Nat.add_one_sub_one] at h_abstract_algMap_eq_zero_x\n    conv_lhs =>\n      rw! [h_abstract_algMap_eq_zero_x]\n    conv_rhs =>\n      rw [algebraMap, Algebra.algebraMap]\n      simp only [BTField.eq_1, CommRing.eq_1, BTFieldIsField.eq_1, instConcreteAlgebra]\n      rw! [h_concrete_algMap_eq_zero_x] -- split algebraMap\n      -- Now change `BinaryTowerAux (i + d' + 1)).fst` back to `BTField (i + d' + 1)`\n      -- for definitional equality, otherwise we can't `rw [ringEquivForwardMapEq]`\n      change (towerEquiv (i + d' + 1)).ringEquiv (join (h_pos:=by omega) 0\n        ((algebraMap (ConcreteBTField i) (ConcreteBTField (i + d'))) r))\n      rw [(towerEquiv (i+d'+1)).ringEquivForwardMapEq]\n      -- now convert to BinaryTower.join_via_add_smul\n      rw [towerRingHomForwardMap_join (k:=i+d'+1) (h_pos:=by omega)]\n      simp only [Nat.add_one_sub_one]\n    -- ⊢ BinaryTower.join_via_add_smul ⋯ = BinaryTower.join_via_add_smul ⋯ =\n    rw [BinaryTower.join_eq_join_iff]\n    constructor\n    · rw [towerRingHomForwardMap_zero]\n    · let h := ih (r:=r)\n      change (BinaryTower.towerAlgebraMap (l:=i) (r:=i+d')\n        (h_le:=by omega)) ((towerEquiv i).ringEquiv r) =\n        towerRingHomForwardMap (i + d') ((concreteTowerAlgebraMap i (i + d') (by omega)) r)\n      rw [h]\n      rw [(towerEquiv (i+d')).ringEquivForwardMapEq]", "nesting_depth": 10, "transitive_dep_count": 306, "subset_aristotle": false, "category": "Applied verif."}
{"id": 7, "thm_name": "AdditiveNTT.intermediateNormVpoly_comp", "thm_stmt": "omit [DecidableEq L] [DecidableEq 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in\ntheorem intermediateNormVpoly_comp (i : Fin ℓ) (k : Fin (ℓ - i + 1))\n  (l : Fin (ℓ - (i.val + k.val) + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k + l, by\n      simp only; omega⟩) =\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i+k, by omega⟩) (k:=⟨l, by\n      simp only; omega⟩)).comp (\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k, by\n      simp only; omega⟩)\n  )", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean", "imports": ["import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import Mathlib.Data.Finsupp.Defs", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Tactic", "import Mathlib.LinearAlgebra.LinearIndependent.Defs"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "context", "module": "Examples.FrankingProtocol"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}], "lib_lemmas": [{"name": "Fin.cast_eq_self", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.coe_cast", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.coe_castSucc", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.coe_ofNat_eq_mod", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.foldl_succ_last", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.foldl_zero", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_last", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.zero_eta", "module": "Init.Data.Fin.Basic"}, {"name": "Nat.add_assoc", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Polynomial.X_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_assoc", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}], "used_local_defs": [{"name": "AdditiveNTT.qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "AdditiveNTT.intermediateNormVpoly", "content": "noncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)"}], "used_local_lemmas": [], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis\n\nimport Mathlib.Tactic\n\nimport Mathlib.Data.Finsupp.Defs\n\nimport Mathlib.LinearAlgebra.LinearIndependent.Defs\n\nopen Polynomial AdditiveNTT Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\nsection IntermediateStructures\n\nnoncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))\n\nnoncomputable section DomainBijection\n\nend DomainBijection\n\nnoncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)", "target_theorem": "omit [DecidableEq L] [DecidableEq 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in\ntheorem intermediateNormVpoly_comp (i : Fin ℓ) (k : Fin (ℓ - i + 1))\n  (l : Fin (ℓ - (i.val + k.val) + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k + l, by\n      simp only; omega⟩) =\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i+k, by omega⟩) (k:=⟨l, by\n      simp only; omega⟩)).comp (\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k, by\n      simp only; omega⟩)\n  ) :=", "ground_truth_proof": ":= by\n  induction l using Fin.succRecOnSameFinType with\n  | zero =>\n    simp only [Fin.coe_ofNat_eq_mod, Nat.zero_mod, add_zero, Fin.eta, Fin.zero_eta]\n    have h_eq_X : intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i + ↑k, by omega⟩ 0 = X := by\n      simp only [intermediateNormVpoly, Fin.coe_ofNat_eq_mod, Nat.zero_mod, Fin.foldl_zero]\n    simp only [h_eq_X, X_comp]\n  | succ j jh p =>\n      -- Inductive case: l = j + 1\n      -- Following the pattern from concreteTowerAlgebraMap_assoc:\n      -- A = |i| --- (k) --- |i+k| --- (j+1) --- |i+k+j+1|\n      -- Proof: A = (j+1) ∘ (k) (direct) = ((1) ∘ (j)) ∘ (k) (succ decomposition)\n      --        = (1) ∘ ((j) ∘ (k)) (associativity) = (1) ∘ (jk) (induction hypothesis)\n      unfold intermediateNormVpoly\n      -- First, rewrite to get the right form for Fin.foldl_succ\n      -- We need Fin.foldl (k + j + 1) which equals Fin.foldl ((k + j) + 1)\n      simp only\n      have h_j_add_1_val: (j + 1).val = j.val + 1 := by\n        rw [Fin.val_add_one']\n        omega\n      simp_rw [h_j_add_1_val]\n      simp_rw [←Nat.add_assoc (n:=k.val) (m:=j.val) (k:=1)]\n      rw [Fin.foldl_succ_last, Fin.foldl_succ_last]\n      simp only [Fin.cast_eq_self, Fin.coe_cast, Fin.val_last, Fin.coe_castSucc]\n      simp_rw [←Nat.add_assoc (n:=i.val) (m:=k.val) (k:=j.val)]\n      rw [comp_assoc]\n      -- ⊢ qMap (i := i + k + j)(...) = qMap (i := i + k + j)(...)\n      congr", "nesting_depth": 5, "transitive_dep_count": 38, "subset_aristotle": false, "category": "Applied verif."}
{"id": 8, "thm_name": "AdditiveNTT.inductive_rec_form_W_comp", "thm_stmt": "omit h_Fq_char_prime hF₂ in\nlemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])\n      (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean", "imports": ["import Mathlib.Algebra.Polynomial.Degree.Definitions", "import ArkLib.Data.Fin.BigOperators", "import ArkLib.Data.Polynomial.MonomialBasis", "import Mathlib.LinearAlgebra.StdBasis", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.Polynomial.Frobenius"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "CommGroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Polynomial.rootMultiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Multiset", "module": "Mathlib.Data.Multiset.Defs"}, {"name": "Multiset.count", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.map", "module": "Mathlib.Data.Multiset.MapFold"}, {"name": "Polynomial.roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "SetLike", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.algEquivOfCompEqX", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "multiplicity", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "EmbeddingLike", "module": "Mathlib.Data.FunLike.Embedding"}, {"name": "CanLift", "module": "Mathlib.Tactic.Lift"}, {"name": "Multiset.filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Finset.val", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Set.InjOn", "module": "Mathlib.Data.Set.Operations"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.val", "module": "Init.Prelude"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}], "lib_lemmas": [{"name": "Fact.out", "module": "Mathlib.Logic.Basic"}, {"name": "Fin.le_zero_iff'", "module": "Mathlib.Data.Fin.SuccPred"}, {"name": "LinearIndependent.injective", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Nat.not_lt_zero", "module": "Init.Prelude"}, {"name": "Set.Ico_eq_empty_iff", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.compl_eq_univ_diff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Set.empty_subset", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.image_empty", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_subset_image_iff", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.subset_compl_singleton_iff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Submodule.span_mono", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "linearIndependent_iff_notMem_span", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Finset.prod_ne_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Polynomial.eval_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_prod", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_sub", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Polynomial.splits_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.splits_prod", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Finset.prod_eq_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Polynomial.X_sub_C_ne_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.X_ne_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_C_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_X_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_sub", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.comp_eq_zero_iff", "module": "Mathlib.Algebra.Polynomial.Degree.Lemmas"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "map_neg", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "sub_eq_neg_self", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "EmbeddingLike.map_eq_zero_iff", "module": "Mathlib.Algebra.Group.Equiv.Defs"}, {"name": "Polynomial.aeval_C", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.aeval_X", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algEquivOfCompEqX_apply", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algebraMap_eq", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.rootMultiplicity_eq_multiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "if_false", "module": "Init.ByCases"}, {"name": "if_true", "module": "Init.ByCases"}, {"name": "map_sub", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "multiplicity_map_eq", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "sub_sub_sub_cancel_right", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Multiset.countP_eq_card_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_map", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.filter_congr", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.count_roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "eq_sub_iff_add_eq", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.image_val_of_injOn", "module": "Mathlib.Data.Finset.Image"}, {"name": "Finset.prod_image", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Polynomial.roots_prod_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Subtype.val_injective", "module": "Mathlib.Data.Subtype"}, {"name": "CanLift.prf", "module": "Mathlib.Tactic.Lift"}, {"name": "Multiset.card_singleton", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.card_zero", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.count_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_singleton", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.count_univ", "module": "Mathlib.Data.Fintype.Basic"}, {"name": "Multiset.count_zero", "module": "Mathlib.Data.Multiset.Count"}, {"name": "SetLike.coe_eq_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "SetLike.mem_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Fin.zero_le", "module": "Init.Data.Fin.Lemmas"}, {"name": "Set.Ico_subset_Ico_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_mono", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_image_of_mem", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.add_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.smul_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.subset_span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "sub_add_cancel", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "Set.Ico_insert_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_singleton", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_union", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Icc", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.union_singleton", "module": "Mathlib.Data.Set.Insert"}, {"name": "Submodule.mem_span_singleton", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.mem_sup", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.smul_mem_iff", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Submodule.span_union", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.sub_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "existsUnique_of_exists_of_unique", "module": "Mathlib.Logic.ExistsUnique"}, {"name": "sub_smul", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "sub_sub_sub_cancel_left", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_const_zero", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.sum_ite_eq'", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise"}, {"name": "Finset.sum_map_val", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Multiset.count_bind", "module": "Mathlib.Data.Multiset.Bind"}, {"name": "Multiset.count_map_eq_count'", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.roots_prod", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "add_left_injective", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "if_false_right", "module": "Init.PropLemmas"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "iff_false", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "Polynomial.monic_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.monic_prod_of_monic", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.Monic.comp", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.Splits.comp_of_degree_le_one", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.degree_X_sub_C_le", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eq_prod_roots_of_monic_of_splits_id", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.natDegree_X", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.natDegree_sub_C", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.comp_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_right_inj", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_add_neg", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_right_inj", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "CommGroupWithZero.mul_inv_cancel", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Fin.mk_lt_of_lt_val", "module": "Init.Data.Fin.Lemmas"}, {"name": "Finset.card_univ", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Finset.prod_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.prod_const", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.prod_mul_distrib", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "NeZero.one_le", "module": "Mathlib.Data.Nat.Cast.NeZero"}, {"name": "Polynomial.C_1", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.C_mul", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_pow", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.mul_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.pow_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.smul_C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.smul_eq_C_mul", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.sub_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_sub_cancel_right", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "algebra_compatible_smul", "module": "Mathlib.Algebra.Algebra.Basic"}, {"name": "map_mul", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_pow", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_pow", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_sub", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "one_pow", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_smul", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "pow_sub₀", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "smul_assoc", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "smul_eq_mul", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "smul_sub", "module": "Mathlib.Algebra.GroupWithZero.Action.Defs"}], "repo_lemmas": [{"name": "Fin.lt_succ'", "content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}, {"name": "Fin.le_succ", "content": "lemma Fin.le_succ (a : Fin r) (h_a_add_1 : a + 1 < r) : a ≤ a + 1"}, {"name": "Fin.le_iff_lt_succ", "content": "lemma Fin.le_iff_lt_succ (a b : Fin r) (h_b : b + 1 < r) : a ≤ b ↔ a < b + 1"}, {"name": "Fin.val_sub_one", "content": "lemma Fin.val_sub_one (a : Fin r) (h_a_sub_1 : a > 0) : (a - 1).val = a.val - 1"}, {"name": "prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L", "content": "theorem prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L\n  (p : L[X]) :\n  (∏ c ∈ (Finset.univ : Finset Fq), (p - Polynomial.C (algebraMap Fq L c))) =\n    p^(Fintype.card Fq) - p"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X_in_L", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X_in_L :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C (algebraMap Fq L c))) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C c)) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "linear_map_of_comp_to_linear_map_of_eval", "content": "theorem linear_map_of_comp_to_linear_map_of_eval (f : L[X])\n  (h_f_linear : IsLinearMap (R := Fq) (M := L[X]) (M₂ := L[X])\n    (f := fun inner_p ↦ f.comp inner_p)) :\n    IsLinearMap (R := Fq) (M := L) (M₂ := L) (f := fun x ↦ f.eval x)"}], "used_local_defs": [{"name": "AdditiveNTT.U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "AdditiveNTT.W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "AdditiveNTT.algEquivAevalXSubC", "content": "@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :="}], "used_local_lemmas": [{"name": "AdditiveNTT.βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "AdditiveNTT.root_U_lift_down", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n  (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n  a ∈ (U 𝔽q β (i+1)) → ∃! x: 𝔽q, a - x • β i ∈ (U 𝔽q β i)"}, {"name": "AdditiveNTT.root_U_lift_up", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime hβ_lin_indep in\ntheorem root_U_lift_up (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) (x : 𝔽q) :\n  a - x • β i ∈ (U 𝔽q β i) → a ∈ (U 𝔽q β (i+1))"}, {"name": "AdditiveNTT.W_monic", "content": "lemma W_monic (i : Fin r) : (W 𝔽q β i).Monic"}, {"name": "AdditiveNTT.W_ne_zero", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W_ne_zero (i : Fin r) : (W 𝔽q β i) ≠ 0"}, {"name": "AdditiveNTT.Wᵢ_eval_βᵢ_neq_zero", "content": "lemma Wᵢ_eval_βᵢ_neq_zero\n    (i : Fin r): (W 𝔽q β i).eval (β i) ≠ 0"}, {"name": "AdditiveNTT.W_splits", "content": "lemma W_splits (i : Fin r) : (W 𝔽q β i).Splits (RingHom.id L)"}, {"name": "AdditiveNTT.roots_W", "content": "lemma roots_W (i : Fin r) : -- converts root Multiset into (univ: Uᵢ.val.map)\n  (W 𝔽q β i).roots = (univ : Finset (U 𝔽q β i)).val.map (fun u => u.val)"}, {"name": "AdditiveNTT.comp_X_sub_C_eq_zero_iff", "content": "omit [Fintype L] [DecidableEq L] in\nlemma comp_X_sub_C_eq_zero_iff (p : L[X]) (a : L) :\n  p.comp (X - C a) = 0 ↔ p = 0"}, {"name": "AdditiveNTT.rootMultiplicity_comp_X_sub_C", "content": "lemma rootMultiplicity_comp_X_sub_C (p : L[X]) (a x : L) :\n    rootMultiplicity x (p.comp (X - C a)) = rootMultiplicity (x - a) p"}, {"name": "AdditiveNTT.roots_comp_X_sub_C", "content": "lemma roots_comp_X_sub_C (p : L[X]) (a : L) :\n    (p.comp (X - C a)).roots = p.roots.map (fun r => r + a)"}, {"name": "AdditiveNTT.Prod_W_comp_X_sub_C_ne_zero", "content": "omit [DecidableEq L] h_Fq_char_prime hF₂ hβ_lin_indep in\nlemma Prod_W_comp_X_sub_C_ne_zero (i : Fin r) :\n    (univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i))) ≠ 0"}, {"name": "AdditiveNTT.rootMultiplicity_W", "content": "lemma rootMultiplicity_W (i : Fin r) (a : L) :\n    rootMultiplicity a (W 𝔽q β i) = if a ∈ (U 𝔽q β i : Set L) then 1 else 0"}, {"name": "AdditiveNTT.rootMultiplicity_prod_W_comp_X_sub_C", "content": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0"}, {"name": "AdditiveNTT.W_prod_comp_decomposition", "content": "lemma W_prod_comp_decomposition\n    (i : Fin r) (hi : i > 0) :\n    (W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))"}, {"name": "AdditiveNTT.comp_sub_C_of_linear_eval", "content": "lemma comp_sub_C_of_linear_eval (p : L[X])\n  (h_lin : IsLinearMap 𝔽q (f := fun inner_p ↦ p.comp inner_p)) (a : L) :\n    p.comp (X - C a) = p - C (eval a p)"}], "local_ctx": "import ArkLib.Data.Nat.Bitwise\n\nimport ArkLib.Data.Polynomial.Frobenius\n\nimport ArkLib.Data.Polynomial.MonomialBasis\n\nimport Mathlib.LinearAlgebra.StdBasis\n\nimport Mathlib.Algebra.Polynomial.Degree.Definitions\n\nopen Polynomial FiniteDimensional Finset Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (h_dim : Module.finrank 𝔽q L = r)\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n\nsection LinearSubspaces\n\ndef U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))\n\nnoncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)\n\nend LinearSubspaces\n\nsection LinearityOfSubspaceVanishingPolynomials\n\n@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :=", "target_theorem": "omit h_Fq_char_prime hF₂ in\nlemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])\n      (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p) :=", "ground_truth_proof": ":= by\n  intro p\n  set W_i := W 𝔽q β i\n  set q := Fintype.card 𝔽q\n  set v := W_i.eval (β i)\n\n  -- First, we must prove that v is non-zero to use its inverse.\n  have hv_ne_zero : v ≠ 0 := by\n    unfold v W_i\n    exact Wᵢ_eval_βᵢ_neq_zero 𝔽q β i\n\n  -- Proof flow:\n  -- `Wᵢ₊₁(X) = ∏_{c ∈ 𝔽q} (Wᵢ ∘ (X - c • βᵢ))` -- from W_prod_comp_decomposition\n    -- `= ∏_{c ∈ 𝔽q} (Wᵢ(X) - c • Wᵢ(βᵢ))` -- linearity of Wᵢ\n    -- `= ∏_{c ∈ 𝔽q} (Wᵢ(X) - c • v)`\n    -- `= v² ∏_{c ∈ 𝔽q} (v⁻¹ • Wᵢ(X) - c)`\n    -- `= v² (v⁻² • Wᵢ(X)² - v⁻¹ • Wᵢ(X))` => FLT (prod_X_sub_C_eq_X_pow_card_sub_X_in_L)\n    -- `= Wᵢ(X)² - v • Wᵢ(X)` => Q.E.D\n\n  have h_scalar_smul_eq_C_v_mul: ∀ s: L, ∀ p: L[X], s • p = C s * p := by\n    intro s p\n    exact smul_eq_C_mul s\n  have h_v_smul_v_inv_eq_one: v • v⁻¹ = 1 := by\n    simp only [smul_eq_mul]\n    exact CommGroupWithZero.mul_inv_cancel v hv_ne_zero\n  have h_v_mul_v_inv_eq_one: v * v⁻¹ = 1 := by\n    exact h_v_smul_v_inv_eq_one\n  -- The main proof using a chain of equalities (the `calc` block).\n  calc\n    (W 𝔽q β (i + 1)).comp p\n    _ = (∏ c: 𝔽q, (W_i).comp (X - C (c • β i))).comp p := by\n      have h_res := W_prod_comp_decomposition 𝔽q β (i+1) (by\n        apply Fin.mk_lt_of_lt_val\n        rw [Fin.val_add_one' (a := i) (h_a_add_1 := h_i_add_1), Nat.zero_mod]\n        omega\n      )\n      rw [h_res]\n      simp only [add_sub_cancel_right]\n      rfl\n    -- Step 2: Apply the linearity property of Wᵢ as a polynomial.\n    _ = (∏ c: 𝔽q, (W_i - C (W_i.eval (c • β i)))).comp p := by\n      congr\n      funext c\n      -- We apply the transformation inside the product for each element `c`.\n      -- apply Finset.prod_congr rfl\n      -- ⊢ W_i.comp (X - C (c • β i)) = W_i - C (eval (c • β i) W_i)\n      exact comp_sub_C_of_linear_eval (p := W_i) (h_lin := h_prev_linear_map) (a := (c • β i))\n    -- Step 3: Apply the linearity of Wᵢ's *evaluation map* to the constant term.\n    -- Hypothesis: `h_prev_linear_map.map_smul`\n    _ = (∏ c: 𝔽q, (W_i - C (c • v))).comp p := by\n      congr\n      funext c\n      -- ⊢ W_i - C (eval (c • β i) W_i) = W_i - C (c • v)\n      congr\n      -- ⊢ eval (c • β i) W_i = c • v\n      -- Use the linearity of the evaluation map, not the composition map\n      have h_eval_linear := Polynomial.linear_map_of_comp_to_linear_map_of_eval (f := (W 𝔽q β i))\n        (h_f_linear := h_prev_linear_map)\n      exact h_eval_linear.map_smul c (β i)\n    -- Step 4: Perform the final algebraic transformation.\n    _ = (C (v^q) * (∏ c: 𝔽q, (C (v⁻¹) * W_i - C (algebraMap 𝔽q L c)))).comp p := by\n      congr\n      calc\n        _ = ∏ c: 𝔽q, (v • (v⁻¹ • W_i - C (algebraMap 𝔽q L c))) := by\n          apply Finset.prod_congr rfl\n          intro c _\n          rw [smul_sub]\n          -- ⊢ W_i - C (c • v) = v • v⁻¹ • W_i - v • C ((algebraMap 𝔽q L) c)\n          rw [smul_C, smul_eq_mul, map_mul]\n          rw [←smul_assoc]\n          rw [h_v_smul_v_inv_eq_one]\n          rw [one_smul]\n          rw [sub_right_inj]\n          -- ⊢ C (c • v) = C v * C ((algebraMap 𝔽q L) c)\n          rw [←C_mul]\n          -- ⊢ C (c • v) = C (v * (algebraMap 𝔽q L) c)\n          have h_c_smul_v: c • v = (algebraMap 𝔽q L c) • v := by\n            exact algebra_compatible_smul L c v\n          rw [h_c_smul_v]\n          rw [mul_comm]\n          rw [smul_eq_mul]\n        _ = ∏ c: 𝔽q, (C v * (v⁻¹ • W_i - C (algebraMap 𝔽q L c))) := by\n          apply Finset.prod_congr rfl\n          intro c _\n          rw [h_scalar_smul_eq_C_v_mul]\n        _ = C (v^q) * (∏ c: 𝔽q, (C v⁻¹ * W_i - C (algebraMap 𝔽q L c))) := by\n          -- rw [Finset.prod_mul_distrib]\n          -- rw [Finset.prod_const, Finset.card_univ]\n          rw [Finset.prod_mul_distrib]\n          conv_lhs =>\n            enter [2]\n            enter [2]\n            rw [h_scalar_smul_eq_C_v_mul]\n          congr\n          -- ⊢ ∏ (x: 𝔽q), C v = C (v ^ q)\n          rw [Finset.prod_const, Finset.card_univ]\n          unfold q\n          exact Eq.symm C_pow\n    _ = (C (v^q) * ((C v⁻¹ * W_i)^q - (C v⁻¹ * W_i))).comp p := by\n      congr\n      -- ⊢ ∏ c, (C v⁻¹ * W_i - C ((algebraMap 𝔽q L) c)) = (C v⁻¹ * W_i) ^ q - C v⁻¹ * W_i\n      rw [Polynomial.prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L (p := C v⁻¹ * W_i)]\n    _ = (C (v^q) * C (v⁻¹^q) * W_i^q - C (v^q) * C v⁻¹ * W_i).comp p := by\n      congr\n      rw [mul_sub]\n      conv_lhs =>\n        rw [mul_pow, ←mul_assoc, ←mul_assoc, ←C_pow]\n    _ = (W_i^q - C (v^(q-1)) * W_i).comp p := by\n      congr\n      · rw [←C_mul, ←mul_pow, h_v_mul_v_inv_eq_one, one_pow, C_1, one_mul]\n      · rw [←C_mul]\n        have h_v_pow_q_minus_1: v^q * v⁻¹ = v^(q-1) := by\n          rw [pow_sub₀ (a := v) (m := q) (n := 1) (ha := hv_ne_zero) (h := by exact NeZero.one_le)]\n          -- ⊢ v ^ q * v⁻¹ = v ^ q * (v ^ 1)⁻¹\n          congr\n          norm_num\n        rw [h_v_pow_q_minus_1]\n    _ = (W_i^q - C (eval (β i) W_i) ^ (q - 1) * W_i).comp p := by\n      simp only [map_pow, W_i, q, v]\n    _ = (W_i^q).comp p - (C (eval (β i) W_i) ^ (q - 1) * W_i).comp p := by\n      rw [sub_comp]\n    _ = (W_i.comp p)^q - (C (eval (β i) W_i) ^ (q - 1)) * (W_i.comp p) := by\n      rw [pow_comp, mul_comp]\n      conv_lhs =>\n        rw [pow_comp]\n        rw [C_comp (a := (eval (β i) W_i)) (p := p)]", "nesting_depth": 6, "transitive_dep_count": 229, "subset_aristotle": false, "category": "Applied verif."}
{"id": 9, "thm_name": "AdditiveNTT.odd_index_intermediate_novel_basis_decomposition", "thm_stmt": "lemma odd_index_intermediate_novel_basis_decomposition\n    (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :\n    intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2 + 1, by\n      apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)\n    ⟩  = X * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by omega⟩ ⟨j, by\n      apply lt_two_pow_of_lt_two_pow_exp_le j (ℓ-i-1) (ℓ-(i+1)) (by omega) (by omega)\n    ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean", "imports": ["import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import Mathlib.Data.Finsupp.Defs", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Tactic", "import ArkLib.Data.Nat.Bitwise", "import Mathlib.LinearAlgebra.LinearIndependent.Defs"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "context", "module": "Examples.FrankingProtocol"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "g", "content": "def g (n : ℕ) (c : ℕ) (x : ℕ) := (x * x + c) % n"}], "lib_lemmas": [{"name": "Polynomial.comp_assoc", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "Fin.coe_ofNat_eq_mod", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.foldl_succ", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Polynomial.X_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_succ", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mul_add_mod_self_right", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Polynomial.pow_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.prod_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "getBit_eq_succ_getBit_of_mul_two_add_one", "content": "lemma getBit_eq_succ_getBit_of_mul_two_add_one {n k : ℕ} : getBit (k+1) (2*n + 1) = getBit k n"}, {"name": "getBit_of_shiftRight", "content": "lemma getBit_of_shiftRight {n p : ℕ}:\n  ∀ k, getBit k (n >>> p) = getBit (k+p) n"}, {"name": "mul_two_add_bit_lt_two_pow", "content": "theorem mul_two_add_bit_lt_two_pow (a b c : ℕ) (i : Fin 2)\n    (h_a : a < 2 ^ b) (h_b : b < c) :\n    a * 2 + i.val < 2^c"}, {"name": "lt_two_pow_of_lt_two_pow_exp_le", "content": "lemma lt_two_pow_of_lt_two_pow_exp_le (x i j: ℕ)\n    (h_x_lt_2_pow_i: x < 2^i) (h_i_le_j: i ≤ j): x < 2^j"}], "used_local_defs": [{"name": "AdditiveNTT.qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "AdditiveNTT.intermediateNormVpoly", "content": "noncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)"}, {"name": "AdditiveNTT.intermediateNovelBasisX", "content": "noncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))"}], "used_local_lemmas": [{"name": "AdditiveNTT.Polynomial.foldl_comp", "content": "omit [Fintype L] [DecidableEq L] in\ntheorem Polynomial.foldl_comp (n : ℕ) (f : Fin n → L[X]) : ∀ initInner initOuter: L[X],\n    Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter.comp initInner)\n    = (Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter)).comp initInner"}, {"name": "AdditiveNTT.intermediateNormVpoly_comp_qmap", "content": "theorem intermediateNormVpoly_comp_qmap (i : Fin (ℓ))\n    (k : Fin (ℓ - i - 1)) : -- corresponds to intermediateNormVpoly_comp"}, {"name": "AdditiveNTT.intermediateNormVpoly_comp_qmap_helper", "content": "theorem intermediateNormVpoly_comp_qmap_helper (i : Fin (ℓ))\n    (k : Fin (ℓ - (↑i + 1))) :\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate\n      ⟨↑i + 1, by omega⟩ (k:=⟨k, by simp only; omega⟩)).comp (qMap 𝔽q β ⟨↑i, by omega⟩) =\n    intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate\n      ⟨↑i, by omega⟩ ⟨k + 1, by simp only; omega⟩"}], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis\n\nimport Mathlib.Tactic\n\nimport Mathlib.Data.Finsupp.Defs\n\nimport Mathlib.LinearAlgebra.LinearIndependent.Defs\n\nopen Polynomial AdditiveNTT Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\nsection IntermediateStructures\n\nnoncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))\n\nnoncomputable section DomainBijection\n\nend DomainBijection\n\nnoncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)\n\nnoncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))", "target_theorem": "lemma odd_index_intermediate_novel_basis_decomposition\n    (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :\n    intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2 + 1, by\n      apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)\n    ⟩  = X * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by omega⟩ ⟨j, by\n      apply lt_two_pow_of_lt_two_pow_exp_le j (ℓ-i-1) (ℓ-(i+1)) (by omega) (by omega)\n    ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩) :=", "ground_truth_proof": ":= by\n  unfold intermediateNovelBasisX\n  rw [prod_comp]\n  -- ∏ k ∈ Fin (ℓ - i), (Wₖ⁽ⁱ⁾(X))^((2j₊₁)ₖ)\n  -- = X * ∏ k ∈ Fin (ℓ - (i+1)), (Wₖ⁽ⁱ⁺¹⁾(X))^((j)ₖ) ∘ q⁽ⁱ⁾(X)\n  simp only [pow_comp]\n\n  conv_rhs =>\n    enter [2]\n    enter [2, x, 1]\n    rw [intermediateNormVpoly_comp_qmap_helper 𝔽q β h_ℓ_add_R_rate\n      ⟨i, by omega⟩ ⟨x, by simp only; omega⟩]\n\n  -- ⊢ ∏ x, intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i, ⋯⟩ x ^ Nat.getBit (↑x) (↑j * 2 + 1) =\n  -- X * ∏ x, intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i, ⋯⟩ ⟨↑x + 1, ⋯⟩ ^ Nat.getBit ↑x ↑j\n\n  set fleft := fun x : Fin (ℓ - ↑i) =>\n    intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩\n      ⟨x, by simp only; omega⟩ ^ Nat.getBit (↑x) (↑j * 2 + 1)\n  have h_n_shift: ℓ - (↑i + 1) + 1 = ℓ - ↑i := by omega\n  have h_fin_n_shift: Fin (ℓ - (↑i + 1) + 1) = Fin (ℓ - ↑i) := by\n    rw [h_n_shift]\n  have h_left_prod_shift :=\n  Fin.prod_univ_succ (M:=L[X]) (n:=ℓ - (↑i + 1)) (f:=fun x => fleft ⟨x, by omega⟩)\n\n  have h_lhs_prod_eq: ∏ x : Fin (ℓ - ↑i),\n    fleft x = ∏ x : Fin (ℓ - (↑i + 1) + 1), fleft ⟨x, by omega⟩ := by\n    exact Eq.symm (Fin.prod_congr' fleft h_n_shift)\n\n  rw [←h_lhs_prod_eq] at h_left_prod_shift\n  rw [h_left_prod_shift]\n\n  have fleft_0_eq_X: fleft ⟨(0: Fin (ℓ - (↑i + 1) + 1)), by omega⟩ = X := by\n    unfold fleft\n    simp only\n    have h_exp: Nat.getBit (0: Fin (ℓ - (↑i + 1) + 1)) (↑j * 2 + 1) = 1 := by\n      simp only [Fin.coe_ofNat_eq_mod, Nat.zero_mod]\n      unfold Nat.getBit\n      simp only [Nat.shiftRight_zero, Nat.and_one_is_mod, Nat.mul_add_mod_self_right, Nat.mod_succ]\n    rw [h_exp]\n    simp only [pow_one, Fin.coe_ofNat_eq_mod, Nat.zero_mod]\n    unfold intermediateNormVpoly\n    simp only [Fin.foldl_zero]\n\n  rw [fleft_0_eq_X]\n  congr -- apply Finset.prod_congr rfl\n  funext x\n  simp only [Fin.val_succ]\n  unfold fleft\n  simp only\n  have h_exp_eq: Nat.getBit (↑x + 1) (↑j * 2 + 1) = Nat.getBit ↑x ↑j := by\n    have h_num_eq: j.val * 2 = 2 * j.val := by omega\n    rw [h_num_eq]\n    apply Nat.getBit_eq_succ_getBit_of_mul_two_add_one (k:=↑x) (n:=↑j)\n\n  rw [h_exp_eq]", "nesting_depth": 5, "transitive_dep_count": 50, "subset_aristotle": false, "category": "Applied verif."}
{"id": 10, "thm_name": "AdditiveNTT.finToBinaryCoeffs_sDomainToFin", "thm_stmt": "omit h_β₀_eq_1 in\nlemma finToBinaryCoeffs_sDomainToFin (i : Fin r) (h_i : i < ℓ + R_rate)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate i) :\n    let pointFinIdx := (sDomainToFin 𝔽q β h_ℓ_add_R_rate i h_i) x\n    finToBinaryCoeffs 𝔽q (i := i) (idx :=pointFinIdx) =\n    (sDomain_basis 𝔽q β\n    h_ℓ_add_R_rate i h_i).repr x", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean", "imports": ["import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import Mathlib.Data.Finsupp.Defs", "import Mathlib.Tactic", "import ArkLib.Data.Nat.Bitwise", "import Mathlib.LinearAlgebra.LinearIndependent.Defs"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Submodule.map", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Module.Basis", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Disjoint", "module": "Mathlib.Order.Disjoint"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "LinearEquiv", "module": "Mathlib.Algebra.Module.Equiv.Defs"}, {"name": "LinearEquiv.ofBijective", "module": "Mathlib.Algebra.Module.Submodule.Equiv"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap.codRestrict", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "LinearMap.ker", "module": "Mathlib.Algebra.Module.Submodule.Ker"}, {"name": "Module.Basis.span", "module": "Mathlib.LinearAlgebra.Basis.Basic"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.subtype", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Subsingleton", "module": "Init.Core"}, {"name": "Units", "module": "Mathlib.Algebra.Group.Units.Defs"}, {"name": "Units.mk0", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "Units.val", "module": "Mathlib.Algebra.Group.Units.Defs"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "polyEvalLinearMap", "content": "noncomputable def polyEvalLinearMap {L 𝔽q : Type*} [Field L] [Field 𝔽q] [Algebra 𝔽q L]\n  (p : L[X]) (hp_add : IsLinearMap 𝔽q (fun x : L => p.eval x)) : L →ₗ[𝔽q] L :=\n{\n  toFun    := fun x => p.eval x,\n  map_add' := hp_add.map_add,\n  map_smul' := hp_add.map_smul\n}"}, {"name": "binaryFinMapToNat", "content": "def binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary : ∀ j: Fin n, m j ≤ 1) : Fin (2^n) :="}], "lib_lemmas": [{"name": "Fintype.card_le_one_iff_subsingleton", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "Fintype.card_units", "module": "Mathlib.Data.Fintype.Units"}, {"name": "Nat.le_of_eq", "module": "Init.Data.Nat.Basic"}, {"name": "Subsingleton.elim", "module": "Init.Core"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "Decidable.not_not", "module": "Init.PropLemmas"}, {"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.is_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "getBit_of_binaryFinMapToNat", "content": "lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :\n    ∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val\n      = if h_k: k < n then m ⟨k, by omega⟩ else 0"}, {"name": "and_two_pow_eq_zero_of_getBit_0", "content": "lemma and_two_pow_eq_zero_of_getBit_0 {n i : ℕ} (h_getBit : getBit i n = 0)\n    : n &&& (2 ^ i) = 0"}, {"name": "and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "getBit_of_lt_two_pow", "content": "lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :\n  getBit k a = if k < n then getBit k a else 0"}, {"name": "getBit_eq_testBit", "content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"}, {"name": "getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "getBit_of_xor", "content": "lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m"}, {"name": "getBit_zero_eq_zero", "content": "lemma getBit_zero_eq_zero {k : Nat} : getBit k 0 = 0"}, {"name": "sum_of_and_eq_zero_is_xor", "content": "lemma sum_of_and_eq_zero_is_xor {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ^^^ m"}, {"name": "sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||"}, {"name": "xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}], "used_local_defs": [{"name": "AdditiveNTT.sDomain", "content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n  let W_i_norm := normalizedW 𝔽q β i\n  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n    AdditiveNTT.normalizedW_is_additive 𝔽q β i\n  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)\n    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)"}, {"name": "AdditiveNTT.sBasis", "content": "def sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=\n  fun k => β ⟨i + k.val, by admit /- proof elided -/\n  ⟩"}, {"name": "AdditiveNTT.sDomain_basis", "content": "noncomputable def sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    Basis (Fin (ℓ + R_rate - i)) 𝔽q (\n      sDomain 𝔽q β h_ℓ_add_R_rate i) :="}, {"name": "AdditiveNTT.splitPointIntoCoeffs", "content": "def splitPointIntoCoeffs (i : Fin r) (h_i : i < ℓ + R_rate)\n  (x : sDomain 𝔽q β h_ℓ_add_R_rate i) :\n  Fin (ℓ + R_rate - i.val) → ℕ := fun j =>\n    if ((sDomain_basis 𝔽q β\n    h_ℓ_add_R_rate i h_i).repr x j = 0) then\n      0 else 1"}, {"name": "AdditiveNTT.sDomainToFin", "content": "noncomputable def sDomainToFin (i : Fin r) (h_i : i < ℓ + R_rate)\n  (x : sDomain 𝔽q β h_ℓ_add_R_rate i) :\n  Fin (2^(ℓ + R_rate - i.val)) :="}, {"name": "AdditiveNTT.finToBinaryCoeffs", "content": "def finToBinaryCoeffs (i : Fin r) (idx : Fin (2 ^ (ℓ + R_rate - i.val))) :\n  Fin (ℓ + R_rate - i.val) → 𝔽q := fun j =>\n    if (Nat.getBit (k:=j) (n:=idx)) = 1 then (1 : 𝔽q) else (0 : 𝔽q)"}], "used_local_lemmas": [{"name": "AdditiveNTT.𝔽q_element_eq_zero_or_eq_one", "content": "omit h_Fq_char_prime in\nlemma 𝔽q_element_eq_zero_or_eq_one : ∀ c: 𝔽q, c = 0 ∨ c = 1"}], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis\n\nimport Mathlib.Tactic\n\nimport Mathlib.Data.Finsupp.Defs\n\nimport Mathlib.LinearAlgebra.LinearIndependent.Defs\n\nopen Polynomial AdditiveNTT Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\nsection IntermediateStructures\n\nnoncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n  let W_i_norm := normalizedW 𝔽q β i\n  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n    AdditiveNTT.normalizedW_is_additive 𝔽q β i\n  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)\n    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)\n\ndef sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=\n  fun k => β ⟨i + k.val, by admit /- proof elided -/\n  ⟩\n\nnoncomputable def sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    Basis (Fin (ℓ + R_rate - i)) 𝔽q (\n      sDomain 𝔽q β h_ℓ_add_R_rate i) :=\n\nnoncomputable section DomainBijection\n\ndef splitPointIntoCoeffs (i : Fin r) (h_i : i < ℓ + R_rate)\n  (x : sDomain 𝔽q β h_ℓ_add_R_rate i) :\n  Fin (ℓ + R_rate - i.val) → ℕ := fun j =>\n    if ((sDomain_basis 𝔽q β\n    h_ℓ_add_R_rate i h_i).repr x j = 0) then\n      0 else 1\n\nnoncomputable def sDomainToFin (i : Fin r) (h_i : i < ℓ + R_rate)\n  (x : sDomain 𝔽q β h_ℓ_add_R_rate i) :\n  Fin (2^(ℓ + R_rate - i.val)) :=\n\ndef finToBinaryCoeffs (i : Fin r) (idx : Fin (2 ^ (ℓ + R_rate - i.val))) :\n  Fin (ℓ + R_rate - i.val) → 𝔽q := fun j =>\n    if (Nat.getBit (k:=j) (n:=idx)) = 1 then (1 : 𝔽q) else (0 : 𝔽q)", "target_theorem": "omit h_β₀_eq_1 in\nlemma finToBinaryCoeffs_sDomainToFin (i : Fin r) (h_i : i < ℓ + R_rate)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate i) :\n    let pointFinIdx :=", "ground_truth_proof": ":= (sDomainToFin 𝔽q β h_ℓ_add_R_rate i h_i) x\n    finToBinaryCoeffs 𝔽q (i := i) (idx :=pointFinIdx) =\n    (sDomain_basis 𝔽q β\n    h_ℓ_add_R_rate i h_i).repr x:= by\n  simp only\n  ext j\n  -- Unfold the definitions to get to the core logic\n  dsimp [sDomainToFin, finToBinaryCoeffs, splitPointIntoCoeffs]\n  -- `Nat.getBit` is the inverse of `Nat.binaryFinMapToNat`\n  rw [Nat.getBit_of_binaryFinMapToNat]\n  -- Let `c` be the j-th coefficient we are considering\n  set c := (sDomain_basis 𝔽q β\n    h_ℓ_add_R_rate i h_i).repr x j\n  -- Since the field has card 2, `c` must be 0 or 1\n  have hc : c = 0 ∨ c = 1 := by exact 𝔽q_element_eq_zero_or_eq_one 𝔽q c\n    -- exact ((Fintype.card_eq_two_iff _).mp h_Fq_card_eq_2).right c\n  -- We can now split on whether c is 0 or 1\n  rcases hc with h_c_zero | h_c_one\n  · -- Case 1: The coefficient is 0\n    simp only [Fin.is_lt, ↓reduceDIte, Fin.eta, h_c_zero, ite_eq_right_iff, one_ne_zero, imp_false,\n      ne_eq]\n    unfold splitPointIntoCoeffs\n    simp only [ite_eq_right_iff, zero_ne_one, imp_false, Decidable.not_not]\n    omega\n  · -- Case 2: The coefficient is 1\n    simp only [Fin.is_lt, ↓reduceDIte, Fin.eta, h_c_one, ite_eq_left_iff, zero_ne_one, imp_false,\n      Decidable.not_not]\n    unfold splitPointIntoCoeffs\n    simp only [ite_eq_right_iff, zero_ne_one, imp_false, ne_eq]\n    change ¬(c) = 0\n    rw [h_c_one]\n    exact one_ne_zero", "nesting_depth": 5, "transitive_dep_count": 84, "subset_aristotle": false, "category": "Applied verif."}
{"id": 11, "thm_name": "AdditiveNTT.sDomain_eq_image_of_upper_span", "thm_stmt": "lemma sDomain_eq_image_of_upper_span (i : Fin r) (h_i : i < ℓ + R_rate) :\n    let V_i := Submodule.span 𝔽q (Set.range (sBasis β h_ℓ_add_R_rate i h_i))\n    let W_i_map := polyEvalLinearMap (normalizedW 𝔽q β i)\n      (normalizedW_is_additive 𝔽q β i)\n    sDomain 𝔽q β h_ℓ_add_R_rate i\n    = Submodule.map W_i_map V_i", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean", "imports": ["import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import Mathlib.Data.Finsupp.Defs", "import Mathlib.Tactic", "import Mathlib.LinearAlgebra.LinearIndependent.Defs"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Submodule.map", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Polynomial.rootMultiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.algEquivOfCompEqX", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}], "used_repo_defs": [{"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "polyEvalLinearMap", "content": "noncomputable def polyEvalLinearMap {L 𝔽q : Type*} [Field L] [Field 𝔽q] [Algebra 𝔽q L]\n  (p : L[X]) (hp_add : IsLinearMap 𝔽q (fun x : L => p.eval x)) : L →ₗ[𝔽q] L :=\n{\n  toFun    := fun x => p.eval x,\n  map_add' := hp_add.map_add,\n  map_smul' := hp_add.map_smul\n}"}, {"name": "g", "content": "def g (n : ℕ) (c : ℕ) (x : ℕ) := (x * x + c) % n"}, {"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}, {"name": "algEquivAevalXSubC", "content": "@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :="}], "lib_lemmas": [{"name": "Fin.mk_le_of_le_val", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.mk_lt_of_lt_val", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.lt_sub_of_add_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Set.mem_Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_range", "module": "Mathlib.Data.Set.Operations"}, {"name": "Fin.lt_trans", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.zero_le", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.le_of_not_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Set.image_union", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_union", "module": "Mathlib.Data.Set.Basic"}, {"name": "Submodule.eq_bot_iff", "module": "Mathlib.Algebra.Module.Submodule.Lattice"}, {"name": "Submodule.map_sup", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Submodule.mem_map", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Submodule.span_union", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "bot_sup_eq", "module": "Mathlib.Order.BoundedOrder.Lattice"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "normalizedWᵢ_vanishing", "content": "lemma normalizedWᵢ_vanishing (i : Fin r) :\n  ∀ u ∈ U 𝔽q β i, (normalizedW 𝔽q β i).eval u = 0"}, {"name": "Wᵢ_vanishing", "content": "lemma Wᵢ_vanishing (i : Fin r) :\n  ∀ u ∈ U 𝔽q β i, (W 𝔽q β i).eval u = 0"}, {"name": "normalizedW_is_additive", "content": "omit hF₂ in\ntheorem normalizedW_is_additive (i : Fin r) :\n  IsLinearMap 𝔽q (f := fun x ↦ (normalizedW 𝔽q β i).eval x)"}, {"name": "normalizedW_is_linear_map", "content": "theorem normalizedW_is_linear_map (i : Fin r) :\n  IsLinearMap 𝔽q (f := fun inner_p ↦ (normalizedW 𝔽q β i).comp inner_p)"}, {"name": "W_linearity", "content": "theorem W_linearity (i : Fin r)\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p)"}, {"name": "inductive_linear_map_W", "content": "omit hF₂ in\nlemma inductive_linear_map_W (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (i + 1)).comp inner_p)"}, {"name": "inductive_rec_form_W_comp", "content": "omit h_Fq_char_prime hF₂ in\nlemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])\n      (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)"}, {"name": "Wᵢ_eval_βᵢ_neq_zero", "content": "lemma Wᵢ_eval_βᵢ_neq_zero\n    (i : Fin r): (W 𝔽q β i).eval (β i) ≠ 0"}, {"name": "βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "W_prod_comp_decomposition", "content": "lemma W_prod_comp_decomposition\n    (i : Fin r) (hi : i > 0) :\n    (W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))"}, {"name": "W_splits", "content": "lemma W_splits (i : Fin r) : (W 𝔽q β i).Splits (RingHom.id L)"}, {"name": "rootMultiplicity_prod_W_comp_X_sub_C", "content": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0"}, {"name": "Prod_W_comp_X_sub_C_ne_zero", "content": "omit [DecidableEq L] h_Fq_char_prime hF₂ hβ_lin_indep in\nlemma Prod_W_comp_X_sub_C_ne_zero (i : Fin r) :\n    (univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i))) ≠ 0"}, {"name": "W_ne_zero", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W_ne_zero (i : Fin r) : (W 𝔽q β i) ≠ 0"}, {"name": "roots_comp_X_sub_C", "content": "lemma roots_comp_X_sub_C (p : L[X]) (a : L) :\n    (p.comp (X - C a)).roots = p.roots.map (fun r => r + a)"}, {"name": "rootMultiplicity_comp_X_sub_C", "content": "lemma rootMultiplicity_comp_X_sub_C (p : L[X]) (a x : L) :\n    rootMultiplicity x (p.comp (X - C a)) = rootMultiplicity (x - a) p"}, {"name": "comp_X_sub_C_eq_zero_iff", "content": "omit [Fintype L] [DecidableEq L] in\nlemma comp_X_sub_C_eq_zero_iff (p : L[X]) (a : L) :\n  p.comp (X - C a) = 0 ↔ p = 0"}, {"name": "rootMultiplicity_W", "content": "lemma rootMultiplicity_W (i : Fin r) (a : L) :\n    rootMultiplicity a (W 𝔽q β i) = if a ∈ (U 𝔽q β i : Set L) then 1 else 0"}, {"name": "roots_W", "content": "lemma roots_W (i : Fin r) : -- converts root Multiset into (univ: Uᵢ.val.map)\n  (W 𝔽q β i).roots = (univ : Finset (U 𝔽q β i)).val.map (fun u => u.val)"}, {"name": "root_U_lift_up", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime hβ_lin_indep in\ntheorem root_U_lift_up (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) (x : 𝔽q) :\n  a - x • β i ∈ (U 𝔽q β i) → a ∈ (U 𝔽q β (i+1))"}, {"name": "root_U_lift_down", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n  (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n  a ∈ (U 𝔽q β (i+1)) → ∃! x: 𝔽q, a - x • β i ∈ (U 𝔽q β i)"}, {"name": "W_monic", "content": "lemma W_monic (i : Fin r) : (W 𝔽q β i).Monic"}, {"name": "comp_sub_C_of_linear_eval", "content": "lemma comp_sub_C_of_linear_eval (p : L[X])\n  (h_lin : IsLinearMap 𝔽q (f := fun inner_p ↦ p.comp inner_p)) (a : L) :\n    p.comp (X - C a) = p - C (eval a p)"}], "used_local_defs": [{"name": "AdditiveNTT.sDomain", "content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n  let W_i_norm := normalizedW 𝔽q β i\n  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n    AdditiveNTT.normalizedW_is_additive 𝔽q β i\n  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)\n    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)"}, {"name": "AdditiveNTT.sBasis", "content": "def sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=\n  fun k => β ⟨i + k.val, by admit /- proof elided -/\n  ⟩"}], "used_local_lemmas": [{"name": "AdditiveNTT.sBasis_range_eq", "content": "omit [NeZero r] [Field L] [Fintype L] [DecidableEq L] [Field 𝔽q] [Algebra 𝔽q L] in\nlemma sBasis_range_eq (i : Fin r) (h_i : i < ℓ + R_rate) :\n    β '' Set.Ico i ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩\n    = Set.range (sBasis β h_ℓ_add_R_rate i h_i)"}], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis\n\nimport Mathlib.Tactic\n\nimport Mathlib.Data.Finsupp.Defs\n\nimport Mathlib.LinearAlgebra.LinearIndependent.Defs\n\nopen Polynomial AdditiveNTT Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\nsection IntermediateStructures\n\nnoncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n  let W_i_norm := normalizedW 𝔽q β i\n  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n    AdditiveNTT.normalizedW_is_additive 𝔽q β i\n  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)\n    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)\n\ndef sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=\n  fun k => β ⟨i + k.val, by admit /- proof elided -/\n  ⟩", "target_theorem": "lemma sDomain_eq_image_of_upper_span (i : Fin r) (h_i : i < ℓ + R_rate) :\n    let V_i :=", "ground_truth_proof": ":= Submodule.span 𝔽q (Set.range (sBasis β h_ℓ_add_R_rate i h_i))\n    let W_i_map := polyEvalLinearMap (normalizedW 𝔽q β i)\n      (normalizedW_is_additive 𝔽q β i)\n    sDomain 𝔽q β h_ℓ_add_R_rate i\n    = Submodule.map W_i_map V_i :=\nby\n  -- Proof: U_{ℓ+R} is the direct sum of Uᵢ and Vᵢ.\n  -- Any x in U_{ℓ+R} can be written as u + v where u ∈ Uᵢ and v ∈ Vᵢ.\n  -- Ŵᵢ(x) = Ŵᵢ(u+v) = Ŵᵢ(u) + Ŵᵢ(v) = 0 + Ŵᵢ(v) = Ŵᵢ(v).\n  -- So the image of U_{ℓ+R} is the same as the image of Vᵢ.\n\n  -- Define V_i and W_i_map for use in the proof\n  set V_i := Submodule.span 𝔽q (Set.range (sBasis β h_ℓ_add_R_rate i h_i))\n  set W_i_map := polyEvalLinearMap (normalizedW 𝔽q β i)\n    (normalizedW_is_additive 𝔽q β i)\n\n  -- First, show that U_{ℓ+R} = U_i ⊔ V_i (direct sum)\n  have h_span_supremum_decomposition : U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩\n    = U 𝔽q β i ⊔ V_i := by\n    unfold U\n    -- U_{ℓ+R} is the span of {β₀, ..., β_{ℓ+R-1}}\n    -- U_i is the span of {β₀, ..., β_{i-1}}\n    -- V_i is the span of {β_i, ..., β_{ℓ+R-1}}\n    have h_ico : Set.Ico 0 ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩\n      = Set.Ico 0 i ∪ Set.Ico i ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩ := by\n      ext k\n      simp only [Set.mem_Ico, Fin.zero_le, true_and, Set.mem_union]\n      constructor\n      · intro h\n        by_cases hk : k < i\n        · left; omega\n        · right; exact ⟨Nat.le_of_not_lt hk, by omega⟩\n      · intro h\n        cases h with\n        | inl h => exact Fin.lt_trans h h_i\n        | inr h => exact h.2\n\n    rw [h_ico, Set.image_union, Submodule.span_union]\n    congr\n    -- ⊢ β '' Set.Ico i (ℓ + R_rate)\n    -- = Set.range (sBasis β (h_ℓ_add_R_rate:=h_ℓ_add_R_rate) i h_i)\n    -- Now how that the image of Set.Ico i (ℓ + R_rate)\n    -- (from the definition of U_{ℓ+R}) is the same as V_i\n    rw [sBasis_range_eq β h_ℓ_add_R_rate i h_i]\n\n  -- Now show that the image of U_{ℓ+R} under W_i_map is the same as the image of V_i\n  rw [sDomain, h_span_supremum_decomposition, Submodule.map_sup]\n\n  -- The image of U_i under W_i_map is {0} because W_i vanishes on U_i\n  have h_U_i_image : Submodule.map W_i_map (U 𝔽q β i) = ⊥ := by\n    -- Show that any element in the image is 0\n    apply (Submodule.eq_bot_iff _).mpr\n    intro x hx\n    -- x ∈ Submodule.map W_i_map (U 𝔽q β i) means x = W_i_map(y) for some y ∈ U_i\n    rcases Submodule.mem_map.mp hx with ⟨y, hy, rfl⟩\n    -- Show that W_i_map y = 0 for any y ∈ U_i\n    have h_eval_zero : (normalizedW 𝔽q β i).eval y = 0 :=\n      normalizedWᵢ_vanishing 𝔽q β i y hy\n    exact h_eval_zero\n\n  -- Combine the results: ⊥ ⊔ V = V\n  rw [h_U_i_image]\n  rw [bot_sup_eq]", "nesting_depth": 11, "transitive_dep_count": 81, "subset_aristotle": false, "category": "Applied verif."}
{"id": 12, "thm_name": "AdditiveNTT.initial_tiled_coeffs_correctness", "thm_stmt": "omit [DecidableEq 𝔽q] hF₂ in\nlemma initial_tiled_coeffs_correctness (h_ℓ : ℓ ≤ r) (a : Fin (2 ^ ℓ) → L) :\n    let b: Fin (2^(ℓ + R_rate)) → L := tileCoeffs a\n    additiveNTTInvariant 𝔽q β h_ℓ_add_R_rate b a (i := ⟨ℓ, by omega⟩)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean", "imports": ["import Mathlib.Data.Finsupp.Defs", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Tactic", "import ArkLib.Data.Nat.Bitwise", "import Mathlib.LinearAlgebra.LinearIndependent.Defs", "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import ArkLib.Data.Polynomial.Frobenius"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "context", "module": "Examples.FrankingProtocol"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Fin.mk", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin.isValue", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "Finset.sum", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Polynomial.rootMultiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.algEquivOfCompEqX", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}], "used_repo_defs": [{"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "getLowBits", "content": "def getLowBits (numLowBits : ℕ) (n : ℕ) := n &&& ((1 <<< numLowBits) - 1)"}, {"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}, {"name": "algEquivAevalXSubC", "content": "@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :="}], "lib_lemmas": [{"name": "Fintype.card_pos", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Polynomial.C_1", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.C_mul", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_pow", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.mul_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.prod_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.sub_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "inv_eq_one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "map_pow", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_inv_cancel₀", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_pow", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_pow_sub_one", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_sub", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_pow", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.mk_eq_mk", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.eval_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "div_one", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Fin.coe_cast", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.coe_castSucc", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.coe_ofNat_eq_mod", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.foldl_succ_last", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.foldl_zero", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_last", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "tsub_zero", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Fin.mk_zero'", "module": "Mathlib.Data.Fin.Basic"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin.val_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.cast_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.prod_congr'", "module": "Mathlib.Algebra.BigOperators.Fin"}, {"name": "Fin.prod_univ_zero", "module": "Mathlib.Algebra.BigOperators.Fin"}, {"name": "Fin.sum_congr'", "module": "Mathlib.Algebra.BigOperators.Fin"}, {"name": "Fin.sum_univ_one", "module": "Mathlib.Algebra.BigOperators.Fin"}, {"name": "Nat.lt_one_iff", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.lt_two_pow_self", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.zero_or", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_shiftLeft", "module": "Init.Data.Nat.Lemmas"}, {"name": "Polynomial.eval_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_mul", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_one", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "pow_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "tsub_self", "module": "Mathlib.Algebra.Order.Sub.Basic"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "W_linear_comp_decomposition", "content": "omit hF₂ in\ntheorem W_linear_comp_decomposition (i : Fin r) (h_i_add_1 : i + 1 < r) :\n    ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)"}, {"name": "W_linearity", "content": "theorem W_linearity (i : Fin r)\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p)"}, {"name": "inductive_linear_map_W", "content": "omit hF₂ in\nlemma inductive_linear_map_W (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (i + 1)).comp inner_p)"}, {"name": "inductive_rec_form_W_comp", "content": "omit h_Fq_char_prime hF₂ in\nlemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])\n      (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)"}, {"name": "Wᵢ_eval_βᵢ_neq_zero", "content": "lemma Wᵢ_eval_βᵢ_neq_zero\n    (i : Fin r): (W 𝔽q β i).eval (β i) ≠ 0"}, {"name": "βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "W_prod_comp_decomposition", "content": "lemma W_prod_comp_decomposition\n    (i : Fin r) (hi : i > 0) :\n    (W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))"}, {"name": "W_splits", "content": "lemma W_splits (i : Fin r) : (W 𝔽q β i).Splits (RingHom.id L)"}, {"name": "rootMultiplicity_prod_W_comp_X_sub_C", "content": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0"}, {"name": "Prod_W_comp_X_sub_C_ne_zero", "content": "omit [DecidableEq L] h_Fq_char_prime hF₂ hβ_lin_indep in\nlemma Prod_W_comp_X_sub_C_ne_zero (i : Fin r) :\n    (univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i))) ≠ 0"}, {"name": "W_ne_zero", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W_ne_zero (i : Fin r) : (W 𝔽q β i) ≠ 0"}, {"name": "roots_comp_X_sub_C", "content": "lemma roots_comp_X_sub_C (p : L[X]) (a : L) :\n    (p.comp (X - C a)).roots = p.roots.map (fun r => r + a)"}, {"name": "rootMultiplicity_comp_X_sub_C", "content": "lemma rootMultiplicity_comp_X_sub_C (p : L[X]) (a x : L) :\n    rootMultiplicity x (p.comp (X - C a)) = rootMultiplicity (x - a) p"}, {"name": "comp_X_sub_C_eq_zero_iff", "content": "omit [Fintype L] [DecidableEq L] in\nlemma comp_X_sub_C_eq_zero_iff (p : L[X]) (a : L) :\n  p.comp (X - C a) = 0 ↔ p = 0"}, {"name": "rootMultiplicity_W", "content": "lemma rootMultiplicity_W (i : Fin r) (a : L) :\n    rootMultiplicity a (W 𝔽q β i) = if a ∈ (U 𝔽q β i : Set L) then 1 else 0"}, {"name": "roots_W", "content": "lemma roots_W (i : Fin r) : -- converts root Multiset into (univ: Uᵢ.val.map)\n  (W 𝔽q β i).roots = (univ : Finset (U 𝔽q β i)).val.map (fun u => u.val)"}, {"name": "root_U_lift_up", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime hβ_lin_indep in\ntheorem root_U_lift_up (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) (x : 𝔽q) :\n  a - x • β i ∈ (U 𝔽q β i) → a ∈ (U 𝔽q β (i+1))"}, {"name": "root_U_lift_down", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n  (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n  a ∈ (U 𝔽q β (i+1)) → ∃! x: 𝔽q, a - x • β i ∈ (U 𝔽q β i)"}, {"name": "W_monic", "content": "lemma W_monic (i : Fin r) : (W 𝔽q β i).Monic"}, {"name": "comp_sub_C_of_linear_eval", "content": "lemma comp_sub_C_of_linear_eval (p : L[X])\n  (h_lin : IsLinearMap 𝔽q (f := fun inner_p ↦ p.comp inner_p)) (a : L) :\n    p.comp (X - C a) = p - C (eval a p)"}, {"name": "prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L", "content": "theorem prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L\n  (p : L[X]) :\n  (∏ c ∈ (Finset.univ : Finset Fq), (p - Polynomial.C (algebraMap Fq L c))) =\n    p^(Fintype.card Fq) - p"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X_in_L", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X_in_L :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C (algebraMap Fq L c))) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C c)) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "W₀_eq_X", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W₀_eq_X : W 𝔽q β 0 = X"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}, {"name": "getLowBits_eq_mod_two_pow", "content": "lemma getLowBits_eq_mod_two_pow {numLowBits : ℕ} (n : ℕ) :\n  getLowBits numLowBits n = n % (2 ^ numLowBits)"}, {"name": "getLowBits_lt_two_pow", "content": "lemma getLowBits_lt_two_pow {n : ℕ} (numLowBits : ℕ) :\n    getLowBits numLowBits n < 2 ^ numLowBits"}], "used_local_defs": [{"name": "AdditiveNTT.qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "AdditiveNTT.qCompositionChain", "content": "noncomputable def qCompositionChain (i : Fin r) : L[X] :=\n  match i with\n  | ⟨0, _⟩ => X\n  | ⟨k + 1, h_k_add_1⟩ => (qMap 𝔽q β ⟨k, by admit /- proof elided -/\n  ⟩).comp (qCompositionChain ⟨k, by admit /- proof elided -/\n  ⟩)"}, {"name": "AdditiveNTT.intermediateNormVpoly", "content": "noncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)"}, {"name": "AdditiveNTT.intermediateNovelBasisX", "content": "noncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))"}, {"name": "AdditiveNTT.intermediateEvaluationPoly", "content": "noncomputable def intermediateEvaluationPoly (i : Fin (ℓ + 1))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i))), C (coeffs ⟨j, by admit /- proof elided -/\n  ⟩) *\n    (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate i ⟨j, by admit /- proof elided -/\n    ⟩)"}, {"name": "AdditiveNTT.evaluationPointω", "content": "noncomputable def evaluationPointω (i : Fin (ℓ + 1))\n    (x : Fin (2 ^ (ℓ + R_rate - i))) : L := \n    \n  ∑ (⟨k, hk⟩: Fin (ℓ + R_rate - i)),\n    if Nat.getBit k x.val = 1 then\n      (normalizedW 𝔽q β ⟨i, by admit /- proof elided -/\n      ⟩).eval (β ⟨i + k, by admit /- proof elided -/\n      ⟩)\n    else\n      0"}, {"name": "AdditiveNTT.tileCoeffs", "content": "def tileCoeffs (a : Fin (2 ^ ℓ) → L) : Fin (2^(ℓ + R_rate)) → L :=\n  fun v => a (Fin.mk (v.val % (2^ℓ)) (Nat.mod_lt v.val (pow_pos (zero_lt_two) ℓ)))"}, {"name": "AdditiveNTT.coeffsBySuffix", "content": "def coeffsBySuffix (a : Fin (2 ^ ℓ) → L) (i : Fin (ℓ + 1)) (v : Fin (2 ^ i.val)) :\n  Fin (2 ^ (ℓ - i)) → L :=\n  fun ⟨j, hj⟩ => by admit /- proof elided -/"}, {"name": "AdditiveNTT.additiveNTTInvariant", "content": "def additiveNTTInvariant (evaluation_buffer : Fin (2 ^ (ℓ + R_rate)) → L)\n    (original_coeffs : Fin (2 ^ ℓ) → L) (i : Fin (ℓ + 1)) : Prop :=\n  ∀ (j : Fin (2^(ℓ + R_rate))),\n    let u_b_v := j.val\n    let v: Fin (2^i.val) := ⟨Nat.getLowBits i.val u_b_v, by admit /- proof elided -/\n    ⟩ \n    let u_b := u_b_v / (2^i.val) \n    have h_u_b : u_b = u_b_v / (2^i.val) := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "AdditiveNTT.qMap_comp_normalizedW", "content": "lemma qMap_comp_normalizedW (i : Fin r) (h_i_add_1 : i + 1 < r) :\n  (qMap 𝔽q β i).comp (normalizedW 𝔽q β i) = normalizedW 𝔽q β (i + 1)"}, {"name": "AdditiveNTT.qCompositionChain_eq_foldl", "content": "lemma qCompositionChain_eq_foldl (i : Fin r) :\n  qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i =\n  Fin.foldl (n:=i) (fun acc j =>\n    (qMap 𝔽q β ⟨j, by omega⟩).comp acc) (X)"}, {"name": "AdditiveNTT.normalizedW_eq_qMap_composition", "content": "lemma normalizedW_eq_qMap_composition (ℓ R_rate : ℕ) (i : Fin r) :\n    normalizedW 𝔽q β i = qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i"}, {"name": "AdditiveNTT.base_intermediateNormVpoly", "content": "theorem base_intermediateNormVpoly\n  (k : Fin (ℓ + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨0, by\n    by_contra ht\n    simp only [not_lt, nonpos_iff_eq_zero] at ht\n    contradiction\n  ⟩ ⟨k, by simp only [tsub_zero]; omega⟩ =\n  normalizedW 𝔽q β ⟨k, by omega⟩"}, {"name": "AdditiveNTT.base_intermediateNovelBasisX", "content": "theorem base_intermediateNovelBasisX (j : Fin (2 ^ ℓ)) :\n  intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨0, by\n    by_contra ht\n    simp only [not_lt, nonpos_iff_eq_zero] at ht\n    contradiction\n  ⟩ j =\n  Xⱼ 𝔽q β ℓ (by omega) j"}], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis\n\nimport Mathlib.Tactic\n\nimport Mathlib.Data.Finsupp.Defs\n\nimport Mathlib.LinearAlgebra.LinearIndependent.Defs\n\nopen Polynomial AdditiveNTT Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\nsection IntermediateStructures\n\nnoncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))\n\nnoncomputable def qCompositionChain (i : Fin r) : L[X] :=\n  match i with\n  | ⟨0, _⟩ => X\n  | ⟨k + 1, h_k_add_1⟩ => (qMap 𝔽q β ⟨k, by admit /- proof elided -/\n  ⟩).comp (qCompositionChain ⟨k, by admit /- proof elided -/\n  ⟩)\n\nnoncomputable section DomainBijection\n\nend DomainBijection\n\nnoncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)\n\nnoncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))\n\nnoncomputable def intermediateEvaluationPoly (i : Fin (ℓ + 1))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i))), C (coeffs ⟨j, by admit /- proof elided -/\n  ⟩) *\n    (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate i ⟨j, by admit /- proof elided -/\n    ⟩)\n\nend IntermediateStructures\n\nsection AlgorithmCorrectness\n\nnoncomputable def evaluationPointω (i : Fin (ℓ + 1))\n    (x : Fin (2 ^ (ℓ + R_rate - i))) : L := \n    \n  ∑ (⟨k, hk⟩: Fin (ℓ + R_rate - i)),\n    if Nat.getBit k x.val = 1 then\n      (normalizedW 𝔽q β ⟨i, by admit /- proof elided -/\n      ⟩).eval (β ⟨i + k, by admit /- proof elided -/\n      ⟩)\n    else\n      0\n\ndef tileCoeffs (a : Fin (2 ^ ℓ) → L) : Fin (2^(ℓ + R_rate)) → L :=\n  fun v => a (Fin.mk (v.val % (2^ℓ)) (Nat.mod_lt v.val (pow_pos (zero_lt_two) ℓ)))\n\ndef coeffsBySuffix (a : Fin (2 ^ ℓ) → L) (i : Fin (ℓ + 1)) (v : Fin (2 ^ i.val)) :\n  Fin (2 ^ (ℓ - i)) → L :=\n  fun ⟨j, hj⟩ => by admit /- proof elided -/\n\ndef additiveNTTInvariant (evaluation_buffer : Fin (2 ^ (ℓ + R_rate)) → L)\n    (original_coeffs : Fin (2 ^ ℓ) → L) (i : Fin (ℓ + 1)) : Prop :=\n  ∀ (j : Fin (2^(ℓ + R_rate))),\n    let u_b_v := j.val\n    let v: Fin (2^i.val) := ⟨Nat.getLowBits i.val u_b_v, by admit /- proof elided -/\n    ⟩ \n    let u_b := u_b_v / (2^i.val) \n    have h_u_b : u_b = u_b_v / (2^i.val) := by admit /- proof elided -/", "target_theorem": "omit [DecidableEq 𝔽q] hF₂ in\nlemma initial_tiled_coeffs_correctness (h_ℓ : ℓ ≤ r) (a : Fin (2 ^ ℓ) → L) :\n    let b: Fin (2^(ℓ + R_rate)) → L :=", "ground_truth_proof": ":= tileCoeffs a\n    additiveNTTInvariant 𝔽q β h_ℓ_add_R_rate b a (i := ⟨ℓ, by omega⟩) := by\n    unfold additiveNTTInvariant\n    simp only\n    intro j\n    unfold coeffsBySuffix\n    simp only [tileCoeffs, evaluationPointω, intermediateEvaluationPoly, Fin.eta]\n    have h_ℓ_sub_ℓ: 2^(ℓ - ℓ) = 1 := by norm_num\n\n    set f_right: Fin (2^(ℓ - ℓ)) → L[X] :=\n      fun ⟨x, hx⟩ => C (a ⟨↑x <<< ℓ ||| Nat.getLowBits ℓ (↑j), by\n        simp only [tsub_self, pow_zero, Nat.lt_one_iff] at hx\n        simp only [hx, Nat.zero_shiftLeft, Nat.zero_or]\n        exact Nat.getLowBits_lt_two_pow (numLowBits:=ℓ) (n:=j.val)\n      ⟩) * intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨ℓ, by omega⟩ ⟨x, by omega⟩\n\n    have h_sum_right : ∑ (x: Fin (2^(ℓ - ℓ))), f_right x =\n      C (a ⟨Nat.getLowBits ℓ (↑j), by exact Nat.getLowBits_lt_two_pow ℓ⟩) *\n    intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨ℓ, by omega⟩ 0 := by\n      have h_sum_eq := Fin.sum_congr' (b:=2^(ℓ - ℓ)) (a:=1) (f:=f_right) (by omega)\n      rw [←h_sum_eq]\n      rw [Fin.sum_univ_one]\n      unfold f_right\n      simp only [Fin.isValue, Fin.cast_zero, Fin.coe_ofNat_eq_mod, tsub_self, pow_zero,\n        Nat.zero_mod, Nat.zero_shiftLeft, Nat.zero_or]\n      congr\n\n    rw [h_sum_right]\n\n    set f_left: Fin (ℓ + R_rate - ℓ) → L := fun x =>\n      if Nat.getBit (x.val) (j.val / 2 ^ ℓ) = 1 then\n        eval (β ⟨ℓ + x.val, by omega⟩) (normalizedW 𝔽q β ⟨ℓ, by omega⟩)\n      else 0\n\n    simp only [eval_mul, eval_C]\n\n    have h_eval : eval (Finset.univ.sum f_left) (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate\n      ⟨ℓ, by omega⟩ 0) = 1 := by\n      have h_base_novel_basis := base_intermediateNovelBasisX 𝔽q β\n        h_ℓ_add_R_rate ⟨ℓ, by exact Nat.lt_two_pow_self⟩\n      simp only [intermediateNovelBasisX, Fin.coe_ofNat_eq_mod, tsub_self, pow_zero,\n        Nat.zero_mod]\n\n      set f_inner : Fin (ℓ - ℓ) → L[X] := fun x => intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate\n        ⟨ℓ, by omega⟩ ⟨x, by simp only; omega⟩ ^ Nat.getBit (x.val) 0\n\n      have h_sum_eq := Fin.prod_congr' (b:=ℓ - ℓ) (a:=0) (f:=f_inner) (by omega)\n      simp_rw [←h_sum_eq, Fin.prod_univ_zero]\n      simp only [eval_one]\n\n    rw [h_eval, mul_one]\n    simp only [Nat.getLowBits_eq_mod_two_pow]", "nesting_depth": 14, "transitive_dep_count": 134, "subset_aristotle": false, "category": "Applied verif."}
{"id": 13, "thm_name": "MlPoly.mobius_apply_zeta_apply_eq_id", "thm_stmt": "theorem mobius_apply_zeta_apply_eq_id (n : ℕ) [NeZero n] (r : Fin n) (l : Fin (r.val + 1))\n    (v : Vector R (2 ^ n)) : lagrangeToMono_segment n r l (monoToLagrange_segment n r l v) = v", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/MlPoly/Basic.lean", "imports": ["import ArkLib.Data.Fin.BigOperators", "import ArkLib.Data.List.Lemmas", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.Vector.Basic", "import Mathlib.RingTheory.MvPolynomial.Basic", "import ToMathlib.General"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.ofFin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.ofFn", "module": "Init.Data.List.OfFn"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Fin.isValue", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "List.foldr", "module": "Init.Data.List.Basic"}, {"name": "List.foldl", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}], "lib_lemmas": [{"name": "List.length_ofFn", "module": "Init.Data.List.OfFn"}, {"name": "List.getElem_ofFn", "module": "Init.Data.List.OfFn"}, {"name": "List.get_eq_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "Fin.is_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.ext_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "List.getElem_dropLast", "module": "Init.Data.List.Lemmas"}, {"name": "List.length_dropLast", "module": "Init.Data.List.Lemmas"}, {"name": "add_tsub_cancel_right", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Fin.coe_ofNat_eq_mod", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.ofFn_succ", "module": "Init.Data.List.OfFn"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "BitVec.getElem_ofFin", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.getLsb_eq_getElem", "module": "Init.Data.BitVec.Basic"}, {"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "Fin.getElem_fin", "module": "Init.GetElem"}, {"name": "Nat.sub_lt_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Vector.getElem_ofFn", "module": "Init.Data.Vector.OfFn"}, {"name": "add_sub_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_self", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin.cast_succ_eq", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.cast_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.coe_cast", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.eq_mk_iff_val_eq", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.mk_zero'", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.val_eq_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.foldl_cons", "module": "Init.Data.List.Basic"}, {"name": "List.foldl_nil", "module": "Init.Data.List.Basic"}, {"name": "List.foldr_cons", "module": "Init.Data.List.Basic"}, {"name": "List.foldr_nil", "module": "Init.Data.List.Basic"}, {"name": "List.ofFn_zero", "module": "Init.Data.List.OfFn"}, {"name": "Nat.lt_of_lt_of_eq", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mod_succ", "module": "Init.Data.Nat.Lemmas"}, {"name": "tsub_self", "module": "Mathlib.Algebra.Order.Sub.Basic"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "testBit_true_eq_getBit_eq_1", "content": "lemma testBit_true_eq_getBit_eq_1 (k n : Nat) : n.testBit k = ((Nat.getBit k n) = 1)"}, {"name": "testBit_false_eq_getBit_eq_0", "content": "lemma testBit_false_eq_getBit_eq_0 (k n : Nat) :\n  (n.testBit k = false) = ((Nat.getBit k n) = 0)"}, {"name": "getBit_of_sub_two_pow_of_bit_1", "content": "lemma getBit_of_sub_two_pow_of_bit_1 {n i j: ℕ} (h_getBit_eq_1: getBit i n = 1) :\n  getBit j (n - 2^i) = (if j = i then 0 else getBit j n)"}, {"name": "and_two_pow_eq_two_pow_of_getBit_1", "content": "lemma and_two_pow_eq_two_pow_of_getBit_1 {n i : ℕ} (h_getBit: getBit i n = 1) :\n    n &&& (2 ^ i) = 2 ^ i"}, {"name": "getBit_of_xor", "content": "lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m"}, {"name": "getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "xor_eq_sub_iff_submask", "content": "lemma xor_eq_sub_iff_submask {n m : ℕ} (h: m ≤ n) : n ^^^ m = n - m ↔ n &&& m = m"}, {"name": "sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||"}, {"name": "xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}, {"name": "foldr_split_inner", "content": "theorem foldr_split_inner {α : Type u} {β : Type v} (f : α → β → β) (init : β)\n    (l : List α) (h : l ≠ []): foldr (f:=f) (init:=init) (l)\n    = foldr (f:=f) (init:=f (l.getLast (by omega)) (init)) (l.dropLast)"}, {"name": "append_getLast_dropLast", "content": "theorem append_getLast_dropLast {α : Type u} (l : List α) (h : l ≠ []) :\n  l.dropLast ++ [l.getLast h] = l"}, {"name": "foldl_split_outer", "content": "theorem foldl_split_outer {α : Type u} {β : Type v} (f : α → β → α) (init : α)\n    (l : List β) (h : l ≠ []): foldl (f:=f) (init:=init) (l)\n    = f (foldl (f:=f) (init:=init) (l.dropLast)) (l.getLast (by omega))"}], "used_local_defs": [{"name": "MlPoly", "content": "@[reducible]\ndef MlPoly (R : Type*) (n : ℕ) := Vector R (2 ^ n)"}, {"name": "MlPoly.monoToLagrangeLevel", "content": "@[inline] def monoToLagrangeLevel {n : ℕ} (j : Fin n) : Vector R (2 ^ n) → Vector R (2 ^ n) :=\n  fun v =>\n    let stride : ℕ := 2 ^ j.val    \n    Vector.ofFn (fun i : Fin (2 ^ n) =>\n      if (BitVec.ofFin i).getLsb j then\n        v[i] + v[i - stride]'(Nat.sub_lt_of_lt i.isLt)\n      else\n        v[i])"}, {"name": "MlPoly.lagrangeToMonoLevel", "content": "@[inline] def lagrangeToMonoLevel {n : ℕ} (j : Fin n) : Vector R (2 ^ n) → Vector R (2 ^ n) :=\n  fun v =>\n    let stride : ℕ := 2 ^ j.val  \n    Vector.ofFn (fun i : Fin (2 ^ n) =>\n      if (BitVec.ofFin i).getLsb j then\n        v[i] - v[i - stride]'(Nat.sub_lt_of_lt i.isLt)\n      else\n        v[i])"}, {"name": "MlPoly.forwardRange", "content": "def forwardRange (n : ℕ) (r : Fin (n)) (l : Fin (r.val + 1)) : List (Fin n) :=\n  let len := r.val - l.val + 1\n  List.ofFn (fun (k : Fin len) =>\n    let val := l.val + k.val\n    have h_bound : val < n := by admit /- proof elided -/\n  )"}, {"name": "MlPoly.monoToLagrange_segment", "content": "def monoToLagrange_segment (n : ℕ) (r : Fin n) (l : Fin (r.val + 1)) :\n    Vector R (2 ^ n) → Vector R (2 ^ n) :=\n  let range := forwardRange n r l\n  (range.foldl (fun acc h => monoToLagrangeLevel h acc))"}, {"name": "MlPoly.lagrangeToMono_segment", "content": "def lagrangeToMono_segment (n : ℕ) (r : Fin n) (l : Fin (r.val + 1)) :\n    Vector R (2 ^ n) → Vector R (2 ^ n) :=\n  let range := forwardRange n r l\n  (range.foldr (fun h acc => lagrangeToMonoLevel h acc))"}], "used_local_lemmas": [{"name": "MlPoly.forwardRange_length", "content": "lemma forwardRange_length (n : ℕ) (r : Fin n) (l : Fin (r.val + 1)) :\n    (forwardRange n r l).length = r.val - l.val + 1"}, {"name": "MlPoly.forwardRange_eq_of_r_eq", "content": "lemma forwardRange_eq_of_r_eq (n : ℕ) (r1 r2 : Fin n) (h_r_eq : r1 = r2) (l : Fin (r1.val + 1)) :\n  forwardRange n r1 l = forwardRange n r2 ⟨l, by omega⟩"}, {"name": "MlPoly.forwardRange_getElem", "content": "lemma forwardRange_getElem (n : ℕ) (r : Fin n) (l : Fin (r.val + 1)) (k : Fin (r.val - l.val + 1)) :\n    (forwardRange n r l).get ⟨k, by\n      rw [forwardRange]; simp only [List.length_ofFn]; omega⟩ = ⟨l.val + k, by omega⟩"}, {"name": "MlPoly.forwardRange_succ_right_ne_empty", "content": "lemma forwardRange_succ_right_ne_empty (n : ℕ) (r : Fin (n - 1)) (l : Fin (r.val + 1)) :\n  forwardRange n ⟨r + 1, by omega⟩ ⟨l, by simp only; omega⟩ ≠ []"}, {"name": "MlPoly.forwardRange_dropLast", "content": "lemma forwardRange_dropLast (n : ℕ) (r : Fin (n - 1)) (l : Fin (r.val + 1)) :\n    (forwardRange n ⟨r + 1, by omega⟩ ⟨l, by simp only; omega⟩).dropLast\n    = forwardRange n ⟨r, by omega⟩ ⟨l, by simp only [Fin.is_lt]⟩"}, {"name": "MlPoly.testBit_of_sub_two_pow_of_bit_1", "content": "lemma testBit_of_sub_two_pow_of_bit_1 {n i : ℕ} (h_testBit_eq_1 : (n).testBit i = true) :\n  (n - 2^i).testBit i = false"}, {"name": "MlPoly.lagrangeToMonoLevel_monoToLagrangeLevel_id", "content": "theorem lagrangeToMonoLevel_monoToLagrangeLevel_id (v : Vector R (2 ^ n)) (i : Fin n) :\n  lagrangeToMonoLevel i (monoToLagrangeLevel i v) = v"}], "local_ctx": "import ArkLib.Data.Nat.Bitwise\n\nimport Mathlib.RingTheory.MvPolynomial.Basic\n\nimport ToMathlib.General\n\nimport ArkLib.Data.Fin.BigOperators\n\nimport ArkLib.Data.List.Lemmas\n\nimport ArkLib.Data.Vector.Basic\n\n@[reducible]\ndef MlPoly (R : Type*) (n : ℕ) := Vector R (2 ^ n) \n\nvariable {R : Type*} {n : ℕ}\n\nnamespace MlPoly\n\nsection MlPolyInstances\n\nend MlPolyInstances\n\nsection MlPolyMonomialBasisAndEvaluations\n\nvariable [CommRing R]\n\nvariable {S : Type*} [CommRing S]\n\nvariable {S : Type*} [CommRing S]\n\nend MlPolyMonomialBasisAndEvaluations\n\nend MlPoly\n\nnamespace MlPolyEval\n\nsection MlPolyEvalInstances\n\nend MlPolyEvalInstances\n\nsection MlPolyLagrangeBasisAndEvaluations\n\nvariable [CommRing R]\n\nvariable {S : Type*} [CommRing S]\n\nvariable {S : Type*} [CommRing S]\n\nend MlPolyLagrangeBasisAndEvaluations\n\nend MlPolyEval\n\nnamespace MlPoly\n\nvariable {R : Type*} [AddCommGroup R]\n\n@[inline] def monoToLagrangeLevel {n : ℕ} (j : Fin n) : Vector R (2 ^ n) → Vector R (2 ^ n) :=\n  fun v =>\n    let stride : ℕ := 2 ^ j.val    \n    Vector.ofFn (fun i : Fin (2 ^ n) =>\n      if (BitVec.ofFin i).getLsb j then\n        v[i] + v[i - stride]'(Nat.sub_lt_of_lt i.isLt)\n      else\n        v[i])\n\n@[inline] def lagrangeToMonoLevel {n : ℕ} (j : Fin n) : Vector R (2 ^ n) → Vector R (2 ^ n) :=\n  fun v =>\n    let stride : ℕ := 2 ^ j.val  \n    Vector.ofFn (fun i : Fin (2 ^ n) =>\n      if (BitVec.ofFin i).getLsb j then\n        v[i] - v[i - stride]'(Nat.sub_lt_of_lt i.isLt)\n      else\n        v[i])\n\ndef forwardRange (n : ℕ) (r : Fin (n)) (l : Fin (r.val + 1)) : List (Fin n) :=\n  let len := r.val - l.val + 1\n  List.ofFn (fun (k : Fin len) =>\n    let val := l.val + k.val\n    have h_bound : val < n := by admit /- proof elided -/\n  )\n\ndef monoToLagrange_segment (n : ℕ) (r : Fin n) (l : Fin (r.val + 1)) :\n    Vector R (2 ^ n) → Vector R (2 ^ n) :=\n  let range := forwardRange n r l\n  (range.foldl (fun acc h => monoToLagrangeLevel h acc))\n\ndef lagrangeToMono_segment (n : ℕ) (r : Fin n) (l : Fin (r.val + 1)) :\n    Vector R (2 ^ n) → Vector R (2 ^ n) :=\n  let range := forwardRange n r l\n  (range.foldr (fun h acc => lagrangeToMonoLevel h acc))", "target_theorem": "theorem mobius_apply_zeta_apply_eq_id (n : ℕ) [NeZero n] (r : Fin n) (l : Fin (r.val + 1))\n    (v : Vector R (2 ^ n)) : lagrangeToMono_segment n r l (monoToLagrange_segment n r l v) = v :=", "ground_truth_proof": ":= by\n  induction r using Fin.succRecOnSameFinType with\n  | zero =>\n    rw [lagrangeToMono_segment, monoToLagrange_segment, forwardRange]\n    simp only [Fin.coe_ofNat_eq_mod, Nat.zero_mod, Fin.val_eq_zero, tsub_self, zero_add,\n      List.ofFn_succ, Fin.isValue, Fin.cast_zero, Nat.mod_succ, add_zero, Fin.mk_zero',\n      Fin.cast_succ_eq, Fin.val_succ, Fin.coe_cast, List.ofFn_zero, List.foldl_cons, List.foldl_nil,\n      List.foldr_cons, List.foldr_nil]\n    exact lagrangeToMonoLevel_monoToLagrangeLevel_id v 0\n  | succ r1 r1_lt_n h_r1 =>\n    unfold lagrangeToMono_segment monoToLagrange_segment\n    if h_l_eq_r: l.val = (r1 + 1).val then\n      rw [forwardRange]\n      simp only [List.ofFn_succ, Fin.coe_ofNat_eq_mod, Nat.zero_mod, add_zero, Fin.val_succ,\n        List.foldl_cons, List.foldr_cons]\n      simp_rw [h_l_eq_r]\n      simp only [Fin.eta, tsub_self, List.ofFn_zero, List.foldl_nil, List.foldr_nil]\n      exact lagrangeToMonoLevel_monoToLagrangeLevel_id v (r1 + 1)\n    else\n      have h_l_lt_r: l.val < (r1 + 1).val := by omega\n      have h_r1_add_1_val: (r1 + 1).val = r1.val + 1 := by\n        rw [Fin.val_add_one']; omega\n      have h_range_ne_empty: forwardRange n (r1 + 1) l ≠ [] := by\n        have h:= forwardRange_succ_right_ne_empty n\n          (r:=⟨r1, by omega⟩) (l:=⟨l, by simp only; omega⟩)\n        simp only [ne_eq] at h\n        have h_r1_add_1: r1 + 1 = ⟨r1.val + 1, by omega⟩ := by\n          exact Fin.eq_mk_iff_val_eq.mpr h_r1_add_1_val\n        rw [forwardRange_eq_of_r_eq (r1:=r1 + 1) (r2:=⟨r1.val + 1, by omega⟩) (h_r_eq:=h_r1_add_1)]\n        exact h\n      rw [List.foldr_split_inner (h:=h_range_ne_empty)]\n      rw [List.foldl_split_outer (h:=h_range_ne_empty)]\n      rw [lagrangeToMonoLevel_monoToLagrangeLevel_id]\n      have h_inductive := h_r1 (l := ⟨l, by exact Nat.lt_of_lt_of_eq h_l_lt_r h_r1_add_1_val⟩)\n      rw [lagrangeToMono_segment, monoToLagrange_segment] at h_inductive\n      simp only at h_inductive\n      have h_range_droplast: (forwardRange n (r1 + 1) l).dropLast\n        = forwardRange n r1 ⟨↑l, by omega⟩ := by\n        have h := forwardRange_dropLast n (r:=⟨r1, by omega⟩) (l:=⟨l, by simp only; omega⟩)\n        simp only [Fin.eta] at h\n        convert h\n      convert h_inductive", "nesting_depth": 7, "transitive_dep_count": 84, "subset_aristotle": false, "category": "Applied verif."}
{"id": 14, "thm_name": "Nat.getLowBits_succ", "thm_stmt": "lemma getLowBits_succ {n: ℕ} (numLowBits: ℕ) :\n    getLowBits (numLowBits + 1) n = getLowBits numLowBits n\n    + (getBit numLowBits n) <<< numLowBits", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/Nat/Bitwise.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Data.Nat.Bitwise", "import Mathlib.Data.Finsupp.Basic", "import Mathlib.Algebra.Order.Ring.Star", "import Mathlib.Data.Nat.Digits.Defs", "import Mathlib.Algebra.BigOperators.Fin"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.and_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat.binaryRec", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bit", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bodd", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.boddDiv2", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.div2", "module": "Mathlib.Data.Nat.Bits"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.eq_of_testBit_eq", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_two_bne_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.one_and_eq_mod_two", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "beq_eq_beq", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.and_assoc", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_comm", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_and_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_and", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_eq_zero_iff_lt", "module": "Init.Data.Nat.Div.Lemmas"}, {"name": "Nat.pow_lt_pow_right", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftLeft_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_add", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_mod_two_eq_zero", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.shiftRight_or_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_two_not_eq_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.testBit_two_pow_sub_one", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "beq_eq_false_iff_ne", "module": "Init.SimpLemmas"}, {"name": "decide_false", "module": "Init.Core"}, {"name": "decide_true", "module": "Init.Core"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "not_le", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "right_eq_ite_iff", "module": "Init.PropLemmas"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}, {"name": "Nat.add_mul_div_left", "module": "Init.Data.Nat.Div.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.div_add_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mul_add_mod_self_right", "module": "Init.Data.Nat.Div.Basic"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.or_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_or", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_xor", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Bool.toNat_lt", "module": "Init.Data.Bool"}, {"name": "Nat.bit_decomp", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.bit_val", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.mul_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.right_distrib", "module": "Init.Data.Nat.Basic"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.zero_shiftLeft", "module": "Init.Data.Nat.Lemmas"}, {"name": "lt_add_iff_pos_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "lt_self_iff_false", "module": "Mathlib.Order.Basic"}, {"name": "zero_lt_one", "module": "Mathlib.Algebra.Order.ZeroLEOne"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Nat.getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "Nat.getLowBits", "content": "def getLowBits (numLowBits : ℕ) (n : ℕ) := n &&& ((1 <<< numLowBits) - 1)"}], "used_local_lemmas": [{"name": "Nat.getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "Nat.eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "Nat.shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "Nat.and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "Nat.getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "Nat.and_two_pow_eq_zero_of_getBit_0", "content": "lemma and_two_pow_eq_zero_of_getBit_0 {n i : ℕ} (h_getBit : getBit i n = 0)\n    : n &&& (2 ^ i) = 0"}, {"name": "Nat.div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "Nat.and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "Nat.xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}, {"name": "Nat.or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||| m = (n1 ||| m1) * 2 + (bn ||| bm)"}, {"name": "Nat.sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "Nat.sum_of_and_eq_zero_is_xor", "content": "lemma sum_of_and_eq_zero_is_xor {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ^^^ m"}, {"name": "Nat.xor_of_and_eq_zero_is_or", "content": "lemma xor_of_and_eq_zero_is_or {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n ^^^ m = n ||| m"}, {"name": "Nat.getBit_of_or", "content": "lemma getBit_of_or {n m k: ℕ} : getBit k (n ||| m) = getBit k n ||| getBit k m"}, {"name": "Nat.getBit_of_and", "content": "lemma getBit_of_and {n m k: ℕ} : getBit k (n &&& m) = getBit k n &&& getBit k m"}, {"name": "Nat.getBit_of_two_pow_sub_one", "content": "lemma getBit_of_two_pow_sub_one {i k: ℕ} : getBit k (2^i - 1) =\n    if k < i then 1 else 0"}, {"name": "Nat.getBit_of_lowBits", "content": "lemma getBit_of_lowBits {n: ℕ} (numLowBits : ℕ) : ∀ k, getBit k (getLowBits numLowBits n) =\n    if k < numLowBits then getBit k n else 0"}], "local_ctx": "import ArkLib.Data.Fin.BigOperators\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nimport Mathlib.Algebra.Order.Ring.Star\n\nimport Mathlib.Data.Nat.Bitwise\n\nimport Mathlib.Data.Nat.Digits.Defs\n\nimport Mathlib.Data.Finsupp.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Fin\n\nnamespace Nat\n\ndef getBit (k n : Nat) : Nat := (n >>> k) &&& 1\n\ndef getLowBits (numLowBits : ℕ) (n : ℕ) := n &&& ((1 <<< numLowBits) - 1)", "target_theorem": "lemma getLowBits_succ {n: ℕ} (numLowBits: ℕ) :\n    getLowBits (numLowBits + 1) n = getLowBits numLowBits n\n    + (getBit numLowBits n) <<< numLowBits :=", "ground_truth_proof": ":= by\n  apply eq_iff_eq_all_getBits.mpr;\n  intro k\n  have h_getBit_lt_numLowBits: getBit numLowBits n < 2 := by exact getBit_lt_2\n  interval_cases h_getBit: getBit numLowBits n\n  · rw [Nat.zero_shiftLeft]\n    simp only [add_zero]\n    -- ⊢ getLowBits n (numLowBits + 1) >>> k &&& 1 = getLowBits n numLowBits >>> k &&& 1\n    change getBit k (getLowBits (numLowBits + 1) n) = getBit k (getLowBits numLowBits n)\n    have getBit_right := getBit_of_lowBits (n := n) (numLowBits := numLowBits) k\n    have getBit_left := getBit_of_lowBits (n := n) (numLowBits := numLowBits + 1) k\n    rw [getBit_right, getBit_left]\n    if h_k: k < numLowBits then\n      simp only [h_k, ↓reduceIte]\n      have h_k_lt: k < numLowBits + 1 := by omega\n      simp only [h_k_lt, ↓reduceIte]\n    else if h_k_eq: k = numLowBits then\n      simp only [h_k_eq]\n      simp only [lt_add_iff_pos_right, zero_lt_one, ↓reduceIte, lt_self_iff_false]\n      omega\n    else\n      have k_ne_lt: ¬(k < numLowBits) := by omega\n      have k_ne_lt_add_1: ¬(k < numLowBits + 1) := by omega\n      simp only [k_ne_lt_add_1, ↓reduceIte, k_ne_lt]\n  · change getBit k (getLowBits (numLowBits + 1) n)\n      = getBit k (getLowBits numLowBits n + 1 <<< numLowBits)\n    have getBit_left := getBit_of_lowBits (n := n) (numLowBits := numLowBits + 1) k\n    have getBit_right := getBit_of_lowBits (n := n) (numLowBits := numLowBits) k\n    rw [getBit_left]\n\n    have h_and_eq_0 := and_two_pow_eq_zero_of_getBit_0 (n:=getLowBits numLowBits n)\n      (i:=numLowBits) (by\n      simp only [getBit_of_lowBits (n := n) (numLowBits := numLowBits) numLowBits,\n        lt_self_iff_false, ↓reduceIte]\n    )\n    rw [←one_mul (a:=2 ^ numLowBits)] at h_and_eq_0\n    rw [←Nat.shiftLeft_eq (a:=1) (b:=numLowBits)] at h_and_eq_0\n    have h_sum_eq_xor := sum_of_and_eq_zero_is_xor (n:=getLowBits numLowBits n)\n      (m:=1 <<< numLowBits) (h_n_AND_m:=h_and_eq_0)\n    have h_sum_eq_or := xor_of_and_eq_zero_is_or (n:=getLowBits numLowBits n)\n      (m:=1 <<< numLowBits) (h_n_AND_m:=h_and_eq_0)\n    rw [h_sum_eq_or] at h_sum_eq_xor\n    rw [h_sum_eq_xor]\n    rw [getBit_of_or]\n    rw [getBit_of_lowBits]\n    conv_rhs =>\n      enter [2, 2]; rw [Nat.shiftLeft_eq, one_mul]\n    rw [getBit_two_pow]\n\n    if h_k: k < numLowBits then\n      have h_k_lt: k < numLowBits + 1 := by omega\n      simp only [h_k_lt, ↓reduceIte, h_k, beq_iff_eq]\n      have h_k_ne_eq: numLowBits ≠ k := by omega\n      simp only [h_k_ne_eq, ↓reduceIte, Nat.or_zero]\n    else if h_k_eq: k = numLowBits then\n      simp only [h_k_eq, lt_add_iff_pos_right, zero_lt_one, ↓reduceIte, lt_self_iff_false, BEq.rfl,\n        Nat.zero_or]\n      omega\n    else\n      have k_ne_lt: ¬(k < numLowBits) := by omega\n      have k_ne_lt_add_1: ¬(k < numLowBits + 1) := by omega\n      simp only [k_ne_lt_add_1, ↓reduceIte, k_ne_lt, beq_iff_eq, Nat.zero_or, right_eq_ite_iff,\n        zero_ne_one, imp_false, ne_eq]\n      omega", "nesting_depth": 4, "transitive_dep_count": 103, "subset_aristotle": true, "category": "Applied verif."}
{"id": 15, "thm_name": "rsum_eq_t1_square_aux", "thm_stmt": "theorem rsum_eq_t1_square_aux\n  {curBTField : Type*} [Field curBTField] -- curBTField ≃ 𝔽_{2^{2^k}}\n  (u : curBTField) -- here u is already lifted to curBTField\n  (k : ℕ)\n  (x_pow_card : ∀ (x : curBTField), x ^ (2 ^ (2 ^ (k))) = x)\n  (u_ne_zero : u ≠ 0)\n  (trace_map_prop : TraceMapProperty curBTField u k):\n   ∑ j ∈ Finset.Icc 1 (2 ^ (k)), u ^ (2 ^ 2 ^ (k) - 2 ^ j) = u", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Prelude.lean", "imports": ["import ArkLib.Data.Fin.BigOperators", "import Mathlib.FieldTheory.Finite.GaloisField", "import ArkLib.Data.Nat.Bitwise", "import Mathlib.LinearAlgebra.StdBasis"], "used_lib_defs": [{"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.pow_le_pow_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.zero_le", "module": "Init.Prelude"}, {"name": "Finset.mem_Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "Finset.mul_sum", "module": "Mathlib.Algebra.BigOperators.Ring.Finset"}, {"name": "Finset.sum_bij'", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Nat.le_pred_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.le_succ_of_le", "module": "Init.Prelude"}, {"name": "Nat.not_le_of_gt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pos_of_ne_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.range_succ_eq_Icc_zero", "module": "Mathlib.Order.Interval.Finset.Nat"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_lt", "module": "Init.Prelude"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_le_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "inv_pow", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "le_refl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_of_le_of_ne", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_right_inj'", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "pow_le_pow_right₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "pow_sub₀", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "zero_le", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}], "used_local_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}], "local_ctx": "import Mathlib.FieldTheory.Finite.GaloisField\n\nimport ArkLib.Data.Fin.BigOperators\n\nimport ArkLib.Data.Nat.Bitwise\n\nimport Mathlib.LinearAlgebra.StdBasis\n\nnoncomputable section Preliminaries\n\nopen Polynomial\n\nopen AdjoinRoot\n\nopen Module\n\nnotation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1\n\nstructure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1", "target_theorem": "theorem rsum_eq_t1_square_aux\n  {curBTField : Type*} [Field curBTField] -- curBTField ≃ 𝔽_{2^{2^k}}\n  (u : curBTField) -- here u is already lifted to curBTField\n  (k : ℕ)\n  (x_pow_card : ∀ (x : curBTField), x ^ (2 ^ (2 ^ (k))) = x)\n  (u_ne_zero : u ≠ 0)\n  (trace_map_prop : TraceMapProperty curBTField u k):\n   ∑ j ∈ Finset.Icc 1 (2 ^ (k)), u ^ (2 ^ 2 ^ (k) - 2 ^ j) = u :=", "ground_truth_proof": ":= by\n\n  have trace_map_icc_t1 : ∑ j ∈ Finset.Icc 0 (2^(k)-1), u ^ (2^j) = 1 := by\n    rw [←Nat.range_succ_eq_Icc_zero (2^(k)-1), Nat.sub_add_cancel (h:=one_le_two_pow_n (k))]\n    exact trace_map_prop.1\n  have trace_map_icc_t1_inv : ∑ j ∈ Finset.Icc 0 (2^(k)-1), u⁻¹ ^ (2^j) = 1 := by\n    rw [←Nat.range_succ_eq_Icc_zero (2^(k)-1), Nat.sub_add_cancel (h:=one_le_two_pow_n (k))]\n    exact trace_map_prop.2\n\n  calc\n    ∑ j ∈ Finset.Icc 1 (2 ^ (k)), u ^ (2 ^ 2 ^ (k) - 2 ^ j)\n      = ∑ j ∈ Finset.Icc 1 (2 ^ (k)), (u ^ (2 ^ 2 ^ (k)) * ((u) ^ 2 ^ j)⁻¹) := by\n      apply Finset.sum_congr rfl (fun j hj => by\n        simp [Finset.mem_Icc] at hj -- hj : 1 ≤ j ∧ j ≤ 2 ^ (k)\n        have h_gte_0_pow : 2 ^ j ≤ 2 ^ 2 ^ (k) := by\n          apply pow_le_pow_right₀ (by decide) (hj.2)\n        rw [pow_sub₀ (a := u) (ha := u_ne_zero) (h := h_gte_0_pow)]\n      )\n    _ = ∑ j ∈ Finset.Icc 1 (2 ^ (k)), ((u) * ((u) ^ 2 ^ j)⁻¹) := by\n      rw [x_pow_card (u)]\n    _ = u * ∑ j ∈ Finset.Icc 1 (2 ^ (k)), ((u) ^ 2 ^ j)⁻¹ := by rw [Finset.mul_sum]\n    _ = u * ∑ j ∈ Finset.Icc 1 (2 ^ (k)), (((u)⁻¹) ^ 2 ^ j) := by\n      congr\n      ext j\n      rw [←inv_pow (a := u) (n := 2 ^ j)]\n    _ = u * ∑ j ∈ Finset.Icc 0 (2 ^ (k) - 1), ((u⁻¹) ^ 2 ^ j) := by\n      rw [mul_right_inj' (a := u) (ha := u_ne_zero)]\n      apply Finset.sum_bij' (fun i _ => if i = 2^(k) then 0 else i)\n        (fun i _ => if i = 0 then 2^(k) else i)\n        -- hi : Maps to Icc 0 (2^(k))\n      · intro i hi\n        simp [Finset.mem_Icc] at hi ⊢\n        by_cases h : i = 2^(k)\n        · simp [h];\n        · simp [h] -- ⊢ i = 0 → 2 ^ (k) = i\n          intro i_eq\n          have this_is_false : False := by\n            have h1 := hi.left  -- h1 : 1 ≤ i\n            rw [i_eq] at h1     -- h1 : 1 ≤ 0\n            have ne_one_le_zero : ¬(1 ≤ 0) := Nat.not_le_of_gt (by decide)\n            exact ne_one_le_zero h1\n          exact False.elim this_is_false\n      -- hj : Maps back\n      · intro i hi\n        simp [Finset.mem_Icc] at hi -- hi : i ≤ 2 ^ (k) - 1\n        by_cases h : i = 0\n        · simp [h];\n        · simp [h];\n          intro i_eq\n          have this_is_false : False := by\n            rw [i_eq] at hi\n            conv at hi =>\n              lhs\n              rw [←add_zero (a:=2^(k))]\n            -- conv at hi =>\n            --   rhs\n            have zero_lt_2_pow_k_plus_1 : 0 < 2^(k) := by\n              norm_num\n            have h_contra : ¬(2^(k) ≤ 2^(k) - 1) := by\n              apply Nat.not_le_of_gt\n              exact Nat.sub_lt zero_lt_2_pow_k_plus_1 (by norm_num)\n            exact h_contra hi\n          exact False.elim this_is_false\n      -- hij : j (i a) = a\n      · intro i hi -- hi : 1 ≤ i ∧ i ≤ 2 ^ (k)\n        simp [Finset.mem_Icc] at hi\n        by_cases h : i = 2^(k)\n        · simp [h]; exact x_pow_card u\n        · simp [h]\n      -- hji : i (j b) = b\n      · intro i hi\n        simp [Finset.mem_Icc] at hi\n        by_cases h : i = 0\n        · simp [h]\n        · simp only [Finset.mem_Icc, zero_le, true_and]; -- hi : 1 ≤ i ∧ i ≤ 2 ^ (k)\n          -- h : ¬i = 0\n          -- ⊢ (if i = 2 ^ (k) then 0 else i) ≤ 2 ^ (k) - 1\n          split_ifs with h2\n          · -- Case : i = 2 ^ (k)\n            -- Goal : 0 ≤ 2 ^ (k) - 1\n            exact Nat.zero_le _\n          · -- Case : i ≠ 2 ^ (k)\n            -- Goal : i ≤ 2 ^ (k) - 1\n            have : i < 2 ^ (k) := by\n              apply lt_of_le_of_ne hi.right h2\n            exact Nat.le_pred_of_lt this\n      -- h_sum : Values match\n      · intro i hi\n        simp [Finset.mem_Icc] at hi\n        rw [Finset.mem_Icc]\n        split_ifs with h2\n        · -- hi : i ≤ 2 ^ (k) - 1, h2 : i = 0\n          -- ⊢ 1 ≤ 2 ^ (k) ∧ 2 ^ (k) ≤ 2 ^ (k)\n          exact ⟨one_le_two_pow_n (k), le_refl _⟩\n        · -- Case : hi : i ≤ 2 ^ (k) - 1, h2 : ¬i = 0\n          -- ⊢ 1 ≤ i ∧ i ≤ 2 ^ (k)\n          have one_le_i : 1 ≤ i := by\n            apply Nat.succ_le_of_lt\n            exact Nat.pos_of_ne_zero h2\n          have tmp : i ≤ 2^(k):= by\n            calc i ≤ (2^(k)-1).succ := Nat.le_succ_of_le hi\n              _ = 2^(k) := by rw [Nat.succ_eq_add_one, Nat.sub_add_cancel\n                (h:=one_le_two_pow_n (k))]\n          exact ⟨one_le_i, tmp⟩\n    _ = u := by rw [trace_map_icc_t1_inv, mul_one]", "nesting_depth": 2, "transitive_dep_count": 35, "subset_aristotle": true, "category": "Applied verif."}
{"id": 16, "thm_name": "AdditiveNTT.rootMultiplicity_prod_W_comp_X_sub_C", "thm_stmt": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean", "imports": ["import Mathlib.Algebra.Polynomial.Degree.Definitions", "import ArkLib.Data.Fin.BigOperators", "import ArkLib.Data.Polynomial.MonomialBasis", "import Mathlib.LinearAlgebra.StdBasis", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.Polynomial.Frobenius"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Polynomial.rootMultiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Multiset", "module": "Mathlib.Data.Multiset.Defs"}, {"name": "Multiset.count", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.map", "module": "Mathlib.Data.Multiset.MapFold"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "SetLike", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Iff", "module": "Init.Core"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.algEquivOfCompEqX", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "multiplicity", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "EmbeddingLike", "module": "Mathlib.Data.FunLike.Embedding"}, {"name": "CanLift", "module": "Mathlib.Tactic.Lift"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Multiset.filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Finset.val", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Set.InjOn", "module": "Mathlib.Data.Set.Operations"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.val", "module": "Init.Prelude"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}], "lib_lemmas": [{"name": "Finset.prod_eq_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Nat.not_lt_zero", "module": "Init.Prelude"}, {"name": "Polynomial.X_sub_C_ne_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Set.Ico_eq_empty_iff", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Polynomial.X_ne_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_C_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_X_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_sub", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.comp_eq_zero_iff", "module": "Mathlib.Algebra.Polynomial.Degree.Lemmas"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "map_neg", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "sub_eq_neg_self", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "EmbeddingLike.map_eq_zero_iff", "module": "Mathlib.Algebra.Group.Equiv.Defs"}, {"name": "Polynomial.aeval_C", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.aeval_X", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algEquivOfCompEqX_apply", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algebraMap_eq", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.rootMultiplicity_eq_multiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "if_false", "module": "Init.ByCases"}, {"name": "if_true", "module": "Init.ByCases"}, {"name": "map_sub", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "multiplicity_map_eq", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "sub_sub_sub_cancel_right", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Multiset.countP_eq_card_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_map", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.filter_congr", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.count_roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "eq_sub_iff_add_eq", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.image_val_of_injOn", "module": "Mathlib.Data.Finset.Image"}, {"name": "Finset.prod_image", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Polynomial.roots_prod_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Subtype.val_injective", "module": "Mathlib.Data.Subtype"}, {"name": "CanLift.prf", "module": "Mathlib.Tactic.Lift"}, {"name": "Multiset.card_singleton", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.card_zero", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.count_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_singleton", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.count_univ", "module": "Mathlib.Data.Fintype.Basic"}, {"name": "Multiset.count_zero", "module": "Mathlib.Data.Multiset.Count"}, {"name": "SetLike.coe_eq_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "SetLike.mem_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Fin.zero_le", "module": "Init.Data.Fin.Lemmas"}, {"name": "Set.Ico_subset_Ico_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_mono", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.mem_image_of_mem", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.add_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.smul_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span_mono", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.subset_span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "sub_add_cancel", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "Fact.out", "module": "Mathlib.Logic.Basic"}, {"name": "Fin.le_zero_iff'", "module": "Mathlib.Data.Fin.SuccPred"}, {"name": "LinearIndependent.injective", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.compl_eq_univ_diff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Set.empty_subset", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.image_empty", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_subset_image_iff", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.subset_compl_singleton_iff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "linearIndependent_iff_notMem_span", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.Ico_insert_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_singleton", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_union", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Icc", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.union_singleton", "module": "Mathlib.Data.Set.Insert"}, {"name": "Submodule.mem_span_singleton", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.mem_sup", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.smul_mem_iff", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Submodule.span_union", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.sub_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "existsUnique_of_exists_of_unique", "module": "Mathlib.Logic.ExistsUnique"}, {"name": "sub_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_smul", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "sub_sub_sub_cancel_left", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_const_zero", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.sum_ite_eq'", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise"}, {"name": "Finset.sum_map_val", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Multiset.count_bind", "module": "Mathlib.Data.Multiset.Bind"}, {"name": "Multiset.count_map_eq_count'", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.roots_prod", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "add_left_injective", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "if_false_right", "module": "Init.PropLemmas"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "iff_false", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "Fin.lt_succ'", "content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}, {"name": "Fin.le_succ", "content": "lemma Fin.le_succ (a : Fin r) (h_a_add_1 : a + 1 < r) : a ≤ a + 1"}, {"name": "Fin.le_iff_lt_succ", "content": "lemma Fin.le_iff_lt_succ (a b : Fin r) (h_b : b + 1 < r) : a ≤ b ↔ a < b + 1"}], "used_local_defs": [{"name": "AdditiveNTT.U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "AdditiveNTT.W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "AdditiveNTT.algEquivAevalXSubC", "content": "@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :="}], "used_local_lemmas": [{"name": "AdditiveNTT.βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "AdditiveNTT.root_U_lift_down", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n  (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n  a ∈ (U 𝔽q β (i+1)) → ∃! x: 𝔽q, a - x • β i ∈ (U 𝔽q β i)"}, {"name": "AdditiveNTT.root_U_lift_up", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime hβ_lin_indep in\ntheorem root_U_lift_up (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) (x : 𝔽q) :\n  a - x • β i ∈ (U 𝔽q β i) → a ∈ (U 𝔽q β (i+1))"}, {"name": "AdditiveNTT.W_ne_zero", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W_ne_zero (i : Fin r) : (W 𝔽q β i) ≠ 0"}, {"name": "AdditiveNTT.roots_W", "content": "lemma roots_W (i : Fin r) : -- converts root Multiset into (univ: Uᵢ.val.map)\n  (W 𝔽q β i).roots = (univ : Finset (U 𝔽q β i)).val.map (fun u => u.val)"}, {"name": "AdditiveNTT.comp_X_sub_C_eq_zero_iff", "content": "omit [Fintype L] [DecidableEq L] in\nlemma comp_X_sub_C_eq_zero_iff (p : L[X]) (a : L) :\n  p.comp (X - C a) = 0 ↔ p = 0"}, {"name": "AdditiveNTT.rootMultiplicity_comp_X_sub_C", "content": "lemma rootMultiplicity_comp_X_sub_C (p : L[X]) (a x : L) :\n    rootMultiplicity x (p.comp (X - C a)) = rootMultiplicity (x - a) p"}, {"name": "AdditiveNTT.roots_comp_X_sub_C", "content": "lemma roots_comp_X_sub_C (p : L[X]) (a : L) :\n    (p.comp (X - C a)).roots = p.roots.map (fun r => r + a)"}, {"name": "AdditiveNTT.Prod_W_comp_X_sub_C_ne_zero", "content": "omit [DecidableEq L] h_Fq_char_prime hF₂ hβ_lin_indep in\nlemma Prod_W_comp_X_sub_C_ne_zero (i : Fin r) :\n    (univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i))) ≠ 0"}, {"name": "AdditiveNTT.rootMultiplicity_W", "content": "lemma rootMultiplicity_W (i : Fin r) (a : L) :\n    rootMultiplicity a (W 𝔽q β i) = if a ∈ (U 𝔽q β i : Set L) then 1 else 0"}], "local_ctx": "import ArkLib.Data.Nat.Bitwise\n\nimport ArkLib.Data.Polynomial.Frobenius\n\nimport ArkLib.Data.Polynomial.MonomialBasis\n\nimport Mathlib.LinearAlgebra.StdBasis\n\nimport Mathlib.Algebra.Polynomial.Degree.Definitions\n\nopen Polynomial FiniteDimensional Finset Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (h_dim : Module.finrank 𝔽q L = r)\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n\nsection LinearSubspaces\n\ndef U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))\n\nnoncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)\n\nend LinearSubspaces\n\nsection LinearityOfSubspaceVanishingPolynomials\n\n@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :=", "target_theorem": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0 :=", "ground_truth_proof": ":= by\n  rw [←Polynomial.count_roots]\n  set f := fun c: 𝔽q => (W 𝔽q β i).comp (X - C (c • β i)) with hf\n  -- ⊢ Multiset.count a (univ.prod f).roots = if a ∈ ↑(U 𝔽q β (i + 1)) then 1 else 0\n  have h_prod_ne_zero: univ.prod f ≠ 0 := Prod_W_comp_X_sub_C_ne_zero 𝔽q β i\n  rw [roots_prod (f := f) (s := univ (α := 𝔽q)) h_prod_ne_zero]\n  set roots_f := fun c: 𝔽q => (f c).roots with hroots_f\n  rw [Multiset.count_bind]\n  -- ⊢ (Multiset.map (fun b ↦ Multiset.count a (roots_f b)) univ.val).sum\n  -- = if a ∈ ↑(U 𝔽q β (i + 1)) then 1 else 0\n  have h_roots_f_eq_roots_W : ∀ b : 𝔽q,\n    roots_f b = (W 𝔽q β i).roots.map (fun r => r + (b • β i)) := by\n    intro b\n    rw [hroots_f, hf]\n    exact roots_comp_X_sub_C (p := (W 𝔽q β i)) (a := (b • β i))\n  simp_rw [h_roots_f_eq_roots_W]\n\n  set shift_up := fun x: 𝔽q => fun r: L => r + x • β i with hshift_up\n  have h_shift_up_all: ∀ x: 𝔽q, ∀ r: L, shift_up x r = r + x • β i := by\n    intro x r\n    rw [hshift_up]\n  simp only [sum_map_val, SetLike.mem_coe]\n  have h_a: ∀ x: 𝔽q, a = shift_up x (a - x • β i) := by\n    intro x\n    rw [hshift_up]\n    simp_all only [ne_eq, implies_true, sub_add_cancel, f, roots_f, shift_up]\n  conv_lhs =>\n    enter [2, x] -- focus on the inner Multiset.count\n    rw [h_a x]\n    enter [2]\n    enter [1]\n    enter [r]\n    rw [←h_shift_up_all x r] -- rewrite to another notation\n  -- ⊢ ∑ x, Multiset.count (shift_up x (a - x • β i)) (Multiset.map (shift_up x) (W 𝔽q β i).roots)\n  -- = if a ∈ ↑(U 𝔽q β (i + 1)) then 1 else 0\n  have h_shift_up_inj: ∀ x: 𝔽q, Function.Injective (shift_up x) := by\n    intro x\n    unfold shift_up\n    exact add_left_injective (x • β i)\n  have h_count_map: ∀ x: 𝔽q,\n    Multiset.count (shift_up x (a - x • β i)) (Multiset.map (shift_up x) (W 𝔽q β i).roots) =\n    Multiset.count (a - x • β i) (W 𝔽q β i).roots := by\n    -- transform to counting (a - x • β i) in the roots of Wᵢ\n    intro x\n    have h_shift_up_inj_x: Function.Injective (shift_up x) := h_shift_up_inj x\n    simp only [Multiset.count_map_eq_count' (hf := h_shift_up_inj_x), count_roots]\n  conv_lhs =>\n    enter [2, x]\n    rw [h_count_map x]\n  -- ⊢ ∑ x, Multiset.count (a - x • β i) (W 𝔽q β i).roots\n  -- = if a ∈ ↑(U 𝔽q β (i + 1)) then 1 else 0\n  have h_root_lift_down := root_U_lift_down 𝔽q β i h_i_add_1 a\n  have h_root_lift_up := root_U_lift_up 𝔽q β i h_i_add_1 a\n  conv_lhs =>\n    enter [2, x]\n    simp only [count_roots]\n    rw [rootMultiplicity_W]\n  by_cases h_a_mem_U_i : a ∈ ↑(U 𝔽q β (i + 1))\n  · -- ⊢ (∑ x, if a - x • β i ∈ ↑(U 𝔽q β i) then 1 else 0)\n    -- = if a ∈ ↑(U 𝔽q β (i + 1)) then 1 else 0\n    have h_true: (a ∈ ↑(U 𝔽q β (i + 1))) = True := by simp only [h_a_mem_U_i]\n    rcases h_root_lift_down h_a_mem_U_i with ⟨x0, hx0, hx0_unique⟩\n    conv =>\n      rhs\n      -- | if a ∈ ↑(U 𝔽q β (i + 1)) then 1 else 0 => reduce this to 1\n      enter [1]\n      exact h_true -- maybe there can be a better way to do this\n    rw [ite_true]\n    classical\n    -- ⊢ (∑ x, if a - x • β i ∈ ↑(U 𝔽q β i) then 1 else 0) = 1\n    have h_true: ∀ x: 𝔽q,\n      if x = x0 then a - x • β i ∈ ↑(U 𝔽q β i) else a - x • β i ∉ ↑(U 𝔽q β i) := by\n      intro x\n      by_cases h_x_eq_x0 : x = x0\n      · rw [if_pos h_x_eq_x0] -- ⊢ a - x • β i ∈ U 𝔽q β i\n        rw [←h_x_eq_x0] at hx0\n        exact hx0\n      · rw [if_neg h_x_eq_x0] -- ⊢ a - x • β i ∉ U 𝔽q β i\n        by_contra h_mem\n        have h1 := hx0_unique x\n        simp only [h_mem, forall_const] at h1\n        contradiction\n\n    have h_true_x: ∀ x: 𝔽q, (a - x • β i ∈ ↑(U 𝔽q β i)) = if x = x0 then True else False := by\n      intro x\n      by_cases h_x_eq_x0 : x = x0\n      · rw [if_pos h_x_eq_x0]\n        rw [←h_x_eq_x0] at hx0\n        simp only [hx0]\n      · rw [if_neg h_x_eq_x0]\n        by_contra h_mem\n        push_neg at h_mem\n        simp only [ne_eq, eq_iff_iff, iff_false, not_not] at h_mem\n        have h2 := hx0_unique x\n        simp only [h_mem, forall_const] at h2\n        contradiction\n    conv =>\n      lhs\n      enter [2, x]\n      simp only [SetLike.mem_coe, h_true_x x, if_false_right, and_true]\n    rw [sum_ite_eq']\n    simp only [mem_univ, ↓reduceIte]\n  · -- ⊢ (∑ x, if a - x • β i ∈ ↑(U 𝔽q β i) then 1 else 0)\n    -- = if a ∈ ↑(U 𝔽q β (i + 1)) then 1 else 0\n    have h_false: (a ∈ ↑(U 𝔽q β (i + 1))) = False := by simp only [h_a_mem_U_i]\n    conv =>\n      rhs -- | if a ∈ ↑(U 𝔽q β (i + 1)) then 1 else 0 => reduce this to 1\n      enter [1]\n      exact h_false -- maybe there can be a better way to do this\n    rw [ite_false]\n\n    have h_zero_x: ∀ x: 𝔽q, (a - x • β i ∈ ↑(U 𝔽q β i)) = False := by\n      intro x\n      by_contra h_mem\n      simp only [eq_iff_iff, iff_false, not_not] at h_mem -- h_mem : a - x • β i ∈ U 𝔽q β i\n      have h_a_mem_U_i := h_root_lift_up x h_mem\n      contradiction\n\n    conv =>\n      lhs\n      enter [2, x]\n      simp only [SetLike.mem_coe, h_zero_x x, if_false_right, and_true]\n    simp only [↓reduceIte, sum_const_zero]", "nesting_depth": 4, "transitive_dep_count": 157, "subset_aristotle": false, "category": "Applied verif."}
{"id": 17, "thm_name": "Binius.BinaryBasefold.is_fiber_iff_generates_quotient_point", "thm_stmt": "theorem is_fiber_iff_generates_quotient_point (i : Fin ℓ) (steps : ℕ)\n    (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :\n    let qMapFiber := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y)\n    let k := pointToIterateQuotientIndex (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := h_i_add_steps) (x := x)\n    y = iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i (k := steps) (h_bound := h_i_add_steps) x ↔\n    qMapFiber k = x", "lean_root": "ArkLib", "rel_path": "ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean", "imports": ["import ArkLib.Data.MvPolynomial.Multilinear", "import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.CodingTheory.ReedSolomon", "import ArkLib.Data.Vector.Basic", "import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound", "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Submodule.map", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Module.Basis", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Disjoint", "module": "Mathlib.Order.Disjoint"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "LinearEquiv", "module": "Mathlib.Algebra.Module.Equiv.Defs"}, {"name": "LinearEquiv.ofBijective", "module": "Mathlib.Algebra.Module.Submodule.Equiv"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap.codRestrict", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "LinearMap.ker", "module": "Mathlib.Algebra.Module.Submodule.Ker"}, {"name": "Module.Basis.span", "module": "Mathlib.LinearAlgebra.Basis.Basic"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.subtype", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.equivFunOnFinite", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Fin.cast", "module": "Init.Data.Fin.Basic"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "context", "module": "Examples.FrankingProtocol"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "sDomain", "content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n  let W_i_norm := normalizedW 𝔽q β i\n  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n    AdditiveNTT.normalizedW_is_additive 𝔽q β i\n  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)\n    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "polyEvalLinearMap", "content": "noncomputable def polyEvalLinearMap {L 𝔽q : Type*} [Field L] [Field 𝔽q] [Algebra 𝔽q L]\n  (p : L[X]) (hp_add : IsLinearMap 𝔽q (fun x : L => p.eval x)) : L →ₗ[𝔽q] L :=\n{\n  toFun    := fun x => p.eval x,\n  map_add' := hp_add.map_add,\n  map_smul' := hp_add.map_smul\n}"}, {"name": "sDomain_basis", "content": "noncomputable def sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    Basis (Fin (ℓ + R_rate - i)) 𝔽q (\n      sDomain 𝔽q β h_ℓ_add_R_rate i) :="}, {"name": "sBasis", "content": "def sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=\n  fun k => β ⟨i + k.val, by admit /- proof elided -/\n  ⟩"}, {"name": "binaryFinMapToNat", "content": "def binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary : ∀ j: Fin n, m j ≤ 1) : Fin (2^n) :="}, {"name": "symm", "content": "def symm (eqv : Equiv pSpec pSpec') : Equiv pSpec' pSpec where\n  round_eq := eqv.round_eq.symm\n  dir_eq := fun i => by admit /- proof elided -/"}, {"name": "Equiv", "content": "@[ext]\nstructure Equiv {m n : ℕ} (pSpec : ProtocolSpec m) (pSpec' : ProtocolSpec n) where\n  round_eq : m = n\n  dir_eq : ∀ i, pSpec.dir i = pSpec'.dir (Fin.cast round_eq i)\n  typeEquiv : ∀ i, pSpec.«Type» i ≃ pSpec'.«Type» (Fin.cast round_eq i)"}, {"name": "ProtocolSpec", "content": "@[ext]\nstructure ProtocolSpec (n : ℕ) where\n   \n  dir : Fin n → Direction\n   \n  «Type» : Fin n → Type\nderiving Inhabited"}, {"name": "Direction", "content": "inductive Direction where\n  | P_to_V  \n  | V_to_P \nderiving DecidableEq, Inhabited, Repr"}, {"name": "iteratedQuotientMap", "content": "noncomputable def iteratedQuotientMap (i : Fin ℓ) (k : ℕ)\n    (h_bound : i.val + k ≤ ℓ) (x : (sDomain 𝔽q β\n    h_ℓ_add_R_rate) ⟨i, by omega⟩) :\n    (sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i.val + k, by omega⟩ :="}, {"name": "intermediateNormVpoly", "content": "noncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)"}, {"name": "qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "qCompositionChain", "content": "noncomputable def qCompositionChain (i : Fin r) : L[X] :=\n  match i with\n  | ⟨0, _⟩ => X\n  | ⟨k + 1, h_k_add_1⟩ => (qMap 𝔽q β ⟨k, by admit /- proof elided -/\n  ⟩).comp (qCompositionChain ⟨k, by admit /- proof elided -/\n  ⟩)"}, {"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}], "lib_lemmas": [{"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Module.Basis.repr_linearCombination", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Module.Basis.repr_symm_apply", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Nat.add_zero", "module": "Init.Core"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}, {"name": "tsub_zero", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_lt_iff_neg_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "add_tsub_cancel_right", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Decidable.not_not", "module": "Init.PropLemmas"}, {"name": "Nat.le_of_not_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_lt_sub_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_sub", "module": "Init.Data.Nat.Basic"}, {"name": "dite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "lt_add_of_pos_right_of_le", "content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"}, {"name": "getSDomainBasisCoeff_of_iteratedQuotientMap", "content": "omit [DecidableEq 𝔽q] hF₂ in\nlemma getSDomainBasisCoeff_of_iteratedQuotientMap\n    [NeZero R_rate] (i : Fin ℓ) (k : ℕ)\n    (h_bound : i.val + k ≤ ℓ) (x : (sDomain 𝔽q β\n    h_ℓ_add_R_rate) ⟨i, by omega⟩) :\n    let y"}, {"name": "base_intermediateNormVpoly", "content": "theorem base_intermediateNormVpoly\n  (k : Fin (ℓ + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨0, by\n    by_contra ht\n    simp only [not_lt, nonpos_iff_eq_zero] at ht\n    contradiction\n  ⟩ ⟨k, by simp only [tsub_zero]; omega⟩ =\n  normalizedW 𝔽q β ⟨k, by omega⟩"}, {"name": "normalizedW_eq_qMap_composition", "content": "lemma normalizedW_eq_qMap_composition (ℓ R_rate : ℕ) (i : Fin r) :\n    normalizedW 𝔽q β i = qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i"}, {"name": "qMap_comp_normalizedW", "content": "lemma qMap_comp_normalizedW (i : Fin r) (h_i_add_1 : i + 1 < r) :\n  (qMap 𝔽q β i).comp (normalizedW 𝔽q β i) = normalizedW 𝔽q β (i + 1)"}, {"name": "qCompositionChain_eq_foldl", "content": "lemma qCompositionChain_eq_foldl (i : Fin r) :\n  qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i =\n  Fin.foldl (n:=i) (fun acc j =>\n    (qMap 𝔽q β ⟨j, by omega⟩).comp acc) (X)"}, {"name": "getSDomainBasisCoeff_of_sum_repr", "content": "omit [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 in\nlemma getSDomainBasisCoeff_of_sum_repr [NeZero R_rate] (i : Fin (ℓ + 1))\n    (x : (sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i, by omega⟩)\n    (x_coeffs : Fin (ℓ + R_rate - i) → 𝔽q)\n    (hx : x = ∑ j_x, (x_coeffs j_x) • (sDomain_basis 𝔽q β\n      h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (h_i := by\n        simp only; apply Nat.lt_add_of_pos_right_of_le; omega) j_x).val) :\n    ∀ (j: Fin (ℓ + R_rate - i)), ((sDomain_basis 𝔽q β\n      h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (h_i := by\n        simp only; apply Nat.lt_add_of_pos_right_of_le; omega)).repr x) j = x_coeffs j"}, {"name": "get_sDomain_basis", "content": "omit [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 in\nlemma get_sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    ∀ (k : Fin (ℓ + R_rate - i)),\n    (sDomain_basis 𝔽q β h_ℓ_add_R_rate\n      i (by omega)) k = eval (β ⟨i + k.val, by omega⟩) (normalizedW 𝔽q β i)"}, {"name": "intermediateNormVpoly_comp", "content": "omit [DecidableEq L] [DecidableEq 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in\ntheorem intermediateNormVpoly_comp (i : Fin ℓ) (k : Fin (ℓ - i + 1))\n  (l : Fin (ℓ - (i.val + k.val) + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k + l, by\n      simp only; omega⟩) =\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i+k, by omega⟩) (k:=⟨l, by\n      simp only; omega⟩)).comp (\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k, by\n      simp only; omega⟩)\n  )"}, {"name": "intermediateNormVpoly_eval_is_linear_map", "content": "omit [DecidableEq L] [DecidableEq 𝔽q] hF₂ hβ_lin_indep h_β₀_eq_1 in\nlemma intermediateNormVpoly_eval_is_linear_map (i : Fin (ℓ + 1)) (k : Fin (ℓ - i + 1)) :\n  IsLinearMap 𝔽q (fun x : L =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i k).eval x)"}, {"name": "qMap_is_linear_map", "content": "theorem qMap_is_linear_map (i : Fin r) :\n  IsLinearMap 𝔽q (f:=fun inner_p ↦ (qMap 𝔽q β i).comp inner_p)"}, {"name": "𝔽q_element_eq_zero_or_eq_one", "content": "omit h_Fq_char_prime in\nlemma 𝔽q_element_eq_zero_or_eq_one : ∀ c: 𝔽q, c = 0 ∨ c = 1"}, {"name": "getBit_of_binaryFinMapToNat", "content": "lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :\n    ∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val\n      = if h_k: k < n then m ⟨k, by omega⟩ else 0"}, {"name": "and_two_pow_eq_zero_of_getBit_0", "content": "lemma and_two_pow_eq_zero_of_getBit_0 {n i : ℕ} (h_getBit : getBit i n = 0)\n    : n &&& (2 ^ i) = 0"}, {"name": "and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "getBit_of_lt_two_pow", "content": "lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :\n  getBit k a = if k < n then getBit k a else 0"}, {"name": "getBit_eq_testBit", "content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"}, {"name": "getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "getBit_of_xor", "content": "lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m"}, {"name": "getBit_zero_eq_zero", "content": "lemma getBit_zero_eq_zero {k : Nat} : getBit k 0 = 0"}, {"name": "sum_of_and_eq_zero_is_xor", "content": "lemma sum_of_and_eq_zero_is_xor {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ^^^ m"}, {"name": "sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||"}, {"name": "xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}], "used_local_defs": [{"name": "Binius.BinaryBasefold.fiber_coeff", "content": "noncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber", "content": "noncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/"}, {"name": "Binius.BinaryBasefold.pointToIterateQuotientIndex", "content": "def pointToIterateQuotientIndex (i : Fin (ℓ + 1)) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩)) : Fin (2 ^ steps) :="}], "used_local_lemmas": [{"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R", "content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n    : i.val + steps < ℓ + 𝓡"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff", "content": "lemma qMap_total_fiber_repr_coeff (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩))\n    (k : Fin (2 ^ steps)) :\n    let x := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y) k\n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)\n      (h_i := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps)\n    let y_coeffs := basis_y.repr y\n    ∀ j, -- j refers to bit index of the fiber point x\n      ((sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (by simp only; omega)).repr x) j\n      = fiber_coeff (i := i) (steps := steps) (j := j) (elementIdx := k)\n        (y_coeffs := y_coeffs)"}, {"name": "Binius.BinaryBasefold.generates_quotient_point_if_is_fiber_of_y", "content": "theorem generates_quotient_point_if_is_fiber_of_y\n    (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩))\n    (hx_is_fiber : ∃ (k : Fin (2 ^ steps)), x = qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := steps) (h_i_add_steps := by\n        simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y) k) :\n    y = iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i (k := steps) (h_bound := h_i_add_steps) x"}], "local_ctx": "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch\n\nimport ArkLib.Data.CodingTheory.ReedSolomon\n\nimport ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT\n\nimport ArkLib.Data.MvPolynomial.Multilinear\n\nimport ArkLib.Data.Vector.Basic\n\nimport ArkLib.ProofSystem.Sumcheck.Spec.SingleRound\n\nnamespace Binius.BinaryBasefold\n\nopen OracleSpec OracleComp ProtocolSpec Finset AdditiveNTT Polynomial MvPolynomial\n  Binius.BinaryBasefold\n\nopen scoped NNReal\n\nopen ReedSolomon Code BerlekampWelch\n\nopen Finset AdditiveNTT Polynomial MvPolynomial Nat Matrix\n\nsection Preliminaries\n\nvariable {L : Type} [CommRing L] (ℓ : ℕ) [NeZero ℓ]\n\nvariable (𝓑 : Fin 2 ↪ L)\n\nend Preliminaries\n\nnoncomputable section       -- expands with 𝔽q in front\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type} [Field L] [Fintype L] [DecidableEq L] [CharP L 2]\n\nvariable (𝔽q : Type) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ 𝓡 ϑ : ℕ} (γ_repetitions : ℕ) [NeZero ℓ] [NeZero 𝓡] [NeZero ϑ] -- Should we allow ℓ = 0?\n\nvariable {h_ℓ_add_R_rate : ℓ + 𝓡 < r} -- ℓ ∈ {1, ..., r-1}\n\nvariable {𝓑 : Fin 2 ↪ L}\n\nsection Essentials\n\nnoncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩\n\nnoncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/\n\ndef pointToIterateQuotientIndex (i : Fin (ℓ + 1)) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩)) : Fin (2 ^ steps) :=", "target_theorem": "theorem is_fiber_iff_generates_quotient_point (i : Fin ℓ) (steps : ℕ)\n    (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :\n    let qMapFiber :=", "ground_truth_proof": ":= qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y)\n    let k := pointToIterateQuotientIndex (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := h_i_add_steps) (x := x)\n    y = iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i (k := steps) (h_bound := h_i_add_steps) x ↔\n    qMapFiber k = x := by\n  let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩\n    (by simp only; omega)\n  let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i.val + steps, by omega⟩\n    (h_i := by apply Nat.lt_add_of_pos_right_of_le; omega)\n  simp only\n  set k := pointToIterateQuotientIndex (i := ⟨i, by omega⟩) (steps := steps)\n    (h_i_add_steps := h_i_add_steps) (x := x)\n  constructor\n  · intro h_x_generates_y\n    -- ⊢ qMap_total_fiber ...` ⟨↑i, ⋯⟩ steps ⋯ y k = x\n    -- We prove that `qMap_total_fiber` with this `k` reconstructs `x` via basis repr\n    apply basis_x.repr.injective\n    ext j\n    let reConstructedX := basis_x.repr (qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := steps) (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y) k)\n    have h_repr_of_reConstructedX := qMap_total_fiber_repr_coeff 𝔽q β i (steps := steps)\n      (h_i_add_steps := by omega) (y := y) (k := k) (j := j)\n    simp only at h_repr_of_reConstructedX\n    -- ⊢ repr of reConstructedX at j = repr of x at j\n    rw [h_repr_of_reConstructedX]; dsimp [k, pointToIterateQuotientIndex, fiber_coeff];\n    rw [getBit_of_binaryFinMapToNat]; simp only [Fin.eta, dite_eq_right_iff, ite_eq_left_iff,\n      one_ne_zero, imp_false, Decidable.not_not]\n    -- Now we only need to do case analysis\n    by_cases h_j : j.val < steps\n    · -- Case 1 : The first `steps` coefficients, determined by `k`.\n      simp only [h_j, ↓reduceDIte, forall_const]\n      by_cases h_coeff_j_of_x : basis_x.repr x j = 0\n      · simp only [basis_x, h_coeff_j_of_x, ↓reduceIte];\n      · simp only [basis_x, h_coeff_j_of_x, ↓reduceIte];\n        have h_coeff := 𝔽q_element_eq_zero_or_eq_one 𝔽q (c := basis_x.repr x j)\n        simp only [h_coeff_j_of_x, false_or] at h_coeff\n        exact id (Eq.symm h_coeff)\n    · -- Case 2 : The remaining coefficients, determined by `y`.\n      simp only [h_j, ↓reduceDIte]\n      simp only [basis_x]\n      -- ⊢ Here we compare coeffs, not the basis elements\n      simp only [h_x_generates_y]\n      have h_res := getSDomainBasisCoeff_of_iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i (k := steps)\n        (h_bound := by omega) x (j := ⟨j - steps, by -- TODO : make this index bound proof cleaner\n          simp only; rw [←Nat.sub_sub]; -- ⊢ ↑j - steps < ℓ + 𝓡 - ↑i - steps\n          apply Nat.sub_lt_sub_right;\n          · exact Nat.le_of_not_lt h_j\n          · exact j.isLt\n        ⟩) -- ⊢ ↑j - steps < ℓ + 𝓡 - (↑i + steps)\n      have h_j_sub_add_steps : j - steps + steps = j := by omega\n      simp only at h_res\n      simp only [h_j_sub_add_steps, Fin.eta] at h_res\n      exact h_res\n  · intro h_x_is_fiber_of_y\n    -- y is the quotient point of x over steps steps\n    apply generates_quotient_point_if_is_fiber_of_y (h_i_add_steps := h_i_add_steps)\n      (x := x) (y := y) (hx_is_fiber := by use k; exact h_x_is_fiber_of_y.symm)", "nesting_depth": 6, "transitive_dep_count": 127, "subset_aristotle": false, "category": "Applied verif."}
{"id": 18, "thm_name": "ConcreteBinaryTower.Z_square_eq", "thm_stmt": "lemma Z_square_eq (k : ℕ) (prevBTFieldProps : ConcreteBTFieldProps (k := k))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI : Field (ConcreteBTField (k + 1)) := mkFieldInstance curBTFieldProps\n  (Z (k + 1)) ^ 2 = 《 Z (k), 1 》", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "AddCommGroup", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.negSucc", "module": "Init.Data.Int.Basic"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "NNRat", "module": "Mathlib.Data.Rat.Init"}, {"name": "NNRat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "Rat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}], "lib_lemmas": [{"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}, {"name": "BitVec.zero_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "pow_two", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.add", "content": "def add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y"}, {"name": "ConcreteBinaryTower.neg", "content": "def neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.ConcreteBTFAddCommGroupProps", "content": "structure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel"}, {"name": "ConcreteBinaryTower.mkAddCommGroupInstance", "content": "def mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}"}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.concrete_mul", "content": "def concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.concrete_inv", "content": "def concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res"}, {"name": "ConcreteBinaryTower.natCast", "content": "def natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.natCast_zero", "content": "def natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :="}, {"name": "ConcreteBinaryTower.natCast_succ", "content": "def natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :="}, {"name": "ConcreteBinaryTower.intCast", "content": "def intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.intCast_ofNat", "content": "def intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :="}, {"name": "ConcreteBinaryTower.intCast_negSucc", "content": "def intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :="}, {"name": "ConcreteBinaryTower.ConcreteBTFRingProps", "content": "structure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c"}, {"name": "ConcreteBinaryTower.ConcreteBTFDivisionRingProps", "content": "structure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldProps", "content": "structure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a"}, {"name": "ConcreteBinaryTower.mkRingInstance", "content": "def mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n"}, {"name": "ConcreteBinaryTower.mkDivisionRingInstance", "content": "def mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)"}, {"name": "ConcreteBinaryTower.mkFieldInstance", "content": "def mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm"}, {"name": "ConcreteBinaryTower.ConcreteBTFStepResult", "content": "structure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))"}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.zero_add", "content": "lemma zero_add {k : ℕ} (a : ConcreteBTField k) : 0 + a = a"}, {"name": "ConcreteBinaryTower.add_zero", "content": "lemma add_zero {k : ℕ} (a : ConcreteBTField k) : a + 0 = a"}, {"name": "ConcreteBinaryTower.zero_is_0", "content": "lemma zero_is_0 {k : ℕ} : (zero (k:=k)) = (0 : ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.one_is_1", "content": "lemma one_is_1 {k : ℕ} : (one (k:=k)) = 1"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.split_of_join", "content": "theorem split_of_join {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_join : x = 《hi_btf, lo_btf》) :\n    (hi_btf, lo_btf) = split h_pos x"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y\n\ndef neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\nstructure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel\n\ndef mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\ndef concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/\n\ndef concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res\n\nsection FieldLemmasOfLevel0\n\nend FieldLemmasOfLevel0\n\nsection NumericCasting\n\ndef natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :=\n\ndef natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :=\n\ndef intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :=\n\ndef intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :=\n\nend NumericCasting\n\nstructure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c\n\nstructure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one\n\nstructure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a\n\ndef mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n\n\ndef mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)\n\ndef mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm\n\nstructure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))\n\nend FieldOperationsAndInstances\n\nsection BTFieldPropsOneLevelLiftingLemmas\n\nvariable {k : ℕ} {h_k : k > 0}\n\nend BTFieldPropsOneLevelLiftingLemmas\n\nsection TowerFieldsConstruction", "target_theorem": "lemma Z_square_eq (k : ℕ) (prevBTFieldProps : ConcreteBTFieldProps (k := k))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI : Field (ConcreteBTField (k + 1)) :=", "ground_truth_proof": ":= mkFieldInstance curBTFieldProps\n  (Z (k + 1)) ^ 2 = 《 Z (k), 1 》 := by\n  letI : Field (ConcreteBTField (k + 1)) := mkFieldInstance curBTFieldProps\n  have hmul : ∀ (a b : ConcreteBTField (k - 1)), concrete_mul a b = a * b := fun a b => rfl\n  rw [pow_two]\n  change concrete_mul (Z (k + 1)) (Z (k + 1)) = 《 Z (k), 1 》\n  have h_split_Z_k_add_1 : split (k:=k+1) (h:=by omega) (Z (k + 1)) = (1, 0) := by\n    exact Eq.symm\n      (split_of_join (by omega) (Z (k + 1)) 1 0 rfl)\n  have h_mul_eq := curBTFieldProps.mul_eq (a:=Z (k+1)) (b:=Z (k+1))\n    (a₁:=1) (a₀:=0) (b₁:=1) (b₀:=0) (h_k:=by omega)\n    (by exact id (Eq.symm h_split_Z_k_add_1)) (by exact id (Eq.symm h_split_Z_k_add_1))\n  rw [h_mul_eq]\n  simp_rw [←zero_is_0, ←one_is_1]\n  simp only [Nat.add_one_sub_one]\n  simp_rw [prevBTFieldProps.mul_zero, prevBTFieldProps.mul_one,\n    prevBTFieldProps.add_zero, prevBTFieldProps.one_mul]\n  simp_rw [prevBTFieldProps.zero_add]", "nesting_depth": 8, "transitive_dep_count": 140, "subset_aristotle": false, "category": "Applied verif."}
{"id": 19, "thm_name": "Binius.BinaryBasefold.qMap_total_fiber_disjoint", "thm_stmt": "theorem qMap_total_fiber_disjoint\n  (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i + steps ≤ ℓ)\n  {y₁ y₂ : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i.val + steps, by omega⟩}\n  (hy_ne : y₁ ≠ y₂) :\n  Disjoint\n    ((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) y₁ '' Set.univ).toFinset)\n    ((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) y₂ '' Set.univ).toFinset)", "lean_root": "ArkLib", "rel_path": "ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean", "imports": ["import ArkLib.Data.MvPolynomial.Multilinear", "import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.CodingTheory.ReedSolomon", "import ArkLib.Data.Vector.Basic", "import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound", "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Submodule.map", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Module.Basis", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Disjoint", "module": "Mathlib.Order.Disjoint"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "LinearEquiv", "module": "Mathlib.Algebra.Module.Equiv.Defs"}, {"name": "LinearEquiv.ofBijective", "module": "Mathlib.Algebra.Module.Submodule.Equiv"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap.codRestrict", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "LinearMap.ker", "module": "Mathlib.Algebra.Module.Submodule.Ker"}, {"name": "Module.Basis.span", "module": "Mathlib.LinearAlgebra.Basis.Basic"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.subtype", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.equivFunOnFinite", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Fin.cast", "module": "Init.Data.Fin.Basic"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "context", "module": "Examples.FrankingProtocol"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "sDomain", "content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n  let W_i_norm := normalizedW 𝔽q β i\n  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n    AdditiveNTT.normalizedW_is_additive 𝔽q β i\n  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)\n    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "polyEvalLinearMap", "content": "noncomputable def polyEvalLinearMap {L 𝔽q : Type*} [Field L] [Field 𝔽q] [Algebra 𝔽q L]\n  (p : L[X]) (hp_add : IsLinearMap 𝔽q (fun x : L => p.eval x)) : L →ₗ[𝔽q] L :=\n{\n  toFun    := fun x => p.eval x,\n  map_add' := hp_add.map_add,\n  map_smul' := hp_add.map_smul\n}"}, {"name": "sDomain_basis", "content": "noncomputable def sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    Basis (Fin (ℓ + R_rate - i)) 𝔽q (\n      sDomain 𝔽q β h_ℓ_add_R_rate i) :="}, {"name": "sBasis", "content": "def sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=\n  fun k => β ⟨i + k.val, by admit /- proof elided -/\n  ⟩"}, {"name": "binaryFinMapToNat", "content": "def binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary : ∀ j: Fin n, m j ≤ 1) : Fin (2^n) :="}, {"name": "symm", "content": "def symm (eqv : Equiv pSpec pSpec') : Equiv pSpec' pSpec where\n  round_eq := eqv.round_eq.symm\n  dir_eq := fun i => by admit /- proof elided -/"}, {"name": "Equiv", "content": "@[ext]\nstructure Equiv {m n : ℕ} (pSpec : ProtocolSpec m) (pSpec' : ProtocolSpec n) where\n  round_eq : m = n\n  dir_eq : ∀ i, pSpec.dir i = pSpec'.dir (Fin.cast round_eq i)\n  typeEquiv : ∀ i, pSpec.«Type» i ≃ pSpec'.«Type» (Fin.cast round_eq i)"}, {"name": "ProtocolSpec", "content": "@[ext]\nstructure ProtocolSpec (n : ℕ) where\n   \n  dir : Fin n → Direction\n   \n  «Type» : Fin n → Type\nderiving Inhabited"}, {"name": "Direction", "content": "inductive Direction where\n  | P_to_V  \n  | V_to_P \nderiving DecidableEq, Inhabited, Repr"}, {"name": "iteratedQuotientMap", "content": "noncomputable def iteratedQuotientMap (i : Fin ℓ) (k : ℕ)\n    (h_bound : i.val + k ≤ ℓ) (x : (sDomain 𝔽q β\n    h_ℓ_add_R_rate) ⟨i, by omega⟩) :\n    (sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i.val + k, by omega⟩ :="}, {"name": "intermediateNormVpoly", "content": "noncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)"}, {"name": "qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "trans", "content": "def trans (eqv : Equiv pSpec pSpec') (eqv' : Equiv pSpec' pSpec'') : Equiv pSpec pSpec'' where\n  round_eq := eqv.round_eq.trans eqv'.round_eq\n  dir_eq := fun i => by admit /- proof elided -/"}, {"name": "qCompositionChain", "content": "noncomputable def qCompositionChain (i : Fin r) : L[X] :=\n  match i with\n  | ⟨0, _⟩ => X\n  | ⟨k + 1, h_k_add_1⟩ => (qMap 𝔽q β ⟨k, by admit /- proof elided -/\n  ⟩).comp (qCompositionChain ⟨k, by admit /- proof elided -/\n  ⟩)"}, {"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}], "lib_lemmas": [{"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Module.Basis.repr_linearCombination", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Module.Basis.repr_symm_apply", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Nat.add_zero", "module": "Init.Core"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}, {"name": "tsub_zero", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_lt_iff_neg_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "add_tsub_cancel_right", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Decidable.not_not", "module": "Init.PropLemmas"}, {"name": "Nat.le_of_not_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_lt_sub_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_sub", "module": "Init.Data.Nat.Basic"}, {"name": "dite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Finset.disjoint_iff_inter_eq_empty", "module": "Mathlib.Data.Finset.Lattice.Lemmas"}, {"name": "Finset.mem_of_mem_inter_left", "module": "Mathlib.Data.Finset.Lattice.Basic"}, {"name": "Finset.mem_of_mem_inter_right", "module": "Mathlib.Data.Finset.Lattice.Basic"}, {"name": "Finset.nonempty_of_ne_empty", "module": "Mathlib.Data.Finset.Empty"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_toFinset", "module": "Mathlib.Data.Fintype.Sets"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "lt_add_of_pos_right_of_le", "content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"}, {"name": "getSDomainBasisCoeff_of_iteratedQuotientMap", "content": "omit [DecidableEq 𝔽q] hF₂ in\nlemma getSDomainBasisCoeff_of_iteratedQuotientMap\n    [NeZero R_rate] (i : Fin ℓ) (k : ℕ)\n    (h_bound : i.val + k ≤ ℓ) (x : (sDomain 𝔽q β\n    h_ℓ_add_R_rate) ⟨i, by omega⟩) :\n    let y"}, {"name": "base_intermediateNormVpoly", "content": "theorem base_intermediateNormVpoly\n  (k : Fin (ℓ + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨0, by\n    by_contra ht\n    simp only [not_lt, nonpos_iff_eq_zero] at ht\n    contradiction\n  ⟩ ⟨k, by simp only [tsub_zero]; omega⟩ =\n  normalizedW 𝔽q β ⟨k, by omega⟩"}, {"name": "normalizedW_eq_qMap_composition", "content": "lemma normalizedW_eq_qMap_composition (ℓ R_rate : ℕ) (i : Fin r) :\n    normalizedW 𝔽q β i = qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i"}, {"name": "qMap_comp_normalizedW", "content": "lemma qMap_comp_normalizedW (i : Fin r) (h_i_add_1 : i + 1 < r) :\n  (qMap 𝔽q β i).comp (normalizedW 𝔽q β i) = normalizedW 𝔽q β (i + 1)"}, {"name": "qCompositionChain_eq_foldl", "content": "lemma qCompositionChain_eq_foldl (i : Fin r) :\n  qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i =\n  Fin.foldl (n:=i) (fun acc j =>\n    (qMap 𝔽q β ⟨j, by omega⟩).comp acc) (X)"}, {"name": "getSDomainBasisCoeff_of_sum_repr", "content": "omit [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 in\nlemma getSDomainBasisCoeff_of_sum_repr [NeZero R_rate] (i : Fin (ℓ + 1))\n    (x : (sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i, by omega⟩)\n    (x_coeffs : Fin (ℓ + R_rate - i) → 𝔽q)\n    (hx : x = ∑ j_x, (x_coeffs j_x) • (sDomain_basis 𝔽q β\n      h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (h_i := by\n        simp only; apply Nat.lt_add_of_pos_right_of_le; omega) j_x).val) :\n    ∀ (j: Fin (ℓ + R_rate - i)), ((sDomain_basis 𝔽q β\n      h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (h_i := by\n        simp only; apply Nat.lt_add_of_pos_right_of_le; omega)).repr x) j = x_coeffs j"}, {"name": "get_sDomain_basis", "content": "omit [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 in\nlemma get_sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    ∀ (k : Fin (ℓ + R_rate - i)),\n    (sDomain_basis 𝔽q β h_ℓ_add_R_rate\n      i (by omega)) k = eval (β ⟨i + k.val, by omega⟩) (normalizedW 𝔽q β i)"}, {"name": "intermediateNormVpoly_comp", "content": "omit [DecidableEq L] [DecidableEq 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in\ntheorem intermediateNormVpoly_comp (i : Fin ℓ) (k : Fin (ℓ - i + 1))\n  (l : Fin (ℓ - (i.val + k.val) + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k + l, by\n      simp only; omega⟩) =\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i+k, by omega⟩) (k:=⟨l, by\n      simp only; omega⟩)).comp (\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k:=⟨k, by\n      simp only; omega⟩)\n  )"}, {"name": "intermediateNormVpoly_eval_is_linear_map", "content": "omit [DecidableEq L] [DecidableEq 𝔽q] hF₂ hβ_lin_indep h_β₀_eq_1 in\nlemma intermediateNormVpoly_eval_is_linear_map (i : Fin (ℓ + 1)) (k : Fin (ℓ - i + 1)) :\n  IsLinearMap 𝔽q (fun x : L =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i k).eval x)"}, {"name": "qMap_is_linear_map", "content": "theorem qMap_is_linear_map (i : Fin r) :\n  IsLinearMap 𝔽q (f:=fun inner_p ↦ (qMap 𝔽q β i).comp inner_p)"}, {"name": "𝔽q_element_eq_zero_or_eq_one", "content": "omit h_Fq_char_prime in\nlemma 𝔽q_element_eq_zero_or_eq_one : ∀ c: 𝔽q, c = 0 ∨ c = 1"}, {"name": "getBit_of_binaryFinMapToNat", "content": "lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :\n    ∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val\n      = if h_k: k < n then m ⟨k, by omega⟩ else 0"}, {"name": "and_two_pow_eq_zero_of_getBit_0", "content": "lemma and_two_pow_eq_zero_of_getBit_0 {n i : ℕ} (h_getBit : getBit i n = 0)\n    : n &&& (2 ^ i) = 0"}, {"name": "and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "getBit_of_lt_two_pow", "content": "lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :\n  getBit k a = if k < n then getBit k a else 0"}, {"name": "getBit_eq_testBit", "content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"}, {"name": "getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "getBit_of_xor", "content": "lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m"}, {"name": "getBit_zero_eq_zero", "content": "lemma getBit_zero_eq_zero {k : Nat} : getBit k 0 = 0"}, {"name": "sum_of_and_eq_zero_is_xor", "content": "lemma sum_of_and_eq_zero_is_xor {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ^^^ m"}, {"name": "sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||"}, {"name": "xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}], "used_local_defs": [{"name": "Binius.BinaryBasefold.fiber_coeff", "content": "noncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber", "content": "noncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/"}, {"name": "Binius.BinaryBasefold.pointToIterateQuotientIndex", "content": "def pointToIterateQuotientIndex (i : Fin (ℓ + 1)) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩)) : Fin (2 ^ steps) :="}], "used_local_lemmas": [{"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R", "content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n    : i.val + steps < ℓ + 𝓡"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff", "content": "lemma qMap_total_fiber_repr_coeff (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩))\n    (k : Fin (2 ^ steps)) :\n    let x := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y) k\n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)\n      (h_i := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps)\n    let y_coeffs := basis_y.repr y\n    ∀ j, -- j refers to bit index of the fiber point x\n      ((sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (by simp only; omega)).repr x) j\n      = fiber_coeff (i := i) (steps := steps) (j := j) (elementIdx := k)\n        (y_coeffs := y_coeffs)"}, {"name": "Binius.BinaryBasefold.generates_quotient_point_if_is_fiber_of_y", "content": "theorem generates_quotient_point_if_is_fiber_of_y\n    (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩))\n    (hx_is_fiber : ∃ (k : Fin (2 ^ steps)), x = qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := steps) (h_i_add_steps := by\n        simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y) k) :\n    y = iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i (k := steps) (h_bound := h_i_add_steps) x"}, {"name": "Binius.BinaryBasefold.is_fiber_iff_generates_quotient_point", "content": "theorem is_fiber_iff_generates_quotient_point (i : Fin ℓ) (steps : ℕ)\n    (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :\n    let qMapFiber := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y)\n    let k := pointToIterateQuotientIndex (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := h_i_add_steps) (x := x)\n    y = iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i (k := steps) (h_bound := h_i_add_steps) x ↔\n    qMapFiber k = x"}], "local_ctx": "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch\n\nimport ArkLib.Data.CodingTheory.ReedSolomon\n\nimport ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT\n\nimport ArkLib.Data.MvPolynomial.Multilinear\n\nimport ArkLib.Data.Vector.Basic\n\nimport ArkLib.ProofSystem.Sumcheck.Spec.SingleRound\n\nnamespace Binius.BinaryBasefold\n\nopen OracleSpec OracleComp ProtocolSpec Finset AdditiveNTT Polynomial MvPolynomial\n  Binius.BinaryBasefold\n\nopen scoped NNReal\n\nopen ReedSolomon Code BerlekampWelch\n\nopen Finset AdditiveNTT Polynomial MvPolynomial Nat Matrix\n\nsection Preliminaries\n\nvariable {L : Type} [CommRing L] (ℓ : ℕ) [NeZero ℓ]\n\nvariable (𝓑 : Fin 2 ↪ L)\n\nend Preliminaries\n\nnoncomputable section       -- expands with 𝔽q in front\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type} [Field L] [Fintype L] [DecidableEq L] [CharP L 2]\n\nvariable (𝔽q : Type) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ 𝓡 ϑ : ℕ} (γ_repetitions : ℕ) [NeZero ℓ] [NeZero 𝓡] [NeZero ϑ] -- Should we allow ℓ = 0?\n\nvariable {h_ℓ_add_R_rate : ℓ + 𝓡 < r} -- ℓ ∈ {1, ..., r-1}\n\nvariable {𝓑 : Fin 2 ↪ L}\n\nsection Essentials\n\nnoncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩\n\nnoncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/\n\ndef pointToIterateQuotientIndex (i : Fin (ℓ + 1)) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (x : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩)) : Fin (2 ^ steps) :=", "target_theorem": "theorem qMap_total_fiber_disjoint\n  (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i + steps ≤ ℓ)\n  {y₁ y₂ : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i.val + steps, by omega⟩}\n  (hy_ne : y₁ ≠ y₂) :\n  Disjoint\n    ((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) y₁ '' Set.univ).toFinset)\n    ((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) y₂ '' Set.univ).toFinset) :=", "ground_truth_proof": ":= by\n -- Proof by contradiction. Assume the intersection is non-empty.\n  rw [Finset.disjoint_iff_inter_eq_empty]\n  by_contra h_nonempty\n  -- Let `x` be an element in the intersection of the two fiber sets.\n  obtain ⟨x, h_x_mem_inter⟩ := Finset.nonempty_of_ne_empty h_nonempty\n  have hx₁ := Finset.mem_of_mem_inter_left h_x_mem_inter\n  have hx₂ := Finset.mem_of_mem_inter_right h_x_mem_inter\n  -- A helper lemma : applying the forward map to a point in a generated fiber returns\n  -- the original quotient point.\n  have iteratedQuotientMap_of_qMap_total_fiber_eq_self\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i.val + steps, by omega⟩)\n    (k : Fin (2 ^ steps)) :\n    iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (k := steps)\n      (h_bound := by omega)\n      (qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n        (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y) k) = y := by\n      have h := generates_quotient_point_if_is_fiber_of_y\n        (h_i_add_steps := h_i_add_steps) (x:=\n        ((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n          (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y) k) :\n          sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩))\n      ) (y := y) (hx_is_fiber := by use k)\n      exact h.symm\n  have h_exists_k₁ : ∃ k, x = qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) y₁ k := by\n    -- convert (x ∈ Finset of the image of the fiber) to statement\n    -- about membership in the Set.\n    rw [Set.mem_toFinset] at hx₁\n    rw [Set.mem_image] at hx₁ -- Set.mem_image gives us t an index that maps to x\n    -- ⊢ `∃ (k : Fin (2 ^ steps)), k ∈ Set.univ ∧ qMap_total_fiber ... y₁ k = x`.\n    rcases hx₁ with ⟨k, _, h_eq⟩\n    use k; exact h_eq.symm\n\n  have h_exists_k₂ : ∃ k, x = qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) y₂ k := by\n    rw [Set.mem_toFinset] at hx₂\n    rw [Set.mem_image] at hx₂ -- Set.mem_image gives us t an index that maps to x\n    rcases hx₂ with ⟨k, _, h_eq⟩\n    use k; exact h_eq.symm\n\n  have h_y₁_eq_quotient_x : y₁ =\n      iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i steps h_i_add_steps x := by\n    apply generates_quotient_point_if_is_fiber_of_y (hx_is_fiber := by exact h_exists_k₁)\n\n  have h_y₂_eq_quotient_x : y₂ =\n      iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i steps h_i_add_steps x := by\n    apply generates_quotient_point_if_is_fiber_of_y (hx_is_fiber := by exact h_exists_k₂)\n\n  let kQuotientIndex := pointToIterateQuotientIndex (i := ⟨i, by omega⟩) (steps := steps)\n    (h_i_add_steps := by omega) (x := x)\n\n  -- Since `x` is in the fiber of `y₁`, applying the forward map to `x` yields `y₁`.\n  have h_map_x_eq_y₁ : iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩)\n      (k := steps) (h_bound := by omega) x = y₁ := by\n    have h := iteratedQuotientMap_of_qMap_total_fiber_eq_self (y := y₁) (k := kQuotientIndex)\n    have hx₁ : x = qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n        (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) y₁ kQuotientIndex := by\n      have h_res := is_fiber_iff_generates_quotient_point 𝔽q β i steps (by omega)\n        (x := x) (y := y₁).mp (h_y₁_eq_quotient_x)\n      exact h_res.symm\n    rw [hx₁]\n    exact iteratedQuotientMap_of_qMap_total_fiber_eq_self y₁ kQuotientIndex\n\n  -- Similarly, since `x` is in the fiber of `y₂`, applying the forward map yields `y₂`.\n  have h_map_x_eq_y₂ : iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩)\n      (k := steps) (h_bound := by omega) x = y₂ := by\n    -- have h := iteratedQuotientMap_of_qMap_total_fiber_eq_self (y := y₂) (k := kQuotientIndex)\n    have hx₂ : x = qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n        (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) y₂ kQuotientIndex := by\n      have h_res := is_fiber_iff_generates_quotient_point 𝔽q β i steps (by omega)\n        (x := x) (y := y₂).mp (h_y₂_eq_quotient_x)\n      exact h_res.symm\n    rw [hx₂]\n    exact iteratedQuotientMap_of_qMap_total_fiber_eq_self y₂ kQuotientIndex\n\n  exact hy_ne (h_map_x_eq_y₁.symm.trans h_map_x_eq_y₂)", "nesting_depth": 6, "transitive_dep_count": 136, "subset_aristotle": false, "category": "Applied verif."}
{"id": 20, "thm_name": "AdditiveNTT.even_index_intermediate_novel_basis_decomposition", "thm_stmt": "lemma even_index_intermediate_novel_basis_decomposition (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :\n  intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2, by\n    apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)\n  ⟩  = (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by omega⟩ ⟨j, by\n    apply lt_two_pow_of_lt_two_pow_exp_le j (ℓ-i-1) (ℓ-(i+1)) (by omega) (by omega)\n  ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean", "imports": ["import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import Mathlib.Data.Finsupp.Defs", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Tactic", "import ArkLib.Data.Nat.Bitwise", "import Mathlib.LinearAlgebra.LinearIndependent.Defs"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "context", "module": "Examples.FrankingProtocol"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "g", "content": "def g (n : ℕ) (c : ℕ) (x : ℕ) := (x * x + c) % n"}], "lib_lemmas": [{"name": "Polynomial.comp_assoc", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "Fin.coe_ofNat_eq_mod", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.foldl_succ", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Polynomial.X_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Finset.prod_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Polynomial.pow_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.prod_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pow_zero", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "getBit_zero_of_two_mul", "content": "lemma getBit_zero_of_two_mul {n : ℕ} : getBit 0 (2*n) = 0"}, {"name": "lt_two_pow_of_lt_two_pow_exp_le", "content": "lemma lt_two_pow_of_lt_two_pow_exp_le (x i j: ℕ)\n    (h_x_lt_2_pow_i: x < 2^i) (h_i_le_j: i ≤ j): x < 2^j"}, {"name": "getBit_eq_succ_getBit_of_mul_two", "content": "lemma getBit_eq_succ_getBit_of_mul_two {n k : ℕ} : getBit (k+1) (2*n) = getBit k n"}, {"name": "getBit_of_shiftRight", "content": "lemma getBit_of_shiftRight {n p : ℕ}:\n  ∀ k, getBit k (n >>> p) = getBit (k+p) n"}, {"name": "mul_two_add_bit_lt_two_pow", "content": "theorem mul_two_add_bit_lt_two_pow (a b c : ℕ) (i : Fin 2)\n    (h_a : a < 2 ^ b) (h_b : b < c) :\n    a * 2 + i.val < 2^c"}], "used_local_defs": [{"name": "AdditiveNTT.qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "AdditiveNTT.intermediateNormVpoly", "content": "noncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)"}, {"name": "AdditiveNTT.intermediateNovelBasisX", "content": "noncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))"}], "used_local_lemmas": [{"name": "AdditiveNTT.Polynomial.foldl_comp", "content": "omit [Fintype L] [DecidableEq L] in\ntheorem Polynomial.foldl_comp (n : ℕ) (f : Fin n → L[X]) : ∀ initInner initOuter: L[X],\n    Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter.comp initInner)\n    = (Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter)).comp initInner"}, {"name": "AdditiveNTT.intermediateNormVpoly_comp_qmap", "content": "theorem intermediateNormVpoly_comp_qmap (i : Fin (ℓ))\n    (k : Fin (ℓ - i - 1)) : -- corresponds to intermediateNormVpoly_comp"}, {"name": "AdditiveNTT.intermediateNormVpoly_comp_qmap_helper", "content": "theorem intermediateNormVpoly_comp_qmap_helper (i : Fin (ℓ))\n    (k : Fin (ℓ - (↑i + 1))) :\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate\n      ⟨↑i + 1, by omega⟩ (k:=⟨k, by simp only; omega⟩)).comp (qMap 𝔽q β ⟨↑i, by omega⟩) =\n    intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate\n      ⟨↑i, by omega⟩ ⟨k + 1, by simp only; omega⟩"}], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis\n\nimport Mathlib.Tactic\n\nimport Mathlib.Data.Finsupp.Defs\n\nimport Mathlib.LinearAlgebra.LinearIndependent.Defs\n\nopen Polynomial AdditiveNTT Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\nsection IntermediateStructures\n\nnoncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))\n\nnoncomputable section DomainBijection\n\nend DomainBijection\n\nnoncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)\n\nnoncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))", "target_theorem": "lemma even_index_intermediate_novel_basis_decomposition (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :\n  intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2, by\n    apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)\n  ⟩  = (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by omega⟩ ⟨j, by\n    apply lt_two_pow_of_lt_two_pow_exp_le j (ℓ-i-1) (ℓ-(i+1)) (by omega) (by omega)\n  ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩) :=", "ground_truth_proof": ":= by\n  unfold intermediateNovelBasisX\n  rw [prod_comp]\n  -- ∏ k ∈ Fin (ℓ - i), (Wₖ⁽ⁱ⁾(X))^((2j)ₖ) = ∏ k ∈ Fin (ℓ - (i+1)), (Wₖ⁽ⁱ⁺¹⁾(X))^((j)ₖ) ∘ q⁽ⁱ⁾(X)\n  simp only [pow_comp]\n  conv_rhs =>\n    enter [2, x]\n    rw [intermediateNormVpoly_comp_qmap_helper 𝔽q]\n\n  -- ⊢ ∏ x, intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i, ⋯⟩ x ^ Nat.getBit (↑x) (↑j * 2) =\n  -- ∏ x, intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i, ⋯⟩ ⟨↑x + 1, ⋯⟩ ^ Nat.getBit ↑x ↑j\n\n  set fleft := fun x : Fin (ℓ - ↑i) =>\n    intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩\n      ⟨x, by simp only; omega⟩ ^ Nat.getBit (↑x) (↑j * 2)\n  have h_n_shift: ℓ - (↑i + 1) + 1 = ℓ - ↑i := by omega\n  have h_fin_n_shift: Fin (ℓ - (↑i + 1) + 1) = Fin (ℓ - ↑i) := by\n    rw [h_n_shift]\n  have h_left_prod_shift :=\n  Fin.prod_univ_succ (M:=L[X]) (n:=ℓ - (↑i + 1)) (f:=fun x => fleft ⟨x, by omega⟩)\n\n  have h_lhs_prod_eq: ∏ x : Fin (ℓ - ↑i),\n    fleft x = ∏ x : Fin (ℓ - (↑i + 1) + 1), fleft ⟨x, by omega⟩ := by\n    exact Eq.symm (Fin.prod_congr' fleft h_n_shift)\n\n  rw [←h_lhs_prod_eq] at h_left_prod_shift\n  rw [h_left_prod_shift]\n\n  have fleft_0_eq_0: fleft ⟨(0: Fin (ℓ - (↑i + 1) + 1)), by omega⟩ = 1 := by\n    unfold fleft\n    simp only\n    have h_exp: Nat.getBit (0: Fin (ℓ - (↑i + 1) + 1)) (↑j * 2) = 0 := by\n      simp only [Fin.coe_ofNat_eq_mod, Nat.zero_mod]\n      have res := Nat.getBit_zero_of_two_mul (n:=j.val)\n      rw [mul_comm] at res\n      exact res\n    rw [h_exp]\n    simp only [pow_zero]\n\n  rw [fleft_0_eq_0, one_mul]\n  apply Finset.prod_congr rfl\n  intro x hx\n  simp only [Fin.val_succ]\n  unfold fleft\n  simp only\n  have h_exp_eq: Nat.getBit (↑x + 1) (↑j * 2) = Nat.getBit ↑x ↑j := by\n    have h_num_eq: j.val * 2 = 2 * j.val := by omega\n    rw [h_num_eq]\n    apply Nat.getBit_eq_succ_getBit_of_mul_two (k:=↑x) (n:=↑j)\n  rw [h_exp_eq]", "nesting_depth": 5, "transitive_dep_count": 50, "subset_aristotle": false, "category": "Applied verif."}
{"id": 21, "thm_name": "ConcreteBinaryTower.split_algebraMap_eq_zero_x", "thm_stmt": "lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x) = (0, x)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "AddCommGroup", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.negSucc", "module": "Init.Data.Int.Basic"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "NNRat", "module": "Mathlib.Data.Rat.Init"}, {"name": "NNRat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "Rat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.toAlgebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "invFun", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "left_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "right_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "absurd", "module": "Init.Prelude"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "instFintypeProd", "module": "Mathlib.Data.Fintype.Prod"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "CommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Algebra.algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "MonoidHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "OneHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}, {"name": "AlgebraTower.toIsScalarTower", "content": "@[simp]\ninstance AlgebraTower.toIsScalarTower (a : AlgebraTower C) {i j k : ι}\n    (h1 : i ≤ j) (h2 : j ≤ k) :\n    letI : Algebra (C i) (C j) :="}, {"name": "AlgebraTower.toAlgebra", "content": "@[simp]\ndef AlgebraTower.toAlgebra {i j : ι} (h : i ≤ j) : Algebra (A i) (A j) :=\n  (AlgebraTower.algebraMap (i:=i) (j:=j) (h:=h)).toAlgebra"}], "lib_lemmas": [{"name": "Ne.dite_eq_left_iff", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.left_eq_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingHom.comp_id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}, {"name": "MonoidHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "Nat.sub_one_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "OneHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "RingHom.coe_mk", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_cast", "module": "Init.PropLemmas"}, {"name": "eqRec_eq_cast", "module": "Batteries.Logic"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.add", "content": "def add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y"}, {"name": "ConcreteBinaryTower.neg", "content": "def neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.ConcreteBTFAddCommGroupProps", "content": "structure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel"}, {"name": "ConcreteBinaryTower.mkAddCommGroupInstance", "content": "def mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}"}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.equivProd", "content": "def equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)"}, {"name": "ConcreteBinaryTower.concrete_mul", "content": "def concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.concrete_inv", "content": "def concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res"}, {"name": "ConcreteBinaryTower.natCast", "content": "def natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.natCast_zero", "content": "def natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :="}, {"name": "ConcreteBinaryTower.natCast_succ", "content": "def natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :="}, {"name": "ConcreteBinaryTower.intCast", "content": "def intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.intCast_ofNat", "content": "def intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :="}, {"name": "ConcreteBinaryTower.intCast_negSucc", "content": "def intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :="}, {"name": "ConcreteBinaryTower.ConcreteBTFRingProps", "content": "structure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c"}, {"name": "ConcreteBinaryTower.ConcreteBTFDivisionRingProps", "content": "structure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldProps", "content": "structure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a"}, {"name": "ConcreteBinaryTower.mkRingInstance", "content": "def mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n"}, {"name": "ConcreteBinaryTower.mkDivisionRingInstance", "content": "def mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)"}, {"name": "ConcreteBinaryTower.mkFieldInstance", "content": "def mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm"}, {"name": "ConcreteBinaryTower.ConcreteBTFStepResult", "content": "structure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))"}, {"name": "ConcreteBinaryTower.liftBTFieldProps", "content": "def liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.liftConcreteBTField", "content": "def liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.concreteCanonicalEmbedding", "content": "def concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :="}, {"name": "ConcreteBinaryTower.instAlgebraLiftConcreteBTField", "content": "instance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))"}, {"name": "ConcreteBinaryTower.getBTFResult", "content": "def getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.canonicalAlgMap", "content": "def canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap", "content": "def concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :="}, {"name": "ConcreteBinaryTower.instAlgebraTowerConcreteBTF", "content": "instance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldAlgebra", "content": "def ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le"}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq", "content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n  ConcreteBTField k = ConcreteBTField m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.cast_join", "content": "lemma cast_join {k n : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) (heq : k = n) :\n  join (k:=k) h_pos hi lo = cast (by rw [heq])\n    (join (k:=n) (by omega) (cast (by subst heq; rfl) hi) (lo:=cast (by subst heq; rfl) lo))"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.split_of_join", "content": "theorem split_of_join {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_join : x = 《hi_btf, lo_btf》) :\n    (hi_btf, lo_btf) = split h_pos x"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_eq_of_dest_eq", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_eq_of_dest_eq (k m n : ℕ) (h_eq : m = n) :\n    (ConcreteBTField k →+* ConcreteBTField m)\n    = (ConcreteBTField k →+* ConcreteBTField n)"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_cast_dest_apply", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_cast_dest_apply (k m n : ℕ) (h_eq : m = n)\n  (f : ConcreteBTField k →+* ConcreteBTField m) (x : ConcreteBTField k) :\n    (cast (ConcreteBTField.RingHom_eq_of_dest_eq (k:=k) (m:=m) (n:=n) h_eq) f) x\n    = cast (by apply cast_ConcreteBTField_eq (h_eq:=h_eq)) (f x)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_id", "content": "lemma concreteTowerAlgebraMap_id (k : ℕ) :\n    concreteTowerAlgebraMap (h_le:=by omega) = RingHom.id (ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_succ_1", "content": "lemma concreteTowerAlgebraMap_succ_1 (k : ℕ) :\n  concreteTowerAlgebraMap (l:=k) (r:=k + 1) (h_le:=by omega) = canonicalAlgMap k"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y\n\ndef neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\nstructure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel\n\ndef mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\ndef equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)\n\ndef concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/\n\ndef concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res\n\nsection FieldLemmasOfLevel0\n\nend FieldLemmasOfLevel0\n\nsection NumericCasting\n\ndef natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :=\n\ndef natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :=\n\ndef intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :=\n\ndef intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :=\n\nend NumericCasting\n\nstructure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c\n\nstructure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one\n\nstructure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a\n\ndef mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n\n\ndef mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)\n\ndef mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm\n\nstructure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))\n\nend FieldOperationsAndInstances\n\nsection BTFieldPropsOneLevelLiftingLemmas\n\nvariable {k : ℕ} {h_k : k > 0}\n\nend BTFieldPropsOneLevelLiftingLemmas\n\nsection TowerFieldsConstruction\n\ndef liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/\n\ndef liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :=\n\ndef concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :=\n\ninstance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))\n\ndef getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/\n\nend TowerFieldsConstruction\n\nsection ConcreteBTFieldAlgebraConstruction\n\ndef canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))\n\ndef concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :=\n\ninstance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/\n\ndef ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le", "target_theorem": "lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra :=", "ground_truth_proof": ":= ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x) = (0, x) := by\n  -- this one is long because of the `cast` stuff, but it should be quite straightforward\n  -- via def of `canonicalAlgMap` and `split_of_join`\n  apply Eq.symm\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  set mappedVal := algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x\n  have h := split_of_join (k:=k) (h_pos:=by omega) (x:=mappedVal)\n    (hi_btf:=zero (k:=k-1)) (lo_btf:=x)\n  apply h\n  -- ⊢ mappedVal = join h_pos zero x\n  unfold mappedVal\n  rw [algebraMap, Algebra.algebraMap]\n  unfold instAlgebra ConcreteBTFieldAlgebra\n  rw [AlgebraTower.toAlgebra, AlgebraTower.algebraMap, instAlgebraTowerConcreteBTF]\n  simp only\n  have h_concrete_embedding_succ_1 := concreteTowerAlgebraMap_succ_1 (k:=k-1)\n  rw! (castMode:=.all) [Nat.sub_one_add_one (by omega)] at h_concrete_embedding_succ_1\n  rw! (castMode:=.all) [h_concrete_embedding_succ_1]\n  rw [eqRec_eq_cast]\n  rw [ConcreteBTField.RingHom_cast_dest_apply (f:=canonicalAlgMap (k - 1))\n    (x:=x) (h_eq:=by omega)]\n  have h_k_sub_1_add_1 : k - 1 + 1 = k := by omega\n  conv_lhs => enter [2]; rw! (castMode:=.all) [h_k_sub_1_add_1]; simp only\n  rw [eqRec_eq_cast, eqRec_eq_cast, cast_cast, cast_eq]\n  rw [ConcreteBTField.RingHom_cast_dest_apply (k:=k - 1) (m:=k - 1 + 1) (n:=k)\n    (h_eq:=by omega) (f:=canonicalAlgMap (k - 1)) (x:=x)]\n  simp only [canonicalAlgMap, concreteCanonicalEmbedding, RingHom.coe_mk, MonoidHom.coe_mk,\n    OneHom.coe_mk]\n  rw [cast_join (k:=k - 1 + 1) (h_pos:=by omega) (n:=k) (heq:=by omega)]\n  simp only [Nat.add_one_sub_one, cast_eq, cast_cast]", "nesting_depth": 8, "transitive_dep_count": 229, "subset_aristotle": false, "category": "Applied verif."}
{"id": 22, "thm_name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "thm_stmt": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}], "used_repo_defs": [{"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}], "lib_lemmas": [{"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/", "target_theorem": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1))) :=", "ground_truth_proof": ":= by\n  have lhs_lo_case := BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat (num_bits:=2 ^ (k - 1))\n    (n:=2 ^ k) (Nat.two_pow_pos (k - 1)) (x:=x)\n  have rhs_hi_case_bitvec_eq := BitVec.extractLsb_eq_shift_ofNat (n:=2 ^ k) (r:=2 ^ k - 1)\n    (l:=2 ^ (k - 1)) (x:=x)\n  constructor\n  · -- Forward direction : split x = (hi_btf, lo_btf) → bitwise operations\n    intro h_split\n    unfold split at h_split\n    have ⟨h_hi, h_lo⟩ := Prod.ext_iff.mp h_split\n    simp only at h_hi h_lo\n    have hi_eq : hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) := by\n      unfold fromNat\n      rw [←h_hi]\n      rw [dcast_symm (h_sub_middle h_pos).symm]\n      rw [rhs_hi_case_bitvec_eq]\n      rw [BitVec.dcast_bitvec_eq]\n    have lo_eq : lo_btf = fromNat (k:=k - 1) (x.toNat &&& ((2 ^ (2 ^ (k - 1)) - 1))) := by\n      unfold fromNat\n      rw [←h_lo]\n      have rhs_lo_case_bitvec_eq :=\n        BitVec.extractLsb_eq_shift_ofNat (n:=2 ^ k) (r:=2 ^ (k - 1) - 1) (l:=0) (x:=x)\n      rw [dcast_symm (h_middle_sub).symm]\n      rw [rhs_lo_case_bitvec_eq]\n      rw [BitVec.dcast_bitvec_eq] -- remove dcast\n      rw [←lhs_lo_case]\n      exact rhs_lo_case_bitvec_eq\n    exact ⟨hi_eq, lo_eq⟩\n  · -- Backward direction : bitwise operations → split x = (hi_btf, lo_btf)\n    intro h_bits\n    unfold split\n    have ⟨h_hi, h_lo⟩ := h_bits\n    have hi_extract_eq : dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1)\n      (lo := 2 ^ (k - 1)) x) = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) := by\n      unfold fromNat\n      rw [dcast_symm (h_sub_middle h_pos).symm]\n      rw [rhs_hi_case_bitvec_eq]\n      rw [BitVec.dcast_bitvec_eq]\n\n    have lo_extract_eq : dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1)\n      (lo := 0) x) = fromNat (k:=k - 1) (x.toNat &&& ((2 ^ (2 ^ (k - 1)) - 1))) := by\n      unfold fromNat\n      rw [lhs_lo_case]\n      rw [BitVec.dcast_bitvec_eq]\n\n    simp only [hi_extract_eq, Nat.sub_zero, lo_extract_eq, Nat.and_two_pow_sub_one_eq_mod, h_hi,\n      h_lo]", "nesting_depth": 4, "transitive_dep_count": 40, "subset_aristotle": false, "category": "Applied verif."}
{"id": 23, "thm_name": "AdditiveNTT.basisVectors_span", "thm_stmt": "theorem basisVectors_span (ℓ : Nat) (h_ℓ : ℓ ≤ r) :\n    Submodule.span L (Set.range (basisVectors 𝔽q β ℓ h_ℓ)) = ⊤", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean", "imports": ["import Mathlib.Algebra.Polynomial.Degree.Definitions", "import ArkLib.Data.Polynomial.MonomialBasis", "import Mathlib.LinearAlgebra.StdBasis", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.Polynomial.Frobenius"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "WithBot", "module": "Mathlib.Order.TypeTags"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "FiniteDimensional", "module": "Mathlib.LinearAlgebra.FiniteDimensional.Defs"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Module.finrank", "module": "Mathlib.LinearAlgebra.Dimension.Finrank"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "Fin.isEmpty'", "module": "Mathlib.Logic.IsEmpty"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "LinearIndepOn", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.InjOn", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.val", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "OrderDual", "module": "Mathlib.Order.Basic"}, {"name": "OrderDual.toDual", "module": "Mathlib.Order.Synonym"}, {"name": "AddHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "Module.Basis", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "AddSubmonoidClass", "module": "Mathlib.Algebra.Group.Submonoid.Defs"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.linearCombination", "module": "Mathlib.LinearAlgebra.Finsupp.LinearCombination"}, {"name": "Finsupp.sum", "module": "Mathlib.Algebra.BigOperators.Finsupp.Basic"}, {"name": "Module.Basis.mk", "module": "Mathlib.LinearAlgebra.Basis.Basic"}, {"name": "Polynomial.coeff", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.degreeLT", "module": "Mathlib.RingTheory.Polynomial.Basic"}, {"name": "Polynomial.monomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "SetLike", "module": "Mathlib.Data.SetLike.Basic"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "finiteDimensional_degreeLT", "content": "instance finiteDimensional_degreeLT {n : ℕ} (h_n_pos : 0 < n) :\n  FiniteDimensional L L⦃< n⦄[X] :="}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "monomialBasisOfDegreeLT", "content": "noncomputable def monomialBasisOfDegreeLT {n : ℕ} : Basis (Fin n) L (L⦃< n⦄[X]) :="}], "lib_lemmas": [{"name": "Fin.card_Ico", "module": "Mathlib.Order.Interval.Finset.Fin"}, {"name": "Fin.coe_ofNat_eq_mod", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fintype.card_ofFinset", "module": "Mathlib.Data.Fintype.Card"}, {"name": "LinearIndependent.injective", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Set.card_image_of_injective", "module": "Mathlib.Data.Set.Finite.Basic"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_image_of_mem", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.range_comp", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.toFinset_card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Subtype.coe_injective", "module": "Mathlib.Data.Subtype"}, {"name": "Subtype.mk_eq_mk", "module": "Mathlib.Data.Subtype"}, {"name": "Subtype.range_coe", "module": "Mathlib.Data.Set.Image"}, {"name": "finrank_span_set_eq_card", "module": "Mathlib.LinearAlgebra.Dimension.Constructions"}, {"name": "tsub_zero", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Module.card_eq_pow_finrank", "module": "Mathlib.FieldTheory.Finiteness"}, {"name": "Finset.card_univ", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Finset.sum_const", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Polynomial.degree_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.degree_prod_of_monic", "module": "Mathlib.Algebra.Polynomial.BigOperators"}, {"name": "Polynomial.monic_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "nsmul_eq_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fact.out", "module": "Mathlib.Logic.Basic"}, {"name": "Fin.le_zero_iff'", "module": "Mathlib.Data.Fin.SuccPred"}, {"name": "Nat.not_lt_zero", "module": "Init.Prelude"}, {"name": "Set.Ico_eq_empty_iff", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.compl_eq_univ_diff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Set.empty_subset", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.image_empty", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_subset_image_iff", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.subset_compl_singleton_iff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Submodule.span_mono", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "linearIndependent_iff_notMem_span", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Finset.prod_ne_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Polynomial.eval_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_prod", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_sub", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Polynomial.degree_C_mul", "module": "Mathlib.Algebra.Polynomial.Degree.Lemmas"}, {"name": "inv_ne_zero", "module": "Mathlib.Algebra.GroupWithZero.NeZero"}, {"name": "one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "CharP.cast_eq_zero", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "Fin.coe_castLE", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.sum_univ_eq_sum_range", "module": "Mathlib.Data.Fintype.BigOperators"}, {"name": "Fin.val_eq_zero_iff", "module": "Init.Data.Fin.Lemmas"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_empty", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.univ_eq_empty", "module": "Mathlib.Data.Finset.BooleanAlgebra"}, {"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.cast_ite", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_ofNat", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_one", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_pow", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Nat.lt_one_iff", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mod_two_ne_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.range_succ_eq_Icc_zero", "module": "Mathlib.Order.Interval.Finset.Nat"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.degree_pow", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.degree_prod", "module": "Mathlib.Algebra.Polynomial.BigOperators"}, {"name": "WithBot.zero_eq_coe", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.WithTop"}, {"name": "if_false", "module": "Init.ByCases"}, {"name": "if_true", "module": "Init.ByCases"}, {"name": "mul_eq_zero_comm", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "pow_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Polynomial.coeff_ne_zero_of_eq_degree", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Matrix.det_of_lowerTriangular", "module": "Mathlib.LinearAlgebra.Matrix.Block"}, {"name": "Matrix.linearIndependent_rows_of_det_ne_zero", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "AddHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "LinearMap.coe_mk", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "OrderDual.toDual_lt_toDual", "module": "Mathlib.Order.Synonym"}, {"name": "Polynomial.coeff_eq_zero_of_natDegree_lt", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.natDegree_eq_of_degree_eq_some", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "LinearIndependent.of_comp", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Fintype.card_fin", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Submodule.eq_top_of_finrank_eq", "module": "Mathlib.LinearAlgebra.FiniteDimensional.Basic"}, {"name": "finrank_span_eq_card", "module": "Mathlib.LinearAlgebra.Dimension.Constructions"}], "repo_lemmas": [{"name": "getBit_repr", "content": "theorem getBit_repr {ℓ : Nat} : ∀ j, j < 2^ℓ →\n  j = ∑ k ∈ Finset.Icc 0 (ℓ-1), (getBit k j) * 2^k"}, {"name": "getBit_of_shiftRight", "content": "lemma getBit_of_shiftRight {n p : ℕ}:\n  ∀ k, getBit k (n >>> p) = getBit (k+p) n"}, {"name": "getBit_eq_zero_or_one", "content": "lemma getBit_eq_zero_or_one {k n : Nat} :\n  getBit k n = 0 ∨ getBit k n = 1"}, {"name": "finrank_degreeLT_n", "content": "theorem finrank_degreeLT_n (n : ℕ) : Module.finrank L (L⦃< n⦄[X]) = n"}], "used_local_defs": [{"name": "AdditiveNTT.U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "AdditiveNTT.W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "AdditiveNTT.normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "AdditiveNTT.X", "content": "noncomputable def Xⱼ (ℓ : ℕ) (h_ℓ : ℓ ≤ r) (j : Fin (2 ^ ℓ)) : L[X] :=\n  (Finset.univ : Finset (Fin ℓ)).prod\n    (fun i => (normalizedW 𝔽q β (Fin.castLE h_ℓ i))^(Nat.getBit i j))"}, {"name": "AdditiveNTT.basisVectors", "content": "noncomputable def basisVectors (ℓ : Nat) (h_ℓ : ℓ ≤ r) :\n  Fin (2 ^ ℓ) → L⦃<2^ℓ⦄[X] :=\n  fun j => ⟨Xⱼ 𝔽q β ℓ h_ℓ j, by admit /- proof elided -/\n  ⟩"}, {"name": "AdditiveNTT.CoeffVecSpace", "content": "abbrev CoeffVecSpace (L : Type u) (ℓ : Nat) := Fin (2^ℓ) → L"}, {"name": "AdditiveNTT.toCoeffsVec", "content": "def toCoeffsVec (ℓ : Nat) : L⦃<2^ℓ⦄[X] →ₗ[L] CoeffVecSpace L ℓ where\n  toFun := fun p => fun i => p.val.coeff i.val\n  map_add' := fun p q => by admit /- proof elided -/"}, {"name": "AdditiveNTT.changeOfBasisMatrix", "content": "noncomputable def changeOfBasisMatrix (ℓ : Nat) (h_ℓ : ℓ ≤ r) : Matrix (Fin (2^ℓ)) (Fin (2^ℓ)) L :=\n    fun j i => (toCoeffsVec (L := L) (ℓ := ℓ) (\n      basisVectors 𝔽q β ℓ h_ℓ j)) i"}], "used_local_lemmas": [{"name": "AdditiveNTT.finrank_U", "content": "omit [Fintype L] [Fintype 𝔽q] h_Fq_char_prime in\nlemma finrank_U (i : Fin r) :\n  Module.finrank 𝔽q (U 𝔽q β i) = i"}, {"name": "AdditiveNTT.U_card", "content": "lemma U_card (i : Fin r) :\n    Fintype.card (U 𝔽q β i) = (Fintype.card 𝔽q)^i.val"}, {"name": "AdditiveNTT.βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "AdditiveNTT.degree_W", "content": "lemma degree_W (i : Fin r) : (W 𝔽q β i).degree = (Fintype.card 𝔽q)^i.val"}, {"name": "AdditiveNTT.Wᵢ_eval_βᵢ_neq_zero", "content": "lemma Wᵢ_eval_βᵢ_neq_zero\n    (i : Fin r): (W 𝔽q β i).eval (β i) ≠ 0"}, {"name": "AdditiveNTT.degree_normalizedW", "content": "lemma degree_normalizedW (i : Fin r) :\n  (normalizedW 𝔽q β i).degree = (Fintype.card 𝔽q)^(i.val)"}, {"name": "AdditiveNTT.X", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "AdditiveNTT.degree_X", "content": "lemma degree_Xⱼ (ℓ : ℕ) (h_ℓ : ℓ ≤ r) (j : Fin (2 ^ ℓ)) :\n  (Xⱼ 𝔽q β ℓ h_ℓ j).degree = j"}, {"name": "AdditiveNTT.linearIndependent_rows_of_lower_triangular_ne_zero_diag", "content": "lemma linearIndependent_rows_of_lower_triangular_ne_zero_diag\n  {n : ℕ} {R : Type*} [Field R] (A : Matrix (Fin n) (Fin n) R)\n  (h_lower_triangular : A.BlockTriangular ⇑OrderDual.toDual) (h_diag : ∀ i, A i i ≠ 0) :\n  LinearIndependent R A"}, {"name": "AdditiveNTT.changeOfBasisMatrix_lower_triangular", "content": "omit h_Fq_char_prime in\ntheorem changeOfBasisMatrix_lower_triangular\n  (ℓ : Nat) (h_ℓ : ℓ ≤ r) :\n  (changeOfBasisMatrix 𝔽q β ℓ h_ℓ).BlockTriangular ⇑OrderDual.toDual"}, {"name": "AdditiveNTT.changeOfBasisMatrix_diag_ne_zero", "content": "omit h_Fq_char_prime in\ntheorem changeOfBasisMatrix_diag_ne_zero\n  (ℓ : Nat) (h_ℓ : ℓ ≤ r) :\n  (∀ i, (changeOfBasisMatrix 𝔽q β ℓ h_ℓ) i i ≠ 0)"}, {"name": "AdditiveNTT.coeff_vectors_linear_independent", "content": "lemma coeff_vectors_linear_independent\n  (ℓ : Nat) (h_ℓ : ℓ ≤ r) :\n    LinearIndependent L (toCoeffsVec (ℓ := ℓ) ∘ (basisVectors 𝔽q β ℓ h_ℓ))"}, {"name": "AdditiveNTT.basisVectors_linear_independent", "content": "theorem basisVectors_linear_independent (ℓ : Nat) (h_ℓ : ℓ ≤ r) :\n    LinearIndependent L (basisVectors 𝔽q β ℓ h_ℓ)"}], "local_ctx": "import ArkLib.Data.Nat.Bitwise\n\nimport ArkLib.Data.Polynomial.Frobenius\n\nimport ArkLib.Data.Polynomial.MonomialBasis\n\nimport Mathlib.LinearAlgebra.StdBasis\n\nimport Mathlib.Algebra.Polynomial.Degree.Definitions\n\nopen Polynomial FiniteDimensional Finset Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (h_dim : Module.finrank 𝔽q L = r)\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n\nsection LinearSubspaces\n\ndef U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))\n\nnoncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)\n\nend LinearSubspaces\n\nsection LinearityOfSubspaceVanishingPolynomials\n\nnoncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i\n\nend LinearityOfSubspaceVanishingPolynomials\n\nsection NovelPolynomialBasisProof\n\nnoncomputable def basisVectors (ℓ : Nat) (h_ℓ : ℓ ≤ r) :\n  Fin (2 ^ ℓ) → L⦃<2^ℓ⦄[X] :=\n  fun j => ⟨Xⱼ 𝔽q β ℓ h_ℓ j, by admit /- proof elided -/\n  ⟩\n\nabbrev CoeffVecSpace (L : Type u) (ℓ : Nat) := Fin (2^ℓ) → L\n\ndef toCoeffsVec (ℓ : Nat) : L⦃<2^ℓ⦄[X] →ₗ[L] CoeffVecSpace L ℓ where\n  toFun := fun p => fun i => p.val.coeff i.val\n  map_add' := fun p q => by admit /- proof elided -/\n\nnoncomputable def changeOfBasisMatrix (ℓ : Nat) (h_ℓ : ℓ ≤ r) : Matrix (Fin (2^ℓ)) (Fin (2^ℓ)) L :=\n    fun j i => (toCoeffsVec (L := L) (ℓ := ℓ) (\n      basisVectors 𝔽q β ℓ h_ℓ j)) i", "target_theorem": "theorem basisVectors_span (ℓ : Nat) (h_ℓ : ℓ ≤ r) :\n    Submodule.span L (Set.range (basisVectors 𝔽q β ℓ h_ℓ)) = ⊤ :=", "ground_truth_proof": ":= by\n  have h_li := basisVectors_linear_independent 𝔽q β ℓ h_ℓ\n  let n := 2 ^ ℓ\n  have h_n: n = 2 ^ ℓ := by omega\n  have h_n_pos: 0 < n := by\n    rw [h_n]\n    exact Nat.two_pow_pos ℓ\n  have h_finrank_eq_n : Module.finrank L (L⦃< n⦄[X]) = n := finrank_degreeLT_n n\n  -- We have `n` linearly independent vectors in an `n`-dimensional space.\n  -- The dimension of their span is `n`.\n  have h_span_finrank : Module.finrank L (Submodule.span L (Set.range (\n    basisVectors 𝔽q β ℓ h_ℓ))) = n := by\n    rw [finrank_span_eq_card h_li, Fintype.card_fin]\n  -- A subspace with the same dimension as the ambient space must be the whole space.\n  rw [←h_finrank_eq_n] at h_span_finrank\n  have inst_finite_dim : FiniteDimensional (K := L) (V := L⦃< n⦄[X]) :=\n    finiteDimensional_degreeLT (h_n_pos := by omega)\n  apply Submodule.eq_top_of_finrank_eq (K := L) (V := L⦃< n⦄[X])\n  exact h_span_finrank", "nesting_depth": 9, "transitive_dep_count": 163, "subset_aristotle": false, "category": "Applied verif."}
{"id": 24, "thm_name": "MlPoly.coeff_of_toMvPolynomial_eq_coeff_of_MlPoly", "thm_stmt": "theorem coeff_of_toMvPolynomial_eq_coeff_of_MlPoly (p : MlPoly R n) (m : Fin n →₀ ℕ) :\n  coeff m (toMvPolynomial p) =\n    if h_binary: (∀ j: Fin n, m j ≤ 1) then\n      let i_of_m: ℕ := Nat.binaryFinMapToNat (m:=m) (h_binary:=h_binary)\n      p[i_of_m]\n    else\n      0", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/MlPoly/Equiv.lean", "imports": ["import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.MlPoly.Basic", "import ArkLib.Data.MvPolynomial.Notation"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.onFinset", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "MvPolynomial.X", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.monomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "Ne", "module": "Init.Core"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "MvPolynomial.coeff", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "MlPoly", "content": "@[reducible]\ndef MlPoly (R : Type*) (n : ℕ) := Vector R (2 ^ n) "}, {"name": "binaryFinMapToNat", "content": "def binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary : ∀ j: Fin n, m j ≤ 1) : Fin (2^n) :="}], "lib_lemmas": [{"name": "Finsupp.onFinset_apply", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Fintype.sum_eq_zero", "module": "Mathlib.Data.Fintype.BigOperators"}, {"name": "MvPolynomial.coeff_monomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_sum", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.mem_restrictDegree", "module": "Mathlib.RingTheory.MvPolynomial.Basic"}, {"name": "MvPolynomial.mem_support_iff", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "congrFun", "module": "Init.Prelude"}, {"name": "Fin.eq_of_val_eq", "module": "Init.Prelude"}, {"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.is_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Finsupp.ext_iff", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_eq_single", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "MvPolynomial.ext", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.notMem_support_iff", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "Nat.not_lt", "module": "Init.Data.Nat.Basic"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "not_le", "module": "Mathlib.Order.Defs.LinearOrder"}], "repo_lemmas": [{"name": "getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "getBit_of_lt_two_pow", "content": "lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :\n  getBit k a = if k < n then getBit k a else 0"}, {"name": "getBit_eq_testBit", "content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"}, {"name": "getBit_of_binaryFinMapToNat", "content": "lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :\n    ∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val\n      = if h_k: k < n then m ⟨k, by omega⟩ else 0"}, {"name": "and_two_pow_eq_zero_of_getBit_0", "content": "lemma and_two_pow_eq_zero_of_getBit_0 {n i : ℕ} (h_getBit : getBit i n = 0)\n    : n &&& (2 ^ i) = 0"}, {"name": "and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "getBit_of_xor", "content": "lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m"}, {"name": "getBit_zero_eq_zero", "content": "lemma getBit_zero_eq_zero {k : Nat} : getBit k 0 = 0"}, {"name": "sum_of_and_eq_zero_is_xor", "content": "lemma sum_of_and_eq_zero_is_xor {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ^^^ m"}, {"name": "sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||"}, {"name": "xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}], "used_local_defs": [{"name": "MlPoly.monomialOfNat", "content": "noncomputable def monomialOfNat (i : ℕ) : (Fin n) →₀ ℕ :=\n  Finsupp.onFinset (s:=Finset.univ (α:=Fin n)) (fun j => Nat.getBit j.val i) (by admit /- proof elided -/\n    )"}, {"name": "MlPoly.toMvPolynomial", "content": "def toMvPolynomial (p : MlPoly R n) : R[X Fin n] :=\n  ∑ i : Fin (2 ^ n), MvPolynomial.monomial (monomialOfNat i) (a:=p[i])"}], "used_local_lemmas": [{"name": "MlPoly.eq_monomialOfNat_iff_eq_bitRepr", "content": "theorem eq_monomialOfNat_iff_eq_bitRepr (m : Fin n →₀ ℕ)\n  (h_binary : ∀ j : Fin n, m j ≤ 1) (i: Fin (2^n)) :\n  monomialOfNat i = m ↔ i = Nat.binaryFinMapToNat m h_binary"}, {"name": "MlPoly.toMvPolynomial_is_multilinear", "content": "theorem toMvPolynomial_is_multilinear (p : MlPoly R n) :\n  (toMvPolynomial p) ∈ R⦃≤ 1⦄[X Fin n]"}], "local_ctx": "import ArkLib.Data.MlPoly.Basic\n\nimport ArkLib.Data.MvPolynomial.Notation\n\nopen MvPolynomial\n\nvariable {R : Type*} [CommRing R] {n : ℕ}\n\nnoncomputable section\n\nnamespace MlPoly\n\nnoncomputable def monomialOfNat (i : ℕ) : (Fin n) →₀ ℕ :=\n  Finsupp.onFinset (s:=Finset.univ (α:=Fin n)) (fun j => Nat.getBit j.val i) (by admit /- proof elided -/\n    ) \n\ndef toMvPolynomial (p : MlPoly R n) : R[X Fin n] :=\n  ∑ i : Fin (2 ^ n), MvPolynomial.monomial (monomialOfNat i) (a:=p[i])", "target_theorem": "theorem coeff_of_toMvPolynomial_eq_coeff_of_MlPoly (p : MlPoly R n) (m : Fin n →₀ ℕ) :\n  coeff m (toMvPolynomial p) =\n    if h_binary: (∀ j: Fin n, m j ≤ 1) then\n      let i_of_m: ℕ :=", "ground_truth_proof": ":= Nat.binaryFinMapToNat (m:=m) (h_binary:=h_binary)\n      p[i_of_m]\n    else\n      0\n  := by\n  if h_binary: (∀ j: Fin n, m j ≤ 1) then\n    unfold toMvPolynomial\n    simp only [h_binary, implies_true, ↓reduceDIte]\n    let i_of_m := Nat.binaryFinMapToNat m h_binary\n    have h_mono_eq : monomialOfNat i_of_m = m := by\n      ext j; simp only [monomialOfNat, Finsupp.onFinset_apply]\n      have h_getBit := Nat.getBit_of_binaryFinMapToNat (n:=n) (m:=m)\n        (h_binary:=h_binary) (k:=j)\n      rw [h_getBit]\n      simp only [j.isLt, ↓reduceDIte, Fin.eta]\n    rw [MvPolynomial.coeff_sum]\n    simp only [MvPolynomial.coeff_monomial]\n    -- ⊢ (∑ x, if monomialOfNat ↑x = m then p[x] else 0) = p[↑(Nat.binaryFinMapToNat ⇑m ⋯)]\n    set f := fun x: Fin (2^n) => if monomialOfNat x.val = m then p[x] else (0: R)\n    -- ⊢ Finset.univ.sum f = p[↑(Nat.binaryFinMapToNat ⇑m ⋯)]\n    rw [Finset.sum_eq_single (a:=⟨i_of_m, by omega⟩)]\n    · -- Goal 1: Prove the main term is correct.\n      simp only [h_mono_eq, ↓reduceIte, Fin.eta, Fin.getElem_fin];\n      rfl\n    · -- Goal 2: Prove all other terms are zero.\n      intro j h_j_mem_univ h_ji_ne\n      -- If `j ≠ i_of_m`, then `monomialOfNat j ≠ monomialOfNat i_of_m` (which is `m`).\n      -- ⊢ (monomial (monomialOfNat ↑j)) p[j] = 0\n      have h_mono_ne : monomialOfNat j.val ≠ m := by\n        intro h_eq_contra\n        have h_j_is_i_of_m := eq_monomialOfNat_iff_eq_bitRepr (m:=m)\n          (h_binary:=h_binary) (i:=j).mp h_eq_contra\n        exact h_ji_ne h_j_is_i_of_m\n      simp only [h_mono_ne, ↓reduceIte]\n    -- Goal 3: Prove `i` is in the summation set.\n    · simp [Finset.mem_univ]\n  else -- this case is similar to the proof of `right_inv` in `equivMvPolynomialDeg1`\n    simp only [h_binary, ↓reduceDIte]\n    -- ⊢ coeff m p.toMvPolynomial = 0\n    have hv := toMvPolynomial_is_multilinear p\n    let vMlPoly: R⦃≤ 1⦄[X Fin n] := ⟨p.toMvPolynomial, hv⟩\n    have h_v_coeff_zero : vMlPoly.val.coeff m = 0 := by\n      refine notMem_support_iff.mp ?_\n      by_contra h_mem_support\n      have hvMlPoly := vMlPoly.2\n      rw [MvPolynomial.mem_restrictDegree] at hvMlPoly\n      have h_deg_le_one: ∀ j: Fin n, (m j) ≤ 1 := by\n        exact fun j ↦ hvMlPoly m h_mem_support j\n      simp only [not_forall, not_le] at h_binary -- h_binary : ∃ x, 1 < m x\n      obtain ⟨j, hj⟩ := h_binary\n      have h_not_1_lt_m_j: ¬(1 < m j) := by exact Nat.not_lt.mpr (hv h_mem_support j)\n      exact h_not_1_lt_m_j hj\n    exact h_v_coeff_zero", "nesting_depth": 6, "transitive_dep_count": 57, "subset_aristotle": false, "category": "Applied verif."}
{"id": 25, "thm_name": "Polynomial.Bivariate.degreeX_mul", "thm_stmt": "@[simp, grind _=_]\nlemma degreeX_mul [IsDomain F] (f g : F[X][Y]) (hf : f ≠ 0) (hg : g ≠ 0) :\n  degreeX (f * g) = degreeX f + degreeX g", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/Polynomial/Bivariate.lean", "imports": ["import ArkLib.Data.Polynomial.Prelims"], "used_lib_defs": [{"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "IsDomain", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "WithBot", "module": "Mathlib.Order.TypeTags"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "OrderBot", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "SemilatticeSup", "module": "Mathlib.Order.Lattice"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Finset.sum_eq_single", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_union", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sup_lt_iff", "module": "Mathlib.Data.Finset.Lattice.Fold"}, {"name": "Polynomial.degree_eq_natDegree", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.degree_lt_degree", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.degree_sum_le", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "WithBot.bot_lt_coe", "module": "Mathlib.Order.WithBot"}, {"name": "lt_of_le_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Finset.exists_mem_eq_sup", "module": "Mathlib.Data.Finset.Lattice.Fold"}, {"name": "Finset.fold_max_le", "module": "Mathlib.Data.Finset.Fold"}, {"name": "Finset.le_sup'_of_le", "module": "Mathlib.Data.Finset.Lattice.Fold"}, {"name": "Finset.max'_mem", "module": "Mathlib.Data.Finset.Max"}, {"name": "Finset.sup_le_iff", "module": "Mathlib.Data.Finset.Lattice.Fold"}, {"name": "Polynomial.coeff_mul", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.mem_support_iff", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.natDegree_sum_le", "module": "Mathlib.Algebra.Polynomial.BigOperators"}, {"name": "Polynomial.natDegree_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.notMem_support_iff", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "le_of_eq", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "mul_eq_zero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "zero_eq_mul", "module": "Mathlib.Algebra.GroupWithZero.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Polynomial.Bivariate.coeff", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "Polynomial.Bivariate.degreeX", "content": "def degreeX (f : F[X][Y]) : ℕ := f.support.sup (fun n => (f.coeff n).natDegree)"}], "used_local_lemmas": [{"name": "Polynomial.Bivariate.natDeg_sum_eq_of_unique", "content": "lemma natDeg_sum_eq_of_unique {α : Type} {s : Finset α} {f : α → F[X]} {deg : ℕ}\n  (mx : α) (h : mx ∈ s) :\n    (f mx).natDegree = deg →\n    (∀ y ∈ s, y ≠ mx → (f y).natDegree < deg ∨ f y = 0) →\n    (∑ x ∈ s, f x).natDegree = deg"}, {"name": "Polynomial.Bivariate.sup_eq_of_le_of_reach", "content": "lemma sup_eq_of_le_of_reach {α β : Type} [SemilatticeSup β] [OrderBot β] {s : Finset α} {f : α → β}\n      (x : α) {y : β} (h : x ∈ s) :\n    f x = y →\n    (∀ x ∈ s, f x ≤ y) →\n    s.sup f = y"}], "local_ctx": "import ArkLib.Data.Polynomial.Prelims\n\nopen Polynomial\n\nopen Polynomial.Bivariate\n\nnamespace Polynomial.Bivariate\n\nnoncomputable section\n\nvariable {F : Type} [Semiring F]\n\ndef coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i\n\ndef degreeX (f : F[X][Y]) : ℕ := f.support.sup (fun n => (f.coeff n).natDegree)\n\nvariable {f : F[X][Y]}\n\nopen Univariate in\n\nopen Classical in", "target_theorem": "@[simp, grind _=_]\nlemma degreeX_mul [IsDomain F] (f g : F[X][Y]) (hf : f ≠ 0) (hg : g ≠ 0) :\n  degreeX (f * g) = degreeX f + degreeX g :=", "ground_truth_proof": ":= by\n  letI s₁ := {n ∈ f.support | (f.coeff n).natDegree = degreeX f}\n  letI s₂ := {n ∈ g.support | (g.coeff n).natDegree = degreeX g}\n  have f_mdeg_nonempty : s₁.Nonempty := by\n    obtain ⟨mfx, _, _⟩ :=\n      Finset.exists_mem_eq_sup _ (show f.support.Nonempty by grind) fun n ↦ (f.coeff n).natDegree\n    use mfx\n    grind [degreeX]\n  have g_mdeg_nonempty : s₂.Nonempty := by\n    obtain ⟨mfx, _, _⟩ :=\n      Finset.exists_mem_eq_sup _ (show g.support.Nonempty by grind) fun n ↦ (g.coeff n).natDegree\n    use mfx\n    grind [degreeX]\n  set mmfx := s₁.max' f_mdeg_nonempty with hmmfx\n  set mmgx := s₂.max' g_mdeg_nonempty with hmmgx\n  have mmfx_def : (f.coeff mmfx).natDegree = degreeX f := by\n    have h := Finset.max'_mem _ f_mdeg_nonempty\n    grind\n  have mmgx_def : (g.coeff mmgx).natDegree = degreeX g := by\n    have h := Finset.max'_mem _ g_mdeg_nonempty\n    grind\n  have h₁ : mmfx ∈ s₁ := Finset.max'_mem _ f_mdeg_nonempty\n  have h₂ : mmgx ∈ s₂ := Finset.max'_mem _ g_mdeg_nonempty\n  have mmfx_neq_0 : f.coeff mmfx ≠ 0 := by grind\n  have mmgx_neq_0 : g.coeff mmgx ≠ 0 := by grind\n  have h₁ {n} : (f.coeff n).natDegree ≤ degreeX f := by\n    have : degreeX f = (f.coeff mmfx).natDegree := by grind\n    by_cases h : n ∈ f.toFinsupp.support\n    · convert Finset.sup_le_iff.mp (le_of_eq this) n h\n    · simp [Polynomial.notMem_support_iff.1 h]\n  have h₂ {n} : (g.coeff n).natDegree ≤ (g.coeff mmgx).natDegree := by\n    have : degreeX g = (g.coeff mmgx).natDegree := by grind\n    by_cases h : n ∈ g.toFinsupp.support\n    · convert Finset.sup_le_iff.mp (le_of_eq this) n h\n    · simp [Polynomial.notMem_support_iff.1 h]\n  have h₁' {n} (h : mmfx < n) :\n    (f.coeff n).natDegree < (f.coeff mmfx).natDegree ∨ f.coeff n = 0 := by\n    suffices f.coeff n ≠ 0 → (f.coeff mmfx).natDegree ≤ (f.coeff n).natDegree → False by grind\n    intros h' contra\n    have : (f.coeff mmfx).natDegree = (f.coeff n).natDegree := by grind\n    have : n ≤ mmfx := Finset.le_sup'_of_le (hb := show n ∈ s₁ by grind) (h := by simp)\n    grind\n  have h₂' {n} (h : mmgx < n) :\n    (g.coeff n).natDegree < (g.coeff mmgx).natDegree ∨ g.coeff n = 0 := by\n    suffices g.coeff n ≠ 0 → (g.coeff mmgx).natDegree ≤ (g.coeff n).natDegree → False by grind\n    intros h' contra\n    have : (g.coeff mmgx).natDegree = (g.coeff n).natDegree := by grind\n    have : n ≤ mmgx := Finset.le_sup'_of_le (hb := show n ∈ s₂ by grind) (h := by simp)\n    grind\n  unfold degreeX\n  have : (fun n ↦ ((f * g).coeff n).natDegree) =\n         fun n ↦ (∑ x ∈ Finset.antidiagonal n, f.coeff x.1 * g.coeff x.2).natDegree := by\n    funext n; rw [Polynomial.coeff_mul]\n  rw [this]\n  have : (∑ x ∈ Finset.antidiagonal (mmfx + mmgx), f.coeff x.1 * g.coeff x.2).natDegree =\n         degreeX f + degreeX g := by\n    apply natDeg_sum_eq_of_unique (mmfx, mmgx) (by simp) (by grind)\n    rintro ⟨y₁, y₂⟩ h h'\n    have : mmfx < y₁ ∨ mmgx < y₂ := by\n      have h_anti : y₁ + y₂ = mmfx + mmgx := by simpa using h\n      grind [mul_eq_zero]\n    grind [mul_eq_zero]\n  apply sup_eq_of_le_of_reach (mmfx + mmgx) _ this\n  swap\n  · rw [Polynomial.mem_support_iff, Polynomial.coeff_mul]\n    by_contra h\n    rw [h, natDegree_zero] at this\n    have : ∑ x ∈ Finset.antidiagonal (mmfx + mmgx), f.coeff x.1 * g.coeff x.2 =\n           f.coeff mmfx * g.coeff mmgx := by\n      apply Finset.sum_eq_single\n              (f := (fun x ↦ f.coeff x.1 * g.coeff x.2)) (mmfx, mmgx) (h₁ := by simp)\n      rintro ⟨b₁, b₂⟩ h h'\n      have : mmfx < b₁ ∨ mmgx < b₂ := by\n        have h_anti : b₁ + b₂ = mmfx + mmgx := by simpa using h\n        have fdegx_eq_0 : degreeX f = 0 := by grind\n        have gdegx_eq_0 : degreeX g = 0 := by grind\n        grind [mul_eq_zero]\n      grind [mul_eq_zero]\n    grind [zero_eq_mul]\n  · intros x h\n    apply le_trans\n      (Polynomial.natDegree_sum_le (Finset.antidiagonal x) (fun x ↦ f.coeff x.1 * g.coeff x.2))\n    rw [Finset.fold_max_le]\n    grind [degreeX]", "nesting_depth": 2, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Applied verif."}
{"id": 26, "thm_name": "Binius.BinaryBasefold.card_qMap_total_fiber", "thm_stmt": "omit [CharP L 2] [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 [NeZero ℓ] in\ntheorem card_qMap_total_fiber (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :\n    Fintype.card (Set.image (qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps)\n      (y := y)) Set.univ) = 2 ^ steps", "lean_root": "ArkLib", "rel_path": "ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean", "imports": ["import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound", "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import ArkLib.Data.MvPolynomial.Multilinear", "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch", "import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.CodingTheory.ReedSolomon", "import ArkLib.Data.Vector.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Submodule.map", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Module.Basis", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Disjoint", "module": "Mathlib.Order.Disjoint"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "LinearEquiv", "module": "Mathlib.Algebra.Module.Equiv.Defs"}, {"name": "LinearEquiv.ofBijective", "module": "Mathlib.Algebra.Module.Submodule.Equiv"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap.codRestrict", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "LinearMap.ker", "module": "Mathlib.Algebra.Module.Submodule.Ker"}, {"name": "Module.Basis.span", "module": "Mathlib.LinearAlgebra.Basis.Basic"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.subtype", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.equivFunOnFinite", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "sDomain", "content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n  let W_i_norm := normalizedW 𝔽q β i\n  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n    AdditiveNTT.normalizedW_is_additive 𝔽q β i\n  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)\n    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "polyEvalLinearMap", "content": "noncomputable def polyEvalLinearMap {L 𝔽q : Type*} [Field L] [Field 𝔽q] [Algebra 𝔽q L]\n  (p : L[X]) (hp_add : IsLinearMap 𝔽q (fun x : L => p.eval x)) : L →ₗ[𝔽q] L :=\n{\n  toFun    := fun x => p.eval x,\n  map_add' := hp_add.map_add,\n  map_smul' := hp_add.map_smul\n}"}, {"name": "sDomain_basis", "content": "noncomputable def sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    Basis (Fin (ℓ + R_rate - i)) 𝔽q (\n      sDomain 𝔽q β h_ℓ_add_R_rate i) :="}, {"name": "sBasis", "content": "def sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=\n  fun k => β ⟨i + k.val, by admit /- proof elided -/\n  ⟩"}], "lib_lemmas": [{"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Module.Basis.repr_linearCombination", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Module.Basis.repr_symm_apply", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Nat.add_zero", "module": "Init.Core"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}, {"name": "tsub_zero", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Decidable.not_not", "module": "Init.PropLemmas"}, {"name": "Fin.eq_of_val_eq", "module": "Init.Prelude"}, {"name": "Fin.is_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fintype.card_fin", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Fintype.card_setUniv", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Set.card_image_of_injective", "module": "Mathlib.Data.Set.Finite.Basic"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "left_eq_ite_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "right_eq_ite_iff", "module": "Init.PropLemmas"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "lt_add_of_pos_right_of_le", "content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"}, {"name": "eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "getBit_of_lt_two_pow", "content": "lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :\n  getBit k a = if k < n then getBit k a else 0"}, {"name": "getBit_eq_testBit", "content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"}, {"name": "getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "getBit_eq_zero_or_one", "content": "lemma getBit_eq_zero_or_one {k n : Nat} :\n  getBit k n = 0 ∨ getBit k n = 1"}], "used_local_defs": [{"name": "Binius.BinaryBasefold.fiber_coeff", "content": "noncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber", "content": "noncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R", "content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n    : i.val + steps < ℓ + 𝓡"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff", "content": "lemma qMap_total_fiber_repr_coeff (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩))\n    (k : Fin (2 ^ steps)) :\n    let x := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y) k\n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)\n      (h_i := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps)\n    let y_coeffs := basis_y.repr y\n    ∀ j, -- j refers to bit index of the fiber point x\n      ((sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (by simp only; omega)).repr x) j\n      = fiber_coeff (i := i) (steps := steps) (j := j) (elementIdx := k)\n        (y_coeffs := y_coeffs)"}], "local_ctx": "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch\n\nimport ArkLib.Data.CodingTheory.ReedSolomon\n\nimport ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT\n\nimport ArkLib.Data.MvPolynomial.Multilinear\n\nimport ArkLib.Data.Vector.Basic\n\nimport ArkLib.ProofSystem.Sumcheck.Spec.SingleRound\n\nnamespace Binius.BinaryBasefold\n\nopen OracleSpec OracleComp ProtocolSpec Finset AdditiveNTT Polynomial MvPolynomial\n  Binius.BinaryBasefold\n\nopen scoped NNReal\n\nopen ReedSolomon Code BerlekampWelch\n\nopen Finset AdditiveNTT Polynomial MvPolynomial Nat Matrix\n\nsection Preliminaries\n\nvariable {L : Type} [CommRing L] (ℓ : ℕ) [NeZero ℓ]\n\nvariable (𝓑 : Fin 2 ↪ L)\n\nend Preliminaries\n\nnoncomputable section       -- expands with 𝔽q in front\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type} [Field L] [Fintype L] [DecidableEq L] [CharP L 2]\n\nvariable (𝔽q : Type) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ 𝓡 ϑ : ℕ} (γ_repetitions : ℕ) [NeZero ℓ] [NeZero 𝓡] [NeZero ϑ] -- Should we allow ℓ = 0?\n\nvariable {h_ℓ_add_R_rate : ℓ + 𝓡 < r} -- ℓ ∈ {1, ..., r-1}\n\nvariable {𝓑 : Fin 2 ↪ L}\n\nsection Essentials\n\nnoncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩\n\nnoncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/", "target_theorem": "omit [CharP L 2] [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 [NeZero ℓ] in\ntheorem card_qMap_total_fiber (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)) :\n    Fintype.card (Set.image (qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps)\n      (y := y)) Set.univ) = 2 ^ steps :=", "ground_truth_proof": ":= by\n  -- The cardinality of the image of a function equals the cardinality of its domain\n  -- if it is injective.\n  rw [Set.card_image_of_injective Set.univ]\n  -- The domain is `Fin (2 ^ steps)`, which has cardinality `2 ^ steps`.\n  · -- ⊢ Fintype.card ↑Set.univ = 2 ^ steps\n    simp only [Fintype.card_setUniv, Fintype.card_fin]\n  · -- prove that `qMap_total_fiber` is an injective function.\n    intro k₁ k₂ h_eq\n    -- Assume two indices `k₁` and `k₂` produce the same point `x`.\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ (by simp only; omega)\n    -- If the points are equal, their basis representations must be equal.\n    set fiberMap := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n      (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y)\n    have h_coeffs_eq : basis_x.repr (fiberMap k₁) = basis_x.repr (fiberMap k₂) := by\n      rw [h_eq]\n    -- The first `steps` coefficients are determined by the bits of `k₁` and `k₂`.\n    -- If the coefficients are equal, the bits must be equal.\n    have h_bits_eq : ∀ j : Fin steps,\n        Nat.getBit (k := j) (n := k₁.val) = Nat.getBit (k := j) (n := k₂.val) := by\n      intro j\n      have h_coeff_j_eq : basis_x.repr (fiberMap k₁) ⟨j, by simp only; omega⟩\n        = basis_x.repr (fiberMap k₂) ⟨j, by simp only; omega⟩ := by rw [h_coeffs_eq]\n      rw [qMap_total_fiber_repr_coeff 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n        (h_i_add_steps := h_i_add_steps) (y := y) (j := ⟨j, by simp only; omega⟩)]\n        at h_coeff_j_eq\n      rw [qMap_total_fiber_repr_coeff 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n        (h_i_add_steps := h_i_add_steps) (y := y) (k := k₂) (j := ⟨j, by simp only; omega⟩)]\n        at h_coeff_j_eq\n      simp only [fiber_coeff, Fin.is_lt, ↓reduceDIte] at h_coeff_j_eq\n      by_cases hbitj_k₁ : Nat.getBit (k := j) (n := k₁.val) = 0\n      · simp only [hbitj_k₁, ↓reduceIte, left_eq_ite_iff, zero_ne_one, imp_false,\n        Decidable.not_not] at ⊢ h_coeff_j_eq\n        simp only [h_coeff_j_eq]\n      · simp only [hbitj_k₁, ↓reduceIte, right_eq_ite_iff, one_ne_zero,\n        imp_false] at ⊢ h_coeff_j_eq\n        have b1 : Nat.getBit (k := j) (n := k₁.val) = 1 := by\n          have h := Nat.getBit_eq_zero_or_one (k := j) (n := k₁.val)\n          simp only [hbitj_k₁, false_or] at h\n          exact h\n        have b2 : Nat.getBit (k := j) (n := k₂.val) = 1 := by\n          have h := Nat.getBit_eq_zero_or_one (k := j) (n := k₂.val)\n          simp only [h_coeff_j_eq, false_or] at h\n          exact h\n        simp only [b1, b2]\n      -- Extract the j-th coefficient from h_coeffs_eq and show it implies the bits are equal.\n    -- If all the bits of two numbers are equal, the numbers themselves are equal.\n    apply Fin.eq_of_val_eq\n    -- ⊢ ∀ {n : ℕ} {i j : Fin n}, ↑i = ↑j → i = j\n    apply eq_iff_eq_all_getBits.mpr\n    intro k\n    by_cases h_k : k < steps\n    · simp only [h_bits_eq ⟨k, by omega⟩]\n    · -- The bits at positions ≥ steps must be deterministic\n      conv_lhs => rw [Nat.getBit_of_lt_two_pow]\n      conv_rhs => rw [Nat.getBit_of_lt_two_pow]\n      simp only [h_k, ↓reduceIte]", "nesting_depth": 5, "transitive_dep_count": 78, "subset_aristotle": false, "category": "Applied verif."}
{"id": 27, "thm_name": "Binius.BinaryBasefold.qMap_total_fiber_one_level_eq", "thm_stmt": "lemma qMap_total_fiber_one_level_eq (i : Fin ℓ) (h_i_add_1 : i.val + 1 ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i + 1, by omega⟩)) (k : Fin 2) :\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ (by simp only; omega)\n    let x : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := 1) (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y) k\n    let y_lifted : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ := sDomain.lift 𝔽q β h_ℓ_add_R_rate\n      (i := ⟨i, by omega⟩) (j := ⟨i.val + 1, by omega⟩)\n      (h_j := by apply Nat.lt_add_of_pos_right_of_le; omega)\n      (h_le := by apply Fin.mk_le_mk.mpr (by omega)) y\n    let free_coeff_term : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ :=\n      (Fin2ToF2 𝔽q k) • (basis_x ⟨0, by simp only; omega⟩)\n    x = free_coeff_term + y_lifted", "lean_root": "ArkLib", "rel_path": "ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean", "imports": ["import ArkLib.Data.MvPolynomial.Multilinear", "import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.CodingTheory.ReedSolomon", "import ArkLib.Data.Vector.Basic", "import ArkLib.ProofSystem.Sumcheck.Spec.SingleRound", "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Submodule.map", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Module.Basis", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Disjoint", "module": "Mathlib.Order.Disjoint"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "LinearEquiv", "module": "Mathlib.Algebra.Module.Equiv.Defs"}, {"name": "LinearEquiv.ofBijective", "module": "Mathlib.Algebra.Module.Submodule.Equiv"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap.codRestrict", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "LinearMap.ker", "module": "Mathlib.Algebra.Module.Submodule.Ker"}, {"name": "Module.Basis.span", "module": "Mathlib.LinearAlgebra.Basis.Basic"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.subtype", "module": "Mathlib.Algebra.Module.Submodule.LinearMap"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.equivFunOnFinite", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Fin.isValue", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "Finsupp.single", "module": "Mathlib.Data.Finsupp.Single"}, {"name": "Function.update", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Nat.reducePow", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Pi.single", "module": "Mathlib.Algebra.Notation.Pi.Basic"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "NeZero", "module": "Init.Data.NeZero"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "sDomain", "content": "noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=\n  let W_i_norm := normalizedW 𝔽q β i\n  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=\n    AdditiveNTT.normalizedW_is_additive 𝔽q β i\n  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)\n    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "polyEvalLinearMap", "content": "noncomputable def polyEvalLinearMap {L 𝔽q : Type*} [Field L] [Field 𝔽q] [Algebra 𝔽q L]\n  (p : L[X]) (hp_add : IsLinearMap 𝔽q (fun x : L => p.eval x)) : L →ₗ[𝔽q] L :=\n{\n  toFun    := fun x => p.eval x,\n  map_add' := hp_add.map_add,\n  map_smul' := hp_add.map_smul\n}"}, {"name": "sDomain_basis", "content": "noncomputable def sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :\n    Basis (Fin (ℓ + R_rate - i)) 𝔽q (\n      sDomain 𝔽q β h_ℓ_add_R_rate i) :="}, {"name": "sBasis", "content": "def sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=\n  fun k => β ⟨i + k.val, by admit /- proof elided -/\n  ⟩"}, {"name": "sDomain.lift", "content": "noncomputable def sDomain.lift (i j : Fin r) (h_j : j < ℓ + R_rate) (h_le : i ≤ j)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate j) :\n    sDomain 𝔽q β h_ℓ_add_R_rate i :="}], "lib_lemmas": [{"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Module.Basis.repr_linearCombination", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Module.Basis.repr_symm_apply", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Nat.add_zero", "module": "Init.Core"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}, {"name": "tsub_zero", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Fin.mk_le_mk", "module": "Init.Data.Fin.Lemmas"}, {"name": "Decidable.not_not", "module": "Init.PropLemmas"}, {"name": "Fin.coe_ofNat_eq_mod", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.val_ne_of_ne", "module": "Init.Data.Fin.Basic"}, {"name": "Finsupp.coe_add", "module": "Mathlib.Algebra.Group.Finsupp"}, {"name": "Finsupp.coe_mk", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.single_eq_same", "module": "Mathlib.Data.Finsupp.Single"}, {"name": "Finsupp.smul_single", "module": "Mathlib.Data.Finsupp.SMulWithZero"}, {"name": "Module.Basis.repr_self", "module": "Mathlib.LinearAlgebra.Basis.Defs"}, {"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.lt_one_iff", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mod_succ", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Pi.add_apply", "module": "Mathlib.Algebra.Notation.Pi.Defs"}, {"name": "Pi.zero_apply", "module": "Mathlib.Algebra.Notation.Pi.Defs"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_tsub_cancel_left", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "left_eq_add", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "map_add", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_smul", "module": "Mathlib.GroupTheory.GroupAction.Hom"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "right_eq_dite_iff", "module": "Init.PropLemmas"}, {"name": "smul_eq_mul", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_lt_one", "module": "Mathlib.Algebra.Order.ZeroLEOne"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "lt_add_of_pos_right_of_le", "content": "@[simp]\nlemma lt_add_of_pos_right_of_le (a b c : ℕ) [NeZero c] (h : a ≤ b) : a < b + c"}, {"name": "basis_repr_of_sDomain_lift", "content": "theorem basis_repr_of_sDomain_lift (i j : Fin r) (h_j : j < ℓ + R_rate) (h_le : i ≤ j)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := j)) :\n    let x₀"}], "used_local_defs": [{"name": "Binius.BinaryBasefold.Fin2ToF2", "content": "def Fin2ToF2 (𝔽q : Type*) [Ring 𝔽q] (k : Fin 2) : 𝔽q :=\n  if k = 0 then 0 else 1"}, {"name": "Binius.BinaryBasefold.fiber_coeff", "content": "noncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber", "content": "noncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "Binius.BinaryBasefold.fin_ℓ_steps_lt_ℓ_add_R", "content": "omit [NeZero ℓ] in\nlemma fin_ℓ_steps_lt_ℓ_add_R (i : Fin ℓ) (steps : ℕ) (h : i.val + steps ≤ ℓ)\n    : i.val + steps < ℓ + 𝓡"}, {"name": "Binius.BinaryBasefold.qMap_total_fiber_repr_coeff", "content": "lemma qMap_total_fiber_repr_coeff (i : Fin ℓ) (steps : ℕ) (h_i_add_steps : i.val + steps ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩))\n    (k : Fin (2 ^ steps)) :\n    let x := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := steps)\n      (h_i_add_steps := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps) (y := y) k\n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by omega⟩)\n      (h_i := by simp only; exact fin_ℓ_steps_lt_ℓ_add_R i steps h_i_add_steps)\n    let y_coeffs := basis_y.repr y\n    ∀ j, -- j refers to bit index of the fiber point x\n      ((sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) (by simp only; omega)).repr x) j\n      = fiber_coeff (i := i) (steps := steps) (j := j) (elementIdx := k)\n        (y_coeffs := y_coeffs)"}], "local_ctx": "import ArkLib.Data.CodingTheory.BerlekampWelch.BerlekampWelch\n\nimport ArkLib.Data.CodingTheory.ReedSolomon\n\nimport ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT\n\nimport ArkLib.Data.MvPolynomial.Multilinear\n\nimport ArkLib.Data.Vector.Basic\n\nimport ArkLib.ProofSystem.Sumcheck.Spec.SingleRound\n\nnamespace Binius.BinaryBasefold\n\nopen OracleSpec OracleComp ProtocolSpec Finset AdditiveNTT Polynomial MvPolynomial\n  Binius.BinaryBasefold\n\nopen scoped NNReal\n\nopen ReedSolomon Code BerlekampWelch\n\nopen Finset AdditiveNTT Polynomial MvPolynomial Nat Matrix\n\nsection Preliminaries\n\nvariable {L : Type} [CommRing L] (ℓ : ℕ) [NeZero ℓ]\n\nvariable (𝓑 : Fin 2 ↪ L)\n\nend Preliminaries\n\nnoncomputable section       -- expands with 𝔽q in front\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type} [Field L] [Fintype L] [DecidableEq L] [CharP L 2]\n\nvariable (𝔽q : Type) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ 𝓡 ϑ : ℕ} (γ_repetitions : ℕ) [NeZero ℓ] [NeZero 𝓡] [NeZero ϑ] -- Should we allow ℓ = 0?\n\nvariable {h_ℓ_add_R_rate : ℓ + 𝓡 < r} -- ℓ ∈ {1, ..., r-1}\n\nvariable {𝓑 : Fin 2 ↪ L}\n\nsection Essentials\n\ndef Fin2ToF2 (𝔽q : Type*) [Ring 𝔽q] (k : Fin 2) : 𝔽q :=\n  if k = 0 then 0 else 1\n\nnoncomputable def fiber_coeff\n    (i : Fin r) (steps : ℕ)\n    (j : Fin (ℓ + 𝓡 - i)) (elementIdx : Fin (2 ^ steps))\n    (y_coeffs : Fin (ℓ + 𝓡 - (i + steps)) →₀ 𝔽q) : 𝔽q :=\n  if hj : j.val < steps then\n    if Nat.getBit (k := j) (n := elementIdx) = 0 then 0 else 1\n  else y_coeffs ⟨j.val - steps, by admit /- proof elided -/\n    ⟩\n\nnoncomputable def qMap_total_fiber\n    \n      (i : Fin r) (steps : ℕ) (h_i_add_steps : i.val + steps < ℓ + 𝓡)\n        (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i.val + steps, by admit /- proof elided -/\n        ⟩)) :\n    Fin (2 ^ steps) → sDomain 𝔽q β h_ℓ_add_R_rate i :=\n  if h_steps : steps = 0 then by\n    \n    subst h_steps\n    simp only [add_zero, Fin.eta] at y\n    exact fun _ => y\n  else by\n    \n    let basis_y := sDomain_basis 𝔽q β h_ℓ_add_R_rate (i := ⟨i+steps,by admit /- proof elided -/\n    ⟩) (by admit /- proof elided -/\n    )\n    let y_coeffs : Fin (ℓ + 𝓡 - (↑i + steps)) →₀ 𝔽q := basis_y.repr y\n\n    let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by admit /- proof elided -/\n    ⟩ (by admit /- proof elided -/\n    )\n    exact fun elementIdx => by admit /- proof elided -/", "target_theorem": "lemma qMap_total_fiber_one_level_eq (i : Fin ℓ) (h_i_add_1 : i.val + 1 ≤ ℓ)\n    (y : sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i + 1, by omega⟩)) (k : Fin 2) :\n    let basis_x :=", "ground_truth_proof": ":= sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ (by simp only; omega)\n    let x : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩)\n      (steps := 1) (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y) k\n    let y_lifted : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ := sDomain.lift 𝔽q β h_ℓ_add_R_rate\n      (i := ⟨i, by omega⟩) (j := ⟨i.val + 1, by omega⟩)\n      (h_j := by apply Nat.lt_add_of_pos_right_of_le; omega)\n      (h_le := by apply Fin.mk_le_mk.mpr (by omega)) y\n    let free_coeff_term : sDomain 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ :=\n      (Fin2ToF2 𝔽q k) • (basis_x ⟨0, by simp only; omega⟩)\n    x = free_coeff_term + y_lifted\n    := by\n  let basis_x := sDomain_basis 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ (by simp only; omega)\n  apply basis_x.repr.injective\n  simp only [map_add, map_smul]\n  simp only [Module.Basis.repr_self, Finsupp.smul_single, smul_eq_mul, mul_one, basis_x]\n  ext j\n  have h_repr_x := qMap_total_fiber_repr_coeff 𝔽q β i (steps := 1) (by omega)\n    (y := y) (k := k) (j := j)\n  simp only [h_repr_x, Finsupp.coe_add, Pi.add_apply]\n  simp only [fiber_coeff, lt_one_iff, reducePow, Fin2ToF2, Fin.isValue]\n\n  by_cases hj : j = ⟨0, by omega⟩\n  · simp only [hj, ↓reduceDIte, Fin.isValue, Finsupp.single_eq_same]\n    by_cases hk : k = 0\n    · simp only [getBit, hk, Fin.isValue, Fin.coe_ofNat_eq_mod, zero_mod, shiftRight_zero,\n      and_one_is_mod, ↓reduceIte, zero_add]\n      -- => Now use basis_repr_of_sDomain_lift\n      simp only [basis_repr_of_sDomain_lift, add_tsub_cancel_left, zero_lt_one, ↓reduceDIte]\n    · have h_k_eq_1 : k = 1 := by omega\n      simp only [getBit, h_k_eq_1, Fin.isValue, Fin.coe_ofNat_eq_mod, mod_succ, shiftRight_zero,\n        Nat.and_self, one_ne_zero, ↓reduceIte, left_eq_add]\n      simp only [basis_repr_of_sDomain_lift, add_tsub_cancel_left, zero_lt_one, ↓reduceDIte]\n  · have hj_ne_zero : j ≠ ⟨0, by omega⟩ := by omega\n    have hj_val_ne_zero : j.val ≠ 0 := by\n      change j.val ≠ ((⟨0, by omega⟩ :  Fin (ℓ + 𝓡 - ↑i)).val)\n      apply Fin.val_ne_of_ne\n      exact hj_ne_zero\n    simp only [hj_val_ne_zero, ↓reduceDIte, Finsupp.single, Fin.isValue, ite_eq_left_iff,\n      one_ne_zero, imp_false, Decidable.not_not, Pi.single, Finsupp.coe_mk, Function.update,\n      hj_ne_zero, Pi.zero_apply, zero_add]\n    simp only [basis_repr_of_sDomain_lift, add_tsub_cancel_left, lt_one_iff, right_eq_dite_iff]\n    intro hj_eq_zero\n    exact False.elim (hj_val_ne_zero hj_eq_zero)", "nesting_depth": 5, "transitive_dep_count": 95, "subset_aristotle": false, "category": "Applied verif."}
{"id": 28, "thm_name": "ReedSolomonCode.minDist", "thm_stmt": "theorem minDist [Field F] [DecidableEq F] (inj : Function.Injective α) [NeZero n] (h : n ≤ m) :\n  minDist ((ReedSolomon.code ⟨α, inj⟩ n) : Set (Fin m → F)) = m - n + 1", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/CodingTheory/ReedSolomon.lean", "imports": ["import ArkLib.Data.CodingTheory.Basic", "import Mathlib.LinearAlgebra.Lagrange", "import ArkLib.Data.MvPolynomial.LinearMvExtension", "import Mathlib.RingTheory.Henselian", "import ArkLib.Data.CodingTheory.Prelims", "import ArkLib.Data.Fin.Lift", "import ArkLib.Data.Polynomial.Interface"], "used_lib_defs": [{"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.degreeLT", "module": "Mathlib.RingTheory.Polynomial.Basic"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Matrix.add", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Multiset", "module": "Mathlib.Data.Multiset.Defs"}, {"name": "Multiset.count", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Set.InjOn", "module": "Mathlib.Data.Set.Operations"}, {"name": "Finset.max", "module": "Mathlib.Data.Finset.Max"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Module.finrank", "module": "Mathlib.LinearAlgebra.Dimension.Finrank"}, {"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Matrix.of", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Finset.min", "module": "Mathlib.Data.Finset.Max"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Matrix.submatrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Matrix.vandermonde", "module": "Mathlib.LinearAlgebra.Vandermonde"}, {"name": "IsDomain", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Matrix.det", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "Matrix.rank", "module": "Mathlib.LinearAlgebra.Matrix.Rank"}, {"name": "Matrix.cRank", "module": "Mathlib.LinearAlgebra.Matrix.Rank"}, {"name": "Fin.val", "module": "Init.Prelude"}, {"name": "Finset.map", "module": "Mathlib.Data.Finset.Image"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap.range", "module": "Mathlib.Algebra.Module.Submodule.Range"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.coeff", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "hammingDist", "module": "Mathlib.InformationTheory.Hamming"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "StrongRankCondition", "module": "Mathlib.LinearAlgebra.InvariantBasisNumber"}], "used_repo_defs": [{"name": "wt", "content": "def wt [Zero F]\n  (v : ι → F) : ℕ := #{i | v i ≠ 0}"}, {"name": "dim", "content": "noncomputable def dim [Semiring F] (LC : LinearCode ι F) : ℕ :=\n  Module.finrank F LC"}, {"name": "LinearCode.{u,", "content": "abbrev LinearCode.{u, v} (ι : Type u) [Fintype ι] (F : Type v) [Semiring F] : Type (max u v) :=\n  Submodule F (ι → F)"}, {"name": "subUpFull", "content": "def subUpFull (U : Matrix (Fin m) (Fin n) F) (r_reindex : Fin n → Fin m) :\n  Matrix (Fin n) (Fin n) F := Matrix.submatrix U r_reindex id"}, {"name": "refl", "content": "@[simps]\ndef refl (pSpec : ProtocolSpec n) : Equiv pSpec pSpec where\n  round_eq := rfl\n  dir_eq := fun _ => rfl\n  typeEquiv := fun _ => _root_.Equiv.refl _"}, {"name": "subLeftFull", "content": "def subLeftFull (U : Matrix (Fin m) (Fin n) F) (c_reindex : Fin m → Fin n) :\n  Matrix (Fin m) (Fin m) F := Matrix.submatrix U id c_reindex"}, {"name": "polynomialOfCoeffs", "content": "def polynomialOfCoeffs (coeffs : Fin deg → F) : F[X] :=\n  ⟨\n    Finset.map ⟨Fin.val, Fin.val_injective⟩ {i | coeffs i ≠ 0},\n    fun i ↦ if h : i < deg then coeffs ⟨i, h⟩ else 0,\n    fun a ↦ by admit /- proof elided -/\n  ⟩"}, {"name": "liftF'", "content": "def liftF' (f : ℕ → α) : Fin n → α :=\n  fun m ↦ f m.1"}, {"name": "fromColGenMat", "content": "noncomputable def fromColGenMat [CommRing F] (G : Matrix ι κ F) : LinearCode ι F :=\n  LinearMap.range G.mulVecLin"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "restrictLinear", "content": "noncomputable def restrictLinear [Semiring F] (S : Finset ι) :\n  (ι → F) →ₗ[F] (S → F) :=\n{ toFun := fun f i => f i.1,\n  map_add' := by admit /- proof elided -/"}, {"name": "minDist", "content": "noncomputable def minDist (C : Set (n → R)) : ℕ :=\n  sInf {d | ∃ u ∈ C, ∃ v ∈ C, u ≠ v ∧ hammingDist u v = d}"}, {"name": "minWtCodewords", "content": "noncomputable def minWtCodewords [Semiring F] (LC : LinearCode ι F) : ℕ :=\n  sInf {w | ∃ c ∈ LC, c ≠ 0 ∧ Code.wt c = w}"}, {"name": "length", "content": "def length [Semiring F] (_ : LinearCode ι F) : ℕ := Fintype.card ι"}, {"name": "minDist", "content": "notation \"Δ\" IC => minDist IC"}], "lib_lemmas": [{"name": "Finset.image_subset_iff", "module": "Mathlib.Data.Finset.Image"}, {"name": "Finset.sum_image", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_le_sum", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "Finset.sum_le_sum_of_subset_of_nonneg", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "Multiset.count_pos", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.toFinset_sum_count_eq", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_of_lt_of_le", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Polynomial.mem_degreeLT", "module": "Mathlib.RingTheory.Polynomial.Basic"}, {"name": "Polynomial.natDegree_lt_iff_degree_lt", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Fin.castLE_injective", "module": "Mathlib.Data.Fin.SuccPred"}, {"name": "Function.Injective.comp", "module": "Init.Data.Function"}, {"name": "Matrix.det_vandermonde_ne_zero_iff", "module": "Mathlib.LinearAlgebra.Vandermonde"}, {"name": "Matrix.mulVecLin_apply", "module": "Mathlib.LinearAlgebra.Matrix.ToLin"}, {"name": "Matrix.mulVec_eq_sum", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Finset.mem_range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_fin_eq_sum_range", "module": "Mathlib.Data.Fintype.BigOperators"}, {"name": "Polynomial.eval_eq_sum_range'", "module": "Mathlib.Algebra.Polynomial.Eval.Degree"}, {"name": "LinearMap.mem_range", "module": "Mathlib.Algebra.Module.Submodule.Range"}, {"name": "Submodule.mem_map", "module": "Mathlib.Algebra.Module.Submodule.Map"}, {"name": "Finset.filter_card_add_filter_neg_card_eq_card", "module": "Mathlib.Data.Finset.Card"}, {"name": "Multiset.count_eq_one_of_mem", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.count_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.nodup_iff_count_eq_one", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Polynomial.card_roots'", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "le_antisymm", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_csInf", "module": "Mathlib.Order.ConditionallyCompleteLattice.Basic"}, {"name": "lt_of_le_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "rank_eq_if_det_ne_zero", "content": "lemma rank_eq_if_det_ne_zero {U : Matrix (Fin n) (Fin n) F} [IsDomain F] :\n  Matrix.det U ≠ 0 → U.rank = n"}, {"name": "rank_eq_if_subUpFull_eq", "content": "lemma rank_eq_if_subUpFull_eq (h : n ≤ m) :\n   (subUpFull U (Fin.castLE h)).rank = n  → U.rank = n"}, {"name": "full_row_rank_via_rank_subLeftFull", "content": "lemma full_row_rank_via_rank_subLeftFull (h : m ≤ n) :\n   (subLeftFull U (Fin.castLE h)).rank = m → U.rank = m"}, {"name": "cRank_rank_conversion", "content": "lemma cRank_rank_conversion :\n  ↑(U.rank) = U.cRank"}, {"name": "liftF'_p_coeff", "content": "@[simp]\nlemma liftF'_p_coeff {p : F[X]} {k : ℕ} {i : Fin k} : liftF' p.coeff i = p.coeff i"}, {"name": "coeff_polynomialOfCoeffs_eq_coeffs", "content": "@[simp]\nlemma coeff_polynomialOfCoeffs_eq_coeffs :\n  Fin.liftF' (polynomialOfCoeffs coeffs).coeff = coeffs"}, {"name": "rank_eq_dim_fromColGenMat", "content": "lemma rank_eq_dim_fromColGenMat [CommRing F] {G : Matrix κ ι F} :\n  G.rank = dim (fromColGenMat G)"}, {"name": "singletonBound", "content": "theorem singletonBound [CommRing F] [StrongRankCondition F]\n  (LC : LinearCode ι F) :\n  dim LC ≤ length LC - Code.minDist (LC : Set (ι → F)) + 1"}, {"name": "dist_UB", "content": "lemma dist_UB [CommRing F] {LC : LinearCode ι F} :\n    Code.minDist (LC : Set (ι → F)) ≤ length LC"}, {"name": "dist_eq_minWtCodewords", "content": "lemma dist_eq_minWtCodewords [CommRing F] {LC : LinearCode ι F} :\n  Code.minDist (LC : Set (ι → F)) = minWtCodewords LC"}, {"name": "hammingDist_eq_wt_sub", "content": "lemma hammingDist_eq_wt_sub [CommRing F] {u v : ι → F} : hammingDist u v = Code.wt (u - v)"}], "used_local_defs": [{"name": "ReedSolomon.evalOnPoints", "content": "def evalOnPoints [Semiring F] : F[X] →ₗ[F] (ι → F) where\n  toFun := fun p => fun x => p.eval (domain x)\n  map_add' := fun x y => by admit /- proof elided -/"}, {"name": "ReedSolomon.code", "content": "def code (deg : ℕ) [Semiring F]: Submodule F (ι → F) :=\n  (Polynomial.degreeLT F deg).map (evalOnPoints domain)"}, {"name": "Vandermonde.nonsquare", "content": "def nonsquare [Semiring F] (ι' : ℕ) (α : ι → F) : Matrix ι (Fin ι') F :=\n  Matrix.of fun i j => (α i) ^ j.1"}, {"name": "ReedSolomonCode.RScodeSet", "content": "abbrev RScodeSet (domain : ι ↪ F) (deg : ℕ) : Set (ι → F) := (ReedSolomon.code domain deg).carrier"}, {"name": "ReedSolomonCode.toFinset", "content": "def toFinset (domain : ι ↪ F) (deg : ℕ) : Finset (ι → F) :=\n  (RScodeSet domain deg).toFinset"}, {"name": "ReedSolomonCode.constantCode", "content": "def constantCode {α : Type*} (x : α) (ι' : Type*) [Fintype ι'] : ι' → α := fun _ ↦ x"}], "used_local_lemmas": [{"name": "Vandermonde.nonsquare_mulVecLin", "content": "lemma nonsquare_mulVecLin [CommSemiring F] {ι' : ℕ} {α₁ : ι ↪ F} {α₂ : Fin ι' → F} {i : ι} :\n  (nonsquare ι' α₁).mulVecLin α₂ i = ∑ x, α₂ x * α₁ i ^ x.1"}, {"name": "Vandermonde.subUpFull_of_vandermonde_is_vandermonde", "content": "lemma subUpFull_of_vandermonde_is_vandermonde (h : n ≤ m) :\n  Matrix.vandermonde (α ∘ Fin.castLE h) =\n  Matrix.subUpFull (nonsquare n α) (Fin.castLE h)"}, {"name": "Vandermonde.subLeftFull_of_vandermonde_is_vandermonde", "content": "lemma subLeftFull_of_vandermonde_is_vandermonde (h : m ≤ n) :\n  Matrix.vandermonde α = Matrix.subLeftFull (nonsquare n α) (Fin.castLE h)"}, {"name": "Vandermonde.rank_nonsquare_eq_deg_of_deg_le", "content": "lemma rank_nonsquare_eq_deg_of_deg_le (inj : Function.Injective α) (h : n ≤ m) :\n  (Vandermonde.nonsquare (ι' := n) α).rank = n"}, {"name": "Vandermonde.rank_nonsquare_eq_deg_of_ι_le", "content": "lemma rank_nonsquare_eq_deg_of_ι_le (inj : Function.Injective α) (h : m ≤ n) :\n  (Vandermonde.nonsquare (ι' := n) α).rank = m"}, {"name": "Vandermonde.rank_nonsquare_rows_eq_min", "content": "@[simp]\nlemma rank_nonsquare_rows_eq_min (inj : Function.Injective α) :\n  (Vandermonde.nonsquare (ι' := n) α).rank = min m n"}, {"name": "Vandermonde.mulVecLin_coeff_vandermondens_eq_eval_matrixOfPolynomials", "content": "theorem mulVecLin_coeff_vandermondens_eq_eval_matrixOfPolynomials\n  {n : ℕ} [NeZero n] {v : ι ↪ F} {p : F[X]} (h_deg : p.natDegree < n) :\n  (Vandermonde.nonsquare (ι' := n) v).mulVecLin (Fin.liftF' p.coeff) =\n  fun i => p.eval (v i)"}, {"name": "ReedSolomonCode.natDegree_lt_of_mem_degreeLT", "content": "lemma natDegree_lt_of_mem_degreeLT [NeZero deg] (h : p ∈ degreeLT F deg) : p.natDegree < deg"}, {"name": "ReedSolomonCode.genMatIsVandermonde", "content": "lemma genMatIsVandermonde [Fintype ι] [Field F] [DecidableEq F] [inst : NeZero m] {α : ι ↪ F} :\n  fromColGenMat (Vandermonde.nonsquare (ι' := m) α) = ReedSolomon.code α m"}, {"name": "ReedSolomonCode.dim_eq_deg_of_le", "content": "lemma dim_eq_deg_of_le [NeZero n] (inj : Function.Injective α) (h : n ≤ m) :\n  dim (ReedSolomon.code ⟨α, inj⟩ n) = n"}, {"name": "ReedSolomonCode.dist_le_length", "content": "@[simp]\nlemma dist_le_length [DecidableEq F] (inj : Function.Injective α) :\n    minDist ((ReedSolomon.code ⟨α, inj⟩ n) : Set (Fin m → F)) ≤ m"}, {"name": "ReedSolomonCode.card_le_card_of_count_inj", "content": "lemma card_le_card_of_count_inj {α β : Type*} [DecidableEq α] [DecidableEq β]\n    {s : Multiset α} {s' : Multiset β}\n  {f : α → β} (inj : Function.Injective f) (h : ∀ a : α, s.count a ≤ s'.count (f a)) :\n  s.card ≤ s'.card"}], "local_ctx": "import ArkLib.Data.MvPolynomial.LinearMvExtension\n\nimport ArkLib.Data.Polynomial.Interface\n\nimport Mathlib.LinearAlgebra.Lagrange\n\nimport Mathlib.RingTheory.Henselian\n\nnamespace ReedSolomon\n\nopen Polynomial NNReal\n\nvariable {F : Type*} {ι : Type*} (domain : ι ↪ F)\n\ndef evalOnPoints [Semiring F] : F[X] →ₗ[F] (ι → F) where\n  toFun := fun p => fun x => p.eval (domain x)\n  map_add' := fun x y => by admit /- proof elided -/\n\ndef code (deg : ℕ) [Semiring F]: Submodule F (ι → F) :=\n  (Polynomial.degreeLT F deg).map (evalOnPoints domain)\n\nvariable [Semiring F]\n\nend ReedSolomon\n\nopen Polynomial Matrix Code LinearCode\n\nvariable {F ι ι' : Type*}\n         {C : Set (ι → F)}\n\nnoncomputable section\n\nnamespace Vandermonde\n\ndef nonsquare [Semiring F] (ι' : ℕ) (α : ι → F) : Matrix ι (Fin ι') F :=\n  Matrix.of fun i j => (α i) ^ j.1\n\nsection\n\nvariable [CommRing F] {m n : ℕ} {α : Fin m → F}\n\nsection\n\nvariable [IsDomain F]\n\nend\n\nend\n\nend Vandermonde\n\nnamespace ReedSolomonCode\n\nsection\n\nopen Finset Function\n\nopen scoped BigOperators\n\nvariable {ι : Type*} [Fintype ι] [Nonempty ι]\n         {F : Type*} [Field F] [Fintype F]\n\nabbrev RScodeSet (domain : ι ↪ F) (deg : ℕ) : Set (ι → F) := (ReedSolomon.code domain deg).carrier\n\nopen Classical in\n\ndef toFinset (domain : ι ↪ F) (deg : ℕ) : Finset (ι → F) :=\n  (RScodeSet domain deg).toFinset\n\nend\n\nsection\n\nvariable {deg m n : ℕ} {α : Fin m → F}\n\nsection\n\nvariable [Semiring F] {p : F[X]}\n\nend\n\nopen LinearCode\n\nsection\n\nopen NNReal\n\nvariable [Field F]\n\nend\n\nsection\n\ndef constantCode {α : Type*} (x : α) (ι' : Type*) [Fintype ι'] : ι' → α := fun _ ↦ x\n\nvariable [Semiring F] {x : F} [Fintype ι] {α : ι ↪ F}\n\nend\n\nopen Finset in", "target_theorem": "theorem minDist [Field F] [DecidableEq F] (inj : Function.Injective α) [NeZero n] (h : n ≤ m) :\n  minDist ((ReedSolomon.code ⟨α, inj⟩ n) : Set (Fin m → F)) = m - n + 1 :=", "ground_truth_proof": ":= by\n  have : NeZero m := by constructor; aesop\n  refine le_antisymm ?p₁ ?p₂\n  case p₁ =>\n    have distUB := singletonBound (LC := ReedSolomon.code ⟨α, inj⟩ n)\n    rw [dim_eq_deg_of_le inj h] at distUB\n    simp at distUB\n    zify [dist_le_length] at distUB\n    omega\n  case p₂ =>\n    rw [dist_eq_minWtCodewords]\n    apply le_csInf (by use m, constantCode 1 _; simp)\n    intro b ⟨msg, ⟨p, p_deg, p_eval_on_α_eq_msg⟩, msg_neq_0, wt_c_eq_b⟩\n    let zeroes : Finset _ := {i | msg i = 0}\n    have eq₁ : zeroes.val.Nodup := by\n      aesop (add simp [Multiset.nodup_iff_count_eq_one, Multiset.count_filter])\n    have msg_zeros_lt_deg : #zeroes < n := by\n      apply lt_of_le_of_lt (b := p.roots.card)\n                           (hbc := lt_of_le_of_lt (Polynomial.card_roots' _)\n                                                  (natDegree_lt_of_mem_degreeLT p_deg))\n      exact card_le_card_of_count_inj inj fun i ↦\n        if h : msg i = 0\n        then suffices 0 < Multiset.count (α i) p.roots by\n                rwa [@Multiset.count_eq_one_of_mem (d := eq₁) (h := by simpa [zeroes])]\n              by aesop\n        else by simp [zeroes, h]\n    have : #zeroes + wt msg = m := by\n      rw [wt, filter_card_add_filter_neg_card_eq_card]\n      simp\n    omega", "nesting_depth": 8, "transitive_dep_count": 118, "subset_aristotle": false, "category": "Applied verif."}
{"id": 29, "thm_name": "Vector.foldl_succ", "thm_stmt": "theorem foldl_succ\n {α β} {n : ℕ} [NeZero n] (f : β → α → β) (init : β) (v : Vector α n) :\n  v.foldl (f:=f) (b:=init) = v.tail.foldl (f:=f) (b:=f init v.head)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/Vector/Basic.lean", "imports": ["import Mathlib.Data.Matrix.Mul", "import Mathlib.Algebra.Order.Sub.Basic", "import Mathlib.Algebra.Order.Star.Basic", "import Mathlib.Algebra.BigOperators.Fin", "import ToMathlib.General"], "used_lib_defs": [{"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "Array.foldl", "module": "Init.Data.Array.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.cons", "module": "Init.Prelude"}, {"name": "List.take", "module": "Init.Data.List.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Vector.foldl", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.head", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.tail", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Array.foldl_toList", "module": "Init.Data.Array.Bootstrap"}, {"name": "Array.toList_extract", "module": "Init.Data.Array.Lemmas"}, {"name": "List.drop_one", "module": "Init.Data.List.TakeDrop"}, {"name": "List.extract_eq_drop_take", "module": "Init.Data.List.Basic"}, {"name": "List.foldl_cons", "module": "Init.Data.List.Basic"}, {"name": "List.getElem_cons_zero", "module": "Init.GetElem"}, {"name": "List.length_nil", "module": "Init.Data.List.Basic"}, {"name": "List.length_tail", "module": "Init.Data.List.Lemmas"}, {"name": "List.tail_cons", "module": "Init.Data.List.Basic"}, {"name": "List.take_eq_self_iff", "module": "Mathlib.Data.List.TakeDrop"}, {"name": "Nat.pos_of_neZero", "module": "Init.Data.Nat.Basic"}, {"name": "Vector.getElem_toList", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.length_toList", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.size_toArray", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.toArray_cast", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.toArray_extract", "module": "Init.Data.Vector.Lemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "le_refl", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Mathlib.Algebra.BigOperators.Fin\n\nimport Mathlib.Algebra.Order.Star.Basic\n\nimport Mathlib.Algebra.Order.Sub.Basic\n\nimport Mathlib.Data.Matrix.Mul\n\nimport ToMathlib.General\n\nnamespace Vector", "target_theorem": "theorem foldl_succ\n {α β} {n : ℕ} [NeZero n] (f : β → α → β) (init : β) (v : Vector α n) :\n  v.foldl (f:=f) (b:=init) = v.tail.foldl (f:=f) (b:=f init v.head) :=", "ground_truth_proof": ":= by\n  simp_rw [Vector.foldl] -- get\n  simp only [size_toArray]\n  have hl_foldl_eq_toList_foldl := Array.foldl_toList (f:=f) (init:=init) (xs:=v.toArray)\n  have hl_foldl_eq: Array.foldl f init v.toArray 0 n = Array.foldl f init v.toArray := by\n    simp only [size_toArray]\n  conv_lhs =>\n    rw [hl_foldl_eq, hl_foldl_eq_toList_foldl.symm]\n  have hr_foldl_eq_toList_foldl_tail := Array.foldl_toList (f:=f) (init:=f init v.head)\n    (xs:=(v.tail.toArray))\n  have hr_foldl_eq: Array.foldl f (f init v.head) v.tail.toArray 0 (n - 1)\n    = Array.foldl f (f init v.head) v.tail.toArray := by\n    simp only [size_toArray] -- Array.foldl_congr\n  conv_rhs =>\n    rw [hr_foldl_eq, hr_foldl_eq_toList_foldl_tail.symm]\n  rw [Vector.head]\n  have h_v_toList_length: 0 < v.toList.length := by\n    simp only [length_toList]\n    exact Nat.pos_of_neZero n\n  rw [←Vector.getElem_toList (h:=h_v_toList_length)]\n  have h_toList_eq: v.toArray.toList = v.toList := rfl\n  rw [Vector.tail]\n  simp only [toArray_cast, toArray_extract, Array.toList_extract, List.extract_eq_drop_take,\n    List.drop_one]\n  simp_rw [h_toList_eq]\n  -- ⊢ List.foldl f init v.toList\n  -- = List.foldl f (f init v.toList[0]) (List.take (n - 1) v.toList.tail)\n  have hTakeTail: List.take (n - 1) v.toList.tail = v.toList.tail := by\n    simp only [List.take_eq_self_iff, List.length_tail, length_toList, le_refl]\n  rw [hTakeTail]\n  have h_v_eq_cons: v.toList = v.head :: (v.toList.tail) := by\n    cases h_list : v.toList with\n    | nil =>\n      have h_len : v.toList.length = 0 := by rw [h_list, List.length_nil]\n      omega\n    | cons hd tl =>\n      have h_v_head: v.head = v.toList[0] := rfl\n      simp_rw [h_v_head]\n      have h_hd: hd = v.toList[0] := by simp only [h_list, List.getElem_cons_zero]\n      simp only [List.tail_cons, List.cons.injEq, and_true]\n      simp_rw [h_hd]\n  conv_lhs => rw [h_v_eq_cons]\n  rw [List.foldl_cons]\n  rfl", "nesting_depth": 1, "transitive_dep_count": 29, "subset_aristotle": true, "category": "Applied verif."}
{"id": 30, "thm_name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "thm_stmt": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}], "lib_lemmas": [{"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=", "target_theorem": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1))) :=", "ground_truth_proof": ":= by\n  -- Idea : derive from theorem join_eq_iff_dcast_extractLsb\n  constructor\n  · -- Forward direction\n    intro h_join\n    have h := join_eq_iff_dcast_extractLsb h_pos x hi_btf lo_btf\n    have ⟨h_hi, h_lo⟩ := h.mp h_join\n    have hi_eq : hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) := by\n      rw [h_hi]\n      have := BitVec.extractLsb_eq_shift_ofNat (n:=2 ^ k) (r:=2 ^ k - 1) (l:=2 ^ (k - 1)) (x:=x)\n      rw [this]\n      unfold fromNat\n      rw [BitVec.dcast_bitvec_eq]\n    have lo_eq : lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)) := by\n      rw [h_lo]\n      have := BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat (num_bits:=2 ^ (k - 1)) (n:=2 ^ k)\n                (Nat.two_pow_pos (k - 1)) (x:=x)\n      rw [this]\n      unfold fromNat\n      rw [BitVec.dcast_bitvec_eq]\n    exact ⟨hi_eq, lo_eq⟩\n  · -- Backward direction\n    intro h_bits\n    have ⟨h_hi, h_lo⟩ := h_bits\n    have h := join_eq_iff_dcast_extractLsb h_pos x hi_btf lo_btf\n    have hi_eq : hi_btf = dcast (h_sub_middle h_pos)\n      (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) := by\n      rw [h_hi]\n      unfold fromNat\n      have := BitVec.extractLsb_eq_shift_ofNat (n:=2 ^ k) (r:=2 ^ k - 1) (l:=2 ^ (k - 1)) (x:=x)\n      rw [this]\n      rw [BitVec.dcast_bitvec_eq]\n    have lo_eq : lo_btf = dcast (h_middle_sub)\n      (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x) := by\n      rw [h_lo]\n      unfold fromNat\n      have := BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat (num_bits:=2 ^ (k - 1)) (n:=2 ^ k)\n                (Nat.two_pow_pos (k - 1)) (x:=x)\n      rw [this]\n      rw [BitVec.dcast_bitvec_eq]\n    exact h.mpr ⟨hi_eq, lo_eq⟩", "nesting_depth": 6, "transitive_dep_count": 94, "subset_aristotle": false, "category": "Applied verif."}
{"id": 31, "thm_name": "ConcreteBinaryTower.split_one", "thm_stmt": "lemma split_one {k : ℕ} (h_k : k > 0) :\n    split h_k (one (k:=k)) = (zero (k:=k - 1), one (k:=k - 1))", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}], "lib_lemmas": [{"name": "Nat.ne_zero_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.one_lt_two_pow_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_eq_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.one_mod_two_pow_eq_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.zero_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.one_mod_two_pow", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.zero_lt_two", "module": "Init.Data.Nat.Basic"}, {"name": "pow_pos", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}], "repo_lemmas": [{"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "zero_lt_pow_n", "content": "theorem zero_lt_pow_n (m : ℕ) (n : ℕ) (h_m : m > 0): 0 < m^n"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq_zero", "content": "theorem BitVec.dcast_bitvec_eq_zero {l r : ℕ} (h_width_eq : l = r) :\n  dcast (h_width_eq) 0#(l) = 0#(r)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.one_bitvec_toNat", "content": "lemma one_bitvec_toNat {width : ℕ} (h_width : width > 0) : (1#width).toNat = 1"}, {"name": "ConcreteBinaryTower.one_bitvec_shiftRight", "content": "lemma one_bitvec_shiftRight {d : ℕ} (h_d : d > 0) : 1 >>> d = 0"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/", "target_theorem": "lemma split_one {k : ℕ} (h_k : k > 0) :\n    split h_k (one (k:=k)) = (zero (k:=k - 1), one (k:=k - 1)) :=", "ground_truth_proof": ":= by\n  rw [split]\n  let lo_bits := BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) (one (k:=k))\n  let hi_bits := BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) (one (k:=k))\n  apply Prod.ext\n  · simp only\n    simp only [BitVec.extractLsb, BitVec.extractLsb']\n    rw [one]\n    have one_toNat_eq := one_bitvec_toNat (width:=2 ^ k)\n      (h_width:=zero_lt_pow_n (m:=2) (n:=k) (h_m:=Nat.zero_lt_two))\n    rw [one_toNat_eq]\n    have one_shiftRight_eq : 1 >>> 2 ^ (k - 1) = 0 :=\n      one_bitvec_shiftRight (d:=2 ^ (k - 1)) (h_d:=by exact Nat.two_pow_pos (k - 1))\n    rw [one_shiftRight_eq]\n    rw [zero, BitVec.zero_eq]\n    have h_sub_middle := sub_middle_of_pow2_with_one_canceled (k:=k) (h_k:=h_k)\n    rw [BitVec.dcast_bitvec_eq_zero]\n  · simp only\n    simp only [BitVec.extractLsb, BitVec.extractLsb']\n    simp only [Nat.sub_zero, one, BitVec.toNat_ofNat, Nat.ofNat_pos, pow_pos, Nat.one_mod_two_pow,\n      Nat.shiftRight_zero] -- converts BitVec.toNat one >>> 0 into 1#(2 ^ (k - 1))\n    rw [BitVec.dcast_bitvec_eq]", "nesting_depth": 4, "transitive_dep_count": 43, "subset_aristotle": false, "category": "Applied verif."}
{"id": 32, "thm_name": "AdditiveNTT.W_prod_comp_decomposition", "thm_stmt": "lemma W_prod_comp_decomposition\n    (i : Fin r) (hi : i > 0) :\n    (W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean", "imports": ["import Mathlib.Algebra.Polynomial.Degree.Definitions", "import ArkLib.Data.Fin.BigOperators", "import ArkLib.Data.Polynomial.MonomialBasis", "import Mathlib.LinearAlgebra.StdBasis", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.Polynomial.Frobenius"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Polynomial.rootMultiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Multiset", "module": "Mathlib.Data.Multiset.Defs"}, {"name": "Multiset.count", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.map", "module": "Mathlib.Data.Multiset.MapFold"}, {"name": "Polynomial.roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "SetLike", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Iff", "module": "Init.Core"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.algEquivOfCompEqX", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "multiplicity", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "EmbeddingLike", "module": "Mathlib.Data.FunLike.Embedding"}, {"name": "CanLift", "module": "Mathlib.Tactic.Lift"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Multiset.filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Finset.val", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Set.InjOn", "module": "Mathlib.Data.Set.Operations"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.val", "module": "Init.Prelude"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}], "lib_lemmas": [{"name": "Polynomial.splits_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.splits_prod", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Finset.prod_eq_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Nat.not_lt_zero", "module": "Init.Prelude"}, {"name": "Polynomial.X_sub_C_ne_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Set.Ico_eq_empty_iff", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Polynomial.X_ne_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_C_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_X_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_sub", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.comp_eq_zero_iff", "module": "Mathlib.Algebra.Polynomial.Degree.Lemmas"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "map_neg", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "sub_eq_neg_self", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "EmbeddingLike.map_eq_zero_iff", "module": "Mathlib.Algebra.Group.Equiv.Defs"}, {"name": "Polynomial.aeval_C", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.aeval_X", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algEquivOfCompEqX_apply", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algebraMap_eq", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.rootMultiplicity_eq_multiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "if_false", "module": "Init.ByCases"}, {"name": "if_true", "module": "Init.ByCases"}, {"name": "map_sub", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "multiplicity_map_eq", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "sub_sub_sub_cancel_right", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Multiset.countP_eq_card_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_map", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.filter_congr", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.count_roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "eq_sub_iff_add_eq", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.image_val_of_injOn", "module": "Mathlib.Data.Finset.Image"}, {"name": "Finset.prod_image", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Polynomial.roots_prod_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Subtype.val_injective", "module": "Mathlib.Data.Subtype"}, {"name": "CanLift.prf", "module": "Mathlib.Tactic.Lift"}, {"name": "Multiset.card_singleton", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.card_zero", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.count_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_singleton", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.count_univ", "module": "Mathlib.Data.Fintype.Basic"}, {"name": "Multiset.count_zero", "module": "Mathlib.Data.Multiset.Count"}, {"name": "SetLike.coe_eq_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "SetLike.mem_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Fin.zero_le", "module": "Init.Data.Fin.Lemmas"}, {"name": "Set.Ico_subset_Ico_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_mono", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.mem_image_of_mem", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.add_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.smul_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span_mono", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.subset_span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "sub_add_cancel", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "Fact.out", "module": "Mathlib.Logic.Basic"}, {"name": "Fin.le_zero_iff'", "module": "Mathlib.Data.Fin.SuccPred"}, {"name": "LinearIndependent.injective", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.compl_eq_univ_diff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Set.empty_subset", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.image_empty", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_subset_image_iff", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.subset_compl_singleton_iff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "linearIndependent_iff_notMem_span", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.Ico_insert_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_singleton", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_union", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Icc", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.union_singleton", "module": "Mathlib.Data.Set.Insert"}, {"name": "Submodule.mem_span_singleton", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.mem_sup", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.smul_mem_iff", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Submodule.span_union", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.sub_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "existsUnique_of_exists_of_unique", "module": "Mathlib.Logic.ExistsUnique"}, {"name": "sub_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_smul", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "sub_sub_sub_cancel_left", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_const_zero", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.sum_ite_eq'", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise"}, {"name": "Finset.sum_map_val", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Multiset.count_bind", "module": "Mathlib.Data.Multiset.Bind"}, {"name": "Multiset.count_map_eq_count'", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.roots_prod", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "add_left_injective", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "if_false_right", "module": "Init.PropLemmas"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "iff_false", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "Polynomial.monic_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.monic_prod_of_monic", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.Monic.comp", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.Splits.comp_of_degree_le_one", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.degree_X_sub_C_le", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eq_prod_roots_of_monic_of_splits_id", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.natDegree_X", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.natDegree_sub_C", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}], "repo_lemmas": [{"name": "Fin.lt_succ'", "content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}, {"name": "Fin.le_succ", "content": "lemma Fin.le_succ (a : Fin r) (h_a_add_1 : a + 1 < r) : a ≤ a + 1"}, {"name": "Fin.le_iff_lt_succ", "content": "lemma Fin.le_iff_lt_succ (a b : Fin r) (h_b : b + 1 < r) : a ≤ b ↔ a < b + 1"}, {"name": "Fin.val_sub_one", "content": "lemma Fin.val_sub_one (a : Fin r) (h_a_sub_1 : a > 0) : (a - 1).val = a.val - 1"}], "used_local_defs": [{"name": "AdditiveNTT.U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "AdditiveNTT.W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "AdditiveNTT.algEquivAevalXSubC", "content": "@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :="}], "used_local_lemmas": [{"name": "AdditiveNTT.βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "AdditiveNTT.root_U_lift_down", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n  (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n  a ∈ (U 𝔽q β (i+1)) → ∃! x: 𝔽q, a - x • β i ∈ (U 𝔽q β i)"}, {"name": "AdditiveNTT.root_U_lift_up", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime hβ_lin_indep in\ntheorem root_U_lift_up (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) (x : 𝔽q) :\n  a - x • β i ∈ (U 𝔽q β i) → a ∈ (U 𝔽q β (i+1))"}, {"name": "AdditiveNTT.W_monic", "content": "lemma W_monic (i : Fin r) : (W 𝔽q β i).Monic"}, {"name": "AdditiveNTT.W_ne_zero", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W_ne_zero (i : Fin r) : (W 𝔽q β i) ≠ 0"}, {"name": "AdditiveNTT.W_splits", "content": "lemma W_splits (i : Fin r) : (W 𝔽q β i).Splits (RingHom.id L)"}, {"name": "AdditiveNTT.roots_W", "content": "lemma roots_W (i : Fin r) : -- converts root Multiset into (univ: Uᵢ.val.map)\n  (W 𝔽q β i).roots = (univ : Finset (U 𝔽q β i)).val.map (fun u => u.val)"}, {"name": "AdditiveNTT.comp_X_sub_C_eq_zero_iff", "content": "omit [Fintype L] [DecidableEq L] in\nlemma comp_X_sub_C_eq_zero_iff (p : L[X]) (a : L) :\n  p.comp (X - C a) = 0 ↔ p = 0"}, {"name": "AdditiveNTT.rootMultiplicity_comp_X_sub_C", "content": "lemma rootMultiplicity_comp_X_sub_C (p : L[X]) (a x : L) :\n    rootMultiplicity x (p.comp (X - C a)) = rootMultiplicity (x - a) p"}, {"name": "AdditiveNTT.roots_comp_X_sub_C", "content": "lemma roots_comp_X_sub_C (p : L[X]) (a : L) :\n    (p.comp (X - C a)).roots = p.roots.map (fun r => r + a)"}, {"name": "AdditiveNTT.Prod_W_comp_X_sub_C_ne_zero", "content": "omit [DecidableEq L] h_Fq_char_prime hF₂ hβ_lin_indep in\nlemma Prod_W_comp_X_sub_C_ne_zero (i : Fin r) :\n    (univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i))) ≠ 0"}, {"name": "AdditiveNTT.rootMultiplicity_W", "content": "lemma rootMultiplicity_W (i : Fin r) (a : L) :\n    rootMultiplicity a (W 𝔽q β i) = if a ∈ (U 𝔽q β i : Set L) then 1 else 0"}, {"name": "AdditiveNTT.rootMultiplicity_prod_W_comp_X_sub_C", "content": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0"}], "local_ctx": "import ArkLib.Data.Nat.Bitwise\n\nimport ArkLib.Data.Polynomial.Frobenius\n\nimport ArkLib.Data.Polynomial.MonomialBasis\n\nimport Mathlib.LinearAlgebra.StdBasis\n\nimport Mathlib.Algebra.Polynomial.Degree.Definitions\n\nopen Polynomial FiniteDimensional Finset Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (h_dim : Module.finrank 𝔽q L = r)\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n\nsection LinearSubspaces\n\ndef U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))\n\nnoncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)\n\nend LinearSubspaces\n\nsection LinearityOfSubspaceVanishingPolynomials\n\n@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :=", "target_theorem": "lemma W_prod_comp_decomposition\n    (i : Fin r) (hi : i > 0) :\n    (W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1))) :=", "ground_truth_proof": ":= by\n  -- ⊢ W 𝔽q β i = ∏ c, (W 𝔽q β (i - 1)).comp (X - C (c • β (i - 1)))\n  -- Define P and Q for clarity\n  set P := W 𝔽q β i\n  set Q := ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))\n\n-- c : 𝔽q => univ\n-- c ∈ finsetX\n\n  -- STRATEGY: Prove P = Q by showing they are monic, split, and have the same roots.\n\n  -- 1. Show P and Q are MONIC.\n  have hP_monic : P.Monic := W_monic (𝔽q := 𝔽q) (β := β) (i :=i)\n  have hQ_monic : Q.Monic := by\n    apply Polynomial.monic_prod_of_monic; intro c _\n    apply Monic.comp\n    · exact W_monic (𝔽q := 𝔽q) (β := β) (i :=(i-1))\n    · -- ⊢ (X - C (c • β (i - 1))).Monic\n      exact Polynomial.monic_X_sub_C (c • β (i - 1))\n    · conv_lhs => rw [natDegree_sub_C, natDegree_X]\n      norm_num\n  -- 2. Show P and Q SPLIT over L.\n  have hP_splits : P.Splits (RingHom.id L) := W_splits 𝔽q β i\n  have hQ_splits : Q.Splits (RingHom.id L) := by\n    apply Polynomial.splits_prod\n    intro c _\n    -- Composition of a splitting polynomial with a linear polynomial also splits.\n    -- ⊢ Splits (RingHom.id L) ((W 𝔽q β (i - 1)).comp (X - C (c • β (i - 1))))\n    apply Splits.comp_of_degree_le_one\n    · exact degree_X_sub_C_le (c • β (i - 1))\n    · -- ⊢ Splits (RingHom.id L) (W 𝔽q β (i - 1))\n      exact W_splits 𝔽q β (i-1)\n\n  -- 3. Show P and Q have the same ROOTS.\n  have h_roots_eq : P.roots = Q.roots := by\n    -- First, characterize the roots of P. They are the elements of Uᵢ.\n    unfold P Q\n    ext u\n    rw [Polynomial.count_roots, Polynomial.count_roots]\n    rw [rootMultiplicity_W]\n    conv_rhs =>\n      rw [rootMultiplicity_prod_W_comp_X_sub_C 𝔽q β (h_i_add_1 := by\n        rw [Fin.val_sub_one (a := i) (h_a_sub_1 := by omega)]\n        omega\n      )]\n    -- ⊢ (if u ∈ ↑(U 𝔽q β i) then 1 else 0) = if u ∈ ↑(U 𝔽q β (i - 1 + 1)) then 1 else 0\n    have h_i : i - 1 + 1 = i := by simp only [sub_add_cancel]\n    rw [h_i]\n\n  -- 4. CONCLUSION: Since P and Q are monic, split, and have the same roots, they are equal.\n  have hP_eq_prod := Polynomial.eq_prod_roots_of_monic_of_splits_id hP_monic hP_splits\n  have hQ_eq_prod := Polynomial.eq_prod_roots_of_monic_of_splits_id hQ_monic hQ_splits\n  rw [hP_eq_prod, hQ_eq_prod, h_roots_eq]", "nesting_depth": 5, "transitive_dep_count": 173, "subset_aristotle": false, "category": "Applied verif."}
{"id": 33, "thm_name": "ConcreteBinaryTower.towerRingHomBackwardMap_forwardMap_eq", "thm_stmt": "lemma towerRingHomBackwardMap_forwardMap_eq (k : ℕ) (x : ConcreteBTField k) :\n  towerRingHomBackwardMap (k:=k) (towerRingHomForwardMap (k:=k) x) = x", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "AddCommGroup", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.negSucc", "module": "Init.Data.Int.Basic"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "NNRat", "module": "Mathlib.Data.Rat.Init"}, {"name": "NNRat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "Rat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.toAlgebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "invFun", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "left_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "right_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "absurd", "module": "Init.Prelude"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "instFintypeProd", "module": "Mathlib.Data.Fintype.Prod"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "CommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "RingEquiv", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "EquivLike", "module": "Mathlib.Data.FunLike.Equiv"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Eq.mpr", "module": "Init.Core"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}, {"name": "Algebra.algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "MonoidHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "OneHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}, {"name": "AlgebraTower.toIsScalarTower", "content": "@[simp]\ninstance AlgebraTower.toIsScalarTower (a : AlgebraTower C) {i j k : ι}\n    (h1 : i ≤ j) (h2 : j ≤ k) :\n    letI : Algebra (C i) (C j) :="}, {"name": "split", "content": "def split (k : ℕ) (h_k : k > 0) (x : BTField k) : BTField (k-1) × BTField (k-1) :="}, {"name": "join_via_add_smul", "content": "def join_via_add_smul {k : ℕ} (h_pos : k > 0) (hi_btf lo_btf : BTField (k - 1)) :\n    BTField k :="}, {"name": "binaryAlgebraTower", "content": "def binaryAlgebraTower {l r : ℕ} (h_le : l ≤ r) : Algebra (BTField l) (BTField r) :="}, {"name": "AlgebraTower.toAlgebra", "content": "@[simp]\ndef AlgebraTower.toAlgebra {i j : ι} (h : i ≤ j) : Algebra (A i) (A j) :=\n  (AlgebraTower.algebraMap (i:=i) (j:=j) (h:=h)).toAlgebra"}, {"name": "Z", "content": "@[simp]\ndef Z (k : ℕ) : BTField k := (BinaryTowerAux k).snd.specialElement "}, {"name": "polyMonic", "content": "instance polyMonic (n : ℕ) : Monic (poly n) := definingPoly_is_monic (Z n)"}, {"name": "poly", "content": "@[simp]\ndef poly (k : ℕ) : Polynomial (BTField k) := definingPoly (s:=(Z k))"}, {"name": "(priority", "content": "instance (priority := 1000) algebra_adjacent_tower (l : ℕ) :\n  Algebra (BTField l) (BTField (l+1)) :="}, {"name": "canonicalEmbedding", "content": "def canonicalEmbedding (k : ℕ) : BTField k →+* BTField (k+1) :=\n  AdjoinRoot.of (poly k)"}, {"name": "towerAlgebraMap", "content": "def towerAlgebraMap (l r : ℕ) (h_le : l ≤ r) : BTField l →+* BTField r :="}], "lib_lemmas": [{"name": "BitVec.cast_ofNat", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ofNat_eq_ofNat", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.eq_zero_or_eq_one", "module": "Init.Data.BitVec.Lemmas"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "BitVec.ofNatLT_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.pow_zero", "module": "Init.Data.Nat.Basic"}, {"name": "cast_cast", "module": "Init.PropLemmas"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "eq_mpr_eq_cast", "module": "Init.PropLemmas"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}, {"name": "BitVec.zero_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Ne.dite_eq_left_iff", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.left_eq_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingHom.comp_id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "MonoidHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "Nat.sub_one_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "OneHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "RingHom.coe_mk", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "eqRec_eq_cast", "module": "Batteries.Logic"}, {"name": "BitVec.ofNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.toNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Equiv.toFun_as_coe", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "EquivLike.coe_coe", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Nat.add_eq_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Prod.mk.eta", "module": "Mathlib.Data.Prod.Basic"}, {"name": "RingEquiv.apply_symm_apply", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "RingEquiv.toEquiv_eq_coe", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "and_false", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}, {"name": "split_join_via_add_smul_eq_iff_split", "content": "lemma split_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)\n    (hi_btf lo_btf : BTField (k - 1)) :\n    split (k:=k) (h_k:=h_pos) (⋘ hi_btf, lo_btf ⋙) = (hi_btf, lo_btf)"}, {"name": "eq_join_via_add_smul_eq_iff_split", "content": "theorem eq_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)\n    (x : BTField k) (hi_btf lo_btf : BTField (k - 1)) :\n    x = ⋘ hi_btf, lo_btf ⋙ ↔\n  split (k:=k) (h_k:=h_pos) x = (hi_btf, lo_btf)"}, {"name": "unique_linear_decomposition_succ", "content": "theorem unique_linear_decomposition_succ (k : ℕ) :\n  ∀ (x : BTField (k+1)), ∃! (p : BTField k × BTField k),\n    x = ⋘ p.1, p.2 ⋙"}, {"name": "algebraMap_adjacent_tower_succ_eq_Adjoin_of", "content": "lemma algebraMap_adjacent_tower_succ_eq_Adjoin_of (k : ℕ) :\n  (algebraMap (BTField k) (BTField (k + 1))) = of (poly k)"}, {"name": "algebraMap_adjacent_tower_def", "content": "lemma algebraMap_adjacent_tower_def (l : ℕ) :\n  (algebraMap (BTField l) (BTField (l + 1))) = canonicalEmbedding l"}, {"name": "towerAlgebraMap_succ_1", "content": "lemma towerAlgebraMap_succ_1 (k : ℕ) :\n  towerAlgebraMap (l:=k) (r:=k+1) (h_le:=by omega) = canonicalEmbedding k"}, {"name": "towerAlgebraMap_id", "content": "lemma towerAlgebraMap_id (k : ℕ) : towerAlgebraMap (h_le:=by omega) = RingHom.id (BTField k)"}, {"name": "binaryTowerAlgebra_def", "content": "lemma binaryTowerAlgebra_def (l r : ℕ) (h_le : l ≤ r) :\n    @binaryAlgebraTower (l:=l) (r:=r) (h_le:=h_le)\n    = (towerAlgebraMap l r h_le).toAlgebra"}, {"name": "poly_natDegree_eq_2", "content": "lemma poly_natDegree_eq_2 (k : ℕ) : (poly (k:=k)).natDegree = 2"}, {"name": "BTField.cast_BTField_eq", "content": "lemma BTField.cast_BTField_eq (k m : ℕ) (h_eq : k = m) :\n  BTField k = BTField m"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.add", "content": "def add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y"}, {"name": "ConcreteBinaryTower.neg", "content": "def neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.ConcreteBTFAddCommGroupProps", "content": "structure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel"}, {"name": "ConcreteBinaryTower.mkAddCommGroupInstance", "content": "def mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}"}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.equivProd", "content": "def equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)"}, {"name": "ConcreteBinaryTower.concrete_mul", "content": "def concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.concrete_inv", "content": "def concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res"}, {"name": "ConcreteBinaryTower.natCast", "content": "def natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.natCast_zero", "content": "def natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :="}, {"name": "ConcreteBinaryTower.natCast_succ", "content": "def natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :="}, {"name": "ConcreteBinaryTower.intCast", "content": "def intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.intCast_ofNat", "content": "def intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :="}, {"name": "ConcreteBinaryTower.intCast_negSucc", "content": "def intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :="}, {"name": "ConcreteBinaryTower.ConcreteBTFRingProps", "content": "structure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c"}, {"name": "ConcreteBinaryTower.ConcreteBTFDivisionRingProps", "content": "structure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldProps", "content": "structure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a"}, {"name": "ConcreteBinaryTower.mkRingInstance", "content": "def mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n"}, {"name": "ConcreteBinaryTower.mkDivisionRingInstance", "content": "def mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)"}, {"name": "ConcreteBinaryTower.mkFieldInstance", "content": "def mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm"}, {"name": "ConcreteBinaryTower.ConcreteBTFStepResult", "content": "structure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))"}, {"name": "ConcreteBinaryTower.liftBTFieldProps", "content": "def liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.liftConcreteBTField", "content": "def liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.concreteCanonicalEmbedding", "content": "def concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :="}, {"name": "ConcreteBinaryTower.instAlgebraLiftConcreteBTField", "content": "instance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))"}, {"name": "ConcreteBinaryTower.getBTFResult", "content": "def getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.canonicalAlgMap", "content": "def canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap", "content": "def concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :="}, {"name": "ConcreteBinaryTower.instAlgebraTowerConcreteBTF", "content": "instance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldAlgebra", "content": "def ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le"}, {"name": "ConcreteBinaryTower.join_via_add_smul", "content": "def join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.towerEquiv_zero", "content": "noncomputable def towerEquiv_zero : RingEquiv (R:=GF(2)) (S:=ConcreteBTField 0) :=  {\n  toFun := fun x => if x = 0 then 0 else 1,\n  invFun := fun x => if x = 0 then 0 else 1,\n  left_inv := fun x => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.towerRingEquiv0", "content": "noncomputable def towerRingEquiv0 : BTField 0 ≃+* ConcreteBTField 0 :="}, {"name": "ConcreteBinaryTower.towerRingEquivFromConcrete0", "content": "noncomputable def towerRingEquivFromConcrete0 : ConcreteBTField 0 ≃+* BTField 0 :="}, {"name": "ConcreteBinaryTower.towerRingHomForwardMap", "content": "noncomputable def towerRingHomForwardMap (k : ℕ) : ConcreteBTField k → BTField k :="}, {"name": "ConcreteBinaryTower.towerRingHomBackwardMap", "content": "noncomputable def towerRingHomBackwardMap (k : ℕ) : BTField k → ConcreteBTField k :="}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq", "content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n  ConcreteBTField k = ConcreteBTField m"}, {"name": "ConcreteBinaryTower.BitVec.bitvec_cast_eq_dcast", "content": "theorem BitVec.bitvec_cast_eq_dcast {n m : Nat} (h : n = m) (bv : BitVec n) :\n  BitVec.cast h bv = DCast.dcast h bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.cast_one", "content": "@[simp] theorem BitVec.cast_one {n m : ℕ} (h : n = m) : BitVec.cast h 1 = 1#m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_zero", "content": "@[simp] theorem BitVec.dcast_zero {n m : ℕ} (h : n = m) : DCast.dcast h (0#n) = 0#m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_one", "content": "@[simp] theorem BitVec.dcast_one {n m : ℕ} (h : n = m) : DCast.dcast h (1#n) = 1#m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.sum_fromNat_eq_from_xor_Nat", "content": "theorem sum_fromNat_eq_from_xor_Nat {k : ℕ} (x y : Nat) :\n  fromNat (k:=k) (x ^^^ y) = fromNat (k:=k) x + fromNat (k:=k) y"}, {"name": "ConcreteBinaryTower.zero_add", "content": "lemma zero_add {k : ℕ} (a : ConcreteBTField k) : 0 + a = a"}, {"name": "ConcreteBinaryTower.add_zero", "content": "lemma add_zero {k : ℕ} (a : ConcreteBTField k) : a + 0 = a"}, {"name": "ConcreteBinaryTower.cast_join", "content": "lemma cast_join {k n : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) (heq : k = n) :\n  join (k:=k) h_pos hi lo = cast (by rw [heq])\n    (join (k:=n) (by omega) (cast (by subst heq; rfl) hi) (lo:=cast (by subst heq; rfl) lo))"}, {"name": "ConcreteBinaryTower.zero_is_0", "content": "lemma zero_is_0 {k : ℕ} : (zero (k:=k)) = (0 : ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.one_is_1", "content": "lemma one_is_1 {k : ℕ} : (one (k:=k)) = 1"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_of_split", "content": "theorem join_of_split {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_split_eq : split h_pos x = (hi_btf, lo_btf)) :\n    x = 《 hi_btf, lo_btf 》"}, {"name": "ConcreteBinaryTower.split_of_join", "content": "theorem split_of_join {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_join : x = 《hi_btf, lo_btf》) :\n    (hi_btf, lo_btf) = split h_pos x"}, {"name": "ConcreteBinaryTower.split_join_eq_split", "content": "lemma split_join_eq_split {k : ℕ} (h_pos : k > 0)\n    (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    split h_pos (《 hi_btf, lo_btf 》) = (hi_btf, lo_btf)"}, {"name": "ConcreteBinaryTower.eq_iff_split_eq", "content": "theorem eq_iff_split_eq {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k) :\n  x₀ = x₁ ↔ (split h_pos x₀ = split h_pos x₁)"}, {"name": "ConcreteBinaryTower.split_sum_eq_sum_split", "content": "theorem split_sum_eq_sum_split {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k)\n  (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1))\n  (h_split_x₀ : split h_pos x₀ = (hi₀, lo₀))\n  (h_split_x₁ : split h_pos x₁ = (hi₁, lo₁)) :\n  split h_pos (x₀ + x₁) = (hi₀ + hi₁, lo₀ + lo₁)"}, {"name": "ConcreteBinaryTower.join_add_join", "content": "theorem join_add_join {k : ℕ} (h_pos : k > 0) (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1)) :\n  《 hi₀, lo₀ 》 + 《 hi₁, lo₁ 》 = 《 hi₀ + hi₁, lo₀ + lo₁ 》"}, {"name": "ConcreteBinaryTower.split_Z", "content": "theorem split_Z {k : ℕ} (h_pos : k > 0) :\n    split h_pos (Z k) = (one (k:=k - 1), zero (k:=k - 1))"}, {"name": "ConcreteBinaryTower.eq_zero_or_eq_one", "content": "theorem eq_zero_or_eq_one {a : ConcreteBTField 0} : a = zero ∨ a = one"}, {"name": "ConcreteBinaryTower.concrete_eq_zero_or_eq_one", "content": "theorem concrete_eq_zero_or_eq_one {k : ℕ} {a : ConcreteBTField k} (h_k_zero : k = 0)\n : a = zero ∨ a = one"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_eq_of_dest_eq", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_eq_of_dest_eq (k m n : ℕ) (h_eq : m = n) :\n    (ConcreteBTField k →+* ConcreteBTField m)\n    = (ConcreteBTField k →+* ConcreteBTField n)"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_cast_dest_apply", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_cast_dest_apply (k m n : ℕ) (h_eq : m = n)\n  (f : ConcreteBTField k →+* ConcreteBTField m) (x : ConcreteBTField k) :\n    (cast (ConcreteBTField.RingHom_eq_of_dest_eq (k:=k) (m:=m) (n:=n) h_eq) f) x\n    = cast (by apply cast_ConcreteBTField_eq (h_eq:=h_eq)) (f x)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_id", "content": "lemma concreteTowerAlgebraMap_id (k : ℕ) :\n    concreteTowerAlgebraMap (h_le:=by omega) = RingHom.id (ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_succ_1", "content": "lemma concreteTowerAlgebraMap_succ_1 (k : ℕ) :\n  concreteTowerAlgebraMap (l:=k) (r:=k + 1) (h_le:=by omega) = canonicalAlgMap k"}, {"name": "ConcreteBinaryTower.split_algebraMap_eq_zero_x", "content": "lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x) = (0, x)"}, {"name": "ConcreteBinaryTower.split_smul_Z_eq_zero_x", "content": "lemma split_smul_Z_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (x • Z k) = (x, 0)"}, {"name": "ConcreteBinaryTower.join_eq_join_via_add_smul", "content": "@[simp]\ntheorem join_eq_join_via_add_smul {k : ℕ} (h_pos : k > 0)\n    (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    《 hi_btf, lo_btf 》 = join_via_add_smul k h_pos hi_btf lo_btf"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y\n\ndef neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\nstructure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel\n\ndef mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\ndef equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)\n\ndef concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/\n\ndef concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res\n\nsection FieldLemmasOfLevel0\n\nend FieldLemmasOfLevel0\n\nsection NumericCasting\n\ndef natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :=\n\ndef natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :=\n\ndef intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :=\n\ndef intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :=\n\nend NumericCasting\n\nstructure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c\n\nstructure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one\n\nstructure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a\n\ndef mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n\n\ndef mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)\n\ndef mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm\n\nstructure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))\n\nend FieldOperationsAndInstances\n\nsection BTFieldPropsOneLevelLiftingLemmas\n\nvariable {k : ℕ} {h_k : k > 0}\n\nend BTFieldPropsOneLevelLiftingLemmas\n\nsection TowerFieldsConstruction\n\ndef liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/\n\ndef liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :=\n\ndef concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :=\n\ninstance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))\n\ndef getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/\n\nend TowerFieldsConstruction\n\nsection ConcreteBTFieldAlgebraConstruction\n\ndef canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))\n\ndef concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :=\n\ninstance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/\n\ndef ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le\n\ndef join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :=\n\nend ConcreteBTFieldAlgebraConstruction\n\nnoncomputable section ConcreteMultilinearBasis\n\nopen Module\n\nend ConcreteMultilinearBasis\n\nsection TowerEquivalence\n\nopen BinaryTower\n\nnoncomputable def towerEquiv_zero : RingEquiv (R:=GF(2)) (S:=ConcreteBTField 0) :=  {\n  toFun := fun x => if x = 0 then 0 else 1,\n  invFun := fun x => if x = 0 then 0 else 1,\n  left_inv := fun x => by admit /- proof elided -/\n\nnoncomputable def towerRingEquiv0 : BTField 0 ≃+* ConcreteBTField 0 :=\n\nnoncomputable def towerRingEquivFromConcrete0 : ConcreteBTField 0 ≃+* BTField 0 :=\n\nnoncomputable def towerRingHomForwardMap (k : ℕ) : ConcreteBTField k → BTField k :=\n\nnoncomputable def towerRingHomBackwardMap (k : ℕ) : BTField k → ConcreteBTField k :=", "target_theorem": "lemma towerRingHomBackwardMap_forwardMap_eq (k : ℕ) (x : ConcreteBTField k) :\n  towerRingHomBackwardMap (k:=k) (towerRingHomForwardMap (k:=k) x) = x :=", "ground_truth_proof": ":= by\n  induction k with\n  | zero =>\n    unfold towerRingHomBackwardMap towerRingHomForwardMap\n    simp only [↓reduceDIte, RingEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe]\n    rcases concrete_eq_zero_or_eq_one (a:=x) (by omega) with x_zero | x_one\n    · rw [x_zero, zero_is_0]\n      unfold towerRingEquivFromConcrete0 -- unfold the inner RingEquiv only\n      simp only [RingEquiv.apply_symm_apply] -- due to definition of `towerRingEquiv0`\n    · rw [x_one, one_is_1]\n      unfold towerRingEquivFromConcrete0 -- unfold the inner RingEquiv only\n      simp only [RingEquiv.apply_symm_apply] -- due to definition of `towerRingEquiv0`\n  | succ k ih =>\n    rw [towerRingHomForwardMap] -- split inner\n    simp only [Nat.add_eq_zero, one_ne_zero, and_false, ↓reduceDIte, Nat.add_one_sub_one]\n    rw [towerRingHomBackwardMap] -- split outer\n    simp only [Nat.add_eq_zero, one_ne_zero, and_false, ↓reduceDIte, Nat.add_one_sub_one]\n    rw [←join_eq_join_via_add_smul]\n    apply Eq.symm\n    apply join_of_split\n    simp only [Nat.add_one_sub_one]\n    rw [BinaryTower.split_join_via_add_smul_eq_iff_split (k:=k + 1)]\n    simp only\n    -- apply induction hypothesis\n    rw [ih, ih]\n    simp only [Prod.mk.eta]", "nesting_depth": 15, "transitive_dep_count": 299, "subset_aristotle": false, "category": "Applied verif."}
{"id": 34, "thm_name": "AdditiveNTT.additiveNTT_correctness", "thm_stmt": "theorem additiveNTT_correctness (h_ℓ : ℓ ≤ r)\n    (original_coeffs : Fin (2 ^ ℓ) → L)\n    (output_buffer : Fin (2 ^ (ℓ + R_rate)) → L)\n    (h_alg : output_buffer = additiveNTT 𝔽q β h_ℓ_add_R_rate original_coeffs) :\n    let P := polynomialFromNovelCoeffs 𝔽q β ℓ h_ℓ original_coeffs\n    ∀ (j : Fin (2^(ℓ + R_rate))),\n      output_buffer j = P.eval (evaluationPointω 𝔽q β h_ℓ_add_R_rate ⟨0, by omega⟩ j)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean", "imports": ["import Mathlib.Data.Finsupp.Defs", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Tactic", "import ArkLib.Data.Nat.Bitwise", "import Mathlib.LinearAlgebra.LinearIndependent.Defs", "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import ArkLib.Data.Polynomial.Frobenius"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "MvPolynomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "MvPolynomial.optionEquivLeft", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "MvPolynomial.renameEquiv", "module": "Mathlib.Algebra.MvPolynomial.Rename"}, {"name": "finSuccEquiv'", "module": "Mathlib.Logic.Equiv.Fin.Basic"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "context", "module": "Examples.FrankingProtocol"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fin.mk", "module": "Init.Prelude"}, {"name": "Fin.isValue", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "AddHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "AddCommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Polynomial.rootMultiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.algEquivOfCompEqX", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.Prime", "module": "Mathlib.Data.Nat.Prime.Defs"}, {"name": "ringChar", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "Finset.sum", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}], "used_repo_defs": [{"name": "notation:70 s:70 \" ^^ \" t:71 => Fintype.piFinset fun (i : t)", "content": "notation:70 s:70 \" ^^ \" t:71 => Fintype.piFinset fun (i : t) ↦ s i"}, {"name": "macro_rules (kind := mvEval)", "content": "macro_rules (kind := mvEval)\n  | `($p⸨$x⸩) => `(MvPolynomial.eval ($x ∘ Fin.cast (by admit /- proof elided -/\n  )) $p)\n  | `($p⸨$x, $y⸩) => `(MvPolynomial.eval (Fin.append $x $y ∘ Fin.cast (by admit /- proof elided -/\n  )) $p)\n  | `($p⸨$x, $y, $z⸩) =>\n      `(MvPolynomial.eval (Fin.append (Fin.append $x $y) $z ∘ Fin.cast (by admit /- proof elided -/\n      )) $p)"}, {"name": "macro_rules (kind := mvEval')", "content": "macro_rules (kind := mvEval')\n  | `($p⸨$x⸩'$h) => `(MvPolynomial.eval ($x ∘ Fin.cast $h) $p)\n  | `($p⸨$x, $y⸩'$h) => `(MvPolynomial.eval (Fin.append $x $y ∘ Fin.cast $h) $p)\n  | `($p⸨$x, $y, $z⸩'$h) =>\n      `(MvPolynomial.eval (Fin.append (Fin.append $x $y) $z ∘ Fin.cast $h) $p)\n\nexample : (X 0 + X 1 * X 2 : ℕ[X Fin 3]) ⸨![1, 2], ![8], ![]⸩ = 17 := by admit /- proof elided -/"}, {"name": "macro_rules (kind := mvEvalToPolynomial)", "content": "macro_rules (kind := mvEvalToPolynomial)\n  | `($p⸨X ⦃$i⦄, $x⸩) =>\n      `(Polynomial.map (MvPolynomial.eval ($x ∘ Fin.cast (by admit /- proof elided -/\n      )))\n        (MvPolynomial.finSuccEquivNth _ $i $p))\n  | `($p⸨X ⦃$i⦄, $x, $y⸩) =>\n      `(Polynomial.map (MvPolynomial.eval (Fin.append $x $y ∘ Fin.cast (by admit /- proof elided -/\n      )))\n        (MvPolynomial.finSuccEquivNth _ $i $p))\n  | `($p⸨X ⦃$i⦄, $x, $y, $z⸩) =>\n      `(Polynomial.map (MvPolynomial.eval (Fin.append (Fin.append $x $y) $z ∘ Fin.cast (by admit /- proof elided -/\n      )))\n        (MvPolynomial.finSuccEquivNth _ $i $p))"}, {"name": "macro_rules (kind := mvEvalToPolynomial')", "content": "macro_rules (kind := mvEvalToPolynomial')\n  | `($p⸨X ⦃$i⦄, $x⸩'$h) =>\n      `(Polynomial.map (MvPolynomial.eval ($x ∘ Fin.cast $h))\n        (MvPolynomial.finSuccEquivNth _ $i $p))\n  | `($p⸨X ⦃$i⦄, $x, $y⸩'$h) =>\n      `(Polynomial.map (MvPolynomial.eval (Fin.append $x $y ∘ Fin.cast $h))\n        (MvPolynomial.finSuccEquivNth _ $i $p))\n  | `($p⸨X ⦃$i⦄, $x, $y, $z⸩'$h) =>\n      `(Polynomial.map (MvPolynomial.eval (Fin.append (Fin.append $x $y) $z ∘ Fin.cast $h))\n        (MvPolynomial.finSuccEquivNth _ $i $p))\n\nexample {a b n : ℕ} (x : Fin a → ℕ) (y : Fin b → ℕ) (p : ℕ[X Fin n]) (h : a + b + 1 = n) :\n  p ⸨x, ![n], y⸩ =\n    MvPolynomial.eval (Fin.append (Fin.append x ![n]) y ∘ Fin.cast (by admit /- proof elided -/\n    )) p := rfl"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "finSuccEquivNth", "content": "def finSuccEquivNth : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] Polynomial (MvPolynomial (Fin n) R) :=\n  (renameEquiv R (_root_.finSuccEquiv' p)).trans (optionEquivLeft R (Fin n))"}, {"name": "getLowBits", "content": "def getLowBits (numLowBits : ℕ) (n : ℕ) := n &&& ((1 <<< numLowBits) - 1)"}, {"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "polynomialFromNovelCoeffs", "content": "noncomputable def polynomialFromNovelCoeffs (ℓ : ℕ) (h_ℓ : ℓ ≤ r)\n  (a : Fin (2 ^ ℓ) → L) : L[X] := ∑ j, C (a j) * (Xⱼ 𝔽q β ℓ h_ℓ j)"}, {"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}, {"name": "getHighBits_no_shl", "content": "def getHighBits_no_shl (numLowBits : ℕ) (n : ℕ) : ℕ := n >>> numLowBits"}, {"name": "getHighBits", "content": "def getHighBits (numLowBits : ℕ) (n : ℕ) : ℕ :=\n  (getHighBits_no_shl numLowBits n) <<< numLowBits"}, {"name": "g", "content": "def g (n : ℕ) (c : ℕ) (x : ℕ) := (x * x + c) % n"}, {"name": "polyEvalLinearMap", "content": "noncomputable def polyEvalLinearMap {L 𝔽q : Type*} [Field L] [Field 𝔽q] [Algebra 𝔽q L]\n  (p : L[X]) (hp_add : IsLinearMap 𝔽q (fun x : L => p.eval x)) : L →ₗ[𝔽q] L :=\n{\n  toFun    := fun x => p.eval x,\n  map_add' := hp_add.map_add,\n  map_smul' := hp_add.map_smul\n}"}, {"name": "algEquivAevalXSubC", "content": "@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :="}], "lib_lemmas": [{"name": "Polynomial.comp_assoc", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "Fin.coe_ofNat_eq_mod", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.foldl_succ", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Polynomial.X_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_succ", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mul_add_mod_self_right", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Polynomial.pow_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.prod_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Finset.prod_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pow_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.is_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.sum_univ_eq_sum_range", "module": "Mathlib.Data.Fintype.BigOperators"}, {"name": "Finset.mul_sum", "module": "Mathlib.Algebra.BigOperators.Ring.Finset"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "Polynomial.C_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.mul_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.sum_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "map_eq_zero", "module": "Mathlib.Algebra.GroupWithZero.Units.Lemmas"}, {"name": "mul_eq_mul_left_iff", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "or_true", "module": "Init.SimpLemmas"}, {"name": "Fintype.card_pos", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Polynomial.C_1", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_mul", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_pow", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.sub_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "inv_eq_one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "map_pow", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_inv_cancel₀", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_pow", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_pow_sub_one", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_sub", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_pow", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.prod_eq_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Polynomial.eval_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_mul", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_prod", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_sub", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "sub_self", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "FiniteField.pow_card", "module": "Mathlib.FieldTheory.Finite.Basic"}, {"name": "Polynomial.C_mul_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.smul_eq_C_mul", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "algebra_compatible_smul", "module": "Mathlib.Algebra.Algebra.Basic"}, {"name": "mul_add", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "smul_pow", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "Nat.add_mod_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Polynomial.eval_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "map_sum", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "map_zero", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.cast_one", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_zero", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.mod_two_not_eq_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.pow_le_pow_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "Nat.pow_lt_pow_right", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_mod_left", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.one_and_eq_mod_two", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_assoc", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_comm", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftLeft_or_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "AddHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "AddHom.toFun_eq_coe", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "LinearMap.coe_mk", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap.coe_toAddHom", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "Nat.add_assoc", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.add_eq_left", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.add_left_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.add_lt_add_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.add_lt_add_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.add_sub_assoc", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.add_zero", "module": "Init.Core"}, {"name": "Nat.div_div_eq_div_mul", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.div_eq_of_lt_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.div_mul_le_self", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.le_add_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.le_add_right_of_le", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.sub_le_sub_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_sub", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.xor_assoc", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_or", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_shiftLeft", "module": "Init.Data.Nat.Lemmas"}, {"name": "Polynomial.eval_add", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "add_right_inj", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "lt_self_iff_false", "module": "Mathlib.Order.Basic"}, {"name": "map_one", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_two", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Fin.mk_eq_mk", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "div_one", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Fin.coe_cast", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.coe_castSucc", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.foldl_succ_last", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.foldl_zero", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_last", "module": "Init.Data.Fin.Lemmas"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "tsub_zero", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Fin.mk_zero'", "module": "Mathlib.Data.Fin.Basic"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin.val_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.cast_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.prod_congr'", "module": "Mathlib.Algebra.BigOperators.Fin"}, {"name": "Fin.prod_univ_zero", "module": "Mathlib.Algebra.BigOperators.Fin"}, {"name": "Fin.sum_congr'", "module": "Mathlib.Algebra.BigOperators.Fin"}, {"name": "Fin.sum_univ_one", "module": "Mathlib.Algebra.BigOperators.Fin"}, {"name": "Nat.lt_one_iff", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.lt_two_pow_self", "module": "Init.Data.Nat.Lemmas"}, {"name": "Polynomial.eval_one", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "tsub_self", "module": "Mathlib.Algebra.Order.Sub.Basic"}, {"name": "Nat.or_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftLeft_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Fin.zero_eta", "module": "Init.Data.Fin.Basic"}, {"name": "Nat.div_add_mod'", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_one", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_right_comm", "module": "Init.Data.Nat.Lemmas"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "getBit_eq_succ_getBit_of_mul_two_add_one", "content": "lemma getBit_eq_succ_getBit_of_mul_two_add_one {n k : ℕ} : getBit (k+1) (2*n + 1) = getBit k n"}, {"name": "getBit_of_shiftRight", "content": "lemma getBit_of_shiftRight {n p : ℕ}:\n  ∀ k, getBit k (n >>> p) = getBit (k+p) n"}, {"name": "mul_two_add_bit_lt_two_pow", "content": "theorem mul_two_add_bit_lt_two_pow (a b c : ℕ) (i : Fin 2)\n    (h_a : a < 2 ^ b) (h_b : b < c) :\n    a * 2 + i.val < 2^c"}, {"name": "lt_two_pow_of_lt_two_pow_exp_le", "content": "lemma lt_two_pow_of_lt_two_pow_exp_le (x i j: ℕ)\n    (h_x_lt_2_pow_i: x < 2^i) (h_i_le_j: i ≤ j): x < 2^j"}, {"name": "getBit_zero_of_two_mul", "content": "lemma getBit_zero_of_two_mul {n : ℕ} : getBit 0 (2*n) = 0"}, {"name": "getBit_eq_succ_getBit_of_mul_two", "content": "lemma getBit_eq_succ_getBit_of_mul_two {n k : ℕ} : getBit (k+1) (2*n) = getBit k n"}, {"name": "Fin.sum_univ_odd_even", "content": "theorem Fin.sum_univ_odd_even {n : ℕ} {M : Type*} [AddCommMonoid M] (f : ℕ → M) :\n    (∑ i : Fin (2 ^ n), f (2 * i)) + (∑ i : Fin (2 ^ n), f (2 * i + 1))\n    = ∑ i: Fin (2 ^ (n+1)), f i"}, {"name": "W_linear_comp_decomposition", "content": "omit hF₂ in\ntheorem W_linear_comp_decomposition (i : Fin r) (h_i_add_1 : i + 1 < r) :\n    ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)"}, {"name": "W_linearity", "content": "theorem W_linearity (i : Fin r)\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p)"}, {"name": "inductive_linear_map_W", "content": "omit hF₂ in\nlemma inductive_linear_map_W (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (i + 1)).comp inner_p)"}, {"name": "inductive_rec_form_W_comp", "content": "omit h_Fq_char_prime hF₂ in\nlemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])\n      (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)"}, {"name": "Wᵢ_eval_βᵢ_neq_zero", "content": "lemma Wᵢ_eval_βᵢ_neq_zero\n    (i : Fin r): (W 𝔽q β i).eval (β i) ≠ 0"}, {"name": "βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "W_prod_comp_decomposition", "content": "lemma W_prod_comp_decomposition\n    (i : Fin r) (hi : i > 0) :\n    (W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))"}, {"name": "W_splits", "content": "lemma W_splits (i : Fin r) : (W 𝔽q β i).Splits (RingHom.id L)"}, {"name": "rootMultiplicity_prod_W_comp_X_sub_C", "content": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0"}, {"name": "Prod_W_comp_X_sub_C_ne_zero", "content": "omit [DecidableEq L] h_Fq_char_prime hF₂ hβ_lin_indep in\nlemma Prod_W_comp_X_sub_C_ne_zero (i : Fin r) :\n    (univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i))) ≠ 0"}, {"name": "W_ne_zero", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W_ne_zero (i : Fin r) : (W 𝔽q β i) ≠ 0"}, {"name": "roots_comp_X_sub_C", "content": "lemma roots_comp_X_sub_C (p : L[X]) (a : L) :\n    (p.comp (X - C a)).roots = p.roots.map (fun r => r + a)"}, {"name": "rootMultiplicity_comp_X_sub_C", "content": "lemma rootMultiplicity_comp_X_sub_C (p : L[X]) (a x : L) :\n    rootMultiplicity x (p.comp (X - C a)) = rootMultiplicity (x - a) p"}, {"name": "comp_X_sub_C_eq_zero_iff", "content": "omit [Fintype L] [DecidableEq L] in\nlemma comp_X_sub_C_eq_zero_iff (p : L[X]) (a : L) :\n  p.comp (X - C a) = 0 ↔ p = 0"}, {"name": "rootMultiplicity_W", "content": "lemma rootMultiplicity_W (i : Fin r) (a : L) :\n    rootMultiplicity a (W 𝔽q β i) = if a ∈ (U 𝔽q β i : Set L) then 1 else 0"}, {"name": "roots_W", "content": "lemma roots_W (i : Fin r) : -- converts root Multiset into (univ: Uᵢ.val.map)\n  (W 𝔽q β i).roots = (univ : Finset (U 𝔽q β i)).val.map (fun u => u.val)"}, {"name": "root_U_lift_up", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime hβ_lin_indep in\ntheorem root_U_lift_up (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) (x : 𝔽q) :\n  a - x • β i ∈ (U 𝔽q β i) → a ∈ (U 𝔽q β (i+1))"}, {"name": "root_U_lift_down", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n  (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n  a ∈ (U 𝔽q β (i+1)) → ∃! x: 𝔽q, a - x • β i ∈ (U 𝔽q β i)"}, {"name": "W_monic", "content": "lemma W_monic (i : Fin r) : (W 𝔽q β i).Monic"}, {"name": "comp_sub_C_of_linear_eval", "content": "lemma comp_sub_C_of_linear_eval (p : L[X])\n  (h_lin : IsLinearMap 𝔽q (f := fun inner_p ↦ p.comp inner_p)) (a : L) :\n    p.comp (X - C a) = p - C (eval a p)"}, {"name": "prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L", "content": "theorem prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L\n  (p : L[X]) :\n  (∏ c ∈ (Finset.univ : Finset Fq), (p - Polynomial.C (algebraMap Fq L c))) =\n    p^(Fintype.card Fq) - p"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X_in_L", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X_in_L :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C (algebraMap Fq L c))) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C c)) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "frobenius_identity_in_algebra", "content": "theorem frobenius_identity_in_algebra [Fact (Nat.Prime (ringChar Fq))]\n  (f g : L[X]) : (f + g)^(Fintype.card Fq) = f^(Fintype.card Fq) + g^(Fintype.card Fq)"}, {"name": "linear_map_of_comp_to_linear_map_of_eval", "content": "theorem linear_map_of_comp_to_linear_map_of_eval (f : L[X])\n  (h_f_linear : IsLinearMap (R := Fq) (M := L[X]) (M₂ := L[X])\n    (f := fun inner_p ↦ f.comp inner_p)) :\n    IsLinearMap (R := Fq) (M := L) (M₂ := L) (f := fun x ↦ f.eval x)"}, {"name": "getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "div_two_pow_lt_two_pow", "content": "theorem div_two_pow_lt_two_pow (x i j : ℕ) (h_x_lt_2_pow_i : x < 2 ^ (i + j)): x / 2^j < 2^(i)"}, {"name": "sum_of_and_eq_zero_is_or", "content": "lemma sum_of_and_eq_zero_is_or {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ||"}, {"name": "sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||"}, {"name": "xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}, {"name": "xor_of_and_eq_zero_is_or", "content": "lemma xor_of_and_eq_zero_is_or {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n ^^^ m = n ||"}, {"name": "eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "and_two_pow_eq_zero_of_getBit_0", "content": "lemma and_two_pow_eq_zero_of_getBit_0 {n i : ℕ} (h_getBit : getBit i n = 0)\n    : n &&& (2 ^ i) = 0"}, {"name": "getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "getLowBits_succ", "content": "lemma getLowBits_succ {n: ℕ} (numLowBits: ℕ) :\n    getLowBits (numLowBits + 1) n = getLowBits numLowBits n\n    + (getBit numLowBits n) <<< numLowBits"}, {"name": "getBit_of_lowBits", "content": "lemma getBit_of_lowBits {n: ℕ} (numLowBits : ℕ) : ∀ k, getBit k (getLowBits numLowBits n) =\n    if k < numLowBits then getBit k n else 0"}, {"name": "getBit_of_and", "content": "lemma getBit_of_and {n m k: ℕ} : getBit k (n &&& m) = getBit k n &&& getBit k m"}, {"name": "getBit_of_two_pow_sub_one", "content": "lemma getBit_of_two_pow_sub_one {i k: ℕ} : getBit k (2^i - 1) =\n    if k < i then 1 else 0"}, {"name": "getBit_of_or", "content": "lemma getBit_of_or {n m k: ℕ} : getBit k (n ||| m) = getBit k n ||"}, {"name": "sum_of_and_eq_zero_is_xor", "content": "lemma sum_of_and_eq_zero_is_xor {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ^^^ m"}, {"name": "xor_eq_sub_iff_submask", "content": "lemma xor_eq_sub_iff_submask {n m : ℕ} (h: m ≤ n) : n ^^^ m = n - m ↔ n &&& m = m"}, {"name": "getBit_of_xor", "content": "lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m"}, {"name": "normalizedWᵢ_eval_βᵢ_eq_1", "content": "lemma normalizedWᵢ_eval_βᵢ_eq_1 {i : Fin r} :\n    (normalizedW (𝔽q := 𝔽q) (β := β) (i :=i)).eval (β i) = 1"}, {"name": "num_eq_highBits_xor_lowBits", "content": "lemma num_eq_highBits_xor_lowBits {n: ℕ} (numLowBits: ℕ) :\n  n = getHighBits numLowBits n ^^^ getLowBits numLowBits n"}, {"name": "and_highBits_lowBits_eq_zero", "content": "theorem and_highBits_lowBits_eq_zero {n : ℕ} (numLowBits : ℕ) :\n    getHighBits numLowBits n &&& getLowBits numLowBits n = 0"}, {"name": "getBit_of_shiftLeft", "content": "lemma getBit_of_shiftLeft {n p : ℕ}:\n  ∀ k, getBit (k) (n <<< p) = if k < p then 0 else getBit (k - p) n"}, {"name": "getBit_of_multiple_of_power_of_two", "content": "lemma getBit_of_multiple_of_power_of_two {n p : ℕ}: ∀ k,\n  getBit (k) (2^p * n) = if k < p then 0 else getBit (k-p) n"}, {"name": "num_eq_highBits_add_lowBits", "content": "lemma num_eq_highBits_add_lowBits {n: ℕ} (numLowBits: ℕ) :\n  n = getHighBits numLowBits n + getLowBits numLowBits n"}, {"name": "getBit_of_add_distrib", "content": "lemma getBit_of_add_distrib {n m k : ℕ}\n  (h_n_AND_m : n &&& m = 0) : getBit k (n + m) = getBit k n + getBit k m"}, {"name": "getLowBits_lt_two_pow", "content": "lemma getLowBits_lt_two_pow {n : ℕ} (numLowBits : ℕ) :\n    getLowBits numLowBits n < 2 ^ numLowBits"}, {"name": "getLowBits_eq_mod_two_pow", "content": "lemma getLowBits_eq_mod_two_pow {numLowBits : ℕ} (n : ℕ) :\n  getLowBits numLowBits n = n % (2 ^ numLowBits)"}, {"name": "and_two_pow_eq_two_pow_of_getBit_1", "content": "lemma and_two_pow_eq_two_pow_of_getBit_1 {n i : ℕ} (h_getBit: getBit i n = 1) :\n    n &&& (2 ^ i) = 2 ^ i"}, {"name": "add_two_pow_of_getBit_eq_zero_lt_two_pow", "content": "lemma add_two_pow_of_getBit_eq_zero_lt_two_pow {n m i : ℕ} (h_n: n < 2^m) (h_i: i < m)\n  (h_getBit_at_i_eq_zero: getBit i n = 0) :\n  n + 2^i < 2^m"}, {"name": "getLowBits_le_self", "content": "lemma getLowBits_le_self {n : ℕ} (numLowBits : ℕ) : getLowBits numLowBits n ≤ n"}, {"name": "W₀_eq_X", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W₀_eq_X : W 𝔽q β 0 = X"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}, {"name": "getLowBits_zero_eq_zero", "content": "lemma getLowBits_zero_eq_zero {n : ℕ} : getLowBits 0 n = 0"}], "used_local_defs": [{"name": "AdditiveNTT.qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "AdditiveNTT.qCompositionChain", "content": "noncomputable def qCompositionChain (i : Fin r) : L[X] :=\n  match i with\n  | ⟨0, _⟩ => X\n  | ⟨k + 1, h_k_add_1⟩ => (qMap 𝔽q β ⟨k, by admit /- proof elided -/\n  ⟩).comp (qCompositionChain ⟨k, by admit /- proof elided -/\n  ⟩)"}, {"name": "AdditiveNTT.intermediateNormVpoly", "content": "noncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)"}, {"name": "AdditiveNTT.intermediateNovelBasisX", "content": "noncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))"}, {"name": "AdditiveNTT.intermediateEvaluationPoly", "content": "noncomputable def intermediateEvaluationPoly (i : Fin (ℓ + 1))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i))), C (coeffs ⟨j, by admit /- proof elided -/\n  ⟩) *\n    (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate i ⟨j, by admit /- proof elided -/\n    ⟩)"}, {"name": "AdditiveNTT.evenRefinement", "content": "noncomputable def evenRefinement (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i-1))), C (coeffs ⟨j*2, by admit /- proof elided -/\n  ⟩) * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by admit /- proof elided -/\n  ⟩ ⟨j, hj⟩)"}, {"name": "AdditiveNTT.oddRefinement", "content": "noncomputable def oddRefinement (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i-1))), C (coeffs ⟨j*2+1, by admit /- proof elided -/\n  ⟩) * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by admit /- proof elided -/\n  ⟩ ⟨j, hj⟩)"}, {"name": "AdditiveNTT.evaluationPointω", "content": "noncomputable def evaluationPointω (i : Fin (ℓ + 1))\n    (x : Fin (2 ^ (ℓ + R_rate - i))) : L := \n    \n  ∑ (⟨k, hk⟩: Fin (ℓ + R_rate - i)),\n    if Nat.getBit k x.val = 1 then\n      (normalizedW 𝔽q β ⟨i, by admit /- proof elided -/\n      ⟩).eval (β ⟨i + k, by admit /- proof elided -/\n      ⟩)\n    else\n      0"}, {"name": "AdditiveNTT.twiddleFactor", "content": "noncomputable def twiddleFactor (i : Fin ℓ) (u : Fin (2 ^ (ℓ + R_rate - i - 1))) : L :=\n  ∑ (⟨k, hk⟩: Fin (ℓ + R_rate - i - 1)),\n    if Nat.getBit k u.val = 1 then\n      \n        \n      (normalizedW 𝔽q β ⟨i, by admit /- proof elided -/\n      ⟩).eval (β ⟨i + 1 + k, by admit /- proof elided -/\n      ⟩)\n    else 0"}, {"name": "AdditiveNTT.tileCoeffs", "content": "def tileCoeffs (a : Fin (2 ^ ℓ) → L) : Fin (2^(ℓ + R_rate)) → L :=\n  fun v => a (Fin.mk (v.val % (2^ℓ)) (Nat.mod_lt v.val (pow_pos (zero_lt_two) ℓ)))"}, {"name": "AdditiveNTT.NTTStage", "content": "noncomputable def NTTStage (i : Fin ℓ) (b : Fin (2 ^ (ℓ + R_rate)) → L) :\n    Fin (2^(ℓ + R_rate)) → L :=\n  have h_2_pow_i_lt_2_pow_ℓ_add_R_rate: 2^i.val < 2^(ℓ + R_rate) := by admit /- proof elided -/"}, {"name": "AdditiveNTT.additiveNTT", "content": "noncomputable def additiveNTT (a : Fin (2 ^ ℓ) → L) : Fin (2^(ℓ + R_rate)) → L :=\n  let b: Fin (2^(ℓ + R_rate)) → L := tileCoeffs a \n  Fin.foldl (n:=ℓ) (f:= fun current_b i  =>\n    NTTStage 𝔽q β h_ℓ_add_R_rate (i := ⟨ℓ - 1 - i, by admit /- proof elided -/\n    ⟩) current_b\n  ) (init:=b)"}, {"name": "AdditiveNTT.coeffsBySuffix", "content": "def coeffsBySuffix (a : Fin (2 ^ ℓ) → L) (i : Fin (ℓ + 1)) (v : Fin (2 ^ i.val)) :\n  Fin (2 ^ (ℓ - i)) → L :=\n  fun ⟨j, hj⟩ => by admit /- proof elided -/"}, {"name": "AdditiveNTT.additiveNTTInvariant", "content": "def additiveNTTInvariant (evaluation_buffer : Fin (2 ^ (ℓ + R_rate)) → L)\n    (original_coeffs : Fin (2 ^ ℓ) → L) (i : Fin (ℓ + 1)) : Prop :=\n  ∀ (j : Fin (2^(ℓ + R_rate))),\n    let u_b_v := j.val\n    let v: Fin (2^i.val) := ⟨Nat.getLowBits i.val u_b_v, by admit /- proof elided -/\n    ⟩ \n    let u_b := u_b_v / (2^i.val) \n    have h_u_b : u_b = u_b_v / (2^i.val) := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "AdditiveNTT.qMap_eval_𝔽q_eq_0", "content": "omit [DecidableEq L] [DecidableEq 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in\ntheorem qMap_eval_𝔽q_eq_0 (i : Fin r) :\n  ∀ c: 𝔽q, (qMap 𝔽q β i).eval (algebraMap 𝔽q L c) = 0"}, {"name": "AdditiveNTT.qMap_comp_normalizedW", "content": "lemma qMap_comp_normalizedW (i : Fin r) (h_i_add_1 : i + 1 < r) :\n  (qMap 𝔽q β i).comp (normalizedW 𝔽q β i) = normalizedW 𝔽q β (i + 1)"}, {"name": "AdditiveNTT.qMap_is_linear_map", "content": "theorem qMap_is_linear_map (i : Fin r) :\n  IsLinearMap 𝔽q (f:=fun inner_p ↦ (qMap 𝔽q β i).comp inner_p)"}, {"name": "AdditiveNTT.qCompositionChain_eq_foldl", "content": "lemma qCompositionChain_eq_foldl (i : Fin r) :\n  qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i =\n  Fin.foldl (n:=i) (fun acc j =>\n    (qMap 𝔽q β ⟨j, by omega⟩).comp acc) (X)"}, {"name": "AdditiveNTT.normalizedW_eq_qMap_composition", "content": "lemma normalizedW_eq_qMap_composition (ℓ R_rate : ℕ) (i : Fin r) :\n    normalizedW 𝔽q β i = qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i"}, {"name": "AdditiveNTT.base_intermediateNormVpoly", "content": "theorem base_intermediateNormVpoly\n  (k : Fin (ℓ + 1)) :\n  intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨0, by\n    by_contra ht\n    simp only [not_lt, nonpos_iff_eq_zero] at ht\n    contradiction\n  ⟩ ⟨k, by simp only [tsub_zero]; omega⟩ =\n  normalizedW 𝔽q β ⟨k, by omega⟩"}, {"name": "AdditiveNTT.Polynomial.foldl_comp", "content": "omit [Fintype L] [DecidableEq L] in\ntheorem Polynomial.foldl_comp (n : ℕ) (f : Fin n → L[X]) : ∀ initInner initOuter: L[X],\n    Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter.comp initInner)\n    = (Fin.foldl (n:=n) (fun acc j => (f j).comp acc) (initOuter)).comp initInner"}, {"name": "AdditiveNTT.intermediateNormVpoly_comp_qmap", "content": "theorem intermediateNormVpoly_comp_qmap (i : Fin (ℓ))\n    (k : Fin (ℓ - i - 1)) : -- corresponds to intermediateNormVpoly_comp"}, {"name": "AdditiveNTT.intermediateNormVpoly_comp_qmap_helper", "content": "theorem intermediateNormVpoly_comp_qmap_helper (i : Fin (ℓ))\n    (k : Fin (ℓ - (↑i + 1))) :\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate\n      ⟨↑i + 1, by omega⟩ (k:=⟨k, by simp only; omega⟩)).comp (qMap 𝔽q β ⟨↑i, by omega⟩) =\n    intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate\n      ⟨↑i, by omega⟩ ⟨k + 1, by simp only; omega⟩"}, {"name": "AdditiveNTT.base_intermediateNovelBasisX", "content": "theorem base_intermediateNovelBasisX (j : Fin (2 ^ ℓ)) :\n  intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨0, by\n    by_contra ht\n    simp only [not_lt, nonpos_iff_eq_zero] at ht\n    contradiction\n  ⟩ j =\n  Xⱼ 𝔽q β ℓ (by omega) j"}, {"name": "AdditiveNTT.even_index_intermediate_novel_basis_decomposition", "content": "lemma even_index_intermediate_novel_basis_decomposition (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :\n  intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2, by\n    apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨0, by omega⟩ (by omega) (by omega)\n  ⟩  = (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by omega⟩ ⟨j, by\n    apply lt_two_pow_of_lt_two_pow_exp_le j (ℓ-i-1) (ℓ-(i+1)) (by omega) (by omega)\n  ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩)"}, {"name": "AdditiveNTT.odd_index_intermediate_novel_basis_decomposition", "content": "lemma odd_index_intermediate_novel_basis_decomposition\n    (i : Fin ℓ) (j : Fin (2 ^ (ℓ - i - 1))) :\n    intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨j * 2 + 1, by\n      apply mul_two_add_bit_lt_two_pow j (ℓ-i-1) (ℓ-i) ⟨1, by omega⟩ (by omega) (by omega)\n    ⟩  = X * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by omega⟩ ⟨j, by\n      apply lt_two_pow_of_lt_two_pow_exp_le j (ℓ-i-1) (ℓ-(i+1)) (by omega) (by omega)\n    ⟩).comp (qMap 𝔽q β ⟨i, by omega⟩)"}, {"name": "AdditiveNTT.evaluation_poly_split_identity", "content": "theorem evaluation_poly_split_identity (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) :\n  let P_i: L[X] := intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ coeffs\n  let P_even_i_plus_1: L[X] := evenRefinement 𝔽q β h_ℓ_add_R_rate i coeffs\n  let P_odd_i_plus_1: L[X] := oddRefinement 𝔽q β h_ℓ_add_R_rate i coeffs\n  let q_i: L[X] := qMap 𝔽q β ⟨i, by omega⟩\n  P_i = (P_even_i_plus_1.comp q_i) + X * (P_odd_i_plus_1.comp q_i)"}, {"name": "AdditiveNTT.intermediate_poly_P_base", "content": "lemma intermediate_poly_P_base (h_ℓ : ℓ ≤ r) (coeffs : Fin (2 ^ ℓ) → L) :\n  intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate ⟨0, by omega⟩ coeffs =\n    polynomialFromNovelCoeffs 𝔽q β ℓ h_ℓ coeffs"}, {"name": "AdditiveNTT.evaluationPointω_eq_twiddleFactor_of_div_2", "content": "omit [DecidableEq L] [DecidableEq 𝔽q] [Fintype 𝔽q] h_Fq_char_prime hF₂ hβ_lin_indep h_β₀_eq_1 in\nlemma evaluationPointω_eq_twiddleFactor_of_div_2 (i : Fin ℓ) (x : Fin (2 ^ (ℓ + R_rate - i))) :\n  evaluationPointω 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ x =\n  twiddleFactor 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨x/2, by\n    have h := div_two_pow_lt_two_pow (x:=x) (i := ℓ + R_rate - i - 1) (j:=1) (by\n      rw [Nat.sub_add_cancel (by omega)]; omega)\n    simp only [pow_one] at h\n    calc _ < 2 ^ (ℓ + R_rate - i - 1) := by omega\n      _ = _"}, {"name": "AdditiveNTT.eval_point_ω_eq_next_twiddleFactor_comp_qmap", "content": "omit [DecidableEq 𝔽q] hF₂ h_β₀_eq_1 in\nlemma eval_point_ω_eq_next_twiddleFactor_comp_qmap\n\n  (i : Fin ℓ) (x : Fin (2 ^ (ℓ + R_rate - (i + 1)))) :\n  -- `j = u||b||v` => x here means u at level i\n  evaluationPointω 𝔽q β h_ℓ_add_R_rate ⟨i.val+1, by omega⟩ x =\n  eval (twiddleFactor 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ ⟨x.val, by\n    calc x.val < 2 ^ (ℓ + R_rate - (i.val + 1)) := by omega\n      _ = 2 ^ (ℓ + R_rate - i.val - 1) := by rfl\n  ⟩) (qMap 𝔽q β ⟨i, by omega⟩)"}, {"name": "AdditiveNTT.base_coeffsBySuffix", "content": "omit [NeZero r] [Field L] [Fintype L] [DecidableEq L] [DecidableEq 𝔽q] [Field 𝔽q] [Algebra 𝔽q L] in\nlemma base_coeffsBySuffix (a : Fin (2 ^ ℓ) → L) :\n  coeffsBySuffix (r:=r) (R_rate := R_rate) a 0 0 = a"}, {"name": "AdditiveNTT.evenRefinement_eq_novel_poly_of_0_leading_suffix", "content": "theorem evenRefinement_eq_novel_poly_of_0_leading_suffix (i : Fin ℓ) (v : Fin (2 ^ i.val))\n    (original_coeffs : Fin (2 ^ ℓ) → L) :\n    have h_v: v.val < 2 ^ (i.val + 1) := by\n      calc v.val < 2 ^ i.val := by omega\n        _ < 2 ^ (i.val + 1) := by apply Nat.pow_lt_pow_right (by omega) (by omega)\n    evenRefinement 𝔽q β h_ℓ_add_R_rate i (coeffsBySuffix (r:=r)\n      (R_rate:=R_rate) (a:=original_coeffs) ⟨i, by omega⟩ v) =\n    intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate ⟨i + 1, by omega⟩\n      (coeffsBySuffix (r:=r) (R_rate:=R_rate) original_coeffs ⟨i + 1, by omega⟩ ⟨v, h_v⟩)"}, {"name": "AdditiveNTT.oddRefinement_eq_novel_poly_of_1_leading_suffix", "content": "theorem oddRefinement_eq_novel_poly_of_1_leading_suffix (i : Fin ℓ) (v : Fin (2 ^ i.val))\n    (original_coeffs : Fin (2 ^ ℓ) → L) :\n    have h_v: v.val ||| (1 <<< i.val) < 2 ^ (i.val + 1)"}, {"name": "AdditiveNTT.initial_tiled_coeffs_correctness", "content": "omit [DecidableEq 𝔽q] hF₂ in\nlemma initial_tiled_coeffs_correctness (h_ℓ : ℓ ≤ r) (a : Fin (2 ^ ℓ) → L) :\n    let b: Fin (2^(ℓ + R_rate)) → L := tileCoeffs a\n    additiveNTTInvariant 𝔽q β h_ℓ_add_R_rate b a (i := ⟨ℓ, by omega⟩)"}, {"name": "AdditiveNTT.NTTStage_correctness", "content": "lemma NTTStage_correctness (i : Fin (ℓ))\n    (input_buffer : Fin (2 ^ (ℓ + R_rate)) → L) (original_coeffs : Fin (2 ^ ℓ) → L) :\n    additiveNTTInvariant 𝔽q β h_ℓ_add_R_rate (evaluation_buffer:=input_buffer)\n      (original_coeffs:=original_coeffs) (i := ⟨i.val+1, by omega⟩) →\n    additiveNTTInvariant 𝔽q β h_ℓ_add_R_rate (evaluation_buffer:=NTTStage 𝔽q β h_ℓ_add_R_rate\n      ⟨i, by omega⟩ input_buffer) (original_coeffs:=original_coeffs) ⟨i, by omega⟩"}, {"name": "AdditiveNTT.foldl_NTTStage_inductive_aux", "content": "lemma foldl_NTTStage_inductive_aux (h_ℓ : ℓ ≤ r) (k : Fin (ℓ + 1))\n    (original_coeffs : Fin (2 ^ ℓ) → L) :\n    additiveNTTInvariant 𝔽q β h_ℓ_add_R_rate\n    (Fin.foldl k (fun current_b i ↦ NTTStage 𝔽q β h_ℓ_add_R_rate\n      ⟨ℓ - i -1, by omega⟩ current_b) (tileCoeffs original_coeffs))\n    original_coeffs ⟨ℓ - k, by omega⟩"}], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis\n\nimport Mathlib.Tactic\n\nimport Mathlib.Data.Finsupp.Defs\n\nimport Mathlib.LinearAlgebra.LinearIndependent.Defs\n\nopen Polynomial AdditiveNTT Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\nsection IntermediateStructures\n\nnoncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))\n\nnoncomputable def qCompositionChain (i : Fin r) : L[X] :=\n  match i with\n  | ⟨0, _⟩ => X\n  | ⟨k + 1, h_k_add_1⟩ => (qMap 𝔽q β ⟨k, by admit /- proof elided -/\n  ⟩).comp (qCompositionChain ⟨k, by admit /- proof elided -/\n  ⟩)\n\nnoncomputable section DomainBijection\n\nend DomainBijection\n\nnoncomputable def intermediateNormVpoly\n    \n    (i: Fin (ℓ+1)) (k : Fin (ℓ - i + 1)) : L[X] :=\n  \n  Fin.foldl (n:=k) (fun acc j =>\n    (qMap 𝔽q β ⟨(i : ℕ) + (j : ℕ), by admit /- proof elided -/\n    ⟩).comp acc) (X)\n\nnoncomputable def intermediateNovelBasisX (i : Fin (ℓ + 1)) (j : Fin (2 ^ (ℓ - i))) : L[X] :=\n  (Finset.univ: Finset (Fin (ℓ - i)) ).prod (fun k =>\n    (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate i (k:=⟨k, by admit /- proof elided -/\n    ⟩)) ^ (Nat.getBit k j))\n\nnoncomputable def intermediateEvaluationPoly (i : Fin (ℓ + 1))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i))), C (coeffs ⟨j, by admit /- proof elided -/\n  ⟩) *\n    (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate i ⟨j, by admit /- proof elided -/\n    ⟩)\n\nnoncomputable def evenRefinement (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i-1))), C (coeffs ⟨j*2, by admit /- proof elided -/\n  ⟩) * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by admit /- proof elided -/\n  ⟩ ⟨j, hj⟩)\n\nnoncomputable def oddRefinement (i : Fin (ℓ))\n    (coeffs : Fin (2 ^ (ℓ - i)) → L) : L[X] :=\n  ∑ (⟨j, hj⟩: Fin (2^(ℓ-i-1))), C (coeffs ⟨j*2+1, by admit /- proof elided -/\n  ⟩) * (intermediateNovelBasisX 𝔽q β h_ℓ_add_R_rate ⟨i+1, by admit /- proof elided -/\n  ⟩ ⟨j, hj⟩)\n\nend IntermediateStructures\n\nsection AlgorithmCorrectness\n\nnoncomputable def evaluationPointω (i : Fin (ℓ + 1))\n    (x : Fin (2 ^ (ℓ + R_rate - i))) : L := \n    \n  ∑ (⟨k, hk⟩: Fin (ℓ + R_rate - i)),\n    if Nat.getBit k x.val = 1 then\n      (normalizedW 𝔽q β ⟨i, by admit /- proof elided -/\n      ⟩).eval (β ⟨i + k, by admit /- proof elided -/\n      ⟩)\n    else\n      0\n\nnoncomputable def twiddleFactor (i : Fin ℓ) (u : Fin (2 ^ (ℓ + R_rate - i - 1))) : L :=\n  ∑ (⟨k, hk⟩: Fin (ℓ + R_rate - i - 1)),\n    if Nat.getBit k u.val = 1 then\n      \n        \n      (normalizedW 𝔽q β ⟨i, by admit /- proof elided -/\n      ⟩).eval (β ⟨i + 1 + k, by admit /- proof elided -/\n      ⟩)\n    else 0\n\ndef tileCoeffs (a : Fin (2 ^ ℓ) → L) : Fin (2^(ℓ + R_rate)) → L :=\n  fun v => a (Fin.mk (v.val % (2^ℓ)) (Nat.mod_lt v.val (pow_pos (zero_lt_two) ℓ)))\n\nnoncomputable def NTTStage (i : Fin ℓ) (b : Fin (2 ^ (ℓ + R_rate)) → L) :\n    Fin (2^(ℓ + R_rate)) → L :=\n  have h_2_pow_i_lt_2_pow_ℓ_add_R_rate: 2^i.val < 2^(ℓ + R_rate) := by admit /- proof elided -/\n\nnoncomputable def additiveNTT (a : Fin (2 ^ ℓ) → L) : Fin (2^(ℓ + R_rate)) → L :=\n  let b: Fin (2^(ℓ + R_rate)) → L := tileCoeffs a \n  Fin.foldl (n:=ℓ) (f:= fun current_b i  =>\n    NTTStage 𝔽q β h_ℓ_add_R_rate (i := ⟨ℓ - 1 - i, by admit /- proof elided -/\n    ⟩) current_b\n  ) (init:=b)\n\ndef coeffsBySuffix (a : Fin (2 ^ ℓ) → L) (i : Fin (ℓ + 1)) (v : Fin (2 ^ i.val)) :\n  Fin (2 ^ (ℓ - i)) → L :=\n  fun ⟨j, hj⟩ => by admit /- proof elided -/\n\ndef additiveNTTInvariant (evaluation_buffer : Fin (2 ^ (ℓ + R_rate)) → L)\n    (original_coeffs : Fin (2 ^ ℓ) → L) (i : Fin (ℓ + 1)) : Prop :=\n  ∀ (j : Fin (2^(ℓ + R_rate))),\n    let u_b_v := j.val\n    let v: Fin (2^i.val) := ⟨Nat.getLowBits i.val u_b_v, by admit /- proof elided -/\n    ⟩ \n    let u_b := u_b_v / (2^i.val) \n    have h_u_b : u_b = u_b_v / (2^i.val) := by admit /- proof elided -/", "target_theorem": "theorem additiveNTT_correctness (h_ℓ : ℓ ≤ r)\n    (original_coeffs : Fin (2 ^ ℓ) → L)\n    (output_buffer : Fin (2 ^ (ℓ + R_rate)) → L)\n    (h_alg : output_buffer = additiveNTT 𝔽q β h_ℓ_add_R_rate original_coeffs) :\n    let P :=", "ground_truth_proof": ":= polynomialFromNovelCoeffs 𝔽q β ℓ h_ℓ original_coeffs\n    ∀ (j : Fin (2^(ℓ + R_rate))),\n      output_buffer j = P.eval (evaluationPointω 𝔽q β h_ℓ_add_R_rate ⟨0, by omega⟩ j) :=\n  by\n  simp only [Fin.zero_eta]\n  intro j\n  simp only [h_alg]\n  unfold additiveNTT\n  set output_foldl := Fin.foldl ℓ (fun current_b i ↦ NTTStage 𝔽q β h_ℓ_add_R_rate\n    ⟨ℓ - i -1, by omega⟩ current_b) (tileCoeffs original_coeffs)\n\n  have output_foldl_correctness : additiveNTTInvariant 𝔽q β h_ℓ_add_R_rate\n    output_foldl original_coeffs ⟨0, by omega⟩ := by\n    have res := foldl_NTTStage_inductive_aux 𝔽q β h_ℓ_add_R_rate\n      h_ℓ\n      (k:=⟨ℓ, by omega⟩) original_coeffs\n    simp only [tsub_self, Fin.zero_eta] at res\n    exact res\n\n  have h_nat_point_ω_eq_j: j.val / 2 * 2 + j.val % 2 = j := by\n    have h_j_mod_2_eq_0: j.val % 2 < 2 := by omega\n    exact Nat.div_add_mod' (↑j) 2\n\n  simp only [additiveNTTInvariant] at output_foldl_correctness\n  have res := output_foldl_correctness j\n  unfold output_foldl at res\n  simp only [Fin.zero_eta, Nat.sub_zero, pow_zero, Nat.div_one, Fin.eta,\n    Nat.pow_zero, Nat.getLowBits_zero_eq_zero (n := j.val), Fin.isValue, base_coeffsBySuffix] at res\n  simp only [←\n    intermediate_poly_P_base 𝔽q β h_ℓ_add_R_rate\n      h_ℓ original_coeffs,\n    Fin.zero_eta]\n  rw [←res]\n  simp_rw [Nat.sub_right_comm] -- ℓ - 1 - ↑i = ℓ - ↑i - 1", "nesting_depth": 14, "transitive_dep_count": 317, "subset_aristotle": false, "category": "Applied verif."}
{"id": 35, "thm_name": "InductiveMerkleTree.functional_completeness", "thm_stmt": "theorem functional_completeness (α : Type) {s : Skeleton}\n  (idx : SkeletonLeafIndex s)\n  (leaf_data_tree : LeafData α s)\n  (hash : α → α → α) :\n  (getPutativeRoot_with_hash\n    idx\n    (leaf_data_tree.get idx)\n    (generateProof\n      (buildMerkleTree_with_hash leaf_data_tree hash) idx)\n    (hash)) =\n  (buildMerkleTree_with_hash leaf_data_tree hash).getRootValue", "lean_root": "ArkLib", "rel_path": "ArkLib/CommitmentScheme/InductiveMerkleTree.lean", "imports": ["import ArkLib.ToMathlib.Data.IndexedBinaryTree.Basic", "import Mathlib.Data.Vector.Snoc", "import ArkLib.CommitmentScheme.Basic", "import VCVio", "import ArkLib.ToVCVio.Oracle"], "used_lib_defs": [{"name": "Repr", "module": "Init.Data.Repr"}, {"name": "List", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "FullData.leftSubtree", "content": "def FullData.leftSubtree {α : Type} {s_left s_right : Skeleton}\n    (tree : FullData α (Skeleton.internal s_left s_right)) :\n    FullData α s_left :=\n  match tree with\n  | FullData.internal _ left _right =>\n    left"}, {"name": "Skeleton", "content": "inductive Skeleton where\n  | leaf : Skeleton\n  | internal : Skeleton → Skeleton → Skeleton"}, {"name": "FullData", "content": "inductive FullData (α : Type) : Skeleton → Type\n  | leaf (value : α) : FullData α Skeleton.leaf\n  | internal {left right} (value : α)\n      (leftData : FullData α left)\n      (rightData : FullData α right) :\n      FullData α (Skeleton.internal left right)\n  deriving Repr"}, {"name": "Skeleton", "content": "inductive Skeleton :\n    (n : ℕ) → (ar : Fin n → ℕ) → Type where\n  | leaf {ar0 : Fin 0 → ℕ} : Skeleton 0 ar0\n  | node {n : ℕ} {ar : Fin (n+1) → ℕ}\n      (children : Fin (ar 0) →\n        Skeleton n (fun i => ar i.succ)) :\n      Skeleton (n+1) ar"}, {"name": "FullData.rightSubtree", "content": "def FullData.rightSubtree {α : Type} {s_left s_right : Skeleton}\n    (tree : FullData α (Skeleton.internal s_left s_right)) :\n    FullData α s_right :=\n  match tree with\n  | FullData.internal _ _left right =>\n    right"}, {"name": "LeafData.rightSubtree", "content": "def LeafData.rightSubtree {α : Type} {s_left s_right : Skeleton}\n    (tree : LeafData α (Skeleton.internal s_left s_right)) :\n    LeafData α s_right :=\n  match tree with\n  | LeafData.internal _left right =>\n    right"}, {"name": "LeafData", "content": "inductive LeafData (α : Type) : Skeleton → Type\n  | leaf (value : α) : LeafData α Skeleton.leaf\n  | internal {left right} (leftData : LeafData α left) (rightData : LeafData α right) :\n      LeafData α (Skeleton.internal left right)\n  deriving Repr"}, {"name": "SkeletonNodeIndex", "content": "inductive SkeletonNodeIndex : Skeleton → Type\n  | ofLeaf : SkeletonNodeIndex Skeleton.leaf\n  | ofInternal {left right} :\n      SkeletonNodeIndex (Skeleton.internal left right)\n  | ofLeft {left right : Skeleton} (idxLeft : SkeletonNodeIndex left) :\n      SkeletonNodeIndex (Skeleton.internal left right)\n  | ofRight {left right : Skeleton} (idxRight : SkeletonNodeIndex right) :\n      SkeletonNodeIndex (Skeleton.internal left right)"}, {"name": "InternalData.rightSubtree", "content": "def InternalData.rightSubtree {α : Type} {s_left s_right : Skeleton}\n    (tree : InternalData α (Skeleton.internal s_left s_right)) :\n    InternalData α s_right :=\n  match tree with\n  | InternalData.internal _ _left right =>\n    right"}, {"name": "InternalData", "content": "inductive InternalData (α : Type) : Skeleton → Type\n  | leaf : InternalData α Skeleton.leaf\n  | internal {left right} (value : α)\n      (leftData : InternalData α left)\n      (rightData : InternalData α right) : InternalData α (Skeleton.internal left right)\n  deriving Repr"}, {"name": "InternalData.leftSubtree", "content": "def InternalData.leftSubtree {α : Type} {s_left s_right : Skeleton}\n    (tree : InternalData α (Skeleton.internal s_left s_right)) :\n    InternalData α s_left :=\n  match tree with\n  | InternalData.internal _ left _right =>\n    left"}, {"name": "SkeletonLeafIndex", "content": "inductive SkeletonLeafIndex : Skeleton → Type\n  | ofLeaf : SkeletonLeafIndex Skeleton.leaf\n  | ofLeft {left right : Skeleton} (idxLeft : SkeletonLeafIndex left) :\n      SkeletonLeafIndex (Skeleton.internal left right)\n  | ofRight {left right : Skeleton} (idxRight : SkeletonLeafIndex right) :\n      SkeletonLeafIndex (Skeleton.internal left right)"}, {"name": "LeafData.leftSubtree", "content": "def LeafData.leftSubtree {α : Type} {s_left s_right : Skeleton}\n    (tree : LeafData α (Skeleton.internal s_left s_right)) :\n    LeafData α s_left :=\n  match tree with\n  | LeafData.internal left _right =>\n    left"}, {"name": "FullData.getRootValue", "content": "def FullData.getRootValue {s} {α : Type} (tree : FullData α s) :=\n  tree.get (getRootIndex s)"}, {"name": "FullData.get", "content": "def FullData.get {s} {α : Type}\n    (tree : FullData α s) (idx : SkeletonNodeIndex s) : α :=\n  match tree, idx with\n  | FullData.leaf value, SkeletonNodeIndex.ofLeaf => value\n  | FullData.internal value _ _, SkeletonNodeIndex.ofInternal => value\n  | FullData.internal _ left _, SkeletonNodeIndex.ofLeft idxLeft =>\n    FullData.get left idxLeft\n  | FullData.internal _ _ right, SkeletonNodeIndex.ofRight idxRight =>\n    FullData.get right idxRight"}, {"name": "InternalData.get", "content": "def InternalData.get {s} {α : Type}\n    (tree : InternalData α s) (idx : SkeletonInternalIndex s) : α :=\n  match tree, idx with\n  | InternalData.internal value _ _, SkeletonInternalIndex.ofInternal => value\n  | InternalData.internal _ left _, SkeletonInternalIndex.ofLeft idxLeft =>\n    InternalData.get left idxLeft\n  | InternalData.internal _ _ right, SkeletonInternalIndex.ofRight idxRight =>\n    InternalData.get right idxRight"}, {"name": "SkeletonInternalIndex", "content": "inductive SkeletonInternalIndex : Skeleton → Type\n  | ofInternal {left right} : SkeletonInternalIndex (Skeleton.internal left right)\n  | ofLeft {left right : Skeleton} (idxLeft : SkeletonInternalIndex left) :\n      SkeletonInternalIndex (Skeleton.internal left right)\n  | ofRight {left right : Skeleton} (idxRight : SkeletonInternalIndex right) :\n      SkeletonInternalIndex (Skeleton.internal left right)"}, {"name": "LeafData.get", "content": "def LeafData.get {s} {α : Type}\n    (tree : LeafData α s) (idx : SkeletonLeafIndex s) : α :=\n  match tree, idx with\n  | LeafData.leaf value, SkeletonLeafIndex.ofLeaf => value\n  | LeafData.internal left _, SkeletonLeafIndex.ofLeft idxLeft =>\n    LeafData.get left idxLeft\n  | LeafData.internal _ right, SkeletonLeafIndex.ofRight idxRight =>\n    LeafData.get right idxRight"}, {"name": "getRootIndex", "content": "def getRootIndex (s : Skeleton) : SkeletonNodeIndex s := match s with\n  | Skeleton.leaf => SkeletonNodeIndex.ofLeaf\n  | Skeleton.internal _ _ =>\n    SkeletonNodeIndex.ofInternal"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "LeafData.rightSubtree_internal", "content": "@[simp]\ntheorem LeafData.rightSubtree_internal {α} {s_left s_right : Skeleton}\n    (left : LeafData α s_left) (right : LeafData α s_right) :\n    (LeafData.internal left right).rightSubtree = right"}, {"name": "LeafData.leftSubtree_internal", "content": "@[simp]\ntheorem LeafData.leftSubtree_internal {α} {s_left s_right : Skeleton}\n    (left : LeafData α s_left) (right : LeafData α s_right) :\n    (LeafData.internal left right).leftSubtree = left"}, {"name": "LeafData.get_ofLeft", "content": "@[simp]\ntheorem LeafData.get_ofLeft {s_left s_right : Skeleton} {α : Type}\n    (tree : LeafData α (Skeleton.internal s_left s_right))\n    (idxLeft : SkeletonLeafIndex s_left) :\n    tree.get (SkeletonLeafIndex.ofLeft idxLeft) =\n      tree.leftSubtree.get idxLeft"}, {"name": "LeafData.get_ofRight", "content": "@[simp]\ntheorem LeafData.get_ofRight {s_left s_right : Skeleton} {α : Type}\n    (tree : LeafData α (Skeleton.internal s_left s_right))\n    (idxRight : SkeletonLeafIndex s_right) :\n    tree.get (SkeletonLeafIndex.ofRight idxRight) =\n      tree.rightSubtree.get idxRight"}, {"name": "FullData.internal_getRootValue", "content": "@[simp]\ntheorem FullData.internal_getRootValue {s_left s_right : Skeleton} {α : Type}\n    (value : α) (left : FullData α s_left) (right : FullData α s_right) :\n    (FullData.internal value left right).getRootValue =\n      value"}], "used_local_defs": [{"name": "InductiveMerkleTree.buildMerkleTree_with_hash", "content": "def buildMerkleTree_with_hash {s} (leaf_tree : LeafData α s) (hashFn : α → α → α) :\n    (FullData α s) :=\n  match leaf_tree with\n  | LeafData.leaf a => FullData.leaf a\n  | LeafData.internal left right =>\n    let leftTree := buildMerkleTree_with_hash left hashFn\n    let rightTree := buildMerkleTree_with_hash right hashFn\n    let rootHash := hashFn (leftTree.getRootValue) (rightTree.getRootValue)\n    FullData.internal rootHash leftTree rightTree"}, {"name": "InductiveMerkleTree.generateProof", "content": "def generateProof {s} (cache_tree : FullData α s) :\n    BinaryTree.SkeletonLeafIndex s → List α\n  | .ofLeaf => []\n  | .ofLeft idxLeft =>\n    (cache_tree.rightSubtree).getRootValue ::\n      (generateProof cache_tree.leftSubtree idxLeft)\n  | .ofRight idxRight =>\n    (cache_tree.leftSubtree).getRootValue ::\n      (generateProof cache_tree.rightSubtree idxRight)"}, {"name": "InductiveMerkleTree.getPutativeRoot_with_hash", "content": "def getPutativeRoot_with_hash {s} (idx : BinaryTree.SkeletonLeafIndex s)\n    (leafValue : α) (proof : List α) (hashFn : α → α → α) : α :=\n  match proof with\n  | [] => leafValue \n  | siblingBelowRootHash :: restProof =>\n    match idx with\n    | BinaryTree.SkeletonLeafIndex.ofLeaf =>\n      \n      \n      \n      leafValue\n    | BinaryTree.SkeletonLeafIndex.ofLeft idxLeft =>\n      \n      hashFn (getPutativeRoot_with_hash idxLeft leafValue restProof hashFn) siblingBelowRootHash\n    | BinaryTree.SkeletonLeafIndex.ofRight idxRight =>\n      \n      hashFn siblingBelowRootHash (getPutativeRoot_with_hash idxRight leafValue restProof hashFn)"}], "used_local_lemmas": [{"name": "InductiveMerkleTree.generateProof_ofLeft", "content": "@[simp]\ntheorem generateProof_ofLeft {sleft sright : Skeleton}\n    (cache_tree : FullData α (Skeleton.internal sleft sright))\n    (idxLeft : SkeletonLeafIndex sleft) :\n    generateProof cache_tree (BinaryTree.SkeletonLeafIndex.ofLeft idxLeft) =\n      (cache_tree.rightSubtree).getRootValue ::\n        (generateProof cache_tree.leftSubtree idxLeft)"}, {"name": "InductiveMerkleTree.generateProof_ofRight", "content": "@[simp]\ntheorem generateProof_ofRight {sleft sright : Skeleton}\n    (cache_tree : FullData α (Skeleton.internal sleft sright))\n    (idxRight : SkeletonLeafIndex sright) :\n    generateProof cache_tree (BinaryTree.SkeletonLeafIndex.ofRight idxRight) =\n      (cache_tree.leftSubtree).getRootValue ::\n        (generateProof cache_tree.rightSubtree idxRight)"}], "local_ctx": "import VCVio\n\nimport ArkLib.ToMathlib.Data.IndexedBinaryTree.Basic\n\nimport ArkLib.CommitmentScheme.Basic\n\nimport Mathlib.Data.Vector.Snoc\n\nimport ArkLib.ToVCVio.Oracle\n\nnamespace InductiveMerkleTree\n\nopen List OracleSpec OracleComp BinaryTree\n\nsection spec\n\nvariable (α : Type)\n\nend spec\n\nvariable {α : Type}\n\ndef buildMerkleTree_with_hash {s} (leaf_tree : LeafData α s) (hashFn : α → α → α) :\n    (FullData α s) :=\n  match leaf_tree with\n  | LeafData.leaf a => FullData.leaf a\n  | LeafData.internal left right =>\n    let leftTree := buildMerkleTree_with_hash left hashFn\n    let rightTree := buildMerkleTree_with_hash right hashFn\n    let rootHash := hashFn (leftTree.getRootValue) (rightTree.getRootValue)\n    FullData.internal rootHash leftTree rightTree\n\ndef generateProof {s} (cache_tree : FullData α s) :\n    BinaryTree.SkeletonLeafIndex s → List α\n  | .ofLeaf => []\n  | .ofLeft idxLeft =>\n    (cache_tree.rightSubtree).getRootValue ::\n      (generateProof cache_tree.leftSubtree idxLeft)\n  | .ofRight idxRight =>\n    (cache_tree.leftSubtree).getRootValue ::\n      (generateProof cache_tree.rightSubtree idxRight)\n\ndef getPutativeRoot_with_hash {s} (idx : BinaryTree.SkeletonLeafIndex s)\n    (leafValue : α) (proof : List α) (hashFn : α → α → α) : α :=\n  match proof with\n  | [] => leafValue \n  | siblingBelowRootHash :: restProof =>\n    match idx with\n    | BinaryTree.SkeletonLeafIndex.ofLeaf =>\n      \n      \n      \n      leafValue\n    | BinaryTree.SkeletonLeafIndex.ofLeft idxLeft =>\n      \n      hashFn (getPutativeRoot_with_hash idxLeft leafValue restProof hashFn) siblingBelowRootHash\n    | BinaryTree.SkeletonLeafIndex.ofRight idxRight =>\n      \n      hashFn siblingBelowRootHash (getPutativeRoot_with_hash idxRight leafValue restProof hashFn)", "target_theorem": "theorem functional_completeness (α : Type) {s : Skeleton}\n  (idx : SkeletonLeafIndex s)\n  (leaf_data_tree : LeafData α s)\n  (hash : α → α → α) :\n  (getPutativeRoot_with_hash\n    idx\n    (leaf_data_tree.get idx)\n    (generateProof\n      (buildMerkleTree_with_hash leaf_data_tree hash) idx)\n    (hash)) =\n  (buildMerkleTree_with_hash leaf_data_tree hash).getRootValue :=", "ground_truth_proof": ":= by\n  induction s with\n  | leaf =>\n    match leaf_data_tree with\n    | LeafData.leaf a =>\n      cases idx with\n      | ofLeaf =>\n        simp [buildMerkleTree_with_hash, getPutativeRoot_with_hash]\n  | internal s_left s_right left_ih right_ih =>\n    match leaf_data_tree with\n    | LeafData.internal left right =>\n      cases idx with\n      | ofLeft idxLeft =>\n        simp_rw [LeafData.get_ofLeft, LeafData.leftSubtree_internal, buildMerkleTree_with_hash,\n          generateProof_ofLeft, FullData.rightSubtree, FullData.leftSubtree,\n          getPutativeRoot_with_hash, left_ih, FullData.internal_getRootValue]\n      | ofRight idxRight =>\n        simp_rw [LeafData.get_ofRight, LeafData.rightSubtree_internal, buildMerkleTree_with_hash,\n          generateProof_ofRight, FullData.leftSubtree, FullData.rightSubtree,\n          getPutativeRoot_with_hash, right_ih, FullData.internal_getRootValue]", "nesting_depth": 4, "transitive_dep_count": 31, "subset_aristotle": false, "category": "Applied verif."}
{"id": 36, "thm_name": "ConcreteBinaryTower.aeval_definingPoly_at_Z_succ", "thm_stmt": "lemma aeval_definingPoly_at_Z_succ (k : ℕ) :\n  (aeval (Z (k + 1))) (definingPoly (s:=Z (k))) = 0", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "AddCommGroup", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.negSucc", "module": "Init.Data.Int.Basic"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "NNRat", "module": "Mathlib.Data.Rat.Init"}, {"name": "NNRat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "Rat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.toAlgebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "invFun", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "left_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "right_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "absurd", "module": "Init.Prelude"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "instFintypeProd", "module": "Mathlib.Data.Fintype.Prod"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Polynomial.aeval", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Algebra.algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}], "lib_lemmas": [{"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}, {"name": "Nat.ne_zero_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.one_lt_two_pow_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_eq_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.one_mod_two_pow_eq_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.zero_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "Nat.one_mod_two_pow", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.zero_lt_two", "module": "Init.Data.Nat.Basic"}, {"name": "pow_pos", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "BitVec.zero_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "pow_two", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "BitVec.ofNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.toNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Ne.dite_eq_left_iff", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.left_eq_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingHom.comp_id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "BitVec.xor_assoc", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_self", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Polynomial.C_mul'", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.aeval_def", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.eval₂_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval₂_X_pow", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval₂_add", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval₂_one", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval₂_smul", "module": "Mathlib.Algebra.Polynomial.Eval.SMul"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}, {"name": "zero_lt_pow_n", "content": "theorem zero_lt_pow_n (m : ℕ) (n : ℕ) (h_m : m > 0): 0 < m^n"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.add", "content": "def add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y"}, {"name": "ConcreteBinaryTower.neg", "content": "def neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.ConcreteBTFAddCommGroupProps", "content": "structure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel"}, {"name": "ConcreteBinaryTower.mkAddCommGroupInstance", "content": "def mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}"}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.equivProd", "content": "def equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)"}, {"name": "ConcreteBinaryTower.concrete_mul", "content": "def concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.concrete_inv", "content": "def concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res"}, {"name": "ConcreteBinaryTower.natCast", "content": "def natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.natCast_zero", "content": "def natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :="}, {"name": "ConcreteBinaryTower.natCast_succ", "content": "def natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :="}, {"name": "ConcreteBinaryTower.intCast", "content": "def intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.intCast_ofNat", "content": "def intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :="}, {"name": "ConcreteBinaryTower.intCast_negSucc", "content": "def intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :="}, {"name": "ConcreteBinaryTower.ConcreteBTFRingProps", "content": "structure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c"}, {"name": "ConcreteBinaryTower.ConcreteBTFDivisionRingProps", "content": "structure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldProps", "content": "structure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a"}, {"name": "ConcreteBinaryTower.mkRingInstance", "content": "def mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n"}, {"name": "ConcreteBinaryTower.mkDivisionRingInstance", "content": "def mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)"}, {"name": "ConcreteBinaryTower.mkFieldInstance", "content": "def mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm"}, {"name": "ConcreteBinaryTower.ConcreteBTFStepResult", "content": "structure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))"}, {"name": "ConcreteBinaryTower.liftBTFieldProps", "content": "def liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.liftConcreteBTField", "content": "def liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.concreteCanonicalEmbedding", "content": "def concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :="}, {"name": "ConcreteBinaryTower.instAlgebraLiftConcreteBTField", "content": "instance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))"}, {"name": "ConcreteBinaryTower.getBTFResult", "content": "def getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.canonicalAlgMap", "content": "def canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap", "content": "def concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :="}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq_zero", "content": "theorem BitVec.dcast_bitvec_eq_zero {l r : ℕ} (h_width_eq : l = r) :\n  dcast (h_width_eq) 0#(l) = 0#(r)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.sum_fromNat_eq_from_xor_Nat", "content": "theorem sum_fromNat_eq_from_xor_Nat {k : ℕ} (x y : Nat) :\n  fromNat (k:=k) (x ^^^ y) = fromNat (k:=k) x + fromNat (k:=k) y"}, {"name": "ConcreteBinaryTower.add_self_cancel", "content": "lemma add_self_cancel {k : ℕ} (a : ConcreteBTField k) : a + a = 0"}, {"name": "ConcreteBinaryTower.add_assoc", "content": "lemma add_assoc {k : ℕ} : ∀ (a b c : ConcreteBTField k), a + b + c = a + (b + c)"}, {"name": "ConcreteBinaryTower.zero_add", "content": "lemma zero_add {k : ℕ} (a : ConcreteBTField k) : 0 + a = a"}, {"name": "ConcreteBinaryTower.add_zero", "content": "lemma add_zero {k : ℕ} (a : ConcreteBTField k) : a + 0 = a"}, {"name": "ConcreteBinaryTower.zero_is_0", "content": "lemma zero_is_0 {k : ℕ} : (zero (k:=k)) = (0 : ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.one_is_1", "content": "lemma one_is_1 {k : ℕ} : (one (k:=k)) = 1"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_of_split", "content": "theorem join_of_split {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_split_eq : split h_pos x = (hi_btf, lo_btf)) :\n    x = 《 hi_btf, lo_btf 》"}, {"name": "ConcreteBinaryTower.split_of_join", "content": "theorem split_of_join {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_join : x = 《hi_btf, lo_btf》) :\n    (hi_btf, lo_btf) = split h_pos x"}, {"name": "ConcreteBinaryTower.eq_iff_split_eq", "content": "theorem eq_iff_split_eq {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k) :\n  x₀ = x₁ ↔ (split h_pos x₀ = split h_pos x₁)"}, {"name": "ConcreteBinaryTower.split_sum_eq_sum_split", "content": "theorem split_sum_eq_sum_split {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k)\n  (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1))\n  (h_split_x₀ : split h_pos x₀ = (hi₀, lo₀))\n  (h_split_x₁ : split h_pos x₁ = (hi₁, lo₁)) :\n  split h_pos (x₀ + x₁) = (hi₀ + hi₁, lo₀ + lo₁)"}, {"name": "ConcreteBinaryTower.join_add_join", "content": "theorem join_add_join {k : ℕ} (h_pos : k > 0) (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1)) :\n  《 hi₀, lo₀ 》 + 《 hi₁, lo₁ 》 = 《 hi₀ + hi₁, lo₀ + lo₁ 》"}, {"name": "ConcreteBinaryTower.one_bitvec_toNat", "content": "lemma one_bitvec_toNat {width : ℕ} (h_width : width > 0) : (1#width).toNat = 1"}, {"name": "ConcreteBinaryTower.one_bitvec_shiftRight", "content": "lemma one_bitvec_shiftRight {d : ℕ} (h_d : d > 0) : 1 >>> d = 0"}, {"name": "ConcreteBinaryTower.split_one", "content": "lemma split_one {k : ℕ} (h_k : k > 0) :\n    split h_k (one (k:=k)) = (zero (k:=k - 1), one (k:=k - 1))"}, {"name": "ConcreteBinaryTower.join_zero_one", "content": "theorem join_zero_one {k : ℕ} (h_k : k > 0) :\n    《 zero (k:=k - 1), one (k:=k - 1) 》 = one (k:=k)"}, {"name": "ConcreteBinaryTower.Z_square_eq", "content": "lemma Z_square_eq (k : ℕ) (prevBTFieldProps : ConcreteBTFieldProps (k := k))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI : Field (ConcreteBTField (k + 1)) := mkFieldInstance curBTFieldProps\n  (Z (k + 1)) ^ 2 = 《 Z (k), 1 》"}, {"name": "ConcreteBinaryTower.Z_square_mul_form", "content": "lemma Z_square_mul_form\n  (k : ℕ)\n  (prev : ConcreteBTFStepResult (k := k)) :\n  letI : Field (ConcreteBTField k) := mkFieldInstance (prev.toConcreteBTFieldProps)\n  letI : Field (ConcreteBTField (k + 1)) := mkFieldInstance (k:=k+1)\n    (props:=liftBTFieldProps (k:=k) (prevBTFResult:=prev))\n  letI : Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n    instAlgebraLiftConcreteBTField k prev\n  Z (k + 1) ^ 2\n    = Z (k + 1)\n      * (algebraMap (ConcreteBTField k) (ConcreteBTField (k + 1))) (Z k)\n      + 1"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_id", "content": "lemma concreteTowerAlgebraMap_id (k : ℕ) :\n    concreteTowerAlgebraMap (h_le:=by omega) = RingHom.id (ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_succ_1", "content": "lemma concreteTowerAlgebraMap_succ_1 (k : ℕ) :\n  concreteTowerAlgebraMap (l:=k) (r:=k + 1) (h_le:=by omega) = canonicalAlgMap k"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y\n\ndef neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\nstructure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel\n\ndef mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\ndef equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)\n\ndef concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/\n\ndef concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res\n\nsection FieldLemmasOfLevel0\n\nend FieldLemmasOfLevel0\n\nsection NumericCasting\n\ndef natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :=\n\ndef natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :=\n\ndef intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :=\n\ndef intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :=\n\nend NumericCasting\n\nstructure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c\n\nstructure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one\n\nstructure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a\n\ndef mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n\n\ndef mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)\n\ndef mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm\n\nstructure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))\n\nend FieldOperationsAndInstances\n\nsection BTFieldPropsOneLevelLiftingLemmas\n\nvariable {k : ℕ} {h_k : k > 0}\n\nend BTFieldPropsOneLevelLiftingLemmas\n\nsection TowerFieldsConstruction\n\ndef liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/\n\ndef liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :=\n\ndef concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :=\n\ninstance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))\n\ndef getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/\n\nend TowerFieldsConstruction\n\nsection ConcreteBTFieldAlgebraConstruction\n\ndef canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))\n\ndef concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :=", "target_theorem": "lemma aeval_definingPoly_at_Z_succ (k : ℕ) :\n  (aeval (Z (k + 1))) (definingPoly (s:=Z (k))) = 0 :=", "ground_truth_proof": ":= by\n  rw [aeval_def]\n  set f := algebraMap (ConcreteBTField k) (ConcreteBTField (k + 1))\n  have h_f_is_canonical_embedding :\n    f = concreteTowerAlgebraMap (l:=k) (r:=k+1) (h_le:=by omega) := by rfl\n  rw [definingPoly, eval₂_add, eval₂_add] -- break down into sum of terms\n  rw [eval₂_X_pow]\n  rw [C_mul']\n  -- ⊢ Z (k + 1) ^ 2 + eval₂ f (Z (k + 1)) (Z k • X) + eval₂ f (Z (k + 1)) 1 = 0\n  simp only [eval₂_one, eval₂_smul, eval₂_X]\n  -- Z_square_mul_form uses instAlgebraLiftConcreteBTField internally\n  rw [Z_square_mul_form (k:=k) (prev:=(getBTFResult (k:=k)))]\n  rw [add_assoc]\n  rw [algebraMap, Algebra.algebraMap, instAlgebraLiftConcreteBTField]\n  simp only\n  -- f uses ConcreteBTFieldAlgebra, it's same as instAlgebraLiftConcreteBTField at step = 1\n  rw [h_f_is_canonical_embedding, concreteTowerAlgebraMap_succ_1]\n  simp only [canonicalAlgMap]; rw [mul_comm]\n  rw [add_self_cancel]", "nesting_depth": 10, "transitive_dep_count": 257, "subset_aristotle": false, "category": "Applied verif."}
{"id": 37, "thm_name": "AdditiveNTT.inductive_linear_map_W", "thm_stmt": "omit hF₂ in\nlemma inductive_linear_map_W (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (i + 1)).comp inner_p)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean", "imports": ["import Mathlib.Algebra.Polynomial.Degree.Definitions", "import ArkLib.Data.Fin.BigOperators", "import ArkLib.Data.Polynomial.MonomialBasis", "import Mathlib.LinearAlgebra.StdBasis", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.Polynomial.Frobenius"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "CommGroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Polynomial.rootMultiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Multiset", "module": "Mathlib.Data.Multiset.Defs"}, {"name": "Multiset.count", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.map", "module": "Mathlib.Data.Multiset.MapFold"}, {"name": "Polynomial.roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "SetLike", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.algEquivOfCompEqX", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "multiplicity", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "EmbeddingLike", "module": "Mathlib.Data.FunLike.Embedding"}, {"name": "CanLift", "module": "Mathlib.Tactic.Lift"}, {"name": "Multiset.filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Finset.val", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Set.InjOn", "module": "Mathlib.Data.Set.Operations"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.val", "module": "Init.Prelude"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.Prime", "module": "Mathlib.Data.Nat.Prime.Defs"}, {"name": "ringChar", "module": "Mathlib.Algebra.CharP.Defs"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}], "lib_lemmas": [{"name": "Fact.out", "module": "Mathlib.Logic.Basic"}, {"name": "Fin.le_zero_iff'", "module": "Mathlib.Data.Fin.SuccPred"}, {"name": "LinearIndependent.injective", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Nat.not_lt_zero", "module": "Init.Prelude"}, {"name": "Set.Ico_eq_empty_iff", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.compl_eq_univ_diff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Set.empty_subset", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.image_empty", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_subset_image_iff", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.subset_compl_singleton_iff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Submodule.span_mono", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "linearIndependent_iff_notMem_span", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Finset.prod_ne_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Polynomial.eval_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_prod", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_sub", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Polynomial.splits_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.splits_prod", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Finset.prod_eq_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Polynomial.X_sub_C_ne_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.X_ne_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_C_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_X_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_sub", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.comp_eq_zero_iff", "module": "Mathlib.Algebra.Polynomial.Degree.Lemmas"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "map_neg", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "sub_eq_neg_self", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "EmbeddingLike.map_eq_zero_iff", "module": "Mathlib.Algebra.Group.Equiv.Defs"}, {"name": "Polynomial.aeval_C", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.aeval_X", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algEquivOfCompEqX_apply", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algebraMap_eq", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.rootMultiplicity_eq_multiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "if_false", "module": "Init.ByCases"}, {"name": "if_true", "module": "Init.ByCases"}, {"name": "map_sub", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "multiplicity_map_eq", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "sub_sub_sub_cancel_right", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Multiset.countP_eq_card_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_map", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.filter_congr", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.count_roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "eq_sub_iff_add_eq", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.image_val_of_injOn", "module": "Mathlib.Data.Finset.Image"}, {"name": "Finset.prod_image", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Polynomial.roots_prod_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Subtype.val_injective", "module": "Mathlib.Data.Subtype"}, {"name": "CanLift.prf", "module": "Mathlib.Tactic.Lift"}, {"name": "Multiset.card_singleton", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.card_zero", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.count_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_singleton", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.count_univ", "module": "Mathlib.Data.Fintype.Basic"}, {"name": "Multiset.count_zero", "module": "Mathlib.Data.Multiset.Count"}, {"name": "SetLike.coe_eq_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "SetLike.mem_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Fin.zero_le", "module": "Init.Data.Fin.Lemmas"}, {"name": "Set.Ico_subset_Ico_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_mono", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_image_of_mem", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.add_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.smul_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.subset_span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "sub_add_cancel", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "Set.Ico_insert_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_singleton", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_union", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Icc", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.union_singleton", "module": "Mathlib.Data.Set.Insert"}, {"name": "Submodule.mem_span_singleton", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.mem_sup", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.smul_mem_iff", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Submodule.span_union", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.sub_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "existsUnique_of_exists_of_unique", "module": "Mathlib.Logic.ExistsUnique"}, {"name": "sub_smul", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "sub_sub_sub_cancel_left", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_const_zero", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.sum_ite_eq'", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise"}, {"name": "Finset.sum_map_val", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Multiset.count_bind", "module": "Mathlib.Data.Multiset.Bind"}, {"name": "Multiset.count_map_eq_count'", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.roots_prod", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "add_left_injective", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "if_false_right", "module": "Init.PropLemmas"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "iff_false", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "Polynomial.monic_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.monic_prod_of_monic", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.Monic.comp", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.Splits.comp_of_degree_le_one", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.degree_X_sub_C_le", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eq_prod_roots_of_monic_of_splits_id", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.natDegree_X", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.natDegree_sub_C", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.comp_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_right_inj", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_add_neg", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_right_inj", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "CommGroupWithZero.mul_inv_cancel", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Fin.mk_lt_of_lt_val", "module": "Init.Data.Fin.Lemmas"}, {"name": "Finset.card_univ", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Finset.prod_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.prod_const", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.prod_mul_distrib", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "NeZero.one_le", "module": "Mathlib.Data.Nat.Cast.NeZero"}, {"name": "Polynomial.C_1", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.C_mul", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_pow", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.mul_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.pow_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.smul_C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.smul_eq_C_mul", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.sub_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_sub_cancel_right", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "algebra_compatible_smul", "module": "Mathlib.Algebra.Algebra.Basic"}, {"name": "map_mul", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_pow", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_pow", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_sub", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "one_pow", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_smul", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "pow_sub₀", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "smul_assoc", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "smul_eq_mul", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "smul_sub", "module": "Mathlib.Algebra.GroupWithZero.Action.Defs"}, {"name": "FiniteField.pow_card", "module": "Mathlib.FieldTheory.Finite.Basic"}, {"name": "algebraMap.coe_pow", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "left_distrib", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_sub_left_distrib", "module": "Mathlib.Algebra.Ring.Defs"}], "repo_lemmas": [{"name": "Fin.lt_succ'", "content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}, {"name": "Fin.le_succ", "content": "lemma Fin.le_succ (a : Fin r) (h_a_add_1 : a + 1 < r) : a ≤ a + 1"}, {"name": "Fin.le_iff_lt_succ", "content": "lemma Fin.le_iff_lt_succ (a b : Fin r) (h_b : b + 1 < r) : a ≤ b ↔ a < b + 1"}, {"name": "Fin.val_sub_one", "content": "lemma Fin.val_sub_one (a : Fin r) (h_a_sub_1 : a > 0) : (a - 1).val = a.val - 1"}, {"name": "prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L", "content": "theorem prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L\n  (p : L[X]) :\n  (∏ c ∈ (Finset.univ : Finset Fq), (p - Polynomial.C (algebraMap Fq L c))) =\n    p^(Fintype.card Fq) - p"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X_in_L", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X_in_L :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C (algebraMap Fq L c))) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C c)) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "linear_map_of_comp_to_linear_map_of_eval", "content": "theorem linear_map_of_comp_to_linear_map_of_eval (f : L[X])\n  (h_f_linear : IsLinearMap (R := Fq) (M := L[X]) (M₂ := L[X])\n    (f := fun inner_p ↦ f.comp inner_p)) :\n    IsLinearMap (R := Fq) (M := L) (M₂ := L) (f := fun x ↦ f.eval x)"}, {"name": "frobenius_identity_in_algebra", "content": "theorem frobenius_identity_in_algebra [Fact (Nat.Prime (ringChar Fq))]\n  (f g : L[X]) : (f + g)^(Fintype.card Fq) = f^(Fintype.card Fq) + g^(Fintype.card Fq)"}], "used_local_defs": [{"name": "AdditiveNTT.U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "AdditiveNTT.W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "AdditiveNTT.algEquivAevalXSubC", "content": "@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :="}], "used_local_lemmas": [{"name": "AdditiveNTT.βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "AdditiveNTT.root_U_lift_down", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n  (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n  a ∈ (U 𝔽q β (i+1)) → ∃! x: 𝔽q, a - x • β i ∈ (U 𝔽q β i)"}, {"name": "AdditiveNTT.root_U_lift_up", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime hβ_lin_indep in\ntheorem root_U_lift_up (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) (x : 𝔽q) :\n  a - x • β i ∈ (U 𝔽q β i) → a ∈ (U 𝔽q β (i+1))"}, {"name": "AdditiveNTT.W_monic", "content": "lemma W_monic (i : Fin r) : (W 𝔽q β i).Monic"}, {"name": "AdditiveNTT.W_ne_zero", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W_ne_zero (i : Fin r) : (W 𝔽q β i) ≠ 0"}, {"name": "AdditiveNTT.Wᵢ_eval_βᵢ_neq_zero", "content": "lemma Wᵢ_eval_βᵢ_neq_zero\n    (i : Fin r): (W 𝔽q β i).eval (β i) ≠ 0"}, {"name": "AdditiveNTT.W_splits", "content": "lemma W_splits (i : Fin r) : (W 𝔽q β i).Splits (RingHom.id L)"}, {"name": "AdditiveNTT.roots_W", "content": "lemma roots_W (i : Fin r) : -- converts root Multiset into (univ: Uᵢ.val.map)\n  (W 𝔽q β i).roots = (univ : Finset (U 𝔽q β i)).val.map (fun u => u.val)"}, {"name": "AdditiveNTT.comp_X_sub_C_eq_zero_iff", "content": "omit [Fintype L] [DecidableEq L] in\nlemma comp_X_sub_C_eq_zero_iff (p : L[X]) (a : L) :\n  p.comp (X - C a) = 0 ↔ p = 0"}, {"name": "AdditiveNTT.rootMultiplicity_comp_X_sub_C", "content": "lemma rootMultiplicity_comp_X_sub_C (p : L[X]) (a x : L) :\n    rootMultiplicity x (p.comp (X - C a)) = rootMultiplicity (x - a) p"}, {"name": "AdditiveNTT.roots_comp_X_sub_C", "content": "lemma roots_comp_X_sub_C (p : L[X]) (a : L) :\n    (p.comp (X - C a)).roots = p.roots.map (fun r => r + a)"}, {"name": "AdditiveNTT.Prod_W_comp_X_sub_C_ne_zero", "content": "omit [DecidableEq L] h_Fq_char_prime hF₂ hβ_lin_indep in\nlemma Prod_W_comp_X_sub_C_ne_zero (i : Fin r) :\n    (univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i))) ≠ 0"}, {"name": "AdditiveNTT.rootMultiplicity_W", "content": "lemma rootMultiplicity_W (i : Fin r) (a : L) :\n    rootMultiplicity a (W 𝔽q β i) = if a ∈ (U 𝔽q β i : Set L) then 1 else 0"}, {"name": "AdditiveNTT.rootMultiplicity_prod_W_comp_X_sub_C", "content": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0"}, {"name": "AdditiveNTT.W_prod_comp_decomposition", "content": "lemma W_prod_comp_decomposition\n    (i : Fin r) (hi : i > 0) :\n    (W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))"}, {"name": "AdditiveNTT.comp_sub_C_of_linear_eval", "content": "lemma comp_sub_C_of_linear_eval (p : L[X])\n  (h_lin : IsLinearMap 𝔽q (f := fun inner_p ↦ p.comp inner_p)) (a : L) :\n    p.comp (X - C a) = p - C (eval a p)"}, {"name": "AdditiveNTT.inductive_rec_form_W_comp", "content": "omit h_Fq_char_prime hF₂ in\nlemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])\n      (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)"}], "local_ctx": "import ArkLib.Data.Nat.Bitwise\n\nimport ArkLib.Data.Polynomial.Frobenius\n\nimport ArkLib.Data.Polynomial.MonomialBasis\n\nimport Mathlib.LinearAlgebra.StdBasis\n\nimport Mathlib.Algebra.Polynomial.Degree.Definitions\n\nopen Polynomial FiniteDimensional Finset Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (h_dim : Module.finrank 𝔽q L = r)\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n\nsection LinearSubspaces\n\ndef U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))\n\nnoncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)\n\nend LinearSubspaces\n\nsection LinearityOfSubspaceVanishingPolynomials\n\n@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :=", "target_theorem": "omit hF₂ in\nlemma inductive_linear_map_W (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (i + 1)).comp inner_p) :=", "ground_truth_proof": ":= by\n\n  have h_rec_form := inductive_rec_form_W_comp\n    (hβ_lin_indep := hβ_lin_indep) (h_prev_linear_map := h_prev_linear_map) (i :=i)\n\n  set q := Fintype.card 𝔽q\n  set v := (W 𝔽q β i).eval (β i)\n\n  -- `∀ f(X), f(X) ∈ L[X]`:\n  constructor\n  · intro f g\n    -- 1. Proof flow\n    -- `Wᵢ₊₁(f(X)+g(X)) = Wᵢ(f(X)+g(X))² - v • Wᵢ(f(X)+g(X))` -- h_rec_form\n    -- `= (Wᵢ(f(X)) + Wᵢ(g(X)))² - v • (Wᵢ(f(X)) + Wᵢ(g(X)))`\n    -- `= (Wᵢ(f(X))² + (Wᵢ(g(X)))² - v • Wᵢ(f(X)) - v • Wᵢ(g(X)))` => Freshman's Dream\n    -- `= (Wᵢ(f(X))² - v • Wᵢ(f(X))) + (Wᵢ(g(X))² - v • Wᵢ(g(X)))` -- h_rec_form\n    -- `= Wᵢ₊₁(f(X)) + Wᵢ₊₁(g(X))` -- Q.E.D.\n\n    -- ⊢ (W 𝔽q β (i + 1)).comp (x + y) = (W 𝔽q β (i + 1)).comp x + (W 𝔽q β (i + 1)).comp y\n    calc\n      _ = ((W 𝔽q β i).comp (f + g))^q - C v ^ (q - 1) * ((W 𝔽q β i).comp (f + g)) := by\n        rw [h_rec_form h_i_add_1]\n      _ = ((W 𝔽q β i).comp f)^q + ((W 𝔽q β i).comp g)^q\n        - C v ^ (q - 1) * ((W 𝔽q β i).comp f) - C v ^ (q - 1) * ((W 𝔽q β i).comp g) := by\n        rw [h_prev_linear_map.map_add]\n        rw [Polynomial.frobenius_identity_in_algebra]\n        rw [left_distrib]\n        unfold q\n        abel_nf\n      _ = (((W 𝔽q β i).comp f)^q - C v ^ (q - 1) * ((W 𝔽q β i).comp f))\n        + (((W 𝔽q β i).comp g)^q - C v ^ (q - 1) * ((W 𝔽q β i).comp g)) := by\n        abel_nf\n      _ = (W 𝔽q β (i+1)).comp f + (W 𝔽q β (i+1)).comp g := by\n        unfold q\n        rw [h_rec_form h_i_add_1 f]\n        rw [h_rec_form h_i_add_1 g]\n  · intro c f\n    -- 2. Proof flow\n    -- `Wᵢ₊₁(c • f(X)) = Wᵢ(c • f(X))² - v • Wᵢ(c • f(X))` -- h_rec_form\n    -- `= c² • Wᵢ(f(X))² - v • c • Wᵢ(f(X))`\n    -- `= c • Wᵢ(f(X))² - v • c • Wᵢ(f(X))` via Fermat's Little Theorem (X^q = X)\n    -- `= c • (Wᵢ(f(X))² - v • Wᵢ(f(X)))` -- h_rec_form\n    -- `= c • Wᵢ₊₁(f(X))` -- Q.E.D.\n    have h_c_smul_to_algebraMap_smul: ∀ t: L[X], c • t = (algebraMap 𝔽q L c) • t := by\n      exact algebra_compatible_smul L c\n    have h_c_smul_to_C_algebraMap_mul: ∀ t: L[X], c • t = C (algebraMap 𝔽q L c) * t := by\n      intro t\n      rw [h_c_smul_to_algebraMap_smul]\n      exact smul_eq_C_mul ((algebraMap 𝔽q L) c)\n    -- ⊢ (W 𝔽q β (i + 1)).comp (c • x) = c • (W 𝔽q β (i + 1)).comp x\n    calc\n      _ = ((W 𝔽q β i).comp (c • f))^q - C v ^ (q - 1) * ((W 𝔽q β i).comp (c • f)) := by\n        rw [h_rec_form h_i_add_1 (c • f)]\n      _ = (C (algebraMap 𝔽q L c) * (W 𝔽q β i).comp f)^q\n        - C v ^ (q - 1) * (C (algebraMap 𝔽q L c) * (W 𝔽q β i).comp f) := by\n        rw [h_prev_linear_map.map_smul]\n        rw [mul_pow]\n        simp_rw [h_c_smul_to_C_algebraMap_mul]\n        congr\n        rw [mul_pow]\n      _ = C (algebraMap 𝔽q L (c^q)) * ((W 𝔽q β i).comp f)^q\n        - C v ^ (q - 1) * (C (algebraMap 𝔽q L c) * (W 𝔽q β i).comp f) := by\n        rw [mul_pow]\n        congr -- ⊢ C ((algebraMap 𝔽q L) c) ^ q = C ((algebraMap 𝔽q L) (c ^ q))\n        rw [←C_pow]\n        simp_rw [algebraMap.coe_pow c q]\n      _ = C (algebraMap 𝔽q L (c^q)) * ((W 𝔽q β i).comp f)^q\n        - C v ^ (q - 1) * (C (algebraMap 𝔽q L c) * (W 𝔽q β i).comp f) := by\n        -- use Fermat's Little Theorem (X^q = X)\n        simp only [map_pow]\n      _ = C (algebraMap 𝔽q L (c)) * ((W 𝔽q β i).comp f)^q\n        - C v ^ (q - 1) * (C (algebraMap 𝔽q L c) * (W 𝔽q β i).comp f) := by\n        rw [FiniteField.pow_card]\n      _ = C (algebraMap 𝔽q L c) * (((W 𝔽q β i).comp f)^q\n        - C v ^ (q - 1) * (W 𝔽q β i).comp f) := by\n        rw [←mul_assoc]\n        conv_lhs => rw [mul_comm (a := C v ^ (q - 1)) (b := C (algebraMap 𝔽q L c))]; rw [mul_assoc]\n        exact\n          Eq.symm\n            (mul_sub_left_distrib (C ((algebraMap 𝔽q L) c)) ((W 𝔽q β i).comp f ^ q)\n              (C v ^ (q - 1) * (W 𝔽q β i).comp f))\n      _ = C (algebraMap 𝔽q L c) * (W 𝔽q β (i + 1)).comp f := by\n        rw [h_rec_form h_i_add_1 f]\n      _ = _ := by\n        rw [h_c_smul_to_C_algebraMap_mul]", "nesting_depth": 7, "transitive_dep_count": 238, "subset_aristotle": false, "category": "Applied verif."}
{"id": 38, "thm_name": "ConcreteBinaryTower.join_eq_join_via_add_smul", "thm_stmt": "@[simp]\ntheorem join_eq_join_via_add_smul {k : ℕ} (h_pos : k > 0)\n    (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    《 hi_btf, lo_btf 》 = join_via_add_smul k h_pos hi_btf lo_btf", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "AddCommGroup", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.negSucc", "module": "Init.Data.Int.Basic"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "NNRat", "module": "Mathlib.Data.Rat.Init"}, {"name": "NNRat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "Rat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.toAlgebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "invFun", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "left_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "right_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "absurd", "module": "Init.Prelude"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "instFintypeProd", "module": "Mathlib.Data.Fintype.Prod"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "CommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}, {"name": "Algebra.algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "MonoidHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "OneHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}, {"name": "AlgebraTower.toIsScalarTower", "content": "@[simp]\ninstance AlgebraTower.toIsScalarTower (a : AlgebraTower C) {i j k : ι}\n    (h1 : i ≤ j) (h2 : j ≤ k) :\n    letI : Algebra (C i) (C j) :="}, {"name": "AlgebraTower.toAlgebra", "content": "@[simp]\ndef AlgebraTower.toAlgebra {i j : ι} (h : i ≤ j) : Algebra (A i) (A j) :=\n  (AlgebraTower.algebraMap (i:=i) (j:=j) (h:=h)).toAlgebra"}], "lib_lemmas": [{"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}, {"name": "BitVec.zero_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Ne.dite_eq_left_iff", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.left_eq_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingHom.comp_id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "MonoidHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "Nat.sub_one_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "OneHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "RingHom.coe_mk", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_cast", "module": "Init.PropLemmas"}, {"name": "eqRec_eq_cast", "module": "Batteries.Logic"}, {"name": "BitVec.ofNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.toNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.add", "content": "def add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y"}, {"name": "ConcreteBinaryTower.neg", "content": "def neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.ConcreteBTFAddCommGroupProps", "content": "structure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel"}, {"name": "ConcreteBinaryTower.mkAddCommGroupInstance", "content": "def mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}"}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.equivProd", "content": "def equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)"}, {"name": "ConcreteBinaryTower.concrete_mul", "content": "def concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.concrete_inv", "content": "def concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res"}, {"name": "ConcreteBinaryTower.natCast", "content": "def natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.natCast_zero", "content": "def natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :="}, {"name": "ConcreteBinaryTower.natCast_succ", "content": "def natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :="}, {"name": "ConcreteBinaryTower.intCast", "content": "def intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.intCast_ofNat", "content": "def intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :="}, {"name": "ConcreteBinaryTower.intCast_negSucc", "content": "def intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :="}, {"name": "ConcreteBinaryTower.ConcreteBTFRingProps", "content": "structure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c"}, {"name": "ConcreteBinaryTower.ConcreteBTFDivisionRingProps", "content": "structure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldProps", "content": "structure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a"}, {"name": "ConcreteBinaryTower.mkRingInstance", "content": "def mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n"}, {"name": "ConcreteBinaryTower.mkDivisionRingInstance", "content": "def mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)"}, {"name": "ConcreteBinaryTower.mkFieldInstance", "content": "def mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm"}, {"name": "ConcreteBinaryTower.ConcreteBTFStepResult", "content": "structure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))"}, {"name": "ConcreteBinaryTower.liftBTFieldProps", "content": "def liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.liftConcreteBTField", "content": "def liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.concreteCanonicalEmbedding", "content": "def concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :="}, {"name": "ConcreteBinaryTower.instAlgebraLiftConcreteBTField", "content": "instance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))"}, {"name": "ConcreteBinaryTower.getBTFResult", "content": "def getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.canonicalAlgMap", "content": "def canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap", "content": "def concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :="}, {"name": "ConcreteBinaryTower.instAlgebraTowerConcreteBTF", "content": "instance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldAlgebra", "content": "def ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le"}, {"name": "ConcreteBinaryTower.join_via_add_smul", "content": "def join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :="}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq", "content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n  ConcreteBTField k = ConcreteBTField m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.sum_fromNat_eq_from_xor_Nat", "content": "theorem sum_fromNat_eq_from_xor_Nat {k : ℕ} (x y : Nat) :\n  fromNat (k:=k) (x ^^^ y) = fromNat (k:=k) x + fromNat (k:=k) y"}, {"name": "ConcreteBinaryTower.zero_add", "content": "lemma zero_add {k : ℕ} (a : ConcreteBTField k) : 0 + a = a"}, {"name": "ConcreteBinaryTower.add_zero", "content": "lemma add_zero {k : ℕ} (a : ConcreteBTField k) : a + 0 = a"}, {"name": "ConcreteBinaryTower.cast_join", "content": "lemma cast_join {k n : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) (heq : k = n) :\n  join (k:=k) h_pos hi lo = cast (by rw [heq])\n    (join (k:=n) (by omega) (cast (by subst heq; rfl) hi) (lo:=cast (by subst heq; rfl) lo))"}, {"name": "ConcreteBinaryTower.zero_is_0", "content": "lemma zero_is_0 {k : ℕ} : (zero (k:=k)) = (0 : ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.one_is_1", "content": "lemma one_is_1 {k : ℕ} : (one (k:=k)) = 1"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_of_split", "content": "theorem join_of_split {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_split_eq : split h_pos x = (hi_btf, lo_btf)) :\n    x = 《 hi_btf, lo_btf 》"}, {"name": "ConcreteBinaryTower.split_of_join", "content": "theorem split_of_join {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_join : x = 《hi_btf, lo_btf》) :\n    (hi_btf, lo_btf) = split h_pos x"}, {"name": "ConcreteBinaryTower.split_join_eq_split", "content": "lemma split_join_eq_split {k : ℕ} (h_pos : k > 0)\n    (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    split h_pos (《 hi_btf, lo_btf 》) = (hi_btf, lo_btf)"}, {"name": "ConcreteBinaryTower.eq_iff_split_eq", "content": "theorem eq_iff_split_eq {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k) :\n  x₀ = x₁ ↔ (split h_pos x₀ = split h_pos x₁)"}, {"name": "ConcreteBinaryTower.split_sum_eq_sum_split", "content": "theorem split_sum_eq_sum_split {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k)\n  (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1))\n  (h_split_x₀ : split h_pos x₀ = (hi₀, lo₀))\n  (h_split_x₁ : split h_pos x₁ = (hi₁, lo₁)) :\n  split h_pos (x₀ + x₁) = (hi₀ + hi₁, lo₀ + lo₁)"}, {"name": "ConcreteBinaryTower.join_add_join", "content": "theorem join_add_join {k : ℕ} (h_pos : k > 0) (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1)) :\n  《 hi₀, lo₀ 》 + 《 hi₁, lo₁ 》 = 《 hi₀ + hi₁, lo₀ + lo₁ 》"}, {"name": "ConcreteBinaryTower.split_Z", "content": "theorem split_Z {k : ℕ} (h_pos : k > 0) :\n    split h_pos (Z k) = (one (k:=k - 1), zero (k:=k - 1))"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_eq_of_dest_eq", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_eq_of_dest_eq (k m n : ℕ) (h_eq : m = n) :\n    (ConcreteBTField k →+* ConcreteBTField m)\n    = (ConcreteBTField k →+* ConcreteBTField n)"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_cast_dest_apply", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_cast_dest_apply (k m n : ℕ) (h_eq : m = n)\n  (f : ConcreteBTField k →+* ConcreteBTField m) (x : ConcreteBTField k) :\n    (cast (ConcreteBTField.RingHom_eq_of_dest_eq (k:=k) (m:=m) (n:=n) h_eq) f) x\n    = cast (by apply cast_ConcreteBTField_eq (h_eq:=h_eq)) (f x)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_id", "content": "lemma concreteTowerAlgebraMap_id (k : ℕ) :\n    concreteTowerAlgebraMap (h_le:=by omega) = RingHom.id (ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_succ_1", "content": "lemma concreteTowerAlgebraMap_succ_1 (k : ℕ) :\n  concreteTowerAlgebraMap (l:=k) (r:=k + 1) (h_le:=by omega) = canonicalAlgMap k"}, {"name": "ConcreteBinaryTower.split_algebraMap_eq_zero_x", "content": "lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x) = (0, x)"}, {"name": "ConcreteBinaryTower.split_smul_Z_eq_zero_x", "content": "lemma split_smul_Z_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (x • Z k) = (x, 0)"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y\n\ndef neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\nstructure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel\n\ndef mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\ndef equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)\n\ndef concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/\n\ndef concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res\n\nsection FieldLemmasOfLevel0\n\nend FieldLemmasOfLevel0\n\nsection NumericCasting\n\ndef natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :=\n\ndef natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :=\n\ndef intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :=\n\ndef intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :=\n\nend NumericCasting\n\nstructure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c\n\nstructure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one\n\nstructure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a\n\ndef mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n\n\ndef mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)\n\ndef mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm\n\nstructure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))\n\nend FieldOperationsAndInstances\n\nsection BTFieldPropsOneLevelLiftingLemmas\n\nvariable {k : ℕ} {h_k : k > 0}\n\nend BTFieldPropsOneLevelLiftingLemmas\n\nsection TowerFieldsConstruction\n\ndef liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/\n\ndef liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :=\n\ndef concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :=\n\ninstance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))\n\ndef getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/\n\nend TowerFieldsConstruction\n\nsection ConcreteBTFieldAlgebraConstruction\n\ndef canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))\n\ndef concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :=\n\ninstance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/\n\ndef ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le\n\ndef join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :=", "target_theorem": "@[simp]\ntheorem join_eq_join_via_add_smul {k : ℕ} (h_pos : k > 0)\n    (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    《 hi_btf, lo_btf 》 = join_via_add_smul k h_pos hi_btf lo_btf :=", "ground_truth_proof": ":= by\n  unfold join_via_add_smul\n  set instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  set hi_lifted := instAlgebra.2 hi_btf with h_hi_lifted\n  -- First, show `hi_btf • Z k` corresponds to `join h_pos hi_btf 0`.\n  have h_hi_term : hi_btf • Z k = 《 hi_btf, 0 》 := by\n    apply join_of_split\n    exact split_smul_Z_eq_zero_x h_pos hi_btf\n  -- Second, show `algebraMap ... lo_btf` corresponds to `join h_pos 0 lo_btf`.\n  have h_lo_term : algebraMap (ConcreteBTField (k-1))\n    (ConcreteBTField k) lo_btf = 《 0, lo_btf 》 := by\n    have h := join_of_split (x := algebraMap (ConcreteBTField (k-1)) (ConcreteBTField k) lo_btf)\n      (h_pos:=by omega) (hi_btf:=zero (k:=k-1)) (lo_btf:=lo_btf)\n    apply h\n    rw [split_algebraMap_eq_zero_x h_pos lo_btf]\n    rfl\n  rw [h_hi_term, h_lo_term]\n   -- ⊢ join h_pos hi_btf lo_btf = join h_pos hi_btf 0 + join h_pos 0 lo_btf\n  rw [join_add_join h_pos hi_btf 0 0 lo_btf]\n  simp only [_root_.add_zero, _root_.zero_add]", "nesting_depth": 14, "transitive_dep_count": 250, "subset_aristotle": false, "category": "Applied verif."}
{"id": 39, "thm_name": "AdditiveNTT.W_linearity", "thm_stmt": "theorem W_linearity (i : Fin r)\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/NovelPolynomialBasis.lean", "imports": ["import Mathlib.Algebra.Polynomial.Degree.Definitions", "import ArkLib.Data.Fin.BigOperators", "import ArkLib.Data.Polynomial.MonomialBasis", "import Mathlib.LinearAlgebra.StdBasis", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.Polynomial.Frobenius"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "ZeroMemClass", "module": "Mathlib.Algebra.Group.Submonoid.Defs"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "CommGroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Polynomial.rootMultiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Multiset", "module": "Mathlib.Data.Multiset.Defs"}, {"name": "Multiset.count", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.map", "module": "Mathlib.Data.Multiset.MapFold"}, {"name": "Polynomial.roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "SetLike", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.algEquivOfCompEqX", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "multiplicity", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "EmbeddingLike", "module": "Mathlib.Data.FunLike.Embedding"}, {"name": "CanLift", "module": "Mathlib.Tactic.Lift"}, {"name": "Multiset.filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Finset.val", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Set.InjOn", "module": "Mathlib.Data.Set.Operations"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.val", "module": "Init.Prelude"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.Prime", "module": "Mathlib.Data.Nat.Prime.Defs"}, {"name": "ringChar", "module": "Mathlib.Algebra.CharP.Defs"}], "used_repo_defs": [{"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}], "lib_lemmas": [{"name": "Fact.out", "module": "Mathlib.Logic.Basic"}, {"name": "Fin.le_zero_iff'", "module": "Mathlib.Data.Fin.SuccPred"}, {"name": "LinearIndependent.injective", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Nat.not_lt_zero", "module": "Init.Prelude"}, {"name": "Set.Ico_eq_empty_iff", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.compl_eq_univ_diff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Set.empty_subset", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.image_empty", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_subset_image_iff", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.subset_compl_singleton_iff", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Submodule.span_mono", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "linearIndependent_iff_notMem_span", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Finset.prod_ne_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Polynomial.eval_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_prod", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_sub", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Polynomial.splits_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.splits_prod", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Finset.prod_eq_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Polynomial.X_sub_C_ne_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.X_ne_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_C_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_X_zero", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.coeff_sub", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.comp_eq_zero_iff", "module": "Mathlib.Algebra.Polynomial.Degree.Lemmas"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "map_neg", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "sub_eq_neg_self", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "EmbeddingLike.map_eq_zero_iff", "module": "Mathlib.Algebra.Group.Equiv.Defs"}, {"name": "Polynomial.aeval_C", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.aeval_X", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algEquivOfCompEqX_apply", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.algebraMap_eq", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial.rootMultiplicity_eq_multiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "if_false", "module": "Init.ByCases"}, {"name": "if_true", "module": "Init.ByCases"}, {"name": "map_sub", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "multiplicity_map_eq", "module": "Mathlib.RingTheory.Multiplicity"}, {"name": "sub_sub_sub_cancel_right", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Multiset.countP_eq_card_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_map", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.filter_congr", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.count_roots", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "eq_sub_iff_add_eq", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.image_val_of_injOn", "module": "Mathlib.Data.Finset.Image"}, {"name": "Finset.prod_image", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Polynomial.roots_prod_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "Subtype.val_injective", "module": "Mathlib.Data.Subtype"}, {"name": "CanLift.prf", "module": "Mathlib.Tactic.Lift"}, {"name": "Multiset.card_singleton", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.card_zero", "module": "Mathlib.Data.Multiset.ZeroCons"}, {"name": "Multiset.count_filter", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Multiset.count_singleton", "module": "Mathlib.Data.Multiset.Count"}, {"name": "Multiset.count_univ", "module": "Mathlib.Data.Fintype.Basic"}, {"name": "Multiset.count_zero", "module": "Mathlib.Data.Multiset.Count"}, {"name": "SetLike.coe_eq_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "SetLike.mem_coe", "module": "Mathlib.Data.SetLike.Basic"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Fin.zero_le", "module": "Init.Data.Fin.Lemmas"}, {"name": "Set.Ico_subset_Ico_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_mono", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_image_of_mem", "module": "Mathlib.Data.Set.Operations"}, {"name": "Submodule.add_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.smul_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.subset_span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "sub_add_cancel", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "Set.Ico_insert_right", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Set.image_singleton", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.image_union", "module": "Mathlib.Data.Set.Image"}, {"name": "Set.mem_Icc", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Set.union_singleton", "module": "Mathlib.Data.Set.Insert"}, {"name": "Submodule.mem_span_singleton", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.mem_sup", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.smul_mem_iff", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Submodule.span_union", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Submodule.sub_mem", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "existsUnique_of_exists_of_unique", "module": "Mathlib.Logic.ExistsUnique"}, {"name": "sub_smul", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "sub_sub_sub_cancel_left", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_const_zero", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.sum_ite_eq'", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise"}, {"name": "Finset.sum_map_val", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Multiset.count_bind", "module": "Mathlib.Data.Multiset.Bind"}, {"name": "Multiset.count_map_eq_count'", "module": "Mathlib.Data.Multiset.Filter"}, {"name": "Polynomial.roots_prod", "module": "Mathlib.Algebra.Polynomial.Roots"}, {"name": "add_left_injective", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "if_false_right", "module": "Init.PropLemmas"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "iff_false", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "Polynomial.monic_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.monic_prod_of_monic", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.Monic.comp", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.Splits.comp_of_degree_le_one", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.degree_X_sub_C_le", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eq_prod_roots_of_monic_of_splits_id", "module": "Mathlib.Algebra.Polynomial.Splits"}, {"name": "Polynomial.natDegree_X", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.natDegree_sub_C", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.comp_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_right_inj", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_add_neg", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_right_inj", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "CommGroupWithZero.mul_inv_cancel", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Fin.mk_lt_of_lt_val", "module": "Init.Data.Fin.Lemmas"}, {"name": "Finset.card_univ", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Finset.prod_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.prod_const", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.prod_mul_distrib", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "NeZero.one_le", "module": "Mathlib.Data.Nat.Cast.NeZero"}, {"name": "Polynomial.C_1", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.C_mul", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_pow", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.mul_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.pow_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.smul_C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.smul_eq_C_mul", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.sub_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "add_sub_cancel_right", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "algebra_compatible_smul", "module": "Mathlib.Algebra.Algebra.Basic"}, {"name": "map_mul", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_pow", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_pow", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_sub", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "one_pow", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_smul", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "pow_sub₀", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "smul_assoc", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "smul_eq_mul", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "smul_sub", "module": "Mathlib.Algebra.GroupWithZero.Action.Defs"}, {"name": "FiniteField.pow_card", "module": "Mathlib.FieldTheory.Finite.Basic"}, {"name": "algebraMap.coe_pow", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "left_distrib", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_sub_left_distrib", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Finset.mem_singleton", "module": "Mathlib.Data.Finset.Insert"}, {"name": "Finset.prod_singleton", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Polynomial.C_0", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Set.Ico_eq_empty", "module": "Mathlib.Order.Interval.Set.Basic"}, {"name": "Submodule.coe_zero", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.mem_bot", "module": "Mathlib.Algebra.Module.Submodule.Lattice"}, {"name": "Submodule.span_empty", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "ZeroMemClass.coe_eq_zero", "module": "Mathlib.Algebra.Group.Submonoid.Defs"}, {"name": "lt_self_iff_false", "module": "Mathlib.Order.Basic"}, {"name": "map_add", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_smul", "module": "Mathlib.GroupTheory.GroupAction.Hom"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "sub_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "true_iff", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "Fin.lt_succ'", "content": "lemma Fin.lt_succ' (a : Fin r) (h_a_add_1 : a + 1 < r) : a < a + 1"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}, {"name": "Fin.le_succ", "content": "lemma Fin.le_succ (a : Fin r) (h_a_add_1 : a + 1 < r) : a ≤ a + 1"}, {"name": "Fin.le_iff_lt_succ", "content": "lemma Fin.le_iff_lt_succ (a b : Fin r) (h_b : b + 1 < r) : a ≤ b ↔ a < b + 1"}, {"name": "Fin.val_sub_one", "content": "lemma Fin.val_sub_one (a : Fin r) (h_a_sub_1 : a > 0) : (a - 1).val = a.val - 1"}, {"name": "prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L", "content": "theorem prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L\n  (p : L[X]) :\n  (∏ c ∈ (Finset.univ : Finset Fq), (p - Polynomial.C (algebraMap Fq L c))) =\n    p^(Fintype.card Fq) - p"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X_in_L", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X_in_L :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C (algebraMap Fq L c))) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C c)) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "linear_map_of_comp_to_linear_map_of_eval", "content": "theorem linear_map_of_comp_to_linear_map_of_eval (f : L[X])\n  (h_f_linear : IsLinearMap (R := Fq) (M := L[X]) (M₂ := L[X])\n    (f := fun inner_p ↦ f.comp inner_p)) :\n    IsLinearMap (R := Fq) (M := L) (M₂ := L) (f := fun x ↦ f.eval x)"}, {"name": "frobenius_identity_in_algebra", "content": "theorem frobenius_identity_in_algebra [Fact (Nat.Prime (ringChar Fq))]\n  (f g : L[X]) : (f + g)^(Fintype.card Fq) = f^(Fintype.card Fq) + g^(Fintype.card Fq)"}], "used_local_defs": [{"name": "AdditiveNTT.U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "AdditiveNTT.W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "AdditiveNTT.algEquivAevalXSubC", "content": "@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :="}], "used_local_lemmas": [{"name": "AdditiveNTT.βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "AdditiveNTT.root_U_lift_down", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n  (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n  a ∈ (U 𝔽q β (i+1)) → ∃! x: 𝔽q, a - x • β i ∈ (U 𝔽q β i)"}, {"name": "AdditiveNTT.root_U_lift_up", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime hβ_lin_indep in\ntheorem root_U_lift_up (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) (x : 𝔽q) :\n  a - x • β i ∈ (U 𝔽q β i) → a ∈ (U 𝔽q β (i+1))"}, {"name": "AdditiveNTT.W_monic", "content": "lemma W_monic (i : Fin r) : (W 𝔽q β i).Monic"}, {"name": "AdditiveNTT.W_ne_zero", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W_ne_zero (i : Fin r) : (W 𝔽q β i) ≠ 0"}, {"name": "AdditiveNTT.Wᵢ_eval_βᵢ_neq_zero", "content": "lemma Wᵢ_eval_βᵢ_neq_zero\n    (i : Fin r): (W 𝔽q β i).eval (β i) ≠ 0"}, {"name": "AdditiveNTT.W_splits", "content": "lemma W_splits (i : Fin r) : (W 𝔽q β i).Splits (RingHom.id L)"}, {"name": "AdditiveNTT.roots_W", "content": "lemma roots_W (i : Fin r) : -- converts root Multiset into (univ: Uᵢ.val.map)\n  (W 𝔽q β i).roots = (univ : Finset (U 𝔽q β i)).val.map (fun u => u.val)"}, {"name": "AdditiveNTT.comp_X_sub_C_eq_zero_iff", "content": "omit [Fintype L] [DecidableEq L] in\nlemma comp_X_sub_C_eq_zero_iff (p : L[X]) (a : L) :\n  p.comp (X - C a) = 0 ↔ p = 0"}, {"name": "AdditiveNTT.rootMultiplicity_comp_X_sub_C", "content": "lemma rootMultiplicity_comp_X_sub_C (p : L[X]) (a x : L) :\n    rootMultiplicity x (p.comp (X - C a)) = rootMultiplicity (x - a) p"}, {"name": "AdditiveNTT.roots_comp_X_sub_C", "content": "lemma roots_comp_X_sub_C (p : L[X]) (a : L) :\n    (p.comp (X - C a)).roots = p.roots.map (fun r => r + a)"}, {"name": "AdditiveNTT.Prod_W_comp_X_sub_C_ne_zero", "content": "omit [DecidableEq L] h_Fq_char_prime hF₂ hβ_lin_indep in\nlemma Prod_W_comp_X_sub_C_ne_zero (i : Fin r) :\n    (univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i))) ≠ 0"}, {"name": "AdditiveNTT.rootMultiplicity_W", "content": "lemma rootMultiplicity_W (i : Fin r) (a : L) :\n    rootMultiplicity a (W 𝔽q β i) = if a ∈ (U 𝔽q β i : Set L) then 1 else 0"}, {"name": "AdditiveNTT.rootMultiplicity_prod_W_comp_X_sub_C", "content": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0"}, {"name": "AdditiveNTT.W_prod_comp_decomposition", "content": "lemma W_prod_comp_decomposition\n    (i : Fin r) (hi : i > 0) :\n    (W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))"}, {"name": "AdditiveNTT.comp_sub_C_of_linear_eval", "content": "lemma comp_sub_C_of_linear_eval (p : L[X])\n  (h_lin : IsLinearMap 𝔽q (f := fun inner_p ↦ p.comp inner_p)) (a : L) :\n    p.comp (X - C a) = p - C (eval a p)"}, {"name": "AdditiveNTT.inductive_rec_form_W_comp", "content": "omit h_Fq_char_prime hF₂ in\nlemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])\n      (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)"}, {"name": "AdditiveNTT.inductive_linear_map_W", "content": "omit hF₂ in\nlemma inductive_linear_map_W (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (i + 1)).comp inner_p)"}], "local_ctx": "import ArkLib.Data.Nat.Bitwise\n\nimport ArkLib.Data.Polynomial.Frobenius\n\nimport ArkLib.Data.Polynomial.MonomialBasis\n\nimport Mathlib.LinearAlgebra.StdBasis\n\nimport Mathlib.Algebra.Polynomial.Degree.Definitions\n\nopen Polynomial FiniteDimensional Finset Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (h_dim : Module.finrank 𝔽q L = r)\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n\nsection LinearSubspaces\n\ndef U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))\n\nnoncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)\n\nend LinearSubspaces\n\nsection LinearityOfSubspaceVanishingPolynomials\n\n@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :=", "target_theorem": "theorem W_linearity (i : Fin r)\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p) :=", "ground_truth_proof": ":= by\n  induction i using Fin.succRecOnSameFinType with\n  | zero =>\n    -- Base Case: i = 0 => Prove W₀ is linear.\n    unfold W\n    have h_U0 : (univ : Finset (U 𝔽q β 0)) = {0} := by\n      ext u -- u : ↥(U 𝔽q β 0)\n      simp only [mem_univ, true_iff, mem_singleton]\n      -- ⊢ u = 0\n      by_contra h\n      have h_u := u.property\n      -- only U and Submodule.span_empty is enough for simp\n      simp only [U, lt_self_iff_false, not_false_eq_true, Set.Ico_eq_empty, Set.image_empty,\n        Submodule.span_empty, Submodule.mem_bot, ZeroMemClass.coe_eq_zero] at h_u\n      contradiction\n\n    rw [h_U0, prod_singleton, Submodule.coe_zero, C_0, sub_zero]\n    -- ⊢ IsLinearMap 𝔽q fun x ↦ eval x X\n    exact { -- can also use `refine` with exact same syntax\n      map_add := fun x y => by\n        rw [X_comp, X_comp, X_comp]\n      map_smul := fun c x => by\n        rw [X_comp, X_comp]\n    }\n  | succ j jh p =>\n    -- Inductive Step: Assume properties hold for `j`, prove for `j+1`.\n    have h_linear_map: (IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (j + 1)).comp inner_p)) := by\n      exact inductive_linear_map_W 𝔽q β (i := j)\n        (h_i_add_1 := by omega) (h_prev_linear_map := p)\n\n    exact h_linear_map", "nesting_depth": 8, "transitive_dep_count": 257, "subset_aristotle": false, "category": "Applied verif."}
{"id": 40, "thm_name": "MvPolynomial.finSuccEquivNth_coeff_coeff", "thm_stmt": "theorem finSuccEquivNth_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :\n    coeff m (Polynomial.coeff (finSuccEquivNth R p f) i) = coeff (m.insertNth p i) f", "lean_root": "ArkLib", "rel_path": "ArkLib/ToMathlib/MvPolynomial/Equiv.lean", "imports": ["import Mathlib.Algebra.MvPolynomial.Equiv", "import ArkLib.ToMathlib.Finsupp.Fin"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "MvPolynomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "MvPolynomial.optionEquivLeft", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "MvPolynomial.renameEquiv", "module": "Mathlib.Algebra.MvPolynomial.Rename"}, {"name": "finSuccEquiv'", "module": "Mathlib.Logic.Equiv.Fin.Basic"}, {"name": "Fin.insertNth", "module": "Mathlib.Data.Fin.Tuple.Basic"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.equivFunOnFinite", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "MvPolynomial.coeff", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "Polynomial.coeff", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "MvPolynomial.C", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.X", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.eval₂Hom", "module": "Mathlib.Algebra.MvPolynomial.Eval"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "AlgEquiv", "module": "Mathlib.Algebra.Algebra.Equiv"}, {"name": "Fin.succAboveCases", "module": "Mathlib.Data.Fin.Tuple.Basic"}, {"name": "Fin.removeNth", "module": "Mathlib.Data.Fin.Tuple.Basic"}, {"name": "Fin.succAbove", "module": "Mathlib.Data.Fin.SuccPred"}], "used_repo_defs": [{"name": "insertNth", "content": "def insertNth (p : Fin (n + 1)) (y : M) (s : Fin n →₀ M) : Fin (n + 1) →₀ M :=\n  Finsupp.equivFunOnFinite.symm (Fin.insertNth p y s : Fin (n + 1) → M)"}, {"name": "removeNth", "content": "def removeNth (p : Fin (n + 1)) (s : Fin (n + 1) →₀ M) : Fin n →₀ M :=\n  Finsupp.equivFunOnFinite.symm (Fin.removeNth p s : Fin n → M)"}], "lib_lemmas": [{"name": "AlgEquiv.coe_trans", "module": "Mathlib.Algebra.Algebra.Equiv"}, {"name": "Function.comp_apply", "module": "Init.Core"}, {"name": "MvPolynomial.aeval_C", "module": "Mathlib.Algebra.MvPolynomial.Eval"}, {"name": "MvPolynomial.coe_eval₂Hom", "module": "Mathlib.Algebra.MvPolynomial.Eval"}, {"name": "MvPolynomial.eval₂_C", "module": "Mathlib.Algebra.MvPolynomial.Eval"}, {"name": "MvPolynomial.ext", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.optionEquivLeft_apply", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "MvPolynomial.renameEquiv_apply", "module": "Mathlib.Algebra.MvPolynomial.Rename"}, {"name": "MvPolynomial.rename_C", "module": "Mathlib.Algebra.MvPolynomial.Rename"}, {"name": "RingHom.coe_coe", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.coe_comp", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Fin.insertNth_apply_same", "module": "Mathlib.Data.Fin.Tuple.Basic"}, {"name": "Fin.insertNth_apply_succAbove", "module": "Mathlib.Data.Fin.Tuple.Basic"}, {"name": "Fin.prod_univ_succAbove", "module": "Mathlib.Algebra.BigOperators.Fin"}, {"name": "Finsupp.prod_pow", "module": "Mathlib.Algebra.BigOperators.Finsupp.Basic"}, {"name": "MvPolynomial.C_1", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_C_mul", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_add", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_monomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_zero", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.eval₂_monomial", "module": "Mathlib.Algebra.MvPolynomial.Eval"}, {"name": "MvPolynomial.induction_on'", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.monomial_eq", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "Polynomial.coeff_C_mul", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.coeff_C_mul_X_pow", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.coeff_add", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "RingHom.map_pow", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "eq_or_ne", "module": "Mathlib.Logic.Basic"}, {"name": "eq_self_iff_true", "module": "Init.Core"}, {"name": "if_false", "module": "Init.ByCases"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "if_true", "module": "Init.ByCases"}, {"name": "map_add", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_prod", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "mul_boole", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ne_or_eq", "module": "Mathlib.Logic.Basic"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "insertNth_self_removeNth", "content": "theorem insertNth_self_removeNth : insertNth p (t p) (removeNth p t) = t"}, {"name": "insertNth_apply_succAbove", "content": "@[simp]\ntheorem insertNth_apply_succAbove : insertNth p y s (p.succAbove i) = s i"}, {"name": "removeNth_apply", "content": "@[simp]\ntheorem removeNth_apply : removeNth p s i = s (p.succAbove i)"}, {"name": "insertNth_apply_same", "content": "@[simp]\ntheorem insertNth_apply_same : insertNth p y s p = y"}], "used_local_defs": [{"name": "MvPolynomial.finSuccEquivNth", "content": "def finSuccEquivNth : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] Polynomial (MvPolynomial (Fin n) R) :=\n  (renameEquiv R (_root_.finSuccEquiv' p)).trans (optionEquivLeft R (Fin n))"}], "used_local_lemmas": [{"name": "MvPolynomial.finSuccEquivNth_eq", "content": "theorem finSuccEquivNth_eq :\n    (finSuccEquivNth R p : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =\n      eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R))\n        (Fin.insertNth p Polynomial.X (Polynomial.C ∘ X))"}, {"name": "MvPolynomial.finSuccEquivNth_apply", "content": "theorem finSuccEquivNth_apply (f : MvPolynomial (Fin (n + 1)) R) :\n    finSuccEquivNth R p f =\n      eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R))\n        (Fin.insertNth p Polynomial.X (Polynomial.C ∘ X)) f"}], "local_ctx": "import Mathlib.Algebra.MvPolynomial.Equiv\n\nimport ArkLib.ToMathlib.Finsupp.Fin\n\nnamespace MvPolynomial\n\nopen Function Finsupp Polynomial\n\nnoncomputable section\n\nsection FinSuccEquivNth\n\nvariable {n : ℕ} {σ : Type*} (R : Type*) [CommSemiring R] (p : Fin (n + 1))\n\ndef finSuccEquivNth : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] Polynomial (MvPolynomial (Fin n) R) :=\n  (renameEquiv R (_root_.finSuccEquiv' p)).trans (optionEquivLeft R (Fin n))\n\nvariable {R} {p}", "target_theorem": "theorem finSuccEquivNth_coeff_coeff (m : Fin n →₀ ℕ) (f : MvPolynomial (Fin (n + 1)) R) (i : ℕ) :\n    coeff m (Polynomial.coeff (finSuccEquivNth R p f) i) = coeff (m.insertNth p i) f :=", "ground_truth_proof": ":= by\n  induction' f using MvPolynomial.induction_on' with u a p q hp hq generalizing i m\n  · simp only [finSuccEquivNth_apply, coe_eval₂Hom, eval₂_monomial, RingHom.coe_comp, comp_apply,\n      prod_pow, Fin.prod_univ_succAbove _ p, Fin.insertNth_apply_same,\n      Fin.insertNth_apply_succAbove, Polynomial.coeff_C_mul, coeff_C_mul, coeff_monomial,\n      ← map_prod, ← RingHom.map_pow]\n    rw [← mul_boole, mul_comm (Polynomial.X ^ u p), Polynomial.coeff_C_mul_X_pow]; congr 1\n    obtain rfl | hjmi := eq_or_ne u (m.insertNth p i)\n    · simpa only [insertNth_apply_same, if_pos rfl, insertNth_apply_succAbove, monomial_eq, C_1,\n        one_mul, prod_pow] using coeff_monomial m m (1 : R)\n    · simp only [hjmi, if_false]\n      obtain hij | rfl := ne_or_eq i (u p)\n      · simp only [hij, if_false, coeff_zero]\n      simp only [eq_self_iff_true, if_true]\n      have hmj : m ≠ u.removeNth p := by\n        rintro rfl\n        rw [insertNth_self_removeNth] at hjmi\n        contradiction\n      simpa only [monomial_eq, C_1, one_mul, prod_pow, Finsupp.removeNth_apply, if_neg hmj.symm]\n        using coeff_monomial m (u.removeNth p) (1 : R)\n  · simp only [map_add, Polynomial.coeff_add, coeff_add, hp, hq]", "nesting_depth": 3, "transitive_dep_count": 70, "subset_aristotle": false, "category": "Applied verif."}
{"id": 41, "thm_name": "ReedSolomonCode.genMatIsVandermonde", "thm_stmt": "lemma genMatIsVandermonde [Fintype ι] [Field F] [DecidableEq F] [inst : NeZero m] {α : ι ↪ F} :\n  fromColGenMat (Vandermonde.nonsquare (ι' := m) α) = ReedSolomon.code α m", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/CodingTheory/ReedSolomon.lean", "imports": ["import Mathlib.LinearAlgebra.Lagrange", "import ArkLib.Data.MvPolynomial.LinearMvExtension", "import Mathlib.RingTheory.Henselian", "import ArkLib.Data.Fin.Lift", "import ArkLib.Data.Polynomial.Interface"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Matrix.of", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.degreeLT", "module": "Mathlib.RingTheory.Polynomial.Basic"}, {"name": "Fin.val", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.map", "module": "Mathlib.Data.Finset.Image"}, {"name": "Finset.max", "module": "Mathlib.Data.Finset.Max"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "LinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "LinearMap.range", "module": "Mathlib.Algebra.Module.Submodule.Range"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Matrix.add", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.coeff", "module": "Mathlib.Algebra.Polynomial.Basic"}], "used_repo_defs": [{"name": "polynomialOfCoeffs", "content": "def polynomialOfCoeffs (coeffs : Fin deg → F) : F[X] :=\n  ⟨\n    Finset.map ⟨Fin.val, Fin.val_injective⟩ {i | coeffs i ≠ 0},\n    fun i ↦ if h : i < deg then coeffs ⟨i, h⟩ else 0,\n    fun a ↦ by admit /- proof elided -/\n  ⟩"}, {"name": "liftF'", "content": "def liftF' (f : ℕ → α) : Fin n → α :=\n  fun m ↦ f m.1"}, {"name": "fromColGenMat", "content": "noncomputable def fromColGenMat [CommRing F] (G : Matrix ι κ F) : LinearCode ι F :=\n  LinearMap.range G.mulVecLin"}, {"name": "LinearCode.{u,", "content": "abbrev LinearCode.{u, v} (ι : Type u) [Fintype ι] (F : Type v) [Semiring F] : Type (max u v) :=\n  Submodule F (ι → F)"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}], "lib_lemmas": [{"name": "Polynomial.mem_degreeLT", "module": "Mathlib.RingTheory.Polynomial.Basic"}, {"name": "Polynomial.natDegree_lt_iff_degree_lt", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Matrix.mulVecLin_apply", "module": "Mathlib.LinearAlgebra.Matrix.ToLin"}, {"name": "Matrix.mulVec_eq_sum", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Finset.mem_range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_fin_eq_sum_range", "module": "Mathlib.Data.Fintype.BigOperators"}, {"name": "Polynomial.eval_eq_sum_range'", "module": "Mathlib.Algebra.Polynomial.Eval.Degree"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "LinearMap.mem_range", "module": "Mathlib.Algebra.Module.Submodule.Range"}, {"name": "Submodule.mem_map", "module": "Mathlib.Algebra.Module.Submodule.Map"}], "repo_lemmas": [{"name": "liftF'_p_coeff", "content": "@[simp]\nlemma liftF'_p_coeff {p : F[X]} {k : ℕ} {i : Fin k} : liftF' p.coeff i = p.coeff i"}, {"name": "coeff_polynomialOfCoeffs_eq_coeffs", "content": "@[simp]\nlemma coeff_polynomialOfCoeffs_eq_coeffs :\n  Fin.liftF' (polynomialOfCoeffs coeffs).coeff = coeffs"}], "used_local_defs": [{"name": "ReedSolomon.evalOnPoints", "content": "def evalOnPoints [Semiring F] : F[X] →ₗ[F] (ι → F) where\n  toFun := fun p => fun x => p.eval (domain x)\n  map_add' := fun x y => by admit /- proof elided -/"}, {"name": "ReedSolomon.code", "content": "def code (deg : ℕ) [Semiring F]: Submodule F (ι → F) :=\n  (Polynomial.degreeLT F deg).map (evalOnPoints domain)"}, {"name": "Vandermonde.nonsquare", "content": "def nonsquare [Semiring F] (ι' : ℕ) (α : ι → F) : Matrix ι (Fin ι') F :=\n  Matrix.of fun i j => (α i) ^ j.1"}], "used_local_lemmas": [{"name": "Vandermonde.nonsquare_mulVecLin", "content": "lemma nonsquare_mulVecLin [CommSemiring F] {ι' : ℕ} {α₁ : ι ↪ F} {α₂ : Fin ι' → F} {i : ι} :\n  (nonsquare ι' α₁).mulVecLin α₂ i = ∑ x, α₂ x * α₁ i ^ x.1"}, {"name": "Vandermonde.mulVecLin_coeff_vandermondens_eq_eval_matrixOfPolynomials", "content": "theorem mulVecLin_coeff_vandermondens_eq_eval_matrixOfPolynomials\n  {n : ℕ} [NeZero n] {v : ι ↪ F} {p : F[X]} (h_deg : p.natDegree < n) :\n  (Vandermonde.nonsquare (ι' := n) v).mulVecLin (Fin.liftF' p.coeff) =\n  fun i => p.eval (v i)"}, {"name": "ReedSolomonCode.natDegree_lt_of_mem_degreeLT", "content": "lemma natDegree_lt_of_mem_degreeLT [NeZero deg] (h : p ∈ degreeLT F deg) : p.natDegree < deg"}], "local_ctx": "import ArkLib.Data.MvPolynomial.LinearMvExtension\n\nimport ArkLib.Data.Polynomial.Interface\n\nimport Mathlib.LinearAlgebra.Lagrange\n\nimport Mathlib.RingTheory.Henselian\n\nnamespace ReedSolomon\n\nopen Polynomial NNReal\n\nvariable {F : Type*} {ι : Type*} (domain : ι ↪ F)\n\ndef evalOnPoints [Semiring F] : F[X] →ₗ[F] (ι → F) where\n  toFun := fun p => fun x => p.eval (domain x)\n  map_add' := fun x y => by admit /- proof elided -/\n\ndef code (deg : ℕ) [Semiring F]: Submodule F (ι → F) :=\n  (Polynomial.degreeLT F deg).map (evalOnPoints domain)\n\nvariable [Semiring F]\n\nend ReedSolomon\n\nopen Polynomial Matrix Code LinearCode\n\nvariable {F ι ι' : Type*}\n         {C : Set (ι → F)}\n\nnoncomputable section\n\nnamespace Vandermonde\n\ndef nonsquare [Semiring F] (ι' : ℕ) (α : ι → F) : Matrix ι (Fin ι') F :=\n  Matrix.of fun i j => (α i) ^ j.1\n\nsection\n\nvariable [CommRing F] {m n : ℕ} {α : Fin m → F}\n\nsection\n\nvariable [IsDomain F]\n\nend\n\nend\n\nend Vandermonde\n\nnamespace ReedSolomonCode\n\nsection\n\nopen Finset Function\n\nopen scoped BigOperators\n\nvariable {ι : Type*} [Fintype ι] [Nonempty ι]\n         {F : Type*} [Field F] [Fintype F]\n\nopen Classical in\n\nend\n\nsection\n\nvariable {deg m n : ℕ} {α : Fin m → F}\n\nsection\n\nvariable [Semiring F] {p : F[X]}\n\nend\n\nopen LinearCode", "target_theorem": "lemma genMatIsVandermonde [Fintype ι] [Field F] [DecidableEq F] [inst : NeZero m] {α : ι ↪ F} :\n  fromColGenMat (Vandermonde.nonsquare (ι' := m) α) = ReedSolomon.code α m :=", "ground_truth_proof": ":= by\n  unfold fromColGenMat ReedSolomon.code\n  ext x; rw [LinearMap.mem_range, Submodule.mem_map]\n  refine ⟨\n    fun ⟨coeffs, h⟩ ↦ ⟨polynomialOfCoeffs coeffs, h.symm ▸ ?p₁⟩,\n    fun ⟨p, h⟩ ↦ ⟨Fin.liftF' p.coeff, ?p₂⟩\n  ⟩\n  · rw [\n      ←coeff_polynomialOfCoeffs_eq_coeffs (coeffs := coeffs),\n      Vandermonde.mulVecLin_coeff_vandermondens_eq_eval_matrixOfPolynomials (by simp)\n    ]\n    simp [ReedSolomon.evalOnPoints]\n  · exact h.2 ▸ Vandermonde.mulVecLin_coeff_vandermondens_eq_eval_matrixOfPolynomials\n                  (natDegree_lt_of_mem_degreeLT h.1)", "nesting_depth": 3, "transitive_dep_count": 47, "subset_aristotle": false, "category": "Applied verif."}
{"id": 42, "thm_name": "UniPoly.toImpl_toPoly_of_canonical", "thm_stmt": "lemma toImpl_toPoly_of_canonical [LawfulBEq R] (p : UniPolyC R) : p.toPoly.toImpl = p", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/UniPoly/Basic.lean", "imports": ["import Mathlib.Algebra.Tropical.Basic", "import ArkLib.Data.Array.Lemmas", "import Mathlib.RingTheory.Polynomial.Basic"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.ofFn", "module": "Mathlib.Algebra.Polynomial.OfFn"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "LawfulBEq", "module": "Init.Core"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.val", "module": "Init.Prelude"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Array.foldl", "module": "Init.Data.Array.Basic"}, {"name": "Array.zipIdx", "module": "Init.Data.Array.Basic"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "Array.zipWith", "module": "Init.Data.Array.Basic"}, {"name": "Array.rightpad", "module": "Init.Data.Array.Basic"}, {"name": "Array.size", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}], "used_repo_defs": [{"name": "findIdxRev?", "content": "def findIdxRev? (cond : α → Bool) (as : Array α) : Option (Fin as.size) :=\n  find ⟨ as.size, Nat.lt_succ_self _ ⟩\nwhere\n  find : Fin (as.size + 1) → Option (Fin as.size)\n    | 0 => none\n    | ⟨ i+1, h ⟩ =>\n      if (cond as[i]) then\n        some ⟨ i, Nat.lt_of_succ_lt_succ h ⟩\n      else\n        find ⟨ i, Nat.lt_of_succ_lt h ⟩"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "matchSize", "content": "@[reducible]\ndef matchSize (a b : Array α) (unit : α) : Array α × Array α :=\n  (a.rightpad (b.size) unit, b.rightpad (a.size) unit)"}, {"name": "findIdxRev?_maximal", "content": "def findIdxRev?_maximal {cond} {as : Array α} {k : Fin as.size} :\n  findIdxRev? cond as = some k → ∀ j : Fin as.size, j > k → ¬ cond as[j] :="}, {"name": "findIdxRev?_def", "content": "def findIdxRev?_def {cond} {as : Array α} {k : Fin as.size} :\n  findIdxRev? cond as = some k → cond as[k] :="}, {"name": "getLast", "content": "def getLast (a : Array α) (h : a.size > 0) : α := a[a.size - 1]"}], "lib_lemmas": [{"name": "Nat.lt_succ_self", "module": "Init.Prelude"}, {"name": "Array.foldl_induction", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.getD_eq_getD_getElem?", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.getElem?_eq_none", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.getElem_zipIdx", "module": "Init.Data.Array.MapIdx"}, {"name": "Nat.lt_trichotomy", "module": "Init.Data.Nat.Basic"}, {"name": "Option.getD_none", "module": "Init.Data.Option.Basic"}, {"name": "Polynomial.coeff_C_mul", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.coeff_X_pow", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "Polynomial.coeff_add", "module": "Mathlib.Algebra.Polynomial.Coeff"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "mul_ite", "module": "Mathlib.Algebra.Notation.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Array.size_rightpad", "module": "Init.Data.Array.Lemmas"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "bne_iff_ne", "module": "Init.SimpLemmas"}, {"name": "bne_self_eq_false", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "Array.getElem?_eq_getElem", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.getElem_extract", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.size_extract", "module": "Init.Data.Array.Lemmas"}, {"name": "Nat.lt_or_ge", "module": "Init.Prelude"}, {"name": "Nat.succ_le_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Option.getD_some", "module": "Init.Data.Option.Basic"}, {"name": "Fin.le_antisymm", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.lt_of_not_ge", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.le_of_not_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.is_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.pred_lt_self", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.le_of_pred_lt", "module": "Init.Data.Nat.Lemmas"}, {"name": "Option.some_inj", "module": "Init.Data.Option.Instances"}, {"name": "Array.eq_empty_of_size_eq_zero", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.ne_empty_of_size_pos", "module": "Init.Data.Array.Lemmas"}, {"name": "Nat.le_antisymm", "module": "Init.Prelude"}, {"name": "Nat.succ_pred_eq_of_pos", "module": "Init.Data.Nat.Basic"}, {"name": "le_refl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Nat.eq_zero_or_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.degree_eq_bot", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.degree_eq_natDegree", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.degree_ne_bot", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.leadingCoeff_ne_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Array.getElem?_ofFn", "module": "Init.Data.Array.Lemmas"}, {"name": "Nat.lt_of_succ_le", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.coeff_eq_zero_of_natDegree_lt", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "symm", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "findIdxRev?_eq_some", "content": "theorem findIdxRev?_eq_some {cond} {as : Array α} (h : ∃ i, ∃ hi : i < as.size, cond as[i]) :\n  ∃ k : Fin as.size, findIdxRev? cond as = some k"}, {"name": "findIdxRev?_eq_none", "content": "theorem findIdxRev?_eq_none {cond} {as : Array α} (h : ∀ i, (hi : i < as.size) → ¬ cond as[i]) :\n  findIdxRev? cond as = none"}, {"name": "findIdxRev?_emtpy_none", "content": "theorem findIdxRev?_emtpy_none {cond} {as : Array α} (h : as = #[]) :\n  findIdxRev? cond as = none"}], "used_local_defs": [{"name": "UniPoly", "content": "@[reducible, inline, specialize]\ndef UniPoly (R : Type*) := Array R"}, {"name": "Polynomial.toImpl", "content": "def Polynomial.toImpl {R : Type*} [Semiring R] (p : R[X]) : UniPoly R :=\n  match p.degree with\n  | ⊥ => #[]\n  | some d  => .ofFn (fun i : Fin (d + 1) => p.coeff i)"}, {"name": "UniPoly.mk", "content": "@[reducible]\ndef mk {R : Type*} (coeffs : Array R) : UniPoly R := coeffs"}, {"name": "UniPoly.coeff", "content": "@[reducible]\ndef coeff (p : UniPoly Q) (i : ℕ) : Q := p.getD i 0"}, {"name": "UniPoly.last_nonzero", "content": "def last_nonzero (p : UniPoly R) : Option (Fin p.size) :=\n  p.findIdxRev? (· != 0)"}, {"name": "UniPoly.trim", "content": "def trim (p : UniPoly R) : UniPoly R :=\n  match p.last_nonzero with\n  | none => #[]\n  | some i => p.extract 0 (i.val + 1)"}, {"name": "UniPoly.degree", "content": "def degree (p : UniPoly R) : Nat :=\n  match p.last_nonzero with\n  | none => 0\n  | some i => i.val + 1"}, {"name": "UniPoly.Trim.last_nonzero_prop", "content": "def last_nonzero_prop {p : UniPoly R} (k : Fin p.size) : Prop :=\n  p[k] ≠ 0 ∧ (∀ j, (hj : j < p.size) → j > k → p[j] = 0)"}, {"name": "UniPoly.Trim.equiv", "content": "def equiv (p q : UniPoly R) : Prop :=\n  ∀ i, p.coeff i = q.coeff i"}, {"name": "UniPoly.UniPolyC", "content": "def UniPolyC (R : Type*) [BEq R] [Ring R] := { p : UniPoly R // p.trim = p }"}, {"name": "UniPoly.eval₂", "content": "def eval₂ [Semiring S] (f : R →+* S) (x : S) (p : UniPoly R) : S :=\n  p.zipIdx.foldl (fun acc ⟨a, i⟩ => acc + f a * x ^ i) 0"}, {"name": "UniPoly.add_raw", "content": "@[inline, specialize]\ndef add_raw (p q : UniPoly R) : UniPoly R :=\n  let ⟨p', q'⟩ := Array.matchSize p q 0\n  .mk (Array.zipWith (· + ·) p' q' )"}, {"name": "UniPoly.canonical", "content": "def canonical (p : UniPoly R) := p.trim = p"}, {"name": "UniPoly.toPoly", "content": "noncomputable def toPoly (p : UniPoly R) : Polynomial R :=\n  p.eval₂ Polynomial.C Polynomial.X"}, {"name": "UniPoly.UniPolyC.toPoly", "content": "noncomputable def UniPolyC.toPoly (p : UniPolyC R) : Polynomial R := p.val.toPoly\n\nalias ofPoly := Polynomial.toImpl"}], "used_local_lemmas": [{"name": "UniPoly.Trim.last_nonzero_none", "content": "theorem last_nonzero_none [LawfulBEq R] {p : UniPoly R} :\n  (∀ i, (hi : i < p.size) → p[i] = 0) → p.last_nonzero = none"}, {"name": "UniPoly.Trim.last_nonzero_some", "content": "theorem last_nonzero_some [LawfulBEq R] {p : UniPoly R} {i} (hi : i < p.size) (h : p[i] ≠ 0) :\n  ∃ k, p.last_nonzero = some k"}, {"name": "UniPoly.Trim.last_nonzero_spec", "content": "theorem last_nonzero_spec [LawfulBEq R] {p : UniPoly R} {k} :\n  p.last_nonzero = some k\n  → p[k] ≠ 0 ∧ (∀ j, (hj : j < p.size) → j > k → p[j] = 0)"}, {"name": "UniPoly.Trim.last_nonzero_unique", "content": "lemma last_nonzero_unique {p : UniPoly Q} {k k' : Fin p.size} :\n  last_nonzero_prop k → last_nonzero_prop k' → k = k'"}, {"name": "UniPoly.Trim.last_nonzero_some_iff", "content": "theorem last_nonzero_some_iff [LawfulBEq R] {p : UniPoly R} {k} :\n  p.last_nonzero = some k ↔ (p[k] ≠ 0 ∧ (∀ j, (hj : j < p.size) → j > k → p[j] = 0))"}, {"name": "UniPoly.Trim.last_nonzero_induct", "content": "theorem last_nonzero_induct [LawfulBEq R] {motive : UniPoly R → Prop}\n  (case1 : ∀ p, p.last_nonzero = none → (∀ i, (hi : i < p.size) → p[i] = 0) → motive p)\n  (case2 : ∀ p : UniPoly R, ∀ k : Fin p.size, p.last_nonzero = some k → p[k] ≠ 0 →\n    (∀ j : ℕ, (hj : j < p.size) → j > k → p[j] = 0) → motive p)\n  (p : UniPoly R) : motive p"}, {"name": "UniPoly.Trim.induct", "content": "theorem induct [LawfulBEq R] {motive : UniPoly R → Prop}\n  (case1 : ∀ p, p.trim = #[] → (∀ i, (hi : i < p.size) → p[i] = 0) → motive p)\n  (case2 : ∀ p : UniPoly R, ∀ k : Fin p.size, p.trim = p.extract 0 (k + 1)\n    → p[k] ≠ 0 → (∀ j : ℕ, (hj : j < p.size) → j > k → p[j] = 0) → motive p)\n  (p : UniPoly R) : motive p"}, {"name": "UniPoly.Trim.size_eq_degree", "content": "theorem size_eq_degree (p : UniPoly R) : p.trim.size = p.degree"}, {"name": "UniPoly.Trim.size_le_size", "content": "theorem size_le_size (p : UniPoly R) : p.trim.size ≤ p.size"}, {"name": "UniPoly.Trim.coeff_eq_getElem_of_lt", "content": "theorem coeff_eq_getElem_of_lt [LawfulBEq R] {p : UniPoly R} {i} (hi : i < p.size) :\n  p.trim.coeff i = p[i]"}, {"name": "UniPoly.Trim.coeff_eq_coeff", "content": "theorem coeff_eq_coeff [LawfulBEq R] (p : UniPoly R) (i : ℕ) :\n  p.trim.coeff i = p.coeff i"}, {"name": "UniPoly.Trim.coeff_eq_getElem", "content": "lemma coeff_eq_getElem {p : UniPoly Q} {i} (hp : i < p.size) :\n  p.coeff i = p[i]"}, {"name": "UniPoly.Trim.coeff_eq_zero", "content": "lemma coeff_eq_zero {p : UniPoly Q} :\n    (∀ i, (hi : i < p.size) → p[i] = 0) ↔ ∀ i, p.coeff i = 0"}, {"name": "UniPoly.Trim.eq_degree_of_equiv", "content": "lemma eq_degree_of_equiv [LawfulBEq R] {p q : UniPoly R} : equiv p q → p.degree = q.degree"}, {"name": "UniPoly.Trim.eq_of_equiv", "content": "theorem eq_of_equiv [LawfulBEq R] {p q : UniPoly R} : equiv p q → p.trim = q.trim"}, {"name": "UniPoly.Trim.canonical_empty", "content": "theorem canonical_empty : (UniPoly.mk (R:=R) #[]).trim = #[]"}, {"name": "UniPoly.Trim.canonical_of_size_zero", "content": "theorem canonical_of_size_zero {p : UniPoly R} : p.size = 0 → p.trim = p"}, {"name": "UniPoly.Trim.canonical_nonempty_iff", "content": "theorem canonical_nonempty_iff [LawfulBEq R] {p : UniPoly R} (hp : p.size > 0) :\n  p.trim = p ↔ p.last_nonzero = some ⟨ p.size - 1, Nat.pred_lt_self hp ⟩"}, {"name": "UniPoly.Trim.last_nonzero_last_iff", "content": "theorem last_nonzero_last_iff [LawfulBEq R] {p : UniPoly R} (hp : p.size > 0) :\n  p.last_nonzero = some ⟨ p.size - 1, Nat.pred_lt_self hp ⟩ ↔ p.getLast hp ≠ 0"}, {"name": "UniPoly.UniPolyC.ext", "content": "@[ext] theorem UniPolyC.ext {p q : UniPolyC R} (h : p.val = q.val) : p = q"}, {"name": "UniPoly.matchSize_size_eq", "content": "lemma matchSize_size_eq {p q : UniPoly Q} :\n    let (p', q') := Array.matchSize p q 0\n    p'.size = q'.size"}, {"name": "UniPoly.matchSize_size", "content": "lemma matchSize_size {p q : UniPoly Q} :\n    let (p', _) := Array.matchSize p q 0\n    p'.size = max p.size q.size"}, {"name": "UniPoly.zipWith_size", "content": "lemma zipWith_size {R} {f : R → R → R} {a b : Array R} (h : a.size = b.size) :\n    (Array.zipWith f a b).size = a.size"}, {"name": "UniPoly.add_size", "content": "theorem add_size {p q : UniPoly Q} : (add_raw p q).size = max p.size q.size"}, {"name": "UniPoly.add_coeff", "content": "theorem add_coeff {p q : UniPoly Q} {i : ℕ} (hi : i < (add_raw p q).size) :\n  (add_raw p q)[i] = p.coeff i + q.coeff i"}, {"name": "UniPoly.zero_add", "content": "theorem zero_add (hp : p.canonical) : 0 + p = p"}, {"name": "UniPoly.coeff_toPoly", "content": "lemma coeff_toPoly {p : UniPoly Q} {n : ℕ} : p.toPoly.coeff n = p.coeff n"}, {"name": "UniPoly.toImpl_elim", "content": "lemma toImpl_elim (p : Q[X]) :\n    (p = 0 ∧ p.toImpl = #[])\n  ∨ (p ≠ 0 ∧ p.toImpl = .ofFn (fun i : Fin (p.natDegree + 1) => p.coeff i))"}, {"name": "UniPoly.toPoly_toImpl", "content": "theorem toPoly_toImpl {p : Q[X]} : p.toImpl.toPoly = p"}, {"name": "UniPoly.trim_toImpl", "content": "theorem trim_toImpl [LawfulBEq R] (p : R[X]) : p.toImpl.trim = p.toImpl"}], "local_ctx": "import Mathlib.Algebra.Tropical.Basic\n\nimport Mathlib.RingTheory.Polynomial.Basic\n\nimport ArkLib.Data.Array.Lemmas\n\nopen Polynomial\n\n@[reducible, inline, specialize]\ndef UniPoly (R : Type*) := Array R\n\ndef Polynomial.toImpl {R : Type*} [Semiring R] (p : R[X]) : UniPoly R :=\n  match p.degree with\n  | ⊥ => #[]\n  | some d  => .ofFn (fun i : Fin (d + 1) => p.coeff i)\n\nnamespace UniPoly\n\n@[reducible]\ndef mk {R : Type*} (coeffs : Array R) : UniPoly R := coeffs\n\nvariable {R : Type*} [Ring R] [BEq R]\n\nvariable {Q : Type*} [Ring Q]\n\n@[reducible]\ndef coeff (p : UniPoly Q) (i : ℕ) : Q := p.getD i 0\n\ndef last_nonzero (p : UniPoly R) : Option (Fin p.size) :=\n  p.findIdxRev? (· != 0)\n\ndef trim (p : UniPoly R) : UniPoly R :=\n  match p.last_nonzero with\n  | none => #[]\n  | some i => p.extract 0 (i.val + 1)\n\ndef degree (p : UniPoly R) : Nat :=\n  match p.last_nonzero with\n  | none => 0\n  | some i => i.val + 1\n\nnamespace Trim\n\ndef last_nonzero_prop {p : UniPoly R} (k : Fin p.size) : Prop :=\n  p[k] ≠ 0 ∧ (∀ j, (hj : j < p.size) → j > k → p[j] = 0)\n\ndef equiv (p q : UniPoly R) : Prop :=\n  ∀ i, p.coeff i = q.coeff i\n\nend Trim\n\ndef UniPolyC (R : Type*) [BEq R] [Ring R] := { p : UniPoly R // p.trim = p }\n\nsection Operations\n\nvariable {S : Type*}\n\ndef eval₂ [Semiring S] (f : R →+* S) (x : S) (p : UniPoly R) : S :=\n  p.zipIdx.foldl (fun acc ⟨a, i⟩ => acc + f a * x ^ i) 0\n\n@[inline, specialize]\ndef add_raw (p q : UniPoly R) : UniPoly R :=\n  let ⟨p', q'⟩ := Array.matchSize p q 0\n  .mk (Array.zipWith (· + ·) p' q' )\n\nvariable (p q r : UniPoly R)\n\ndef canonical (p : UniPoly R) := p.trim = p\n\nend Operations\n\nnamespace OperationsC\n\nvariable {R : Type*} [Ring R] [BEq R] [LawfulBEq R]\n\nvariable (p q r : UniPolyC R)\n\nend OperationsC\n\nsection ToPoly\n\nnoncomputable def toPoly (p : UniPoly R) : Polynomial R :=\n  p.eval₂ Polynomial.C Polynomial.X\n\nnoncomputable def UniPolyC.toPoly (p : UniPolyC R) : Polynomial R := p.val.toPoly\n\nalias ofPoly := Polynomial.toImpl", "target_theorem": "lemma toImpl_toPoly_of_canonical [LawfulBEq R] (p : UniPolyC R) : p.toPoly.toImpl = p :=", "ground_truth_proof": ":= by\n  -- we will change something slightly more general: `toPoly` is injective on canonical polynomials\n  suffices h_inj : ∀ q : UniPolyC R, p.toPoly = q.toPoly → p = q by\n    have : p.toPoly = p.toPoly.toImpl.toPoly := by rw [toPoly_toImpl]\n    exact h_inj ⟨ p.toPoly.toImpl, trim_toImpl p.toPoly ⟩ this |> congrArg Subtype.val |>.symm\n  intro q hpq\n  apply UniPolyC.ext\n  apply Trim.canonical_ext p.property q.property\n  intro i\n  rw [← coeff_toPoly, ← coeff_toPoly]\n  exact hpq |> congrArg (fun p => p.coeff i)", "nesting_depth": 8, "transitive_dep_count": 128, "subset_aristotle": false, "category": "Applied verif."}
{"id": 43, "thm_name": "ConcreteBinaryTower.split_sum_eq_sum_split", "thm_stmt": "theorem split_sum_eq_sum_split {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k)\n  (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1))\n  (h_split_x₀ : split h_pos x₀ = (hi₀, lo₀))\n  (h_split_x₁ : split h_pos x₁ = (hi₁, lo₁)) :\n  split h_pos (x₀ + x₁) = (hi₀ + hi₁, lo₀ + lo₁)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "Prod", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}], "lib_lemmas": [{"name": "BitVec.ofNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}, {"name": "Prod.mk.injEq", "module": "Init.Core"}, {"name": "BitVec.toNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}, {"name": "dcast_trans", "content": "@[simp]\ntheorem dcast_trans (h : a = a') (h' : a' = a'') :\n    dcast h' (dcast h b) = dcast (h.trans h') b"}, {"name": "heq_of_dcast", "content": "theorem heq_of_dcast (ha : a = a') (hb : dcast ha b = b') : HEq b b'"}, {"name": "--", "content": "-- Note: For BitVec, dcast h is just BitVec.cast h (see BitVec.instDCast).\n-- This means for any h : n = m, we have: dcast h (x ^^^ y) = dcast h x ^^^ dcast h y\n-- This can be proved by: subst h; rfl\n-- Similarly, BitVec.extractLsb distributes over xor because shift and mask distribute over xor."}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.add", "content": "def add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.sum_fromNat_eq_from_xor_Nat", "content": "theorem sum_fromNat_eq_from_xor_Nat {k : ℕ} (x y : Nat) :\n  fromNat (k:=k) (x ^^^ y) = fromNat (k:=k) x + fromNat (k:=k) y"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_of_split", "content": "theorem join_of_split {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_split_eq : split h_pos x = (hi_btf, lo_btf)) :\n    x = 《 hi_btf, lo_btf 》"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y\n\ninstance (k : ℕ) : HAdd (ConcreteBTField k) (ConcreteBTField k) (ConcreteBTField k) where\n  hAdd := add\n\ninstance (k : ℕ) : Add (ConcreteBTField k) where\n  add := add\n\n-- split extracts the high and low halves of a bitvector using BitVec.extractLsb,\n-- then casts them to the correct width using dcast. It returns (hi, lo).\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n  have h_hi : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n  let lo : ConcreteBTField (k - 1) := dcast h_lo lo_bits\n  let hi : ConcreteBTField (k - 1) := dcast h_hi hi_bits\n  (hi, lo)\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=", "target_theorem": "theorem split_sum_eq_sum_split {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k)\n  (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1))\n  (h_split_x₀ : split h_pos x₀ = (hi₀, lo₀))\n  (h_split_x₁ : split h_pos x₁ = (hi₁, lo₁)) :\n  split h_pos (x₀ + x₁) = (hi₀ + hi₁, lo₀ + lo₁) :=", "ground_truth_proof": ":= by\n  have h_x₀ := join_of_split h_pos x₀ hi₀ lo₀ h_split_x₀\n  have h_x₁ := join_of_split h_pos x₁ hi₁ lo₁ h_split_x₁\n  -- Approach : convert equation to Nat realm for simple proof\n  have h₀ := (split_bitvec_eq_iff_fromNat (k:=k) (h_pos:=h_pos) x₀ hi₀ lo₀).mp h_split_x₀\n  have h₁ := (split_bitvec_eq_iff_fromNat (k:=k) (h_pos:=h_pos) x₁ hi₁ lo₁).mp h_split_x₁\n  have h_sum_hi : (hi₀ + hi₁) = fromNat (BitVec.toNat (x₀ + x₁) >>> 2 ^ (k - 1)) := by\n    rw [h₀.1, h₁.1]\n    rw [←sum_fromNat_eq_from_xor_Nat]\n    have h_nat_eq : BitVec.toNat x₀ >>> 2 ^ (k - 1) ^^^ BitVec.toNat x₁ >>> 2 ^ (k - 1)\n      = BitVec.toNat (x₀ + x₁) >>> 2 ^ (k - 1) := by\n      -- unfold Concrete BTF addition into BitVec.xor\n      simp only [instHAddConcreteBTField, add, BitVec.xor_eq]\n      rw [Nat.shiftRight_xor_distrib.symm]\n      rw [BitVec.toNat_xor] -- distribution of BitVec.xor over BitVec.toNat\n    rw [h_nat_eq]\n  have h_sum_lo : (lo₀ + lo₁) = fromNat (BitVec.toNat (x₀ + x₁) &&& 2 ^ 2 ^ (k - 1) - 1) := by\n    rw [h₀.2, h₁.2]\n    rw [←sum_fromNat_eq_from_xor_Nat]\n    have h_nat_eq : BitVec.toNat x₀ &&& 2 ^ 2 ^ (k - 1) - 1 ^^^ BitVec.toNat x₁\n      &&& 2 ^ 2 ^ (k - 1) - 1 = BitVec.toNat (x₀ + x₁) &&& 2 ^ 2 ^ (k - 1) - 1 := by\n      simp only [instHAddConcreteBTField, add, BitVec.xor_eq]\n      rw [BitVec.toNat_xor]\n      rw [Nat.and_xor_distrib_right.symm]\n    rw [h_nat_eq]\n  have h_sum_hi_lo : (hi₀ + hi₁, lo₀ + lo₁) = split h_pos (x₀ + x₁) := by\n    rw [(split_bitvec_eq_iff_fromNat (k:=k) (h_pos:=h_pos) (x₀ + x₁)\n      (hi₀ + hi₁) (lo₀ + lo₁)).mpr ⟨h_sum_hi, h_sum_lo⟩]\n  exact h_sum_hi_lo.symm", "nesting_depth": 8, "transitive_dep_count": 106, "subset_aristotle": false, "category": "Applied verif."}
{"id": 44, "thm_name": "ConcreteBinaryTower.concrete_eq_zero_or_eq_one", "thm_stmt": "theorem concrete_eq_zero_or_eq_one {k : ℕ} {a : ConcreteBTField k} (h_k_zero : k = 0)\n : a = zero ∨ a = one", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "Eq.mpr", "module": "Init.Core"}, {"name": "cast", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}], "lib_lemmas": [{"name": "BitVec.cast_ofNat", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ofNat_eq_ofNat", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.eq_zero_or_eq_one", "module": "Init.Data.BitVec.Lemmas"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "BitVec.ofNatLT_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.pow_zero", "module": "Init.Data.Nat.Basic"}, {"name": "cast_cast", "module": "Init.PropLemmas"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "eq_mpr_eq_cast", "module": "Init.PropLemmas"}], "repo_lemmas": [{"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.BitVec.bitvec_cast_eq_dcast", "content": "theorem BitVec.bitvec_cast_eq_dcast {n m : Nat} (h : n = m) (bv : BitVec n) :\n  BitVec.cast h bv = DCast.dcast h bv"}, {"name": "ConcreteBinaryTower.BitVec.cast_one", "content": "@[simp] theorem BitVec.cast_one {n m : ℕ} (h : n = m) : BitVec.cast h 1 = 1#m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_zero", "content": "@[simp] theorem BitVec.dcast_zero {n m : ℕ} (h : n = m) : DCast.dcast h (0#n) = 0#m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_one", "content": "@[simp] theorem BitVec.dcast_one {n m : ℕ} (h : n = m) : DCast.dcast h (1#n) = 1#m"}, {"name": "ConcreteBinaryTower.eq_zero_or_eq_one", "content": "theorem eq_zero_or_eq_one {a : ConcreteBTField 0} : a = zero ∨ a = one"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)", "target_theorem": "theorem concrete_eq_zero_or_eq_one {k : ℕ} {a : ConcreteBTField k} (h_k_zero : k = 0)\n : a = zero ∨ a = one :=", "ground_truth_proof": ":= by\n  if h_k_zero : k = 0 then\n    have h_2_pow_k_eq_1 : 2 ^ k = 1 := by rw [h_k_zero]; norm_num\n    let a0 : ConcreteBTField 0 := Eq.mp (congrArg ConcreteBTField h_k_zero) a\n    have a0_is_eq_mp_a : a0 = Eq.mp (congrArg ConcreteBTField h_k_zero) a := by rfl\n    -- Approach : convert to BitVec.cast and derive equality of the cast for 0 and 1\n    rcases eq_zero_or_eq_one (a := a0) with (ha0 | ha1)\n    · -- a0 = zero\n      left\n      -- Transport equality back to ConcreteBTField k\n      have : a = Eq.mpr (congrArg ConcreteBTField h_k_zero) a0 := by\n        simp only [a0_is_eq_mp_a, eq_mp_eq_cast, eq_mpr_eq_cast, cast_cast, cast_eq]\n      rw [this, ha0]\n      -- zero (k:=k) = Eq.mpr ... (zero (k:=0))\n      have : zero = Eq.mpr (congrArg ConcreteBTField h_k_zero) (zero (k:=0)) := by\n        simp only [zero, eq_mpr_eq_cast, BitVec.zero]\n        rw [←dcast_eq_root_cast]\n        simp only [BitVec.ofNatLT_zero, Nat.pow_zero]\n        rw [BitVec.dcast_zero] -- ⊢ 1 = 2 ^ k\n        exact h_2_pow_k_eq_1.symm\n      rw [this]\n    · -- a0 = one\n      right\n      have : a = Eq.mpr (congrArg ConcreteBTField h_k_zero) a0 := by\n        simp only [a0_is_eq_mp_a, eq_mp_eq_cast, eq_mpr_eq_cast, cast_cast, cast_eq]\n      rw [this, ha1]\n      have : one = Eq.mpr (congrArg ConcreteBTField h_k_zero) (one (k:=0)) := by\n        simp only [one, eq_mpr_eq_cast]\n        rw [←dcast_eq_root_cast]\n        simp only [Nat.pow_zero]\n        rw [BitVec.dcast_one] -- ⊢ 1 = 2 ^ k\n        exact h_2_pow_k_eq_1.symm\n      rw [this]\n  else\n    contradiction", "nesting_depth": 4, "transitive_dep_count": 32, "subset_aristotle": false, "category": "Applied verif."}
{"id": 45, "thm_name": "ConcreteBinaryTower.concrete_mul_left_distrib0", "thm_stmt": "lemma concrete_mul_left_distrib0 (a b c : ConcreteBTField 0) :\n  concrete_mul a (b + c) = concrete_mul a b + concrete_mul a c", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}], "lib_lemmas": [{"name": "BitVec.xor_self", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.eq_zero_or_eq_one", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_eq_zero_iff", "module": "Init.Data.BitVec.Lemmas"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.concrete_mul", "content": "def concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.add_self_cancel", "content": "lemma add_self_cancel {k : ℕ} (a : ConcreteBTField k) : a + a = 0"}, {"name": "ConcreteBinaryTower.add_eq_zero_iff_eq", "content": "lemma add_eq_zero_iff_eq {k : ℕ} (a b : ConcreteBTField k) : a + b = 0 ↔ a = b"}, {"name": "ConcreteBinaryTower.zero_is_0", "content": "lemma zero_is_0 {k : ℕ} : (zero (k:=k)) = (0 : ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.one_is_1", "content": "lemma one_is_1 {k : ℕ} : (one (k:=k)) = 1"}, {"name": "ConcreteBinaryTower.eq_zero_or_eq_one", "content": "theorem eq_zero_or_eq_one {a : ConcreteBTField 0} : a = zero ∨ a = one"}, {"name": "ConcreteBinaryTower.add_eq_one_iff", "content": "lemma add_eq_one_iff (a b : ConcreteBTField 0) :\n  a + b = 1 ↔ (a = 0 ∧ b = 1) ∨ (a = 1 ∧ b = 0)"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\ndef concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/\n\nsection FieldLemmasOfLevel0", "target_theorem": "lemma concrete_mul_left_distrib0 (a b c : ConcreteBTField 0) :\n  concrete_mul a (b + c) = concrete_mul a b + concrete_mul a c :=", "ground_truth_proof": ":= by\n  rcases eq_zero_or_eq_one (a := a) with (ha | ha)\n  · simp [ha, concrete_mul, zero_is_0]  -- a = zero\n  · simp [ha, concrete_mul, zero_is_0, one_is_1];\n    rcases eq_zero_or_eq_one (a := b + c) with (hb_add_c | hb_add_c)\n    · simp [hb_add_c, zero_is_0];\n      rw [zero_is_0] at hb_add_c\n      have b_eq_c : b = c := (add_eq_zero_iff_eq b c).mp hb_add_c\n      simp only [b_eq_c, add_self_cancel]\n    · simp [hb_add_c, one_is_1];\n      have c_cases := (add_eq_one_iff b c).mp hb_add_c\n      rcases eq_zero_or_eq_one (a := b) with (hb | hb)\n      · simp [hb, zero_is_0];\n        rw [one_is_1] at hb_add_c\n        rw [zero_is_0] at hb\n        simp [hb] at c_cases\n        have c_ne_0 : c ≠ 0 := by\n          simp only [c_cases, ne_eq, one_ne_zero, not_false_eq_true]\n        rw [if_neg c_ne_0]\n        exact c_cases.symm\n      · rw [one_is_1] at hb; simp [hb];\n        simp [hb] at c_cases\n        exact c_cases", "nesting_depth": 5, "transitive_dep_count": 32, "subset_aristotle": false, "category": "Applied verif."}
{"id": 46, "thm_name": "coeffs_of_comp_minus_x", "thm_stmt": "theorem coeffs_of_comp_minus_x {f : Polynomial F} {n : ℕ} :\n    (f.comp (-X)).coeff n = if Even n then f.coeff n else -f.coeff n", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/NonBinaryField/Basic.lean", "imports": ["import Mathlib.Tactic.FieldSimp", "import Mathlib.Algebra.Polynomial.FieldDivision", "import Mathlib.Tactic.LinearCombination"], "used_lib_defs": [{"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Even", "module": "Mathlib.Algebra.Group.Even"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}], "lib_lemmas": [{"name": "Nat.even_add_one", "module": "Mathlib.Algebra.Group.Nat.Even"}, {"name": "Nat.even_iff", "module": "Mathlib.Algebra.Group.Nat.Even"}, {"name": "Polynomial.coeff_X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.degree_pos_induction_on", "module": "Mathlib.Algebra.Polynomial.Inductions"}, {"name": "Polynomial.natDegree_eq_zero", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "coeffs_of_comp_minus_x_pos_degree", "content": "private lemma coeffs_of_comp_minus_x_pos_degree {f : Polynomial F} {n : ℕ} (h : 0 < f.degree) :\n    (f.comp (-X)).coeff n = if Even n then f.coeff n else -f.coeff n"}], "local_ctx": "import Mathlib.Algebra.Polynomial.FieldDivision\n\nimport Mathlib.Tactic.FieldSimp\n\nimport Mathlib.Tactic.LinearCombination\n\nsection NonBinaryField\n\nvariable {F : Type*} [NonBinaryField F]\n\nend NonBinaryField\n\nsection\n\nvariable {F : Type*} [Field F]\n\nopen Polynomial", "target_theorem": "theorem coeffs_of_comp_minus_x {f : Polynomial F} {n : ℕ} :\n    (f.comp (-X)).coeff n = if Even n then f.coeff n else -f.coeff n :=", "ground_truth_proof": ":= by\n    by_cases hpos : 0 < f.degree\n    · rw [coeffs_of_comp_minus_x_pos_degree hpos]\n    · have : f.natDegree = 0 := by aesop (add simp natDegree_pos_iff_degree_pos.symm)\n      cases n <;> aesop (add simp natDegree_eq_zero)", "nesting_depth": 2, "transitive_dep_count": 12, "subset_aristotle": false, "category": "Applied verif."}
{"id": 47, "thm_name": "UniPoly.Trim.eq_degree_of_equiv", "thm_stmt": "lemma eq_degree_of_equiv [LawfulBEq R] {p q : UniPoly R} : equiv p q → p.degree = q.degree", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/UniPoly/Basic.lean", "imports": ["import Mathlib.Algebra.Tropical.Basic", "import ArkLib.Data.Array.Lemmas", "import Mathlib.RingTheory.Polynomial.Basic"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "LawfulBEq", "module": "Init.Core"}], "used_repo_defs": [{"name": "findIdxRev?", "content": "def findIdxRev? (cond : α → Bool) (as : Array α) : Option (Fin as.size) :=\n  find ⟨ as.size, Nat.lt_succ_self _ ⟩\nwhere\n  find : Fin (as.size + 1) → Option (Fin as.size)\n    | 0 => none\n    | ⟨ i+1, h ⟩ =>\n      if (cond as[i]) then\n        some ⟨ i, Nat.lt_of_succ_lt_succ h ⟩\n      else\n        find ⟨ i, Nat.lt_of_succ_lt h ⟩"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "findIdxRev?_maximal", "content": "def findIdxRev?_maximal {cond} {as : Array α} {k : Fin as.size} :\n  findIdxRev? cond as = some k → ∀ j : Fin as.size, j > k → ¬ cond as[j] :="}, {"name": "findIdxRev?_def", "content": "def findIdxRev?_def {cond} {as : Array α} {k : Fin as.size} :\n  findIdxRev? cond as = some k → cond as[k] :="}], "lib_lemmas": [{"name": "Nat.lt_succ_self", "module": "Init.Prelude"}, {"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "bne_iff_ne", "module": "Init.SimpLemmas"}, {"name": "bne_self_eq_false", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "Fin.le_antisymm", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.lt_of_not_ge", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.lt_or_ge", "module": "Init.Prelude"}, {"name": "Nat.le_of_not_lt", "module": "Init.Data.Nat.Basic"}], "repo_lemmas": [{"name": "findIdxRev?_eq_some", "content": "theorem findIdxRev?_eq_some {cond} {as : Array α} (h : ∃ i, ∃ hi : i < as.size, cond as[i]) :\n  ∃ k : Fin as.size, findIdxRev? cond as = some k"}, {"name": "findIdxRev?_eq_none", "content": "theorem findIdxRev?_eq_none {cond} {as : Array α} (h : ∀ i, (hi : i < as.size) → ¬ cond as[i]) :\n  findIdxRev? cond as = none"}], "used_local_defs": [{"name": "UniPoly", "content": "@[reducible, inline, specialize]\ndef UniPoly (R : Type*) := Array R"}, {"name": "UniPoly.coeff", "content": "@[reducible]\ndef coeff (p : UniPoly Q) (i : ℕ) : Q := p.getD i 0"}, {"name": "UniPoly.last_nonzero", "content": "def last_nonzero (p : UniPoly R) : Option (Fin p.size) :=\n  p.findIdxRev? (· != 0)"}, {"name": "UniPoly.degree", "content": "def degree (p : UniPoly R) : Nat :=\n  match p.last_nonzero with\n  | none => 0\n  | some i => i.val + 1"}, {"name": "UniPoly.Trim.last_nonzero_prop", "content": "def last_nonzero_prop {p : UniPoly R} (k : Fin p.size) : Prop :=\n  p[k] ≠ 0 ∧ (∀ j, (hj : j < p.size) → j > k → p[j] = 0)"}, {"name": "UniPoly.Trim.equiv", "content": "def equiv (p q : UniPoly R) : Prop :=\n  ∀ i, p.coeff i = q.coeff i"}], "used_local_lemmas": [{"name": "UniPoly.Trim.last_nonzero_none", "content": "theorem last_nonzero_none [LawfulBEq R] {p : UniPoly R} :\n  (∀ i, (hi : i < p.size) → p[i] = 0) → p.last_nonzero = none"}, {"name": "UniPoly.Trim.last_nonzero_some", "content": "theorem last_nonzero_some [LawfulBEq R] {p : UniPoly R} {i} (hi : i < p.size) (h : p[i] ≠ 0) :\n  ∃ k, p.last_nonzero = some k"}, {"name": "UniPoly.Trim.last_nonzero_spec", "content": "theorem last_nonzero_spec [LawfulBEq R] {p : UniPoly R} {k} :\n  p.last_nonzero = some k\n  → p[k] ≠ 0 ∧ (∀ j, (hj : j < p.size) → j > k → p[j] = 0)"}, {"name": "UniPoly.Trim.last_nonzero_unique", "content": "lemma last_nonzero_unique {p : UniPoly Q} {k k' : Fin p.size} :\n  last_nonzero_prop k → last_nonzero_prop k' → k = k'"}, {"name": "UniPoly.Trim.last_nonzero_some_iff", "content": "theorem last_nonzero_some_iff [LawfulBEq R] {p : UniPoly R} {k} :\n  p.last_nonzero = some k ↔ (p[k] ≠ 0 ∧ (∀ j, (hj : j < p.size) → j > k → p[j] = 0))"}, {"name": "UniPoly.Trim.last_nonzero_induct", "content": "theorem last_nonzero_induct [LawfulBEq R] {motive : UniPoly R → Prop}\n  (case1 : ∀ p, p.last_nonzero = none → (∀ i, (hi : i < p.size) → p[i] = 0) → motive p)\n  (case2 : ∀ p : UniPoly R, ∀ k : Fin p.size, p.last_nonzero = some k → p[k] ≠ 0 →\n    (∀ j : ℕ, (hj : j < p.size) → j > k → p[j] = 0) → motive p)\n  (p : UniPoly R) : motive p"}, {"name": "UniPoly.Trim.coeff_eq_zero", "content": "lemma coeff_eq_zero {p : UniPoly Q} :\n    (∀ i, (hi : i < p.size) → p[i] = 0) ↔ ∀ i, p.coeff i = 0"}], "local_ctx": "import Mathlib.Algebra.Tropical.Basic\n\nimport Mathlib.RingTheory.Polynomial.Basic\n\nimport ArkLib.Data.Array.Lemmas\n\nopen Polynomial\n\n@[reducible, inline, specialize]\ndef UniPoly (R : Type*) := Array R\n\nnamespace UniPoly\n\nvariable {R : Type*} [Ring R] [BEq R]\n\nvariable {Q : Type*} [Ring Q]\n\n@[reducible]\ndef coeff (p : UniPoly Q) (i : ℕ) : Q := p.getD i 0\n\ndef last_nonzero (p : UniPoly R) : Option (Fin p.size) :=\n  p.findIdxRev? (· != 0)\n\ndef degree (p : UniPoly R) : Nat :=\n  match p.last_nonzero with\n  | none => 0\n  | some i => i.val + 1\n\nnamespace Trim\n\ndef last_nonzero_prop {p : UniPoly R} (k : Fin p.size) : Prop :=\n  p[k] ≠ 0 ∧ (∀ j, (hj : j < p.size) → j > k → p[j] = 0)\n\ndef equiv (p q : UniPoly R) : Prop :=\n  ∀ i, p.coeff i = q.coeff i", "target_theorem": "lemma eq_degree_of_equiv [LawfulBEq R] {p q : UniPoly R} : equiv p q → p.degree = q.degree :=", "ground_truth_proof": ":= by\n  unfold equiv degree\n  intro h_equiv\n  induction p using last_nonzero_induct with\n  | case1 p h_none_p h_all_zero =>\n    have h_zero_p : ∀ i, p.coeff i = 0 := coeff_eq_zero.mp h_all_zero\n    have h_zero_q : ∀ i, q.coeff i = 0 := by intro i; rw [← h_equiv, h_zero_p]\n    have h_none_q : q.last_nonzero = none := last_nonzero_none (coeff_eq_zero.mpr h_zero_q)\n    rw [h_none_p, h_none_q]\n  | case2 p k h_some_p h_nonzero_p h_max_p =>\n    have h_equiv_k := h_equiv k\n    have k_lt_q : k < q.size := by\n      by_contra h_not_lt\n      have h_ge := Nat.le_of_not_lt h_not_lt\n      simp [h_ge] at h_equiv_k\n      contradiction\n    simp [k_lt_q] at h_equiv_k\n    have h_nonzero_q : q[k.val] ≠ 0 := by rwa [← h_equiv_k]\n    have h_max_q : ∀ j, (hj : j < q.size) → j > k → q[j] = 0 := by\n      intro j hj j_gt_k\n      have h_eq := h_equiv j\n      simp [hj] at h_eq\n      rw [← h_eq]\n      rcases Nat.lt_or_ge j p.size with hj | hj\n      · simp [hj, h_max_p j hj j_gt_k]\n      · simp [hj]\n    have h_some_q : q.last_nonzero = some ⟨ k, k_lt_q ⟩ :=\n      last_nonzero_some_iff.mpr ⟨ h_nonzero_q, h_max_q ⟩\n    rw [h_some_p, h_some_q]", "nesting_depth": 3, "transitive_dep_count": 36, "subset_aristotle": false, "category": "Applied verif."}
{"id": 48, "thm_name": "ConcreteBinaryTower.towerRingHomForwardMap_Z", "thm_stmt": "lemma towerRingHomForwardMap_Z (k : ℕ) :\n  towerRingHomForwardMap k (Z k) = BinaryTower.Z k", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "EquivLike", "module": "Mathlib.Data.FunLike.Equiv"}, {"name": "RingEquiv", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}, {"name": "join_via_add_smul", "content": "def join_via_add_smul {k : ℕ} (h_pos : k > 0) (hi_btf lo_btf : BTField (k - 1)) :\n    BTField k :="}, {"name": "binaryAlgebraTower", "content": "def binaryAlgebraTower {l r : ℕ} (h_le : l ≤ r) : Algebra (BTField l) (BTField r) :="}, {"name": "AlgebraTower.toAlgebra", "content": "@[simp]\ndef AlgebraTower.toAlgebra {i j : ι} (h : i ≤ j) : Algebra (A i) (A j) :=\n  (AlgebraTower.algebraMap (i:=i) (j:=j) (h:=h)).toAlgebra"}, {"name": "Z", "content": "@[simp]\ndef Z (k : ℕ) : BTField k := (BinaryTowerAux k).snd.specialElement"}], "lib_lemmas": [{"name": "BitVec.extractLsb_ofNat", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.zero_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.zero_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.zero_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Equiv.toFun_as_coe", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "EquivLike.coe_coe", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Nat.add_eq_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "RingEquiv.toEquiv_eq_coe", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Nat.ne_zero_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.one_lt_two_pow_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_eq_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.one_mod_two_pow_eq_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.one_mod_two_pow", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.zero_lt_two", "module": "Init.Data.Nat.Basic"}, {"name": "pow_pos", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}], "repo_lemmas": [{"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "join_via_add_smul_zero", "content": "lemma join_via_add_smul_zero {k : ℕ} (h_pos : k > 0) :\n  ⋘ 0, 0 ⋙ = 0"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "zero_lt_pow_n", "content": "theorem zero_lt_pow_n (m : ℕ) (n : ℕ) (h_m : m > 0): 0 < m^n"}, {"name": "join_via_add_smul_one", "content": "lemma join_via_add_smul_one {k : ℕ} (h_pos : k > 0) :\n  ⋘ 0, 1 ⋙ = 1"}, {"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}, {"name": "join_via_add_smul_one_zero_eq_Z", "content": "lemma join_via_add_smul_one_zero_eq_Z {k : ℕ} (h_pos : k > 0) :\n  ⋘ 1, 0 ⋙ = Z k"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.towerRingEquivFromConcrete0", "content": "noncomputable def towerRingEquivFromConcrete0 : ConcreteBTField 0 ≃+* BTField 0 :="}, {"name": "ConcreteBinaryTower.towerRingHomForwardMap", "content": "noncomputable def towerRingHomForwardMap (k : ℕ) : ConcreteBTField k → BTField k :="}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq_zero", "content": "theorem BitVec.dcast_bitvec_eq_zero {l r : ℕ} (h_width_eq : l = r) :\n  dcast (h_width_eq) 0#(l) = 0#(r)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.zero_is_0", "content": "lemma zero_is_0 {k : ℕ} : (zero (k:=k)) = (0 : ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.one_is_1", "content": "lemma one_is_1 {k : ℕ} : (one (k:=k)) = 1"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.split_of_join", "content": "theorem split_of_join {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_join : x = 《hi_btf, lo_btf》) :\n    (hi_btf, lo_btf) = split h_pos x"}, {"name": "ConcreteBinaryTower.split_zero", "content": "theorem split_zero {k : ℕ} (h_pos : k > 0) : split h_pos zero = (zero, zero)"}, {"name": "ConcreteBinaryTower.split_Z", "content": "theorem split_Z {k : ℕ} (h_pos : k > 0) :\n    split h_pos (Z k) = (one (k:=k - 1), zero (k:=k - 1))"}, {"name": "ConcreteBinaryTower.one_bitvec_toNat", "content": "lemma one_bitvec_toNat {width : ℕ} (h_width : width > 0) : (1#width).toNat = 1"}, {"name": "ConcreteBinaryTower.one_bitvec_shiftRight", "content": "lemma one_bitvec_shiftRight {d : ℕ} (h_d : d > 0) : 1 >>> d = 0"}, {"name": "ConcreteBinaryTower.split_one", "content": "lemma split_one {k : ℕ} (h_k : k > 0) :\n    split h_k (one (k:=k)) = (zero (k:=k - 1), one (k:=k - 1))"}, {"name": "ConcreteBinaryTower.towerRingHomForwardMap_zero", "content": "lemma towerRingHomForwardMap_zero {k : ℕ} :\n  (towerRingHomForwardMap k) 0 = 0"}, {"name": "ConcreteBinaryTower.towerRingHomForwardMap_one", "content": "lemma towerRingHomForwardMap_one {k : ℕ} :\n  (towerRingHomForwardMap k) 1 = 1"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\nsection FieldLemmasOfLevel0\n\nend FieldLemmasOfLevel0\n\nsection NumericCasting\n\nend NumericCasting\n\nend FieldOperationsAndInstances\n\nsection BTFieldPropsOneLevelLiftingLemmas\n\nvariable {k : ℕ} {h_k : k > 0}\n\nend BTFieldPropsOneLevelLiftingLemmas\n\nsection TowerFieldsConstruction\n\nend TowerFieldsConstruction\n\nsection ConcreteBTFieldAlgebraConstruction\n\nend ConcreteBTFieldAlgebraConstruction\n\nnoncomputable section ConcreteMultilinearBasis\n\nopen Module\n\nend ConcreteMultilinearBasis\n\nsection TowerEquivalence\n\nopen BinaryTower\n\nnoncomputable def towerRingEquivFromConcrete0 : ConcreteBTField 0 ≃+* BTField 0 :=\n\nnoncomputable def towerRingHomForwardMap (k : ℕ) : ConcreteBTField k → BTField k :=", "target_theorem": "lemma towerRingHomForwardMap_Z (k : ℕ) :\n  towerRingHomForwardMap k (Z k) = BinaryTower.Z k :=", "ground_truth_proof": ":= by\n  induction k with\n  | zero =>\n    unfold towerRingHomForwardMap\n    simp only [RingEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe, ↓reduceDIte,\n      towerRingEquivFromConcrete0]\n    rfl\n  | succ k ih =>\n    unfold towerRingHomForwardMap\n    simp only [Nat.add_eq_zero, one_ne_zero, and_false, ↓reduceDIte,\n      Nat.add_one_sub_one]\n    rw! [split_Z]\n    simp only [Nat.add_one_sub_one, one_is_1, zero_is_0]\n    rw! [towerRingHomForwardMap_zero, towerRingHomForwardMap_one]\n    exact BinaryTower.join_via_add_smul_one_zero_eq_Z (k:=k+1) (h_pos:=by omega)", "nesting_depth": 9, "transitive_dep_count": 196, "subset_aristotle": false, "category": "Applied verif."}
{"id": 49, "thm_name": "Nat.num_eq_highBits_add_lowBits", "thm_stmt": "lemma num_eq_highBits_add_lowBits {n: ℕ} (numLowBits: ℕ) :\n  n = getHighBits numLowBits n + getLowBits numLowBits n", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/Nat/Bitwise.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Data.Nat.Bitwise", "import Mathlib.Data.Finsupp.Basic", "import Mathlib.Algebra.Order.Ring.Star", "import Mathlib.Data.Nat.Digits.Defs", "import Mathlib.Algebra.BigOperators.Fin"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.and_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat.binaryRec", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bit", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bodd", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.boddDiv2", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.div2", "module": "Mathlib.Data.Nat.Bits"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.shiftRight_add", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.eq_of_testBit_eq", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_two_bne_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.one_and_eq_mod_two", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "beq_eq_beq", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.and_assoc", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_comm", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_and_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_two_not_eq_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.testBit_two_pow_sub_one", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "beq_eq_false_iff_ne", "module": "Init.SimpLemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "decide_false", "module": "Init.Core"}, {"name": "decide_true", "module": "Init.Core"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "Nat.and_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "not_le", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "right_eq_ite_iff", "module": "Init.PropLemmas"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}, {"name": "Bool.false_and", "module": "Init.SimpLemmas"}, {"name": "Bool.true_and", "module": "Init.SimpLemmas"}, {"name": "Nat.testBit_two_pow_mul", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.add_mul_div_left", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.and_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.div_add_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.div_eq_zero_iff_lt", "module": "Init.Data.Nat.Div.Lemmas"}, {"name": "Nat.mul_add_mod_self_right", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.zero_and", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.or_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_or", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_xor", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Bool.toNat_lt", "module": "Init.Data.Bool"}, {"name": "Nat.bit_decomp", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.bit_val", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.mul_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.right_distrib", "module": "Init.Data.Nat.Basic"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.zero_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_or_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "IsEmpty.forall_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Nat.getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "Nat.getLowBits", "content": "def getLowBits (numLowBits : ℕ) (n : ℕ) := n &&& ((1 <<< numLowBits) - 1)"}, {"name": "Nat.getHighBits_no_shl", "content": "def getHighBits_no_shl (numLowBits : ℕ) (n : ℕ) : ℕ := n >>> numLowBits"}, {"name": "Nat.getHighBits", "content": "def getHighBits (numLowBits : ℕ) (n : ℕ) : ℕ :=\n  (getHighBits_no_shl numLowBits n) <<< numLowBits"}], "used_local_lemmas": [{"name": "Nat.getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "Nat.eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "Nat.shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "Nat.and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "Nat.div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "Nat.and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "Nat.xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}, {"name": "Nat.or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||| m = (n1 ||| m1) * 2 + (bn ||| bm)"}, {"name": "Nat.sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "Nat.xor_of_and_eq_zero_is_or", "content": "lemma xor_of_and_eq_zero_is_or {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n ^^^ m = n ||| m"}, {"name": "Nat.sum_of_and_eq_zero_is_or", "content": "lemma sum_of_and_eq_zero_is_or {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ||| m"}, {"name": "Nat.getBit_of_multiple_of_power_of_two", "content": "lemma getBit_of_multiple_of_power_of_two {n p : ℕ}: ∀ k,\n  getBit (k) (2^p * n) = if k < p then 0 else getBit (k-p) n"}, {"name": "Nat.getBit_of_shiftLeft", "content": "lemma getBit_of_shiftLeft {n p : ℕ}:\n  ∀ k, getBit (k) (n <<< p) = if k < p then 0 else getBit (k - p) n"}, {"name": "Nat.getBit_of_shiftRight", "content": "lemma getBit_of_shiftRight {n p : ℕ}:\n  ∀ k, getBit k (n >>> p) = getBit (k+p) n"}, {"name": "Nat.getBit_of_and", "content": "lemma getBit_of_and {n m k: ℕ} : getBit k (n &&& m) = getBit k n &&& getBit k m"}, {"name": "Nat.getBit_of_two_pow_sub_one", "content": "lemma getBit_of_two_pow_sub_one {i k: ℕ} : getBit k (2^i - 1) =\n    if k < i then 1 else 0"}, {"name": "Nat.getBit_of_lowBits", "content": "lemma getBit_of_lowBits {n: ℕ} (numLowBits : ℕ) : ∀ k, getBit k (getLowBits numLowBits n) =\n    if k < numLowBits then getBit k n else 0"}, {"name": "Nat.and_highBits_lowBits_eq_zero", "content": "theorem and_highBits_lowBits_eq_zero {n : ℕ} (numLowBits : ℕ) :\n    getHighBits numLowBits n &&& getLowBits numLowBits n = 0"}], "local_ctx": "import ArkLib.Data.Fin.BigOperators\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nimport Mathlib.Algebra.Order.Ring.Star\n\nimport Mathlib.Data.Nat.Bitwise\n\nimport Mathlib.Data.Nat.Digits.Defs\n\nimport Mathlib.Data.Finsupp.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Fin\n\nnamespace Nat\n\ndef getBit (k n : Nat) : Nat := (n >>> k) &&& 1\n\ndef getLowBits (numLowBits : ℕ) (n : ℕ) := n &&& ((1 <<< numLowBits) - 1)\n\ndef getHighBits_no_shl (numLowBits : ℕ) (n : ℕ) : ℕ := n >>> numLowBits\n\ndef getHighBits (numLowBits : ℕ) (n : ℕ) : ℕ :=\n  (getHighBits_no_shl numLowBits n) <<< numLowBits", "target_theorem": "lemma num_eq_highBits_add_lowBits {n: ℕ} (numLowBits: ℕ) :\n  n = getHighBits numLowBits n + getLowBits numLowBits n :=", "ground_truth_proof": ":= by\n  apply eq_iff_eq_all_getBits.mpr; unfold getBit\n  intro k\n  --- use 2 getBit extractions to get the condition for getLowBits of ((n >>> numLowBits) <<<\n  --  numLowBits)\n  set highBits_no_shl := n >>> numLowBits\n  have h_getBit_highBits_shl := getBit_of_shiftLeft (n := highBits_no_shl) (p := numLowBits)\n  have h_getBit_lowBits := getBit_of_lowBits (n := n) (numLowBits := numLowBits)\n  -- AND of highBitss & lowBitss is 0 => we use this to convert the sum into OR\n  have h_and := and_highBits_lowBits_eq_zero (n := n) (numLowBits := numLowBits)\n  rw [sum_of_and_eq_zero_is_or h_and]\n  --- now reason on bitwise operations only\n  rw [Nat.shiftRight_or_distrib, Nat.and_distrib_right]\n  change getBit k n = getBit k ((n >>> numLowBits) <<< numLowBits)\n    ||| getBit k (getLowBits numLowBits n)\n  rw [h_getBit_highBits_shl, h_getBit_lowBits]\n  if h_k: k < numLowBits then\n    simp only [h_k, ↓reduceIte, Nat.zero_or] at *\n  else\n    have h_ne: ¬(k < numLowBits) := by omega\n    have h_num_le_k: numLowBits ≤ k := by omega\n    simp only [h_ne, not_false_eq_true, ↓reduceIte, Nat.or_zero] at *\n    rw [getBit_of_shiftRight]\n    congr\n    rw [Nat.sub_add_cancel (n:=k) (m:=numLowBits) (by omega)]", "nesting_depth": 4, "transitive_dep_count": 103, "subset_aristotle": false, "category": "Applied verif."}
{"id": 50, "thm_name": "BerlekampWelch.elocPolyF_deg", "thm_stmt": "@[simp]\nlemma elocPolyF_deg {ωs f : Fin n → F} : (ElocPolyF ωs f p).natDegree = Δ₀(f, p.eval ∘ ωs)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/CodingTheory/BerlekampWelch/ElocPoly.lean", "imports": ["import ArkLib.Data.CodingTheory.Basic", "import Init.Data.List.FinRange", "import ArkLib.Data.Fin.Lift", "import Mathlib.Data.Finset.Insert", "import Mathlib.Data.Fintype.Card", "import Mathlib.Algebra.Polynomial.FieldDivision", "import Mathlib.Data.Matrix.Mul", "import Mathlib.Algebra.Field.Basic", "import Mathlib.Algebra.Polynomial.Basic", "import Mathlib.Algebra.Polynomial.Degree.Definitions"], "used_lib_defs": [{"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.prod", "module": "Batteries.Data.List.Basic"}, {"name": "List.range", "module": "Init.Data.List.Basic"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Finset.max", "module": "Mathlib.Data.Finset.Max"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Module.finrank", "module": "Mathlib.LinearAlgebra.Dimension.Finrank"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "hammingDist", "module": "Mathlib.InformationTheory.Hamming"}, {"name": "Fin.add", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.elim0", "module": "Init.Data.Fin.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Polynomial.erase", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "hammingDist", "content": "notation \"Δ₀(\" u \", \" v \")\" => hammingDist u v"}, {"name": "distFromCode", "content": "notation \"Δ₀(\" u \", \" C \")\" => distFromCode u C"}, {"name": "scoped macro_rules", "content": "scoped macro_rules\n  | `(ρ $t:term) => `(LinearCode.rate $t)"}, {"name": "liftF", "content": "def liftF (f : Fin n → α) : ℕ → α :=\n  fun m ↦ if h : m < n then f ⟨m, h⟩ else 0"}, {"name": "rate", "content": "noncomputable def rate [Semiring F] (LC : LinearCode ι F) : ℚ≥0 :=\n  (dim LC : ℚ≥0) / length LC"}, {"name": "dim", "content": "noncomputable def dim [Semiring F] (LC : LinearCode ι F) : ℕ :=\n  Module.finrank F LC"}, {"name": "LinearCode.{u,", "content": "abbrev LinearCode.{u, v} (ι : Type u) [Fintype ι] (F : Type v) [Semiring F] : Type (max u v) :=\n  Submodule F (ι → F)"}, {"name": "length", "content": "def length [Semiring F] (_ : LinearCode ι F) : ℕ := Fintype.card ι"}, {"name": "contract", "content": "abbrev contract (m : ℕ) (f : Fin n → α) := liftF (liftF' (n := m) (liftF f))"}, {"name": "liftF'", "content": "def liftF' (f : ℕ → α) : Fin n → α :=\n  fun m ↦ f m.1"}], "lib_lemmas": [{"name": "List.mem_range", "module": "Init.Data.List.Nat.Range"}, {"name": "List.pmap_eq_map", "module": "Init.Data.List.Attach"}, {"name": "List.pmap_eq_map_attach", "module": "Init.Data.List.Attach"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "List.map_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.prod_append", "module": "Batteries.Data.List.Lemmas"}, {"name": "List.range_succ", "module": "Init.Data.List.Range"}, {"name": "Nat.lt_add_one", "module": "Init.Prelude"}, {"name": "Finset.card_filter", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_fin_eq_sum_range", "module": "Mathlib.Data.Fintype.BigOperators"}, {"name": "Finset.sum_range_succ", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Polynomial.X_ne_C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.natDegree_mul", "module": "Mathlib.Algebra.Polynomial.Degree.Domain"}, {"name": "Polynomial.natDegree_one", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "funext_iff", "module": "Init.Ext"}, {"name": "hamming_zero_eq_dist", "module": "Mathlib.InformationTheory.Hamming"}, {"name": "sub_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}], "repo_lemmas": [{"name": "contract_eq_liftF_of_lt", "content": "lemma contract_eq_liftF_of_lt {k : ℕ} (h₁ : k < m) :\n  contract m f' k = liftF f' k"}, {"name": "liftF_succ", "content": "@[simp]\nlemma liftF_succ {f : Fin (n + 1) → α} : liftF f n = f ⟨n, Nat.lt_add_one _⟩"}], "used_local_defs": [{"name": "BerlekampWelch.ElocPoly", "content": "protected noncomputable def ElocPoly (n : ℕ) (ωs f : ℕ → F) (p : Polynomial F) : Polynomial F :=\n  List.prod <| (List.range n).map fun i =>\n    if f i = p.eval (ωs i)\n    then 1\n    else X - C (ωs i)"}, {"name": "BerlekampWelch.ElocPolyF", "content": "noncomputable def ElocPolyF (ωs f : Fin n → F) (p : Polynomial F) : Polynomial F :=\n  ElocPoly n (liftF ωs) (liftF f) p"}], "used_local_lemmas": [{"name": "BerlekampWelch.elocPoly_zero", "content": "@[simp]\nprotected lemma elocPoly_zero : ElocPoly 0 ωs f p = 1"}, {"name": "BerlekampWelch.elocPoly_succ", "content": "@[simp]\nprotected lemma elocPoly_succ :\n  ElocPoly (n + 1) ωs f p =\n  ElocPoly n ωs f p *\n    if f n = p.eval (ωs n)\n    then 1\n    else X - C (ωs n)"}, {"name": "BerlekampWelch.elocPoly_congr", "content": "protected lemma elocPoly_congr {ωs' f' : ℕ → F}\n  (h₁ : ∀ {m}, m < n → ωs m = ωs' m) (h₂ : ∀ {m}, m < n → f m = f' m) :\n  ElocPoly n ωs f = ElocPoly n ωs' f'"}, {"name": "BerlekampWelch.elocPolyF_eq_elocPoly'", "content": "@[simp]\nprotected lemma elocPolyF_eq_elocPoly' {ωs f : Fin n → F} :\n  ElocPolyF ωs f p = ElocPoly n (liftF ωs) (liftF f) p"}, {"name": "BerlekampWelch.elocPoly_leftF_leftF_eq_contract", "content": "protected lemma elocPoly_leftF_leftF_eq_contract {ωs f : Fin m → F} :\n  ElocPoly n (liftF ωs) (liftF f) =\n  ElocPoly n (contract n ωs) (contract n f)"}], "local_ctx": "import Init.Data.List.FinRange\n\nimport Mathlib.Algebra.Field.Basic\n\nimport Mathlib.Algebra.Polynomial.Basic\n\nimport Mathlib.Algebra.Polynomial.Degree.Definitions\n\nimport Mathlib.Algebra.Polynomial.FieldDivision\n\nimport Mathlib.Data.Finset.Insert\n\nimport Mathlib.Data.Fintype.Card\n\nimport Mathlib.Data.Matrix.Mul\n\nimport ArkLib.Data.CodingTheory.Basic\n\nimport ArkLib.Data.Fin.Lift\n\nnamespace BerlekampWelch\n\nvariable {F : Type} [Field F]\n         {m n : ℕ} {p : Polynomial F}\n\nvariable [DecidableEq F]\n\nsection ElocPoly\n\nopen Polynomial\n\nprotected noncomputable def ElocPoly (n : ℕ) (ωs f : ℕ → F) (p : Polynomial F) : Polynomial F :=\n  List.prod <| (List.range n).map fun i =>\n    if f i = p.eval (ωs i)\n    then 1\n    else X - C (ωs i)\n\nsection\n\nopen BerlekampWelch (ElocPoly)\n\nvariable {ωs f : ℕ → F}\n\nopen BerlekampWelch (elocPoly_succ) in\n\nsection\n\nopen Fin\n\nopen BerlekampWelch (elocPoly_congr)\n\nnoncomputable def ElocPolyF (ωs f : Fin n → F) (p : Polynomial F) : Polynomial F :=\n  ElocPoly n (liftF ωs) (liftF f) p\n\nopen BerlekampWelch\n  (elocPolyF_eq_elocPoly' elocPoly_leftF_leftF_eq_contract\n   elocPoly_zero elocPoly_succ)\n\nopen Fin", "target_theorem": "@[simp]\nlemma elocPolyF_deg {ωs f : Fin n → F} : (ElocPolyF ωs f p).natDegree = Δ₀(f, p.eval ∘ ωs) :=", "ground_truth_proof": ":= by\n  rw [elocPolyF_eq_elocPoly']\n  induction' n with n ih\n  · simp only [elocPoly_zero, natDegree_one, hamming_zero_eq_dist]\n    exact funext_iff.2 (Fin.elim0 ·)\n  · rw [\n      elocPoly_succ,\n      natDegree_mul (by simp)\n                    (by aesop (erase simp liftF_succ)\n                              (add simp [sub_eq_zero])\n                              (add safe forward (X_ne_C (liftF ωs n)))),\n      elocPoly_leftF_leftF_eq_contract\n    ]\n    aesop (config := {warnOnNonterminal := false}) (add simp [\n      hammingDist.eq_def, Finset.card_filter, Finset.sum_fin_eq_sum_range, Finset.sum_range_succ\n    ]) <;> (apply Finset.sum_congr rfl; aesop (add safe (by omega)))", "nesting_depth": 4, "transitive_dep_count": 42, "subset_aristotle": false, "category": "Applied verif."}
{"id": 51, "thm_name": "Fin.zero_dappend", "thm_stmt": "@[simp]\ntheorem zero_dappend {motive : Fin (0 + n) → Sort u} {u : (i : Fin 0) → motive (castAdd n i)}\n    (v : (i : Fin n) → motive (natAdd 0 i)) :\n    dappend (motive := motive) u v = fun i => cast (by simp) (v (i.cast (by omega)))", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/Fin/Tuple/Lemmas.lean", "imports": ["import ArkLib.Data.Fin.Tuple.Notation"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.castAdd", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.natAdd", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.castSucc", "module": "Init.Data.Fin.Basic"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "Fin.cast", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.elim0", "module": "Init.Data.Fin.Basic"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Fin.snoc", "module": "Mathlib.Data.Fin.Tuple.Basic"}, {"name": "id", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "dappend", "content": "@[elab_as_elim]\ndef dappend {m n : ℕ} {motive : Fin (m + n) → Sort u}\n    (u : (i : Fin m) → motive (Fin.castAdd n i))\n    (v : (i : Fin n) → motive (Fin.natAdd m i))\n    (i : Fin (m + n)) : motive i :=\n  match n with\n  | 0 => u i\n  | k + 1 => dconcat (dappend u (fun i => v (Fin.castSucc i))) (v (Fin.last k)) i"}, {"name": "dconcat", "content": "@[elab_as_elim]\ndef dconcat {n : ℕ} {motive : Fin (n + 1) → Sort u}\n    (u : (i : Fin n) → motive i.castSucc) (a : motive (last n)) (i : Fin (n + 1)) : motive i :=\n  match n with\n  | 0 => match i with | 0 => a\n  | _ + 1 => dcons (u 0) (dconcat (fun i => u (Fin.succ i)) a) i"}, {"name": "dcons", "content": "@[elab_as_elim]\ndef dcons {n : ℕ} {motive : Fin (n + 1) → Sort u} (a : motive 0)\n    (b : (i : Fin n) → motive i.succ) (i : Fin (n + 1)) : motive i :=\n  match n with\n  | 0 => match i with | 0 => a\n  | _ + 1 => match i with\n    | 0 => a\n    | ⟨k + 1, hk⟩ => b ⟨k, Nat.succ_lt_succ_iff.mp hk⟩"}, {"name": "vcons", "content": "def vcons {n : ℕ} (a : α) (v : Fin n → α) : Fin (n + 1) → α :=\n  dcons a v"}, {"name": "hconcat", "content": "def hconcat {n : ℕ} {α : Fin n → Sort u} {β : Sort u} (u : (i : Fin n) → α i) (a : β) :\n    (i : Fin (n + 1)) → Fin.vconcat α β i :=\n  fconcat (F := id) u a"}, {"name": "fconcat", "content": "def fconcat {A : Sort u} {F : A → Sort v} {n : ℕ} {α : Fin n → A} {β : A}\n    (u : (i : Fin n) → F (α i)) (a : F β) : (i : Fin (n + 1)) → F (Fin.vconcat α β i) :=\n  match n with\n  | 0 => fun i => match i with | 0 => a\n  | _ + 1 => fcons (u 0) (fconcat (fun i => u (Fin.succ i)) a)"}, {"name": "fcons", "content": "def fcons {A : Sort u} {F : A → Sort v} {n : ℕ} {α : A} {β : Fin n → A}\n    (a : F α) (b : (i : Fin n) → F (β i)) : (i : Fin (n + 1)) → F (Fin.vcons α β i) :=\n  match n with\n  | 0 => fun i => match i with | 0 => a\n  | _ + 1 => fun i => match i with\n    | 0 => a\n    | ⟨k + 1, hk⟩ => b ⟨k, Nat.succ_lt_succ_iff.mp hk⟩"}, {"name": "vconcat", "content": "def vconcat {n : ℕ} (v : Fin n → α) (a : α) : Fin (n + 1) → α :=\n  dconcat v a"}, {"name": "happend", "content": "def happend {m n : ℕ} {α : Fin m → Sort u} {β : Fin n → Sort u}\n    (u : (i : Fin m) → α i) (v : (i : Fin n) → β i) : (i : Fin (m + n)) → Fin.vappend α β i :=\n  fappend (F := id) u v"}, {"name": "vappend", "content": "def vappend {m n : ℕ} {α : Sort*} (u : Fin m → α) (v : Fin n → α) : Fin (m + n) → α :=\n  dappend u v"}, {"name": "fappend", "content": "def fappend {A : Sort u} {F : A → Sort v} {m n : ℕ} {α : Fin m → A} {β : Fin n → A}\n    (u : (i : Fin m) → F (α i)) (v : (i : Fin n) → F (β i)) :\n    (i : Fin (m + n)) → F (Fin.vappend α β i) :=\n  match n with\n  | 0 => u\n  | k + 1 => fconcat (fappend u (fun i => v (Fin.castSucc i))) (v (Fin.last k))"}, {"name": "vempty", "content": "abbrev vempty {α : Sort*} : Fin 0 → α := Fin.elim0"}, {"name": "dempty", "content": "def dempty {α : Fin 0 → Sort u} : (i : Fin 0) → α i := fun i => Fin.elim0 i"}, {"name": "hcons", "content": "def hcons {n : ℕ} {α : Sort u} {β : Fin n → Sort u} (a : α) (b : (i : Fin n) → β i) :\n    (i : Fin (n + 1)) → Fin.vcons α β i :=\n  fcons (F := id) a b"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:67 \" ::ᵛ \" => Fin.vcons"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixl:65 \" :+ᵛ \" => Fin.vconcat"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:67 \" ::ʰ \" => Fin.hcons"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixl:65 \" :+ʰ \" => Fin.hconcat"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:67 \" ::ᵈ \" => Fin.dcons"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixl:65 \" :+ᵈ \" => Fin.dconcat"}, {"name": "infixl:65 \" ++ᵛ \" => Fin.vappend", "content": "infixl:65 \" ++ᵛ \" => Fin.vappend"}, {"name": "infixl:65 \" ++ᵈ \" => Fin.dappend", "content": "infixl:65 \" ++ᵈ \" => Fin.dappend"}, {"name": "infixl:65 \" ++ʰ \" => Fin.happend", "content": "infixl:65 \" ++ʰ \" => Fin.happend"}], "lib_lemmas": [{"name": "Fin.ext", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.snoc_castSucc", "module": "Mathlib.Data.Fin.Tuple.Basic"}, {"name": "Fin.snoc_last", "module": "Mathlib.Data.Fin.Tuple.Basic"}, {"name": "Fin.forall_fin_zero_pi", "module": "Mathlib.Data.Fin.Tuple.Basic"}, {"name": "Fin.val_last", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.add_eq", "module": "Init.Data.Nat.Basic"}, {"name": "lt_self_iff_false", "module": "Mathlib.Order.Basic"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Fin.dconcat_last", "content": "@[simp]\ntheorem dconcat_last {motive : Fin (n + 1) → Sort u} (v : (i : Fin n) → motive (castSucc i))\n    (a : motive (last n)) : (v :+ᵈ⟨motive⟩ a) (last n) = a"}, {"name": "Fin.dconcat_castSucc", "content": "@[simp]\ntheorem dconcat_castSucc {motive : Fin (n + 1) → Sort u} (v : (i : Fin n) → motive (castSucc i))\n    (a : motive (last n)) (i : Fin n) : (v :+ᵈ⟨motive⟩ a) (castSucc i) = v i"}, {"name": "Fin.dconcat_eq_snoc", "content": "@[csimp]\ntheorem dconcat_eq_snoc : @dconcat = @snoc"}], "local_ctx": "import ArkLib.Data.Fin.Tuple.Notation\n\nnamespace Fin\n\nvariable {m n : ℕ} {α : Sort u}", "target_theorem": "@[simp]\ntheorem zero_dappend {motive : Fin (0 + n) → Sort u} {u : (i : Fin 0) → motive (castAdd n i)}\n    (v : (i : Fin n) → motive (natAdd 0 i)) :\n    dappend (motive := motive) u v = fun i => cast (by simp) (v (i.cast (by omega))) :=", "ground_truth_proof": ":= by\n  induction n with\n  | zero => ext i; exact Fin.elim0 i\n  | succ n ih =>\n    simp [dappend, ih, dconcat_eq_snoc, Fin.cast, last]\n    ext i\n    by_cases h : i.val < n\n    · have : i = Fin.castSucc ⟨i.val, by simp [h]⟩ := by ext; simp\n      rw [this, snoc_castSucc]\n      simp\n    · have : i.val = n := by omega\n      have : i = Fin.last _ := by ext; simp [this]\n      rw! [this]\n      subst this\n      simp_all only [forall_fin_zero_pi, Nat.add_eq, val_last, zero_add,\n                    lt_self_iff_false, not_false_eq_true, snoc_last]\n      grind only [cases Or]", "nesting_depth": 5, "transitive_dep_count": 27, "subset_aristotle": false, "category": "Applied verif."}
{"id": 52, "thm_name": "BerlekampWelch.solutionToQ_zero", "thm_stmt": "@[simp]\nlemma solutionToQ_zero {v : Fin (2 * 0 + 0) → F} :\n  solutionToQ (F := F) 0 0 v = 0 := rfl", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/CodingTheory/BerlekampWelch/Condition.lean", "imports": ["import Mathlib.Data.Matrix.Reflection", "import ArkLib.Data.CodingTheory.Basic", "import ArkLib.Data.CodingTheory.BerlekampWelch.Sorries", "import Init.Data.List.FinRange", "import Mathlib.Data.Finset.Insert", "import ArkLib.Data.Polynomial.Interface", "import Mathlib.Data.Fintype.Card", "import Mathlib.Algebra.Polynomial.FieldDivision", "import Mathlib.Data.Matrix.Mul", "import Mathlib.Algebra.Field.Basic", "import Mathlib.Algebra.Polynomial.Basic", "import Mathlib.Algebra.Polynomial.Degree.Definitions", "import ArkLib.Data.CodingTheory.BerlekampWelch.ElocPoly"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.add", "module": "Mathlib.Algebra.Group.Pointwise.Finset.Basic"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}], "used_repo_defs": [{"name": "liftF", "content": "def liftF (f : Fin n → α) : ℕ → α :=\n  fun m ↦ if h : m < n then f ⟨m, h⟩ else 0"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "BerlekampWelch.solutionToQ", "content": "def solutionToQ (e k : ℕ) (v : Fin (2 * e + k) → F) : Polynomial F :=\n  ⟨\n    (Finset.range (e + k)).filter (fun x => liftF v (e + x) ≠ 0),\n    fun i => if i < e + k then liftF v (e + i) else 0,\n    by admit /- proof elided -/\n  ⟩"}], "used_local_lemmas": [], "local_ctx": "import Init.Data.List.FinRange\n\nimport Mathlib.Algebra.Field.Basic\n\nimport Mathlib.Algebra.Polynomial.Basic\n\nimport Mathlib.Algebra.Polynomial.Degree.Definitions\n\nimport Mathlib.Algebra.Polynomial.FieldDivision\n\nimport Mathlib.Data.Finset.Insert\n\nimport Mathlib.Data.Fintype.Card\n\nimport Mathlib.Data.Matrix.Mul\n\nimport Mathlib.Data.Matrix.Reflection\n\nimport ArkLib.Data.CodingTheory.Basic\n\nimport ArkLib.Data.Polynomial.Interface\n\nimport ArkLib.Data.CodingTheory.BerlekampWelch.ElocPoly\n\nimport ArkLib.Data.CodingTheory.BerlekampWelch.Sorries\n\nnamespace BerlekampWelch\n\nvariable {α : Type} {F : Type} [Field F]\n         {n e k : ℕ}\n         {i : Fin n}\n         {j : Fin (2 * e + k)}\n         {ωs f : Fin n → F}\n         {v : Fin (2 * e + k) → F}\n         {E Q : Polynomial F}\n         {p : Polynomial F}\n\nsection\n\nopen Polynomial Finset in\n\nopen Fin\n\nopen Polynomial\n\nvariable [DecidableEq F]\n\ndef solutionToQ (e k : ℕ) (v : Fin (2 * e + k) → F) : Polynomial F :=\n  ⟨\n    (Finset.range (e + k)).filter (fun x => liftF v (e + x) ≠ 0),\n    fun i => if i < e + k then liftF v (e + i) else 0,\n    by admit /- proof elided -/\n  ⟩", "target_theorem": "@[simp]\nlemma solutionToQ_zero {v : Fin (2 * 0 + 0) → F} :\n  solutionToQ (F := F) 0 0 v = 0 :=", "ground_truth_proof": ":= rfl", "nesting_depth": 2, "transitive_dep_count": 6, "subset_aristotle": false, "category": "Applied verif."}
{"id": 53, "thm_name": "BinaryTower.eq_join_via_add_smul_eq_iff_split", "thm_stmt": "theorem eq_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)\n    (x : BTField k) (hi_btf lo_btf : BTField (k - 1)) :\n    x = ⋘ hi_btf, lo_btf ⋙ ↔\n  split (k:=k) (h_k:=h_pos) x = (hi_btf, lo_btf)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Basic.lean", "imports": ["import Mathlib.Tactic.DepRewrite", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.RingTheory.AlgebraTower"], "used_lib_defs": [{"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "PowerBasis", "module": "Mathlib.RingTheory.PowerBasis"}, {"name": "Fin", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}, {"name": "AlgebraTower.toAlgebra", "content": "@[simp]\ndef AlgebraTower.toAlgebra {i j : ι} (h : i ≤ j) : Algebra (A i) (A j) :=\n  (AlgebraTower.algebraMap (i:=i) (j:=j) (h:=h)).toAlgebra"}], "lib_lemmas": [{"name": "Ne.dite_eq_left_iff", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.left_eq_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingHom.comp_id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Polynomial.natDegree_eq_of_degree_eq_some", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Algebra.smul_def", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "eqRec_eq_cast", "module": "Batteries.Logic"}], "repo_lemmas": [{"name": "degree_definingPoly", "content": "lemma degree_definingPoly {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (definingPoly s).degree = 2"}, {"name": "degree_s_smul_X_add_1", "content": "lemma degree_s_smul_X_add_1 {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (C s * (X : (F)[X]) + 1).degree = 1"}, {"name": "degree_s_smul_X", "content": "lemma degree_s_smul_X {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (C s * (X : (F)[X])).degree = 1"}, {"name": "unique_linear_form_of_elements_in_adjoined_commring", "content": "theorem unique_linear_form_of_elements_in_adjoined_commring\n    {R : Type*} [CommRing R] (f : R[X]) (hf_deg : f.natDegree = 2)\n    (hf_monic : Monic f) (c1 : AdjoinRoot f) :\n    ∃! p : R × R, c1 = (AdjoinRoot.of f) p.1 * root f + (AdjoinRoot.of f) p.2"}, {"name": "unique_linear_sum_repr", "content": "theorem unique_linear_sum_repr (R : Type*) [CommRing R] (S : Type*) [CommRing S] [Algebra R S]\n    (pb : PowerBasis R S) (h_dim : pb.dim = 2) (s : S) :\n    ∃! p : R × R, s = p.fst • pb.gen + algebraMap R S p.snd"}, {"name": "unique_repr", "content": "theorem unique_repr {R : Type*} [CommRing R] {S : Type*} [CommRing S] [Algebra R S]\n    (pb : PowerBasis R S) (repr1 repr2 : Fin pb.dim →₀ R)\n    (h : ∑ i : Fin pb.dim, repr1 i • pb.basis i = ∑ i : Fin pb.dim, repr2 i • pb.basis i) :\n    repr1 = repr2"}], "used_local_defs": [{"name": "BinaryTower.BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "BinaryTower.BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTower.binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "BinaryTower.BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTower.BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTower.Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "BinaryTower.sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "BinaryTower.Z", "content": "@[simp]\ndef Z (k : ℕ) : BTField k := (BinaryTowerAux k).snd.specialElement"}, {"name": "BinaryTower.poly", "content": "@[simp]\ndef poly (k : ℕ) : Polynomial (BTField k) := definingPoly (s:=(Z k))"}, {"name": "BinaryTower.polyMonic", "content": "instance polyMonic (n : ℕ) : Monic (poly n) := definingPoly_is_monic (Z n)"}, {"name": "BinaryTower.canonicalEmbedding", "content": "def canonicalEmbedding (k : ℕ) : BTField k →+* BTField (k+1) :=\n  AdjoinRoot.of (poly k)"}, {"name": "BinaryTower.towerAlgebraMap", "content": "def towerAlgebraMap (l r : ℕ) (h_le : l ≤ r) : BTField l →+* BTField r :="}, {"name": "BinaryTower.binaryAlgebraTower", "content": "def binaryAlgebraTower {l r : ℕ} (h_le : l ≤ r) : Algebra (BTField l) (BTField r) :="}, {"name": "BinaryTower.algebra_adjacent_tower", "content": "instance (priority := 1000) algebra_adjacent_tower (l : ℕ) :\n  Algebra (BTField l) (BTField (l+1)) :="}, {"name": "BinaryTower.join_via_add_smul", "content": "def join_via_add_smul {k : ℕ} (h_pos : k > 0) (hi_btf lo_btf : BTField (k - 1)) :\n    BTField k :="}, {"name": "BinaryTower.split", "content": "def split (k : ℕ) (h_k : k > 0) (x : BTField k) : BTField (k-1) × BTField (k-1) :="}], "used_local_lemmas": [{"name": "BinaryTower.poly_natDegree_eq_2", "content": "lemma poly_natDegree_eq_2 (k : ℕ) : (poly (k:=k)).natDegree = 2"}, {"name": "BinaryTower.BTField.cast_BTField_eq", "content": "lemma BTField.cast_BTField_eq (k m : ℕ) (h_eq : k = m) :\n  BTField k = BTField m"}, {"name": "BinaryTower.towerAlgebraMap_id", "content": "lemma towerAlgebraMap_id (k : ℕ) : towerAlgebraMap (h_le:=by omega) = RingHom.id (BTField k)"}, {"name": "BinaryTower.towerAlgebraMap_succ_1", "content": "lemma towerAlgebraMap_succ_1 (k : ℕ) :\n  towerAlgebraMap (l:=k) (r:=k+1) (h_le:=by omega) = canonicalEmbedding k"}, {"name": "BinaryTower.binaryTowerAlgebra_def", "content": "lemma binaryTowerAlgebra_def (l r : ℕ) (h_le : l ≤ r) :\n    @binaryAlgebraTower (l:=l) (r:=r) (h_le:=h_le)\n    = (towerAlgebraMap l r h_le).toAlgebra"}, {"name": "BinaryTower.algebraMap_adjacent_tower_def", "content": "lemma algebraMap_adjacent_tower_def (l : ℕ) :\n  (algebraMap (BTField l) (BTField (l + 1))) = canonicalEmbedding l"}, {"name": "BinaryTower.algebraMap_adjacent_tower_succ_eq_Adjoin_of", "content": "lemma algebraMap_adjacent_tower_succ_eq_Adjoin_of (k : ℕ) :\n  (algebraMap (BTField k) (BTField (k + 1))) = of (poly k)"}, {"name": "BinaryTower.unique_linear_decomposition_succ", "content": "theorem unique_linear_decomposition_succ (k : ℕ) :\n  ∀ (x : BTField (k+1)), ∃! (p : BTField k × BTField k),\n    x = ⋘ p.1, p.2 ⋙"}], "local_ctx": "import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude\n\nimport ArkLib.Data.RingTheory.AlgebraTower\n\nimport Mathlib.Tactic.DepRewrite\n\nnamespace BinaryTower\n\nnoncomputable section\n\nopen Polynomial AdjoinRoot Module\n\nsection BTFieldDefs\n\nstructure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k\n\nstructure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0\n\ndef binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :=\n\ndef BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/\n\n@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1\n\n@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)\n\n@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq\n\n@[simp]\ndef Z (k : ℕ) : BTField k := (BinaryTowerAux k).snd.specialElement \n\n@[simp]\ndef poly (k : ℕ) : Polynomial (BTField k) := definingPoly (s:=(Z k))\n\ninstance polyMonic (n : ℕ) : Monic (poly n) := definingPoly_is_monic (Z n)\n\nend BTFieldDefs\n\nsection BinaryAlgebraTower\n\ndef canonicalEmbedding (k : ℕ) : BTField k →+* BTField (k+1) :=\n  AdjoinRoot.of (poly k)\n\ndef towerAlgebraMap (l r : ℕ) (h_le : l ≤ r) : BTField l →+* BTField r :=\n\ndef binaryAlgebraTower {l r : ℕ} (h_le : l ≤ r) : Algebra (BTField l) (BTField r) :=\n\ninstance (priority := 1000) algebra_adjacent_tower (l : ℕ) :\n  Algebra (BTField l) (BTField (l+1)) :=\n\nend BinaryAlgebraTower\n\nnoncomputable section MultilinearBasis\n\ndef join_via_add_smul {k : ℕ} (h_pos : k > 0) (hi_btf lo_btf : BTField (k - 1)) :\n    BTField k :=\n\ndef split (k : ℕ) (h_k : k > 0) (x : BTField k) : BTField (k-1) × BTField (k-1) :=", "target_theorem": "theorem eq_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)\n    (x : BTField k) (hi_btf lo_btf : BTField (k - 1)) :\n    x = ⋘ hi_btf, lo_btf ⋙ ↔\n  split (k:=k) (h_k:=h_pos) x = (hi_btf, lo_btf) :=", "ground_truth_proof": ":= by\n  have h_k_sub_1_add_1_eq_k : k - 1 + 1 = k := by omega\n  have h_BTField_eq := BTField.cast_BTField_eq (k:=k) (m:=k-1+1) (h_eq:=by omega)\n  set p := unique_linear_decomposition_succ (k:=(k-1)) (x:=(Eq.mp (h:=h_BTField_eq) x)) with hp\n  -- -- ⊢ x = join_via_add_smul k h_pos hi lo\n  have h_p_satisfy := p.choose_spec\n  set xPair := p.choose\n  constructor\n  · intro h_x_eq_join\n    -- Due to `unique_linear_decomposition_succ`, there must be exactly one pair\n    -- `(hi, lo)` that satisfies the equation : `x = join_via_add_smul k h_pos hi lo`\n    -- Now we prove `⟨hi_btf, lo_btf⟩` is `Exists.choose` of `unique_linear_decomposition_succ`\n    -- which is actually same as the definition of the `split` function\n    have h_must_eq := h_p_satisfy.2 (⟨hi_btf, lo_btf⟩)\n    simp only [eq_mp_eq_cast] at h_must_eq\n    have h_hibtf_lobtf_eq_xPair := h_must_eq (by\n      rw! (castMode := .all) [h_k_sub_1_add_1_eq_k]\n      simp only [cast_eq]\n      convert h_x_eq_join\n      · rw [eqRec_eq_cast]; rfl\n      · rw [eqRec_eq_cast]; rfl\n    )\n    have h_split_eq_xPair : split k h_pos x = xPair := by rfl\n    rw [h_split_eq_xPair, h_hibtf_lobtf_eq_xPair.symm]\n  · intro h_split_eq\n    unfold split at h_split_eq\n    have h_hibtf_lobtf_eq_xPair : ⟨hi_btf, lo_btf⟩ = xPair := by rw [←h_split_eq]\n    have h_xPair_satisfy_join_via_add_smul := h_p_satisfy.1\n    rw [←h_hibtf_lobtf_eq_xPair] at h_xPair_satisfy_join_via_add_smul\n    rw [eq_mp_eq_cast] at h_xPair_satisfy_join_via_add_smul\n    rw! (castMode := .all) [h_k_sub_1_add_1_eq_k] at h_xPair_satisfy_join_via_add_smul\n    simp only [cast_eq] at h_xPair_satisfy_join_via_add_smul\n    convert h_xPair_satisfy_join_via_add_smul\n    · rw [eqRec_eq_cast]; rfl\n    · rw [eqRec_eq_cast]; rfl", "nesting_depth": 6, "transitive_dep_count": 98, "subset_aristotle": false, "category": "Applied verif."}
{"id": 54, "thm_name": "BinaryTower.algebraMap_eq_zero_x", "thm_stmt": "lemma algebraMap_eq_zero_x {i j : ℕ} (h_le : i < j) (x : BTField i) :\n    letI instAlgebra := binaryAlgebraTower (l:=i) (r:=j) (h_le:=by omega)\n    letI instAlgebraPred := binaryAlgebraTower (l:=i) (r:=j-1) (h_le:=by omega)\n    algebraMap (BTField i) (BTField j) x\n      = ⋘ 0, algebraMap (BTField i) (BTField (j-1)) x ⋙", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Basic.lean", "imports": ["import Mathlib.Tactic.DepRewrite", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.RingTheory.AlgebraTower"], "used_lib_defs": [{"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Algebra.algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "PowerBasis", "module": "Mathlib.RingTheory.PowerBasis"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "cast", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}, {"name": "AlgebraTower.toAlgebra", "content": "@[simp]\ndef AlgebraTower.toAlgebra {i j : ι} (h : i ≤ j) : Algebra (A i) (A j) :=\n  (AlgebraTower.algebraMap (i:=i) (j:=j) (h:=h)).toAlgebra"}], "lib_lemmas": [{"name": "Ne.dite_eq_left_iff", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.left_eq_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingHom.comp_id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Polynomial.natDegree_eq_of_degree_eq_some", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Algebra.smul_def", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "eqRec_eq_cast", "module": "Batteries.Logic"}, {"name": "Algebra.smul_def'", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Nat.sub_one_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "cast_cast", "module": "Init.PropLemmas"}, {"name": "map_zero", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Function.comp_apply", "module": "Init.Core"}, {"name": "Polynomial.ext", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "RingHom.coe_comp", "module": "Mathlib.Algebra.Ring.Hom.Defs"}], "repo_lemmas": [{"name": "degree_definingPoly", "content": "lemma degree_definingPoly {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (definingPoly s).degree = 2"}, {"name": "degree_s_smul_X_add_1", "content": "lemma degree_s_smul_X_add_1 {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (C s * (X : (F)[X]) + 1).degree = 1"}, {"name": "degree_s_smul_X", "content": "lemma degree_s_smul_X {F : Type*} [Field F] [Fintype F] (s : F) [NeZero s] :\n  (C s * (X : (F)[X])).degree = 1"}, {"name": "unique_linear_form_of_elements_in_adjoined_commring", "content": "theorem unique_linear_form_of_elements_in_adjoined_commring\n    {R : Type*} [CommRing R] (f : R[X]) (hf_deg : f.natDegree = 2)\n    (hf_monic : Monic f) (c1 : AdjoinRoot f) :\n    ∃! p : R × R, c1 = (AdjoinRoot.of f) p.1 * root f + (AdjoinRoot.of f) p.2"}, {"name": "unique_linear_sum_repr", "content": "theorem unique_linear_sum_repr (R : Type*) [CommRing R] (S : Type*) [CommRing S] [Algebra R S]\n    (pb : PowerBasis R S) (h_dim : pb.dim = 2) (s : S) :\n    ∃! p : R × R, s = p.fst • pb.gen + algebraMap R S p.snd"}, {"name": "unique_repr", "content": "theorem unique_repr {R : Type*} [CommRing R] {S : Type*} [CommRing S] [Algebra R S]\n    (pb : PowerBasis R S) (repr1 repr2 : Fin pb.dim →₀ R)\n    (h : ∑ i : Fin pb.dim, repr1 i • pb.basis i = ∑ i : Fin pb.dim, repr2 i • pb.basis i) :\n    repr1 = repr2"}], "used_local_defs": [{"name": "BinaryTower.BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "BinaryTower.BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTower.binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "BinaryTower.BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTower.BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTower.Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "BinaryTower.sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "BinaryTower.Z", "content": "@[simp]\ndef Z (k : ℕ) : BTField k := (BinaryTowerAux k).snd.specialElement"}, {"name": "BinaryTower.poly", "content": "@[simp]\ndef poly (k : ℕ) : Polynomial (BTField k) := definingPoly (s:=(Z k))"}, {"name": "BinaryTower.polyMonic", "content": "instance polyMonic (n : ℕ) : Monic (poly n) := definingPoly_is_monic (Z n)"}, {"name": "BinaryTower.canonicalEmbedding", "content": "def canonicalEmbedding (k : ℕ) : BTField k →+* BTField (k+1) :=\n  AdjoinRoot.of (poly k)"}, {"name": "BinaryTower.towerAlgebraMap", "content": "def towerAlgebraMap (l r : ℕ) (h_le : l ≤ r) : BTField l →+* BTField r :="}, {"name": "BinaryTower.binaryAlgebraTower", "content": "def binaryAlgebraTower {l r : ℕ} (h_le : l ≤ r) : Algebra (BTField l) (BTField r) :="}, {"name": "BinaryTower.algebra_adjacent_tower", "content": "instance (priority := 1000) algebra_adjacent_tower (l : ℕ) :\n  Algebra (BTField l) (BTField (l+1)) :="}, {"name": "BinaryTower.join_via_add_smul", "content": "def join_via_add_smul {k : ℕ} (h_pos : k > 0) (hi_btf lo_btf : BTField (k - 1)) :\n    BTField k :="}, {"name": "BinaryTower.split", "content": "def split (k : ℕ) (h_k : k > 0) (x : BTField k) : BTField (k-1) × BTField (k-1) :="}], "used_local_lemmas": [{"name": "BinaryTower.poly_natDegree_eq_2", "content": "lemma poly_natDegree_eq_2 (k : ℕ) : (poly (k:=k)).natDegree = 2"}, {"name": "BinaryTower.BTField.cast_BTField_eq", "content": "lemma BTField.cast_BTField_eq (k m : ℕ) (h_eq : k = m) :\n  BTField k = BTField m"}, {"name": "BinaryTower.BTField.RingHom_eq_of_dest_eq", "content": "@[simp]\ntheorem BTField.RingHom_eq_of_dest_eq (k m n : ℕ) (h_eq : m = n) :\n  (BTField k →+* BTField m) = (BTField k →+* BTField n)"}, {"name": "BinaryTower.BTField.RingHom_cast_dest_apply", "content": "@[simp]\ntheorem BTField.RingHom_cast_dest_apply (k m n : ℕ) (h_eq : m = n)\n  (f : BTField k →+* BTField m) (x : BTField k) :\n    (cast (BTField.RingHom_eq_of_dest_eq (k:=k) (m:=m) (n:=n) h_eq) f) x\n    = cast (by apply cast_BTField_eq (h_eq:=h_eq)) (f x)"}, {"name": "BinaryTower.towerAlgebraMap_id", "content": "lemma towerAlgebraMap_id (k : ℕ) : towerAlgebraMap (h_le:=by omega) = RingHom.id (BTField k)"}, {"name": "BinaryTower.towerAlgebraMap_succ_1", "content": "lemma towerAlgebraMap_succ_1 (k : ℕ) :\n  towerAlgebraMap (l:=k) (r:=k+1) (h_le:=by omega) = canonicalEmbedding k"}, {"name": "BinaryTower.towerAlgebraMap_succ", "content": "lemma towerAlgebraMap_succ (l r : ℕ) (h_le : l ≤ r) :\n  towerAlgebraMap (l:=l) (r:=r+1) (h_le:=by omega) =\n  (towerAlgebraMap (l:=r) (r:=r+1) (h_le:=by omega)).comp\n  (towerAlgebraMap (l:=l) (r:=r) (h_le:=by omega))"}, {"name": "BinaryTower.binaryTowerAlgebra_def", "content": "lemma binaryTowerAlgebra_def (l r : ℕ) (h_le : l ≤ r) :\n    @binaryAlgebraTower (l:=l) (r:=r) (h_le:=h_le)\n    = (towerAlgebraMap l r h_le).toAlgebra"}, {"name": "BinaryTower.algebraMap_adjacent_tower_def", "content": "lemma algebraMap_adjacent_tower_def (l : ℕ) :\n  (algebraMap (BTField l) (BTField (l + 1))) = canonicalEmbedding l"}, {"name": "BinaryTower.algebraMap_adjacent_tower_succ_eq_Adjoin_of", "content": "lemma algebraMap_adjacent_tower_succ_eq_Adjoin_of (k : ℕ) :\n  (algebraMap (BTField k) (BTField (k + 1))) = of (poly k)"}, {"name": "BinaryTower.unique_linear_decomposition_succ", "content": "theorem unique_linear_decomposition_succ (k : ℕ) :\n  ∀ (x : BTField (k+1)), ∃! (p : BTField k × BTField k),\n    x = ⋘ p.1, p.2 ⋙"}, {"name": "BinaryTower.eq_join_via_add_smul_eq_iff_split", "content": "theorem eq_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)\n    (x : BTField k) (hi_btf lo_btf : BTField (k - 1)) :\n    x = ⋘ hi_btf, lo_btf ⋙ ↔\n  split (k:=k) (h_k:=h_pos) x = (hi_btf, lo_btf)"}, {"name": "BinaryTower.split_algebraMap_eq_zero_x", "content": "lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : BTField (k - 1)) :\n  letI instAlgebra := binaryAlgebraTower (l:=k-1) (r:=k) (h_le:=by omega)\n  split (k:=k) (h_k:=h_pos) (algebraMap (BTField (k - 1)) (BTField k) x) = (0, x)"}, {"name": "BinaryTower.algebraMap_succ_eq_zero_x", "content": "lemma algebraMap_succ_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : BTField (k - 1)) :\n  letI instAlgebra := binaryAlgebraTower (l:=k-1) (r:=k) (h_le:=by omega)\n  algebraMap (BTField (k - 1)) (BTField k) x = ⋘ 0, x ⋙"}], "local_ctx": "import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude\n\nimport ArkLib.Data.RingTheory.AlgebraTower\n\nimport Mathlib.Tactic.DepRewrite\n\nnamespace BinaryTower\n\nnoncomputable section\n\nopen Polynomial AdjoinRoot Module\n\nsection BTFieldDefs\n\nstructure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k\n\nstructure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0\n\ndef binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :=\n\ndef BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/\n\n@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1\n\n@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)\n\n@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq\n\n@[simp]\ndef Z (k : ℕ) : BTField k := (BinaryTowerAux k).snd.specialElement \n\n@[simp]\ndef poly (k : ℕ) : Polynomial (BTField k) := definingPoly (s:=(Z k))\n\ninstance polyMonic (n : ℕ) : Monic (poly n) := definingPoly_is_monic (Z n)\n\nend BTFieldDefs\n\nsection BinaryAlgebraTower\n\ndef canonicalEmbedding (k : ℕ) : BTField k →+* BTField (k+1) :=\n  AdjoinRoot.of (poly k)\n\ndef towerAlgebraMap (l r : ℕ) (h_le : l ≤ r) : BTField l →+* BTField r :=\n\ndef binaryAlgebraTower {l r : ℕ} (h_le : l ≤ r) : Algebra (BTField l) (BTField r) :=\n\ninstance (priority := 1000) algebra_adjacent_tower (l : ℕ) :\n  Algebra (BTField l) (BTField (l+1)) :=\n\nend BinaryAlgebraTower\n\nnoncomputable section MultilinearBasis\n\ndef join_via_add_smul {k : ℕ} (h_pos : k > 0) (hi_btf lo_btf : BTField (k - 1)) :\n    BTField k :=\n\ndef split (k : ℕ) (h_k : k > 0) (x : BTField k) : BTField (k-1) × BTField (k-1) :=", "target_theorem": "lemma algebraMap_eq_zero_x {i j : ℕ} (h_le : i < j) (x : BTField i) :\n    letI instAlgebra :=", "ground_truth_proof": ":= binaryAlgebraTower (l:=i) (r:=j) (h_le:=by omega)\n    letI instAlgebraPred := binaryAlgebraTower (l:=i) (r:=j-1) (h_le:=by omega)\n    algebraMap (BTField i) (BTField j) x\n      = ⋘ 0, algebraMap (BTField i) (BTField (j-1)) x ⋙ := by\n  set d := j - i with d_eq\n  induction hd : d with\n  | zero =>\n    have h_i_eq_j : i = j := by omega\n    have h_i_ne_j : i ≠ j := by omega\n    contradiction\n  | succ d' => -- this one does not even use inductive hypothesis\n    have h_j_eq : j = i + d' + 1 := by omega\n    change (towerAlgebraMap (l:=i) (r:=j) (h_le:=by omega)) x =\n      join_via_add_smul (h_pos:=by omega) 0 ((towerAlgebraMap (l:=i) (r:=j-1) (h_le:=by omega)) x)\n    rw! [h_j_eq]\n    rw [towerAlgebraMap_succ (l:=i) (r:=i+d') (h_le:=by omega)]\n    simp only [RingHom.coe_comp, Function.comp_apply, Nat.add_one_sub_one]\n    set r := towerAlgebraMap (l:=i) (r:=i+d') (h_le:=by omega) x with h_r\n    have h := algebraMap_succ_eq_zero_x (k:=i+d'+1) (h_pos:=by omega) r\n    simp only [Nat.add_one_sub_one] at h\n    rw [←h]\n    rfl", "nesting_depth": 8, "transitive_dep_count": 114, "subset_aristotle": false, "category": "Applied verif."}
{"id": 55, "thm_name": "Nat.getBit_of_sub_two_pow_of_bit_1", "thm_stmt": "lemma getBit_of_sub_two_pow_of_bit_1 {n i j: ℕ} (h_getBit_eq_1: getBit i n = 1) :\n  getBit j (n - 2^i) = (if j = i then 0 else getBit j n)", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/Nat/Bitwise.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Data.Nat.Bitwise", "import Mathlib.Data.Finsupp.Basic", "import Mathlib.Algebra.Order.Ring.Star", "import Mathlib.Data.Nat.Digits.Defs", "import Mathlib.Algebra.BigOperators.Fin"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.binaryRec", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bit", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bodd", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.boddDiv2", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.div2", "module": "Mathlib.Data.Nat.Bits"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Bool.toNat_true", "module": "Init.Data.Bool"}, {"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_two_pow", "module": "Mathlib.Data.Nat.Bitwise"}, {"name": "Nat.mod_two_bne_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.one_and_eq_mod_two", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.add_mul_div_left", "module": "Init.Data.Nat.Div.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.div_add_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.div_eq_zero_iff_lt", "module": "Init.Data.Nat.Div.Lemmas"}, {"name": "Nat.mul_add_mod_self_right", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.zero_and", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.or_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_or", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_xor", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Bool.toNat_lt", "module": "Init.Data.Bool"}, {"name": "Nat.bit_decomp", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.bit_val", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.mul_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.right_distrib", "module": "Init.Data.Nat.Basic"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.add_left_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.add_zero", "module": "Init.Core"}, {"name": "Nat.and_assoc", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mul_eq_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_eq_of_eq_add", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.xor_assoc", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_eq_zero", "module": "Mathlib.Data.Nat.Bitwise"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.pow_lt_pow_right", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftLeft_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_add", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.two_pow_mod_two_eq_zero", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "Nat.ge_two_pow_of_testBit", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "if_neg", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Nat.getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}], "used_local_lemmas": [{"name": "Nat.testBit_true_eq_getBit_eq_1", "content": "lemma testBit_true_eq_getBit_eq_1 (k n : Nat) : n.testBit k = ((Nat.getBit k n) = 1)"}, {"name": "Nat.getBit_two_pow", "content": "lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)"}, {"name": "Nat.and_two_pow_eq_two_pow_of_getBit_1", "content": "lemma and_two_pow_eq_two_pow_of_getBit_1 {n i : ℕ} (h_getBit: getBit i n = 1) :\n    n &&& (2 ^ i) = 2 ^ i"}, {"name": "Nat.div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "Nat.and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "Nat.xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}, {"name": "Nat.or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||| m = (n1 ||| m1) * 2 + (bn ||| bm)"}, {"name": "Nat.sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "Nat.xor_eq_sub_iff_submask", "content": "lemma xor_eq_sub_iff_submask {n m : ℕ} (h: m ≤ n) : n ^^^ m = n - m ↔ n &&& m = m"}, {"name": "Nat.getBit_of_xor", "content": "lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m"}], "local_ctx": "import ArkLib.Data.Fin.BigOperators\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nimport Mathlib.Algebra.Order.Ring.Star\n\nimport Mathlib.Data.Nat.Bitwise\n\nimport Mathlib.Data.Nat.Digits.Defs\n\nimport Mathlib.Data.Finsupp.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Fin\n\nnamespace Nat\n\ndef getBit (k n : Nat) : Nat := (n >>> k) &&& 1", "target_theorem": "lemma getBit_of_sub_two_pow_of_bit_1 {n i j: ℕ} (h_getBit_eq_1: getBit i n = 1) :\n  getBit j (n - 2^i) = (if j = i then 0 else getBit j n) :=", "ground_truth_proof": ":= by\n  have h_2_pow_i_lt_n: 2^i ≤ n := by\n    apply Nat.ge_two_pow_of_testBit\n    rw [Nat.testBit_true_eq_getBit_eq_1]\n    exact h_getBit_eq_1\n  have h_xor_eq_sub := (Nat.xor_eq_sub_iff_submask (n:=n) (m:=2^i) (h_2_pow_i_lt_n)).mpr (by\n    exact and_two_pow_eq_two_pow_of_getBit_1 h_getBit_eq_1)\n  rw [h_xor_eq_sub.symm]\n  rw [Nat.getBit_of_xor]\n  if h_j_eq_i: j = i then\n    rw [h_j_eq_i]\n    rw [h_getBit_eq_1]\n    rw [Nat.getBit_two_pow]\n    simp only [BEq.rfl, ↓reduceIte, Nat.xor_self]\n  else\n    rw [Nat.getBit_two_pow]\n    simp only [beq_iff_eq]\n    simp only [h_j_eq_i, ↓reduceIte]\n    push_neg at h_j_eq_i\n    simp only [if_neg h_j_eq_i.symm, xor_zero]", "nesting_depth": 4, "transitive_dep_count": 78, "subset_aristotle": false, "category": "Applied verif."}
{"id": 56, "thm_name": "Binius.BinaryBasefold.toOutCodewordsCount_succ_eq", "thm_stmt": "lemma toOutCodewordsCount_succ_eq (i : Fin ℓ) :\n  (toOutCodewordsCount ℓ ϑ i.succ) =\n    if isCommitmentRound ℓ ϑ i then (toOutCodewordsCount ℓ ϑ i.castSucc) + 1\n    else (toOutCodewordsCount ℓ ϑ i.castSucc)", "lean_root": "ArkLib", "rel_path": "ArkLib/ProofSystem/Binius/BinaryBasefold/Basic.lean", "imports": ["import ArkLib.ProofSystem.Binius.BinaryBasefold.Prelude"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Ne", "module": "Init.Core"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Matrix.neg", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.succ_div_of_dvd", "module": "Init.Data.Nat.Div.Lemmas"}, {"name": "Nat.succ_div_of_not_dvd", "module": "Init.Data.Nat.Div.Lemmas"}, {"name": "Fin.coe_castSucc", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.val_pos_iff", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.add_le_add_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.add_mul", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_of_lt_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.div_eq_zero_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.le_of_add_le_add_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.le_of_dvd", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "Nat.mul_div_eq_iff_dvd", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.sub_one_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "NeZero.ne'", "module": "Init.Data.NeZero"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_pos_iff", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "lt_add_iff_pos_left", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "or_true", "module": "Init.SimpLemmas"}, {"name": "zero_lt_one", "module": "Mathlib.Algebra.Order.ZeroLEOne"}, {"name": "Decidable.not_not", "module": "Init.PropLemmas"}, {"name": "Fact.out", "module": "Mathlib.Logic.Basic"}, {"name": "Fin.is_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.add_eq_left", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.add_left_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "left_eq_ite_iff", "module": "Init.PropLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_le", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Binius.BinaryBasefold.toOutCodewordsCount", "content": "def toOutCodewordsCount (i : Fin (ℓ + 1)) : ℕ :="}, {"name": "Binius.BinaryBasefold.isCommitmentRound", "content": "def isCommitmentRound (i : Fin ℓ) : Prop :=\n  ϑ ∣ i.val + 1 ∧ i.val + 1 ≠ ℓ"}], "used_local_lemmas": [{"name": "Binius.BinaryBasefold.div_add_one_eq_if_dvd", "content": "lemma div_add_one_eq_if_dvd (i ϑ : ℕ) [NeZero ϑ] :\n    (i + 1) / ϑ = if ϑ ∣ i + 1 then i / ϑ + 1 else i / ϑ"}, {"name": "Binius.BinaryBasefold.toOutCodewordsCount_succ_eq_add_one_iff", "content": "omit hdiv in\nlemma toOutCodewordsCount_succ_eq_add_one_iff (i : Fin ℓ) :\n    isCommitmentRound ℓ ϑ i ↔\n    (toOutCodewordsCount ℓ ϑ i.castSucc) + 1 = toOutCodewordsCount ℓ ϑ i.succ"}], "local_ctx": "import ArkLib.ProofSystem.Binius.BinaryBasefold.Prelude\n\nnoncomputable section\n\nnamespace Binius.BinaryBasefold\n\nopen OracleSpec OracleComp ProtocolSpec Finset AdditiveNTT Polynomial MvPolynomial\n  Binius.BinaryBasefold\n\nopen scoped NNReal\n\nopen ReedSolomon Code BerlekampWelch\n\nopen Finset AdditiveNTT Polynomial MvPolynomial Nat Matrix\n\nvariable {L : Type} [CommRing L] (ℓ : ℕ) [NeZero ℓ]\n\nvariable (𝓑 : Fin 2 ↪ L)\n\nsection OracleStatementIndex\n\nvariable (ℓ : ℕ) (ϑ : ℕ) [NeZero ℓ] [NeZero ϑ] [hdiv : Fact (ϑ ∣ ℓ)]\n\ndef toOutCodewordsCount (i : Fin (ℓ + 1)) : ℕ :=\n\ndef isCommitmentRound (i : Fin ℓ) : Prop :=\n  ϑ ∣ i.val + 1 ∧ i.val + 1 ≠ ℓ\n\nopen Classical in", "target_theorem": "lemma toOutCodewordsCount_succ_eq (i : Fin ℓ) :\n  (toOutCodewordsCount ℓ ϑ i.succ) =\n    if isCommitmentRound ℓ ϑ i then (toOutCodewordsCount ℓ ϑ i.castSucc) + 1\n    else (toOutCodewordsCount ℓ ϑ i.castSucc) :=", "ground_truth_proof": ":= by\n  have h_succ_val: i.succ.val = i.val + 1 := rfl\n  by_cases hv: ϑ ∣ i.val + 1 ∧ i.val + 1 ≠ ℓ\n  · have h_succ := (toOutCodewordsCount_succ_eq_add_one_iff ℓ ϑ i).mp hv\n    rw [←h_succ];\n    simp only [left_eq_ite_iff, Nat.add_eq_left, one_ne_zero, imp_false, Decidable.not_not]\n    exact hv\n  · rw [isCommitmentRound]\n    simp [ne_eq, hv, ↓reduceIte]\n    unfold toOutCodewordsCount\n    have h_i_lt_ℓ: i.castSucc.val < ℓ := by\n      change i.val < ℓ\n      omega\n    simp only [Fin.val_succ, Fin.coe_castSucc, Fin.is_lt, ↓reduceIte]\n    rw [div_add_one_eq_if_dvd]\n    by_cases hv_div_succ: ϑ ∣ i.val + 1\n    · simp only [hv_div_succ, ↓reduceIte, Nat.add_eq_left, ite_eq_right_iff, one_ne_zero,\n      imp_false, not_lt, ge_iff_le]\n      simp only [hv_div_succ, ne_eq, true_and, Decidable.not_not] at hv\n      have h_eq: i.succ.val = ℓ := by\n        change i.succ.val = (⟨ℓ, by omega⟩: Fin (ℓ + 1)).val\n        exact hv\n      omega\n    · simp only [hv_div_succ, ↓reduceIte, Nat.add_left_cancel_iff, ite_eq_left_iff, not_lt,\n      zero_ne_one, imp_false, not_le, gt_iff_lt]\n      if hi_succ_lt: i.succ.val < ℓ then\n        omega\n      else\n        simp only [Fin.val_succ, not_lt] at hi_succ_lt\n        have hi_succ_le_ℓ: i.succ.val ≤ ℓ := by omega\n        have hi_succ_eq_ℓ: i.val + 1 = ℓ := by omega\n        rw [hi_succ_eq_ℓ] at hv_div_succ\n        exact False.elim (hv_div_succ (hdiv.out))", "nesting_depth": 3, "transitive_dep_count": 53, "subset_aristotle": false, "category": "Applied verif."}
{"id": 57, "thm_name": "AdditiveNTT.evalWAt_eq_W", "thm_stmt": "theorem evalWAt_eq_W (i : Fin r) (x : L) :\n  evalWAt (β := β) (ℓ := ℓ) (R_rate := R_rate) (i := i) x =\n    (W (𝔽q := 𝔽q) (β := β) (i := i)).eval x", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Impl", "import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT", "import ArkLib.Data.Nat.Bitwise", "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "CommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Finset.sum", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "LinearIndependent", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Nat.reduceBEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.mk", "module": "Init.Prelude"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Module.finrank", "module": "Mathlib.LinearAlgebra.Dimension.Finrank"}], "used_repo_defs": [{"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "BitVec.instDCast", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "ConcreteBTField.instDCast_local", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}], "lib_lemmas": [{"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "Fact.out", "module": "Mathlib.Logic.Basic"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_sub_distrib", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Fintype.bijective_iff_injective_and_card", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "Fintype.card_fin", "module": "Mathlib.Data.Fintype.Card"}, {"name": "LinearIndependent.comp", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "Nat.le_of_lt_succ", "module": "Init.Prelude"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "linearIndependent_iff'", "module": "Mathlib.LinearAlgebra.LinearIndependent.Defs"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "sub_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_smul", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "sub_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}, {"name": "Finset.prod_bij", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "List.map_map", "module": "Init.Data.List.Lemmas"}, {"name": "Polynomial.eval_C", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_prod", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.eval_sub", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "getBit_of_lt_two_pow", "content": "lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :\n  getBit k a = if k < n then getBit k a else 0"}, {"name": "getBit_eq_testBit", "content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"}, {"name": "getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "U_card", "content": "lemma U_card (i : Fin r) :\n    Fintype.card (U 𝔽q β i) = (Fintype.card 𝔽q)^i.val"}, {"name": "finrank_U", "content": "omit [Fintype L] [Fintype 𝔽q] h_Fq_char_prime in\nlemma finrank_U (i : Fin r) :\n  Module.finrank 𝔽q (U 𝔽q β i) = i"}], "used_local_defs": [{"name": "AdditiveNTT.bitsToU", "content": "def bitsToU (i : Fin r) (k : Fin (2 ^ i.val)) :\n    AdditiveNTT.U (L := L) (𝔽q := 𝔽q) (β := β) i :=\n  let val := (Finset.univ : Finset (Fin i)).sum fun j =>\n    if (Nat.getBit (n := k.val) (k := j.val) == 1) then\n      β ⟨j, by admit /- proof elided -/\n      ⟩\n    else 0\n\n  \n  ⟨val, by admit /- proof elided -/\n  ⟩"}, {"name": "AdditiveNTT.getUElements", "content": "def getUElements (i : Fin r) : List L :=\n  (List.finRange (2^i.val)).map fun k =>\n    (Finset.univ : Finset (Fin i)).sum fun j =>\n      if Nat.getBit (n := k.val) (k := j.val) == 1 then\n        β ⟨j.val, by admit /- proof elided -/\n        ⟩\n      else 0"}, {"name": "AdditiveNTT.evalWAt", "content": "def evalWAt (i : Fin r) (x : L) : L :=\n  ((getUElements (β := β) (ℓ := ℓ) (R_rate := R_rate) i).map (fun u => x - u)).prod"}], "used_local_lemmas": [{"name": "AdditiveNTT.List.prod_finRange_eq_finset_prod", "content": "lemma List.prod_finRange_eq_finset_prod {M : Type*} [CommMonoid M] {n : ℕ} (f : Fin n → M) :\n    ((List.finRange n).map f).prod = ∏ i : Fin n, f i"}, {"name": "AdditiveNTT.bitsToU_bijective", "content": "theorem bitsToU_bijective (i : Fin r) :\n  Function.Bijective (bitsToU (𝔽q := 𝔽q) (β := β) (ℓ := ℓ) (R_rate := R_rate) i)"}], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.AdditiveNTT\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Impl\n\nnamespace AdditiveNTT\n\nopen ConcreteBinaryTower\n\nsection HelperFunctions\n\nend HelperFunctions\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type} [Field L] [Fintype L] [DecidableEq L]\n\nvariable {𝔽q : Type} [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n\nvariable [hFq_card : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n\nvariable [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nsection Algorithm\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\ndef bitsToU (i : Fin r) (k : Fin (2 ^ i.val)) :\n    AdditiveNTT.U (L := L) (𝔽q := 𝔽q) (β := β) i :=\n  let val := (Finset.univ : Finset (Fin i)).sum fun j =>\n    if (Nat.getBit (n := k.val) (k := j.val) == 1) then\n      β ⟨j, by admit /- proof elided -/\n      ⟩\n    else 0\n\n  \n  ⟨val, by admit /- proof elided -/\n  ⟩\n\ndef getUElements (i : Fin r) : List L :=\n  (List.finRange (2^i.val)).map fun k =>\n    (Finset.univ : Finset (Fin i)).sum fun j =>\n      if Nat.getBit (n := k.val) (k := j.val) == 1 then\n        β ⟨j.val, by admit /- proof elided -/\n        ⟩\n      else 0\n\ndef evalWAt (i : Fin r) (x : L) : L :=\n  ((getUElements (β := β) (ℓ := ℓ) (R_rate := R_rate) i).map (fun u => x - u)).prod", "target_theorem": "theorem evalWAt_eq_W (i : Fin r) (x : L) :\n  evalWAt (β := β) (ℓ := ℓ) (R_rate := R_rate) (i := i) x =\n    (W (𝔽q := 𝔽q) (β := β) (i := i)).eval x :=", "ground_truth_proof": ":= by\n  -- 1. Convert implementation to mathematical product over Fin(2^i)\n  unfold evalWAt getUElements\n  rw [List.map_map]\n  rw [List.prod_finRange_eq_finset_prod] -- Now the pattern matches!\n  -- 2. Prepare RHS\n  rw [AdditiveNTT.W, Polynomial.eval_prod]\n  simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C]\n  -- 3. Use Finset.prod_bij to show equality via the bijection\n  -- LHS: ∏ k : Fin(2^i), (x - bitsToU k), RHS: ∏ u : U i,      (x - u)\n  apply Finset.prod_bij (s := ((Finset.univ (α := (Fin (2^(i.val)))))))\n    (t := (Finset.univ : Finset (U 𝔽q β i)))\n    (i := fun k _ =>\n      bitsToU (𝔽q := 𝔽q) (β := β) (ℓ := ℓ) (r := r) (R_rate := R_rate) (L := L) (i := i) k)\n    (hi := by\n      -- Proof that the map lands in the target set (Finset.univ)\n      intro a _\n      exact Finset.mem_univ _)\n    (i_inj := by\n      -- Proof of Injectivity (uses our previous theorem)\n      intro a₁ _ a₂ _ h_eq\n      exact (bitsToU_bijective (𝔽q := 𝔽q) (β := β) (ℓ := ℓ)\n        (r := r) (R_rate := R_rate) (L := L) (i := i)).1 h_eq)\n    (i_surj := by\n      -- Proof of Surjectivity (uses our previous theorem)\n      intro b _\n      -- We need to provide the element 'a' and the proof it is in the set\n      obtain ⟨a, ha_eq⟩ := (bitsToU_bijective (𝔽q := 𝔽q)\n        (β := β) (ℓ := ℓ) (r := r) (R_rate := R_rate) (L := L) (i := i)).2 b\n      use a\n      constructor\n      · exact ha_eq\n      · exact Finset.mem_univ a\n    )\n    (h := by -- Proof that the values are equal: (x - bitsToU k) = (x - (bitsToU k))\n      intro a ha_univ\n      rfl\n    )", "nesting_depth": 4, "transitive_dep_count": 73, "subset_aristotle": false, "category": "Applied verif."}
{"id": 58, "thm_name": "AdditiveNTT.normalizedW_eq_qMap_composition", "thm_stmt": "lemma normalizedW_eq_qMap_composition (ℓ R_rate : ℕ) (i : Fin r) :\n    normalizedW 𝔽q β i = qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/AdditiveNTT/AdditiveNTT.lean", "imports": ["import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis", "import Mathlib.Data.Finsupp.Defs", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Tactic", "import ArkLib.Data.Polynomial.Frobenius", "import Mathlib.LinearAlgebra.LinearIndependent.Defs"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Subspace", "module": "Mathlib.Algebra.Module.Submodule.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ico", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Submodule", "module": "Mathlib.Algebra.Module.Submodule.Defs"}, {"name": "Submodule.span", "module": "Mathlib.LinearAlgebra.Span.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Polynomial.eval", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "IsLinearMap", "module": "Mathlib.Algebra.Module.LinearMap.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Polynomial.rootMultiplicity", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "CommRing", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Polynomial.algEquivOfCompEqX", "module": "Mathlib.Algebra.Polynomial.AlgebraMap"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}], "used_repo_defs": [{"name": "W", "content": "noncomputable def W (i : Fin r) : L[X] :=\n  ∏ u : U 𝔽q β i, (X - C u.val)"}, {"name": "U", "content": "def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))"}, {"name": "normalizedW", "content": "noncomputable def normalizedW (i : Fin r) : L[X] :=\n  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i"}, {"name": "getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "Fin.succRecOnSameFinType", "content": "@[elab_as_elim] def Fin.succRecOnSameFinType {motive : Fin r → Sort _}\n    (zero : motive (0 : Fin r))\n    (succ : ∀ i : Fin r, i + 1 < r → motive i → motive (i + 1)) : ∀ (i : Fin r), motive i\n  | ⟨0, _⟩ => by admit /- proof elided -/\n  | ⟨Nat.succ i_val, h⟩ => by admit /- proof elided -/"}, {"name": "algEquivAevalXSubC", "content": "@[simps!]\nnoncomputable def algEquivAevalXSubC {R : Type*} [CommRing R] (t : R) : R[X] ≃ₐ[R] R[X] :="}], "lib_lemmas": [{"name": "Fintype.card_pos", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Polynomial.C_1", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.C_mul", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_pow", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.comp_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.mul_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.prod_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Polynomial.sub_comp", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "inv_eq_one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "map_pow", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_inv_cancel₀", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_pow", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_pow_sub_one", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_sub", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_pow", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin.eta", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.mk_eq_mk", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.eval_X", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "div_one", "module": "Mathlib.Algebra.Group.Basic"}], "repo_lemmas": [{"name": "Xⱼ_zero_eq_one", "content": "lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :\n  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1"}, {"name": "W_linear_comp_decomposition", "content": "omit hF₂ in\ntheorem W_linear_comp_decomposition (i : Fin r) (h_i_add_1 : i + 1 < r) :\n    ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)"}, {"name": "W_linearity", "content": "theorem W_linearity (i : Fin r)\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p)"}, {"name": "inductive_linear_map_W", "content": "omit hF₂ in\nlemma inductive_linear_map_W (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : IsLinearMap 𝔽q (f := fun inner_p ↦ (W 𝔽q β (i + 1)).comp inner_p)"}, {"name": "inductive_rec_form_W_comp", "content": "omit h_Fq_char_prime hF₂ in\nlemma inductive_rec_form_W_comp (i : Fin r) (h_i_add_1 : i + 1 < r)\n    (h_prev_linear_map : IsLinearMap (R := 𝔽q) (M := L[X]) (M₂ := L[X])\n      (f := fun inner_p ↦ (W 𝔽q β i).comp inner_p))\n    : ∀ p: L[X], (W 𝔽q β (i + 1)).comp p =\n      ((W 𝔽q β i).comp p) ^ Fintype.card 𝔽q -\n        C (eval (β i) (W 𝔽q β i)) ^ (Fintype.card 𝔽q - 1) * ((W 𝔽q β i).comp p)"}, {"name": "Wᵢ_eval_βᵢ_neq_zero", "content": "lemma Wᵢ_eval_βᵢ_neq_zero\n    (i : Fin r): (W 𝔽q β i).eval (β i) ≠ 0"}, {"name": "βᵢ_not_in_Uᵢ", "content": "lemma βᵢ_not_in_Uᵢ (i : Fin r) :\n    β i ∉ U 𝔽q β i"}, {"name": "W_prod_comp_decomposition", "content": "lemma W_prod_comp_decomposition\n    (i : Fin r) (hi : i > 0) :\n    (W 𝔽q β i) = ∏ c: 𝔽q, (W 𝔽q β (i-1)).comp (X - C (c • β (i-1)))"}, {"name": "W_splits", "content": "lemma W_splits (i : Fin r) : (W 𝔽q β i).Splits (RingHom.id L)"}, {"name": "rootMultiplicity_prod_W_comp_X_sub_C", "content": "omit h_Fq_char_prime hF₂ in\nlemma rootMultiplicity_prod_W_comp_X_sub_C\n    (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n    rootMultiplicity a ((univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i)))) =\n    if a ∈ (U 𝔽q β (i+1) : Set L) then 1 else 0"}, {"name": "Prod_W_comp_X_sub_C_ne_zero", "content": "omit [DecidableEq L] h_Fq_char_prime hF₂ hβ_lin_indep in\nlemma Prod_W_comp_X_sub_C_ne_zero (i : Fin r) :\n    (univ : Finset 𝔽q).prod (fun c => (W 𝔽q β i).comp (X - C (c • β i))) ≠ 0"}, {"name": "W_ne_zero", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W_ne_zero (i : Fin r) : (W 𝔽q β i) ≠ 0"}, {"name": "roots_comp_X_sub_C", "content": "lemma roots_comp_X_sub_C (p : L[X]) (a : L) :\n    (p.comp (X - C a)).roots = p.roots.map (fun r => r + a)"}, {"name": "rootMultiplicity_comp_X_sub_C", "content": "lemma rootMultiplicity_comp_X_sub_C (p : L[X]) (a x : L) :\n    rootMultiplicity x (p.comp (X - C a)) = rootMultiplicity (x - a) p"}, {"name": "comp_X_sub_C_eq_zero_iff", "content": "omit [Fintype L] [DecidableEq L] in\nlemma comp_X_sub_C_eq_zero_iff (p : L[X]) (a : L) :\n  p.comp (X - C a) = 0 ↔ p = 0"}, {"name": "rootMultiplicity_W", "content": "lemma rootMultiplicity_W (i : Fin r) (a : L) :\n    rootMultiplicity a (W 𝔽q β i) = if a ∈ (U 𝔽q β i : Set L) then 1 else 0"}, {"name": "roots_W", "content": "lemma roots_W (i : Fin r) : -- converts root Multiset into (univ: Uᵢ.val.map)\n  (W 𝔽q β i).roots = (univ : Finset (U 𝔽q β i)).val.map (fun u => u.val)"}, {"name": "root_U_lift_up", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime hβ_lin_indep in\ntheorem root_U_lift_up (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) (x : 𝔽q) :\n  a - x • β i ∈ (U 𝔽q β i) → a ∈ (U 𝔽q β (i+1))"}, {"name": "root_U_lift_down", "content": "omit [Fintype L] [DecidableEq L] [Fintype 𝔽q] h_Fq_char_prime in\ntheorem root_U_lift_down\n  (i : Fin r) (h_i_add_1 : i + 1 < r) (a : L) :\n  a ∈ (U 𝔽q β (i+1)) → ∃! x: 𝔽q, a - x • β i ∈ (U 𝔽q β i)"}, {"name": "W_monic", "content": "lemma W_monic (i : Fin r) : (W 𝔽q β i).Monic"}, {"name": "comp_sub_C_of_linear_eval", "content": "lemma comp_sub_C_of_linear_eval (p : L[X])\n  (h_lin : IsLinearMap 𝔽q (f := fun inner_p ↦ p.comp inner_p)) (a : L) :\n    p.comp (X - C a) = p - C (eval a p)"}, {"name": "prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L", "content": "theorem prod_poly_sub_C_eq_poly_pow_card_sub_poly_in_L\n  (p : L[X]) :\n  (∏ c ∈ (Finset.univ : Finset Fq), (p - Polynomial.C (algebraMap Fq L c))) =\n    p^(Fintype.card Fq) - p"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X_in_L", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X_in_L :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C (algebraMap Fq L c))) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "prod_X_sub_C_eq_X_pow_card_sub_X", "content": "theorem prod_X_sub_C_eq_X_pow_card_sub_X :\n  (∏ c ∈ (Finset.univ : Finset Fq), (Polynomial.X - Polynomial.C c)) =\n    Polynomial.X^(Fintype.card Fq) - Polynomial.X"}, {"name": "W₀_eq_X", "content": "omit [DecidableEq L] [Fintype 𝔽q] hβ_lin_indep in\nlemma W₀_eq_X : W 𝔽q β 0 = X"}, {"name": "Fin.val_add_one'", "content": "lemma Fin.val_add_one' (a : Fin r) (h_a_add_1 : a + 1 < r) : (a + 1).val = a.val + 1"}], "used_local_defs": [{"name": "AdditiveNTT.qMap", "content": "noncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))"}, {"name": "AdditiveNTT.qCompositionChain", "content": "noncomputable def qCompositionChain (i : Fin r) : L[X] :=\n  match i with\n  | ⟨0, _⟩ => X\n  | ⟨k + 1, h_k_add_1⟩ => (qMap 𝔽q β ⟨k, by admit /- proof elided -/\n  ⟩).comp (qCompositionChain ⟨k, by admit /- proof elided -/\n  ⟩)"}], "used_local_lemmas": [{"name": "AdditiveNTT.qMap_comp_normalizedW", "content": "lemma qMap_comp_normalizedW (i : Fin r) (h_i_add_1 : i + 1 < r) :\n  (qMap 𝔽q β i).comp (normalizedW 𝔽q β i) = normalizedW 𝔽q β (i + 1)"}], "local_ctx": "import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis\n\nimport Mathlib.Tactic\n\nimport Mathlib.Data.Finsupp.Defs\n\nimport Mathlib.LinearAlgebra.LinearIndependent.Defs\n\nopen Polynomial AdditiveNTT Module\n\nnamespace AdditiveNTT\n\nvariable {r : ℕ} [NeZero r]\n\nvariable {L : Type u} [Field L] [Fintype L] [DecidableEq L]\n\nvariable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]\n  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]\n\nvariable [Algebra 𝔽q L]\n\nvariable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]\n  [h_β₀_eq_1 : Fact (β 0 = 1)]\n\nvariable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}\n\nsection IntermediateStructures\n\nnoncomputable def qMap (i : Fin r) : L[X] :=\n  let constMultiplier := ((W 𝔽q β i).eval (β i))^(Fintype.card 𝔽q)\n    / ((W 𝔽q β (i + 1)).eval (β (i + 1)))\n  C constMultiplier * ∏ c: 𝔽q, (X - C (algebraMap 𝔽q L c))\n\nnoncomputable def qCompositionChain (i : Fin r) : L[X] :=\n  match i with\n  | ⟨0, _⟩ => X\n  | ⟨k + 1, h_k_add_1⟩ => (qMap 𝔽q β ⟨k, by admit /- proof elided -/\n  ⟩).comp (qCompositionChain ⟨k, by admit /- proof elided -/\n  ⟩)", "target_theorem": "lemma normalizedW_eq_qMap_composition (ℓ R_rate : ℕ) (i : Fin r) :\n    normalizedW 𝔽q β i = qCompositionChain 𝔽q β (ℓ:=ℓ) (R_rate:=R_rate) i :=", "ground_truth_proof": ":=\nby\n  -- We proceed by induction on i.\n  induction i using Fin.succRecOnSameFinType with\n  | zero =>\n    -- Base case: i = 0\n    -- We need to show `normalizedW ... 0 = qCompositionChain 0`.\n    -- The RHS is `X` by definition of the chain.\n    rw [qCompositionChain.eq_def]\n    -- The LHS is `C (1 / eval (β 0) (W ... 0)) * (W ... 0)`.\n    rw [normalizedW, W₀_eq_X, eval_X, h_β₀_eq_1.out, div_one, C_1, one_mul]\n    rfl\n  | succ k k_h i_h =>\n    -- Inductive step: Assume the property holds for k, prove for k+1.\n    -- The goal is `normalizedW ... (k+1) = qCompositionChain (k+1)`.\n    -- The RHS is `(qMap k).comp (qCompositionChain k)` by definition.\n    rw [qCompositionChain.eq_def]\n    -- From Lemma 4.2, we know `normalizedW ... (k+1) = (qMap k).comp (normalizedW ... k)`.\n    -- How to choose the rhs?\n    have h_eq: ⟨k.val.succ, k_h⟩ = k + 1 := by\n      rw [Fin.mk_eq_mk]\n      rw [Fin.val_add_one']\n      exact k_h\n    simp only [h_eq.symm, Nat.succ_eq_add_one, Fin.eta]\n    have h_res := qMap_comp_normalizedW 𝔽q β k k_h\n    -- ⊢ normalizedW 𝔽q β ⟨↑k + 1, k_h⟩ = (qMap 𝔽q β k).comp (qCompositionChain 𝔽q β k)\n    rw [←i_h]\n    rw [h_res]\n    simp only [h_eq]", "nesting_depth": 11, "transitive_dep_count": 86, "subset_aristotle": false, "category": "Applied verif."}
{"id": 59, "thm_name": "Nat.getHighBits_no_shl_joinBits", "thm_stmt": "lemma getHighBits_no_shl_joinBits {n m : ℕ} (low : Fin (2 ^ n)) (high : Fin (2 ^ m)) :\n  getHighBits_no_shl n (joinBits low high).val = high.val", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/Nat/Bitwise.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import ArkLib.Data.Fin.BigOperators", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Data.Nat.Bitwise", "import Mathlib.Data.Finsupp.Basic", "import Mathlib.Algebra.Order.Ring.Star", "import Mathlib.Data.Nat.Digits.Defs", "import Mathlib.Algebra.BigOperators.Fin"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat.binaryRec", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bit", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.bodd", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.boddDiv2", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.div2", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.and_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.add_mul_div_left", "module": "Init.Data.Nat.Div.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.and_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.div_add_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.div_eq_zero_iff_lt", "module": "Init.Data.Nat.Div.Lemmas"}, {"name": "Nat.mul_add_mod_self_right", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.zero_and", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.or_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.or_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_or", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_div_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_mod_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_self", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_xor", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Bool.toNat_lt", "module": "Init.Data.Bool"}, {"name": "Nat.bit_decomp", "module": "Mathlib.Data.Nat.Bits"}, {"name": "Nat.bit_val", "module": "Mathlib.Data.Nat.BinaryRec"}, {"name": "Nat.mul_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.right_distrib", "module": "Init.Data.Nat.Basic"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.shiftRight_and_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.zero_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.and_one_is_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "Nat.mod_two_bne_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.one_and_eq_mod_two", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "Bool.not_eq_true", "module": "Init.SimpLemmas"}, {"name": "Nat.pow_le_pow_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.testBit_eq_false_of_lt", "module": "Mathlib.Data.Nat.Bitwise"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "Bool.false_and", "module": "Init.SimpLemmas"}, {"name": "Bool.true_and", "module": "Init.SimpLemmas"}, {"name": "Nat.and_comm", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_two_not_eq_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.testBit_two_pow_mul", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "beq_eq_beq", "module": "Mathlib.Logic.Basic"}, {"name": "beq_eq_false_iff_ne", "module": "Init.SimpLemmas"}, {"name": "decide_false", "module": "Init.Core"}, {"name": "decide_true", "module": "Init.Core"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.eq_of_testBit_eq", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_assoc", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_or_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftLeft_shiftRight", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Nat.getBit", "content": "def getBit (k n : Nat) : Nat := (n >>> k) &&& 1"}, {"name": "Nat.getHighBits_no_shl", "content": "def getHighBits_no_shl (numLowBits : ℕ) (n : ℕ) : ℕ := n >>> numLowBits"}, {"name": "Nat.joinBits", "content": "def joinBits {n m : ℕ} (low : Fin (2 ^ n)) (high : Fin (2 ^ m)) : Fin (2 ^ (m+n)) :=\n  ⟨(high.val <<< n) ||| low.val, by admit /- proof elided -/\n  ⟩"}], "used_local_lemmas": [{"name": "Nat.getBit_lt_2", "content": "lemma getBit_lt_2 {k n : Nat} : getBit k n < 2"}, {"name": "Nat.getBit_eq_testBit", "content": "lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0"}, {"name": "Nat.eq_iff_eq_all_getBits", "content": "lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m"}, {"name": "Nat.shiftRight_and_one_distrib", "content": "lemma shiftRight_and_one_distrib {n m k : ℕ} :\n    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m"}, {"name": "Nat.and_eq_zero_iff_and_each_getBit_eq_zero", "content": "lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :\n    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0"}, {"name": "Nat.div_2_form", "content": "lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):\n  (nD2 * 2 + b) / 2 = nD2"}, {"name": "Nat.and_by_split_lowBits", "content": "lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)"}, {"name": "Nat.xor_by_split_lowBits", "content": "lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)"}, {"name": "Nat.or_by_split_lowBits", "content": "lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)\n  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):\n  n ||| m = (n1 ||| m1) * 2 + (bn ||| bm)"}, {"name": "Nat.sum_eq_xor_plus_twice_and", "content": "lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)"}, {"name": "Nat.add_shiftRight_distrib", "content": "lemma add_shiftRight_distrib {n m k : ℕ} (h_and_zero : n &&& m = 0):\n  (n + m) >>> k = (n >>> k) + (m >>> k)"}, {"name": "Nat.xor_of_and_eq_zero_is_or", "content": "lemma xor_of_and_eq_zero_is_or {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n ^^^ m = n ||| m"}, {"name": "Nat.sum_of_and_eq_zero_is_or", "content": "lemma sum_of_and_eq_zero_is_or {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ||| m"}, {"name": "Nat.getBit_of_multiple_of_power_of_two", "content": "lemma getBit_of_multiple_of_power_of_two {n p : ℕ}: ∀ k,\n  getBit (k) (2^p * n) = if k < p then 0 else getBit (k-p) n"}, {"name": "Nat.getBit_of_shiftLeft", "content": "lemma getBit_of_shiftLeft {n p : ℕ}:\n  ∀ k, getBit (k) (n <<< p) = if k < p then 0 else getBit (k - p) n"}, {"name": "Nat.getBit_of_lt_two_pow", "content": "lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :\n  getBit k a = if k < n then getBit k a else 0"}, {"name": "Nat.and_shl_eq_zero_of_lt_two_pow", "content": "lemma and_shl_eq_zero_of_lt_two_pow {a n b : ℕ} (hb : b < 2 ^ n) : (a <<< n) &&& b = 0"}], "local_ctx": "import ArkLib.Data.Fin.BigOperators\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nimport Mathlib.Algebra.Order.Ring.Star\n\nimport Mathlib.Data.Nat.Bitwise\n\nimport Mathlib.Data.Nat.Digits.Defs\n\nimport Mathlib.Data.Finsupp.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Fin\n\nnamespace Nat\n\ndef getBit (k n : Nat) : Nat := (n >>> k) &&& 1\n\ndef getHighBits_no_shl (numLowBits : ℕ) (n : ℕ) : ℕ := n >>> numLowBits\n\ndef joinBits {n m : ℕ} (low : Fin (2 ^ n)) (high : Fin (2 ^ m)) : Fin (2 ^ (m+n)) :=\n  ⟨(high.val <<< n) ||| low.val, by admit /- proof elided -/\n  ⟩", "target_theorem": "lemma getHighBits_no_shl_joinBits {n m : ℕ} (low : Fin (2 ^ n)) (high : Fin (2 ^ m)) :\n  getHighBits_no_shl n (joinBits low high).val = high.val :=", "ground_truth_proof": ":= by\n  unfold joinBits getHighBits_no_shl\n  dsimp\n  have h_and_zero := and_shl_eq_zero_of_lt_two_pow (a := high.val) (b := low.val) (hb := low.isLt)\n  rw [←Nat.sum_of_and_eq_zero_is_or h_and_zero]\n  rw [Nat.add_shiftRight_distrib h_and_zero]\n  rw [Nat.shiftLeft_shiftRight]\n  rw [Nat.shiftRight_eq_div_pow]\n  have h: low.val/2^n = 0 := by\n    apply Nat.div_eq_zero_iff_lt (x:=low) (k:=2^n) (h:=by exact Nat.two_pow_pos n).mpr (by omega)\n  simp only [h, add_zero]", "nesting_depth": 4, "transitive_dep_count": 97, "subset_aristotle": false, "category": "Applied verif."}
{"id": 60, "thm_name": "ConcreteBinaryTower.towerRingHomForwardMap_backwardMap_eq", "thm_stmt": "lemma towerRingHomForwardMap_backwardMap_eq (k : ℕ) (x : BTField k) :\n  towerRingHomForwardMap (k:=k) (towerRingHomBackwardMap (k:=k) x) = x", "lean_root": "ArkLib", "rel_path": "ArkLib/Data/FieldTheory/BinaryField/Tower/Impl.lean", "imports": ["import ArkLib.Data.FieldTheory.BinaryField.Tower.Prelude", "import ArkLib.Data.FieldTheory.BinaryField.Tower.Basic", "import ArkLib.Data.Classes.DCast"], "used_lib_defs": [{"name": "Eq", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.append", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.extractLsb", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "AddCommGroup", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.negSucc", "module": "Init.Data.Int.Basic"}, {"name": "Ring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "DivisionRing", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "NNRat", "module": "Mathlib.Data.Rat.Init"}, {"name": "NNRat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "Rat.castRec", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot.instField", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Irreducible", "module": "Mathlib.Algebra.Group.Irreducible.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "AdjoinRoot", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.of", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "AdjoinRoot.root", "module": "Mathlib.RingTheory.AdjoinRoot"}, {"name": "Eq.mp", "module": "Init.Core"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "DivisionSemiring", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Finset.Icc", "module": "Mathlib.Order.Interval.Finset.Defs"}, {"name": "GroupWithZero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "Ne", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Equiv.ofBijective", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Fintype.ofEquiv", "module": "Mathlib.Data.Fintype.OfMap"}, {"name": "Function.Bijective", "module": "Mathlib.Logic.Function.Defs"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.Surjective", "module": "Init.Data.Function"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Fintype.ofFinite", "module": "Mathlib.Data.Fintype.EquivFin"}, {"name": "List.Vector.cons", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "toFun", "module": "ToMathlib.Control.Monad.Hom"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.toAlgebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "invFun", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "left_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "right_inv", "module": "ToMathlib.Control.Monad.Equiv"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "absurd", "module": "Init.Prelude"}, {"name": "instAlgebra", "module": "Mathlib.LinearAlgebra.TensorAlgebra.Basic"}, {"name": "instFintypeProd", "module": "Mathlib.Data.Fintype.Prod"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "RingHom.id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "CommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "RingEquiv", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "EquivLike", "module": "Mathlib.Data.FunLike.Equiv"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.extractLsb'", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth", "module": "Init.Data.BitVec.Basic"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "pos", "module": "ToMathlib.Control.Comonad.Instances"}, {"name": "BitVec.toNat", "module": "Init.Prelude"}, {"name": "Algebra.algebraMap", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "MonoidHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "OneHom", "module": "Mathlib.Algebra.Group.Hom.Defs"}], "used_repo_defs": [{"name": "GaloisField", "content": "notation : 10 \"GF(\" term : 10 \")\" => GaloisField term 1"}, {"name": "DCast", "content": "class DCast (α : Sort*) (β : α → Sort*) where\n  dcast : ∀ {a a' : α}, a = a' → β a → β a'\n  dcast_id : ∀ {a : α}, dcast (Eq.refl a) = id"}, {"name": "sumZeroIffEq", "content": "@[simp]\ndef sumZeroIffEq (k : ℕ) : ∀ (x y : BTField k),\n  x + y = 0 ↔ x = y := (BinaryTowerAux k).2.sumZeroIffEq"}, {"name": "BTField", "content": "@[simp]\ndef BTField (k : ℕ) := (BinaryTowerAux k).1"}, {"name": "BinaryTowerAux", "content": "def BinaryTowerAux (k : ℕ) : (Σ' (F : Type 0), BinaryTowerResult F k) :=\n  match k with\n  | 0 => \n    let curBTField := GF(2)\n    let newList : List.Vector (GF(2)) 1 := List.Vector.cons (1 : GF(2)) List.Vector.nil\n    let specialElement : GF(2) := newList.1.headI\n    let firstElementOfVecIsSpecialElement : newList.1.headI = specialElement := rfl\n    let specialElementIs1 : specialElement = 1 := by admit /- proof elided -/"}, {"name": "BinaryTowerInductiveStepResult", "content": "structure BinaryTowerInductiveStepResult (k : ℕ) (prevBTField : Type _)\n  (prevBTResult : BinaryTowerResult prevBTField k) [instPrevBTFieldIsField : Field prevBTField]\n  (prevPoly : Polynomial prevBTField) (F : Type _) where\n  binaryTowerResult : BinaryTowerResult F (k+1)\n  eq_adjoin : F = AdjoinRoot prevPoly\n  u_is_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement = AdjoinRoot.root prevPoly\n  eval_defining_poly_at_root : Eq.mp (eq_adjoin) binaryTowerResult.specialElement^2 +\n    Eq.mp (eq_adjoin) binaryTowerResult.specialElement * (of prevPoly) prevBTResult.specialElement\n    + 1 = 0"}, {"name": "BinaryTowerResult", "content": "structure BinaryTowerResult (F : Type _) (k : ℕ) where\n  vec : (List.Vector F (k + 1))\n  instField : (Field F)\n  instFintype : Fintype F\n  specialElement : F\n  specialElementNeZero : NeZero specialElement\n  firstElementOfVecIsSpecialElement [Inhabited F] : vec.1.headI = specialElement\n  instIrreduciblePoly : (Irreducible (p := (definingPoly specialElement)))\n  sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y\n  fieldFintypeCard : Fintype.card F = 2^(2^k)\n  traceMapEvalAtRootsIs1 : TraceMapProperty F specialElement k"}, {"name": "Inhabited", "content": "@[simp]\ninstance Inhabited (k : ℕ) : Inhabited (BTField k) where\n  default := (0 : BTField k)"}, {"name": "TraceMapProperty", "content": "structure TraceMapProperty (F : Type*) [Field F] (u : F) (k : ℕ) : Prop where\n  element_trace : ∑ i ∈ Finset.range (2 ^ k), u ^ (2 ^ i) = 1\n  inverse_trace : ∑ i ∈ Finset.range (2 ^ k), (u⁻¹) ^ (2 ^ i) = 1"}, {"name": "definingPoly", "content": "noncomputable def definingPoly {F : Type*} [instField : Field F] (s : F)\n  := X ^ 2 + C s * X + 1"}, {"name": "binary_tower_inductive_step", "content": "def binary_tower_inductive_step\n  (k : Nat)\n  (prevBTField : Type _) [Field prevBTField]\n  (prevBTResult : BinaryTowerResult prevBTField k)\n: Σ' (F : Type _), BinaryTowerInductiveStepResult (k:=k) (prevBTField:=prevBTField)\n  (prevBTResult:=prevBTResult) (prevPoly:=definingPoly (F:=prevBTField)\n    (instField:=prevBTResult.instField) (s:=prevBTResult.specialElement)) (F:=F)\n  (instPrevBTFieldIsField:=prevBTResult.instField) :="}, {"name": "AlgebraTowerEquiv.symm", "content": "def AlgebraTowerEquiv.symm (e : AlgebraTowerEquiv A B) : AlgebraTowerEquiv B A where\n  toRingEquiv := fun i => (e.toRingEquiv i).symm\n  commutesLeft' := fun i j h r => by admit /- proof elided -/"}, {"name": "AlgebraTowerEquiv", "content": "structure AlgebraTowerEquiv (A : ι → Type*) [∀ i, CommSemiring (A i)] [a : AlgebraTower A]\n  (B : ι → Type*) [∀ i, CommSemiring (B i)] [b : AlgebraTower B]\n  where\n    toRingEquiv : ∀ i, (A i ≃+* B i)\n    commutesLeft' : ∀ (i j : ι) (h : i ≤ j) (r : A i),\n      (b.algebraMap (i:=i) (j:=j) (h:=h)) ((toRingEquiv i) r) =\n      (toRingEquiv j) (a.algebraMap (i:=i) (j:=j) (h:=h) r)"}, {"name": "SpecialElementRelation", "content": "structure SpecialElementRelation {F_prev : Type*} [Field F_prev] (t1 : F_prev)\n  {F_cur : Type*} [Field F_cur] (u : F_cur) [Algebra F_prev F_cur] : Prop where\n    sum_inv_eq : u + u⁻¹ = algebraMap F_prev F_cur t1\n    h_u_square : u^2 = u * (algebraMap F_prev F_cur t1) + 1"}, {"name": "irreducible_quadratic_defining_poly_of_traceMap_eq_1", "content": "instance irreducible_quadratic_defining_poly_of_traceMap_eq_1\n  {F : Type*} [Field F] [Fintype F] [CharP F 2] (s : F) [NeZero s] (k : ℕ)\n  (trace_map_prop : TraceMapProperty F s k)\n  (fintypeCard : Fintype.card F = 2 ^ (2 ^ k))\n  : Irreducible (definingPoly s) :="}, {"name": "charP_eq_2_of_add_self_eq_zero", "content": "instance charP_eq_2_of_add_self_eq_zero {F : Type*} [Field F]\n    (sumZeroIffEq : ∀ (x y : F), x + y = 0 ↔ x = y) : CharP F 2 :=\n  have h_two : (2 : (F)) = 0 := by admit /- proof elided -/"}, {"name": "coeff.{u}", "content": "def coeff.{u} {F : Type u} [Semiring F] (f : F[X][Y]) (i j : ℕ) : F := (f.coeff j).coeff i"}, {"name": "GF_2_fintype", "content": "instance GF_2_fintype : Fintype (GF(2)) := Fintype.ofFinite (GF(2))"}, {"name": "AlgebraTower.toIsScalarTower", "content": "@[simp]\ninstance AlgebraTower.toIsScalarTower (a : AlgebraTower C) {i j k : ι}\n    (h1 : i ≤ j) (h2 : j ≤ k) :\n    letI : Algebra (C i) (C j) :="}, {"name": "split", "content": "def split (k : ℕ) (h_k : k > 0) (x : BTField k) : BTField (k-1) × BTField (k-1) :="}, {"name": "join_via_add_smul", "content": "def join_via_add_smul {k : ℕ} (h_pos : k > 0) (hi_btf lo_btf : BTField (k - 1)) :\n    BTField k :="}, {"name": "binaryAlgebraTower", "content": "def binaryAlgebraTower {l r : ℕ} (h_le : l ≤ r) : Algebra (BTField l) (BTField r) :="}, {"name": "AlgebraTower.toAlgebra", "content": "@[simp]\ndef AlgebraTower.toAlgebra {i j : ι} (h : i ≤ j) : Algebra (A i) (A j) :=\n  (AlgebraTower.algebraMap (i:=i) (j:=j) (h:=h)).toAlgebra"}, {"name": "Z", "content": "@[simp]\ndef Z (k : ℕ) : BTField k := (BinaryTowerAux k).snd.specialElement "}, {"name": "polyMonic", "content": "instance polyMonic (n : ℕ) : Monic (poly n) := definingPoly_is_monic (Z n)"}, {"name": "poly", "content": "@[simp]\ndef poly (k : ℕ) : Polynomial (BTField k) := definingPoly (s:=(Z k))"}, {"name": "(priority", "content": "instance (priority := 1000) algebra_adjacent_tower (l : ℕ) :\n  Algebra (BTField l) (BTField (l+1)) :="}, {"name": "canonicalEmbedding", "content": "def canonicalEmbedding (k : ℕ) : BTField k →+* BTField (k+1) :=\n  AdjoinRoot.of (poly k)"}, {"name": "towerAlgebraMap", "content": "def towerAlgebraMap (l r : ℕ) (h_le : l ≤ r) : BTField l →+* BTField r :="}], "lib_lemmas": [{"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_zero", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec.ofNat_toNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.toNat_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.and_two_pow_sub_one_eq_mod", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_zero", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_two", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_succ", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pred_mul_two", "module": "Init.Data.Nat.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_add_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.extractLsb'_append_eq_of_le", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_refl", "module": "Init.Prelude"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "eq_mp_eq_cast", "module": "Init.PropLemmas"}, {"name": "BitVec.append_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.setWidth_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_of_lt_mul", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.div_mul_cancel", "module": "Init.Data.Nat.Dvd"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.shiftLeft_eq", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.shiftRight_eq_div_pow", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.shiftLeft_add_eq_or_of_lt", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "BitVec.eq_of_toNat_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.isLt", "module": "Init.Data.BitVec.BasicAux"}, {"name": "BitVec.toNat_append", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "OfNat.ofNat_ne_one", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "Prod.ext_iff", "module": "Init.Ext"}, {"name": "BitVec.zero_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Ne.dite_eq_left_iff", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.left_eq_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingHom.comp_id", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_eq", "module": "Init.Core"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "MonoidHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "Nat.sub_one_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "OneHom.coe_mk", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "RingHom.coe_mk", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "cast_cast", "module": "Init.PropLemmas"}, {"name": "eqRec_eq_cast", "module": "Batteries.Logic"}, {"name": "BitVec.ofNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.xor_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.toNat_xor", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.and_xor_distrib_right", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.shiftRight_xor_distrib", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Equiv.toFun_as_coe", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "EquivLike.coe_coe", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Nat.add_eq_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "RingEquiv.symm_apply_apply", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "RingEquiv.toEquiv_eq_coe", "module": "Mathlib.Algebra.Ring.Equiv"}, {"name": "and_false", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "one_le_two_pow_n", "content": "theorem one_le_two_pow_n (n : ℕ) : 1 ≤ 2 ^ n"}, {"name": "dcast_eq", "content": "@[simp]\ntheorem dcast_eq : dcast (Eq.refl a) b = b"}, {"name": "one_le_sub_consecutive_two_pow", "content": "theorem one_le_sub_consecutive_two_pow (n : ℕ): 1 ≤ 2^(n+1) - 2^n"}, {"name": "dcast_eq_root_cast", "content": "theorem dcast_eq_root_cast (h : a = a') : dcast h b = _root_.cast (congrArg β h) b"}, {"name": "dcast_symm", "content": "theorem dcast_symm (ha : a = a') (hb : dcast ha b = b') : dcast (ha.symm) b' = b"}, {"name": "dcast_eq_dcast_iff", "content": "theorem dcast_eq_dcast_iff (h : a = a'') (h' : a' = a'') :\n    dcast h b = dcast h' b' ↔ b = dcast (h'.trans h.symm) b'"}, {"name": "GF_2_value_eq_zero_or_one", "content": "theorem GF_2_value_eq_zero_or_one (x : GF(2)) : x = 0 ∨ x = 1"}, {"name": "eq_join_via_add_smul_eq_iff_split", "content": "theorem eq_join_via_add_smul_eq_iff_split (k : ℕ) (h_pos : k > 0)\n    (x : BTField k) (hi_btf lo_btf : BTField (k - 1)) :\n    x = ⋘ hi_btf, lo_btf ⋙ ↔\n  split (k:=k) (h_k:=h_pos) x = (hi_btf, lo_btf)"}, {"name": "BTField.cast_BTField_eq", "content": "lemma BTField.cast_BTField_eq (k m : ℕ) (h_eq : k = m) :\n  BTField k = BTField m"}, {"name": "unique_linear_decomposition_succ", "content": "theorem unique_linear_decomposition_succ (k : ℕ) :\n  ∀ (x : BTField (k+1)), ∃! (p : BTField k × BTField k),\n    x = ⋘ p.1, p.2 ⋙"}, {"name": "algebraMap_adjacent_tower_succ_eq_Adjoin_of", "content": "lemma algebraMap_adjacent_tower_succ_eq_Adjoin_of (k : ℕ) :\n  (algebraMap (BTField k) (BTField (k + 1))) = of (poly k)"}, {"name": "algebraMap_adjacent_tower_def", "content": "lemma algebraMap_adjacent_tower_def (l : ℕ) :\n  (algebraMap (BTField l) (BTField (l + 1))) = canonicalEmbedding l"}, {"name": "towerAlgebraMap_succ_1", "content": "lemma towerAlgebraMap_succ_1 (k : ℕ) :\n  towerAlgebraMap (l:=k) (r:=k+1) (h_le:=by omega) = canonicalEmbedding k"}, {"name": "towerAlgebraMap_id", "content": "lemma towerAlgebraMap_id (k : ℕ) : towerAlgebraMap (h_le:=by omega) = RingHom.id (BTField k)"}, {"name": "binaryTowerAlgebra_def", "content": "lemma binaryTowerAlgebra_def (l r : ℕ) (h_le : l ≤ r) :\n    @binaryAlgebraTower (l:=l) (r:=r) (h_le:=h_le)\n    = (towerAlgebraMap l r h_le).toAlgebra"}, {"name": "poly_natDegree_eq_2", "content": "lemma poly_natDegree_eq_2 (k : ℕ) : (poly (k:=k)).natDegree = 2"}], "used_local_defs": [{"name": "ConcreteBinaryTower.ConcreteBTField", "content": "def ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)"}, {"name": "ConcreteBinaryTower.BitVec", "content": "instance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.fromNat", "content": "def fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n"}, {"name": "ConcreteBinaryTower.ConcreteBTField", "content": "instance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.zero", "content": "def zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)"}, {"name": "ConcreteBinaryTower.one", "content": "def one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)"}, {"name": "ConcreteBinaryTower.add", "content": "def add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y"}, {"name": "ConcreteBinaryTower.neg", "content": "def neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x"}, {"name": "ConcreteBinaryTower.split", "content": "def split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.join", "content": "def join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.ConcreteBTFAddCommGroupProps", "content": "structure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel"}, {"name": "ConcreteBinaryTower.mkAddCommGroupInstance", "content": "def mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}"}, {"name": "ConcreteBinaryTower.Z", "content": "def Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》"}, {"name": "ConcreteBinaryTower.equivProd", "content": "def equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)"}, {"name": "ConcreteBinaryTower.concrete_mul", "content": "def concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.concrete_inv", "content": "def concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res"}, {"name": "ConcreteBinaryTower.natCast", "content": "def natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.natCast_zero", "content": "def natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :="}, {"name": "ConcreteBinaryTower.natCast_succ", "content": "def natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :="}, {"name": "ConcreteBinaryTower.intCast", "content": "def intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one"}, {"name": "ConcreteBinaryTower.intCast_ofNat", "content": "def intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :="}, {"name": "ConcreteBinaryTower.intCast_negSucc", "content": "def intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :="}, {"name": "ConcreteBinaryTower.ConcreteBTFRingProps", "content": "structure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c"}, {"name": "ConcreteBinaryTower.ConcreteBTFDivisionRingProps", "content": "structure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldProps", "content": "structure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a"}, {"name": "ConcreteBinaryTower.mkRingInstance", "content": "def mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n"}, {"name": "ConcreteBinaryTower.mkDivisionRingInstance", "content": "def mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)"}, {"name": "ConcreteBinaryTower.mkFieldInstance", "content": "def mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm"}, {"name": "ConcreteBinaryTower.ConcreteBTFStepResult", "content": "structure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))"}, {"name": "ConcreteBinaryTower.liftBTFieldProps", "content": "def liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.liftConcreteBTField", "content": "def liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :="}, {"name": "ConcreteBinaryTower.concreteCanonicalEmbedding", "content": "def concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :="}, {"name": "ConcreteBinaryTower.instAlgebraLiftConcreteBTField", "content": "instance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))"}, {"name": "ConcreteBinaryTower.getBTFResult", "content": "def getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.canonicalAlgMap", "content": "def canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap", "content": "def concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :="}, {"name": "ConcreteBinaryTower.instAlgebraTowerConcreteBTF", "content": "instance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.ConcreteBTFieldAlgebra", "content": "def ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le"}, {"name": "ConcreteBinaryTower.join_via_add_smul", "content": "def join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :="}, {"name": "ConcreteBinaryTower.towerEquiv_zero", "content": "noncomputable def towerEquiv_zero : RingEquiv (R:=GF(2)) (S:=ConcreteBTField 0) :=  {\n  toFun := fun x => if x = 0 then 0 else 1,\n  invFun := fun x => if x = 0 then 0 else 1,\n  left_inv := fun x => by admit /- proof elided -/"}, {"name": "ConcreteBinaryTower.towerRingEquiv0", "content": "noncomputable def towerRingEquiv0 : BTField 0 ≃+* ConcreteBTField 0 :="}, {"name": "ConcreteBinaryTower.towerRingEquivFromConcrete0", "content": "noncomputable def towerRingEquivFromConcrete0 : ConcreteBTField 0 ≃+* BTField 0 :="}, {"name": "ConcreteBinaryTower.towerRingHomForwardMap", "content": "noncomputable def towerRingHomForwardMap (k : ℕ) : ConcreteBTField k → BTField k :="}, {"name": "ConcreteBinaryTower.towerRingHomBackwardMap", "content": "noncomputable def towerRingHomBackwardMap (k : ℕ) : BTField k → ConcreteBTField k :="}], "used_local_lemmas": [{"name": "ConcreteBinaryTower.cast_ConcreteBTField_eq", "content": "lemma cast_ConcreteBTField_eq (k m : ℕ) (h_eq : k = m) :\n  ConcreteBTField k = ConcreteBTField m"}, {"name": "ConcreteBinaryTower.BitVec.dcast_id", "content": "theorem BitVec.dcast_id {n : Nat} (bv : BitVec n) :\n  DCast.dcast (Eq.refl n) bv = bv"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_eq", "content": "theorem BitVec.dcast_bitvec_eq {l r val : ℕ} (h_width_eq : l = r) :\n    dcast h_width_eq (BitVec.ofNat l val) = BitVec.ofNat r val"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_toNat_eq", "content": "theorem BitVec.dcast_bitvec_toNat_eq {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2) :\n    BitVec.toNat x = BitVec.toNat (dcast (h_width_eq) x)"}, {"name": "ConcreteBinaryTower.BitVec.dcast_bitvec_extractLsb_eq", "content": "theorem BitVec.dcast_bitvec_extractLsb_eq {w hi1 lo1 hi2 lo2 : ℕ}\n    (x : BitVec w) (h_lo_eq : lo1 = lo2)\n    (h_width_eq : hi1 - lo1 + 1 = hi2 - lo2 + 1) :\n    dcast h_width_eq (BitVec.extractLsb (hi:=hi1) (lo:=lo1) x)\n      = BitVec.extractLsb (hi:=hi2) (lo:=lo2) (x)"}, {"name": "ConcreteBinaryTower.BitVec.eq_mp_eq_dcast", "content": "theorem BitVec.eq_mp_eq_dcast {w w2 : ℕ} (x : BitVec w) (h_width_eq : w = w2)\n  (h_bitvec_eq : BitVec w = BitVec w2 := by rw [h_width_eq]) :\n  Eq.mp (h:=h_bitvec_eq) (a:=x) = dcast (h_width_eq) (x)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_hi", "content": "theorem BitVec.extractLsb_concat_hi {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_hi : hi_size > 0) :\n  BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (by omega), Nat.sub_add_cancel (by omega), Nat.add_sub_cancel]\n  ) hi"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_concat_lo", "content": "theorem BitVec.extractLsb_concat_lo {hi_size lo_size : ℕ} (hi : BitVec hi_size)\n  (lo : BitVec lo_size) (h_lo : lo_size > 0) : BitVec.extractLsb (hi:=lo_size - 1) (lo:=0)\n  (BitVec.append (msbs:=hi) (lsbs:=lo)) = dcast (by\n    rw [←Nat.sub_add_comm (h:=by omega), Nat.sub_add_cancel (h:=by omega), Nat.sub_zero]\n  ) lo"}, {"name": "ConcreteBinaryTower.Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo", "content": "theorem Nat.shiftRight_lo_mod_2_pow_hi_shiftLeft_lo (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n  (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) = (n - n % 2 ^ lo_len)"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts (n hi_len lo_len : ℕ)\n    (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) + (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.Nat.reconstruct_from_hi_and_lo_parts_or_ver", "content": "theorem Nat.reconstruct_from_hi_and_lo_parts_or_ver (n hi_len lo_len : ℕ)\n  (h_n : n < 2 ^ (hi_len + lo_len)) :\n    n = (((n >>> lo_len) % (2 ^ hi_len)) <<< lo_len) ||| (n % (2 ^ lo_len))"}, {"name": "ConcreteBinaryTower.BitVec.eq_append_iff_extract", "content": "theorem BitVec.eq_append_iff_extract {lo_size hi_size : ℕ} (lo : BitVec lo_size)\n  (hi : BitVec hi_size) (h_hi_gt_0 : hi_size > 0) (h_lo_gt_0 : lo_size > 0)\n  (x : BitVec (hi_size + lo_size)) : x = dcast (by rfl) (BitVec.append (msbs:=hi) (lsbs:=lo)) ↔\n  hi = dcast (by omega) (BitVec.extractLsb (hi:=hi_size + lo_size - 1) (lo:=lo_size) x) ∧\n  lo = dcast (by omega) (BitVec.extractLsb (hi:=lo_size - 1) (lo:=0) x)"}, {"name": "ConcreteBinaryTower.one_le_sub_middle_of_pow2", "content": "lemma one_le_sub_middle_of_pow2 {k : ℕ} (h_k : 1 ≤ k) : 1 ≤ 2 ^ k - 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.sub_middle_of_pow2_with_one_canceled", "content": "lemma sub_middle_of_pow2_with_one_canceled {k : ℕ} (h_k : 1 ≤ k) : 2 ^ k - 1 - 2 ^ (k - 1) + 1\n  = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sub_middle", "content": "lemma h_sub_middle {k : ℕ} (h_pos : k > 0) : 2 ^ k - 1 - 2 ^ (k - 1) + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_middle_sub", "content": "lemma h_middle_sub {k : ℕ} : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1)"}, {"name": "ConcreteBinaryTower.h_sum_two_same_pow2", "content": "lemma h_sum_two_same_pow2 {k : ℕ} (h_pos : k > 0) : 2 ^ (k - 1) + 2 ^ (k - 1) = 2 ^ k"}, {"name": "ConcreteBinaryTower.sum_fromNat_eq_from_xor_Nat", "content": "theorem sum_fromNat_eq_from_xor_Nat {k : ℕ} (x y : Nat) :\n  fromNat (k:=k) (x ^^^ y) = fromNat (k:=k) x + fromNat (k:=k) y"}, {"name": "ConcreteBinaryTower.zero_add", "content": "lemma zero_add {k : ℕ} (a : ConcreteBTField k) : 0 + a = a"}, {"name": "ConcreteBinaryTower.add_zero", "content": "lemma add_zero {k : ℕ} (a : ConcreteBTField k) : a + 0 = a"}, {"name": "ConcreteBinaryTower.cast_join", "content": "lemma cast_join {k n : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) (heq : k = n) :\n  join (k:=k) h_pos hi lo = cast (by rw [heq])\n    (join (k:=n) (by omega) (cast (by subst heq; rfl) hi) (lo:=cast (by subst heq; rfl) lo))"}, {"name": "ConcreteBinaryTower.zero_is_0", "content": "lemma zero_is_0 {k : ℕ} : (zero (k:=k)) = (0 : ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.one_is_1", "content": "lemma one_is_1 {k : ℕ} : (one (k:=k)) = 1"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_shift_ofNat", "content": "theorem BitVec.extractLsb_eq_shift_ofNat {n : Nat} (x : BitVec n) (l r : Nat) :\n    BitVec.extractLsb r l x = BitVec.ofNat (r - l + 1) (x.toNat >>> l)"}, {"name": "ConcreteBinaryTower.setWidth_eq_ofNat_mod", "content": "theorem setWidth_eq_ofNat_mod {n num_bits : Nat} (x : BitVec n) :\n  BitVec.setWidth num_bits x = BitVec.ofNat num_bits (x.toNat % 2 ^ num_bits)"}, {"name": "ConcreteBinaryTower.BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat", "content": "theorem BitVec.extractLsb_eq_and_pow_2_minus_1_ofNat {n num_bits : Nat}\n  (h_num_bits : num_bits > 0) (x : BitVec n) :\n  BitVec.extractLsb (hi:= num_bits - 1) (lo := 0) x =\n    BitVec.ofNat (num_bits - 1 - 0 + 1) (x.toNat &&& (2 ^ num_bits - 1))"}, {"name": "ConcreteBinaryTower.split_bitvec_eq_iff_fromNat", "content": "theorem split_bitvec_eq_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  split h_pos x = (hi_btf, lo_btf) ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_eq_iff_dcast_extractLsb", "content": "theorem join_eq_iff_dcast_extractLsb {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = dcast (h_sub_middle h_pos) (BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x) ∧\n  lo_btf = dcast (h_middle_sub) (BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x))"}, {"name": "ConcreteBinaryTower.join_eq_bitvec_iff_fromNat", "content": "theorem join_eq_bitvec_iff_fromNat {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n  (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n  x = 《 hi_btf, lo_btf 》 ↔\n  (hi_btf = fromNat (k:=k - 1) (x.toNat >>> 2 ^ (k - 1)) ∧\n  lo_btf = fromNat (k:=k - 1) (x.toNat &&& (2 ^ (2 ^ (k - 1)) - 1)))"}, {"name": "ConcreteBinaryTower.join_of_split", "content": "theorem join_of_split {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_split_eq : split h_pos x = (hi_btf, lo_btf)) :\n    x = 《 hi_btf, lo_btf 》"}, {"name": "ConcreteBinaryTower.split_of_join", "content": "theorem split_of_join {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField k)\n    (hi_btf lo_btf : ConcreteBTField (k - 1))\n    (h_join : x = 《hi_btf, lo_btf》) :\n    (hi_btf, lo_btf) = split h_pos x"}, {"name": "ConcreteBinaryTower.split_join_eq_split", "content": "lemma split_join_eq_split {k : ℕ} (h_pos : k > 0)\n    (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    split h_pos (《 hi_btf, lo_btf 》) = (hi_btf, lo_btf)"}, {"name": "ConcreteBinaryTower.eq_iff_split_eq", "content": "theorem eq_iff_split_eq {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k) :\n  x₀ = x₁ ↔ (split h_pos x₀ = split h_pos x₁)"}, {"name": "ConcreteBinaryTower.split_sum_eq_sum_split", "content": "theorem split_sum_eq_sum_split {k : ℕ} (h_pos : k > 0) (x₀ x₁ : ConcreteBTField k)\n  (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1))\n  (h_split_x₀ : split h_pos x₀ = (hi₀, lo₀))\n  (h_split_x₁ : split h_pos x₁ = (hi₁, lo₁)) :\n  split h_pos (x₀ + x₁) = (hi₀ + hi₁, lo₀ + lo₁)"}, {"name": "ConcreteBinaryTower.join_add_join", "content": "theorem join_add_join {k : ℕ} (h_pos : k > 0) (hi₀ lo₀ hi₁ lo₁ : ConcreteBTField (k - 1)) :\n  《 hi₀, lo₀ 》 + 《 hi₁, lo₁ 》 = 《 hi₀ + hi₁, lo₀ + lo₁ 》"}, {"name": "ConcreteBinaryTower.split_Z", "content": "theorem split_Z {k : ℕ} (h_pos : k > 0) :\n    split h_pos (Z k) = (one (k:=k - 1), zero (k:=k - 1))"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_eq_of_dest_eq", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_eq_of_dest_eq (k m n : ℕ) (h_eq : m = n) :\n    (ConcreteBTField k →+* ConcreteBTField m)\n    = (ConcreteBTField k →+* ConcreteBTField n)"}, {"name": "ConcreteBinaryTower.ConcreteBTField.RingHom_cast_dest_apply", "content": "@[simp]\ntheorem ConcreteBTField.RingHom_cast_dest_apply (k m n : ℕ) (h_eq : m = n)\n  (f : ConcreteBTField k →+* ConcreteBTField m) (x : ConcreteBTField k) :\n    (cast (ConcreteBTField.RingHom_eq_of_dest_eq (k:=k) (m:=m) (n:=n) h_eq) f) x\n    = cast (by apply cast_ConcreteBTField_eq (h_eq:=h_eq)) (f x)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_id", "content": "lemma concreteTowerAlgebraMap_id (k : ℕ) :\n    concreteTowerAlgebraMap (h_le:=by omega) = RingHom.id (ConcreteBTField k)"}, {"name": "ConcreteBinaryTower.concreteTowerAlgebraMap_succ_1", "content": "lemma concreteTowerAlgebraMap_succ_1 (k : ℕ) :\n  concreteTowerAlgebraMap (l:=k) (r:=k + 1) (h_le:=by omega) = canonicalAlgMap k"}, {"name": "ConcreteBinaryTower.split_algebraMap_eq_zero_x", "content": "lemma split_algebraMap_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (algebraMap (ConcreteBTField (k - 1)) (ConcreteBTField k) x) = (0, x)"}, {"name": "ConcreteBinaryTower.split_smul_Z_eq_zero_x", "content": "lemma split_smul_Z_eq_zero_x {k : ℕ} (h_pos : k > 0) (x : ConcreteBTField (k - 1)) :\n  letI instAlgebra := ConcreteBTFieldAlgebra (l:=k-1) (r:=k) (h_le:=by omega)\n  split h_pos (x • Z k) = (x, 0)"}, {"name": "ConcreteBinaryTower.join_eq_join_via_add_smul", "content": "@[simp]\ntheorem join_eq_join_via_add_smul {k : ℕ} (h_pos : k > 0)\n    (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    《 hi_btf, lo_btf 》 = join_via_add_smul k h_pos hi_btf lo_btf"}, {"name": "ConcreteBinaryTower.split_join_via_add_smul_eq_iff_split", "content": "lemma split_join_via_add_smul_eq_iff_split {k : ℕ} (h_pos : k > 0)\n    (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    split (k:=k) (h:=by omega) (x:=join_via_add_smul (k:=k) (h_pos:=h_pos) hi_btf lo_btf) =\n      (hi_btf, lo_btf)"}], "local_ctx": "import ArkLib.Data.Classes.DCast\n\nimport ArkLib.Data.FieldTheory.BinaryField.Tower.Basic\n\nnamespace ConcreteBinaryTower\n\nopen Polynomial\n\ndef ConcreteBTField : ℕ → Type := fun k => BitVec (2 ^ k)\n\nsection BitVecDCast\n\ninstance BitVec.instDCast : DCast Nat BitVec where\n  dcast h := BitVec.cast h\n  dcast_id := by admit /- proof elided -/\n\nend BitVecDCast\n\nsection ConversionUtils\n\ndef fromNat {k : ℕ} (n : Nat) : ConcreteBTField k :=\n  BitVec.ofNat (2 ^ k) n\n\ninstance ConcreteBTField.instDCast_local : DCast ℕ ConcreteBTField where\n  dcast h_k_eq term_k1 := BitVec.cast (congrArg (fun n => 2 ^ n) h_k_eq) term_k1\n  dcast_id := by admit /- proof elided -/\n\nend ConversionUtils\n\nsection NumericLemmas\n\nend NumericLemmas\n\nsection FieldOperationsAndInstances\n\ndef zero {k : ℕ} : ConcreteBTField k := BitVec.zero (2 ^ k)\n\ndef one {k : ℕ} : ConcreteBTField k := 1#(2 ^ k)\n\ndef add {k : ℕ} (x y : ConcreteBTField k) : ConcreteBTField k := BitVec.xor x y\n\ndef neg {k : ℕ} (x : ConcreteBTField k) : ConcreteBTField k := x\n\ndef split {k : ℕ} (h : k > 0) (x : ConcreteBTField k) :\n    ConcreteBTField (k - 1) × ConcreteBTField (k - 1) :=\n  let lo_bits : BitVec (2 ^ (k - 1) - 1 - 0 + 1) :=\n    BitVec.extractLsb (hi := 2 ^ (k - 1) - 1) (lo := 0) x\n  let hi_bits : BitVec (2 ^ k - 1 - 2 ^ (k - 1) + 1) :=\n    BitVec.extractLsb (hi := 2 ^ k - 1) (lo := 2 ^ (k - 1)) x\n  have h_lo : 2 ^ (k - 1) - 1 - 0 + 1 = 2 ^ (k - 1) := by admit /- proof elided -/\n\ndef join {k : ℕ} (h_pos : k > 0) (hi lo : ConcreteBTField (k - 1)) : ConcreteBTField k :=\n\nstructure ConcreteBTFAddCommGroupProps (k : ℕ) where\n  add_assoc : ∀ a b c : ConcreteBTField k, (a + b) + c = a + (b + c) := add_assoc\n  add_comm : ∀ a b : ConcreteBTField k, a + b = b + a := add_comm\n  add_zero : ∀ a : ConcreteBTField k, a + zero = a := add_zero\n  zero_add : ∀ a : ConcreteBTField k, zero + a = a := zero_add\n  add_neg : ∀ a : ConcreteBTField k, a + (neg a) = zero := neg_add_cancel\n\ndef mkAddCommGroupInstance {k : ℕ} : AddCommGroup (ConcreteBTField k) := {\n  zero := zero\n  neg := neg\n  sub := fun x y => add x y\n  add_assoc := add_assoc\n  add_comm := add_comm\n  zero_add := zero_add\n  add_zero := add_zero\n  nsmul := fun n x => if n % 2 = (0 : ℕ) then zero else x\n  zsmul := fun (n : ℤ) x => if n % 2 = 0 then zero else x  \n  neg_add_cancel := neg_add_cancel\n  nsmul_succ := nsmul_succ\n  zsmul_succ' := fun n a => zsmul_succ n a\n  add := add\n  zsmul_neg' := zsmul_neg' (k := k)\n}\n\ndef Z (k : ℕ) : ConcreteBTField k :=\n  if h_k : k = 0 then one\n  else\n    《 one (k:=k-1), zero (k:=k-1) 》\n\ndef equivProd {k : ℕ} (h_k_pos : k > 0) :\n  ConcreteBTField k ≃ ConcreteBTField (k - 1) × ConcreteBTField (k - 1) where\n  toFun := split h_k_pos\n  invFun := fun (hi, lo) => 《 hi, lo 》\n  left_inv := fun x => Eq.symm (join_of_split h_k_pos x _ _ rfl)\n  right_inv := fun ⟨hi, lo⟩ => Eq.symm (split_of_join h_k_pos _ hi lo rfl)\n\ndef concrete_mul {k : ℕ} (a b : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = zero then zero\n    else if b = zero then zero\n    else if a = one then b\n    else if b = one then a\n    else zero \n  else\n    have h_k_gt_0 : k > 0 := by admit /- proof elided -/\n\ndef concrete_inv {k : ℕ} (a : ConcreteBTField k) : ConcreteBTField k :=\n  if h_k_zero : k = 0 then\n    if a = 0 then 0 else 1\n  else\n    if h_a_zero : a = 0 then 0\n    else if h_a_one : a = 1 then 1\n    else\n      let h_k_gt_0 : k > 0 := Nat.zero_lt_of_ne_zero h_k_zero\n      let (a_hi, a_lo) := split (k:=k) (h:=h_k_gt_0) a\n      let prevZ := Z (k - 1)\n      let a_lo_next := a_lo + concrete_mul a_hi prevZ\n      let delta := concrete_mul a_lo a_lo_next + concrete_mul a_hi a_hi\n      let delta_inverse := concrete_inv delta\n      let out_hi := concrete_mul delta_inverse a_hi\n      let out_lo := concrete_mul delta_inverse a_lo_next\n      let res := 《 out_hi, out_lo 》\n      res\n\nsection FieldLemmasOfLevel0\n\nend FieldLemmasOfLevel0\n\nsection NumericCasting\n\ndef natCast {k : ℕ} (n : ℕ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef natCast_zero {k : ℕ} : natCast (k:=k) 0 = zero :=\n\ndef natCast_succ {k : ℕ} (n : ℕ) : natCast (k:=k) (n + 1) = natCast (k:=k) n + 1 :=\n\ndef intCast {k : ℕ} (n : ℤ) : ConcreteBTField k := if n % 2 = 0 then zero else one\n\ndef intCast_ofNat {k : ℕ} (n : ℕ) : intCast (k:=k) (n : ℤ) = natCast n :=\n\ndef intCast_negSucc {k : ℕ} (n : ℕ) : intCast (k:=k) (Int.negSucc n)\n  = - (↑(n + 1) : ConcreteBTField k) :=\n\nend NumericCasting\n\nstructure ConcreteBTFRingProps (k : ℕ) extends (ConcreteBTFAddCommGroupProps k) where\n  \n  mul_eq : ∀ (a b : ConcreteBTField k) (h_k : k > 0)\n    {a₁ a₀ b₁ b₀ : ConcreteBTField (k - 1)}\n    (_h_a : (a₁, a₀) = split h_k a) (_h_b : (b₁, b₀) = split h_k b),\n    concrete_mul a b =\n      《 concrete_mul a₀ b₁ + concrete_mul b₀ a₁ + concrete_mul (concrete_mul a₁ b₁) (Z (k - 1)),\n      concrete_mul a₀ b₀ + concrete_mul a₁ b₁ 》\n\n  \n  zero_mul : ∀ a : ConcreteBTField k, concrete_mul zero a = zero\n  zero_mul' : ∀ a : ConcreteBTField k, concrete_mul 0 a = 0\n  mul_zero : ∀ a : ConcreteBTField k, concrete_mul a zero = zero\n  mul_zero' : ∀ a : ConcreteBTField k, concrete_mul a 0 = 0\n  one_mul : ∀ a : ConcreteBTField k, concrete_mul one a = a\n  mul_one : ∀ a : ConcreteBTField k, concrete_mul a one = a\n\n  \n  mul_assoc : ∀ a b c : ConcreteBTField k, concrete_mul (concrete_mul a b) c\n    = concrete_mul a (concrete_mul b c)\n  mul_left_distrib : ∀ a b c : ConcreteBTField k, concrete_mul a (b + c)\n    = concrete_mul a b + concrete_mul a c\n  mul_right_distrib : ∀ a b c : ConcreteBTField k, concrete_mul (a + b) c\n    = concrete_mul a c + concrete_mul b c\n\nstructure ConcreteBTFDivisionRingProps (k : ℕ) extends (ConcreteBTFRingProps k) where\n  \n  mul_inv_cancel : ∀ a : ConcreteBTField k, a ≠ zero → concrete_mul a (concrete_inv a) = one\n\nstructure ConcreteBTFieldProps (k : ℕ) extends (ConcreteBTFDivisionRingProps k) where\n  \n  mul_comm : ∀ a b : ConcreteBTField k, concrete_mul a b = concrete_mul b a\n\ndef mkRingInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Ring (ConcreteBTField k) where\n  toAddCommGroup := mkAddCommGroupInstance\n  toOne := inferInstance\n  mul := concrete_mul\n  mul_assoc := props.mul_assoc\n  one_mul := props.one_mul\n  mul_one := props.mul_one\n  left_distrib := props.mul_left_distrib\n  right_distrib := props.mul_right_distrib\n  zero_mul := props.zero_mul\n  mul_zero := props.mul_zero\n\n  natCast n := natCast n\n  natCast_zero := natCast_zero\n  natCast_succ n := natCast_succ n\n  intCast n := intCast n\n  intCast_ofNat n := intCast_ofNat n\n  intCast_negSucc n := intCast_negSucc n\n\ndef mkDivisionRingInstance {k : ℕ} (props : ConcreteBTFieldProps k)\n    : DivisionRing (ConcreteBTField k) where\n  toRing := mkRingInstance (k:=k) props\n  inv := concrete_inv\n  exists_pair_ne := concrete_exists_pair_ne (k := k)\n  mul_inv_cancel := props.mul_inv_cancel\n  inv_zero := concrete_inv_zero\n  qsmul := (Rat.castRec · * ·)\n  nnqsmul := (NNRat.castRec · * ·)\n\ndef mkFieldInstance {k : ℕ} (props : ConcreteBTFieldProps k) : Field (ConcreteBTField k) where\n  toDivisionRing := mkDivisionRingInstance (k:=k) props\n  mul_comm := props.mul_comm\n\nstructure ConcreteBTFStepResult (k : ℕ) extends (ConcreteBTFieldProps k) where\n  instFintype : Fintype (ConcreteBTField k)\n  fieldFintypeCard : Fintype.card (ConcreteBTField k) = 2^(2^k)\n  \n  sumZeroIffEq : ∀ (x y : ConcreteBTField k), x + y = 0 ↔ x = y\n  traceMapEvalAtRootsIs1 :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    TraceMapProperty (ConcreteBTField k) (u:=Z k) k\n  instIrreduciblePoly :\n    letI := mkFieldInstance (k:=k) (props:=toConcreteBTFieldProps)\n    (Irreducible (p := (definingPoly (s:=(Z k)))))\n\nend FieldOperationsAndInstances\n\nsection BTFieldPropsOneLevelLiftingLemmas\n\nvariable {k : ℕ} {h_k : k > 0}\n\nend BTFieldPropsOneLevelLiftingLemmas\n\nsection TowerFieldsConstruction\n\ndef liftBTFieldProps (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  ConcreteBTFieldProps (k + 1) := {\n    zero_mul := concrete_zero_mul (prevBTFResult.toConcreteBTFieldProps),\n    zero_mul' := fun a => by admit /- proof elided -/\n\ndef liftConcreteBTField (k : ℕ) (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  Field (ConcreteBTField (k + 1)) :=\n\ndef concreteCanonicalEmbedding (k : ℕ)\n    (prevBTFieldProps : ConcreteBTFieldProps (k := (k)))\n    (curBTFieldProps : ConcreteBTFieldProps (k := (k + 1))) :\n  letI := mkFieldInstance prevBTFieldProps\n  letI := mkFieldInstance curBTFieldProps\n  ConcreteBTField k →+* ConcreteBTField (k + 1) :=\n\ninstance instAlgebraLiftConcreteBTField (k : ℕ)\n  (prevBTFResult : ConcreteBTFStepResult (k := k)) :\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  Algebra (ConcreteBTField k) (ConcreteBTField (k + 1)) :=\n  letI := mkFieldInstance (prevBTFResult.toConcreteBTFieldProps)\n  letI := liftConcreteBTField (k:=k) prevBTFResult\n  RingHom.toAlgebra (R:=ConcreteBTField k) (S:=ConcreteBTField (k + 1))\n    (i:=(concreteCanonicalEmbedding (k:=k)\n      (prevBTFieldProps:=prevBTFResult.toConcreteBTFieldProps)\n      (curBTFieldProps:=liftBTFieldProps (k:=k) (prevBTFResult:=prevBTFResult))))\n\ndef getBTFResult (k : ℕ) : ConcreteBTFStepResult k :=\n  match k with\n  | 0 =>\n    let base : ConcreteBTFieldProps 0 := {\n      mul_eq := fun a b h_k _ _ _ _ _ _ => by admit /- proof elided -/\n| c1_one\n        · \n          rw [c1_zero] at h_mul\n          \n          simp at h_mul\n        · \n          rcases c2_cases with c2_zero | c2_one\n          · \n            rw [c2_zero] at h_mul\n            \n            simp at h_mul\n          · \n            \n            exact ⟨c1_one, c2_one⟩\n      \n      have specialElement_eq_zero : specialElement = 0 := by admit /- proof elided -/\n\nend TowerFieldsConstruction\n\nsection ConcreteBTFieldAlgebraConstruction\n\ndef canonicalAlgMap (k : ℕ) := concreteCanonicalEmbedding (k:=k)\n  (prevBTFieldProps:= ((getBTFResult k).toConcreteBTFieldProps))\n  (curBTFieldProps:= ((getBTFResult (k + 1)).toConcreteBTFieldProps))\n\ndef concreteTowerAlgebraMap (l r : ℕ) (h_le : l ≤ r) :\n    ConcreteBTField l →+* ConcreteBTField r :=\n\ninstance instAlgebraTowerConcreteBTF : AlgebraTower (ConcreteBTField) where\n  algebraMap := concreteTowerAlgebraMap\n  commutes' := by admit /- proof elided -/\n\ndef ConcreteBTFieldAlgebra {l r : ℕ} (h_le : l ≤ r) :\n    Algebra (ConcreteBTField l) (ConcreteBTField r) := instAlgebraTowerConcreteBTF.toAlgebra h_le\n\ndef join_via_add_smul (k : ℕ) (h_pos : k > 0) (hi_btf lo_btf : ConcreteBTField (k - 1)) :\n    ConcreteBTField k :=\n\nend ConcreteBTFieldAlgebraConstruction\n\nnoncomputable section ConcreteMultilinearBasis\n\nopen Module\n\nend ConcreteMultilinearBasis\n\nsection TowerEquivalence\n\nopen BinaryTower\n\nnoncomputable def towerEquiv_zero : RingEquiv (R:=GF(2)) (S:=ConcreteBTField 0) :=  {\n  toFun := fun x => if x = 0 then 0 else 1,\n  invFun := fun x => if x = 0 then 0 else 1,\n  left_inv := fun x => by admit /- proof elided -/\n\nnoncomputable def towerRingEquiv0 : BTField 0 ≃+* ConcreteBTField 0 :=\n\nnoncomputable def towerRingEquivFromConcrete0 : ConcreteBTField 0 ≃+* BTField 0 :=\n\nnoncomputable def towerRingHomForwardMap (k : ℕ) : ConcreteBTField k → BTField k :=\n\nnoncomputable def towerRingHomBackwardMap (k : ℕ) : BTField k → ConcreteBTField k :=", "target_theorem": "lemma towerRingHomForwardMap_backwardMap_eq (k : ℕ) (x : BTField k) :\n  towerRingHomForwardMap (k:=k) (towerRingHomBackwardMap (k:=k) x) = x :=", "ground_truth_proof": ":= by\n  induction k with\n  | zero =>\n    unfold towerRingHomForwardMap towerRingHomBackwardMap\n    simp only [↓reduceDIte, RingEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe]\n    rcases GF_2_value_eq_zero_or_one x with x_zero | x_one\n    · rw [x_zero];\n      unfold towerRingEquivFromConcrete0 -- ⊢ towerRingEquiv0.symm (towerRingEquiv0 0) = 0\n      exact RingEquiv.symm_apply_apply towerRingEquiv0 0\n    · rw [x_one];\n      unfold towerRingEquivFromConcrete0 -- ⊢ towerRingEquiv0.symm (towerRingEquiv0 1) = 1\n      exact RingEquiv.symm_apply_apply towerRingEquiv0 1\n  | succ k ih =>\n    rw [towerRingHomBackwardMap] -- split inner\n    simp only [Nat.add_eq_zero, one_ne_zero, and_false, ↓reduceDIte,\n      Nat.add_one_sub_one]\n    rw [towerRingHomForwardMap] -- split outer\n    simp only [Nat.add_eq_zero, one_ne_zero, and_false, ↓reduceDIte,\n      Nat.add_one_sub_one]\n    apply Eq.symm\n    rw! [split_join_via_add_smul_eq_iff_split (k:=k + 1)]\n    simp only\n    -- apply induction hypothesis\n    rw [ih, ih]\n    rw [BinaryTower.eq_join_via_add_smul_eq_iff_split]", "nesting_depth": 15, "transitive_dep_count": 285, "subset_aristotle": false, "category": "Applied verif."}
{"id": 61, "thm_name": "Capless.preservation", "thm_stmt": "theorem preservation\n  (hr : Reduce state state')\n  (ht : TypedState state Γ E) :\n  Preserve Γ E state'", "lean_root": "capless-lean", "rel_path": "Capless/Soundness/Preservation.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Subst.Type.Typing", "import Capless.Renaming.Capture.Typing", "import Capless.Weakening.TypedCont.Term", "import Capless.Basic", "import Capless.Typing.Basic", "import Capless.CaptureSet", "import Capless.Store", "import Capless.Narrowing.Typing", "import Capless.Type.Basic", "import Capless.Weakening.Subtyping", "import Capless.Subst.Term.Typing", "import Capless.Inversion.Typing", "import Capless.Typing.Boundary", "import Capless.Subst.Capture.Typing", "import Capless.Renaming.Capture.Subcapturing", "import Capless.Weakening.TypedCont.Type", "import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Term.Typing", "import Capless.Renaming.Type.Subcapturing", "import Capless.Narrowing.Subtyping", "import Capless.WellScoped.Basic", "import Capless.Renaming.Term.Subcapturing", "import Capless.Weakening.TypedCont.Capture", "import Capless.Inversion.Lookup", "import Capless.Narrowing.TypedCont", "import Capless.Renaming.Type.Subtyping", "import Capless.Tactics", "import Capless.Renaming.Type.Typing", "import Capless.Subtyping.Basic", "import Capless.Weakening.TypedCont", "import Capless.Weakening.Typing", "import Capless.Renaming.Capture.Subtyping", "import Capless.Type", "import Capless.Reduction"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "abs", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}], "used_repo_defs": [{"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "macro \"split_and\" : tactic => `(tactic| repeat any_goals app", "content": "macro \"split_and\" : tactic => `(tactic| repeat any_goals apply And.intro)"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "TypedStore", "content": "inductive TypedStore : Store n m k -> Context n m k -> Prop where\n| empty : TypedStore Store.empty Context.empty\n| val :\n  TypedStore σ Γ ->\n  Typed Γ t (EType.type E) Ct ->\n  (hv : t.IsValue) ->\n  TypedStore (Store.val σ t hv) (Γ.var E)\n| tval :\n  TypedStore σ Γ ->\n  TypedStore (Store.tval σ S) (Γ.tvar (TBinding.inst S))\n| cval :\n  TypedStore σ Γ ->\n  TypedStore (Store.cval σ C) (Γ.cvar (CBinding.inst C))\n| label :\n  TypedStore σ Γ ->\n  TypedStore (Store.label σ S) (Γ.label S)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "TypedState", "content": "inductive TypedState : State n m k -> Context n m k -> EType n m k -> Prop where\n| mk :\n  TypedStore σ Γ ->\n  Typed Γ t E Ct ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedState (State.mk σ cont t) Γ E'"}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "Reduce", "content": "inductive Reduce : State n m k -> State n' m' k' -> Prop where\n| apply {σ : Store n m k} :\n  σ.Bound x (Term.lam T t) ->\n  Reduce ⟨σ | cont | Term.app x y⟩ ⟨σ | cont | t.open y⟩\n| tapply {σ : Store n m k} :\n  σ.Bound x (Term.tlam S t) ->\n  Reduce ⟨σ | cont | Term.tapp x X⟩ ⟨σ | cont | t.topen X⟩\n| capply {σ : Store n m k} :\n  σ.Bound x (Term.clam B t) ->\n  Reduce ⟨σ | cont | Term.capp x c⟩ ⟨σ | cont | t.copen c⟩\n| enter :\n  Reduce\n    ⟨σ | cont | boundary:S in t⟩\n    ⟨(σ.label S).cval {x=0} | cont.weaken.cweaken.scope 0 | t⟩\n| leave_var :\n  Reduce\n    ⟨σ | cont.scope x | Term.var y⟩\n    ⟨σ | cont | Term.var y⟩\n| leave_val {v : Term n m k} :\n  (hv : Term.IsValue v) ->\n  Reduce\n    ⟨σ | cont.scope x | v⟩\n    ⟨σ | cont | v⟩\n| invoke {σ : Store n m k} {cont : Cont n m k} :\n  σ.LBound x S ->\n  cont.HasLabel x tail ->\n  Reduce\n    ⟨σ | cont | Term.invoke x y⟩\n    ⟨σ | tail | Term.var y⟩\n| push :\n  Reduce\n    ⟨σ | cont | Term.letin t u⟩\n    ⟨σ | Cont.cons u cont | t⟩\n| push_ex :\n  Reduce\n    ⟨σ | cont | Term.letex t u⟩\n    ⟨σ | Cont.conse u cont | t⟩\n| rename :\n  Reduce\n    ⟨σ | Cont.cons u cont | Term.var x⟩\n    ⟨σ | cont | u.open x⟩\n| lift :\n  (hv : Term.IsValue v) ->\n  Reduce\n    ⟨σ | Cont.cons u cont | v⟩\n    ⟨σ.val v hv | cont.weaken | u⟩\n| lift_ex :\n  Reduce\n    ⟨σ | Cont.conse u cont | Term.pack C x⟩\n    ⟨σ.cval C | cont.cweaken | u.open x⟩\n| tlift :\n  Reduce\n    ⟨σ | cont | Term.bindt S t⟩\n    ⟨σ.tval S | cont.tweaken | t⟩\n| clift :\n  Reduce\n    ⟨σ | cont | Term.bindc C t⟩\n    ⟨σ.cval C | cont.cweaken | t⟩"}, {"name": "infix:30 \" ", "content": "infix:30 \" "}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "VarSubst.narrow", "content": "def VarSubst.narrow\n  (hs : CSubtyp Γ T' T) :\n  VarSubst (Γ.var T) FinFun.id (Γ.var T') :="}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "CType.open", "content": "def CType.open (C : CType (n+1) m k) (x : Fin n) : CType n m k :=\n  C.rename (FinFun.open x)"}, {"name": "CVarSubst.open", "content": "def CVarSubst.open :\n  CVarSubst\n    (Γ.cvar (CBinding.bound (CBound.upper {c=c})))\n    (FinFun.open c)\n    Γ :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "CVarSubst.narrow", "content": "def CVarSubst.narrow\n  (hs : Subbound Γ B' B) :\n  CVarSubst\n    (Γ,c<:B)\n    FinFun.id\n    (Γ,c<:B') :="}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "CaptureSet.copen", "content": "def CaptureSet.copen (C : CaptureSet n (k+1)) (x : Fin k) : CaptureSet n k :=\n  C.crename (FinFun.open x)"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "CaptureSet.open", "content": "def CaptureSet.open (C : CaptureSet (n+1) k) (x : Fin n) : CaptureSet n k :=\n  C.rename (FinFun.open x)"}, {"name": "Term.open", "content": "def Term.open (t : Term (n+1) m k) (x : Fin n) : Term n m k :=\n  t.rename (FinFun.open x)"}, {"name": "SType.open", "content": "def SType.open (S : SType (n+1) m k) (x : Fin n) : SType n m k :=\n  S.rename (FinFun.open x)"}, {"name": "TVarSubst.open", "content": "def TVarSubst.open :\n  TVarSubst\n    (Γ.tvar (TBinding.bound (SType.tvar X)))\n    (FinFun.open X)\n    Γ :=\n  { map := fun x E hb => by admit /- proof elided -/"}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "TVarSubst.narrow", "content": "def TVarSubst.narrow\n  (hs : SSubtyp Γ S' S) :\n  TVarSubst\n    (Γ.tvar (TBinding.bound S))\n    FinFun.id\n    (Γ.tvar (TBinding.bound S')) :="}, {"name": "VarSubst.open", "content": "def VarSubst.open\n  (hx : Typed Γ (Term.var x) (EType.type T) Cx) :\n  VarSubst (Γ.var T) (FinFun.open x) Γ :="}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "Subcapt.tweaken", "content": "def Subcapt.tweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.tvar b) ⊢ C1 <:c C2 :="}, {"name": "EType.cweaken_type", "content": "@[simp]\ndef EType.cweaken_type :\n  (EType.type T).cweaken = EType.type (T.cweaken) :="}, {"name": "Typed.cweaken_cext_ext", "content": "def Typed.cweaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) t.cweaken1 E.cweaken1 Ct.cweaken1 :="}, {"name": "CVarMap.weaken_cext_ext", "content": "def CVarMap.weaken_cext_ext {Γ : Context n m k} :\n  CVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken.ext\n    (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "Typed.cweaken_ext", "content": "def Typed.cweaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.cvar b).var T.cweaken) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "CVarMap.weaken_ext", "content": "def CVarMap.weaken_ext {Γ : Context n m k} :\n  CVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.cvar b).var T.cweaken) :="}, {"name": "VarMap.weaken_ext", "content": "def VarMap.weaken_ext {Γ : Context n m k} :\n  VarMap\n    (Γ.var T)\n    FinFun.weaken.ext\n    ((Γ.var P).var T.weaken) :="}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "VarMap.weaken_cext_ext", "content": "def VarMap.weaken_cext_ext {Γ : Context n m k} :\n  VarMap\n    ((Γ.cvar (CBinding.bound b)).var T)\n    FinFun.weaken.ext\n    (((Γ.var P).cvar (CBinding.bound b.weaken)).var T.weaken) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.weaken_ext", "content": "def TVarMap.weaken_ext {Γ : Context n m k} :\n  TVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.tvar b).var T.tweaken) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "TVarMap.weaken_cext_ext", "content": "def TVarMap.weaken_cext_ext {Γ : Context n m k} :\n  TVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken\n    (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CType.copen", "content": "def CType.copen (C : CType n m (k+1)) (x : Fin k) : CType n m k :=\n  C.crename (FinFun.open x)"}, {"name": "SType.copen", "content": "def SType.copen (S : SType n m (k+1)) (x : Fin k) : SType n m k :=\n  S.crename (FinFun.open x)"}, {"name": "Term.copen", "content": "def Term.copen (t : Term n m (k+1)) (c : Fin k) : Term n m k :=\n  t.crename (FinFun.open c)"}, {"name": "EType.copen", "content": "def EType.copen (E : EType n m (k+1)) (x : Fin k) : EType n m k :=\n  E.crename (FinFun.open x)"}, {"name": "SType.topen", "content": "def SType.topen (S : SType n (m+1) k) (X : Fin m) : SType n m k :=\n  S.trename (FinFun.open X)"}, {"name": "CType.topen", "content": "def CType.topen (C : CType n (m+1) k) (X : Fin m) : CType n m k :=\n  C.trename (FinFun.open X)"}, {"name": "Term.topen", "content": "def Term.topen (t : Term n (m+1) k) (X : Fin m) : Term n m k :=\n  t.trename (FinFun.open X)"}, {"name": "EType.topen", "content": "def EType.topen (E : EType n (m+1) k) (X : Fin m) : EType n m k :=\n  E.trename (FinFun.open X)"}, {"name": "CVarRename.boundary", "content": "def CVarRename.boundary {Γ : Context n m k} {S : SType n m k} :\n  CVarMap\n    (((Γ.label S),c<:*),x:(Label[S.weaken.cweaken])^{c=0})\n    FinFun.weaken.ext\n    ((((Γ.label S),c:={x=0}),c<:*),x:(Label[S.weaken.cweaken.cweaken])^{c=0}) :="}, {"name": "CType.weaken_capt", "content": "@[simp]\ndef CType.weaken_capt :\n  (CType.capt C S).weaken = CType.capt C.weaken S.weaken :="}, {"name": "VarRename.boundary", "content": "def VarRename.boundary {Γ : Context n m k} {S : SType n m k} :\n  VarMap\n    ((Γ,c<:*),x:(Label[S.cweaken])^{c=0})\n    FinFun.weaken.ext\n    (((Γ.label S),c<:*),x:(Label[S.weaken.cweaken])^{c=0}) :="}, {"name": "CVarSubst.boundary", "content": "def CVarSubst.boundary {Γ : Context n m k} {S : SType n m k} :\n  CVarSubst\n    ((((Γ.label S),c:={x=0}),c<:*),x:(Label[S.weaken.cweaken.cweaken])^{c=0})\n    (FinFun.open 0)\n    (((Γ.label S),c:={x=0}),x:(Label[S.weaken.cweaken])^{c=0}) :="}, {"name": "VarSubst.boundary", "content": "def VarSubst.boundary {Γ : Context n m k} {S : SType n m k} :\n  VarSubst\n    (((Γ.label S),c:={x=0}),x:(Label[S.weaken.cweaken])^{c=0})\n    (FinFun.open 0)\n    ((Γ.label S),c:={x=0}) :="}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "VarMap.lweaken_ext", "content": "def VarMap.lweaken_ext {Γ : Context n m k} :\n  VarMap\n    (Γ.var T)\n    FinFun.weaken.ext\n    ((Γ.label P).var T.weaken) :="}, {"name": "VarMap.lweaken", "content": "def VarMap.lweaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.label S) :="}, {"name": "VarMap.lweaken_cext_ext", "content": "def VarMap.lweaken_cext_ext {Γ : Context n m k} :\n  VarMap\n    ((Γ.cvar (CBinding.bound b)).var T)\n    FinFun.weaken.ext\n    (((Γ.label P).cvar (CBinding.bound b.weaken)).var T.weaken) :="}, {"name": "CVarSubst.instantiate", "content": "def CVarSubst.instantiate {Γ : Context n m k} :\n  CVarSubst\n    (Γ.cvar (CBinding.bound CBound.star))\n    FinFun.id\n    (Γ.cvar (CBinding.inst C)) :="}, {"name": "CBound.cweaken_upper", "content": "@[simp]\ndef CBound.cweaken_upper :\n  (CBound.upper C).cweaken = CBound.upper C.cweaken :="}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "Store.lookup_inv_bound", "content": "def Store.lookup_inv_bound\n  (hl : Store.Bound σ x v)\n  (ht : TypedStore σ Γ)\n  (hb : Context.Bound Γ x T) :\n  ∃ Cv, Typed Γ v (EType.type T) Cv :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Subcapt.refl", "content": "theorem Subcapt.refl :\n  Subcapt Γ C C"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "TypedCont.cweaken", "content": "theorem TypedCont.cweaken\n  (h : TypedCont Γ E t E' Ct) :\n  TypedCont (Γ.cvar b) E.cweaken t.cweaken E'.cweaken Ct.cweaken"}, {"name": "Cont.HasLabel.cweaken", "content": "theorem Cont.HasLabel.cweaken\n  (h : Cont.HasLabel cont l tail) :\n  Cont.HasLabel (cont.cweaken) l tail.cweaken"}, {"name": "WellScoped.cweaken", "content": "theorem WellScoped.cweaken\n  (h : WellScoped Γ E Ct) :\n  WellScoped (Γ.cvar b) E.cweaken Ct.cweaken"}, {"name": "CaptureSet.cweaken1_cweaken", "content": "theorem CaptureSet.cweaken1_cweaken (C : CaptureSet n k) :\n  C.cweaken.cweaken1 = C.cweaken.cweaken"}, {"name": "CaptureSet.cweaken1_weaken", "content": "theorem CaptureSet.cweaken1_weaken (C : CaptureSet n (k+1)) :\n  C.weaken.cweaken1 = C.cweaken1.weaken"}, {"name": "EType.cweaken1_cweaken", "content": "theorem EType.cweaken1_cweaken (E : EType n m k) :\n  E.cweaken.cweaken1 = E.cweaken.cweaken"}, {"name": "EType.cweaken1_weaken", "content": "theorem EType.cweaken1_weaken (E : EType n m (k+1)) :\n  E.weaken.cweaken1 = E.cweaken1.weaken"}, {"name": "EType.cweaken_ex", "content": "theorem EType.cweaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).cweaken = EType.ex T.cweaken1"}, {"name": "EType.cweaken_weaken", "content": "theorem EType.cweaken_weaken (E : EType n m k) :\n  E.weaken.cweaken = E.cweaken.weaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "WellScoped.weaken", "content": "theorem WellScoped.weaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.var T) cont.weaken Ct.weaken"}, {"name": "Cont.HasLabel.weaken", "content": "theorem Cont.HasLabel.weaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.weaken x.succ tail.weaken"}, {"name": "TypedCont.weaken", "content": "theorem TypedCont.weaken\n  (h : TypedCont Γ E t E' C0) :\n  TypedCont (Γ.var T) E.weaken t.weaken E'.weaken C0.weaken"}, {"name": "CaptureSet.weaken_cweaken", "content": "theorem CaptureSet.weaken_cweaken (C : CaptureSet n k) :\n  C.cweaken.weaken = C.weaken.cweaken"}, {"name": "CaptureSet.weaken1_weaken", "content": "theorem CaptureSet.weaken1_weaken (C : CaptureSet n k) :\n  C.weaken.weaken1 = C.weaken.weaken"}, {"name": "EType.weaken_ex", "content": "theorem EType.weaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).weaken = EType.ex T.weaken"}, {"name": "EType.weaken1_weaken", "content": "theorem EType.weaken1_weaken (E : EType n m k) :\n  E.weaken.weaken1 = E.weaken.weaken"}, {"name": "EType.weaken_cweaken", "content": "theorem EType.weaken_cweaken (E : EType n m k) :\n  E.cweaken.weaken = E.weaken.cweaken"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "Typed.narrow", "content": "theorem Typed.narrow\n  (h : Typed (Γ,x: T) t E Ct)\n  (hs : CSubtyp Γ T' T) :\n  Typed (Γ,x: T') t E Ct"}, {"name": "Subbound.tweaken", "content": "theorem Subbound.tweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.tvar b) B1 B2"}, {"name": "WellScoped.var_inv", "content": "theorem WellScoped.var_inv\n  (hsc : WellScoped Γ cont {x=x})\n  (hbx : Γ.Bound x (S^C)) :\n  WellScoped Γ cont C"}, {"name": "CaptureSet.crename_id", "content": "theorem CaptureSet.crename_id {C : CaptureSet n k} :\n  C.crename FinFun.id = C"}, {"name": "ESubtyp.narrow", "content": "theorem ESubtyp.narrow\n  (h : ESubtyp (Γ.var T) E1 E2)\n  (hs : CSubtyp Γ T' T) :\n  ESubtyp (Γ.var T') E1 E2"}, {"name": "Typed.capp_inv", "content": "theorem Typed.capp_inv\n  (h : Typed Γ (Term.capp x c) E Ct0) :\n  ∃ Cf F E0,\n    Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.cforall (CBound.upper {c=c}) F))) {x=x} ∧\n    E0 = F.copen c ∧\n    ESubtyp Γ E0 E"}, {"name": "Typed.capp_inv'", "content": "theorem Typed.capp_inv'\n  (he : t0 = Term.capp x c)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ Cf F E0,\n    Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.cforall (CBound.upper {c=c}) F))) {x=x} ∧\n    E0 = F.copen c ∧\n    ESubtyp Γ E0 E"}, {"name": "Typed.tapp_inv", "content": "theorem Typed.tapp_inv\n  (h : Typed Γ (Term.tapp x X) E Ct) :\n  ∃ Cf F E0,\n    Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.tforall (SType.tvar X) F))) {x=x}\n    ∧ E0 = F.topen X\n    ∧ ESubtyp Γ E0 E"}, {"name": "Typed.tapp_inv'", "content": "theorem Typed.tapp_inv'\n  (he : t0 = Term.tapp x X)\n  (h : Typed Γ t0 E Ct) :\n  ∃ Cf F E0,\n    Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.tforall (SType.tvar X) F))) {x=x}\n    ∧ E0 = F.topen X\n    ∧ ESubtyp Γ E0 E"}, {"name": "Typed.val_precise_cv", "content": "theorem Typed.val_precise_cv\n  (ht : Typed Γ t (EType.type T) Ct)\n  (hv : t.IsValue) :\n  Typed Γ t (EType.type T) {}"}, {"name": "Typed.val_precise_cv'", "content": "theorem Typed.val_precise_cv'\n  (he : E0 = EType.type T)\n  (ht : Typed Γ t E0 Ct)\n  (hv : t.IsValue) :\n  Typed Γ t E0 {}"}, {"name": "Typed.boundary_body_typing", "content": "theorem Typed.boundary_body_typing {Γ : Context n m k} {S : SType n m k}\n  (ht : Typed ((Γ,c<:*),x:(Label[S.cweaken])^{c=0}) t E Ct) :\n  Typed ((Γ.label S),c:={x=0}) t E Ct"}, {"name": "CaptureSet.copen_cweaken_ext", "content": "theorem CaptureSet.copen_cweaken_ext {C : CaptureSet n (k+1)} :\n  (C.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = C"}, {"name": "CaptureSet.open_weaken_ext", "content": "theorem CaptureSet.open_weaken_ext {C : CaptureSet (n+1) k} :\n  (C.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = C"}, {"name": "EType.copen_cweaken_ext", "content": "theorem EType.copen_cweaken_ext {E : EType n m (k+1)} :\n  (E.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = E"}, {"name": "EType.open_weaken_ext", "content": "theorem EType.open_weaken_ext {E : EType (n+1) m k} :\n  (E.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = E"}, {"name": "Term.copen_cweaken_ext", "content": "theorem Term.copen_cweaken_ext {t : Term n m (k+1)} :\n  (t.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = t"}, {"name": "Term.open_weaken_ext", "content": "theorem Term.open_weaken_ext {t : Term (n+1) m k} :\n  (t.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = t"}, {"name": "Typed.topen", "content": "theorem Typed.topen\n  (h : Typed (Γ,X<: (SType.tvar X)) t E Ct) :\n  Typed Γ (t.topen X) (E.topen X) Ct"}, {"name": "Typed.tsubst", "content": "theorem Typed.tsubst\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (σ : TVarSubst Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "Typed.invoke_inv", "content": "theorem Typed.invoke_inv {Γ : Context n m k}\n  (ht : Typed Γ (Term.invoke x y) E Ct) :\n  ∃ S0 C0,\n    Typed Γ (Term.var x) (Label[S0]^C0) {x=x} ∧\n    Typed Γ (Term.var y) (EType.type (S0^{})) {x=y}"}, {"name": "Typed.invoke_inv'", "content": "theorem Typed.invoke_inv' {Γ : Context n m k}\n  (he : t0 = Term.invoke x y)\n  (ht : Typed Γ t0 E Ct) :\n  ∃ S0 C0,\n    Typed Γ (Term.var x) (Label[S0]^C0) {x=x} ∧\n    Typed Γ (Term.var y) (EType.type (S0^{})) {x=y}"}, {"name": "WellScoped.scope", "content": "theorem WellScoped.scope\n  (hsc : WellScoped Γ cont C) :\n  WellScoped Γ (Cont.scope x cont) C"}, {"name": "Typed.letin_inv", "content": "theorem Typed.letin_inv {Γ : Context n m k}\n  (h : Typed Γ (Term.letin t u) E Ct) :\n  ∃ T E0,\n    Typed Γ t (EType.type T) Ct ∧\n    Typed (Γ.var T) u E0.weaken Ct.weaken ∧\n    ESubtyp Γ E0 E"}, {"name": "Typed.letin_inv'", "content": "theorem Typed.letin_inv' {Γ : Context n m k}\n  (he : t0 = Term.letin t u)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ T E0,\n    Typed Γ t (EType.type T) Ct0 ∧\n    Typed (Γ.var T) u E0.weaken Ct0.weaken ∧\n    ESubtyp Γ E0 E"}, {"name": "Cont.HasLabel.tweaken", "content": "theorem Cont.HasLabel.tweaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.tweaken x tail.tweaken"}, {"name": "TypedCont.lweaken", "content": "theorem TypedCont.lweaken\n  (h : TypedCont Γ E cont E' Ct) :\n  TypedCont (Γ.label S) E.weaken cont.weaken E'.weaken Ct.weaken"}, {"name": "Cont.HasLabel.lweaken", "content": "theorem Cont.HasLabel.lweaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.weaken x.succ tail.weaken"}, {"name": "WellScoped.lweaken", "content": "theorem WellScoped.lweaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.label S) cont.weaken Ct.weaken"}, {"name": "CSubtyp.tweaken", "content": "theorem CSubtyp.tweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.tvar b) E1.tweaken E2.tweaken"}, {"name": "Typed.canonical_form_pack", "content": "theorem Typed.canonical_form_pack\n  (ht : Γ.IsTight)\n  (h : Typed Γ (Term.pack C x) (EType.ex T) Ct) :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x}"}, {"name": "Typed.canonical_form_pack'", "content": "theorem Typed.canonical_form_pack'\n  (ht : Γ.IsTight)\n  (he1 : t0 = Term.pack C x)\n  (he2 : E0 = EType.ex T)\n  (h : Typed Γ t0 E0 Ct) :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x}"}, {"name": "EType.rename_rename", "content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (E.rename f).rename g = E.rename (g ∘ f)"}, {"name": "CType.rename_rename", "content": "theorem CType.rename_rename (T : CType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (T.rename f).rename g = T.rename (g ∘ f)"}, {"name": "SType.rename_rename", "content": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f)"}, {"name": "CBound.rename_rename", "content": "theorem CBound.rename_rename {b : CBound n k} :\n  (b.rename f).rename g = b.rename (g ∘ f)"}, {"name": "CaptureSet.rename_id", "content": "theorem CaptureSet.rename_id {C : CaptureSet n k} :\n  C.rename FinFun.id = C"}, {"name": "CaptureSet.rename_rename", "content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n  (C.rename f).rename g = C.rename (g ∘ f)"}, {"name": "WellScoped.tweaken", "content": "theorem WellScoped.tweaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.tvar b) cont.tweaken Ct"}, {"name": "TypedCont.tweaken", "content": "theorem TypedCont.tweaken\n  (h : TypedCont Γ E t E' C0) :\n  TypedCont (Γ.tvar S) E.tweaken t.tweaken E'.tweaken C0"}, {"name": "EType.tweaken_ex", "content": "theorem EType.tweaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).tweaken = EType.ex T.tweaken"}, {"name": "EType.tweaken_weaken", "content": "theorem EType.tweaken_weaken (E : EType n m k) :\n  E.weaken.tweaken = E.tweaken.weaken"}, {"name": "EType.tweaken_cweaken", "content": "theorem EType.tweaken_cweaken (E : EType n m k) :\n  E.cweaken.tweaken = E.tweaken.cweaken"}, {"name": "Typed.bindt_inv", "content": "theorem Typed.bindt_inv {Γ : Context n m k}\n  (h : Typed Γ (let X=T in t) E Ct) :\n  ∃ E0,\n    Typed (Γ,X:=T) t E0.tweaken Ct ∧\n    (Γ ⊢ E0 <:e E)"}, {"name": "Typed.bindt_inv'", "content": "theorem Typed.bindt_inv' {Γ : Context n m k}\n  (he : t0 = Term.bindt T t)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ E0,\n    Typed (Γ.tvar (TBinding.inst T)) t E0.tweaken Ct0 ∧\n    ESubtyp Γ E0 E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}, {"name": "Typed.cinstantiate_extvar", "content": "theorem Typed.cinstantiate_extvar {Γ : Context n m k}\n  (h : Typed ((Γ,c<:CBound.star).var P) t E Ct) :\n  Typed ((Γ,c:=C).var P) t E Ct"}, {"name": "Typed.csubst", "content": "theorem Typed.csubst\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (σ : CVarSubst Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "Typed.tapp_inv_capt", "content": "theorem Typed.tapp_inv_capt\n  (ht : Typed Γ (Term.tapp x X) E Ct) :\n  Γ ⊢ ({x=x}) <:c Ct"}, {"name": "Typed.tapp_inv_capt'", "content": "theorem Typed.tapp_inv_capt'\n  (he : t0 = Term.tapp x X)\n  (ht : Typed Γ t0 E Ct) :\n  Γ ⊢ ({x=x}) <:c Ct"}, {"name": "WellScoped.conse", "content": "theorem WellScoped.conse\n  (hsc : WellScoped Γ cont C) :\n  WellScoped Γ (Cont.conse u cont) C"}, {"name": "Typed.boundary_inv", "content": "theorem Typed.boundary_inv {Γ : Context n m k} {S : SType n m k}\n  (ht : Typed Γ (boundary:S in t) E Ct) :\n  Typed\n    ((Γ,c<:*),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{})\n    (Ct.cweaken.weaken ∪ {c=0} ∪ {x=0}) ∧\n    (Γ ⊢ (S^{}) <:e E)"}, {"name": "Typed.boundary_inv'", "content": "theorem Typed.boundary_inv' {Γ : Context n m k} {S : SType n m k}\n  (he : t0 = (boundary:S in t))\n  (ht : Typed Γ t0 E Ct) :\n  Typed\n    ((Γ,c<:*),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{})\n    (Ct.cweaken.weaken ∪ {c=0} ∪ {x=0}) ∧\n    (Γ ⊢ (S^{}) <:e E)"}, {"name": "EType.rename_id", "content": "theorem EType.rename_id {E : EType n m k} :\n  E.rename FinFun.id = E"}, {"name": "CType.rename_id", "content": "theorem CType.rename_id {T : CType n m k} :\n  T.rename FinFun.id = T"}, {"name": "SType.rename_id", "content": "theorem SType.rename_id {S : SType n m k} :\n  S.rename FinFun.id = S"}, {"name": "CBound.rename_id", "content": "theorem CBound.rename_id {b : CBound n k} :\n  b.rename FinFun.id = b"}, {"name": "TypedCont.narrow", "content": "theorem TypedCont.narrow\n  (h : TypedCont Γ E1 cont E C0)\n  (hsub : ESubtyp Γ E2 E1) :\n  TypedCont Γ E2 cont E C0"}, {"name": "Typed.capp_inv_capt", "content": "theorem Typed.capp_inv_capt\n  (ht : Typed Γ (Term.capp x c) E Ct) :\n  Γ ⊢ ({x=x}) <:c Ct"}, {"name": "Typed.capp_inv_capt'", "content": "theorem Typed.capp_inv_capt'\n  (he : t0 = Term.capp x c)\n  (ht : Typed Γ t0 E Ct) :\n  Γ ⊢ ({x=x}) <:c Ct"}, {"name": "Typed.open", "content": "theorem Typed.open\n  (h : Typed (Γ,x: P) t E Ct)\n  (hx : Typed Γ (Term.var x) (EType.type P) Cx) :\n  Typed Γ (t.open x) (E.open x) (Ct.open x)"}, {"name": "Typed.subst", "content": "theorem Typed.subst\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (σ : VarSubst Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "Typed.letex_inv", "content": "theorem Typed.letex_inv {Γ : Context n m k}\n  (h : Typed Γ (Term.letex t u) E Ct) :\n  ∃ T E0,\n    Typed Γ t (EType.ex T) Ct ∧\n    Typed ((Γ,c<:*).var T) u E0.cweaken.weaken Ct.cweaken.weaken ∧\n    ESubtyp Γ E0 E"}, {"name": "Typed.letex_inv'", "content": "theorem Typed.letex_inv' {Γ : Context n m k}\n  (he : t0 = Term.letex t u)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ T E0,\n    Typed Γ t (EType.ex T) Ct0 ∧\n    Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) u E0.cweaken.weaken Ct0.cweaken.weaken ∧\n    ESubtyp Γ E0 E"}, {"name": "Typed.app_inv", "content": "theorem Typed.app_inv\n  (h : Typed Γ (Term.app x y) E Ct) :\n  ∃ T Cf F E0, Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.forall T F))) {x=x}\n    ∧ Typed Γ (Term.var y) (EType.type T) {x=y}\n    ∧ E0 = F.open y\n    ∧ ESubtyp Γ E0 E"}, {"name": "Typed.app_inv'", "content": "theorem Typed.app_inv'\n  (he : t0 = Term.app x y)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ T Cf F E0, Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.forall T F))) {x=x}\n    ∧ Typed Γ (Term.var y) (EType.type T) {x=y}\n    ∧ E0 = F.open y\n    ∧ ESubtyp Γ E0 E"}, {"name": "Typed.precise_cv", "content": "theorem Typed.precise_cv\n  (h : Typed Γ (Term.var x) E C0) :\n  Typed Γ (Term.var x) E {x=x}"}, {"name": "Typed.precise_cv'", "content": "theorem Typed.precise_cv'\n  (he : t0 = Term.var x)\n  (h : Typed Γ t0 E C0) :\n  Typed Γ (Term.var x) E {x=x}"}, {"name": "Typed.canonical_form_clam", "content": "theorem Typed.canonical_form_clam\n  (ht : Γ.IsTight)\n  (h : Typed Γ (Term.clam B t) (EType.type ((∀[c<:B']E)^Cf)) Ct) :\n  Subbound Γ B' B ∧\n    Typed (Γ,c<:B') t E Cf.cweaken"}, {"name": "Typed.canonical_form_clam'", "content": "theorem Typed.canonical_form_clam'\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.cforall B' E))\n  (he1 : t0 = Term.clam B t)\n  (he2 : E0 = EType.type (CType.capt Cf S0))\n  (h : Typed Γ t0 E0 Ct0) :\n  Subbound Γ B' B ∧ Typed (Γ.cvar (CBinding.bound B')) t E Cf.cweaken"}, {"name": "SSubtyp.tweaken", "content": "theorem SSubtyp.tweaken\n  (h : SSubtyp Γ S1 S2) :\n  SSubtyp (Γ.tvar b) S1.tweaken S2.tweaken"}, {"name": "Store.lookup_inv_typing", "content": "theorem Store.lookup_inv_typing\n  (hl : Store.Bound σ x v)\n  (ht : TypedStore σ Γ)\n  (hx : Typed Γ (Term.var x) (EType.type (S^C)) Cx) :\n  ∃ S0 C0 Cv0,\n    Typed Γ v (EType.type (S0^C0)) Cv0 ∧\n    Γ.Bound x (S0^C0) ∧\n    (Γ ⊢ (S0^{x=x}) <: (S^C))"}, {"name": "Store.bound_type", "content": "theorem Store.bound_type\n  (hl : Store.Bound σ x v)\n  (ht : TypedStore σ Γ) :\n  ∃ T0, Context.Bound Γ x T0"}, {"name": "FinFun.open_comp_weaken", "content": "theorem FinFun.open_comp_weaken :\n  (FinFun.open x) ∘ weaken = id"}, {"name": "TypedStore.is_tight", "content": "theorem TypedStore.is_tight\n  (h : TypedStore σ Γ) :\n  Γ.IsTight"}, {"name": "Typed.copen", "content": "theorem Typed.copen\n  (h : Typed (Γ,c<:CBound.upper {c=c}) t E Ct) :\n  Typed Γ (t.copen c) (E.copen c) (Ct.copen c)"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "Typed.label_inv_sub", "content": "theorem Typed.label_inv_sub\n  (ht : Typed Γ (Term.var x) (Label[S]^C) Ct) (hb : Γ.LBound x S1)\n  (hg : Γ.IsTight) :\n  ∃ S0, Γ.LBound x S0 ∧ (Γ ⊢ S <:s S0)"}, {"name": "Typed.label_inv", "content": "theorem Typed.label_inv\n  (ht : Typed Γ (Term.var x) (EType.type T) Ct) (hb : Γ.LBound x S1) :\n  ∃ S0, Γ.LBound x S0 ∧ (Γ ⊢ (Label[S0]^{x=x}) <: T)"}, {"name": "Typed.label_inv'", "content": "theorem Typed.label_inv'\n  (he1 : t0 = Term.var x)\n  (he2 : E0 = EType.type T)\n  (ht : Typed Γ t0 E0 Ct) (hb : Γ.LBound x S1) :\n  ∃ S0, Γ.LBound x S0 ∧ (Γ ⊢ (Label[S0]^{x=x}) <: T)"}, {"name": "CaptureSet.crename_crename", "content": "theorem CaptureSet.crename_crename {C : CaptureSet n k} :\n  (C.crename f).crename g = C.crename (g ∘ f)"}, {"name": "WellScoped.subcapt", "content": "theorem WellScoped.subcapt\n  (hsc : WellScoped Γ cont C)\n  (hs : Γ ⊢ C' <:c C) :\n  WellScoped Γ cont C'"}, {"name": "WellScoped.subset", "content": "theorem WellScoped.subset\n  (hsc : WellScoped Γ cont C)\n  (hs : C' ⊆ C) :\n  WellScoped Γ cont C'"}, {"name": "Typed.app_inv_capt", "content": "theorem Typed.app_inv_capt\n  (ht : Typed Γ (Term.app x y) E Ct) :\n  Γ ⊢ ({x=x}∪{x=y}) <:c Ct"}, {"name": "Typed.app_inv_capt'", "content": "theorem Typed.app_inv_capt'\n  (he : t0 = Term.app x y)\n  (ht : Typed Γ t0 E Ct) :\n  Γ ⊢ ({x=x}∪{x=y}) <:c Ct"}, {"name": "WellScoped.cons", "content": "theorem WellScoped.cons\n  (hsc : WellScoped Γ cont C) :\n  WellScoped Γ (Cont.cons u cont) C"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "Typed.canonical_form_lam", "content": "theorem Typed.canonical_form_lam\n  (ht : Γ.IsTight)\n  (h : Typed Γ (Term.lam T t) (EType.type ((∀(x:T')E)^Cf)) Ct) :\n  CSubtyp Γ T' T ∧\n  Typed (Γ.var T') t E (Cf.weaken ∪ {x=0})"}, {"name": "Typed.canonical_form_lam'", "content": "theorem Typed.canonical_form_lam'\n  (ht : Γ.IsTight)\n  (he1 : t0 = Term.lam T t) (hd2 : SType.Dealias Γ S0 (SType.forall T' E))\n  (he2 : E0 = EType.type (CType.capt Cf S0))\n  (h : Typed Γ t0 E0 Ct0) :\n  CSubtyp Γ T' T ∧\n  Typed (Γ.var T') t E (Cf.weaken ∪ {x=0})"}, {"name": "Typed.canonical_form_tlam", "content": "theorem Typed.canonical_form_tlam\n  (ht : Γ.IsTight)\n  (h : Typed Γ (Term.tlam S t) (EType.type ((∀[X<:S']E)^Cf)) Ct0) :\n  SSubtyp Γ S' S ∧\n  Typed (Γ,X<:S') t E Cf"}, {"name": "Typed.canonical_form_tlam'", "content": "theorem Typed.canonical_form_tlam'\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.tforall S' E))\n  (he1 : t0 = Term.tlam S t)\n  (he2 : E0 = EType.type (CType.capt Cf S0))\n  (h : Typed Γ t0 E0 Ct0) :\n  SSubtyp Γ S' S ∧\n  Typed (Γ.tvar (TBinding.bound S')) t E Cf"}, {"name": "Typed.bindc_inv", "content": "theorem Typed.bindc_inv {Γ : Context n m k}\n  (h : Typed Γ (let c=C in t) E Ct) :\n  ∃ E0,\n    Typed (Γ,c:=C) t E0.cweaken Ct.cweaken ∧\n    (Γ ⊢ E0 <:e E)"}, {"name": "Typed.bindc_inv'", "content": "theorem Typed.bindc_inv' {Γ : Context n m k}\n  (he : t0 = Term.bindc C t)\n  (h : Typed Γ t0 E Ct) :\n  ∃ E0,\n    Typed (Γ.cvar (CBinding.inst C)) t E0.cweaken Ct.cweaken ∧\n    ESubtyp Γ E0 E"}, {"name": "ESubtyp.tweaken", "content": "theorem ESubtyp.tweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.tvar b) E1.tweaken E2.tweaken"}, {"name": "Store.bound_label", "content": "theorem Store.bound_label\n  (hl : Store.LBound σ x S)\n  (ht : TypedStore σ Γ) :\n  Γ.LBound x S"}, {"name": "Cont.has_label_tail_inv", "content": "theorem Cont.has_label_tail_inv\n  (htc : TypedCont Γ E1 cont E2 Ct)\n  (hb : Γ.LBound x S0)\n  (hh : cont.HasLabel x tail) :\n  ∃ Ct1, TypedCont Γ (S0^{}) tail E2 Ct1"}, {"name": "Typed.var_inv_cs", "content": "theorem Typed.var_inv_cs\n  (hx : Typed Γ (Term.var x) (EType.type (S^C)) Cx) :\n  Γ ⊢ ({x=x}) <:c C"}, {"name": "Typed.var_inv_cs'", "content": "theorem Typed.var_inv_cs'\n  (he1 : t0 = Term.var x)\n  (he2 : E0 = EType.type (S^C))\n  (hx : Typed Γ t0 E0 Cx) :\n  Γ ⊢ ({x=x}) <:c C"}], "used_local_defs": [{"name": "Capless.Preserve", "content": "inductive Preserve : Context n m k -> EType n m k -> State n' m' k' -> Prop where\n| mk :\n  TypedState state Γ E ->\n  Preserve Γ E state\n| mk_weaken :\n  TypedState state (Γ.var P) E.weaken ->\n  Preserve Γ E state\n| mk_tweaken :\n  TypedState state (Γ.tvar b) E.tweaken ->\n  Preserve Γ E state\n| mk_cweaken :\n  TypedState state (Γ.cvar b) E.cweaken ->\n  Preserve Γ E state\n| mk_enter :\n  TypedState state ((Γ.label S).cvar b) E.weaken.cweaken ->\n  Preserve Γ E state"}], "used_local_lemmas": [{"name": "Capless.value_typing_widen", "content": "theorem value_typing_widen\n  (hv : Typed Γ v (EType.type (S^C)) Cv)\n  (hs : Γ ⊢ (S^C1) <: (S'^C2)) :\n  Typed Γ v (S'^C) Cv"}, {"name": "Capless.EType.weaken_cweaken_helper", "content": "theorem EType.weaken_cweaken_helper {S : SType n m k} :\n  (EType.type (S^{})).weaken.cweaken = EType.type (S.weaken.cweaken^{})"}], "local_ctx": "import Capless.Store\n\nimport Capless.Type\n\nimport Capless.Reduction\n\nimport Capless.Inversion.Typing\n\nimport Capless.Inversion.Lookup\n\nimport Capless.Renaming.Term.Subtyping\n\nimport Capless.Renaming.Type.Subtyping\n\nimport Capless.Renaming.Capture.Subtyping\n\nimport Capless.Subst.Term.Typing\n\nimport Capless.Subst.Type.Typing\n\nimport Capless.Subst.Capture.Typing\n\nimport Capless.Weakening.TypedCont\n\nimport Capless.Tactics\n\nimport Capless.WellScoped.Basic\n\nimport Capless.Narrowing.TypedCont\n\nimport Capless.Typing.Boundary\n\nnamespace Capless\n\ninductive Preserve : Context n m k -> EType n m k -> State n' m' k' -> Prop where\n| mk :\n  TypedState state Γ E ->\n  Preserve Γ E state\n| mk_weaken :\n  TypedState state (Γ.var P) E.weaken ->\n  Preserve Γ E state\n| mk_tweaken :\n  TypedState state (Γ.tvar b) E.tweaken ->\n  Preserve Γ E state\n| mk_cweaken :\n  TypedState state (Γ.cvar b) E.cweaken ->\n  Preserve Γ E state\n| mk_enter :\n  TypedState state ((Γ.label S).cvar b) E.weaken.cweaken ->\n  Preserve Γ E state", "target_theorem": "theorem preservation\n  (hr : Reduce state state')\n  (ht : TypedState state Γ E) :\n  Preserve Γ E state' :=", "ground_truth_proof": ":= by\n  cases hr\n  case apply hl =>\n    cases ht\n    case mk hs hsc ht hc =>\n      have hg := TypedStore.is_tight hs\n      have ⟨T0, Cf, F0, E0, hx, hy, he1, hs1⟩:= Typed.app_inv ht\n      have ⟨Sv, Cv, Cv0, hv, hbx, hvs⟩ := Store.lookup_inv_typing hl hs hx\n      have hv' := value_typing_widen hv hvs\n      have ⟨hcfs, hcft⟩ := Typed.canonical_form_lam hg hv'\n      constructor\n      constructor\n      { easy }\n      { apply Typed.sub\n        { apply Typed.open (h := hcft)\n          exact hy }\n        { apply Subcapt.refl }\n        { subst he1\n          easy } }\n      { have h1 := Typed.app_inv_capt ht\n        have h2 := WellScoped.subcapt hsc h1\n        simp [CaptureSet.open]\n        simp [FinFun.open, CaptureSet.weaken, CaptureSet.rename_rename]\n        simp [FinFun.open_comp_weaken, CaptureSet.rename_id]\n        cases h2; rename_i h2 h3\n        apply WellScoped.union\n        { apply WellScoped.var_inv\n          exact h2; easy }\n        { easy } }\n      { easy }\n  case tapply hl =>\n    cases ht\n    case mk hs hsc ht hc =>\n      have hg := TypedStore.is_tight hs\n      have ⟨Cf, F, E0, hx, he0, hs0⟩ := Typed.tapp_inv ht\n      have ⟨Sv, Cv, Cv0, hv, hbx, hvs⟩ := Store.lookup_inv_typing hl hs hx\n      have hv' := value_typing_widen hv hvs\n      have ⟨hs1, hft⟩ := Typed.canonical_form_tlam hg hv'\n      constructor\n      constructor\n      { easy }\n      { apply Typed.sub\n        { apply Typed.topen (h := hft) }\n        { apply Subcapt.refl }\n        { subst he0\n          easy } }\n      { have h1 := Typed.tapp_inv_capt ht\n        have h2 := WellScoped.subcapt hsc h1\n        apply WellScoped.var_inv\n        exact h2\n        easy }\n      easy\n  case capply hl =>\n    cases ht\n    case mk hs hsc ht hc =>\n      have hg := TypedStore.is_tight hs\n      have ⟨Cf, F, E0, hx, he1, hs1⟩ := Typed.capp_inv ht\n      have ⟨Sv, Cv, Cv0, hv, hbx, hvs⟩ := Store.lookup_inv_typing hl hs hx\n      have hv' := value_typing_widen hv hvs\n      have ⟨hsb, hct⟩ := Typed.canonical_form_clam hg hv'\n      constructor\n      constructor\n      { easy }\n      { apply Typed.sub\n        { apply Typed.copen hct }\n        { apply Subcapt.refl }\n        { subst he1\n          exact hs1 } }\n      { have h1 := Typed.capp_inv_capt ht\n        have h2 := WellScoped.subcapt hsc h1\n        simp [CaptureSet.cweaken, CaptureSet.copen, CaptureSet.crename_crename]\n        simp [FinFun.open_comp_weaken, CaptureSet.crename_id]\n        apply WellScoped.var_inv\n        exact h2\n        easy }\n      easy\n  case push =>\n    cases ht\n    case mk hs hsc ht hc =>\n      have ⟨T, E0, htt, htu, hsub⟩ := Typed.letin_inv ht\n      constructor\n      constructor\n      { easy }\n      { exact htt }\n      { apply WellScoped.cons; easy }\n      { constructor\n        apply Typed.sub <;> try easy\n        apply Subcapt.refl\n        apply ESubtyp.weaken; easy\n        { easy }\n        easy }\n  case push_ex =>\n    cases ht\n    case mk hs hsc ht hc =>\n      have ⟨T, E0, htt, htu, hsub⟩ := Typed.letex_inv ht\n      constructor\n      constructor\n      { exact hs }\n      { exact htt }\n      { apply WellScoped.conse; easy }\n      { constructor\n        apply Typed.sub; exact htu; apply Subcapt.refl\n        apply ESubtyp.weaken\n        apply ESubtyp.cweaken; exact hsub\n        { easy }\n        exact hc }\n  case rename =>\n    cases ht\n    case mk hs hsc hx hc =>\n      cases hc\n      case cons hu hsc0 hc0 =>\n        have hu1 := hu.open hx\n        simp [EType.weaken, EType.open] at hu1\n        simp [EType.rename_rename] at hu1\n        simp [FinFun.open_comp_weaken] at hu1\n        simp [EType.rename_id] at hu1\n        constructor\n        constructor <;> try easy\n        simp [CaptureSet.weaken, CaptureSet.open]\n        simp [CaptureSet.rename_rename]\n        simp [FinFun.open_comp_weaken, CaptureSet.rename_id]\n        easy\n  case lift_ex =>\n    cases ht\n    case mk hs hsc ht hc =>\n      cases hc\n      case conse hu hsc hc0 =>\n        have hg := TypedStore.is_tight hs\n        have hx := Typed.canonical_form_pack hg ht\n        rename_i C _ _ _ _ _ _ _\n        have hu1 := hu.cinstantiate_extvar (C := C)\n        have hu2 := hu1.open hx\n        simp [EType.weaken, EType.open, EType.rename_rename] at hu2\n        simp [FinFun.open_comp_weaken] at hu2\n        simp [EType.rename_id] at hu2\n        apply Preserve.mk_cweaken\n        constructor\n        { constructor; exact hs }\n        { exact hu2 }\n        { simp [CaptureSet.weaken, CaptureSet.open]\n          simp [CaptureSet.rename_rename, FinFun.open_comp_weaken]\n          simp [CaptureSet.rename_id]\n          apply hsc.cweaken }\n        { apply TypedCont.cweaken; exact hc0 }\n  case lift hv =>\n    cases ht\n    case mk hs hsc ht hc =>\n      cases hc\n      case cons hu hsc0 hc0 =>\n        apply Preserve.mk_weaken\n        constructor\n        { constructor; exact hs; exact ht }\n        { exact hu }\n        { apply hsc0.weaken }\n        { apply TypedCont.weaken; exact hc0 }\n  case tlift =>\n    cases ht\n    case mk hs hsc ht hc =>\n      apply Preserve.mk_tweaken\n      have ⟨E0, ht, hsub⟩ := Typed.bindt_inv ht\n      constructor\n      { constructor; exact hs }\n      { apply Typed.sub\n        exact ht; apply Subcapt.refl\n        apply ESubtyp.tweaken; exact hsub }\n      { apply hsc.tweaken }\n      { apply TypedCont.tweaken; exact hc }\n  case clift =>\n    cases ht\n    case mk hs hsc ht hc =>\n      apply Preserve.mk_cweaken\n      have ⟨E0, ht, hsub⟩ := Typed.bindc_inv ht\n      constructor\n      { constructor; exact hs }\n      { apply Typed.sub\n        exact ht; apply Subcapt.refl\n        apply ESubtyp.cweaken; exact hsub }\n      { apply hsc.cweaken }\n      { apply TypedCont.cweaken; exact hc }\n  case enter =>\n    cases ht\n    case mk hs hsc ht hc =>\n      have ⟨ht0, hsub0⟩ := Typed.boundary_inv ht\n      apply Preserve.mk_enter\n      constructor\n      { constructor; constructor; easy }\n      { apply Typed.boundary_body_typing ht0 }\n      { repeat any_goals apply WellScoped.union\n        { rw [CaptureSet.weaken_cweaken]\n          apply WellScoped.scope\n          apply WellScoped.cweaken\n          apply WellScoped.lweaken; easy }\n        { constructor; constructor\n          simp\n          apply WellScoped.label; repeat constructor }\n        { apply WellScoped.label; repeat constructor } }\n      { constructor; constructor; constructor\n        rw [<- EType.weaken_cweaken_helper]\n        apply TypedCont.cweaken\n        apply TypedCont.lweaken\n        apply TypedCont.narrow; easy; easy\n        simp [SType.cweaken, SType.weaken]\n        rw [SType.crename_rename_comm]\n        apply CSubtyp.refl }\n  case leave_var =>\n    cases ht\n    case mk hs hsc ht hc =>\n      have ht1 := Typed.precise_cv ht\n      apply Preserve.mk\n      cases hc\n      rename_i hsub hbl hc0\n      constructor\n      { easy }\n      { apply Typed.sub\n        { exact ht1 }\n        { apply Subcapt.refl }\n        { constructor; easy } }\n      { have ht1 := Typed.sub ht Subcapt.refl (ESubtyp.type hsub)\n        have hy := Typed.var_inv_cs ht1\n        apply WellScoped.subcapt\n        apply WellScoped.empty\n        easy }\n      { easy }\n  case leave_val =>\n    cases ht\n    case mk hs hsc ht hc =>\n      rename_i hv _ _ _\n      cases hc\n      case scope hsub hbl hc0 =>\n        have ht1 := Typed.sub ht Subcapt.refl (ESubtyp.type hsub)\n        have ht2 := Typed.val_precise_cv ht1 hv\n        apply Preserve.mk\n        constructor\n        { easy }\n        { apply Typed.sub\n          { exact ht2 }\n          { apply Subcapt.refl }\n          { apply ESubtyp.refl } }\n        { constructor }\n        { easy }\n  case invoke hl hhl =>\n    cases ht\n    case mk hs hsc ht hc =>\n      have hg := TypedStore.is_tight hs\n      have ⟨S0, C0, hx, hy⟩ := Typed.invoke_inv ht\n      have h1 := Store.bound_label hl hs\n      have ⟨S0, hbx, hsub⟩ := Typed.label_inv_sub hx h1 hg\n      have ⟨Ct1, hc1⟩ := Cont.has_label_tail_inv hc hbx hhl\n      apply Preserve.mk\n      constructor\n      { easy }\n      { exact hy }\n      { have hy1 := Typed.var_inv_cs hy\n        apply WellScoped.subcapt\n        apply WellScoped.empty\n        easy }\n      { apply hc1.narrow\n        constructor; constructor\n        apply Subcapt.refl; easy }", "nesting_depth": 7, "transitive_dep_count": 334, "subset_aristotle": false, "category": "Type systems"}
{"id": 62, "thm_name": "Capless.Typed.rename", "thm_stmt": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)", "lean_root": "capless-lean", "rel_path": "Capless/Renaming/Term/Typing.lean", "imports": ["import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Term.Subcapturing", "import Capless.Typing", "import Capless.Type.Basic", "import Capless.CaptureSet", "import Capless.Renaming.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "EType.copen", "content": "def EType.copen (E : EType n m (k+1)) (x : Fin k) : EType n m k :=\n  E.crename (FinFun.open x)"}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "EType.topen", "content": "def EType.topen (E : EType n (m+1) k) (X : Fin m) : EType n m k :=\n  E.trename (FinFun.open X)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "EType.cweaken_rename_comm", "content": "theorem EType.cweaken_rename_comm {E : EType n m k} :\n  E.cweaken.rename f = (E.rename f).cweaken"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "EType.rename_copen", "content": "theorem EType.rename_copen :\n  (EType.copen E c).rename f = (E.rename f).copen c"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "CaptureSet.ext_rename_singleton_zero", "content": "theorem CaptureSet.ext_rename_singleton_zero {f : FinFun n n'} :\n  ({x=0} : CaptureSet (n+1) k).rename f.ext = {x=0}"}, {"name": "EType.weaken_rename", "content": "theorem EType.weaken_rename {E : EType n m k} :\n  (E.rename f).weaken = E.weaken.rename f.ext"}, {"name": "EType.rename_rename", "content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (E.rename f).rename g = E.rename (g ∘ f)"}, {"name": "CType.rename_rename", "content": "theorem CType.rename_rename (T : CType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (T.rename f).rename g = T.rename (g ∘ f)"}, {"name": "SType.rename_rename", "content": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f)"}, {"name": "CBound.rename_rename", "content": "theorem CBound.rename_rename {b : CBound n k} :\n  (b.rename f).rename g = b.rename (g ∘ f)"}, {"name": "SType.cweaken_rename_comm", "content": "theorem SType.cweaken_rename_comm {S : SType n m k} :\n  S.cweaken.rename f = (S.rename f).cweaken"}, {"name": "EType.tweaken_rename", "content": "theorem EType.tweaken_rename {E : EType n m k} :\n  E.tweaken.rename f = (E.rename f).tweaken"}, {"name": "EType.trename_rename_comm", "content": "theorem EType.trename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (E.trename g).rename f = (E.rename f).trename g"}, {"name": "CType.trename_rename_comm", "content": "theorem CType.trename_rename_comm (T : CType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (T.trename g).rename f = (T.rename f).trename g"}, {"name": "SType.trename_rename_comm", "content": "theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (S.trename g).rename f = (S.rename f).trename g"}, {"name": "CaptureSet.weaken_rename", "content": "theorem CaptureSet.weaken_rename {C : CaptureSet n k} :\n  (C.rename f).weaken = C.weaken.rename f.ext"}, {"name": "CaptureSet.rename_rename", "content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n  (C.rename f).rename g = C.rename (g ∘ f)"}, {"name": "CaptureSet.cweaken_rename_comm", "content": "theorem CaptureSet.cweaken_rename_comm {C : CaptureSet n k} {f : FinFun n n'} :\n  (C.cweaken).rename f = (C.rename f).cweaken"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "EType.rename_topen", "content": "theorem EType.rename_topen :\n  (EType.topen E X).rename f = (E.rename f).topen X"}, {"name": "SType.weaken_rename", "content": "theorem SType.weaken_rename {S : SType n m k} :\n  (S.rename f).weaken = S.weaken.rename f.ext"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Typing\n\nimport Capless.Renaming.Basic\n\nimport Capless.Renaming.Term.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f) :=", "ground_truth_proof": ":= by\n  induction h generalizing n'\n  case var hb =>\n    simp [Term.rename, EType.rename, CType.rename]\n    apply Typed.var\n    have hb1 := ρ.map _ _ hb\n    simp [CType.rename] at hb1\n    trivial\n  case pack ih =>\n    simp [Term.rename, EType.rename]\n    apply Typed.pack\n    have ih := ih (ρ.cext _)\n    simp [Term.rename, EType.rename] at ih\n    exact ih\n  case sub hsc hs ih =>\n    apply Typed.sub\n    apply ih; trivial\n    apply! hsc.rename\n    apply! hs.rename\n  case abs iht =>\n    simp [Term.rename, EType.rename, CType.rename, SType.rename]\n    apply Typed.abs\n    rw [CaptureSet.weaken_rename]\n    rw [<- CaptureSet.ext_rename_singleton_zero (f := f)]\n    apply? iht\n    apply ρ.ext\n  case tabs iht =>\n    simp [Term.rename, EType.rename, CType.rename, SType.rename]\n    apply Typed.tabs\n    apply? iht\n    apply ρ.text\n  case cabs iht =>\n    simp [Term.rename, EType.rename, CType.rename, SType.rename]\n    apply Typed.cabs\n    rw [<- CaptureSet.cweaken_rename_comm]\n    apply? iht\n    apply ρ.cext\n  case app ih1 ih2 =>\n    simp [Term.rename]\n    simp [EType.rename_open]\n    apply Typed.app\n    have ih1 := ih1 ρ\n    simp [Term.rename, EType.rename, CType.rename, SType.rename] at ih1\n    exact ih1\n    have ih2 := ih2 ρ\n    simp [Term.rename, EType.rename] at ih2\n    exact ih2\n  case tapp ih =>\n    simp [Term.rename]\n    simp [EType.rename_topen]\n    apply Typed.tapp\n    have ih := ih ρ\n    simp [Term.rename, EType.rename, CType.rename, SType.rename] at ih\n    trivial\n  case capp ih =>\n    simp [Term.rename, EType.rename_copen]\n    apply Typed.capp\n    have ih := ih ρ\n    simp [Term.rename, EType.rename, CType.rename, SType.rename] at ih\n    trivial\n  case letin ih1 ih2 =>\n    simp [Term.rename]\n    apply Typed.letin\n    have ih1 := ih1 ρ\n    simp [EType.rename] at ih1\n    exact ih1\n    have ih2 := ih2 (ρ.ext _)\n    rw [<- EType.weaken_rename] at ih2\n    rw [CaptureSet.weaken_rename]\n    trivial\n  case letex ih1 ih2 =>\n    simp [Term.rename]\n    apply letex\n    have ih1 := ih1 ρ\n    simp [EType.rename] at ih1\n    exact ih1\n    have ih2 := ih2 ((ρ.cext _).ext _)\n    rw [<- EType.cweaken_rename_comm]\n    rw [EType.weaken_rename]\n    rw [<- CaptureSet.cweaken_rename_comm]\n    rw [CaptureSet.weaken_rename]\n    trivial\n  case bindt ih =>\n    simp [Term.rename]\n    apply Typed.bindt\n    have ih := ih (ρ.text _)\n    simp [Term.rename, TBinding.rename, EType.rename, CType.rename] at ih\n    rw [EType.tweaken_rename] at ih\n    trivial\n  case bindc ih =>\n    simp [Term.rename]\n    apply Typed.bindc\n    have ih := ih (ρ.cext _)\n    simp [Term.rename, CBinding.rename] at ih\n    rw [EType.cweaken_rename_comm] at ih\n    rw [<- CaptureSet.cweaken_rename_comm]\n    trivial\n  case label =>\n    simp [Term.rename, EType.rename, CType.rename, SType.rename]\n    apply label\n    have h := ρ.lmap\n    aesop\n  case invoke ih1 ih2 =>\n    simp [Term.rename]\n    apply Typed.invoke\n    simp [Term.rename, EType.rename, CType.rename, SType.rename] at ih1\n    apply ih1; trivial\n    simp [Term.rename, EType.rename, CType.rename, SType.rename] at ih2\n    apply ih2; trivial\n  case boundary ih =>\n    simp [Term.rename, EType.rename, CType.rename]\n    apply Typed.boundary\n    have ih := ih ((ρ.cext _).ext _)\n    simp [CBinding.rename, FinFun.ext, CType.rename, SType.rename] at ih\n    rw\n      [ <- SType.cweaken_rename_comm\n      , SType.weaken_rename\n      , <- CaptureSet.cweaken_rename_comm\n      , CaptureSet.weaken_rename ]\n    simp [CBound.rename, EType.rename, CType.rename] at ih\n    exact ih", "nesting_depth": 4, "transitive_dep_count": 111, "subset_aristotle": false, "category": "Type systems"}
{"id": 63, "thm_name": "Capless.Typed.subst", "thm_stmt": "theorem Typed.subst\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (σ : VarSubst Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)", "lean_root": "capless-lean", "rel_path": "Capless/Subst/Term/Typing.lean", "imports": ["import Capless.Renaming.Capture.Typing", "import Capless.Renaming.Term.Typing", "import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Type.Subcapturing", "import Capless.Subst.Basic", "import Capless.Subst.Term.Subcapturing", "import Capless.Typing.Basic", "import Capless.Renaming.Term.Subcapturing", "import Capless.CaptureSet", "import Capless.Subst.Term.Subtyping", "import Capless.Renaming.Type.Subtyping", "import Capless.Typing", "import Capless.Renaming.Type.Typing", "import Capless.Type.Basic", "import Capless.Renaming.Capture.Subtyping", "import Capless.Renaming.Capture.Subcapturing"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "abs", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}], "used_repo_defs": [{"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "SSubtyp.subst_motive3", "content": "def SSubtyp.subst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.subst_motive2", "content": "def SSubtyp.subst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "SSubtyp.subst_motive1", "content": "def SSubtyp.subst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "EType.copen", "content": "def EType.copen (E : EType n m (k+1)) (x : Fin k) : EType n m k :=\n  E.crename (FinFun.open x)"}, {"name": "EType.topen", "content": "def EType.topen (E : EType n (m+1) k) (X : Fin m) : EType n m k :=\n  E.trename (FinFun.open X)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "Subbound.subst", "content": "theorem Subbound.subst\n  (h : Subbound Γ B1 B2)\n  (σ : VarSubst Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.subst", "content": "theorem ESubtyp.subst\n  (h : ESubtyp Γ E1 E2)\n  (σ : VarSubst Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CSubtyp.subst", "content": "theorem CSubtyp.subst\n  (h : CSubtyp Γ T1 T2)\n  (σ : VarSubst Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.subst", "content": "theorem SSubtyp.subst\n  (h : SSubtyp Γ S1 S2)\n  (σ : VarSubst Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "EType.cweaken_rename_comm", "content": "theorem EType.cweaken_rename_comm {E : EType n m k} :\n  E.cweaken.rename f = (E.rename f).cweaken"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "EType.rename_copen", "content": "theorem EType.rename_copen :\n  (EType.copen E c).rename f = (E.rename f).copen c"}, {"name": "Subcapt.subst", "content": "theorem Subcapt.subst\n  (h : Subcapt Γ C1 C2)\n  (σ : VarSubst Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "CaptureSet.ext_rename_singleton_zero", "content": "theorem CaptureSet.ext_rename_singleton_zero {f : FinFun n n'} :\n  ({x=0} : CaptureSet (n+1) k).rename f.ext = {x=0}"}, {"name": "EType.weaken_rename", "content": "theorem EType.weaken_rename {E : EType n m k} :\n  (E.rename f).weaken = E.weaken.rename f.ext"}, {"name": "EType.rename_rename", "content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (E.rename f).rename g = E.rename (g ∘ f)"}, {"name": "CType.rename_rename", "content": "theorem CType.rename_rename (T : CType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (T.rename f).rename g = T.rename (g ∘ f)"}, {"name": "SType.rename_rename", "content": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f)"}, {"name": "CBound.rename_rename", "content": "theorem CBound.rename_rename {b : CBound n k} :\n  (b.rename f).rename g = b.rename (g ∘ f)"}, {"name": "SType.cweaken_rename_comm", "content": "theorem SType.cweaken_rename_comm {S : SType n m k} :\n  S.cweaken.rename f = (S.rename f).cweaken"}, {"name": "EType.tweaken_rename", "content": "theorem EType.tweaken_rename {E : EType n m k} :\n  E.tweaken.rename f = (E.rename f).tweaken"}, {"name": "EType.trename_rename_comm", "content": "theorem EType.trename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (E.trename g).rename f = (E.rename f).trename g"}, {"name": "CType.trename_rename_comm", "content": "theorem CType.trename_rename_comm (T : CType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (T.trename g).rename f = (T.rename f).trename g"}, {"name": "SType.trename_rename_comm", "content": "theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (S.trename g).rename f = (S.rename f).trename g"}, {"name": "CaptureSet.weaken_rename", "content": "theorem CaptureSet.weaken_rename {C : CaptureSet n k} :\n  (C.rename f).weaken = C.weaken.rename f.ext"}, {"name": "CaptureSet.rename_rename", "content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n  (C.rename f).rename g = C.rename (g ∘ f)"}, {"name": "Typed.precise_capture", "content": "theorem Typed.precise_capture\n  (h : Typed Γ (Term.var x) (EType.type (CType.capt C S)) C0) :\n  Typed Γ (Term.var x) (EType.type (CType.capt {x=x} S)) {x=x}"}, {"name": "Typed.precise_capture'", "content": "theorem Typed.precise_capture'\n  (he1 : t0 = Term.var x)\n  (he2 : E0 = EType.type (CType.capt C S))\n  (h : Typed Γ t0 E0 C0) :\n  Typed Γ (Term.var x) (EType.type (CType.capt {x=x} S)) {x=x}"}, {"name": "CaptureSet.cweaken_rename_comm", "content": "theorem CaptureSet.cweaken_rename_comm {C : CaptureSet n k} {f : FinFun n n'} :\n  (C.cweaken).rename f = (C.rename f).cweaken"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "EType.rename_topen", "content": "theorem EType.rename_topen :\n  (EType.topen E X).rename f = (E.rename f).topen X"}, {"name": "SType.weaken_rename", "content": "theorem SType.weaken_rename {S : SType n m k} :\n  (S.rename f).weaken = S.weaken.rename f.ext"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Typing\n\nimport Capless.Subst.Basic\n\nimport Capless.Subst.Term.Subtyping\n\nimport Capless.Renaming.Term.Typing\n\nnamespace Capless", "target_theorem": "theorem Typed.subst\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (σ : VarSubst Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f) :=", "ground_truth_proof": ":= by\n  induction h generalizing n'\n  case var hb =>\n    simp [Term.rename, EType.rename, CType.rename]\n    have hb1 := σ.map _ _ hb\n    simp [CType.rename] at hb1\n    apply Typed.precise_capture\n    trivial\n  case pack ih =>\n    simp [Term.rename, EType.rename]\n    apply pack\n    have ih := ih σ.cext\n    simp [EType.rename] at ih\n    exact ih\n  case sub hsc hs ih =>\n    apply sub\n    { apply ih; trivial }\n    { apply! hsc.subst }\n    { apply! hs.subst }\n  case abs ih =>\n    simp [Term.rename, EType.rename, CType.rename, SType.rename]\n    apply abs\n    { rw [CaptureSet.weaken_rename]\n      rw [<- CaptureSet.ext_rename_singleton_zero (f := f)]\n      apply ih\n      apply σ.ext }\n  case tabs ih =>\n    simp [Term.rename, EType.rename, CType.rename, SType.rename]\n    apply tabs\n    { apply ih\n      apply σ.text }\n  case cabs ih =>\n    simp [Term.rename, EType.rename, CType.rename, SType.rename]\n    apply cabs\n    { rw [<- CaptureSet.cweaken_rename_comm]\n      apply ih\n      apply σ.cext }\n  case app ih1 ih2 =>\n    simp [Term.rename]\n    rw [EType.rename_open]\n    apply app\n    { have ih1 := ih1 σ\n      simp [Term.rename, EType.rename, CType.rename, SType.rename] at ih1\n      exact ih1 }\n    { have ih2 := ih2 σ\n      simp [Term.rename, EType.rename, CType.rename, SType.rename] at ih2\n      exact ih2 }\n  case tapp ih =>\n    simp [Term.rename]\n    rw [EType.rename_topen]\n    apply tapp\n    have ih1 := ih σ\n    simp [Term.rename, EType.rename, CType.rename, SType.rename] at ih1\n    exact ih1\n  case capp ih =>\n    simp [Term.rename]\n    rw [EType.rename_copen]\n    apply capp\n    have ih1 := ih σ\n    simp [Term.rename, EType.rename, CType.rename, SType.rename] at ih1\n    exact ih1\n  case letin ih1 ih2 =>\n    simp [Term.rename]\n    apply letin\n    { have ih1 := ih1 σ\n      simp [EType.rename] at ih1\n      exact ih1 }\n    { have ih2 := ih2 (σ.ext _)\n      rw [<- EType.weaken_rename] at ih2\n      rw [CaptureSet.weaken_rename]\n      exact ih2 }\n  case letex ih1 ih2 =>\n    simp [Term.rename]\n    apply letex\n    { have ih1 := ih1 σ\n      simp [EType.rename] at ih1\n      exact ih1 }\n    { have ih2 := ih2 (σ.cext.ext _)\n      rw [<- EType.weaken_rename] at ih2\n      rw [EType.cweaken_rename_comm] at ih2\n      rw [<- CaptureSet.cweaken_rename_comm]\n      rw [CaptureSet.weaken_rename]\n      exact ih2 }\n  case bindt ih =>\n    simp [Term.rename]\n    apply bindt\n    have ih := ih σ.text\n    rw [EType.tweaken_rename] at ih\n    simp [TBinding.rename] at ih\n    exact ih\n  case bindc ih =>\n    simp [Term.rename]\n    apply bindc\n    have ih := ih σ.cext\n    rw [EType.cweaken_rename_comm] at ih\n    simp [CBinding.rename] at ih\n    rw [<- CaptureSet.cweaken_rename_comm]\n    exact ih\n  case label hb =>\n    have hb1 := σ.lmap _ _ hb\n    simp [Term.rename, EType.rename, CType.rename, SType.rename]\n    apply label\n    aesop\n  case invoke ih1 ih2 =>\n    simp [Term.rename]\n    simp [EType.rename, CType.rename, SType.rename] at *\n    apply invoke\n    apply ih1; assumption\n    apply ih2; assumption\n  case boundary ih =>\n    simp [Term.rename]\n    simp [EType.rename, CType.rename, SType.rename] at *\n    apply boundary\n    have ih := ih (σ.cext.ext _)\n    simp\n      [ CBinding.rename\n      , EType.rename\n      , CType.rename\n      , SType.rename\n      , <- SType.weaken_rename\n      , SType.cweaken_rename_comm\n      , <- CaptureSet.weaken_rename\n      , CaptureSet.cweaken_rename_comm\n      , FinFun.ext ] at ih\n    exact ih", "nesting_depth": 5, "transitive_dep_count": 190, "subset_aristotle": false, "category": "Type systems"}
{"id": 64, "thm_name": "Capless.Typed.csubst", "thm_stmt": "theorem Typed.csubst\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (σ : CVarSubst Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)", "lean_root": "capless-lean", "rel_path": "Capless/Subst/Capture/Typing.lean", "imports": ["import Capless.Renaming.Capture.Typing", "import Capless.Renaming.Term.Typing", "import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Type.Subcapturing", "import Capless.Subst.Basic", "import Capless.Renaming.Term.Subcapturing", "import Capless.CaptureSet", "import Capless.Subst.Capture.Subcapturing", "import Capless.Subst.Capture.Subtyping", "import Capless.Renaming.Type.Subtyping", "import Capless.Typing", "import Capless.Renaming.Type.Typing", "import Capless.Type.Basic", "import Capless.Renaming.Capture.Subtyping", "import Capless.Renaming.Capture.Subcapturing"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "SType.topen", "content": "def SType.topen (S : SType n (m+1) k) (X : Fin m) : SType n m k :=\n  S.trename (FinFun.open X)"}, {"name": "CType.topen", "content": "def CType.topen (C : CType n (m+1) k) (X : Fin m) : CType n m k :=\n  C.trename (FinFun.open X)"}, {"name": "EType.topen", "content": "def EType.topen (E : EType n (m+1) k) (X : Fin m) : EType n m k :=\n  E.trename (FinFun.open X)"}, {"name": "SSubtyp.csubst_motive3", "content": "def SSubtyp.csubst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.csubst_motive1", "content": "def SSubtyp.csubst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SSubtyp.csubst_motive2", "content": "def SSubtyp.csubst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "CType.copen", "content": "def CType.copen (C : CType n m (k+1)) (x : Fin k) : CType n m k :=\n  C.crename (FinFun.open x)"}, {"name": "CaptureSet.copen", "content": "def CaptureSet.copen (C : CaptureSet n (k+1)) (x : Fin k) : CaptureSet n k :=\n  C.crename (FinFun.open x)"}, {"name": "SType.copen", "content": "def SType.copen (S : SType n m (k+1)) (x : Fin k) : SType n m k :=\n  S.crename (FinFun.open x)"}, {"name": "EType.copen", "content": "def EType.copen (E : EType n m (k+1)) (x : Fin k) : EType n m k :=\n  E.crename (FinFun.open x)"}, {"name": "SType.open", "content": "def SType.open (S : SType (n+1) m k) (x : Fin n) : SType n m k :=\n  S.rename (FinFun.open x)"}, {"name": "CType.open", "content": "def CType.open (C : CType (n+1) m k) (x : Fin n) : CType n m k :=\n  C.rename (FinFun.open x)"}, {"name": "CaptureSet.open", "content": "def CaptureSet.open (C : CaptureSet (n+1) k) (x : Fin n) : CaptureSet n k :=\n  C.rename (FinFun.open x)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "SType.cweaken_crename", "content": "theorem SType.cweaken_crename {S : SType n m k} :\n  (S.crename f).cweaken = S.cweaken.crename f.ext"}, {"name": "SType.crename_crename", "content": "theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (S.crename f).crename g = S.crename (g ∘ f)"}, {"name": "CBound.crename_crename", "content": "theorem CBound.crename_crename {b : CBound n k} :\n  (b.crename f).crename g = b.crename (g ∘ f)"}, {"name": "EType.crename_crename", "content": "theorem EType.crename_crename (E : EType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (E.crename f).crename g = E.crename (g ∘ f)"}, {"name": "CType.crename_crename", "content": "theorem CType.crename_crename (T : CType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (T.crename f).crename g = T.crename (g ∘ f)"}, {"name": "EType.weaken_crename", "content": "theorem EType.weaken_crename {E : EType n m k} :\n  (E.crename f).weaken = E.weaken.crename f"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "EType.crename_topen", "content": "theorem EType.crename_topen {E : EType n (m+1) k} :\n  (E.topen X).crename f = (E.crename f).topen X"}, {"name": "EType.crename_trename_comm", "content": "theorem EType.crename_trename_comm (E : EType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (E.crename f).trename g = (E.trename g).crename f"}, {"name": "CType.crename_trename_comm", "content": "theorem CType.crename_trename_comm (T : CType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (T.crename f).trename g = (T.trename g).crename f"}, {"name": "SType.crename_trename_comm", "content": "theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (S.crename f).trename g = (S.trename g).crename f"}, {"name": "Subbound.csubst", "content": "theorem Subbound.csubst\n  (h : Subbound Γ B1 B2)\n  (σ : CVarSubst Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "SSubtyp.csubst", "content": "theorem SSubtyp.csubst\n  (h : SSubtyp Γ S1 S2)\n  (σ : CVarSubst Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "CSubtyp.csubst", "content": "theorem CSubtyp.csubst\n  (h : CSubtyp Γ T1 T2)\n  (σ : CVarSubst Γ f Δ) :\n  CSubtyp Δ (T1.crename f) (T2.crename f)"}, {"name": "ESubtyp.csubst", "content": "theorem ESubtyp.csubst\n  (h : ESubtyp Γ E1 E2)\n  (σ : CVarSubst Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "Subcapt.csubst", "content": "theorem Subcapt.csubst\n  (h : Subcapt Γ C1 C2)\n  (σ : CVarSubst Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "CaptureSet.weaken_crename", "content": "theorem CaptureSet.weaken_crename {C : CaptureSet n k} :\n  (C.crename f).weaken = C.weaken.crename f"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "CaptureSet.cweaken_crename", "content": "theorem CaptureSet.cweaken_crename {C : CaptureSet n k} :\n  (C.crename f).cweaken = C.cweaken.crename f.ext"}, {"name": "CaptureSet.crename_crename", "content": "theorem CaptureSet.crename_crename {C : CaptureSet n k} :\n  (C.crename f).crename g = C.crename (g ∘ f)"}, {"name": "SType.weaken_crename", "content": "theorem SType.weaken_crename {S : SType n m k} :\n  (S.crename f).weaken = S.weaken.crename f"}, {"name": "EType.crename_copen", "content": "theorem EType.crename_copen {E : EType n m (k+1)} :\n  (E.copen c).crename f = (E.crename f.ext).copen (f c)"}, {"name": "EType.tweaken_crename", "content": "theorem EType.tweaken_crename {E : EType n m k} :\n  (E.crename f).tweaken = E.tweaken.crename f"}, {"name": "EType.crename_open", "content": "theorem EType.crename_open {E : EType (n+1) m k} :\n  (E.open x).crename f = (E.crename f).open x"}, {"name": "EType.cweaken_crename", "content": "theorem EType.cweaken_crename {E : EType n m k} :\n  (E.crename f).cweaken = E.cweaken.crename f.ext"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Subst.Basic\n\nimport Capless.Subst.Capture.Subtyping\n\nimport Capless.Typing\n\nnamespace Capless", "target_theorem": "theorem Typed.csubst\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (σ : CVarSubst Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f) :=", "ground_truth_proof": ":= by\n    induction h generalizing k'\n    case var hb =>\n      simp [Term.crename, EType.crename, CType.crename]\n      have hb1 := σ.map _ _ hb\n      simp [CType.crename] at hb1\n      apply Typed.var; trivial\n    case pack ih =>\n      simp [Term.crename, EType.crename]\n      apply pack\n      have ih := ih σ.cext\n      simp [EType.crename] at ih\n      exact ih\n    case sub hsc hs ih =>\n      apply sub\n      { apply ih; trivial }\n      { apply! hsc.csubst }\n      { apply! hs.csubst }\n    case abs ih =>\n      simp [Term.crename, EType.crename, CType.crename, SType.crename]\n      apply abs\n      { rw [CaptureSet.weaken_crename]\n        apply ih\n        apply σ.ext }\n    case tabs ih =>\n      simp [Term.crename, EType.crename, CType.crename, SType.crename]\n      apply tabs\n      { apply ih\n        apply σ.text }\n    case cabs ih =>\n      simp [Term.crename, EType.crename, CType.crename, SType.crename]\n      apply cabs\n      { rw [CaptureSet.cweaken_crename]\n        apply ih\n        apply σ.cext }\n    case app ih1 ih2 =>\n      simp [Term.crename]\n      rw [EType.crename_open]\n      apply app\n      { have ih1 := ih1 σ\n        simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih1\n        exact ih1 }\n      { have ih2 := ih2 σ\n        simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih2\n        exact ih2 }\n    case tapp ih =>\n      simp [Term.crename]\n      rw [EType.crename_topen]\n      apply tapp\n      have ih1 := ih σ\n      simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih1\n      exact ih1\n    case capp ih =>\n      simp [Term.crename]\n      rw [EType.crename_copen]\n      apply capp\n      have ih1 := ih σ\n      simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih1\n      exact ih1\n    case letin ih1 ih2 =>\n      simp [Term.crename]\n      apply letin\n      { have ih1 := ih1 σ\n        simp [EType.crename] at ih1\n        exact ih1 }\n      { have ih2 := ih2 (σ.ext _)\n        rw [<- EType.weaken_crename] at ih2\n        rw [CaptureSet.weaken_crename]\n        exact ih2 }\n    case letex ih1 ih2 =>\n      simp [Term.crename]\n      apply letex\n      { have ih1 := ih1 σ\n        simp [EType.crename] at ih1\n        exact ih1 }\n      { have ih2 := ih2 (σ.cext.ext _)\n        rw [<- EType.weaken_crename] at ih2\n        rw [<- EType.cweaken_crename] at ih2\n        rw [CaptureSet.cweaken_crename]\n        rw [CaptureSet.weaken_crename]\n        exact ih2 }\n    case bindt ih =>\n      simp [Term.crename]\n      apply bindt\n      have ih := ih σ.text\n      rw [<- EType.tweaken_crename] at ih\n      simp [TBinding.crename] at ih\n      exact ih\n    case bindc ih =>\n      simp [Term.crename]\n      apply bindc\n      have ih := ih σ.cext\n      rw [<- EType.cweaken_crename] at ih\n      rw [CaptureSet.cweaken_crename]\n      trivial\n    case label =>\n      simp [Term.crename, EType.crename, CType.crename, SType.crename]\n      apply label\n      have h := σ.lmap\n      aesop\n    case invoke ih1 ih2 =>\n      simp [Term.crename]\n      simp [EType.crename, CType.crename, SType.crename] at ih1 ih2\n      apply invoke\n      apply ih1; assumption\n      apply ih2; assumption\n    case boundary ih =>\n      simp [Term.crename]\n      simp [EType.crename, CType.crename, SType.crename]\n      apply boundary\n      have ih := ih (σ.cext.ext _)\n      simp [CBinding.crename, EType.crename, CType.crename, SType.crename, FinFun.ext] at ih\n      rw [ <- SType.cweaken_crename\n         , <- SType.weaken_crename\n         , <- SType.cweaken_crename\n         , <- CaptureSet.weaken_crename\n         , <- CaptureSet.cweaken_crename ] at ih\n      aesop", "nesting_depth": 5, "transitive_dep_count": 195, "subset_aristotle": false, "category": "Type systems"}
{"id": 65, "thm_name": "Capless.Typed.crename", "thm_stmt": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)", "lean_root": "capless-lean", "rel_path": "Capless/Renaming/Capture/Typing.lean", "imports": ["import Capless.Typing", "import Capless.Renaming.Capture.Subtyping", "import Capless.Type.Basic", "import Capless.CaptureSet", "import Capless.Renaming.Capture.Subcapturing", "import Capless.Renaming.Basic"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "SType.topen", "content": "def SType.topen (S : SType n (m+1) k) (X : Fin m) : SType n m k :=\n  S.trename (FinFun.open X)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "CType.topen", "content": "def CType.topen (C : CType n (m+1) k) (X : Fin m) : CType n m k :=\n  C.trename (FinFun.open X)"}, {"name": "EType.topen", "content": "def EType.topen (E : EType n (m+1) k) (X : Fin m) : EType n m k :=\n  E.trename (FinFun.open X)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CType.copen", "content": "def CType.copen (C : CType n m (k+1)) (x : Fin k) : CType n m k :=\n  C.crename (FinFun.open x)"}, {"name": "CaptureSet.copen", "content": "def CaptureSet.copen (C : CaptureSet n (k+1)) (x : Fin k) : CaptureSet n k :=\n  C.crename (FinFun.open x)"}, {"name": "SType.copen", "content": "def SType.copen (S : SType n m (k+1)) (x : Fin k) : SType n m k :=\n  S.crename (FinFun.open x)"}, {"name": "EType.copen", "content": "def EType.copen (E : EType n m (k+1)) (x : Fin k) : EType n m k :=\n  E.crename (FinFun.open x)"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "SType.open", "content": "def SType.open (S : SType (n+1) m k) (x : Fin n) : SType n m k :=\n  S.rename (FinFun.open x)"}, {"name": "CType.open", "content": "def CType.open (C : CType (n+1) m k) (x : Fin n) : CType n m k :=\n  C.rename (FinFun.open x)"}, {"name": "CaptureSet.open", "content": "def CaptureSet.open (C : CaptureSet (n+1) k) (x : Fin n) : CaptureSet n k :=\n  C.rename (FinFun.open x)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "SType.cweaken_crename", "content": "theorem SType.cweaken_crename {S : SType n m k} :\n  (S.crename f).cweaken = S.cweaken.crename f.ext"}, {"name": "SType.crename_crename", "content": "theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (S.crename f).crename g = S.crename (g ∘ f)"}, {"name": "CBound.crename_crename", "content": "theorem CBound.crename_crename {b : CBound n k} :\n  (b.crename f).crename g = b.crename (g ∘ f)"}, {"name": "EType.crename_crename", "content": "theorem EType.crename_crename (E : EType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (E.crename f).crename g = E.crename (g ∘ f)"}, {"name": "CType.crename_crename", "content": "theorem CType.crename_crename (T : CType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (T.crename f).crename g = T.crename (g ∘ f)"}, {"name": "EType.weaken_crename", "content": "theorem EType.weaken_crename {E : EType n m k} :\n  (E.crename f).weaken = E.weaken.crename f"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "EType.crename_topen", "content": "theorem EType.crename_topen {E : EType n (m+1) k} :\n  (E.topen X).crename f = (E.crename f).topen X"}, {"name": "EType.crename_trename_comm", "content": "theorem EType.crename_trename_comm (E : EType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (E.crename f).trename g = (E.trename g).crename f"}, {"name": "CType.crename_trename_comm", "content": "theorem CType.crename_trename_comm (T : CType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (T.crename f).trename g = (T.trename g).crename f"}, {"name": "SType.crename_trename_comm", "content": "theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (S.crename f).trename g = (S.trename g).crename f"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "CaptureSet.weaken_crename", "content": "theorem CaptureSet.weaken_crename {C : CaptureSet n k} :\n  (C.crename f).weaken = C.weaken.crename f"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "CaptureSet.cweaken_crename", "content": "theorem CaptureSet.cweaken_crename {C : CaptureSet n k} :\n  (C.crename f).cweaken = C.cweaken.crename f.ext"}, {"name": "CaptureSet.crename_crename", "content": "theorem CaptureSet.crename_crename {C : CaptureSet n k} :\n  (C.crename f).crename g = C.crename (g ∘ f)"}, {"name": "SType.weaken_crename", "content": "theorem SType.weaken_crename {S : SType n m k} :\n  (S.crename f).weaken = S.weaken.crename f"}, {"name": "EType.crename_copen", "content": "theorem EType.crename_copen {E : EType n m (k+1)} :\n  (E.copen c).crename f = (E.crename f.ext).copen (f c)"}, {"name": "EType.tweaken_crename", "content": "theorem EType.tweaken_crename {E : EType n m k} :\n  (E.crename f).tweaken = E.tweaken.crename f"}, {"name": "EType.crename_open", "content": "theorem EType.crename_open {E : EType (n+1) m k} :\n  (E.open x).crename f = (E.crename f).open x"}, {"name": "EType.cweaken_crename", "content": "theorem EType.cweaken_crename {E : EType n m k} :\n  (E.crename f).cweaken = E.cweaken.crename f.ext"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Typing\n\nimport Capless.Renaming.Basic\n\nimport Capless.Renaming.Capture.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f) :=", "ground_truth_proof": ":= by\n  induction h generalizing k'\n  case var hb =>\n    simp [Term.crename, EType.crename, CType.crename]\n    apply var\n    have hb1 := ρ.map _ _ hb\n    simp [CType.crename] at hb1\n    exact hb1\n  case pack ih =>\n    simp [Term.crename, EType.crename]\n    apply pack\n    have ih := ih (ρ.cext _)\n    simp [Term.crename, EType.crename] at ih\n    exact ih\n  case sub hsc hsub ih =>\n    apply sub\n    apply ih ρ\n    apply! hsc.crename\n    apply! ESubtyp.crename hsub\n  case abs ih =>\n    simp [Term.crename, EType.crename, CType.crename, SType.crename]\n    apply abs\n    rw [CaptureSet.weaken_crename]\n    apply ih\n    apply ρ.ext\n  case tabs hc ih =>\n    simp [Term.crename, EType.crename, CType.crename, SType.crename]\n    apply tabs\n    apply ih\n    apply ρ.text\n  case cabs hc ih =>\n    simp [Term.crename, EType.crename, CType.crename, SType.crename]\n    apply cabs\n    rw [CaptureSet.cweaken_crename]\n    apply ih\n    apply ρ.cext\n  case app ih1 ih2 =>\n    simp [Term.crename, EType.crename_open]\n    apply app\n    have ih1 := ih1 ρ\n    simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih1\n    exact ih1\n    have ih2 := ih2 ρ\n    simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih2\n    exact ih2\n  case tapp ih1 =>\n    simp [Term.crename, EType.crename_topen]\n    apply tapp\n    have ih1 := ih1 ρ\n    simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih1\n    exact ih1\n  case capp ih1 =>\n    simp [Term.crename, EType.crename_copen]\n    apply capp\n    have ih1 := ih1 ρ\n    simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih1\n    exact ih1\n  case letin ih1 ih2 =>\n    simp [Term.crename]\n    apply letin\n    have ih1 := ih1 ρ\n    simp [EType.crename] at ih1\n    exact ih1\n    have ih2 := ih2 (ρ.ext _)\n    rw [<- EType.weaken_crename] at ih2\n    rw [CaptureSet.weaken_crename]\n    exact ih2\n  case letex ih1 ih2 =>\n    simp [Term.crename]\n    apply letex\n    have ih1 := ih1 ρ\n    simp [EType.crename] at ih1\n    exact ih1\n    have ih2 := ih2 ((ρ.cext _).ext _)\n    rw [EType.cweaken_crename]\n    rw [EType.weaken_crename]\n    rw [CaptureSet.cweaken_crename, CaptureSet.weaken_crename]\n    exact ih2\n  case bindt ih =>\n    simp [Term.crename]\n    apply bindt\n    have ih := ih (ρ.text _)\n    rw [<- EType.tweaken_crename] at ih\n    exact ih\n  case bindc ih =>\n    simp [Term.crename]\n    apply bindc\n    have ih := ih (ρ.cext _)\n    rw [<- EType.cweaken_crename] at ih\n    rw [CaptureSet.cweaken_crename]\n    exact ih\n  case label =>\n    simp [Term.crename, EType.crename, CType.crename, SType.crename]\n    apply label\n    have h := ρ.lmap\n    aesop\n  case invoke ih1 ih2 =>\n    simp [Term.crename]\n    apply invoke\n    simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih1\n    apply ih1; assumption\n    simp [Term.crename, EType.crename, CType.crename, SType.crename] at ih2\n    apply ih2; assumption\n  case boundary ih =>\n    simp [Term.crename, EType.crename, CType.crename, SType.crename]\n    apply boundary\n    have ih := ih ((ρ.cext _).ext _)\n    simp [CBinding.crename,\n          TBinding.crename,\n          CType.crename, EType.crename,\n          FinFun.ext,\n          SType.crename] at ih\n    rw [<- SType.cweaken_crename,\n        <- SType.weaken_crename,\n        <- SType.cweaken_crename,\n        <- CaptureSet.weaken_crename,\n        <- CaptureSet.cweaken_crename] at ih\n    exact ih", "nesting_depth": 3, "transitive_dep_count": 119, "subset_aristotle": false, "category": "Type systems"}
{"id": 66, "thm_name": "Capless.Typed.trename", "thm_stmt": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct", "lean_root": "capless-lean", "rel_path": "Capless/Renaming/Type/Typing.lean", "imports": ["import Capless.Renaming.Type.Subtyping", "import Capless.Typing", "import Capless.Type.Basic", "import Capless.Renaming.Type.Subcapturing", "import Capless.Renaming.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "TVarMap.weaken_ext", "content": "def TVarMap.weaken_ext {Γ : Context n m k} :\n  TVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.tvar b).var T.tweaken) :="}, {"name": "TVarMap.weaken_cext_ext", "content": "def TVarMap.weaken_cext_ext {Γ : Context n m k} :\n  TVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken\n    (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) :=\n    \nstructure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "SType.topen", "content": "def SType.topen (S : SType n (m+1) k) (X : Fin m) : SType n m k :=\n  S.trename (FinFun.open X)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "CType.topen", "content": "def CType.topen (C : CType n (m+1) k) (X : Fin m) : CType n m k :=\n  C.trename (FinFun.open X)"}, {"name": "EType.topen", "content": "def EType.topen (E : EType n (m+1) k) (X : Fin m) : EType n m k :=\n  E.trename (FinFun.open X)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "CType.copen", "content": "def CType.copen (C : CType n m (k+1)) (x : Fin k) : CType n m k :=\n  C.crename (FinFun.open x)"}, {"name": "CaptureSet.copen", "content": "def CaptureSet.copen (C : CaptureSet n (k+1)) (x : Fin k) : CaptureSet n k :=\n  C.crename (FinFun.open x)"}, {"name": "SType.copen", "content": "def SType.copen (S : SType n m (k+1)) (x : Fin k) : SType n m k :=\n  S.crename (FinFun.open x)"}, {"name": "EType.copen", "content": "def EType.copen (E : EType n m (k+1)) (x : Fin k) : EType n m k :=\n  E.crename (FinFun.open x)"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "CType.open", "content": "def CType.open (C : CType (n+1) m k) (x : Fin n) : CType n m k :=\n  C.rename (FinFun.open x)"}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "CaptureSet.open", "content": "def CaptureSet.open (C : CaptureSet (n+1) k) (x : Fin n) : CaptureSet n k :=\n  C.rename (FinFun.open x)"}, {"name": "SType.open", "content": "def SType.open (S : SType (n+1) m k) (x : Fin n) : SType n m k :=\n  S.rename (FinFun.open x)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "EType.trename_topen", "content": "theorem EType.trename_topen {E : EType n (m+1) k} :\n  (E.topen X).trename f = (E.trename f.ext).topen (f X)"}, {"name": "EType.trename_trename", "content": "theorem EType.trename_trename (E : EType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (E.trename f).trename g = E.trename (g ∘ f)"}, {"name": "CType.trename_trename", "content": "theorem CType.trename_trename (T : CType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (T.trename f).trename g = T.trename (g ∘ f)"}, {"name": "SType.trename_trename", "content": "theorem SType.trename_trename (S : SType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (S.trename f).trename g = S.trename (g ∘ f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "EType.weaken_trename", "content": "theorem EType.weaken_trename {E : EType n m k} :\n  (E.trename f).weaken = E.weaken.trename f"}, {"name": "CType.weaken_trename", "content": "theorem CType.weaken_trename {C : CType n m k} :\n  (C.trename f).weaken = C.weaken.trename f"}, {"name": "EType.trename_rename_comm", "content": "theorem EType.trename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (E.trename g).rename f = (E.rename f).trename g"}, {"name": "CType.trename_rename_comm", "content": "theorem CType.trename_rename_comm (T : CType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (T.trename g).rename f = (T.rename f).trename g"}, {"name": "SType.trename_rename_comm", "content": "theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (S.trename g).rename f = (S.rename f).trename g"}, {"name": "SType.weaken_trename", "content": "theorem SType.weaken_trename {S : SType n m k} :\n  (S.trename f).weaken = S.weaken.trename f"}, {"name": "TBinding.weaken_trename", "content": "theorem TBinding.weaken_trename {b : TBinding n m k} :\n  (b.trename f).weaken = b.weaken.trename f"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "EType.cweaken_trename", "content": "theorem EType.cweaken_trename {E : EType n m k} :\n  (E.trename f).cweaken = E.cweaken.trename f"}, {"name": "EType.crename_trename_comm", "content": "theorem EType.crename_trename_comm (E : EType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (E.crename f).trename g = (E.trename g).crename f"}, {"name": "CType.crename_trename_comm", "content": "theorem CType.crename_trename_comm (T : CType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (T.crename f).trename g = (T.trename g).crename f"}, {"name": "SType.crename_trename_comm", "content": "theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (S.crename f).trename g = (S.trename g).crename f"}, {"name": "EType.trename_copen", "content": "theorem EType.trename_copen {E : EType n m (k+1)} :\n  (E.copen c).trename f = (E.trename f).copen c"}, {"name": "SType.trename_copen", "content": "theorem SType.trename_copen {S : SType n m (k+1)} :\n  (S.copen c).trename f = (S.trename f).copen c"}, {"name": "CType.trename_copen", "content": "theorem CType.trename_copen {C : CType n m (k+1)} :\n  (C.copen c).trename f = (C.trename f).copen c"}, {"name": "EType.tweaken_trename", "content": "theorem EType.tweaken_trename {E : EType n m k} :\n  (E.trename f).tweaken = E.tweaken.trename f.ext"}, {"name": "SType.cweaken_trename", "content": "theorem SType.cweaken_trename {S : SType n m k} :\n  (S.trename f).cweaken = S.cweaken.trename f"}, {"name": "EType.trename_open", "content": "theorem EType.trename_open {E : EType (n+1) m k} :\n  (E.open x).trename f = (E.trename f).open x"}, {"name": "SType.cweaken_copen_id", "content": "theorem SType.cweaken_copen_id {S : SType n m k} :\n  S.cweaken.crename (FinFun.open x) = S"}, {"name": "CType.cweaken_copen_id", "content": "theorem CType.cweaken_copen_id {T : CType n m k} :\n  T.cweaken.crename (FinFun.open x) = T"}, {"name": "TBinding.cweaken_copen_id", "content": "theorem TBinding.cweaken_copen_id {b : TBinding n m k} :\n  b.cweaken.crename (FinFun.open x) = b"}, {"name": "Term.copen_cweaken_ext", "content": "theorem Term.copen_cweaken_ext {t : Term n m (k+1)} :\n  (t.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = t"}, {"name": "EType.copen_cweaken_ext", "content": "theorem EType.copen_cweaken_ext {E : EType n m (k+1)} :\n  (E.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = E"}, {"name": "CaptureSet.copen_cweaken_ext", "content": "theorem CaptureSet.copen_cweaken_ext {C : CaptureSet n (k+1)} :\n  (C.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = C"}, {"name": "Term.open_weaken_ext", "content": "theorem Term.open_weaken_ext {t : Term (n+1) m k} :\n  (t.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = t"}, {"name": "EType.open_weaken_ext", "content": "theorem EType.open_weaken_ext {E : EType (n+1) m k} :\n  (E.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = E"}, {"name": "CaptureSet.open_weaken_ext", "content": "theorem CaptureSet.open_weaken_ext {C : CaptureSet (n+1) k} :\n  (C.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = C"}, {"name": "Typed.boundary_body_typing", "content": "theorem Typed.boundary_body_typing {Γ : Context n m k} {S : SType n m k}\n  (ht : Typed ((Γ,c<:*),x:(Label[S.cweaken])^{c=0}) t E Ct) :\n  Typed ((Γ.label S),c:={x=0}) t E Ct"}, {"name": "Typed.narrow", "content": "theorem Typed.narrow\n  (h : Typed (Γ,x: T) t E Ct)\n  (hs : CSubtyp Γ T' T) :\n  Typed (Γ,x: T') t E Ct"}, {"name": "Typed.tnarrow", "content": "theorem Typed.tnarrow\n  (h : Typed (Γ,X<: S) t E Ct)\n  (hs : SSubtyp Γ S' S) :\n  Typed (Γ,X<: S') t E Ct"}, {"name": "Typed.cnarrow", "content": "theorem Typed.cnarrow\n  (h : Typed (Γ,c<:B) t E Ct)\n  (hs : Subbound Γ B' B) :\n  Typed (Γ,c<:B') t E Ct"}, {"name": "TypedCont.narrow", "content": "theorem TypedCont.narrow\n  (h : TypedCont Γ E1 cont E C0)\n  (hsub : ESubtyp Γ E2 E1) :\n  TypedCont Γ E2 cont E C0"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Typing\n\nimport Capless.Renaming.Basic\n\nimport Capless.Renaming.Type.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct :=", "ground_truth_proof": ":= by\n  induction h generalizing m'\n  case var =>\n    simp [Term.trename, EType.trename, CType.trename]\n    apply var\n    rename_i hb\n    have hb1 := ρ.map _ _ hb\n    simp [CType.trename] at hb1\n    trivial\n  case pack ih =>\n    simp [Term.trename, EType.trename]\n    apply pack\n    have ih := ih (ρ.cext _)\n    simp [Term.trename, EType.trename] at ih\n    trivial\n  case sub hsc hs ih =>\n    apply sub\n    aesop\n    apply! hsc.trename\n    apply! ESubtyp.trename\n  case abs ih =>\n    simp [Term.trename, EType.trename, CType.trename, SType.trename]\n    apply abs\n    apply? ih\n    apply! TVarMap.ext\n  case app ih1 ih2 =>\n    simp [Term.trename]\n    rw [EType.trename_open]\n    apply app\n    have ih1 := ih1 ρ\n    simp [EType.trename, CType.trename, SType.trename, Term.trename] at ih1\n    trivial\n    have ih2 := ih2 ρ\n    simp [Term.trename, EType.trename] at ih2\n    trivial\n  case tabs ih =>\n    simp [Term.trename, EType.trename, CType.trename, SType.trename]\n    apply tabs\n    apply? ih\n    apply! TVarMap.text\n  case cabs ih =>\n    simp [Term.trename, EType.trename, CType.trename, SType.trename]\n    apply cabs\n    have ih1 := ih (ρ.cext _)\n    trivial\n  case tapp ih =>\n    simp [Term.trename]\n    rw [EType.trename_topen]\n    apply tapp\n    have ih := ih ρ\n    simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih\n    trivial\n  case capp ih =>\n    simp [Term.trename]\n    rw [EType.trename_copen]\n    apply capp\n    have ih := ih ρ\n    simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih\n    trivial\n  case letin ih1 ih2 =>\n    simp [Term.trename]\n    apply letin\n    simp [EType.trename] at ih1\n    aesop\n    have ih2 := ih2 (ρ.ext _)\n    rw [<- EType.weaken_trename] at ih2\n    trivial\n  case letex ih1 ih2 =>\n    simp [Term.trename]\n    apply letex\n    simp [EType.trename] at ih1\n    aesop\n    have ih2 := ih2 ((ρ.cext _).ext _)\n    rw [<- EType.weaken_trename] at ih2\n    rw [<- EType.cweaken_trename] at ih2\n    trivial\n  case bindt ih =>\n    simp [Term.trename]\n    apply bindt\n    have ih := ih (ρ.text _)\n    rw [EType.tweaken_trename]\n    trivial\n  case bindc ih =>\n    simp [Term.trename]\n    apply bindc\n    have ih := ih (ρ.cext _)\n    rw [EType.cweaken_trename]\n    trivial\n  case label =>\n    simp [Term.trename, EType.trename, CType.trename, SType.trename]\n    apply label\n    have h := ρ.lmap\n    aesop\n  case invoke ih1 ih2 =>\n    simp [Term.trename]\n    apply invoke\n    simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih1\n    apply ih1; trivial\n    simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih2\n    apply ih2; trivial\n  case boundary ih =>\n    simp [Term.trename, EType.trename, CType.trename]\n    apply boundary\n    have ih := ih ((ρ.cext _).ext _)\n    simp [FinFun.ext, CType.trename, SType.trename] at ih\n    rw [ SType.cweaken_trename\n       , SType.weaken_trename ]\n    simp [EType.trename, CType.trename] at ih\n    exact ih", "nesting_depth": 3, "transitive_dep_count": 111, "subset_aristotle": false, "category": "Type systems"}
{"id": 67, "thm_name": "Capless.Typed.tsubst", "thm_stmt": "theorem Typed.tsubst\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (σ : TVarSubst Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct", "lean_root": "capless-lean", "rel_path": "Capless/Subst/Type/Typing.lean", "imports": ["import Capless.Renaming.Type.Subtyping", "import Capless.Renaming.Term.Typing", "import Capless.Typing", "import Capless.Renaming.Type.Typing", "import Capless.Renaming.Capture.Typing", "import Capless.Renaming.Type.Subcapturing", "import Capless.Renaming.Term.Subtyping", "import Capless.Subst.Type.Subcapturing", "import Capless.Type.Basic", "import Capless.Subst.Basic", "import Capless.Subst.Type.Subtyping", "import Capless.Renaming.Term.Subcapturing", "import Capless.Renaming.Capture.Subtyping", "import Capless.Renaming.Capture.Subcapturing"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "SType.topen", "content": "def SType.topen (S : SType n (m+1) k) (X : Fin m) : SType n m k :=\n  S.trename (FinFun.open X)"}, {"name": "CType.topen", "content": "def CType.topen (C : CType n (m+1) k) (X : Fin m) : CType n m k :=\n  C.trename (FinFun.open X)"}, {"name": "EType.topen", "content": "def EType.topen (E : EType n (m+1) k) (X : Fin m) : EType n m k :=\n  E.trename (FinFun.open X)"}, {"name": "SSubtyp.tsubst_motive1", "content": "def SSubtyp.tsubst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.tsubst_motive2", "content": "def SSubtyp.tsubst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n  CSubtyp Δ (C1.trename f) (C2.trename f)"}, {"name": "SSubtyp.tsubst_motive3", "content": "def SSubtyp.tsubst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "CType.copen", "content": "def CType.copen (C : CType n m (k+1)) (x : Fin k) : CType n m k :=\n  C.crename (FinFun.open x)"}, {"name": "CaptureSet.copen", "content": "def CaptureSet.copen (C : CaptureSet n (k+1)) (x : Fin k) : CaptureSet n k :=\n  C.crename (FinFun.open x)"}, {"name": "SType.copen", "content": "def SType.copen (S : SType n m (k+1)) (x : Fin k) : SType n m k :=\n  S.crename (FinFun.open x)"}, {"name": "EType.copen", "content": "def EType.copen (E : EType n m (k+1)) (x : Fin k) : EType n m k :=\n  E.crename (FinFun.open x)"}, {"name": "CType.open", "content": "def CType.open (C : CType (n+1) m k) (x : Fin n) : CType n m k :=\n  C.rename (FinFun.open x)"}, {"name": "CaptureSet.open", "content": "def CaptureSet.open (C : CaptureSet (n+1) k) (x : Fin n) : CaptureSet n k :=\n  C.rename (FinFun.open x)"}, {"name": "SType.open", "content": "def SType.open (S : SType (n+1) m k) (x : Fin n) : SType n m k :=\n  S.rename (FinFun.open x)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "EType.trename_topen", "content": "theorem EType.trename_topen {E : EType n (m+1) k} :\n  (E.topen X).trename f = (E.trename f.ext).topen (f X)"}, {"name": "EType.trename_trename", "content": "theorem EType.trename_trename (E : EType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (E.trename f).trename g = E.trename (g ∘ f)"}, {"name": "CType.trename_trename", "content": "theorem CType.trename_trename (T : CType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (T.trename f).trename g = T.trename (g ∘ f)"}, {"name": "SType.trename_trename", "content": "theorem SType.trename_trename (S : SType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (S.trename f).trename g = S.trename (g ∘ f)"}, {"name": "SSubtyp.tsubst", "content": "theorem SSubtyp.tsubst\n  (h : SSubtyp Γ S1 S2)\n  (σ : TVarSubst Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "Subbound.tsubst", "content": "theorem Subbound.tsubst\n  (h : Subbound Γ B1 B2)\n  (σ : TVarSubst Γ f Δ) :\n  Subbound Δ B1 B2"}, {"name": "Subcapt.tsubst", "content": "theorem Subcapt.tsubst\n  (h : Subcapt Γ C1 C2)\n  (σ : TVarSubst Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "EType.weaken_trename", "content": "theorem EType.weaken_trename {E : EType n m k} :\n  (E.trename f).weaken = E.weaken.trename f"}, {"name": "EType.trename_rename_comm", "content": "theorem EType.trename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (E.trename g).rename f = (E.rename f).trename g"}, {"name": "CType.trename_rename_comm", "content": "theorem CType.trename_rename_comm (T : CType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (T.trename g).rename f = (T.rename f).trename g"}, {"name": "SType.trename_rename_comm", "content": "theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (S.trename g).rename f = (S.rename f).trename g"}, {"name": "ESubtyp.tsubst", "content": "theorem ESubtyp.tsubst\n  (h : ESubtyp Γ E1 E2)\n  (σ : TVarSubst Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "CSubtyp.tsubst", "content": "theorem CSubtyp.tsubst\n  (h : CSubtyp Γ T1 T2)\n  (σ : TVarSubst Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SType.weaken_trename", "content": "theorem SType.weaken_trename {S : SType n m k} :\n  (S.trename f).weaken = S.weaken.trename f"}, {"name": "EType.cweaken_trename", "content": "theorem EType.cweaken_trename {E : EType n m k} :\n  (E.trename f).cweaken = E.cweaken.trename f"}, {"name": "EType.crename_trename_comm", "content": "theorem EType.crename_trename_comm (E : EType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (E.crename f).trename g = (E.trename g).crename f"}, {"name": "CType.crename_trename_comm", "content": "theorem CType.crename_trename_comm (T : CType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (T.crename f).trename g = (T.trename g).crename f"}, {"name": "SType.crename_trename_comm", "content": "theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (S.crename f).trename g = (S.trename g).crename f"}, {"name": "EType.trename_copen", "content": "theorem EType.trename_copen {E : EType n m (k+1)} :\n  (E.copen c).trename f = (E.trename f).copen c"}, {"name": "EType.tweaken_trename", "content": "theorem EType.tweaken_trename {E : EType n m k} :\n  (E.trename f).tweaken = E.tweaken.trename f.ext"}, {"name": "SType.cweaken_trename", "content": "theorem SType.cweaken_trename {S : SType n m k} :\n  (S.trename f).cweaken = S.cweaken.trename f"}, {"name": "EType.trename_open", "content": "theorem EType.trename_open {E : EType (n+1) m k} :\n  (E.open x).trename f = (E.trename f).open x"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Subst.Basic\n\nimport Capless.Subst.Type.Subtyping\n\nimport Capless.Typing\n\nnamespace Capless", "target_theorem": "theorem Typed.tsubst\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (σ : TVarSubst Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct :=", "ground_truth_proof": ":= by\n    induction h generalizing m'\n    case var hb =>\n      simp [Term.trename, EType.trename, CType.trename]\n      have hb1 := σ.map _ _ hb\n      simp [CType.trename] at hb1\n      apply Typed.var; trivial\n    case pack ih =>\n      simp [Term.trename, EType.trename]\n      apply pack\n      have ih := ih σ.cext\n      simp [EType.trename] at ih\n      exact ih\n    case sub hsc hs ih =>\n      apply sub\n      { apply ih; trivial }\n      { apply! hsc.tsubst  }\n      { apply! hs.tsubst }\n    case abs ih =>\n      simp [Term.trename, EType.trename, CType.trename, SType.trename]\n      apply abs\n      { apply ih\n        apply σ.ext }\n    case tabs ih =>\n      simp [Term.trename, EType.trename, CType.trename, SType.trename]\n      apply tabs\n      { apply ih\n        apply σ.text }\n    case cabs ih =>\n      simp [Term.trename, EType.trename, CType.trename, SType.trename]\n      apply cabs\n      { apply ih\n        apply σ.cext }\n    case app ih1 ih2 =>\n      simp [Term.trename]\n      rw [EType.trename_open]\n      apply app\n      { have ih1 := ih1 σ\n        simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih1\n        exact ih1 }\n      { have ih2 := ih2 σ\n        simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih2\n        exact ih2 }\n    case tapp ih =>\n      simp [Term.trename]\n      rw [EType.trename_topen]\n      apply tapp\n      have ih1 := ih σ\n      simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih1\n      exact ih1\n    case capp ih =>\n      simp [Term.trename]\n      rw [EType.trename_copen]\n      apply capp\n      have ih1 := ih σ\n      simp [Term.trename, EType.trename, CType.trename, SType.trename] at ih1\n      exact ih1\n    case letin ih1 ih2 =>\n      simp [Term.trename]\n      apply letin\n      { have ih1 := ih1 σ\n        simp [EType.trename] at ih1\n        exact ih1 }\n      { have ih2 := ih2 (σ.ext _)\n        rw [<- EType.weaken_trename] at ih2\n        exact ih2 }\n    case letex ih1 ih2 =>\n      simp [Term.trename]\n      apply letex\n      { have ih1 := ih1 σ\n        simp [EType.trename] at ih1\n        exact ih1 }\n      { have ih2 := ih2 (σ.cext.ext _)\n        rw [<-EType.weaken_trename] at ih2\n        rw [<-EType.cweaken_trename] at ih2\n        exact ih2 }\n    case bindt ih =>\n      simp [Term.trename]\n      apply bindt\n      have ih := ih (σ.text _)\n      rw [<-EType.tweaken_trename] at ih\n      simp [TBinding.trename] at ih\n      exact ih\n    case bindc ih =>\n      simp [Term.trename]\n      apply bindc\n      have ih := ih σ.cext\n      rw [<-EType.cweaken_trename] at ih\n      trivial\n    case label hb =>\n      simp [Term.trename, EType.trename, CType.trename, SType.trename]\n      have hb1 := σ.lmap _ _ hb\n      apply label; assumption\n    case invoke ih1 ih2 =>\n      simp [Term.trename]\n      simp [EType.trename, CType.trename, SType.trename] at ih1 ih2\n      apply invoke\n      apply ih1; assumption\n      apply ih2; assumption\n    case boundary ih =>\n      simp [Term.trename]\n      simp [EType.trename, CType.trename, SType.trename]\n      apply boundary\n      have ih := ih (σ.cext.ext _)\n      simp [EType.trename, CType.trename, SType.trename] at ih\n      rw [ <- SType.cweaken_trename\n         , <- SType.weaken_trename\n         , <- SType.cweaken_trename ] at ih\n      aesop", "nesting_depth": 5, "transitive_dep_count": 189, "subset_aristotle": false, "category": "Type systems"}
{"id": 68, "thm_name": "Capless.SSubtyp.rename", "thm_stmt": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)", "lean_root": "capless-lean", "rel_path": "Capless/Renaming/Term/Subtyping.lean", "imports": ["import Capless.Renaming.Term.Subcapturing", "import Capless.Subtyping", "import Capless.Renaming.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}], "used_local_defs": [{"name": "Capless.SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "Capless.SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "Capless.SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}], "used_local_lemmas": [{"name": "Capless.Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}], "local_ctx": "import Capless.Subtyping\n\nimport Capless.Renaming.Basic\n\nimport Capless.Renaming.Term.Subcapturing\n\nnamespace Capless\n\ndef SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)\n\ndef SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)\n\ndef SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)", "target_theorem": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f) :=", "ground_truth_proof": ":= by\n  apply SSubtyp.rec\n    (motive_1 := fun Γ E1 E2 h => SSubtyp.rename_motive1 Γ E1 E2)\n    (motive_2 := fun Γ C1 C2 h => SSubtyp.rename_motive2 Γ C1 C2)\n    (motive_3 := fun Γ S1 S2 h => SSubtyp.rename_motive3 Γ S1 S2)\n    (t := h) (ρ := ρ)\n  case exist ih =>\n    unfold SSubtyp.rename_motive1 SSubtyp.rename_motive2\n    intros; intros\n    simp [EType.rename]\n    apply ESubtyp.exist\n    rename_i ih _ _ _ _\n    apply ih; try assumption\n    apply VarMap.cext; trivial\n  case type ih =>\n    unfold rename_motive1 rename_motive2\n    repeat intro\n    simp [EType.rename]\n    apply ESubtyp.type\n    aesop\n  case capt =>\n    unfold rename_motive2 rename_motive3\n    repeat intro\n    simp [CType.rename]\n    apply CSubtyp.capt\n    apply Subcapt.rename <;> assumption\n    aesop\n  case top =>\n    unfold rename_motive3\n    repeat intro\n    simp [SType.rename]\n    constructor\n  case refl =>\n    unfold rename_motive3\n    repeat intro\n    constructor\n  case trans =>\n    unfold rename_motive3\n    repeat intro\n    rename_i ih1 ih2 _ _ _ _\n    apply trans <;> aesop\n  case tvar =>\n    unfold rename_motive3\n    repeat intro\n    simp [SType.rename]\n    apply SSubtyp.tvar\n    rename_i hb _ _ _ ρ\n    have hb1 := ρ.tmap _ _ hb\n    simp [TBinding.rename] at hb1\n    assumption\n  case tinstl =>\n    unfold rename_motive3\n    repeat intro\n    simp [SType.rename]\n    apply SSubtyp.tinstl\n    rename_i hb _ _ _ ρ\n    have hb1 := ρ.tmap _ _ hb\n    simp [TBinding.rename] at hb1\n    assumption\n  case tinstr =>\n    unfold rename_motive3\n    repeat intro\n    simp [SType.rename]\n    apply SSubtyp.tinstr\n    rename_i hb _ _ _ ρ\n    have hb1 := ρ.tmap _ _ hb\n    simp [TBinding.rename] at hb1\n    assumption\n  case boxed =>\n    unfold rename_motive3\n    repeat intro\n    simp [SType.rename]\n    apply SSubtyp.boxed\n    aesop\n  case label =>\n    unfold rename_motive3\n    repeat intro\n    simp [SType.rename]\n    apply SSubtyp.label\n    aesop\n  case xforall =>\n    unfold rename_motive3 rename_motive1\n    repeat intro\n    simp [SType.rename]\n    apply SSubtyp.xforall\n    aesop\n    rename_i ih _ _ _ _\n    apply ih; try assumption\n    apply VarMap.ext; trivial\n  case cforall =>\n    unfold rename_motive1 rename_motive3\n    repeat intro\n    simp [SType.rename]\n    apply SSubtyp.cforall\n    { apply Subbound.rename <;> easy }\n    { rename_i ih _ _ _ _\n      apply ih\n      apply VarMap.cext; trivial }\n  case tforall =>\n    unfold rename_motive1 rename_motive3\n    repeat intro\n    simp [SType.rename]\n    apply SSubtyp.tforall\n    aesop\n    rename_i ih1 ih2 _ _ _ _\n    apply ih2; try assumption\n    apply VarMap.text; trivial", "nesting_depth": 4, "transitive_dep_count": 49, "subset_aristotle": false, "category": "Type systems"}
{"id": 69, "thm_name": "Capless.SSubtyp.subst", "thm_stmt": "theorem SSubtyp.subst\n  (h : SSubtyp Γ S1 S2)\n  (σ : VarSubst Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)", "lean_root": "capless-lean", "rel_path": "Capless/Subst/Term/Subtyping.lean", "imports": ["import Capless.Subst.Term.Subcapturing", "import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Term.Typing", "import Capless.Renaming.Term.Subcapturing", "import Capless.Subtyping", "import Capless.Subst.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "abs", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}], "used_repo_defs": [{"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "Subcapt.subst", "content": "theorem Subcapt.subst\n  (h : Subcapt Γ C1 C2)\n  (σ : VarSubst Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}], "used_local_defs": [{"name": "Capless.SSubtyp.subst_motive1", "content": "def SSubtyp.subst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "Capless.SSubtyp.subst_motive2", "content": "def SSubtyp.subst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "Capless.SSubtyp.subst_motive3", "content": "def SSubtyp.subst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}], "used_local_lemmas": [{"name": "Capless.Subbound.subst", "content": "theorem Subbound.subst\n  (h : Subbound Γ B1 B2)\n  (σ : VarSubst Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}], "local_ctx": "import Capless.Subst.Basic\n\nimport Capless.Subtyping\n\nimport Capless.Subst.Term.Subcapturing\n\nnamespace Capless\n\ndef SSubtyp.subst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)\n\ndef SSubtyp.subst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)\n\ndef SSubtyp.subst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)", "target_theorem": "theorem SSubtyp.subst\n  (h : SSubtyp Γ S1 S2)\n  (σ : VarSubst Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f) :=", "ground_truth_proof": ":= by\n  apply SSubtyp.rec\n    (motive_1 := fun Γ E1 E2 h => SSubtyp.subst_motive1 Γ E1 E2)\n    (motive_2 := fun Γ C1 C2 h => SSubtyp.subst_motive2 Γ C1 C2)\n    (motive_3 := fun Γ S1 S2 h => SSubtyp.subst_motive3 Γ S1 S2)\n    (t := h) (ρ := σ)\n  case exist =>\n    unfold subst_motive1 subst_motive2\n    repeat intro\n    simp [EType.rename]\n    apply ESubtyp.exist\n    rename_i ih _ _ _ _\n    apply ih\n    apply VarSubst.cext; trivial\n  case type =>\n    unfold subst_motive1 subst_motive2\n    repeat intro\n    simp [EType.rename]\n    apply ESubtyp.type\n    aesop\n  case capt =>\n    unfold subst_motive2 subst_motive3\n    repeat intro\n    simp [CType.rename]\n    apply CSubtyp.capt\n    apply Subcapt.subst <;> trivial\n    aesop\n  case top =>\n    unfold subst_motive3\n    repeat intro\n    simp [SType.rename]\n    apply top\n  case refl =>\n    unfold subst_motive3\n    repeat intro\n    apply refl\n  case trans =>\n    unfold subst_motive3\n    repeat intro\n    apply trans\n    { aesop }\n    { aesop }\n  case tvar =>\n    unfold subst_motive3\n    repeat intro\n    simp [SType.rename]\n    apply tvar\n    rename_i hb _ _ _ σ\n    have hb1 := σ.tmap _ _ hb\n    simp [TBinding.rename] at hb1\n    exact hb1\n  case tinstl =>\n    unfold subst_motive3\n    repeat intro\n    simp [SType.rename]\n    apply tinstl\n    rename_i hb _ _ _ σ\n    have hb1 := σ.tmap _ _ hb\n    simp [TBinding.rename] at hb1\n    exact hb1\n  case tinstr =>\n    unfold subst_motive3\n    repeat intro\n    simp [SType.rename]\n    apply tinstr\n    rename_i hb _ _ _ σ\n    have hb1 := σ.tmap _ _ hb\n    simp [TBinding.rename] at hb1\n    exact hb1\n  case boxed =>\n    unfold subst_motive2 subst_motive3\n    repeat intro\n    simp [SType.rename]\n    apply boxed\n    aesop\n  case label =>\n    unfold subst_motive3\n    repeat intro\n    simp\n    apply label\n    aesop\n  case xforall =>\n    unfold subst_motive1 subst_motive2 subst_motive3\n    repeat intro\n    simp [SType.rename]\n    apply xforall\n    { aesop }\n    { rename_i ih _ _ _ σ\n      apply ih\n      apply VarSubst.ext; trivial }\n  case tforall =>\n    unfold subst_motive1 subst_motive3\n    repeat intro\n    simp [SType.rename]\n    apply tforall\n    { aesop }\n    { rename_i ih _ _ _ σ\n      apply ih\n      apply VarSubst.text; trivial }\n  case cforall =>\n    unfold subst_motive1 subst_motive3\n    repeat intro\n    simp [SType.rename]\n    apply cforall\n    { apply Subbound.subst <;> easy }\n    { rename_i ih _ _ _ σ\n      apply ih\n      apply VarSubst.cext; trivial }", "nesting_depth": 6, "transitive_dep_count": 122, "subset_aristotle": false, "category": "Type systems"}
{"id": 70, "thm_name": "Capless.SSubtyp.csubst", "thm_stmt": "theorem SSubtyp.csubst\n  (h : SSubtyp Γ S1 S2)\n  (σ : CVarSubst Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)", "lean_root": "capless-lean", "rel_path": "Capless/Subst/Capture/Subtyping.lean", "imports": ["import Capless.Renaming.Capture.Typing", "import Capless.Subst.Basic", "import Capless.Renaming.Capture.Subtyping", "import Capless.Subst.Capture.Subcapturing", "import Capless.Context", "import Capless.Subtyping", "import Capless.Renaming.Capture.Subcapturing"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "Subcapt.csubst", "content": "theorem Subcapt.csubst\n  (h : Subcapt Γ C1 C2)\n  (σ : CVarSubst Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "TBinding.crename_bound", "content": "lemma TBinding.crename_bound: (TBinding.bound T).crename f = TBinding.bound (T.crename f)"}], "used_local_defs": [{"name": "Capless.SSubtyp.csubst_motive1", "content": "def SSubtyp.csubst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "Capless.SSubtyp.csubst_motive2", "content": "def SSubtyp.csubst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "Capless.SSubtyp.csubst_motive3", "content": "def SSubtyp.csubst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}], "used_local_lemmas": [{"name": "Capless.Subbound.csubst", "content": "theorem Subbound.csubst\n  (h : Subbound Γ B1 B2)\n  (σ : CVarSubst Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}], "local_ctx": "import Capless.Subtyping\n\nimport Capless.Subst.Basic\n\nimport Capless.Subst.Capture.Subcapturing\n\nnamespace Capless\n\ndef SSubtyp.csubst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)\n\ndef SSubtyp.csubst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)\n\ndef SSubtyp.csubst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)", "target_theorem": "theorem SSubtyp.csubst\n  (h : SSubtyp Γ S1 S2)\n  (σ : CVarSubst Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f) :=", "ground_truth_proof": ":= by\n    apply SSubtyp.rec\n      (motive_1 := fun Γ E1 E2 _ => SSubtyp.csubst_motive1 Γ E1 E2)\n      (motive_2 := fun Γ C1 C2 _ => SSubtyp.csubst_motive2 Γ C1 C2)\n      (motive_3 := fun Γ S1 S2 _ => SSubtyp.csubst_motive3 Γ S1 S2)\n      (t := h) (ρ := σ)\n    case exist =>\n      unfold csubst_motive1 csubst_motive2\n      repeat intro\n      simp [EType.crename]\n      apply ESubtyp.exist\n      rename_i ih _ _ _ ρ\n      apply ih ; try assumption\n      apply CVarSubst.cext; trivial\n    case type =>\n      unfold csubst_motive1 csubst_motive2\n      repeat intro\n      simp [EType.crename]\n      apply ESubtyp.type\n      aesop\n    case capt =>\n      unfold csubst_motive2 csubst_motive3\n      repeat intro\n      simp [CType.crename]\n      apply CSubtyp.capt\n      apply Subcapt.csubst <;> trivial\n      aesop\n    case top =>\n      unfold csubst_motive3\n      repeat intro\n      simp [SType.crename]\n      apply top\n    case refl =>\n      unfold csubst_motive3\n      repeat intro\n      apply refl\n    case trans =>\n      unfold csubst_motive3\n      repeat intro\n      apply trans\n      { aesop }\n      { aesop }\n    case tvar =>\n      unfold csubst_motive3\n      repeat intro\n      rename_i hb _ _ _ σ\n      have hb1 := σ.tmap _ _ hb\n      simp [SType.crename]\n      apply tvar\n      trivial\n    case tinstl =>\n      unfold csubst_motive3\n      repeat intro\n      rename_i hb _ _ Δ σ\n      have hb1 := σ.tmap _ _ hb\n      simp [SType.crename]\n      apply SSubtyp.tinstl\n      trivial\n    case tinstr =>\n      unfold csubst_motive3\n      repeat intro\n      rename_i hb _ _ Δ σ\n      have hb1 := σ.tmap _ _ hb\n      simp [SType.crename]\n      apply SSubtyp.tinstr\n      trivial\n    case boxed =>\n      unfold csubst_motive2 csubst_motive3\n      repeat intro\n      simp [SType.crename]\n      apply boxed\n      aesop\n    case label =>\n      unfold csubst_motive3\n      repeat intro\n      simp [SType.crename]\n      apply SSubtyp.label\n      aesop\n    case xforall =>\n      unfold csubst_motive1 csubst_motive2 csubst_motive3\n      repeat intro\n      simp [SType.crename]\n      apply xforall\n      { aesop }\n      { rename_i ih _ _ _ σ\n        apply ih ; try assumption\n        apply CVarSubst.ext; trivial }\n    case tforall =>\n      unfold csubst_motive1 csubst_motive3\n      repeat intro\n      simp [SType.crename]\n      apply tforall\n      { aesop }\n      { rename_i ih _ _ _ σ\n        apply ih ; try assumption\n        rw [<-TBinding.crename_bound]\n        apply CVarSubst.text; trivial }\n    case cforall =>\n      unfold csubst_motive1 csubst_motive3\n      repeat intro\n      simp [SType.crename]\n      apply cforall\n      { apply Subbound.csubst <;> easy }\n      { rename_i ih _ _ _ σ\n        apply ih ; try assumption\n        apply CVarSubst.cext; trivial\n      }", "nesting_depth": 6, "transitive_dep_count": 112, "subset_aristotle": false, "category": "Type systems"}
{"id": 71, "thm_name": "Capless.SSubtyp.crename", "thm_stmt": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)", "lean_root": "capless-lean", "rel_path": "Capless/Renaming/Capture/Subtyping.lean", "imports": ["import Capless.Tactics", "import Capless.Subtyping", "import Capless.Renaming.Capture.Subcapturing", "import Capless.Renaming.Basic"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}], "used_local_defs": [{"name": "Capless.SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "Capless.SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "Capless.SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}], "used_local_lemmas": [{"name": "Capless.Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}], "local_ctx": "import Capless.Tactics\n\nimport Capless.Subtyping\n\nimport Capless.Renaming.Basic\n\nimport Capless.Renaming.Capture.Subcapturing\n\nnamespace Capless\n\ndef SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)\n\ndef SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)\n\ndef SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)", "target_theorem": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f) :=", "ground_truth_proof": ":= by\n  apply SSubtyp.rec\n    (motive_1 := fun Γ E1 E2 h => SSubtyp.crename_motive1 Γ E1 E2)\n    (motive_2 := fun Γ C1 C2 h => SSubtyp.crename_motive2 Γ C1 C2)\n    (motive_3 := fun Γ S1 S2 h => SSubtyp.crename_motive3 Γ S1 S2)\n    (t := h) (ρ := ρ)\n  case exist =>\n    unfold SSubtyp.crename_motive1 SSubtyp.crename_motive2\n    repeat intro\n    simp [EType.crename]\n    apply ESubtyp.exist\n    rename_i ih _ _ _ _\n    apply ih; try assumption\n    apply CVarMap.cext; trivial\n  case type =>\n    unfold crename_motive2 crename_motive1\n    repeat intro\n    simp [EType.crename]\n    apply ESubtyp.type\n    aesop\n  case capt =>\n    unfold crename_motive3 crename_motive2\n    repeat intro\n    simp [CType.crename]\n    apply CSubtyp.capt\n    apply Subcapt.crename <;> aesop\n    aesop\n  case top =>\n    unfold crename_motive3\n    repeat intro\n    simp [SType.crename]\n    apply SSubtyp.top\n  case refl =>\n    unfold crename_motive3\n    repeat intro\n    constructor\n  case trans =>\n    unfold crename_motive3\n    repeat intro\n    apply SSubtyp.trans\n    aesop\n    aesop\n  case tvar =>\n    unfold crename_motive3\n    repeat intro\n    simp [SType.crename]\n    apply SSubtyp.tvar\n    rename_i hb _ _ _ ρ\n    have hb1 := ρ.tmap _ _ hb\n    simp [TBinding.crename] at hb1\n    trivial\n  case tinstl =>\n    unfold crename_motive3\n    repeat intro\n    simp [SType.crename]\n    apply SSubtyp.tinstl\n    rename_i hb _ _ _ ρ\n    have hb1 := ρ.tmap _ _ hb\n    simp [TBinding.crename] at hb1\n    assumption\n  case tinstr =>\n    unfold crename_motive3\n    repeat intro\n    simp [SType.crename]\n    apply SSubtyp.tinstr\n    rename_i hb _ _ _ ρ\n    have hb1 := ρ.tmap _ _ hb\n    simp [TBinding.crename] at hb1\n    assumption\n  case boxed =>\n    unfold crename_motive3 crename_motive2\n    repeat intro\n    simp [SType.crename]\n    apply SSubtyp.boxed\n    aesop\n  case label =>\n    unfold crename_motive3\n    repeat intro\n    simp [SType.crename]\n    apply SSubtyp.label\n    aesop\n  case xforall =>\n    unfold crename_motive1 crename_motive3\n    repeat intro\n    simp [SType.crename]\n    apply SSubtyp.xforall\n    aesop\n    rename_i ih _ _ _ _\n    apply ih; try assumption\n    apply CVarMap.ext; trivial\n  case tforall =>\n    unfold crename_motive1 crename_motive3\n    repeat intro\n    simp [SType.crename]\n    apply SSubtyp.tforall\n    aesop\n    rename_i ih1 ih2 _ _ _ _\n    apply ih2; try easy\n    apply CVarMap.text; easy\n  case cforall =>\n    unfold crename_motive1 crename_motive3\n    repeat intro\n    simp [SType.crename]\n    apply SSubtyp.cforall\n    { apply Subbound.crename <;> easy }\n    { rename_i ih _ _ _ _\n      apply ih\n      apply CVarMap.cext; easy }", "nesting_depth": 6, "transitive_dep_count": 60, "subset_aristotle": false, "category": "Type systems"}
{"id": 72, "thm_name": "Capless.SSubtyp.tsubst", "thm_stmt": "theorem SSubtyp.tsubst\n  (h : SSubtyp Γ S1 S2)\n  (σ : TVarSubst Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)", "lean_root": "capless-lean", "rel_path": "Capless/Subst/Type/Subtyping.lean", "imports": ["import Capless.Renaming.Type.Subtyping", "import Capless.Renaming.Type.Typing", "import Capless.Renaming.Type.Subcapturing", "import Capless.Subst.Type.Subcapturing", "import Capless.Subtyping", "import Capless.Subst.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "Subcapt.tsubst", "content": "theorem Subcapt.tsubst\n  (h : Subcapt Γ C1 C2)\n  (σ : TVarSubst Γ f Δ) :\n  Subcapt Δ C1 C2"}], "used_local_defs": [{"name": "Capless.SSubtyp.tsubst_motive1", "content": "def SSubtyp.tsubst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Capless.SSubtyp.tsubst_motive2", "content": "def SSubtyp.tsubst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n  CSubtyp Δ (C1.trename f) (C2.trename f)"}, {"name": "Capless.SSubtyp.tsubst_motive3", "content": "def SSubtyp.tsubst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}], "used_local_lemmas": [{"name": "Capless.Subbound.tsubst", "content": "theorem Subbound.tsubst\n  (h : Subbound Γ B1 B2)\n  (σ : TVarSubst Γ f Δ) :\n  Subbound Δ B1 B2"}], "local_ctx": "import Capless.Subst.Basic\n\nimport Capless.Subtyping\n\nimport Capless.Subst.Type.Subcapturing\n\nnamespace Capless\n\ndef SSubtyp.tsubst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)\n\ndef SSubtyp.tsubst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n  CSubtyp Δ (C1.trename f) (C2.trename f)\n\ndef SSubtyp.tsubst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarSubst Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)", "target_theorem": "theorem SSubtyp.tsubst\n  (h : SSubtyp Γ S1 S2)\n  (σ : TVarSubst Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f) :=", "ground_truth_proof": ":= by\n    apply SSubtyp.rec\n      (motive_1 := fun Γ E1 E2 _ => SSubtyp.tsubst_motive1 Γ E1 E2)\n      (motive_2 := fun Γ C1 C2 _ => SSubtyp.tsubst_motive2 Γ C1 C2)\n      (motive_3 := fun Γ S1 S2 _ => SSubtyp.tsubst_motive3 Γ S1 S2)\n      (t := h) (ρ := σ)\n    case exist =>\n      unfold tsubst_motive1 tsubst_motive2\n      repeat intro\n      simp [EType.trename]\n      apply ESubtyp.exist\n      rename_i ih _ _ _ ρ\n      apply ih ; try assumption\n      apply TVarSubst.cext; trivial\n    case type =>\n      unfold tsubst_motive1 tsubst_motive2\n      repeat intro\n      simp [EType.trename]\n      apply ESubtyp.type\n      aesop\n    case capt =>\n      unfold tsubst_motive2 tsubst_motive3\n      repeat intro\n      simp [CType.trename]\n      apply CSubtyp.capt\n      apply Subcapt.tsubst <;> trivial\n      aesop\n    case top =>\n      unfold tsubst_motive3\n      repeat intro\n      simp [SType.trename]\n      apply top\n    case refl =>\n      unfold tsubst_motive3\n      repeat intro\n      apply refl\n    case trans =>\n      unfold tsubst_motive3\n      repeat intro\n      apply trans\n      { aesop }\n      { aesop }\n    case tvar =>\n      unfold tsubst_motive3\n      repeat intro\n      rename_i hb _ _ _ σ\n      have hb1 := σ.tmap _ _ hb\n      simp [SType.trename]\n      trivial\n    case tinstl =>\n      unfold tsubst_motive3\n      repeat intro\n      rename_i hb _ _ Δ σ\n      have hb1 := σ.tmap_inst _ _ hb\n      simp [SType.trename]\n      apply SSubtyp.tinstl\n      trivial\n    case tinstr =>\n      unfold tsubst_motive3\n      repeat intro\n      rename_i hb _ _ Δ σ\n      have hb1 := σ.tmap_inst _ _ hb\n      simp [SType.trename]\n      apply SSubtyp.tinstr\n      trivial\n    case boxed =>\n      unfold tsubst_motive2 tsubst_motive3\n      repeat intro\n      simp [SType.trename]\n      apply boxed\n      aesop\n    case label =>\n      unfold tsubst_motive3\n      repeat intro\n      simp [SType.trename]\n      apply label\n      aesop\n    case xforall =>\n      unfold tsubst_motive1 tsubst_motive2 tsubst_motive3\n      repeat intro\n      simp [SType.trename]\n      apply xforall\n      { aesop }\n      { rename_i ih _ _ _ σ\n        apply ih ; try assumption\n        apply TVarSubst.ext; trivial }\n    case tforall =>\n      unfold tsubst_motive1 tsubst_motive3\n      repeat intro\n      simp [SType.trename]\n      apply tforall\n      { aesop }\n      { rename_i ih _ _ _ σ\n        apply ih ; try assumption\n        apply TVarSubst.text; trivial }\n    case cforall =>\n      unfold tsubst_motive1 tsubst_motive3\n      repeat intro\n      simp [SType.trename]\n      apply cforall\n      { apply Subbound.tsubst <;> easy }\n      { rename_i ih _ _ _ σ\n        apply ih ; try assumption\n        apply TVarSubst.cext; trivial }", "nesting_depth": 5, "transitive_dep_count": 121, "subset_aristotle": false, "category": "Type systems"}
{"id": 73, "thm_name": "Capless.SSubtyp.trename", "thm_stmt": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)", "lean_root": "capless-lean", "rel_path": "Capless/Renaming/Type/Subtyping.lean", "imports": ["import Capless.Tactics", "import Capless.Renaming.Type.Subcapturing", "import Capless.Subtyping", "import Capless.Renaming.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}], "used_local_defs": [{"name": "Capless.SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Capless.SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "Capless.SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}], "used_local_lemmas": [{"name": "Capless.Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}], "local_ctx": "import Capless.Tactics\n\nimport Capless.Subtyping\n\nimport Capless.Renaming.Basic\n\nimport Capless.Renaming.Type.Subcapturing\n\nnamespace Capless\n\ndef SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)\n\ndef SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)\n\ndef SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)", "target_theorem": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f) :=", "ground_truth_proof": ":= by\n  apply SSubtyp.rec\n    (motive_1 := fun Γ E1 E2 h => SSubtyp.trename_motive1 Γ E1 E2)\n    (motive_2 := fun Γ C1 C2 h => SSubtyp.trename_motive2 Γ C1 C2)\n    (motive_3 := fun Γ S1 S2 h => SSubtyp.trename_motive3 Γ S1 S2)\n    (t := h) (ρ := ρ)\n  case exist =>\n    unfold trename_motive1 trename_motive2\n    repeat intro\n    simp [EType.trename]\n    apply ESubtyp.exist\n    rename_i ih _ _ _ _\n    apply ih; apply TVarMap.cext; trivial\n  case type =>\n    unfold trename_motive1 trename_motive2\n    repeat intro\n    simp [EType.trename]\n    apply ESubtyp.type\n    aesop\n  case capt =>\n    unfold trename_motive2 trename_motive3\n    repeat intro\n    simp [CType.trename]\n    apply CSubtyp.capt\n    apply Subcapt.trename <;> trivial\n    aesop\n  case top =>\n    unfold trename_motive3\n    repeat intro\n    simp [SType.trename]\n    apply SSubtyp.top\n  case refl =>\n    unfold trename_motive3\n    repeat intro\n    apply refl\n  case trans =>\n    unfold trename_motive3\n    repeat intro\n    apply trans <;> aesop\n  case tvar =>\n    unfold trename_motive3\n    repeat intro\n    simp [SType.trename]\n    apply tvar\n    rename_i hb _ _ _ ρ\n    have hb1 := ρ.tmap _ _ hb\n    simp [TBinding.trename] at hb1\n    exact hb1\n  case tinstl =>\n    unfold trename_motive3\n    repeat intro\n    simp [SType.trename]\n    apply tinstl\n    rename_i hb _ _ _ ρ\n    have hb1 := ρ.tmap _ _ hb\n    simp [TBinding.trename] at hb1\n    exact hb1\n  case tinstr =>\n    unfold trename_motive3\n    repeat intro\n    simp [SType.trename]\n    apply tinstr\n    rename_i hb _ _ _ ρ\n    have hb1 := ρ.tmap _ _ hb\n    simp [TBinding.trename] at hb1\n    exact hb1\n  case boxed =>\n    unfold trename_motive2 trename_motive3\n    repeat intro\n    simp [SType.trename]\n    apply boxed\n    aesop\n  case label =>\n    unfold trename_motive3\n    repeat intro\n    simp [SType.trename]\n    apply label\n    aesop\n  case xforall =>\n    unfold trename_motive1 trename_motive3\n    repeat intro\n    simp [SType.trename]\n    apply xforall\n    aesop\n    rename_i ih2 _ _ _ _\n    apply ih2; apply TVarMap.ext; easy\n  case tforall =>\n    unfold trename_motive1 trename_motive3\n    repeat intro\n    simp [SType.trename]\n    apply tforall\n    aesop\n    rename_i ih2 _ _ _ _\n    apply ih2; apply TVarMap.text; easy\n  case cforall =>\n    unfold trename_motive1 trename_motive3\n    repeat intro\n    simp [SType.trename]\n    apply cforall\n    { apply Subbound.trename <;> easy }\n    { rename_i ih2 _ _ _ _\n      apply ih2; apply TVarMap.cext; easy }", "nesting_depth": 6, "transitive_dep_count": 45, "subset_aristotle": false, "category": "Type systems"}
{"id": 74, "thm_name": "Capless.SSubtyp.sub_dealias_cforall_inv", "thm_stmt": "theorem SSubtyp.sub_dealias_cforall_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.cforall B1 E1))\n  (h2 : SType.Dealias Γ S2 (SType.cforall B2 E2))\n  (hs : SSubtyp Γ S1 S2) :\n  Subbound Γ B2 B1 ∧ ESubtyp (Γ.cvar (CBinding.bound B2)) E1 E2", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Subtyping.lean", "imports": ["import Capless.Subtyping.Basic", "import Capless.Narrowing.Subtyping", "import Capless.Inversion.Basic", "import Capless.Narrowing", "import Capless.Subtyping", "import Capless.Store", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Exists", "module": "Init.Core"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CVarSubst.narrow", "content": "def CVarSubst.narrow\n  (hs : Subbound Γ B' B) :\n  CVarSubst\n    (Γ,c<:B)\n    FinFun.id\n    (Γ,c<:B') :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "refl", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "Context.tbound_inj", "content": "theorem Context.tbound_inj\n  (h1 : Context.TBound Γ X b1)\n  (h2 : Context.TBound Γ X b2) : b1 = b2"}, {"name": "Context.tvar_tbound_succ_inv", "content": "theorem Context.tvar_tbound_succ_inv\n  (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Context.tvar_tbound_succ_inv'", "content": "theorem Context.tvar_tbound_succ_inv'\n  (he1 : Γ0 = Γ.tvar p) (he2 : X0 = Fin.succ X)\n  (hb : Context.TBound Γ0 X0 b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Context.tight_bound_tvar_absurd", "content": "theorem Context.tight_bound_tvar_absurd\n  (ht : Context.IsTight Γ)\n  (hb : Context.TBound Γ X (TBinding.bound S)) : False"}, {"name": "Context.cvar_tbound_inv_bound", "content": "theorem Context.cvar_tbound_inv_bound\n  (hb : Context.TBound (Γ.cvar p) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = S0.cweaken"}, {"name": "Context.cvar_tbound_inv", "content": "theorem Context.cvar_tbound_inv\n  (hb : Context.TBound (Γ.cvar p) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.cvar_tbound_inv'", "content": "theorem Context.cvar_tbound_inv'\n  (he : Γ0 = Γ.cvar p)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.label_tbound_inv_bound", "content": "theorem Context.label_tbound_inv_bound\n  (hb : Context.TBound (Γ.label l) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.label_tbound_inv", "content": "theorem Context.label_tbound_inv\n  (hb : Context.TBound (Γ.label l) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.label_tbound_inv'", "content": "theorem Context.label_tbound_inv'\n  (he : Γ0 = Γ.label l)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.tinst_tbound_bound_inv", "content": "theorem Context.tinst_tbound_bound_inv\n  (hb : Context.TBound (Γ.tvar (TBinding.inst P)) X (TBinding.bound S)) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.tinst_tbound_bound_inv'", "content": "theorem Context.tinst_tbound_bound_inv'\n  (he1 : Γ0 = Γ.tvar (TBinding.inst P))\n  (he2 : b0 = TBinding.bound S)\n  (hb : Context.TBound Γ0 X b0) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.var_tbound_inv_bound", "content": "theorem Context.var_tbound_inv_bound\n  (hb : Context.TBound (Γ.var P) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.var_tbound_inv", "content": "theorem Context.var_tbound_inv\n  (hb : Context.TBound (Γ.var P) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.var_tbound_inv'", "content": "theorem Context.var_tbound_inv'\n  (he : Γ0 = Γ.var P)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "ESubtyp.trans", "content": "theorem ESubtyp.trans\n  (h1 : ESubtyp Γ E1 E2)\n  (h2 : ESubtyp Γ E2 E3) :\n  ESubtyp Γ E1 E3"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}, {"name": "ESubtyp.cnarrow", "content": "theorem ESubtyp.cnarrow\n  (h : ESubtyp (Γ,c<:B) E1 E2)\n  (hs : Subbound Γ B' B) :\n  ESubtyp (Γ,c<:B') E1 E2"}, {"name": "Subbound.trans", "content": "theorem Subbound.trans\n  (h1 : Subbound Γ B1 B2)\n  (h2 : Subbound Γ B2 B3) :\n  Subbound Γ B1 B3"}, {"name": "Subbound.refl", "content": "theorem Subbound.refl {B : CBound n k} :\n  Subbound Γ B B"}], "used_local_defs": [{"name": "Capless.SSubtyp.dealias_right_cforall.emotive", "content": "def SSubtyp.dealias_right_cforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_cforall.cmotive", "content": "def SSubtyp.dealias_right_cforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_cforall.smotive", "content": "def SSubtyp.dealias_right_cforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {B2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)),\n    ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}, {"name": "Capless.SSubtyp.dealias_cforall_inv.emotive", "content": "def SSubtyp.dealias_cforall_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_cforall_inv.cmotive", "content": "def SSubtyp.dealias_cforall_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_cforall_inv.smotive", "content": "def SSubtyp.dealias_cforall_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {B1 E1 B2 E2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.cforall B1 E1))\n    (h2 : SType.Dealias Γ S2 (SType.cforall B2 E2)),\n    Subbound Γ B2 B1 ∧ ESubtyp (Γ.cvar (CBinding.bound B2)) E1 E2"}], "used_local_lemmas": [{"name": "Capless.SSubtyp.dealias_right_cforall", "content": "theorem SSubtyp.dealias_right_cforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)) :\n  ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}, {"name": "Capless.SType.dealias_cforall_inj'", "content": "theorem SType.dealias_cforall_inj'\n  (he1 : S1 = SType.cforall B1 E1) (he2 : S2 = SType.cforall B2 E2)\n  (h1 : SType.Dealias Γ S S1)\n  (h2 : SType.Dealias Γ S S2) :\n  B1 = B2 ∧ E1 = E2"}, {"name": "Capless.SType.dealias_cforall_inj", "content": "theorem SType.dealias_cforall_inj\n  (h1 : SType.Dealias Γ S (SType.cforall B1 E1))\n  (h2 : SType.Dealias Γ S (SType.cforall B2 E2)) :\n  B1 = B2 ∧ E1 = E2"}], "local_ctx": "import Capless.Subtyping\n\nimport Capless.Store\n\nimport Capless.Inversion.Basic\n\nimport Capless.Inversion.Context\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Narrowing\n\nnamespace Capless\n\ndef SSubtyp.dealias_right_cforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_cforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_cforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {B2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)),\n    ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)\n\ntheorem SSubtyp.dealias_right_cforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)) :\n  ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)\n\ndef SSubtyp.dealias_cforall_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_cforall_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_cforall_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {B1 E1 B2 E2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.cforall B1 E1))\n    (h2 : SType.Dealias Γ S2 (SType.cforall B2 E2)),\n    Subbound Γ B2 B1 ∧ ESubtyp (Γ.cvar (CBinding.bound B2)) E1 E2", "target_theorem": "theorem SSubtyp.sub_dealias_cforall_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.cforall B1 E1))\n  (h2 : SType.Dealias Γ S2 (SType.cforall B2 E2))\n  (hs : SSubtyp Γ S1 S2) :\n  Subbound Γ B2 B1 ∧ ESubtyp (Γ.cvar (CBinding.bound B2)) E1 E2 :=", "ground_truth_proof": ":= by\n  apply SSubtyp.rec\n    (motive_1 := fun Γ E1 E2 h => SSubtyp.dealias_cforall_inv.emotive Γ E1 E2)\n    (motive_2 := fun Γ C1 C2 h => SSubtyp.dealias_cforall_inv.cmotive Γ C1 C2)\n    (motive_3 := fun Γ S1 S2 h => SSubtyp.dealias_cforall_inv.smotive Γ S1 S2)\n    (t := hs) (h1 := h1) (h2 := h2) (ht := ht)\n  case exist => aesop\n  case type => aesop\n  case capt => unfold dealias_cforall_inv.cmotive; aesop\n  case top =>\n    unfold dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd2\n    cases hd2\n  case refl =>\n    unfold dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    have h := SType.dealias_cforall_inj hd1 hd2\n    cases h; subst_vars\n    apply And.intro\n    { apply Subbound.refl }\n    { apply ESubtyp.refl }\n  case trans =>\n    unfold dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hs2 ih1 ih2 B1 E1 B2 E2 ht hd1 hd2\n    have h := SSubtyp.dealias_right_cforall hs2 ht hd2\n    have ⟨B3, E3, hd3⟩ := h\n    have ⟨he11, he12⟩ := ih1 ht hd1 hd3\n    have ⟨he21, he22⟩ := ih2 ht hd3 hd2\n    constructor\n    { apply Subbound.trans <;> easy }\n    { apply ESubtyp.trans\n      { apply ESubtyp.cnarrow <;> easy }\n      { easy } }\n  case tinstl =>\n    unfold dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n    rename_i hb1 _ _ _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_cforall_inj hd1 hd2\n    cases h\n    subst_vars\n    apply And.intro\n    { apply Subbound.refl }\n    { apply ESubtyp.refl }\n  case tinstr =>\n    unfold dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_cforall_inj hd1 hd2\n    cases h\n    subst_vars\n    apply And.intro\n    { apply Subbound.refl }\n    { apply ESubtyp.refl }\n  case tvar =>\n    unfold dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n  case boxed =>\n    unfold dealias_cforall_inv.cmotive dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case xforall =>\n    unfold dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case tforall =>\n    unfold dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case cforall =>\n    unfold dealias_cforall_inv.emotive dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    cases hd1; cases hd2\n    rename_i ih _ _\n    trivial\n  case label =>\n    unfold dealias_cforall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd", "nesting_depth": 7, "transitive_dep_count": 120, "subset_aristotle": false, "category": "Type systems"}
{"id": 75, "thm_name": "Capless.SSubtyp.sub_dealias_forall_inv", "thm_stmt": "theorem SSubtyp.sub_dealias_forall_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.forall T1 E1))\n  (h2 : SType.Dealias Γ S2 (SType.forall T2 E2))\n  (hs : SSubtyp Γ S1 S2) :\n  CSubtyp Γ T2 T1 ∧ ESubtyp (Γ.var T2) E1 E2", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Subtyping.lean", "imports": ["import Capless.Narrowing.Typing", "import Capless.Subtyping.Basic", "import Capless.Narrowing.Subtyping", "import Capless.Narrowing.TypedCont", "import Capless.Inversion.Basic", "import Capless.Narrowing", "import Capless.Subtyping", "import Capless.Store", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CVarSubst.narrow", "content": "def CVarSubst.narrow\n  (hs : Subbound Γ B' B) :\n  CVarSubst\n    (Γ,c<:B)\n    FinFun.id\n    (Γ,c<:B') :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "TVarSubst.narrow", "content": "def TVarSubst.narrow\n  (hs : SSubtyp Γ S' S) :\n  TVarSubst\n    (Γ.tvar (TBinding.bound S))\n    FinFun.id\n    (Γ.tvar (TBinding.bound S')) :="}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "VarSubst.narrow", "content": "def VarSubst.narrow\n  (hs : CSubtyp Γ T' T) :\n  VarSubst (Γ.var T) FinFun.id (Γ.var T') :="}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "refl", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "Context.tbound_inj", "content": "theorem Context.tbound_inj\n  (h1 : Context.TBound Γ X b1)\n  (h2 : Context.TBound Γ X b2) : b1 = b2"}, {"name": "Context.tvar_tbound_succ_inv", "content": "theorem Context.tvar_tbound_succ_inv\n  (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Context.tvar_tbound_succ_inv'", "content": "theorem Context.tvar_tbound_succ_inv'\n  (he1 : Γ0 = Γ.tvar p) (he2 : X0 = Fin.succ X)\n  (hb : Context.TBound Γ0 X0 b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Context.tight_bound_tvar_absurd", "content": "theorem Context.tight_bound_tvar_absurd\n  (ht : Context.IsTight Γ)\n  (hb : Context.TBound Γ X (TBinding.bound S)) : False"}, {"name": "Context.cvar_tbound_inv_bound", "content": "theorem Context.cvar_tbound_inv_bound\n  (hb : Context.TBound (Γ.cvar p) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = S0.cweaken"}, {"name": "Context.cvar_tbound_inv", "content": "theorem Context.cvar_tbound_inv\n  (hb : Context.TBound (Γ.cvar p) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.cvar_tbound_inv'", "content": "theorem Context.cvar_tbound_inv'\n  (he : Γ0 = Γ.cvar p)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.label_tbound_inv_bound", "content": "theorem Context.label_tbound_inv_bound\n  (hb : Context.TBound (Γ.label l) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.label_tbound_inv", "content": "theorem Context.label_tbound_inv\n  (hb : Context.TBound (Γ.label l) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.label_tbound_inv'", "content": "theorem Context.label_tbound_inv'\n  (he : Γ0 = Γ.label l)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.tinst_tbound_bound_inv", "content": "theorem Context.tinst_tbound_bound_inv\n  (hb : Context.TBound (Γ.tvar (TBinding.inst P)) X (TBinding.bound S)) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.tinst_tbound_bound_inv'", "content": "theorem Context.tinst_tbound_bound_inv'\n  (he1 : Γ0 = Γ.tvar (TBinding.inst P))\n  (he2 : b0 = TBinding.bound S)\n  (hb : Context.TBound Γ0 X b0) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.var_tbound_inv_bound", "content": "theorem Context.var_tbound_inv_bound\n  (hb : Context.TBound (Γ.var P) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.var_tbound_inv", "content": "theorem Context.var_tbound_inv\n  (hb : Context.TBound (Γ.var P) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.var_tbound_inv'", "content": "theorem Context.var_tbound_inv'\n  (he : Γ0 = Γ.var P)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Typed.narrow", "content": "theorem Typed.narrow\n  (h : Typed (Γ,x: T) t E Ct)\n  (hs : CSubtyp Γ T' T) :\n  Typed (Γ,x: T') t E Ct"}, {"name": "ESubtyp.narrow", "content": "theorem ESubtyp.narrow\n  (h : ESubtyp (Γ.var T) E1 E2)\n  (hs : CSubtyp Γ T' T) :\n  ESubtyp (Γ.var T') E1 E2"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "TypedCont.narrow", "content": "theorem TypedCont.narrow\n  (h : TypedCont Γ E1 cont E C0)\n  (hsub : ESubtyp Γ E2 E1) :\n  TypedCont Γ E2 cont E C0"}, {"name": "ESubtyp.trans", "content": "theorem ESubtyp.trans\n  (h1 : ESubtyp Γ E1 E2)\n  (h2 : ESubtyp Γ E2 E3) :\n  ESubtyp Γ E1 E3"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}], "used_local_defs": [{"name": "Capless.SSubtyp.dealias_right_forall.emotive", "content": "def SSubtyp.dealias_right_forall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_forall.cmotive", "content": "def SSubtyp.dealias_right_forall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_forall.smotive", "content": "def SSubtyp.dealias_right_forall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.forall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}, {"name": "Capless.SSubtyp.dealias_forall_inv.emotive", "content": "def SSubtyp.dealias_forall_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_forall_inv.cmotive", "content": "def SSubtyp.dealias_forall_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_forall_inv.smotive", "content": "def SSubtyp.dealias_forall_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 E1 T2 E2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.forall T1 E1))\n    (h2 : SType.Dealias Γ S2 (SType.forall T2 E2)),\n    CSubtyp Γ T2 T1 ∧ ESubtyp (Γ.var T2) E1 E2"}], "used_local_lemmas": [{"name": "Capless.SSubtyp.dealias_right_forall", "content": "theorem SSubtyp.dealias_right_forall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.forall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}, {"name": "Capless.SType.dealias_forall_inj'", "content": "theorem SType.dealias_forall_inj'\n  (he1 : S1 = SType.forall T1 E1) (he2 : S2 = SType.forall T2 E2)\n  (h1 : SType.Dealias Γ S S1)\n  (h2 : SType.Dealias Γ S S2) :\n  T1 = T2 ∧ E1 = E2"}, {"name": "Capless.SType.dealias_forall_inj", "content": "theorem SType.dealias_forall_inj\n  (h1 : SType.Dealias Γ S (SType.forall T1 E1))\n  (h2 : SType.Dealias Γ S (SType.forall T2 E2)) :\n  T1 = T2 ∧ E1 = E2"}], "local_ctx": "import Capless.Subtyping\n\nimport Capless.Store\n\nimport Capless.Inversion.Basic\n\nimport Capless.Inversion.Context\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Narrowing\n\nnamespace Capless\n\ndef SSubtyp.dealias_right_forall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_forall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_forall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.forall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)\n\ntheorem SSubtyp.dealias_right_forall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.forall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)\n\ndef SSubtyp.dealias_forall_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_forall_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_forall_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 E1 T2 E2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.forall T1 E1))\n    (h2 : SType.Dealias Γ S2 (SType.forall T2 E2)),\n    CSubtyp Γ T2 T1 ∧ ESubtyp (Γ.var T2) E1 E2", "target_theorem": "theorem SSubtyp.sub_dealias_forall_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.forall T1 E1))\n  (h2 : SType.Dealias Γ S2 (SType.forall T2 E2))\n  (hs : SSubtyp Γ S1 S2) :\n  CSubtyp Γ T2 T1 ∧ ESubtyp (Γ.var T2) E1 E2 :=", "ground_truth_proof": ":= by\n  apply SSubtyp.rec\n    (motive_1 := fun Γ E1 E2 h => SSubtyp.dealias_forall_inv.emotive Γ E1 E2)\n    (motive_2 := fun Γ C1 C2 h => SSubtyp.dealias_forall_inv.cmotive Γ C1 C2)\n    (motive_3 := fun Γ S1 S2 h => SSubtyp.dealias_forall_inv.smotive Γ S1 S2)\n    (t := hs) (h1 := h1) (h2 := h2) (ht := ht)\n  case exist => aesop\n  case type => aesop\n  case capt => unfold dealias_forall_inv.cmotive; aesop\n  case top =>\n    unfold dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd2\n    cases hd2\n  case refl =>\n    unfold dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    have h := SType.dealias_forall_inj hd1 hd2\n    cases h; subst_vars\n    constructor\n    { apply CSubtyp.refl }\n    { apply ESubtyp.refl }\n  case xforall =>\n    unfold dealias_forall_inv.emotive dealias_forall_inv.cmotive dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    cases hd1; cases hd2\n    aesop\n  case trans =>\n    unfold dealias_forall_inv.smotive\n    repeat intro\n    rename_i hs2 ih1 ih2 T1 E1 T2 E2 ht hd1 hd2\n    have h := SSubtyp.dealias_right_forall hs2 ht hd2\n    have ⟨T3, E3, hd3⟩ := h\n    have ⟨hc1, he1⟩ := ih1 ht hd1 hd3\n    have ⟨hc2, he2⟩ := ih2 ht hd3 hd2\n    have he1' := he1.narrow hc2\n    constructor\n    { apply CSubtyp.trans <;> trivial }\n    { apply ESubtyp.trans <;> trivial }\n  case tinstl =>\n    unfold dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n    rename_i hb1 _ _ _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_forall_inj hd1 hd2\n    cases h\n    subst_vars\n    constructor\n    { apply CSubtyp.refl }\n    { apply ESubtyp.refl }\n  case tinstr =>\n    unfold dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_forall_inj hd1 hd2\n    cases h\n    subst_vars\n    constructor\n    { apply CSubtyp.refl }\n    { apply ESubtyp.refl }\n  case tvar =>\n    unfold dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n  case boxed =>\n    unfold dealias_forall_inv.cmotive dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case label =>\n    unfold dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case tforall =>\n    unfold dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case cforall =>\n    unfold dealias_forall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd", "nesting_depth": 5, "transitive_dep_count": 128, "subset_aristotle": false, "category": "Type systems"}
{"id": 76, "thm_name": "Capless.SSubtyp.sub_dealias_tforall_inv", "thm_stmt": "theorem SSubtyp.sub_dealias_tforall_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.tforall T1 E1))\n  (h2 : SType.Dealias Γ S2 (SType.tforall T2 E2))\n  (hs : SSubtyp Γ S1 S2) :\n  SSubtyp Γ T2 T1 ∧ ESubtyp (Γ.tvar (TBinding.bound T2)) E1 E2", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Subtyping.lean", "imports": ["import Capless.Narrowing.Typing", "import Capless.Subtyping.Basic", "import Capless.Narrowing.Subtyping", "import Capless.Inversion.Basic", "import Capless.Narrowing", "import Capless.Subtyping", "import Capless.Store", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Exists", "module": "Init.Core"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarSubst.narrow", "content": "def TVarSubst.narrow\n  (hs : SSubtyp Γ S' S) :\n  TVarSubst\n    (Γ.tvar (TBinding.bound S))\n    FinFun.id\n    (Γ.tvar (TBinding.bound S')) :="}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "refl", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "Context.tbound_inj", "content": "theorem Context.tbound_inj\n  (h1 : Context.TBound Γ X b1)\n  (h2 : Context.TBound Γ X b2) : b1 = b2"}, {"name": "Context.tvar_tbound_succ_inv", "content": "theorem Context.tvar_tbound_succ_inv\n  (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Context.tvar_tbound_succ_inv'", "content": "theorem Context.tvar_tbound_succ_inv'\n  (he1 : Γ0 = Γ.tvar p) (he2 : X0 = Fin.succ X)\n  (hb : Context.TBound Γ0 X0 b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Context.tight_bound_tvar_absurd", "content": "theorem Context.tight_bound_tvar_absurd\n  (ht : Context.IsTight Γ)\n  (hb : Context.TBound Γ X (TBinding.bound S)) : False"}, {"name": "Context.cvar_tbound_inv_bound", "content": "theorem Context.cvar_tbound_inv_bound\n  (hb : Context.TBound (Γ.cvar p) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = S0.cweaken"}, {"name": "Context.cvar_tbound_inv", "content": "theorem Context.cvar_tbound_inv\n  (hb : Context.TBound (Γ.cvar p) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.cvar_tbound_inv'", "content": "theorem Context.cvar_tbound_inv'\n  (he : Γ0 = Γ.cvar p)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.label_tbound_inv_bound", "content": "theorem Context.label_tbound_inv_bound\n  (hb : Context.TBound (Γ.label l) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.label_tbound_inv", "content": "theorem Context.label_tbound_inv\n  (hb : Context.TBound (Γ.label l) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.label_tbound_inv'", "content": "theorem Context.label_tbound_inv'\n  (he : Γ0 = Γ.label l)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.tinst_tbound_bound_inv", "content": "theorem Context.tinst_tbound_bound_inv\n  (hb : Context.TBound (Γ.tvar (TBinding.inst P)) X (TBinding.bound S)) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.tinst_tbound_bound_inv'", "content": "theorem Context.tinst_tbound_bound_inv'\n  (he1 : Γ0 = Γ.tvar (TBinding.inst P))\n  (he2 : b0 = TBinding.bound S)\n  (hb : Context.TBound Γ0 X b0) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.var_tbound_inv_bound", "content": "theorem Context.var_tbound_inv_bound\n  (hb : Context.TBound (Γ.var P) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.var_tbound_inv", "content": "theorem Context.var_tbound_inv\n  (hb : Context.TBound (Γ.var P) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.var_tbound_inv'", "content": "theorem Context.var_tbound_inv'\n  (he : Γ0 = Γ.var P)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Typed.tnarrow", "content": "theorem Typed.tnarrow\n  (h : Typed (Γ,X<: S) t E Ct)\n  (hs : SSubtyp Γ S' S) :\n  Typed (Γ,X<: S') t E Ct"}, {"name": "ESubtyp.tnarrow", "content": "theorem ESubtyp.tnarrow\n  (h : ESubtyp (Γ.tvar (TBinding.bound S)) E1 E2)\n  (hs : SSubtyp Γ S' S) :\n  ESubtyp (Γ.tvar (TBinding.bound S')) E1 E2"}, {"name": "ESubtyp.trans", "content": "theorem ESubtyp.trans\n  (h1 : ESubtyp Γ E1 E2)\n  (h2 : ESubtyp Γ E2 E3) :\n  ESubtyp Γ E1 E3"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}], "used_local_defs": [{"name": "Capless.SSubtyp.dealias_right_tforall.emotive", "content": "def SSubtyp.dealias_right_tforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_tforall.cmotive", "content": "def SSubtyp.dealias_right_tforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_tforall.smotive", "content": "def SSubtyp.dealias_right_tforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}, {"name": "Capless.SSubtyp.dealias_tforall_inv.emotive", "content": "def SSubtyp.dealias_tforall_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_tforall_inv.cmotive", "content": "def SSubtyp.dealias_tforall_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_tforall_inv.smotive", "content": "def SSubtyp.dealias_tforall_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 E1 T2 E2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.tforall T1 E1))\n    (h2 : SType.Dealias Γ S2 (SType.tforall T2 E2)),\n    SSubtyp Γ T2 T1 ∧ ESubtyp (Γ.tvar (TBinding.bound T2)) E1 E2"}], "used_local_lemmas": [{"name": "Capless.SSubtyp.dealias_right_tforall", "content": "theorem SSubtyp.dealias_right_tforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}, {"name": "Capless.SType.dealias_tforall_inj'", "content": "theorem SType.dealias_tforall_inj'\n  (he1 : S1 = SType.tforall T1 E1) (he2 : S2 = SType.tforall T2 E2)\n  (h1 : SType.Dealias Γ S S1)\n  (h2 : SType.Dealias Γ S S2) :\n  T1 = T2 ∧ E1 = E2"}, {"name": "Capless.SType.dealias_tforall_inj", "content": "theorem SType.dealias_tforall_inj\n  (h1 : SType.Dealias Γ S (SType.tforall T1 E1))\n  (h2 : SType.Dealias Γ S (SType.tforall T2 E2)) :\n  T1 = T2 ∧ E1 = E2"}], "local_ctx": "import Capless.Subtyping\n\nimport Capless.Store\n\nimport Capless.Inversion.Basic\n\nimport Capless.Inversion.Context\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Narrowing\n\nnamespace Capless\n\ndef SSubtyp.dealias_right_tforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_tforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_tforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)\n\ntheorem SSubtyp.dealias_right_tforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)\n\ndef SSubtyp.dealias_tforall_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_tforall_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_tforall_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 E1 T2 E2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.tforall T1 E1))\n    (h2 : SType.Dealias Γ S2 (SType.tforall T2 E2)),\n    SSubtyp Γ T2 T1 ∧ ESubtyp (Γ.tvar (TBinding.bound T2)) E1 E2", "target_theorem": "theorem SSubtyp.sub_dealias_tforall_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.tforall T1 E1))\n  (h2 : SType.Dealias Γ S2 (SType.tforall T2 E2))\n  (hs : SSubtyp Γ S1 S2) :\n  SSubtyp Γ T2 T1 ∧ ESubtyp (Γ.tvar (TBinding.bound T2)) E1 E2 :=", "ground_truth_proof": ":= by\n  apply SSubtyp.rec\n    (motive_1 := fun Γ E1 E2 h => SSubtyp.dealias_tforall_inv.emotive Γ E1 E2)\n    (motive_2 := fun Γ C1 C2 h => SSubtyp.dealias_tforall_inv.cmotive Γ C1 C2)\n    (motive_3 := fun Γ S1 S2 h => SSubtyp.dealias_tforall_inv.smotive Γ S1 S2)\n    (t := hs) (h1 := h1) (h2 := h2) (ht := ht)\n  case exist => aesop\n  case type => aesop\n  case capt => unfold dealias_tforall_inv.cmotive; aesop\n  case top =>\n    unfold dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd2\n    cases hd2\n  case refl =>\n    unfold dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    have h := SType.dealias_tforall_inj hd1 hd2\n    cases h; subst_vars\n    constructor\n    { apply SSubtyp.refl }\n    { apply ESubtyp.refl }\n  case trans =>\n    unfold dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hs1 hs2 ih1 ih2 T1 E1 T2 E2 ht hd1 hd2\n    have h := SSubtyp.dealias_right_tforall hs2 ht hd2\n    have ⟨T3, E3, hd3⟩ := h\n    have ⟨hs1, he1⟩ := ih1 ht hd1 hd3\n    have ⟨hs2, he2⟩ := ih2 ht hd3 hd2\n    apply And.intro\n    { apply! SSubtyp.trans }\n    { apply? ESubtyp.trans\n      apply? he1.tnarrow }\n  case tvar =>\n    unfold dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n  case tinstl =>\n    unfold dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n    rename_i hb1 _ _ _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_tforall_inj hd1 hd2\n    cases h\n    subst_vars\n    constructor\n    { apply SSubtyp.refl }\n    { apply ESubtyp.refl }\n  case tinstr =>\n    unfold dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_tforall_inj hd1 hd2\n    cases h\n    subst_vars\n    constructor\n    { apply SSubtyp.refl }\n    { apply ESubtyp.refl }\n  case boxed =>\n    unfold dealias_tforall_inv.cmotive dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case label =>\n    unfold dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case xforall =>\n    unfold dealias_tforall_inv.emotive dealias_tforall_inv.cmotive dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case tforall =>\n    unfold dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    cases hd1; cases hd2\n    aesop\n  case cforall =>\n    unfold dealias_tforall_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    cases hd1", "nesting_depth": 5, "transitive_dep_count": 121, "subset_aristotle": false, "category": "Type systems"}
{"id": 77, "thm_name": "Capless.SSubtyp.sub_dealias_boxed_inv", "thm_stmt": "theorem SSubtyp.sub_dealias_boxed_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.box T1))\n  (h2 : SType.Dealias Γ S2 (SType.box T2))\n  (hs : SSubtyp Γ S1 S2) :\n  CSubtyp Γ T1 T2", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Subtyping.lean", "imports": ["import Capless.Inversion.Basic", "import Capless.Narrowing", "import Capless.Subtyping.Basic", "import Capless.Subtyping", "import Capless.Store", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "refl", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "Context.tbound_inj", "content": "theorem Context.tbound_inj\n  (h1 : Context.TBound Γ X b1)\n  (h2 : Context.TBound Γ X b2) : b1 = b2"}, {"name": "Context.tvar_tbound_succ_inv", "content": "theorem Context.tvar_tbound_succ_inv\n  (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Context.tvar_tbound_succ_inv'", "content": "theorem Context.tvar_tbound_succ_inv'\n  (he1 : Γ0 = Γ.tvar p) (he2 : X0 = Fin.succ X)\n  (hb : Context.TBound Γ0 X0 b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Context.tight_bound_tvar_absurd", "content": "theorem Context.tight_bound_tvar_absurd\n  (ht : Context.IsTight Γ)\n  (hb : Context.TBound Γ X (TBinding.bound S)) : False"}, {"name": "Context.cvar_tbound_inv_bound", "content": "theorem Context.cvar_tbound_inv_bound\n  (hb : Context.TBound (Γ.cvar p) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = S0.cweaken"}, {"name": "Context.cvar_tbound_inv", "content": "theorem Context.cvar_tbound_inv\n  (hb : Context.TBound (Γ.cvar p) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.cvar_tbound_inv'", "content": "theorem Context.cvar_tbound_inv'\n  (he : Γ0 = Γ.cvar p)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.label_tbound_inv_bound", "content": "theorem Context.label_tbound_inv_bound\n  (hb : Context.TBound (Γ.label l) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.label_tbound_inv", "content": "theorem Context.label_tbound_inv\n  (hb : Context.TBound (Γ.label l) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.label_tbound_inv'", "content": "theorem Context.label_tbound_inv'\n  (he : Γ0 = Γ.label l)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.tinst_tbound_bound_inv", "content": "theorem Context.tinst_tbound_bound_inv\n  (hb : Context.TBound (Γ.tvar (TBinding.inst P)) X (TBinding.bound S)) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.tinst_tbound_bound_inv'", "content": "theorem Context.tinst_tbound_bound_inv'\n  (he1 : Γ0 = Γ.tvar (TBinding.inst P))\n  (he2 : b0 = TBinding.bound S)\n  (hb : Context.TBound Γ0 X b0) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.var_tbound_inv_bound", "content": "theorem Context.var_tbound_inv_bound\n  (hb : Context.TBound (Γ.var P) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.var_tbound_inv", "content": "theorem Context.var_tbound_inv\n  (hb : Context.TBound (Γ.var P) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.var_tbound_inv'", "content": "theorem Context.var_tbound_inv'\n  (he : Γ0 = Γ.var P)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}], "used_local_defs": [{"name": "Capless.SSubtyp.dealias_right_boxed.emotive", "content": "def SSubtyp.dealias_right_boxed.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_boxed.cmotive", "content": "def SSubtyp.dealias_right_boxed.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_boxed.smotive", "content": "def SSubtyp.dealias_right_boxed.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.box T2)),\n    ∃ T1, SType.Dealias Γ S1 (SType.box T1)"}, {"name": "Capless.SSubtyp.dealias_boxed_inv.emotive", "content": "def SSubtyp.dealias_boxed_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_boxed_inv.cmotive", "content": "def SSubtyp.dealias_boxed_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_boxed_inv.smotive", "content": "def SSubtyp.dealias_boxed_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 T2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.box T1))\n    (h2 : SType.Dealias Γ S2 (SType.box T2)),\n    CSubtyp Γ T1 T2"}], "used_local_lemmas": [{"name": "Capless.SSubtyp.dealias_right_boxed", "content": "theorem SSubtyp.dealias_right_boxed\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.box T2)) :\n  ∃ T1, SType.Dealias Γ S1 (SType.box T1)"}, {"name": "Capless.SType.dealias_boxed_inj'", "content": "theorem SType.dealias_boxed_inj'\n  (he1 : S1 = SType.box T1) (he2 : S2 = SType.box T2)\n  (h1 : SType.Dealias Γ S S1)\n  (h2 : SType.Dealias Γ S S2) :\n  T1 = T2"}, {"name": "Capless.SType.dealias_boxed_inj", "content": "theorem SType.dealias_boxed_inj\n  (h1 : SType.Dealias Γ S (SType.box T1))\n  (h2 : SType.Dealias Γ S (SType.box T2)) :\n  T1 = T2"}], "local_ctx": "import Capless.Subtyping\n\nimport Capless.Store\n\nimport Capless.Inversion.Basic\n\nimport Capless.Inversion.Context\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Narrowing\n\nnamespace Capless\n\ndef SSubtyp.dealias_right_boxed.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_boxed.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_boxed.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.box T2)),\n    ∃ T1, SType.Dealias Γ S1 (SType.box T1)\n\ntheorem SSubtyp.dealias_right_boxed\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.box T2)) :\n  ∃ T1, SType.Dealias Γ S1 (SType.box T1)\n\ndef SSubtyp.dealias_boxed_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_boxed_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_boxed_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 T2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.box T1))\n    (h2 : SType.Dealias Γ S2 (SType.box T2)),\n    CSubtyp Γ T1 T2", "target_theorem": "theorem SSubtyp.sub_dealias_boxed_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.box T1))\n  (h2 : SType.Dealias Γ S2 (SType.box T2))\n  (hs : SSubtyp Γ S1 S2) :\n  CSubtyp Γ T1 T2 :=", "ground_truth_proof": ":= by\n  apply SSubtyp.rec\n    (motive_1 := fun Γ E1 E2 _ => SSubtyp.dealias_boxed_inv.emotive Γ E1 E2)\n    (motive_2 := fun Γ C1 C2 _ => SSubtyp.dealias_boxed_inv.cmotive Γ C1 C2)\n    (motive_3 := fun Γ S1 S2 _ => SSubtyp.dealias_boxed_inv.smotive Γ S1 S2)\n    (t := hs) (h1 := h1) (h2 := h2) (ht := ht)\n  case exist => aesop\n  case type => aesop\n  case capt => unfold dealias_boxed_inv.cmotive; aesop\n  case top =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd2\n    cases hd2\n  case refl =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    have h := SType.dealias_boxed_inj hd1 hd2\n    cases h\n    apply CSubtyp.refl\n  case trans =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hs2 ih1 ih2 T1 T2 ht hd1 hd2\n    have h := SSubtyp.dealias_right_boxed hs2 ht hd2\n    have ⟨T3, hd3⟩ := h\n    have hc1 := ih1 ht hd1 hd3\n    have hc2 := ih2 ht hd3 hd2\n    apply CSubtyp.trans <;> trivial\n  case tinstl =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n    rename_i hb1 _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_boxed_inj hd1 hd2\n    cases h\n    apply CSubtyp.refl\n  case tinstr =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_boxed_inj hd1 hd2\n    cases h\n    apply CSubtyp.refl\n  case tvar =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n  case boxed =>\n    unfold dealias_boxed_inv.cmotive dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    cases hd1; cases hd2\n    rename_i ih _ _\n    trivial\n  case xforall =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case tforall =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case cforall =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case label =>\n    unfold dealias_boxed_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd", "nesting_depth": 7, "transitive_dep_count": 109, "subset_aristotle": false, "category": "Type systems"}
{"id": 78, "thm_name": "Capless.SSubtyp.sub_dealias_label_inv", "thm_stmt": "theorem SSubtyp.sub_dealias_label_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.label T1))\n  (h2 : SType.Dealias Γ S2 (SType.label T2))\n  (hs : SSubtyp Γ S1 S2) :\n  SSubtyp Γ T2 T1", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Subtyping.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Subtyping.Basic", "import Capless.Inversion.Basic", "import Capless.Narrowing", "import Capless.Subtyping", "import Capless.Store", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Exists", "module": "Init.Core"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "refl", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}, {"name": "Context.tbound_inj", "content": "theorem Context.tbound_inj\n  (h1 : Context.TBound Γ X b1)\n  (h2 : Context.TBound Γ X b2) : b1 = b2"}, {"name": "Context.tvar_tbound_succ_inv", "content": "theorem Context.tvar_tbound_succ_inv\n  (hb : Context.TBound (Γ.tvar p) (Fin.succ X) b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Context.tvar_tbound_succ_inv'", "content": "theorem Context.tvar_tbound_succ_inv'\n  (he1 : Γ0 = Γ.tvar p) (he2 : X0 = Fin.succ X)\n  (hb : Context.TBound Γ0 X0 b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.tweaken"}, {"name": "Subcapt.refl", "content": "theorem Subcapt.refl :\n  Subcapt Γ C C"}, {"name": "Subbound.refl", "content": "theorem Subbound.refl {B : CBound n k} :\n  Subbound Γ B B"}, {"name": "Context.tight_bound_tvar_absurd", "content": "theorem Context.tight_bound_tvar_absurd\n  (ht : Context.IsTight Γ)\n  (hb : Context.TBound Γ X (TBinding.bound S)) : False"}, {"name": "Context.cvar_tbound_inv_bound", "content": "theorem Context.cvar_tbound_inv_bound\n  (hb : Context.TBound (Γ.cvar p) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = S0.cweaken"}, {"name": "Context.cvar_tbound_inv", "content": "theorem Context.cvar_tbound_inv\n  (hb : Context.TBound (Γ.cvar p) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.cvar_tbound_inv'", "content": "theorem Context.cvar_tbound_inv'\n  (he : Γ0 = Γ.cvar p)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.label_tbound_inv_bound", "content": "theorem Context.label_tbound_inv_bound\n  (hb : Context.TBound (Γ.label l) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.label_tbound_inv", "content": "theorem Context.label_tbound_inv\n  (hb : Context.TBound (Γ.label l) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.label_tbound_inv'", "content": "theorem Context.label_tbound_inv'\n  (he : Γ0 = Γ.label l)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.tinst_tbound_bound_inv", "content": "theorem Context.tinst_tbound_bound_inv\n  (hb : Context.TBound (Γ.tvar (TBinding.inst P)) X (TBinding.bound S)) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.tinst_tbound_bound_inv'", "content": "theorem Context.tinst_tbound_bound_inv'\n  (he1 : Γ0 = Γ.tvar (TBinding.inst P))\n  (he2 : b0 = TBinding.bound S)\n  (hb : Context.TBound Γ0 X b0) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Context.var_tbound_inv_bound", "content": "theorem Context.var_tbound_inv_bound\n  (hb : Context.TBound (Γ.var P) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Context.var_tbound_inv", "content": "theorem Context.var_tbound_inv\n  (hb : Context.TBound (Γ.var P) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Context.var_tbound_inv'", "content": "theorem Context.var_tbound_inv'\n  (he : Γ0 = Γ.var P)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}], "used_local_defs": [{"name": "Capless.SSubtyp.dealias_right_label.emotive", "content": "def SSubtyp.dealias_right_label.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_label.cmotive", "content": "def SSubtyp.dealias_right_label.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_right_label.smotive", "content": "def SSubtyp.dealias_right_label.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.label T2)),\n    ∃ T1, SType.Dealias Γ S1 (SType.label T1)"}, {"name": "Capless.SSubtyp.dealias_label_inv.emotive", "content": "def SSubtyp.dealias_label_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_label_inv.cmotive", "content": "def SSubtyp.dealias_label_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Capless.SSubtyp.dealias_label_inv.smotive", "content": "def SSubtyp.dealias_label_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 T2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.label T1))\n    (h2 : SType.Dealias Γ S2 (SType.label T2)),\n    SSubtyp Γ T2 T1"}], "used_local_lemmas": [{"name": "Capless.SSubtyp.dealias_right_label", "content": "theorem SSubtyp.dealias_right_label\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.label T2)) :\n  ∃ T1, SType.Dealias Γ S1 (SType.label T1)"}, {"name": "Capless.SType.dealias_label_inj'", "content": "theorem SType.dealias_label_inj'\n  (he1 : S1 = SType.label T1) (he2 : S2 = SType.label T2)\n  (h1 : SType.Dealias Γ S S1)\n  (h2 : SType.Dealias Γ S S2) :\n  T1 = T2"}, {"name": "Capless.SType.dealias_label_inj", "content": "theorem SType.dealias_label_inj\n  (h1 : SType.Dealias Γ S (SType.label T1))\n  (h2 : SType.Dealias Γ S (SType.label T2)) :\n  T1 = T2"}], "local_ctx": "import Capless.Subtyping\n\nimport Capless.Store\n\nimport Capless.Inversion.Basic\n\nimport Capless.Inversion.Context\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Narrowing\n\nnamespace Capless\n\ndef SSubtyp.dealias_right_label.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_label.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_right_label.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.label T2)),\n    ∃ T1, SType.Dealias Γ S1 (SType.label T1)\n\ntheorem SSubtyp.dealias_right_label\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.label T2)) :\n  ∃ T1, SType.Dealias Γ S1 (SType.label T1)\n\ndef SSubtyp.dealias_label_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_label_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True\n\ndef SSubtyp.dealias_label_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 T2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.label T1))\n    (h2 : SType.Dealias Γ S2 (SType.label T2)),\n    SSubtyp Γ T2 T1", "target_theorem": "theorem SSubtyp.sub_dealias_label_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.label T1))\n  (h2 : SType.Dealias Γ S2 (SType.label T2))\n  (hs : SSubtyp Γ S1 S2) :\n  SSubtyp Γ T2 T1 :=", "ground_truth_proof": ":= by\n  apply SSubtyp.rec\n    (motive_1 := fun Γ E1 E2 _ => SSubtyp.dealias_label_inv.emotive Γ E1 E2)\n    (motive_2 := fun Γ C1 C2 _ => SSubtyp.dealias_label_inv.cmotive Γ C1 C2)\n    (motive_3 := fun Γ S1 S2 _ => SSubtyp.dealias_label_inv.smotive Γ S1 S2)\n    (t := hs) (h1 := h1) (h2 := h2) (ht := ht)\n  case exist => aesop\n  case type => aesop\n  case capt => unfold dealias_label_inv.cmotive; aesop\n  case top =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hd2\n    cases hd2\n  case refl =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    have h := SType.dealias_label_inj hd1 hd2\n    cases h\n    apply SSubtyp.refl\n  case trans =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hs2 ih1 ih2 T1 T2 ht hd1 hd2\n    have h := SSubtyp.dealias_right_label hs2 ht hd2\n    have ⟨T3, hd3⟩ := h\n    have hs1 := ih1 ht hd1 hd3\n    have hs2 := ih2 ht hd3 hd2\n    apply SSubtyp.trans <;> trivial\n  case tinstl =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n    rename_i hb1 _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_label_inj hd1 hd2\n    cases h\n    apply SSubtyp.refl\n  case tinstr =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n    rename_i hd1 hd2\n    have h := SType.dealias_label_inj hd1 hd2\n    cases h\n    apply SSubtyp.refl\n  case tvar =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hd _\n    cases hd\n    rename_i hb1 _ _ _ _ _ hb2 _\n    have h := Context.tbound_inj hb1 hb2\n    cases h\n  case boxed =>\n    unfold dealias_label_inv.cmotive dealias_label_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case xforall =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case tforall =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case cforall =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hd\n    cases hd\n  case label =>\n    unfold dealias_label_inv.smotive\n    repeat intro\n    rename_i hd1 hd2\n    cases hd1; cases hd2\n    rename_i ih _ _\n    trivial", "nesting_depth": 5, "transitive_dep_count": 112, "subset_aristotle": false, "category": "Type systems"}
{"id": 79, "thm_name": "Capless.progress", "thm_stmt": "theorem progress\n  (ht : TypedState state Γ E) :\n  Progress state", "lean_root": "capless-lean", "rel_path": "Capless/Soundness/Progress.lean", "imports": ["import Capless.Inversion.Context", "import Capless.Weakening.IsValue", "import Mathlib.Data.Fin.Basic", "import Capless.WellScoped.Basic", "import Capless.Inversion.Subtyping", "import Capless.Inversion.Lookup", "import Capless.Inversion.Typing", "import Capless.Store", "import Capless.Reduction", "import Capless.Narrowing.TypedCont"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "abs", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "Fin.elim0", "module": "Init.Data.Fin.Basic"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Or.inl", "module": "Init.Prelude"}, {"name": "Or.inr", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "TypedStore", "content": "inductive TypedStore : Store n m k -> Context n m k -> Prop where\n| empty : TypedStore Store.empty Context.empty\n| val :\n  TypedStore σ Γ ->\n  Typed Γ t (EType.type E) Ct ->\n  (hv : t.IsValue) ->\n  TypedStore (Store.val σ t hv) (Γ.var E)\n| tval :\n  TypedStore σ Γ ->\n  TypedStore (Store.tval σ S) (Γ.tvar (TBinding.inst S))\n| cval :\n  TypedStore σ Γ ->\n  TypedStore (Store.cval σ C) (Γ.cvar (CBinding.inst C))\n| label :\n  TypedStore σ Γ ->\n  TypedStore (Store.label σ S) (Γ.label S)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Reduce", "content": "inductive Reduce : State n m k -> State n' m' k' -> Prop where\n| apply {σ : Store n m k} :\n  σ.Bound x (Term.lam T t) ->\n  Reduce ⟨σ | cont | Term.app x y⟩ ⟨σ | cont | t.open y⟩\n| tapply {σ : Store n m k} :\n  σ.Bound x (Term.tlam S t) ->\n  Reduce ⟨σ | cont | Term.tapp x X⟩ ⟨σ | cont | t.topen X⟩\n| capply {σ : Store n m k} :\n  σ.Bound x (Term.clam B t) ->\n  Reduce ⟨σ | cont | Term.capp x c⟩ ⟨σ | cont | t.copen c⟩\n| enter :\n  Reduce\n    ⟨σ | cont | boundary:S in t⟩\n    ⟨(σ.label S).cval {x=0} | cont.weaken.cweaken.scope 0 | t⟩\n| leave_var :\n  Reduce\n    ⟨σ | cont.scope x | Term.var y⟩\n    ⟨σ | cont | Term.var y⟩\n| leave_val {v : Term n m k} :\n  (hv : Term.IsValue v) ->\n  Reduce\n    ⟨σ | cont.scope x | v⟩\n    ⟨σ | cont | v⟩\n| invoke {σ : Store n m k} {cont : Cont n m k} :\n  σ.LBound x S ->\n  cont.HasLabel x tail ->\n  Reduce\n    ⟨σ | cont | Term.invoke x y⟩\n    ⟨σ | tail | Term.var y⟩\n| push :\n  Reduce\n    ⟨σ | cont | Term.letin t u⟩\n    ⟨σ | Cont.cons u cont | t⟩\n| push_ex :\n  Reduce\n    ⟨σ | cont | Term.letex t u⟩\n    ⟨σ | Cont.conse u cont | t⟩\n| rename :\n  Reduce\n    ⟨σ | Cont.cons u cont | Term.var x⟩\n    ⟨σ | cont | u.open x⟩\n| lift :\n  (hv : Term.IsValue v) ->\n  Reduce\n    ⟨σ | Cont.cons u cont | v⟩\n    ⟨σ.val v hv | cont.weaken | u⟩\n| lift_ex :\n  Reduce\n    ⟨σ | Cont.conse u cont | Term.pack C x⟩\n    ⟨σ.cval C | cont.cweaken | u.open x⟩\n| tlift :\n  Reduce\n    ⟨σ | cont | Term.bindt S t⟩\n    ⟨σ.tval S | cont.tweaken | t⟩\n| clift :\n  Reduce\n    ⟨σ | cont | Term.bindc C t⟩\n    ⟨σ.cval C | cont.cweaken | t⟩"}, {"name": "infix:30 \" ", "content": "infix:30 \" "}, {"name": "TypedState", "content": "inductive TypedState : State n m k -> Context n m k -> EType n m k -> Prop where\n| mk :\n  TypedStore σ Γ ->\n  Typed Γ t E Ct ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedState (State.mk σ cont t) Γ E'"}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "CType.IsValue", "content": "inductive CType.IsValue : CType n m k -> Prop where\n| capt :\n  S.IsValue ->\n  CType.IsValue (S^C)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "SSubtyp.dealias_right_cforall.cmotive", "content": "def SSubtyp.dealias_right_cforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_cforall.smotive", "content": "def SSubtyp.dealias_right_cforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {B2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)),\n    ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "SSubtyp.dealias_right_cforall.emotive", "content": "def SSubtyp.dealias_right_cforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_tforall.smotive", "content": "def SSubtyp.dealias_right_tforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}, {"name": "SSubtyp.dealias_right_tforall.emotive", "content": "def SSubtyp.dealias_right_tforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_tforall.cmotive", "content": "def SSubtyp.dealias_right_tforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "SSubtyp.dealias_right_boxed.smotive", "content": "def SSubtyp.dealias_right_boxed.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.box T2)),\n    ∃ T1, SType.Dealias Γ S1 (SType.box T1)"}, {"name": "SSubtyp.dealias_right_boxed.emotive", "content": "def SSubtyp.dealias_right_boxed.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_boxed.cmotive", "content": "def SSubtyp.dealias_right_boxed.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.emotive", "content": "def SSubtyp.dealias_right_forall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.cmotive", "content": "def SSubtyp.dealias_right_forall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.smotive", "content": "def SSubtyp.dealias_right_forall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.forall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}, {"name": "SSubtyp.dealias_right_label.smotive", "content": "def SSubtyp.dealias_right_label.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.label T2)),\n    ∃ T1, SType.Dealias Γ S1 (SType.label T1)"}, {"name": "SSubtyp.dealias_right_label.cmotive", "content": "def SSubtyp.dealias_right_label.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_label.emotive", "content": "def SSubtyp.dealias_right_label.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "Store.lookup_inv_bound", "content": "def Store.lookup_inv_bound\n  (hl : Store.Bound σ x v)\n  (ht : TypedStore σ Γ)\n  (hb : Context.Bound Γ x T) :\n  ∃ Cv, Typed Γ v (EType.type T) Cv :="}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "VarMap.lweaken", "content": "def VarMap.lweaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.label S) :="}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "Subcapt.tweaken", "content": "def Subcapt.tweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.tvar b) ⊢ C1 <:c C2 :="}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Term.IsValue.weaken", "content": "theorem Term.IsValue.weaken\n  (hv : Term.IsValue t) :\n  Term.IsValue t.weaken"}, {"name": "Term.IsValue.tweaken", "content": "theorem Term.IsValue.tweaken\n  (hv : Term.IsValue t) :\n  Term.IsValue t.tweaken"}, {"name": "Term.IsValue.cweaken", "content": "theorem Term.IsValue.cweaken\n  (hv : Term.IsValue t) :\n  Term.IsValue t.cweaken"}, {"name": "Typed.label_inv", "content": "theorem Typed.label_inv\n  (ht : Typed Γ (Term.var x) (EType.type T) Ct) (hb : Γ.LBound x S1) :\n  ∃ S0, Γ.LBound x S0 ∧ (Γ ⊢ (Label[S0]^{x=x}) <: T)"}, {"name": "Typed.label_inv'", "content": "theorem Typed.label_inv'\n  (he1 : t0 = Term.var x)\n  (he2 : E0 = EType.type T)\n  (ht : Typed Γ t0 E0 Ct) (hb : Γ.LBound x S1) :\n  ∃ S0, Γ.LBound x S0 ∧ (Γ ⊢ (Label[S0]^{x=x}) <: T)"}, {"name": "SSubtyp.dealias_right_cforall", "content": "theorem SSubtyp.dealias_right_cforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)) :\n  ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}, {"name": "SSubtyp.dealias_right_tforall", "content": "theorem SSubtyp.dealias_right_tforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}, {"name": "Context.lbound_inj", "content": "theorem Context.lbound_inj\n  (hb1 : Context.LBound Γ x S1)\n  (hb2 : Context.LBound Γ x S2) : S1 = S2"}, {"name": "Context.label_lbound_succ_inv", "content": "theorem Context.label_lbound_succ_inv\n  (hb : Context.LBound (Γ.label l) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.label_lbound_succ_inv'", "content": "theorem Context.label_lbound_succ_inv'\n  (he1 : Γ0 = Γ.label l) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.var_lbound_succ_inv", "content": "theorem Context.var_lbound_succ_inv\n  (hb : Context.LBound (Γ.var T) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.var_lbound_succ_inv'", "content": "theorem Context.var_lbound_succ_inv'\n  (he1 : Γ0 = Γ.var T) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "SSubtyp.dealias_right_boxed", "content": "theorem SSubtyp.dealias_right_boxed\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.box T2)) :\n  ∃ T1, SType.Dealias Γ S1 (SType.box T1)"}, {"name": "Store.bound_label", "content": "theorem Store.bound_label\n  (hl : Store.LBound σ x S)\n  (ht : TypedStore σ Γ) :\n  Γ.LBound x S"}, {"name": "SSubtyp.dealias_right_forall", "content": "theorem SSubtyp.dealias_right_forall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.forall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}, {"name": "TypedStore.is_tight", "content": "theorem TypedStore.is_tight\n  (h : TypedStore σ Γ) :\n  Γ.IsTight"}, {"name": "SSubtyp.dealias_right_label", "content": "theorem SSubtyp.dealias_right_label\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.label T2)) :\n  ∃ T1, SType.Dealias Γ S1 (SType.label T1)"}, {"name": "Store.lookup_inv_typing_alt", "content": "theorem Store.lookup_inv_typing_alt\n  (hl : Store.Bound σ x v)\n  (ht : TypedStore σ Γ)\n  (hx : Typed Γ (Term.var x) (EType.type (S^C)) Cx) :\n  ∃ C0 Cv0, Typed Γ v (EType.type (S^C0)) Cv0"}, {"name": "Store.lookup_inv_typing", "content": "theorem Store.lookup_inv_typing\n  (hl : Store.Bound σ x v)\n  (ht : TypedStore σ Γ)\n  (hx : Typed Γ (Term.var x) (EType.type (S^C)) Cx) :\n  ∃ S0 C0 Cv0,\n    Typed Γ v (EType.type (S0^C0)) Cv0 ∧\n    Γ.Bound x (S0^C0) ∧\n    (Γ ⊢ (S0^{x=x}) <: (S^C))"}, {"name": "Store.bound_type", "content": "theorem Store.bound_type\n  (hl : Store.Bound σ x v)\n  (ht : TypedStore σ Γ) :\n  ∃ T0, Context.Bound Γ x T0"}, {"name": "Typed.cforall_inv", "content": "theorem Typed.cforall_inv {v : Term n m k}\n  (hg : Γ.IsTight)\n  (hv : v.IsValue)\n  (ht : Typed Γ v (EType.type (CType.capt Cv (SType.cforall B E))) Ct) :\n  ∃ B0 t, v = Term.clam B0 t"}, {"name": "Typed.cforall_inv'", "content": "theorem Typed.cforall_inv' {v : Term n m k}\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.cforall B E))\n  (he : E0 = EType.type (CType.capt Cv S0))\n  (hv : v.IsValue)\n  (ht : Typed Γ v E0 Ct) :\n  ∃ B0 t, v = Term.clam B0 t"}, {"name": "WellScoped.subcapt", "content": "theorem WellScoped.subcapt\n  (hsc : WellScoped Γ cont C)\n  (hs : Γ ⊢ C' <:c C) :\n  WellScoped Γ cont C'"}, {"name": "WellScoped.subset", "content": "theorem WellScoped.subset\n  (hsc : WellScoped Γ cont C)\n  (hs : C' ⊆ C) :\n  WellScoped Γ cont C'"}, {"name": "Typed.forall_inv", "content": "theorem Typed.forall_inv {v : Term n m k}\n  (hg : Γ.IsTight)\n  (hv : v.IsValue)\n  (ht : Typed Γ v (EType.type (CType.capt Cv (SType.forall T E))) Ct) :\n  ∃ T0 t, v = Term.lam T0 t"}, {"name": "Typed.forall_inv'", "content": "theorem Typed.forall_inv' {v : Term n m k}\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.forall T E))\n  (he : E0 = EType.type (CType.capt Cv S0))\n  (hv : v.IsValue)\n  (ht : Typed Γ v E0 Ct) :\n  ∃ T0 t, v = Term.lam T0 t"}, {"name": "Typed.tforall_inv", "content": "theorem Typed.tforall_inv {v : Term n m k}\n  (hg : Γ.IsTight)\n  (hv : v.IsValue)\n  (ht : Typed Γ v (EType.type (CType.capt Cv (SType.tforall X E))) Ct) :\n  ∃ X t, v = Term.tlam X t"}, {"name": "Typed.tforall_inv'", "content": "theorem Typed.tforall_inv' {v : Term n m k}\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.tforall X E))\n  (he : E0 = EType.type (CType.capt Cv S0))\n  (hv : v.IsValue)\n  (ht : Typed Γ v E0 Ct) :\n  ∃ X t, v = Term.tlam X t"}, {"name": "WellScoped.label_inv", "content": "theorem WellScoped.label_inv\n  (hsc : WellScoped Γ cont {x=x})\n  (hbl : Γ.LBound x S) :\n  ∃ tail, cont.HasLabel x tail"}, {"name": "TypedCont.narrow", "content": "theorem TypedCont.narrow\n  (h : TypedCont Γ E1 cont E C0)\n  (hsub : ESubtyp Γ E2 E1) :\n  TypedCont Γ E2 cont E C0"}], "used_local_defs": [{"name": "Capless.Progress", "content": "inductive Progress : State n m k -> Prop where\n| halt_var :\n  Progress ⟨σ, Cont.none, Term.var x⟩\n| halt_value {t : Term n m k} :\n  t.IsValue ->\n  Progress ⟨σ, Cont.none, t⟩\n| step :\n  Reduce state state' ->\n  Progress state"}], "used_local_lemmas": [{"name": "Capless.Store.lookup_exists", "content": "theorem Store.lookup_exists {σ : Store n m k} {x : Fin n} :\n  (∃ v, Store.Bound σ x v ∧ v.IsValue) ∨ (∃ S, Store.LBound σ x S)"}, {"name": "Capless.Store.val_lookup_exists", "content": "theorem Store.val_lookup_exists {σ : Store n m k} {x : Fin n}\n  (hs : TypedStore σ Γ) (hx : Typed Γ (Term.var x) (EType.type T) Cx)\n  (hvt : T.IsValue) :\n  ∃ v, Store.Bound σ x v ∧ v.IsValue"}, {"name": "Capless.Store.value_typing_label_absurd'", "content": "theorem Store.value_typing_label_absurd'\n  (hg : Γ.IsTight)\n  (he : E0 = EType.type (S0^C))\n  (hd : SType.Dealias Γ S0 (Label[S]))\n  (ht : Typed Γ v E0 Cv)\n  (hv : v.IsValue) : False"}, {"name": "Capless.Store.value_typing_label_absurd", "content": "theorem Store.value_typing_label_absurd\n  (hg : Γ.IsTight)\n  (ht : Typed Γ v (EType.type (Label[S]^C)) Cv)\n  (hv : v.IsValue) : False"}, {"name": "Capless.Store.label_lookup_exists", "content": "theorem Store.label_lookup_exists {σ : Store n m k} {x : Fin n}\n  (hs : TypedStore σ Γ)\n  (hx : Typed Γ (Term.var x) (EType.type (Label[S]^C)) Cx) :\n  ∃ S0, Store.LBound σ x S0"}], "local_ctx": "import Mathlib.Data.Fin.Basic\n\nimport Capless.Reduction\n\nimport Capless.Narrowing.TypedCont\n\nimport Capless.Inversion.Lookup\n\nimport Capless.Inversion.Typing\n\nimport Capless.Weakening.IsValue\n\nimport Capless.WellScoped.Basic\n\nnamespace Capless\n\ninductive Progress : State n m k -> Prop where\n| halt_var :\n  Progress ⟨σ, Cont.none, Term.var x⟩\n| halt_value {t : Term n m k} :\n  t.IsValue ->\n  Progress ⟨σ, Cont.none, t⟩\n| step :\n  Reduce state state' ->\n  Progress state", "target_theorem": "theorem progress\n  (ht : TypedState state Γ E) :\n  Progress state :=", "ground_truth_proof": ":= by\n  cases ht\n  case mk hs ht hsc hc =>\n    induction ht\n    case var =>\n      cases hc <;> aesop\n    case label =>\n      cases hc <;> aesop\n    case pack =>\n      cases hc <;> aesop\n    case sub hsub ih _ _ _ =>\n      apply ih <;> try easy\n      apply WellScoped.subcapt; easy; easy\n      apply! TypedCont.narrow\n    case abs => cases hc <;> aesop\n    case tabs => cases hc <;> aesop\n    case cabs => cases hc <;> aesop\n    case app =>\n      rename_i x _ _ _ _ hx _ _ _ σ _ _\n      have hg := TypedStore.is_tight hs\n      have ⟨v0, hb0, hv0⟩ := Store.val_lookup_exists (σ := σ) (x := x) hs hx (by aesop)\n      have ⟨Cv, Cv0, htv⟩ := Store.lookup_inv_typing_alt hb0 hs hx\n      have ⟨U0, t0, he⟩ := Typed.forall_inv hg hv0 htv\n      aesop\n    case tapp x _ _ _ hx _ σ _ _ =>\n      have hg := TypedStore.is_tight hs\n      have ⟨v0, hb0, hv0⟩ := Store.val_lookup_exists (σ := σ) (x := x) hs hx (by aesop)\n      have ⟨Cv, Cv0, htv⟩ := Store.lookup_inv_typing_alt hb0 hs hx\n      have ⟨U0, t0, he⟩ := Typed.tforall_inv hg hv0 htv\n      aesop\n    case capp x _ _ _ hx _ σ _ _ =>\n      have hg := TypedStore.is_tight hs\n      have ⟨v0, hb0, hv0⟩ := Store.val_lookup_exists (σ := σ) (x := x) hs hx (by aesop)\n      have ⟨Cv, Ct0, htv⟩ := Store.lookup_inv_typing_alt hb0 hs hx\n      have ⟨t0, he⟩ := Typed.cforall_inv hg hv0 htv\n      aesop\n    case letin => aesop\n    case letex => aesop\n    case bindt => aesop\n    case bindc => aesop\n    case invoke hx hy _ _ σ cont Ct =>\n      cases hsc; rename_i hsc _\n      have hg := TypedStore.is_tight hs\n      have ⟨S0, hl⟩ := Store.label_lookup_exists hs hx\n      have hl := Store.bound_label hl hs\n      have ⟨_, hsl⟩ := WellScoped.label_inv hsc hl\n      aesop\n    case boundary => aesop", "nesting_depth": 8, "transitive_dep_count": 164, "subset_aristotle": false, "category": "Type systems"}
{"id": 80, "thm_name": "Capless.TypedCont.lweaken", "thm_stmt": "theorem TypedCont.lweaken\n  (h : TypedCont Γ E cont E' Ct) :\n  TypedCont (Γ.label S) E.weaken cont.weaken E'.weaken Ct.weaken", "lean_root": "capless-lean", "rel_path": "Capless/Weakening/TypedCont/Term.lean", "imports": ["import Capless.Type.Basic", "import Capless.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.CaptureSet", "import Capless.Weakening.Subcapturing", "import Capless.Store"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "VarMap.lweaken_ext", "content": "def VarMap.lweaken_ext {Γ : Context n m k} :\n  VarMap\n    (Γ.var T)\n    FinFun.weaken.ext\n    ((Γ.label P).var T.weaken) :="}, {"name": "VarMap.lweaken", "content": "def VarMap.lweaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.label S) :="}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "VarMap.lweaken_cext_ext", "content": "def VarMap.lweaken_cext_ext {Γ : Context n m k} :\n  VarMap\n    ((Γ.cvar (CBinding.bound b)).var T)\n    FinFun.weaken.ext\n    (((Γ.label P).cvar (CBinding.bound b.weaken)).var T.weaken) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "VarMap.weaken_ext", "content": "def VarMap.weaken_ext {Γ : Context n m k} :\n  VarMap\n    (Γ.var T)\n    FinFun.weaken.ext\n    ((Γ.var P).var T.weaken) :="}, {"name": "CVarMap.weaken_ext", "content": "def CVarMap.weaken_ext {Γ : Context n m k} :\n  CVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.cvar b).var T.cweaken) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken_cext_ext", "content": "def CVarMap.weaken_cext_ext {Γ : Context n m k} :\n  CVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken.ext\n    (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarMap.weaken_cext_ext", "content": "def VarMap.weaken_cext_ext {Γ : Context n m k} :\n  VarMap\n    ((Γ.cvar (CBinding.bound b)).var T)\n    FinFun.weaken.ext\n    (((Γ.var P).cvar (CBinding.bound b.weaken)).var T.weaken) :="}, {"name": "TVarMap.weaken_ext", "content": "def TVarMap.weaken_ext {Γ : Context n m k} :\n  TVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.tvar b).var T.tweaken) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TVarMap.weaken_cext_ext", "content": "def TVarMap.weaken_cext_ext {Γ : Context n m k} :\n  TVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken\n    (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "FinFun.comp_weaken", "content": "theorem FinFun.comp_weaken {f : FinFun n n'} :\n  weaken ∘ f = f.ext ∘ weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "CaptureSet.rename_rename", "content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n  (C.rename f).rename g = C.rename (g ∘ f)"}, {"name": "EType.rename_rename", "content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (E.rename f).rename g = E.rename (g ∘ f)"}, {"name": "CType.rename_rename", "content": "theorem CType.rename_rename (T : CType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (T.rename f).rename g = T.rename (g ∘ f)"}, {"name": "SType.rename_rename", "content": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f)"}, {"name": "CBound.rename_rename", "content": "theorem CBound.rename_rename {b : CBound n k} :\n  (b.rename f).rename g = b.rename (g ∘ f)"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "Typed.weaken_ext", "content": "theorem Typed.weaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.var P).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "Typed.weaken_cext_ext", "content": "theorem Typed.weaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.var P).cvar (CBinding.bound B.weaken)).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "ESubtyp.lweaken", "content": "theorem ESubtyp.lweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.label S) E1.weaken E2.weaken"}, {"name": "CSubtyp.lweaken", "content": "theorem CSubtyp.lweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.label S) E1.weaken E2.weaken"}, {"name": "Typed.lweaken", "content": "theorem Typed.lweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.label S) t.weaken E.weaken Ct.weaken"}, {"name": "SSubtyp.lweaken", "content": "theorem SSubtyp.lweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ S, SSubtyp (Γ.label S) S1.weaken S2.weaken"}, {"name": "Typed.lweaken_cext_ext", "content": "theorem Typed.lweaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.label P).cvar (CBinding.bound B.weaken)).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "Typed.lweaken_ext", "content": "theorem Typed.lweaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.label P).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.weaken1_weaken", "content": "theorem EType.weaken1_weaken (E : EType n m k) :\n  E.weaken.weaken1 = E.weaken.weaken"}, {"name": "Capless.CaptureSet.weaken1_weaken", "content": "theorem CaptureSet.weaken1_weaken (C : CaptureSet n k) :\n  C.weaken.weaken1 = C.weaken.weaken"}, {"name": "Capless.EType.weaken_ex", "content": "theorem EType.weaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).weaken = EType.ex T.weaken"}, {"name": "Capless.EType.weaken_cweaken", "content": "theorem EType.weaken_cweaken (E : EType n m k) :\n  E.cweaken.weaken = E.weaken.cweaken"}, {"name": "Capless.CaptureSet.weaken_cweaken", "content": "theorem CaptureSet.weaken_cweaken (C : CaptureSet n k) :\n  C.cweaken.weaken = C.weaken.cweaken"}, {"name": "Capless.Cont.HasLabel.weaken", "content": "theorem Cont.HasLabel.weaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.weaken x.succ tail.weaken"}, {"name": "Capless.WellScoped.weaken", "content": "theorem WellScoped.weaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.var T) cont.weaken Ct.weaken"}, {"name": "Capless.TypedCont.weaken", "content": "theorem TypedCont.weaken\n  (h : TypedCont Γ E t E' C0) :\n  TypedCont (Γ.var T) E.weaken t.weaken E'.weaken C0.weaken"}, {"name": "Capless.Cont.HasLabel.lweaken", "content": "theorem Cont.HasLabel.lweaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.weaken x.succ tail.weaken"}, {"name": "Capless.WellScoped.lweaken", "content": "theorem WellScoped.lweaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.label S) cont.weaken Ct.weaken"}], "local_ctx": "import Capless.Store\n\nimport Capless.Weakening.Typing\n\nimport Capless.Weakening.Subtyping\n\nimport Capless.Weakening.Subcapturing\n\nnamespace Capless", "target_theorem": "theorem TypedCont.lweaken\n  (h : TypedCont Γ E cont E' Ct) :\n  TypedCont (Γ.label S) E.weaken cont.weaken E'.weaken Ct.weaken :=", "ground_truth_proof": ":= by\n  induction h\n  case none =>\n    simp [Cont.weaken]\n    apply none\n    apply? ESubtyp.lweaken\n  case cons ih =>\n    simp [Cont.weaken]\n    have heq : ∀ {n m k} {T0 : CType n m k}, (EType.type T0).weaken = EType.type T0.weaken := by\n      intro T0\n      simp [EType.weaken, EType.rename, CType.weaken]\n    -- rw [heq]\n    apply cons\n    { rename_i ht _ _\n      have ht1 := ht.lweaken_ext (P := S)\n      rw [EType.weaken1_weaken] at ht1\n      rw [CaptureSet.weaken1_weaken] at ht1\n      exact ht1 }\n    { apply WellScoped.lweaken; assumption }\n    { exact ih }\n  case conse ih =>\n    simp [Cont.weaken, EType.weaken_ex]\n    apply conse\n    { rename_i ht _ _\n      have ht1 := ht.lweaken_cext_ext (P := S)\n      rw [EType.weaken1_weaken] at ht1\n      rw [EType.weaken_cweaken] at ht1\n      rw [CaptureSet.weaken1_weaken] at ht1\n      rw [CaptureSet.weaken_cweaken] at ht1\n      exact ht1 }\n    { apply WellScoped.lweaken; aesop }\n    { exact ih }\n  case scope hs ih =>\n    simp [Cont.weaken]\n    apply scope\n    { constructor; aesop }\n    { aesop }\n    { have h1 := hs.lweaken (S:=S)\n      aesop }", "nesting_depth": 7, "transitive_dep_count": 140, "subset_aristotle": false, "category": "Type systems"}
{"id": 81, "thm_name": "Capless.TypedCont.weaken", "thm_stmt": "theorem TypedCont.weaken\n  (h : TypedCont Γ E t E' C0) :\n  TypedCont (Γ.var T) E.weaken t.weaken E'.weaken C0.weaken", "lean_root": "capless-lean", "rel_path": "Capless/Weakening/TypedCont/Term.lean", "imports": ["import Capless.Type.Basic", "import Capless.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.CaptureSet", "import Capless.Weakening.Subcapturing", "import Capless.Store"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "VarMap.weaken_ext", "content": "def VarMap.weaken_ext {Γ : Context n m k} :\n  VarMap\n    (Γ.var T)\n    FinFun.weaken.ext\n    ((Γ.var P).var T.weaken) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "CVarMap.weaken_ext", "content": "def CVarMap.weaken_ext {Γ : Context n m k} :\n  CVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.cvar b).var T.cweaken) :="}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CVarMap.weaken_cext_ext", "content": "def CVarMap.weaken_cext_ext {Γ : Context n m k} :\n  CVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken.ext\n    (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "VarMap.weaken_cext_ext", "content": "def VarMap.weaken_cext_ext {Γ : Context n m k} :\n  VarMap\n    ((Γ.cvar (CBinding.bound b)).var T)\n    FinFun.weaken.ext\n    (((Γ.var P).cvar (CBinding.bound b.weaken)).var T.weaken) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.weaken_ext", "content": "def TVarMap.weaken_ext {Γ : Context n m k} :\n  TVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.tvar b).var T.tweaken) :="}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken_cext_ext", "content": "def TVarMap.weaken_cext_ext {Γ : Context n m k} :\n  TVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken\n    (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "FinFun.comp_weaken", "content": "theorem FinFun.comp_weaken {f : FinFun n n'} :\n  weaken ∘ f = f.ext ∘ weaken"}, {"name": "EType.rename_rename", "content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (E.rename f).rename g = E.rename (g ∘ f)"}, {"name": "CType.rename_rename", "content": "theorem CType.rename_rename (T : CType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (T.rename f).rename g = T.rename (g ∘ f)"}, {"name": "SType.rename_rename", "content": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f)"}, {"name": "CBound.rename_rename", "content": "theorem CBound.rename_rename {b : CBound n k} :\n  (b.rename f).rename g = b.rename (g ∘ f)"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "Typed.weaken_ext", "content": "theorem Typed.weaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.var P).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "Typed.weaken_cext_ext", "content": "theorem Typed.weaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.var P).cvar (CBinding.bound B.weaken)).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "CaptureSet.rename_rename", "content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n  (C.rename f).rename g = C.rename (g ∘ f)"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.weaken1_weaken", "content": "theorem EType.weaken1_weaken (E : EType n m k) :\n  E.weaken.weaken1 = E.weaken.weaken"}, {"name": "Capless.CaptureSet.weaken1_weaken", "content": "theorem CaptureSet.weaken1_weaken (C : CaptureSet n k) :\n  C.weaken.weaken1 = C.weaken.weaken"}, {"name": "Capless.EType.weaken_ex", "content": "theorem EType.weaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).weaken = EType.ex T.weaken"}, {"name": "Capless.EType.weaken_cweaken", "content": "theorem EType.weaken_cweaken (E : EType n m k) :\n  E.cweaken.weaken = E.weaken.cweaken"}, {"name": "Capless.CaptureSet.weaken_cweaken", "content": "theorem CaptureSet.weaken_cweaken (C : CaptureSet n k) :\n  C.cweaken.weaken = C.weaken.cweaken"}, {"name": "Capless.Cont.HasLabel.weaken", "content": "theorem Cont.HasLabel.weaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.weaken x.succ tail.weaken"}, {"name": "Capless.WellScoped.weaken", "content": "theorem WellScoped.weaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.var T) cont.weaken Ct.weaken"}], "local_ctx": "import Capless.Store\n\nimport Capless.Weakening.Typing\n\nimport Capless.Weakening.Subtyping\n\nimport Capless.Weakening.Subcapturing\n\nnamespace Capless", "target_theorem": "theorem TypedCont.weaken\n  (h : TypedCont Γ E t E' C0) :\n  TypedCont (Γ.var T) E.weaken t.weaken E'.weaken C0.weaken :=", "ground_truth_proof": ":= by\n  induction h\n  case none =>\n    simp [Cont.weaken]\n    apply none\n    apply? ESubtyp.weaken\n  case cons ih =>\n    simp [Cont.weaken]\n    have heq : ∀ {n m k} {T0 : CType n m k}, (EType.type T0).weaken = EType.type T0.weaken := by\n      intro T0\n      simp [EType.weaken, EType.rename, CType.weaken]\n    -- rw [heq]\n    apply cons\n    { rename_i ht _ _\n      have ht1 := ht.weaken_ext (P := T)\n      rw [EType.weaken1_weaken] at ht1\n      rw [CaptureSet.weaken1_weaken] at ht1\n      exact ht1 }\n    { apply WellScoped.weaken; assumption }\n    { exact ih }\n  case conse ih =>\n    simp [Cont.weaken, EType.weaken_ex]\n    apply conse\n    { rename_i ht _ _\n      have ht1 := ht.weaken_cext_ext (P := T)\n      rw [EType.weaken1_weaken] at ht1\n      rw [EType.weaken_cweaken] at ht1\n      rw [CaptureSet.weaken1_weaken] at ht1\n      rw [CaptureSet.weaken_cweaken] at ht1\n      exact ht1 }\n    { apply WellScoped.weaken; aesop }\n    { exact ih }\n  case scope hs ih =>\n    simp [Cont.weaken]\n    apply scope\n    { constructor; aesop }\n    { aesop }\n    { have h1 := hs.weaken (T:=T)\n      aesop }", "nesting_depth": 5, "transitive_dep_count": 128, "subset_aristotle": false, "category": "Type systems"}
{"id": 82, "thm_name": "Capless.TypedCont.cweaken", "thm_stmt": "theorem TypedCont.cweaken\n  (h : TypedCont Γ E t E' Ct) :\n  TypedCont (Γ.cvar b) E.cweaken t.cweaken E'.cweaken Ct.cweaken", "lean_root": "capless-lean", "rel_path": "Capless/Weakening/TypedCont/Capture.lean", "imports": ["import Capless.Type.Basic", "import Capless.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.CaptureSet", "import Capless.Weakening.Subcapturing", "import Capless.Store"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "EType.cweaken_type", "content": "@[simp]\ndef EType.cweaken_type :\n  (EType.type T).cweaken = EType.type (T.cweaken) :="}, {"name": "Typed.cweaken_cext_ext", "content": "def Typed.cweaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) t.cweaken1 E.cweaken1 Ct.cweaken1 :="}, {"name": "CVarMap.weaken_cext_ext", "content": "def CVarMap.weaken_cext_ext {Γ : Context n m k} :\n  CVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken.ext\n    (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "Typed.cweaken_ext", "content": "def Typed.cweaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.cvar b).var T.cweaken) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "CVarMap.weaken_ext", "content": "def CVarMap.weaken_ext {Γ : Context n m k} :\n  CVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.cvar b).var T.cweaken) :="}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "EType.crename_crename", "content": "theorem EType.crename_crename (E : EType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (E.crename f).crename g = E.crename (g ∘ f)"}, {"name": "CType.crename_crename", "content": "theorem CType.crename_crename (T : CType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (T.crename f).crename g = T.crename (g ∘ f)"}, {"name": "SType.crename_crename", "content": "theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (S.crename f).crename g = S.crename (g ∘ f)"}, {"name": "CBound.crename_crename", "content": "theorem CBound.crename_crename {b : CBound n k} :\n  (b.crename f).crename g = b.crename (g ∘ f)"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "FinFun.comp_weaken", "content": "theorem FinFun.comp_weaken {f : FinFun n n'} :\n  weaken ∘ f = f.ext ∘ weaken"}, {"name": "CaptureSet.weaken_crename", "content": "theorem CaptureSet.weaken_crename {C : CaptureSet n k} :\n  (C.crename f).weaken = C.weaken.crename f"}, {"name": "CaptureSet.crename_crename", "content": "theorem CaptureSet.crename_crename {C : CaptureSet n k} :\n  (C.crename f).crename g = C.crename (g ∘ f)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.cweaken_ex", "content": "theorem EType.cweaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).cweaken = EType.ex T.cweaken1"}, {"name": "Capless.EType.cweaken_weaken", "content": "theorem EType.cweaken_weaken (E : EType n m k) :\n  E.weaken.cweaken = E.cweaken.weaken"}, {"name": "Capless.EType.cweaken1_weaken", "content": "theorem EType.cweaken1_weaken (E : EType n m (k+1)) :\n  E.weaken.cweaken1 = E.cweaken1.weaken"}, {"name": "Capless.CaptureSet.cweaken1_weaken", "content": "theorem CaptureSet.cweaken1_weaken (C : CaptureSet n (k+1)) :\n  C.weaken.cweaken1 = C.cweaken1.weaken"}, {"name": "Capless.EType.cweaken1_cweaken", "content": "theorem EType.cweaken1_cweaken (E : EType n m k) :\n  E.cweaken.cweaken1 = E.cweaken.cweaken"}, {"name": "Capless.CaptureSet.cweaken1_cweaken", "content": "theorem CaptureSet.cweaken1_cweaken (C : CaptureSet n k) :\n  C.cweaken.cweaken1 = C.cweaken.cweaken"}, {"name": "Capless.Cont.HasLabel.cweaken", "content": "theorem Cont.HasLabel.cweaken\n  (h : Cont.HasLabel cont l tail) :\n  Cont.HasLabel (cont.cweaken) l tail.cweaken"}, {"name": "Capless.WellScoped.cweaken", "content": "theorem WellScoped.cweaken\n  (h : WellScoped Γ E Ct) :\n  WellScoped (Γ.cvar b) E.cweaken Ct.cweaken"}], "local_ctx": "import Capless.Store\n\nimport Capless.Weakening.Typing\n\nimport Capless.Weakening.Subtyping\n\nimport Capless.Weakening.Subcapturing\n\nnamespace Capless", "target_theorem": "theorem TypedCont.cweaken\n  (h : TypedCont Γ E t E' Ct) :\n  TypedCont (Γ.cvar b) E.cweaken t.cweaken E'.cweaken Ct.cweaken :=", "ground_truth_proof": ":= by\n  induction h\n  case none =>\n    simp [Cont.cweaken]\n    apply none\n    apply? ESubtyp.cweaken\n  case cons ht hs _ ih =>\n    simp [Cont.cweaken, EType.cweaken_type]\n    apply cons\n    { have ht1 := ht.cweaken_ext (b := b)\n      rw [EType.cweaken_weaken] at ht1\n      rw [CaptureSet.weaken_crename]\n      exact ht1 }\n    { apply hs.cweaken }\n    { exact ih }\n  case conse ht hs _ ih =>\n    simp [Cont.cweaken, EType.cweaken_ex]\n    apply conse\n    { have ht1 := ht.cweaken_cext_ext (b := b)\n      rw [EType.cweaken1_weaken] at ht1\n      rw [EType.cweaken1_cweaken] at ht1\n      rw [CaptureSet.cweaken1_weaken] at ht1\n      rw [CaptureSet.cweaken1_cweaken] at ht1\n      exact ht1 }\n    { apply hs.cweaken }\n    { exact ih }\n  case scope hb _ hs ih =>\n    simp [Cont.cweaken]\n    apply scope\n    have hb1 := Context.LBound.there_cvar (b := b) hb\n    exact hb1\n    simp at ih\n    apply ih\n    have h := hs.cweaken (b:=b)\n    aesop", "nesting_depth": 5, "transitive_dep_count": 118, "subset_aristotle": false, "category": "Type systems"}
{"id": 83, "thm_name": "Capless.Subcapt.rename", "thm_stmt": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)", "lean_root": "capless-lean", "rel_path": "Capless/Renaming/Term/Subcapturing.lean", "imports": ["import Capless.Subcapturing", "import Mathlib.Data.Finset.Image", "import Capless.Renaming.Basic", "import Capless.CaptureSet"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CaptureSet.rename_union", "content": "theorem CaptureSet.rename_union {C1 C2 : CaptureSet n k} {f : FinFun n n'} :\n  (C1 ∪ C2).rename f = C1.rename f ∪ C2.rename f"}, {"name": "CaptureSet.rename_singleton", "content": "theorem CaptureSet.rename_singleton {x : Fin n} {f : FinFun n n'} :\n  ({x=x} : CaptureSet n k).rename f = {x=f x}"}, {"name": "CaptureSet.rename_csingleton", "content": "theorem CaptureSet.rename_csingleton {x : Fin k} {f : FinFun n n'} :\n  {c=x}.rename f = {c=x}"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}], "local_ctx": "import Capless.Subcapturing\n\nimport Capless.Renaming.Basic\n\nimport Mathlib.Data.Finset.Image\n\nnamespace Capless", "target_theorem": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f) :=", "ground_truth_proof": ":= by\n  induction h\n  case trans ih1 ih2 => apply trans <;> aesop\n  case subset hsub =>\n    apply subset\n    apply CaptureSet.Subset.rename; trivial\n  case union ih1 ih2 =>\n    simp [CaptureSet.rename_union]\n    apply union <;> aesop\n  case var hb =>\n    simp [CaptureSet.rename_singleton]\n    apply var\n    have hb1 := ρ.map _ _ hb\n    simp [EType.rename, CType.rename] at hb1\n    assumption\n  case cinstl hb =>\n    simp [CaptureSet.rename_csingleton]\n    have hb1 := ρ.cmap _ _ hb\n    simp [CBinding.rename] at hb1\n    apply cinstl\n    assumption\n  case cinstr hb =>\n    simp [CaptureSet.rename_csingleton]\n    have hb1 := ρ.cmap _ _ hb\n    simp [CBinding.rename] at hb1\n    apply cinstr\n    assumption\n  case cbound hb =>\n    simp [CaptureSet.rename_csingleton]\n    have hb1 := ρ.cmap _ _ hb\n    simp [CBinding.rename, CBound.rename] at hb1\n    apply cbound\n    easy", "nesting_depth": 3, "transitive_dep_count": 36, "subset_aristotle": false, "category": "Type systems"}
{"id": 84, "thm_name": "Capless.Store.val_lookup_exists", "thm_stmt": "theorem Store.val_lookup_exists {σ : Store n m k} {x : Fin n}\n  (hs : TypedStore σ Γ) (hx : Typed Γ (Term.var x) (EType.type T) Cx)\n  (hvt : T.IsValue) :\n  ∃ v, Store.Bound σ x v ∧ v.IsValue", "lean_root": "capless-lean", "rel_path": "Capless/Soundness/Progress.lean", "imports": ["import Capless.Inversion.Context", "import Capless.Weakening.IsValue", "import Mathlib.Data.Fin.Basic", "import Capless.WellScoped.Basic", "import Capless.Inversion.Subtyping", "import Capless.Inversion.Lookup", "import Capless.Inversion.Typing", "import Capless.Store", "import Capless.Reduction", "import Capless.Narrowing.TypedCont"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "abs", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "Fin.elim0", "module": "Init.Data.Fin.Basic"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Or.inl", "module": "Init.Prelude"}, {"name": "Or.inr", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType.IsValue", "content": "inductive CType.IsValue : CType n m k -> Prop where\n| capt :\n  S.IsValue ->\n  CType.IsValue (S^C)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "TypedStore", "content": "inductive TypedStore : Store n m k -> Context n m k -> Prop where\n| empty : TypedStore Store.empty Context.empty\n| val :\n  TypedStore σ Γ ->\n  Typed Γ t (EType.type E) Ct ->\n  (hv : t.IsValue) ->\n  TypedStore (Store.val σ t hv) (Γ.var E)\n| tval :\n  TypedStore σ Γ ->\n  TypedStore (Store.tval σ S) (Γ.tvar (TBinding.inst S))\n| cval :\n  TypedStore σ Γ ->\n  TypedStore (Store.cval σ C) (Γ.cvar (CBinding.inst C))\n| label :\n  TypedStore σ Γ ->\n  TypedStore (Store.label σ S) (Γ.label S)"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Reduce", "content": "inductive Reduce : State n m k -> State n' m' k' -> Prop where\n| apply {σ : Store n m k} :\n  σ.Bound x (Term.lam T t) ->\n  Reduce ⟨σ | cont | Term.app x y⟩ ⟨σ | cont | t.open y⟩\n| tapply {σ : Store n m k} :\n  σ.Bound x (Term.tlam S t) ->\n  Reduce ⟨σ | cont | Term.tapp x X⟩ ⟨σ | cont | t.topen X⟩\n| capply {σ : Store n m k} :\n  σ.Bound x (Term.clam B t) ->\n  Reduce ⟨σ | cont | Term.capp x c⟩ ⟨σ | cont | t.copen c⟩\n| enter :\n  Reduce\n    ⟨σ | cont | boundary:S in t⟩\n    ⟨(σ.label S).cval {x=0} | cont.weaken.cweaken.scope 0 | t⟩\n| leave_var :\n  Reduce\n    ⟨σ | cont.scope x | Term.var y⟩\n    ⟨σ | cont | Term.var y⟩\n| leave_val {v : Term n m k} :\n  (hv : Term.IsValue v) ->\n  Reduce\n    ⟨σ | cont.scope x | v⟩\n    ⟨σ | cont | v⟩\n| invoke {σ : Store n m k} {cont : Cont n m k} :\n  σ.LBound x S ->\n  cont.HasLabel x tail ->\n  Reduce\n    ⟨σ | cont | Term.invoke x y⟩\n    ⟨σ | tail | Term.var y⟩\n| push :\n  Reduce\n    ⟨σ | cont | Term.letin t u⟩\n    ⟨σ | Cont.cons u cont | t⟩\n| push_ex :\n  Reduce\n    ⟨σ | cont | Term.letex t u⟩\n    ⟨σ | Cont.conse u cont | t⟩\n| rename :\n  Reduce\n    ⟨σ | Cont.cons u cont | Term.var x⟩\n    ⟨σ | cont | u.open x⟩\n| lift :\n  (hv : Term.IsValue v) ->\n  Reduce\n    ⟨σ | Cont.cons u cont | v⟩\n    ⟨σ.val v hv | cont.weaken | u⟩\n| lift_ex :\n  Reduce\n    ⟨σ | Cont.conse u cont | Term.pack C x⟩\n    ⟨σ.cval C | cont.cweaken | u.open x⟩\n| tlift :\n  Reduce\n    ⟨σ | cont | Term.bindt S t⟩\n    ⟨σ.tval S | cont.tweaken | t⟩\n| clift :\n  Reduce\n    ⟨σ | cont | Term.bindc C t⟩\n    ⟨σ.cval C | cont.cweaken | t⟩"}, {"name": "infix:30 \" ", "content": "infix:30 \" "}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "SSubtyp.dealias_right_cforall.cmotive", "content": "def SSubtyp.dealias_right_cforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_cforall.smotive", "content": "def SSubtyp.dealias_right_cforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {B2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)),\n    ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "SSubtyp.dealias_right_cforall.emotive", "content": "def SSubtyp.dealias_right_cforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp.dealias_right_tforall.smotive", "content": "def SSubtyp.dealias_right_tforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}, {"name": "SSubtyp.dealias_right_tforall.emotive", "content": "def SSubtyp.dealias_right_tforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_tforall.cmotive", "content": "def SSubtyp.dealias_right_tforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "SSubtyp.dealias_right_boxed.smotive", "content": "def SSubtyp.dealias_right_boxed.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.box T2)),\n    ∃ T1, SType.Dealias Γ S1 (SType.box T1)"}, {"name": "SSubtyp.dealias_right_boxed.emotive", "content": "def SSubtyp.dealias_right_boxed.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_boxed.cmotive", "content": "def SSubtyp.dealias_right_boxed.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.emotive", "content": "def SSubtyp.dealias_right_forall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.cmotive", "content": "def SSubtyp.dealias_right_forall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.smotive", "content": "def SSubtyp.dealias_right_forall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.forall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Term.IsValue.weaken", "content": "theorem Term.IsValue.weaken\n  (hv : Term.IsValue t) :\n  Term.IsValue t.weaken"}, {"name": "Term.IsValue.tweaken", "content": "theorem Term.IsValue.tweaken\n  (hv : Term.IsValue t) :\n  Term.IsValue t.tweaken"}, {"name": "Term.IsValue.cweaken", "content": "theorem Term.IsValue.cweaken\n  (hv : Term.IsValue t) :\n  Term.IsValue t.cweaken"}, {"name": "Typed.label_inv", "content": "theorem Typed.label_inv\n  (ht : Typed Γ (Term.var x) (EType.type T) Ct) (hb : Γ.LBound x S1) :\n  ∃ S0, Γ.LBound x S0 ∧ (Γ ⊢ (Label[S0]^{x=x}) <: T)"}, {"name": "Typed.label_inv'", "content": "theorem Typed.label_inv'\n  (he1 : t0 = Term.var x)\n  (he2 : E0 = EType.type T)\n  (ht : Typed Γ t0 E0 Ct) (hb : Γ.LBound x S1) :\n  ∃ S0, Γ.LBound x S0 ∧ (Γ ⊢ (Label[S0]^{x=x}) <: T)"}, {"name": "SSubtyp.dealias_right_cforall", "content": "theorem SSubtyp.dealias_right_cforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)) :\n  ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}, {"name": "SSubtyp.dealias_right_tforall", "content": "theorem SSubtyp.dealias_right_tforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}, {"name": "Context.lbound_inj", "content": "theorem Context.lbound_inj\n  (hb1 : Context.LBound Γ x S1)\n  (hb2 : Context.LBound Γ x S2) : S1 = S2"}, {"name": "Context.label_lbound_succ_inv", "content": "theorem Context.label_lbound_succ_inv\n  (hb : Context.LBound (Γ.label l) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.label_lbound_succ_inv'", "content": "theorem Context.label_lbound_succ_inv'\n  (he1 : Γ0 = Γ.label l) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.var_lbound_succ_inv", "content": "theorem Context.var_lbound_succ_inv\n  (hb : Context.LBound (Γ.var T) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.var_lbound_succ_inv'", "content": "theorem Context.var_lbound_succ_inv'\n  (he1 : Γ0 = Γ.var T) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "SSubtyp.dealias_right_boxed", "content": "theorem SSubtyp.dealias_right_boxed\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.box T2)) :\n  ∃ T1, SType.Dealias Γ S1 (SType.box T1)"}, {"name": "Store.bound_label", "content": "theorem Store.bound_label\n  (hl : Store.LBound σ x S)\n  (ht : TypedStore σ Γ) :\n  Γ.LBound x S"}, {"name": "SSubtyp.dealias_right_forall", "content": "theorem SSubtyp.dealias_right_forall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.forall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}, {"name": "TypedStore.is_tight", "content": "theorem TypedStore.is_tight\n  (h : TypedStore σ Γ) :\n  Γ.IsTight"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.Store.lookup_exists", "content": "theorem Store.lookup_exists {σ : Store n m k} {x : Fin n} :\n  (∃ v, Store.Bound σ x v ∧ v.IsValue) ∨ (∃ S, Store.LBound σ x S)"}], "local_ctx": "import Mathlib.Data.Fin.Basic\n\nimport Capless.Reduction\n\nimport Capless.Narrowing.TypedCont\n\nimport Capless.Inversion.Lookup\n\nimport Capless.Inversion.Typing\n\nimport Capless.Weakening.IsValue\n\nimport Capless.WellScoped.Basic\n\nnamespace Capless", "target_theorem": "theorem Store.val_lookup_exists {σ : Store n m k} {x : Fin n}\n  (hs : TypedStore σ Γ) (hx : Typed Γ (Term.var x) (EType.type T) Cx)\n  (hvt : T.IsValue) :\n  ∃ v, Store.Bound σ x v ∧ v.IsValue :=", "ground_truth_proof": ":= by\n  have hg := TypedStore.is_tight hs\n  have h := Store.lookup_exists (σ := σ) (x := x)\n  cases h\n  case inl h => easy\n  case inr h =>\n    have ⟨S, hl⟩ := h\n    have hb := Store.bound_label hl hs\n    have ⟨S0, hb0, hsub⟩ := Typed.label_inv hx hb\n    have h := Context.lbound_inj hb hb0\n    subst_vars\n    cases hvt\n    case capt hvt =>\n      cases hsub; rename_i hsub\n      cases hvt\n      case xforall =>\n        have ⟨_, _, hd1⟩ := SSubtyp.dealias_right_forall hsub hg (by constructor)\n        cases hd1\n      case tforall =>\n        have ⟨_, _, hd1⟩ := SSubtyp.dealias_right_tforall hsub hg (by constructor)\n        cases hd1\n      case cforall =>\n        have ⟨_, _, hd1⟩ := SSubtyp.dealias_right_cforall hsub hg (by constructor)\n        cases hd1\n      case box =>\n        have ⟨_, hd1⟩ := SSubtyp.dealias_right_boxed hsub hg (by constructor)\n        cases hd1", "nesting_depth": 4, "transitive_dep_count": 102, "subset_aristotle": false, "category": "Type systems"}
{"id": 85, "thm_name": "Capless.Typed.canonical_form_tlam'", "thm_stmt": "theorem Typed.canonical_form_tlam'\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.tforall S' E))\n  (he1 : t0 = Term.tlam S t)\n  (he2 : E0 = EType.type (CType.capt Cf S0))\n  (h : Typed Γ t0 E0 Ct0) :\n  SSubtyp Γ S' S ∧\n  Typed (Γ.tvar (TBinding.bound S')) t E Cf", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Narrowing.Typing", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Narrowing.Subtyping", "import Capless.Weakening.Subtyping", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Subcapt.tweaken", "content": "def Subcapt.tweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.tvar b) ⊢ C1 <:c C2 :="}, {"name": "TVarSubst.narrow", "content": "def TVarSubst.narrow\n  (hs : SSubtyp Γ S' S) :\n  TVarSubst\n    (Γ.tvar (TBinding.bound S))\n    FinFun.id\n    (Γ.tvar (TBinding.bound S')) :="}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "SSubtyp.dealias_tforall_inv.smotive", "content": "def SSubtyp.dealias_tforall_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 E1 T2 E2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.tforall T1 E1))\n    (h2 : SType.Dealias Γ S2 (SType.tforall T2 E2)),\n    SSubtyp Γ T2 T1 ∧ ESubtyp (Γ.tvar (TBinding.bound T2)) E1 E2"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "SSubtyp.dealias_tforall_inv.cmotive", "content": "def SSubtyp.dealias_tforall_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_tforall_inv.emotive", "content": "def SSubtyp.dealias_tforall_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_tforall.smotive", "content": "def SSubtyp.dealias_right_tforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}, {"name": "SSubtyp.dealias_right_tforall.emotive", "content": "def SSubtyp.dealias_right_tforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_tforall.cmotive", "content": "def SSubtyp.dealias_right_tforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Typed.tnarrow", "content": "theorem Typed.tnarrow\n  (h : Typed (Γ,X<: S) t E Ct)\n  (hs : SSubtyp Γ S' S) :\n  Typed (Γ,X<: S') t E Ct"}, {"name": "SSubtyp.sub_dealias_tforall_inv", "content": "theorem SSubtyp.sub_dealias_tforall_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.tforall T1 E1))\n  (h2 : SType.Dealias Γ S2 (SType.tforall T2 E2))\n  (hs : SSubtyp Γ S1 S2) :\n  SSubtyp Γ T2 T1 ∧ ESubtyp (Γ.tvar (TBinding.bound T2)) E1 E2"}, {"name": "SSubtyp.dealias_right_tforall", "content": "theorem SSubtyp.dealias_right_tforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}, {"name": "SType.dealias_tforall_inj", "content": "theorem SType.dealias_tforall_inj\n  (h1 : SType.Dealias Γ S (SType.tforall T1 E1))\n  (h2 : SType.Dealias Γ S (SType.tforall T2 E2)) :\n  T1 = T2 ∧ E1 = E2"}, {"name": "SType.dealias_tforall_inj'", "content": "theorem SType.dealias_tforall_inj'\n  (he1 : S1 = SType.tforall T1 E1) (he2 : S2 = SType.tforall T2 E2)\n  (h1 : SType.Dealias Γ S S1)\n  (h2 : SType.Dealias Γ S S2) :\n  T1 = T2 ∧ E1 = E2"}, {"name": "ESubtyp.tnarrow", "content": "theorem ESubtyp.tnarrow\n  (h : ESubtyp (Γ.tvar (TBinding.bound S)) E1 E2)\n  (hs : SSubtyp Γ S' S) :\n  ESubtyp (Γ.tvar (TBinding.bound S')) E1 E2"}, {"name": "ESubtyp.tweaken", "content": "theorem ESubtyp.tweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.tvar b) E1.tweaken E2.tweaken"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}, {"name": "Subbound.tweaken", "content": "theorem Subbound.tweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.tvar b) B1 B2"}, {"name": "SSubtyp.tweaken", "content": "theorem SSubtyp.tweaken\n  (h : SSubtyp Γ S1 S2) :\n  SSubtyp (Γ.tvar b) S1.tweaken S2.tweaken"}, {"name": "CSubtyp.tweaken", "content": "theorem CSubtyp.tweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.tvar b) E1.tweaken E2.tweaken"}, {"name": "Subcapt.refl", "content": "theorem Subcapt.refl :\n  Subcapt Γ C C"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.canonical_form_tlam'\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.tforall S' E))\n  (he1 : t0 = Term.tlam S t)\n  (he2 : E0 = EType.type (CType.capt Cf S0))\n  (h : Typed Γ t0 E0 Ct0) :\n  SSubtyp Γ S' S ∧\n  Typed (Γ.tvar (TBinding.bound S')) t E Cf :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he1 | cases he2)\n  case tabs =>\n    cases he1; cases he2\n    cases hd\n    constructor\n    apply SSubtyp.refl\n    trivial\n  case sub hs ih =>\n    subst he2\n    cases hs\n    rename_i hs\n    cases hs\n    rename_i hsc hs\n    have ⟨S1, E1, hd3⟩ := SSubtyp.dealias_right_tforall hs ht hd\n    have ih := ih ht hd3 he1 rfl\n    have h := SSubtyp.sub_dealias_tforall_inv ht hd3 hd hs\n    have ⟨hs1, ht1⟩ := ih\n    have ⟨hs2, ht2⟩ := h\n    apply And.intro\n    { apply! SSubtyp.trans }\n    { constructor\n      apply? Typed.sub\n      apply ht1.tnarrow; assumption; apply Subcapt.refl\n      apply hsc.tweaken\n      apply ESubtyp.refl }", "nesting_depth": 5, "transitive_dep_count": 68, "subset_aristotle": false, "category": "Type systems"}
{"id": 86, "thm_name": "Capless.Subcapt.crename", "thm_stmt": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)", "lean_root": "capless-lean", "rel_path": "Capless/Renaming/Capture/Subcapturing.lean", "imports": ["import Capless.Subcapturing", "import Mathlib.Data.Finset.Image", "import Capless.Renaming.Basic", "import Capless.CaptureSet"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CaptureSet.crename_csingleton", "content": "theorem CaptureSet.crename_csingleton {x : Fin k} {f : FinFun k k'} :\n  ({c=x} : CaptureSet n k).crename f = {c=f x}"}, {"name": "CaptureSet.crename_union", "content": "theorem CaptureSet.crename_union {C1 C2 : CaptureSet n k} {f : FinFun k k'} :\n  (C1 ∪ C2).crename f = C1.crename f ∪ C2.crename f"}, {"name": "CaptureSet.crename_singleton", "content": "theorem CaptureSet.crename_singleton {x : Fin n} {f : FinFun k k'} :\n  {x=x}.crename f = {x=x}"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}], "local_ctx": "import Capless.Subcapturing\n\nimport Capless.Renaming.Basic\n\nimport Mathlib.Data.Finset.Image\n\nnamespace Capless", "target_theorem": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f) :=", "ground_truth_proof": ":= by\n  induction h\n  case trans ih1 ih2 => apply trans <;> aesop\n  case subset hsub =>\n    apply subset\n    apply CaptureSet.Subset.crename; trivial\n  case union ih1 ih2 =>\n    simp [CaptureSet.crename_union]\n    apply union <;> aesop\n  case var hb =>\n    simp [CaptureSet.crename_singleton]\n    apply var\n    have hb1 := ρ.map _ _ hb\n    simp [EType.crename, CType.crename] at hb1\n    assumption\n  case cinstl hb =>\n    simp [CaptureSet.crename_csingleton]\n    have hb1 := ρ.cmap _ _ hb\n    simp [CBinding.rename] at hb1\n    apply cinstl\n    assumption\n  case cinstr hb =>\n    simp [CaptureSet.crename_csingleton]\n    have hb1 := ρ.cmap _ _ hb\n    simp [CBinding.rename] at hb1\n    apply cinstr\n    assumption\n  case cbound hb =>\n    simp [CaptureSet.crename_csingleton]\n    have hb1 := ρ.cmap _ _ hb\n    simp [CBinding.rename] at hb1\n    apply cbound\n    assumption", "nesting_depth": 3, "transitive_dep_count": 43, "subset_aristotle": false, "category": "Type systems"}
{"id": 87, "thm_name": "Capless.Typed.boundary_body_typing", "thm_stmt": "theorem Typed.boundary_body_typing {Γ : Context n m k} {S : SType n m k}\n  (ht : Typed ((Γ,c<:*),x:(Label[S.cweaken])^{c=0}) t E Ct) :\n  Typed ((Γ.label S),c:={x=0}) t E Ct", "lean_root": "capless-lean", "rel_path": "Capless/Typing/Boundary.lean", "imports": ["import Capless.Renaming.Capture.Typing", "import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Term.Typing", "import Capless.Renaming.Type.Subcapturing", "import Capless.Basic", "import Capless.Subst.Term.Subcapturing", "import Capless.Renaming.Term.Subcapturing", "import Capless.CaptureSet", "import Capless.Subst.Capture.Subcapturing", "import Capless.Term", "import Capless.Subst.Capture.Subtyping", "import Capless.Narrowing.Typing", "import Capless.Subst.Term.Subtyping", "import Capless.Renaming.Type.Subtyping", "import Capless.Typing", "import Capless.Type.Basic", "import Capless.Renaming.Type.Typing", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.Subst.Term.Typing", "import Capless.Renaming.Capture.Subtyping", "import Capless.Subst.Capture.Typing", "import Capless.Renaming.Capture.Subcapturing"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "abs", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "TypedStore", "content": "inductive TypedStore : Store n m k -> Context n m k -> Prop where\n| empty : TypedStore Store.empty Context.empty\n| val :\n  TypedStore σ Γ ->\n  Typed Γ t (EType.type E) Ct ->\n  (hv : t.IsValue) ->\n  TypedStore (Store.val σ t hv) (Γ.var E)\n| tval :\n  TypedStore σ Γ ->\n  TypedStore (Store.tval σ S) (Γ.tvar (TBinding.inst S))\n| cval :\n  TypedStore σ Γ ->\n  TypedStore (Store.cval σ C) (Γ.cvar (CBinding.inst C))\n| label :\n  TypedStore σ Γ ->\n  TypedStore (Store.label σ S) (Γ.label S)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CType.weaken_capt", "content": "@[simp]\ndef CType.weaken_capt :\n  (CType.capt C S).weaken = CType.capt C.weaken S.weaken :="}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "SSubtyp.subst_motive3", "content": "def SSubtyp.subst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.subst_motive2", "content": "def SSubtyp.subst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "SSubtyp.subst_motive1", "content": "def SSubtyp.subst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarSubst Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "SSubtyp.csubst_motive3", "content": "def SSubtyp.csubst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.csubst_motive1", "content": "def SSubtyp.csubst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SSubtyp.csubst_motive2", "content": "def SSubtyp.csubst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CaptureSet.crename_crename", "content": "theorem CaptureSet.crename_crename {C : CaptureSet n k} :\n  (C.crename f).crename g = C.crename (g ∘ f)"}, {"name": "CaptureSet.crename_id", "content": "theorem CaptureSet.crename_id {C : CaptureSet n k} :\n  C.crename FinFun.id = C"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "FinFun.open_zero_comp_weaken_ext", "content": "theorem FinFun.open_zero_comp_weaken_ext :\n  (FinFun.open 0) ∘ (weaken.ext : FinFun (n+1) (n+2)) = id"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "CaptureSet.rename_rename", "content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n  (C.rename f).rename g = C.rename (g ∘ f)"}, {"name": "CaptureSet.rename_id", "content": "theorem CaptureSet.rename_id {C : CaptureSet n k} :\n  C.rename FinFun.id = C"}, {"name": "Term.rename_id", "content": "theorem Term.rename_id {t : Term n m k} :\n  t.rename FinFun.id = t"}, {"name": "Term.rename_rename", "content": "theorem Term.rename_rename {t : Term n m k} {f : FinFun n n'} {g : FinFun n' n''} :\n  (t.rename f).rename g = t.rename (g.comp f)"}, {"name": "EType.rename_id", "content": "theorem EType.rename_id {E : EType n m k} :\n  E.rename FinFun.id = E"}, {"name": "CType.rename_id", "content": "theorem CType.rename_id {T : CType n m k} :\n  T.rename FinFun.id = T"}, {"name": "SType.rename_id", "content": "theorem SType.rename_id {S : SType n m k} :\n  S.rename FinFun.id = S"}, {"name": "CBound.rename_id", "content": "theorem CBound.rename_id {b : CBound n k} :\n  b.rename FinFun.id = b"}, {"name": "EType.rename_rename", "content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (E.rename f).rename g = E.rename (g ∘ f)"}, {"name": "CType.rename_rename", "content": "theorem CType.rename_rename (T : CType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (T.rename f).rename g = T.rename (g ∘ f)"}, {"name": "SType.rename_rename", "content": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f)"}, {"name": "CBound.rename_rename", "content": "theorem CBound.rename_rename {b : CBound n k} :\n  (b.rename f).rename g = b.rename (g ∘ f)"}, {"name": "Term.crename_crename", "content": "theorem Term.crename_crename {t : Term n m k} {f : FinFun k k'} {g : FinFun k' k''} :\n  (t.crename f).crename g = t.crename (g.comp f)"}, {"name": "Term.crename_id", "content": "theorem Term.crename_id {t : Term n m k} :\n  t.crename FinFun.id = t"}, {"name": "EType.crename_crename", "content": "theorem EType.crename_crename (E : EType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (E.crename f).crename g = E.crename (g ∘ f)"}, {"name": "CType.crename_crename", "content": "theorem CType.crename_crename (T : CType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (T.crename f).crename g = T.crename (g ∘ f)"}, {"name": "SType.crename_crename", "content": "theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (S.crename f).crename g = S.crename (g ∘ f)"}, {"name": "CBound.crename_crename", "content": "theorem CBound.crename_crename {b : CBound n k} :\n  (b.crename f).crename g = b.crename (g ∘ f)"}, {"name": "EType.crename_id", "content": "theorem EType.crename_id {E : EType n m k} :\n  E.crename FinFun.id = E"}, {"name": "CType.crename_id", "content": "theorem CType.crename_id {T : CType n m k} :\n  T.crename FinFun.id = T"}, {"name": "SType.crename_id", "content": "theorem SType.crename_id {S : SType n m k} :\n  S.crename FinFun.id = S"}, {"name": "CBound.crename_id", "content": "theorem CBound.crename_id {b : CBound n k} :\n  b.crename FinFun.id = b"}, {"name": "Subbound.subst", "content": "theorem Subbound.subst\n  (h : Subbound Γ B1 B2)\n  (σ : VarSubst Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.subst", "content": "theorem ESubtyp.subst\n  (h : ESubtyp Γ E1 E2)\n  (σ : VarSubst Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CSubtyp.subst", "content": "theorem CSubtyp.subst\n  (h : CSubtyp Γ T1 T2)\n  (σ : VarSubst Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.subst", "content": "theorem SSubtyp.subst\n  (h : SSubtyp Γ S1 S2)\n  (σ : VarSubst Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Typed.csubst", "content": "theorem Typed.csubst\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (σ : CVarSubst Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "Subcapt.subst", "content": "theorem Subcapt.subst\n  (h : Subcapt Γ C1 C2)\n  (σ : VarSubst Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Subbound.csubst", "content": "theorem Subbound.csubst\n  (h : Subbound Γ B1 B2)\n  (σ : CVarSubst Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "SSubtyp.csubst", "content": "theorem SSubtyp.csubst\n  (h : SSubtyp Γ S1 S2)\n  (σ : CVarSubst Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "CSubtyp.csubst", "content": "theorem CSubtyp.csubst\n  (h : CSubtyp Γ T1 T2)\n  (σ : CVarSubst Γ f Δ) :\n  CSubtyp Δ (T1.crename f) (T2.crename f)"}, {"name": "ESubtyp.csubst", "content": "theorem ESubtyp.csubst\n  (h : ESubtyp Γ E1 E2)\n  (σ : CVarSubst Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "Subcapt.csubst", "content": "theorem Subcapt.csubst\n  (h : Subcapt Γ C1 C2)\n  (σ : CVarSubst Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.subst", "content": "theorem Typed.subst\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (σ : VarSubst Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}], "used_local_defs": [{"name": "Capless.VarRename.boundary", "content": "def VarRename.boundary {Γ : Context n m k} {S : SType n m k} :\n  VarMap\n    ((Γ,c<:*),x:(Label[S.cweaken])^{c=0})\n    FinFun.weaken.ext\n    (((Γ.label S),c<:*),x:(Label[S.weaken.cweaken])^{c=0}) :="}, {"name": "Capless.CVarRename.boundary", "content": "def CVarRename.boundary {Γ : Context n m k} {S : SType n m k} :\n  CVarMap\n    (((Γ.label S),c<:*),x:(Label[S.weaken.cweaken])^{c=0})\n    FinFun.weaken.ext\n    ((((Γ.label S),c:={x=0}),c<:*),x:(Label[S.weaken.cweaken.cweaken])^{c=0}) :="}, {"name": "Capless.CVarSubst.boundary", "content": "def CVarSubst.boundary {Γ : Context n m k} {S : SType n m k} :\n  CVarSubst\n    ((((Γ.label S),c:={x=0}),c<:*),x:(Label[S.weaken.cweaken.cweaken])^{c=0})\n    (FinFun.open 0)\n    (((Γ.label S),c:={x=0}),x:(Label[S.weaken.cweaken])^{c=0}) :="}, {"name": "Capless.VarSubst.boundary", "content": "def VarSubst.boundary {Γ : Context n m k} {S : SType n m k} :\n  VarSubst\n    (((Γ.label S),c:={x=0}),x:(Label[S.weaken.cweaken])^{c=0})\n    (FinFun.open 0)\n    ((Γ.label S),c:={x=0}) :="}], "used_local_lemmas": [{"name": "Capless.Term.copen_cweaken_ext", "content": "theorem Term.copen_cweaken_ext {t : Term n m (k+1)} :\n  (t.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = t"}, {"name": "Capless.EType.copen_cweaken_ext", "content": "theorem EType.copen_cweaken_ext {E : EType n m (k+1)} :\n  (E.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = E"}, {"name": "Capless.CaptureSet.copen_cweaken_ext", "content": "theorem CaptureSet.copen_cweaken_ext {C : CaptureSet n (k+1)} :\n  (C.crename (FinFun.weaken.ext)).crename (FinFun.open 0) = C"}, {"name": "Capless.Term.open_weaken_ext", "content": "theorem Term.open_weaken_ext {t : Term (n+1) m k} :\n  (t.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = t"}, {"name": "Capless.EType.open_weaken_ext", "content": "theorem EType.open_weaken_ext {E : EType (n+1) m k} :\n  (E.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = E"}, {"name": "Capless.CaptureSet.open_weaken_ext", "content": "theorem CaptureSet.open_weaken_ext {C : CaptureSet (n+1) k} :\n  (C.rename (FinFun.weaken.ext)).rename (FinFun.open 0) = C"}], "local_ctx": "import Capless.Typing\n\nimport Capless.Weakening.Typing\n\nimport Capless.Narrowing.Typing\n\nnamespace Capless\n\ndef VarRename.boundary {Γ : Context n m k} {S : SType n m k} :\n  VarMap\n    ((Γ,c<:*),x:(Label[S.cweaken])^{c=0})\n    FinFun.weaken.ext\n    (((Γ.label S),c<:*),x:(Label[S.weaken.cweaken])^{c=0}) :=\n\ndef CVarRename.boundary {Γ : Context n m k} {S : SType n m k} :\n  CVarMap\n    (((Γ.label S),c<:*),x:(Label[S.weaken.cweaken])^{c=0})\n    FinFun.weaken.ext\n    ((((Γ.label S),c:={x=0}),c<:*),x:(Label[S.weaken.cweaken.cweaken])^{c=0}) :=\n\ndef CVarSubst.boundary {Γ : Context n m k} {S : SType n m k} :\n  CVarSubst\n    ((((Γ.label S),c:={x=0}),c<:*),x:(Label[S.weaken.cweaken.cweaken])^{c=0})\n    (FinFun.open 0)\n    (((Γ.label S),c:={x=0}),x:(Label[S.weaken.cweaken])^{c=0}) :=\n\ndef VarSubst.boundary {Γ : Context n m k} {S : SType n m k} :\n  VarSubst\n    (((Γ.label S),c:={x=0}),x:(Label[S.weaken.cweaken])^{c=0})\n    (FinFun.open 0)\n    ((Γ.label S),c:={x=0}) :=", "target_theorem": "theorem Typed.boundary_body_typing {Γ : Context n m k} {S : SType n m k}\n  (ht : Typed ((Γ,c<:*),x:(Label[S.cweaken])^{c=0}) t E Ct) :\n  Typed ((Γ.label S),c:={x=0}) t E Ct :=", "ground_truth_proof": ":= by\n  have h := ht.rename VarRename.boundary\n  have h := h.crename CVarRename.boundary\n  have h := h.csubst CVarSubst.boundary\n  simp [Term.copen_cweaken_ext, EType.copen_cweaken_ext, CaptureSet.copen_cweaken_ext] at h\n  have h := h.subst VarSubst.boundary\n  simp [Term.open_weaken_ext, EType.open_weaken_ext, CaptureSet.open_weaken_ext] at h\n  easy", "nesting_depth": 5, "transitive_dep_count": 220, "subset_aristotle": false, "category": "Type systems"}
{"id": 88, "thm_name": "Capless.Typed.canonical_form_lam'", "thm_stmt": "theorem Typed.canonical_form_lam'\n  (ht : Γ.IsTight)\n  (he1 : t0 = Term.lam T t) (hd2 : SType.Dealias Γ S0 (SType.forall T' E))\n  (he2 : E0 = EType.type (CType.capt Cf S0))\n  (h : Typed Γ t0 E0 Ct0) :\n  CSubtyp Γ T' T ∧\n  Typed (Γ.var T') t E (Cf.weaken ∪ {x=0})", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Narrowing.Typing", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Narrowing.Subtyping", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.Narrowing.TypedCont", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CVarSubst.narrow", "content": "def CVarSubst.narrow\n  (hs : Subbound Γ B' B) :\n  CVarSubst\n    (Γ,c<:B)\n    FinFun.id\n    (Γ,c<:B') :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "TVarSubst.narrow", "content": "def TVarSubst.narrow\n  (hs : SSubtyp Γ S' S) :\n  TVarSubst\n    (Γ.tvar (TBinding.bound S))\n    FinFun.id\n    (Γ.tvar (TBinding.bound S')) :="}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "VarSubst.narrow", "content": "def VarSubst.narrow\n  (hs : CSubtyp Γ T' T) :\n  VarSubst (Γ.var T) FinFun.id (Γ.var T') :="}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "SSubtyp.dealias_right_forall.emotive", "content": "def SSubtyp.dealias_right_forall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.cmotive", "content": "def SSubtyp.dealias_right_forall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.smotive", "content": "def SSubtyp.dealias_right_forall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.forall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "SSubtyp.dealias_forall_inv.smotive", "content": "def SSubtyp.dealias_forall_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T1 E1 T2 E2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.forall T1 E1))\n    (h2 : SType.Dealias Γ S2 (SType.forall T2 E2)),\n    CSubtyp Γ T2 T1 ∧ ESubtyp (Γ.var T2) E1 E2"}, {"name": "SSubtyp.dealias_forall_inv.emotive", "content": "def SSubtyp.dealias_forall_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_forall_inv.cmotive", "content": "def SSubtyp.dealias_forall_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Typed.narrow", "content": "theorem Typed.narrow\n  (h : Typed (Γ,x: T) t E Ct)\n  (hs : CSubtyp Γ T' T) :\n  Typed (Γ,x: T') t E Ct"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "TypedCont.narrow", "content": "theorem TypedCont.narrow\n  (h : TypedCont Γ E1 cont E C0)\n  (hsub : ESubtyp Γ E2 E1) :\n  TypedCont Γ E2 cont E C0"}, {"name": "Subcapt.refl", "content": "theorem Subcapt.refl :\n  Subcapt Γ C C"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "ESubtyp.narrow", "content": "theorem ESubtyp.narrow\n  (h : ESubtyp (Γ.var T) E1 E2)\n  (hs : CSubtyp Γ T' T) :\n  ESubtyp (Γ.var T') E1 E2"}, {"name": "SSubtyp.dealias_right_forall", "content": "theorem SSubtyp.dealias_right_forall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.forall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}, {"name": "SSubtyp.sub_dealias_forall_inv", "content": "theorem SSubtyp.sub_dealias_forall_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.forall T1 E1))\n  (h2 : SType.Dealias Γ S2 (SType.forall T2 E2))\n  (hs : SSubtyp Γ S1 S2) :\n  CSubtyp Γ T2 T1 ∧ ESubtyp (Γ.var T2) E1 E2"}, {"name": "SType.dealias_forall_inj", "content": "theorem SType.dealias_forall_inj\n  (h1 : SType.Dealias Γ S (SType.forall T1 E1))\n  (h2 : SType.Dealias Γ S (SType.forall T2 E2)) :\n  T1 = T2 ∧ E1 = E2"}, {"name": "SType.dealias_forall_inj'", "content": "theorem SType.dealias_forall_inj'\n  (he1 : S1 = SType.forall T1 E1) (he2 : S2 = SType.forall T2 E2)\n  (h1 : SType.Dealias Γ S S1)\n  (h2 : SType.Dealias Γ S S2) :\n  T1 = T2 ∧ E1 = E2"}, {"name": "Subcapt.join", "content": "theorem Subcapt.join\n  (h1 : Γ ⊢ C1 <:c D1)\n  (h2 : Γ ⊢ C2 <:c D2) :\n  Γ ⊢ C1 ∪ C2 <:c D1 ∪ D2"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.canonical_form_lam'\n  (ht : Γ.IsTight)\n  (he1 : t0 = Term.lam T t) (hd2 : SType.Dealias Γ S0 (SType.forall T' E))\n  (he2 : E0 = EType.type (CType.capt Cf S0))\n  (h : Typed Γ t0 E0 Ct0) :\n  CSubtyp Γ T' T ∧\n  Typed (Γ.var T') t E (Cf.weaken ∪ {x=0}) :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he1 | cases he2)\n  case abs =>\n    cases he1; cases he2\n    cases hd2\n    constructor\n    { apply CSubtyp.refl }\n    { aesop }\n  case sub hs ih =>\n    subst he2\n    cases hs\n    rename_i hs\n    cases hs\n    rename_i hsc hs\n    have ⟨T1, E1, hd3⟩ := SSubtyp.dealias_right_forall hs ht hd2\n    have ih := ih ht he1 hd3 rfl\n    have h := SSubtyp.sub_dealias_forall_inv ht hd3 hd2 hs\n    have ⟨hs1, ht1⟩ := ih\n    have ⟨hs2, ht2⟩ := h\n    apply And.intro\n    { apply! CSubtyp.trans }\n    { apply Typed.sub <;> try easy\n      apply ht1.narrow\n      assumption\n      apply Subcapt.join\n      { apply hsc.weaken }\n      { apply Subcapt.refl } }", "nesting_depth": 4, "transitive_dep_count": 106, "subset_aristotle": false, "category": "Type systems"}
{"id": 89, "thm_name": "Capless.Typed.canonical_form_clam'", "thm_stmt": "theorem Typed.canonical_form_clam'\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.cforall B' E))\n  (he1 : t0 = Term.clam B t)\n  (he2 : E0 = EType.type (CType.capt Cf S0))\n  (h : Typed Γ t0 E0 Ct0) :\n  Subbound Γ B' B ∧ Typed (Γ.cvar (CBinding.bound B')) t E Cf.cweaken", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Narrowing.Typing", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Narrowing.Subtyping", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "SSubtyp.dealias_right_cforall.cmotive", "content": "def SSubtyp.dealias_right_cforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_cforall.smotive", "content": "def SSubtyp.dealias_right_cforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {B2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)),\n    ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "SSubtyp.dealias_right_cforall.emotive", "content": "def SSubtyp.dealias_right_cforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "CVarSubst.narrow", "content": "def CVarSubst.narrow\n  (hs : Subbound Γ B' B) :\n  CVarSubst\n    (Γ,c<:B)\n    FinFun.id\n    (Γ,c<:B') :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "SSubtyp.dealias_cforall_inv.emotive", "content": "def SSubtyp.dealias_cforall_inv.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_cforall_inv.smotive", "content": "def SSubtyp.dealias_cforall_inv.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {B1 E1 B2 E2}\n    (ht : Γ.IsTight)\n    (h1 : SType.Dealias Γ S1 (SType.cforall B1 E1))\n    (h2 : SType.Dealias Γ S2 (SType.cforall B2 E2)),\n    Subbound Γ B2 B1 ∧ ESubtyp (Γ.cvar (CBinding.bound B2)) E1 E2"}, {"name": "SSubtyp.dealias_cforall_inv.cmotive", "content": "def SSubtyp.dealias_cforall_inv.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "SSubtyp.dealias_right_cforall", "content": "theorem SSubtyp.dealias_right_cforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)) :\n  ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}, {"name": "Typed.cnarrow", "content": "theorem Typed.cnarrow\n  (h : Typed (Γ,c<:B) t E Ct)\n  (hs : Subbound Γ B' B) :\n  Typed (Γ,c<:B') t E Ct"}, {"name": "SSubtyp.sub_dealias_cforall_inv", "content": "theorem SSubtyp.sub_dealias_cforall_inv\n  (ht : Γ.IsTight)\n  (h1 : SType.Dealias Γ S1 (SType.cforall B1 E1))\n  (h2 : SType.Dealias Γ S2 (SType.cforall B2 E2))\n  (hs : SSubtyp Γ S1 S2) :\n  Subbound Γ B2 B1 ∧ ESubtyp (Γ.cvar (CBinding.bound B2)) E1 E2"}, {"name": "SType.dealias_cforall_inj", "content": "theorem SType.dealias_cforall_inj\n  (h1 : SType.Dealias Γ S (SType.cforall B1 E1))\n  (h2 : SType.Dealias Γ S (SType.cforall B2 E2)) :\n  B1 = B2 ∧ E1 = E2"}, {"name": "SType.dealias_cforall_inj'", "content": "theorem SType.dealias_cforall_inj'\n  (he1 : S1 = SType.cforall B1 E1) (he2 : S2 = SType.cforall B2 E2)\n  (h1 : SType.Dealias Γ S S1)\n  (h2 : SType.Dealias Γ S S2) :\n  B1 = B2 ∧ E1 = E2"}, {"name": "ESubtyp.cnarrow", "content": "theorem ESubtyp.cnarrow\n  (h : ESubtyp (Γ,c<:B) E1 E2)\n  (hs : Subbound Γ B' B) :\n  ESubtyp (Γ,c<:B') E1 E2"}, {"name": "Subbound.trans", "content": "theorem Subbound.trans\n  (h1 : Subbound Γ B1 B2)\n  (h2 : Subbound Γ B2 B3) :\n  Subbound Γ B1 B3"}, {"name": "Subbound.refl", "content": "theorem Subbound.refl {B : CBound n k} :\n  Subbound Γ B B"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.canonical_form_clam'\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.cforall B' E))\n  (he1 : t0 = Term.clam B t)\n  (he2 : E0 = EType.type (CType.capt Cf S0))\n  (h : Typed Γ t0 E0 Ct0) :\n  Subbound Γ B' B ∧ Typed (Γ.cvar (CBinding.bound B')) t E Cf.cweaken :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he1 | cases he2)\n  case cabs =>\n    cases he1; cases he2\n    cases hd\n    apply And.intro\n    { apply Subbound.refl }\n    { trivial }\n  case sub hs ih =>\n    subst he2\n    cases hs\n    rename_i hs\n    cases hs\n    rename_i hsc hs\n    have ⟨B1, E1, hd3⟩ := SSubtyp.dealias_right_cforall hs ht hd\n    have ⟨ih1, ih2⟩ := ih ht hd3 he1 rfl\n    have ⟨h1, h2⟩ := SSubtyp.sub_dealias_cforall_inv ht hd3 hd hs\n    constructor\n    { apply Subbound.trans <;> easy }\n    apply Typed.sub\n    { apply ih2.cnarrow; easy }\n    { apply Subcapt.cweaken; easy }\n    { easy }", "nesting_depth": 4, "transitive_dep_count": 56, "subset_aristotle": false, "category": "Type systems"}
{"id": 90, "thm_name": "Capless.TypedCont.tweaken", "thm_stmt": "theorem TypedCont.tweaken\n  (h : TypedCont Γ E t E' C0) :\n  TypedCont (Γ.tvar S) E.tweaken t.tweaken E'.tweaken C0", "lean_root": "capless-lean", "rel_path": "Capless/Weakening/TypedCont/Type.lean", "imports": ["import Capless.Type.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subcapturing", "import Capless.Store", "import Capless.Weakening.Subtyping"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Subcapt.tweaken", "content": "def Subcapt.tweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.tvar b) ⊢ C1 <:c C2 :="}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "TVarMap.weaken_ext", "content": "def TVarMap.weaken_ext {Γ : Context n m k} :\n  TVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.tvar b).var T.tweaken) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "TVarMap.weaken_cext_ext", "content": "def TVarMap.weaken_cext_ext {Γ : Context n m k} :\n  TVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken\n    (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "Subbound.tweaken", "content": "theorem Subbound.tweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.tvar b) B1 B2"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.tweaken", "content": "theorem SSubtyp.tweaken\n  (h : SSubtyp Γ S1 S2) :\n  SSubtyp (Γ.tvar b) S1.tweaken S2.tweaken"}, {"name": "CSubtyp.tweaken", "content": "theorem CSubtyp.tweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.tvar b) E1.tweaken E2.tweaken"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "ESubtyp.tweaken", "content": "theorem ESubtyp.tweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.tvar b) E1.tweaken E2.tweaken"}, {"name": "EType.trename_rename_comm", "content": "theorem EType.trename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (E.trename g).rename f = (E.rename f).trename g"}, {"name": "CType.trename_rename_comm", "content": "theorem CType.trename_rename_comm (T : CType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (T.trename g).rename f = (T.rename f).trename g"}, {"name": "SType.trename_rename_comm", "content": "theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (S.trename g).rename f = (S.rename f).trename g"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "EType.crename_trename_comm", "content": "theorem EType.crename_trename_comm (E : EType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (E.crename f).trename g = (E.trename g).crename f"}, {"name": "CType.crename_trename_comm", "content": "theorem CType.crename_trename_comm (T : CType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (T.crename f).trename g = (T.trename g).crename f"}, {"name": "SType.crename_trename_comm", "content": "theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (S.crename f).trename g = (S.trename g).crename f"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Typed.tweaken_ext", "content": "theorem Typed.tweaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.tvar b).var T.tweaken) t.tweaken E.tweaken Ct"}, {"name": "Typed.tweaken_cext_ext", "content": "theorem Typed.tweaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) t.tweaken E.tweaken Ct"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.tweaken_ex", "content": "theorem EType.tweaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).tweaken = EType.ex T.tweaken"}, {"name": "Capless.EType.tweaken_weaken", "content": "theorem EType.tweaken_weaken (E : EType n m k) :\n  E.weaken.tweaken = E.tweaken.weaken"}, {"name": "Capless.EType.tweaken_cweaken", "content": "theorem EType.tweaken_cweaken (E : EType n m k) :\n  E.cweaken.tweaken = E.tweaken.cweaken"}, {"name": "Capless.Cont.HasLabel.tweaken", "content": "theorem Cont.HasLabel.tweaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.tweaken x tail.tweaken"}, {"name": "Capless.WellScoped.tweaken", "content": "theorem WellScoped.tweaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.tvar b) cont.tweaken Ct"}], "local_ctx": "import Capless.Store\n\nimport Capless.Weakening.Typing\n\nimport Capless.Weakening.Subtyping\n\nimport Capless.Weakening.Subcapturing\n\nnamespace Capless", "target_theorem": "theorem TypedCont.tweaken\n  (h : TypedCont Γ E t E' C0) :\n  TypedCont (Γ.tvar S) E.tweaken t.tweaken E'.tweaken C0 :=", "ground_truth_proof": ":= by\n  induction h\n  case none =>\n    simp [Cont.tweaken]\n    apply none\n    apply? ESubtyp.tweaken\n  case cons ht hs _ ih =>\n    simp [Cont.tweaken]\n    -- simp [EType.tweaken_type]\n    apply cons\n    { have ht1 := ht.tweaken_ext (b := S)\n      rw [EType.tweaken_weaken] at ht1\n      exact ht1 }\n    { apply hs.tweaken }\n    { exact ih }\n  case conse ht hs _ ih =>\n    simp [Cont.tweaken]\n    simp [EType.tweaken_ex]\n    apply conse\n    { have ht1 := ht.tweaken_cext_ext (b := S)\n      rw [EType.tweaken_weaken] at ht1\n      rw [EType.tweaken_cweaken] at ht1\n      exact ht1 }\n    { apply hs.tweaken }\n    { exact ih }\n  case scope hb _ hs ih =>\n    simp [Cont.tweaken]\n    apply scope\n    have hb1 := Context.LBound.there_tvar (b := S) hb\n    exact hb1\n    simp at ih\n    apply ih\n    have h := hs.tweaken (b:=S)\n    aesop", "nesting_depth": 5, "transitive_dep_count": 125, "subset_aristotle": false, "category": "Type systems"}
{"id": 91, "thm_name": "Capless.SType.crename_rename_comm", "thm_stmt": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f :=\n  match S with\n  | SType.top => by simp [SType.rename, SType.crename]\n  | SType.tvar X => by simp [SType.rename, SType.crename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.crename_rename_comm E1 f g\n    have ih2 := EType.crename_rename_comm E2 f.ext g\n    simp [SType.rename, SType.crename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.crename_rename_comm S f g\n    have ih2 := EType.crename_rename_comm E f g\n    simp [SType.rename, SType.crename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.crename_rename_comm E f g.ext\n    simp [SType.rename, CBound.crename_rename_comm, SType.crename, ih]\n  | SType.box T => by\n    have ih := CType.crename_rename_comm T f g\n    simp [SType.rename, SType.crename, ih]\n  | SType.label S => by\n    have ih := SType.crename_rename_comm S f g\n    simp [SType.rename, SType.crename, ih]", "lean_root": "capless-lean", "rel_path": "Capless/Type/Basic.lean", "imports": ["import Capless.Type.Renaming", "import Capless.Type.Core", "import Capless.CaptureSet"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "Capless.EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "Capless.CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}], "local_ctx": "import Capless.Type.Core\n\nimport Capless.Type.Renaming\n\nnamespace Capless", "target_theorem": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f :=", "ground_truth_proof": ":=\n  match S with\n  | SType.top => by simp [SType.rename, SType.crename]\n  | SType.tvar X => by simp [SType.rename, SType.crename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.crename_rename_comm E1 f g\n    have ih2 := EType.crename_rename_comm E2 f.ext g\n    simp [SType.rename, SType.crename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.crename_rename_comm S f g\n    have ih2 := EType.crename_rename_comm E f g\n    simp [SType.rename, SType.crename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.crename_rename_comm E f g.ext\n    simp [SType.rename, CBound.crename_rename_comm, SType.crename, ih]\n  | SType.box T => by\n    have ih := CType.crename_rename_comm T f g\n    simp [SType.rename, SType.crename, ih]\n  | SType.label S => by\n    have ih := SType.crename_rename_comm S f g\n    simp [SType.rename, SType.crename, ih]", "nesting_depth": 5, "transitive_dep_count": 24, "subset_aristotle": false, "category": "Type systems"}
{"id": 92, "thm_name": "Capless.SType.rename_rename", "thm_stmt": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f) :=\n  match S with\n  | SType.top => by simp [SType.rename]\n  | SType.tvar X => by simp [SType.rename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.rename_rename E1 f g\n    have ih2 := EType.rename_rename E2 f.ext g.ext\n    simp [SType.rename, ih1, ih2, FinFun.ext_comp_ext]\n  | SType.tforall S E => by\n    have ih1 := SType.rename_rename S f g\n    have ih2 := EType.rename_rename E f g\n    simp [SType.rename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.rename_rename E f g\n    simp [SType.rename, CBound.rename_rename, ih]\n  | SType.box T => by\n    have ih := CType.rename_rename T f g\n    simp [SType.rename, ih]\n  | SType.label S => by\n    have ih := SType.rename_rename S f g\n    simp [SType.rename, ih]", "lean_root": "capless-lean", "rel_path": "Capless/Type/Basic.lean", "imports": ["import Capless.Basic", "import Capless.Type.Renaming", "import Capless.Type.Core", "import Capless.CaptureSet"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CaptureSet.rename_rename", "content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n  (C.rename f).rename g = C.rename (g ∘ f)"}, {"name": "FinFun.ext_comp_ext", "content": "theorem FinFun.ext_comp_ext {f : FinFun n n'} {g : FinFun n' n''} :\n  g.ext ∘ f.ext = FinFun.ext (g ∘ f)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.CBound.rename_rename", "content": "theorem CBound.rename_rename {b : CBound n k} :\n  (b.rename f).rename g = b.rename (g ∘ f)"}, {"name": "Capless.EType.rename_rename", "content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (E.rename f).rename g = E.rename (g ∘ f)"}, {"name": "Capless.CType.rename_rename", "content": "theorem CType.rename_rename (T : CType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (T.rename f).rename g = T.rename (g ∘ f)"}], "local_ctx": "import Capless.Type.Core\n\nimport Capless.Type.Renaming\n\nnamespace Capless\n\nend", "target_theorem": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f) :=", "ground_truth_proof": ":=\n  match S with\n  | SType.top => by simp [SType.rename]\n  | SType.tvar X => by simp [SType.rename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.rename_rename E1 f g\n    have ih2 := EType.rename_rename E2 f.ext g.ext\n    simp [SType.rename, ih1, ih2, FinFun.ext_comp_ext]\n  | SType.tforall S E => by\n    have ih1 := SType.rename_rename S f g\n    have ih2 := EType.rename_rename E f g\n    simp [SType.rename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.rename_rename E f g\n    simp [SType.rename, CBound.rename_rename, ih]\n  | SType.box T => by\n    have ih := CType.rename_rename T f g\n    simp [SType.rename, ih]\n  | SType.label S => by\n    have ih := SType.rename_rename S f g\n    simp [SType.rename, ih]", "nesting_depth": 4, "transitive_dep_count": 20, "subset_aristotle": false, "category": "Type systems"}
{"id": 93, "thm_name": "Capless.SType.trename_rename_comm", "thm_stmt": "theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (S.trename g).rename f = (S.rename f).trename g :=\n  match S with\n  | SType.top => by simp [SType.trename, SType.rename]\n  | SType.tvar X => by simp [SType.trename, SType.rename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.trename_rename_comm E1 f g\n    have ih2 := EType.trename_rename_comm E2 f.ext g\n    simp [SType.trename, SType.rename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.trename_rename_comm S f g\n    have ih2 := EType.trename_rename_comm E f g.ext\n    simp [SType.trename, SType.rename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.trename_rename_comm E f g\n    simp [SType.trename, SType.rename, ih]\n  | SType.box T => by\n    have ih := CType.trename_rename_comm T f g\n    simp [SType.trename, SType.rename, ih]\n  | SType.label S => by\n    have ih := SType.trename_rename_comm S f g\n    simp [SType.trename, SType.rename, ih]", "lean_root": "capless-lean", "rel_path": "Capless/Type/Basic.lean", "imports": ["import Capless.Type.Renaming", "import Capless.Type.Core"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.trename_rename_comm", "content": "theorem EType.trename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (E.trename g).rename f = (E.rename f).trename g"}, {"name": "Capless.CType.trename_rename_comm", "content": "theorem CType.trename_rename_comm (T : CType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (T.trename g).rename f = (T.rename f).trename g"}], "local_ctx": "import Capless.Type.Core\n\nimport Capless.Type.Renaming\n\nnamespace Capless\n\nend\n\nend", "target_theorem": "theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (S.trename g).rename f = (S.rename f).trename g :=", "ground_truth_proof": ":=\n  match S with\n  | SType.top => by simp [SType.trename, SType.rename]\n  | SType.tvar X => by simp [SType.trename, SType.rename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.trename_rename_comm E1 f g\n    have ih2 := EType.trename_rename_comm E2 f.ext g\n    simp [SType.trename, SType.rename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.trename_rename_comm S f g\n    have ih2 := EType.trename_rename_comm E f g.ext\n    simp [SType.trename, SType.rename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.trename_rename_comm E f g\n    simp [SType.trename, SType.rename, ih]\n  | SType.box T => by\n    have ih := CType.trename_rename_comm T f g\n    simp [SType.trename, SType.rename, ih]\n  | SType.label S => by\n    have ih := SType.trename_rename_comm S f g\n    simp [SType.trename, SType.rename, ih]", "nesting_depth": 4, "transitive_dep_count": 20, "subset_aristotle": false, "category": "Type systems"}
{"id": 94, "thm_name": "Capless.SType.rename_id", "thm_stmt": "theorem SType.rename_id {S : SType n m k} :\n  S.rename FinFun.id = S :=\n  match S with\n  | SType.top => by simp [SType.rename]\n  | SType.tvar X => by simp [SType.rename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.rename_id (T := E1)\n    have ih2 := EType.rename_id (E := E2)\n    simp [SType.rename, FinFun.id_ext, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.rename_id (S := S)\n    have ih2 := EType.rename_id (E := E)\n    simp [SType.rename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.rename_id (E := E)\n    simp [SType.rename, CBound.rename_id, ih]\n  | SType.box T => by\n    have ih := CType.rename_id (T := T)\n    simp [SType.rename, ih]\n  | SType.label S => by\n    have ih := SType.rename_id (S := S)\n    simp [SType.rename, ih]", "lean_root": "capless-lean", "rel_path": "Capless/Type/Basic.lean", "imports": ["import Capless.Basic", "import Capless.Type.Renaming", "import Capless.Type.Core", "import Capless.CaptureSet"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CaptureSet.rename_id", "content": "theorem CaptureSet.rename_id {C : CaptureSet n k} :\n  C.rename FinFun.id = C"}, {"name": "FinFun.id_ext", "content": "theorem FinFun.id_ext :\n  (FinFun.ext (n := n) id) = id"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.CBound.rename_id", "content": "theorem CBound.rename_id {b : CBound n k} :\n  b.rename FinFun.id = b"}, {"name": "Capless.EType.rename_id", "content": "theorem EType.rename_id {E : EType n m k} :\n  E.rename FinFun.id = E"}, {"name": "Capless.CType.rename_id", "content": "theorem CType.rename_id {T : CType n m k} :\n  T.rename FinFun.id = T"}], "local_ctx": "import Capless.Type.Core\n\nimport Capless.Type.Renaming\n\nnamespace Capless\n\nend\n\nend\n\nend\n\nend\n\nend\n\nend", "target_theorem": "theorem SType.rename_id {S : SType n m k} :\n  S.rename FinFun.id = S :=", "ground_truth_proof": ":=\n  match S with\n  | SType.top => by simp [SType.rename]\n  | SType.tvar X => by simp [SType.rename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.rename_id (T := E1)\n    have ih2 := EType.rename_id (E := E2)\n    simp [SType.rename, FinFun.id_ext, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.rename_id (S := S)\n    have ih2 := EType.rename_id (E := E)\n    simp [SType.rename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.rename_id (E := E)\n    simp [SType.rename, CBound.rename_id, ih]\n  | SType.box T => by\n    have ih := CType.rename_id (T := T)\n    simp [SType.rename, ih]\n  | SType.label S => by\n    have ih := SType.rename_id (S := S)\n    simp [SType.rename, ih]", "nesting_depth": 4, "transitive_dep_count": 21, "subset_aristotle": false, "category": "Type systems"}
{"id": 95, "thm_name": "Capless.Context.cvar_bound_cvar_inst_inv'", "thm_stmt": "theorem Context.cvar_bound_cvar_inst_inv' {Γ : Context n m k}\n  (he1 : Γ' = Context.cvar Γ (CBinding.bound b0))\n  (he2 : b' = CBinding.inst C)\n  (hb : Context.CBound Γ' c b') :\n  ∃ c0 C0, c = c0.succ ∧ C = C0.cweaken ∧ Context.CBound Γ c0 (CBinding.inst C0)", "lean_root": "capless-lean", "rel_path": "Capless/Context.lean", "imports": ["import Capless.Type", "import Capless.CaptureSet"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Capless.TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Capless.CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "Capless.Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Capless.TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "Capless.CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "Capless.TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Capless.CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Capless.Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Capless.Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}], "used_local_lemmas": [{"name": "Capless.CBinding.eq_inst_cweaken_inv", "content": "theorem CBinding.eq_inst_cweaken_inv {b : CBinding n k}\n  (h : CBinding.inst C = b.cweaken) :\n  ∃ C0, b = CBinding.inst C0"}], "local_ctx": "import Capless.Type\n\nimport Capless.CaptureSet\n\nnamespace Capless\n\ninductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k\n\ninductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k\n\ninductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)\n\nnotation:30 Γ \",x:\" T => Context.var Γ T\n\nnotation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)\n\nnotation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)\n\nnotation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)\n\nnotation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)\n\nnotation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)\n\ndef TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)\n\ndef CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)\n\ndef TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken\n\ndef CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken\n\ninductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken\n\ninductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken", "target_theorem": "theorem Context.cvar_bound_cvar_inst_inv' {Γ : Context n m k}\n  (he1 : Γ' = Context.cvar Γ (CBinding.bound b0))\n  (he2 : b' = CBinding.inst C)\n  (hb : Context.CBound Γ' c b') :\n  ∃ c0 C0, c = c0.succ ∧ C = C0.cweaken ∧ Context.CBound Γ c0 (CBinding.inst C0) :=", "ground_truth_proof": ":= by\n  cases hb <;> try (solve | cases he1)\n  case here =>\n    have h := CBinding.eq_inst_cweaken_inv (Eq.symm he2)\n    have ⟨C0, h⟩ := h\n    subst h; cases he1\n  case there_cvar =>\n    have ⟨C0, h⟩ := CBinding.eq_inst_cweaken_inv (Eq.symm he2)\n    subst h; simp [CBinding.cweaken, CBinding.crename] at he2\n    rename_i x0 _ _ _\n    exists x0, C0; simp [CaptureSet.cweaken]; aesop", "nesting_depth": 3, "transitive_dep_count": 32, "subset_aristotle": false, "category": "Type systems"}
{"id": 96, "thm_name": "Capless.SType.crename_crename", "thm_stmt": "theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (S.crename f).crename g = S.crename (g ∘ f) :=\n  match S with\n  | SType.top => by simp [SType.crename]\n  | SType.tvar X => by simp [SType.crename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.crename_crename E1 f g\n    have ih2 := EType.crename_crename E2 f g\n    simp [SType.crename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.crename_crename S f g\n    have ih2 := EType.crename_crename E f g\n    simp [SType.crename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.crename_crename E f.ext g.ext\n    simp [SType.crename, ih, FinFun.ext_comp_ext, CBound.crename_crename]\n  | SType.box T => by\n    have ih := CType.crename_crename T f g\n    simp [SType.crename, ih]\n  | SType.label S => by\n    have ih := SType.crename_crename S f g\n    simp [SType.crename, ih]", "lean_root": "capless-lean", "rel_path": "Capless/Type/Basic.lean", "imports": ["import Capless.Basic", "import Capless.Type.Renaming", "import Capless.Type.Core", "import Capless.CaptureSet"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CaptureSet.crename_crename", "content": "theorem CaptureSet.crename_crename {C : CaptureSet n k} :\n  (C.crename f).crename g = C.crename (g ∘ f)"}, {"name": "FinFun.ext_comp_ext", "content": "theorem FinFun.ext_comp_ext {f : FinFun n n'} {g : FinFun n' n''} :\n  g.ext ∘ f.ext = FinFun.ext (g ∘ f)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.CBound.crename_crename", "content": "theorem CBound.crename_crename {b : CBound n k} :\n  (b.crename f).crename g = b.crename (g ∘ f)"}, {"name": "Capless.EType.crename_crename", "content": "theorem EType.crename_crename (E : EType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (E.crename f).crename g = E.crename (g ∘ f)"}, {"name": "Capless.CType.crename_crename", "content": "theorem CType.crename_crename (T : CType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (T.crename f).crename g = T.crename (g ∘ f)"}], "local_ctx": "import Capless.Type.Core\n\nimport Capless.Type.Renaming\n\nnamespace Capless\n\nend\n\nend\n\nend", "target_theorem": "theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (S.crename f).crename g = S.crename (g ∘ f) :=", "ground_truth_proof": ":=\n  match S with\n  | SType.top => by simp [SType.crename]\n  | SType.tvar X => by simp [SType.crename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.crename_crename E1 f g\n    have ih2 := EType.crename_crename E2 f g\n    simp [SType.crename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.crename_crename S f g\n    have ih2 := EType.crename_crename E f g\n    simp [SType.crename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.crename_crename E f.ext g.ext\n    simp [SType.crename, ih, FinFun.ext_comp_ext, CBound.crename_crename]\n  | SType.box T => by\n    have ih := CType.crename_crename T f g\n    simp [SType.crename, ih]\n  | SType.label S => by\n    have ih := SType.crename_crename S f g\n    simp [SType.crename, ih]", "nesting_depth": 4, "transitive_dep_count": 20, "subset_aristotle": false, "category": "Type systems"}
{"id": 97, "thm_name": "Capless.SType.crename_trename_comm", "thm_stmt": "theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (S.crename f).trename g = (S.trename g).crename f :=\n  match S with\n  | SType.top => by simp [SType.crename, SType.trename]\n  | SType.tvar X => by simp [SType.crename, SType.trename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.crename_trename_comm E1 f g\n    have ih2 := EType.crename_trename_comm E2 f g\n    simp [SType.crename, SType.trename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.crename_trename_comm S f g\n    have ih2 := EType.crename_trename_comm E f g.ext\n    simp [SType.crename, SType.trename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.crename_trename_comm E f.ext g\n    simp [SType.crename, SType.trename, ih]\n  | SType.box T => by\n    have ih := CType.crename_trename_comm T f g\n    simp [SType.crename, SType.trename, ih]\n  | SType.label S => by\n    have ih := SType.crename_trename_comm S f g\n    simp [SType.crename, SType.trename, ih]", "lean_root": "capless-lean", "rel_path": "Capless/Type/Basic.lean", "imports": ["import Capless.Type.Renaming", "import Capless.Type.Core"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.crename_trename_comm", "content": "theorem EType.crename_trename_comm (E : EType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (E.crename f).trename g = (E.trename g).crename f"}, {"name": "Capless.CType.crename_trename_comm", "content": "theorem CType.crename_trename_comm (T : CType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (T.crename f).trename g = (T.trename g).crename f"}], "local_ctx": "import Capless.Type.Core\n\nimport Capless.Type.Renaming\n\nnamespace Capless\n\nend\n\nend\n\nend\n\nend", "target_theorem": "theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (S.crename f).trename g = (S.trename g).crename f :=", "ground_truth_proof": ":=\n  match S with\n  | SType.top => by simp [SType.crename, SType.trename]\n  | SType.tvar X => by simp [SType.crename, SType.trename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.crename_trename_comm E1 f g\n    have ih2 := EType.crename_trename_comm E2 f g\n    simp [SType.crename, SType.trename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.crename_trename_comm S f g\n    have ih2 := EType.crename_trename_comm E f g.ext\n    simp [SType.crename, SType.trename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.crename_trename_comm E f.ext g\n    simp [SType.crename, SType.trename, ih]\n  | SType.box T => by\n    have ih := CType.crename_trename_comm T f g\n    simp [SType.crename, SType.trename, ih]\n  | SType.label S => by\n    have ih := SType.crename_trename_comm S f g\n    simp [SType.crename, SType.trename, ih]", "nesting_depth": 3, "transitive_dep_count": 20, "subset_aristotle": false, "category": "Type systems"}
{"id": 98, "thm_name": "Capless.SType.crename_id", "thm_stmt": "theorem SType.crename_id {S : SType n m k} :\n  S.crename FinFun.id = S :=\n  match S with\n  | SType.top => by simp [SType.crename]\n  | SType.tvar X => by simp [SType.crename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.crename_id (T := E1)\n    have ih2 := EType.crename_id (E := E2)\n    simp [SType.crename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.crename_id (S := S)\n    have ih2 := EType.crename_id (E := E)\n    simp [SType.crename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.crename_id (E := E)\n    simp [SType.crename, CBound.crename_id, FinFun.id_ext, ih]\n  | SType.box T => by\n    have ih := CType.crename_id (T := T)\n    simp [SType.crename, ih]\n  | SType.label S => by\n    have ih := SType.crename_id (S := S)\n    simp [SType.crename, ih]", "lean_root": "capless-lean", "rel_path": "Capless/Type/Basic.lean", "imports": ["import Capless.Basic", "import Capless.Type.Renaming", "import Capless.Type.Core", "import Capless.CaptureSet"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CaptureSet.crename_id", "content": "theorem CaptureSet.crename_id {C : CaptureSet n k} :\n  C.crename FinFun.id = C"}, {"name": "FinFun.id_ext", "content": "theorem FinFun.id_ext :\n  (FinFun.ext (n := n) id) = id"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.CBound.crename_id", "content": "theorem CBound.crename_id {b : CBound n k} :\n  b.crename FinFun.id = b"}, {"name": "Capless.EType.crename_id", "content": "theorem EType.crename_id {E : EType n m k} :\n  E.crename FinFun.id = E"}, {"name": "Capless.CType.crename_id", "content": "theorem CType.crename_id {T : CType n m k} :\n  T.crename FinFun.id = T"}], "local_ctx": "import Capless.Type.Core\n\nimport Capless.Type.Renaming\n\nnamespace Capless\n\nend\n\nend\n\nend\n\nend\n\nend\n\nend\n\nend\n\nend", "target_theorem": "theorem SType.crename_id {S : SType n m k} :\n  S.crename FinFun.id = S :=", "ground_truth_proof": ":=\n  match S with\n  | SType.top => by simp [SType.crename]\n  | SType.tvar X => by simp [SType.crename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.crename_id (T := E1)\n    have ih2 := EType.crename_id (E := E2)\n    simp [SType.crename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.crename_id (S := S)\n    have ih2 := EType.crename_id (E := E)\n    simp [SType.crename, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.crename_id (E := E)\n    simp [SType.crename, CBound.crename_id, FinFun.id_ext, ih]\n  | SType.box T => by\n    have ih := CType.crename_id (T := T)\n    simp [SType.crename, ih]\n  | SType.label S => by\n    have ih := SType.crename_id (S := S)\n    simp [SType.crename, ih]", "nesting_depth": 5, "transitive_dep_count": 21, "subset_aristotle": false, "category": "Type systems"}
{"id": 99, "thm_name": "Capless.SType.trename_trename", "thm_stmt": "theorem SType.trename_trename (S : SType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (S.trename f).trename g = S.trename (g ∘ f) :=\n  match S with\n  | SType.top => by simp [SType.trename]\n  | SType.tvar X => by simp [SType.trename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.trename_trename E1 f g\n    have ih2 := EType.trename_trename E2 f g\n    simp [SType.trename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.trename_trename S f g\n    have ih2 := EType.trename_trename E f.ext g.ext\n    simp [SType.trename, ih1, ih2, FinFun.ext_comp_ext]\n  | SType.cforall B E => by\n    have ih := EType.trename_trename E f g\n    simp [SType.trename, ih]\n  | SType.box T => by\n    have ih := CType.trename_trename T f g\n    simp [SType.trename, ih]\n  | SType.label S => by\n    have ih := SType.trename_trename S f g\n    simp [SType.trename, ih]", "lean_root": "capless-lean", "rel_path": "Capless/Type/Basic.lean", "imports": ["import Capless.Basic", "import Capless.Type.Renaming", "import Capless.Type.Core"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "FinFun.ext_comp_ext", "content": "theorem FinFun.ext_comp_ext {f : FinFun n n'} {g : FinFun n' n''} :\n  g.ext ∘ f.ext = FinFun.ext (g ∘ f)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.trename_trename", "content": "theorem EType.trename_trename (E : EType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (E.trename f).trename g = E.trename (g ∘ f)"}, {"name": "Capless.CType.trename_trename", "content": "theorem CType.trename_trename (T : CType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (T.trename f).trename g = T.trename (g ∘ f)"}], "local_ctx": "import Capless.Type.Core\n\nimport Capless.Type.Renaming\n\nnamespace Capless\n\nend\n\nend\n\nend\n\nend\n\nend", "target_theorem": "theorem SType.trename_trename (S : SType n m k) (f : FinFun m m') (g : FinFun m' m'') :\n  (S.trename f).trename g = S.trename (g ∘ f) :=", "ground_truth_proof": ":=\n  match S with\n  | SType.top => by simp [SType.trename]\n  | SType.tvar X => by simp [SType.trename]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.trename_trename E1 f g\n    have ih2 := EType.trename_trename E2 f g\n    simp [SType.trename, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.trename_trename S f g\n    have ih2 := EType.trename_trename E f.ext g.ext\n    simp [SType.trename, ih1, ih2, FinFun.ext_comp_ext]\n  | SType.cforall B E => by\n    have ih := EType.trename_trename E f g\n    simp [SType.trename, ih]\n  | SType.box T => by\n    have ih := CType.trename_trename T f g\n    simp [SType.trename, ih]\n  | SType.label S => by\n    have ih := SType.trename_trename S f g\n    simp [SType.trename, ih]", "nesting_depth": 4, "transitive_dep_count": 15, "subset_aristotle": false, "category": "Type systems"}
{"id": 100, "thm_name": "Capless.Typed.letex_inv'", "thm_stmt": "theorem Typed.letex_inv' {Γ : Context n m k}\n  (he : t0 = Term.letex t u)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ T E0,\n    Typed Γ t (EType.ex T) Ct0 ∧\n    Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) u E0.cweaken.weaken Ct0.cweaken.weaken ∧\n    ESubtyp Γ E0 E", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Exists", "module": "Init.Core"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "ESubtyp.trans", "content": "theorem ESubtyp.trans\n  (h1 : ESubtyp Γ E1 E2)\n  (h2 : ESubtyp Γ E2 E3) :\n  ESubtyp Γ E1 E3"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.letex_inv' {Γ : Context n m k}\n  (he : t0 = Term.letex t u)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ T E0,\n    Typed Γ t (EType.ex T) Ct0 ∧\n    Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) u E0.cweaken.weaken Ct0.cweaken.weaken ∧\n    ESubtyp Γ E0 E :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he)\n  case letex =>\n    cases he\n    repeat apply Exists.intro\n    constructor; trivial\n    constructor; trivial\n    apply ESubtyp.refl\n  case sub hs ih =>\n    have ih := ih he\n    obtain ⟨T, E0, ht, hu, hs0⟩ := ih\n    have hs1 := ESubtyp.trans hs0 hs\n    repeat apply Exists.intro\n    repeat any_goals apply And.intro\n    { apply Typed.sub\n      easy\n      easy\n      apply ESubtyp.refl }\n    { apply Typed.sub\n      easy\n      apply Subcapt.weaken; apply Subcapt.cweaken; easy\n      apply ESubtyp.refl }\n    { easy }", "nesting_depth": 3, "transitive_dep_count": 92, "subset_aristotle": false, "category": "Type systems"}
{"id": 101, "thm_name": "Capless.SType.trename_id", "thm_stmt": "theorem SType.trename_id {S : SType n m k} :\n  S.trename FinFun.id = S :=\n  match S with\n  | SType.top => by simp [SType.trename]\n  | SType.tvar X => by simp [SType.trename, FinFun.id]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.trename_id (T := E1)\n    have ih2 := EType.trename_id (E := E2)\n    simp [SType.trename, FinFun.id_ext, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.trename_id (S := S)\n    have ih2 := EType.trename_id (E := E)\n    simp [SType.trename, FinFun.id_ext, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.trename_id (E := E)\n    simp [SType.trename, ih]\n  | SType.box T => by\n    have ih := CType.trename_id (T := T)\n    simp [SType.trename, ih]\n  | SType.label S => by\n    have ih := SType.trename_id (S := S)\n    simp [SType.trename, ih]", "lean_root": "capless-lean", "rel_path": "Capless/Type/Basic.lean", "imports": ["import Capless.Basic", "import Capless.Type.Renaming", "import Capless.Type.Core"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "FinFun.id_ext", "content": "theorem FinFun.id_ext :\n  (FinFun.ext (n := n) id) = id"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.trename_id", "content": "theorem EType.trename_id {E : EType n m k} :\n  E.trename FinFun.id = E"}, {"name": "Capless.CType.trename_id", "content": "theorem CType.trename_id {T : CType n m k} :\n  T.trename FinFun.id = T"}], "local_ctx": "import Capless.Type.Core\n\nimport Capless.Type.Renaming\n\nnamespace Capless\n\nend\n\nend\n\nend\n\nend\n\nend\n\nend\n\nend", "target_theorem": "theorem SType.trename_id {S : SType n m k} :\n  S.trename FinFun.id = S :=", "ground_truth_proof": ":=\n  match S with\n  | SType.top => by simp [SType.trename]\n  | SType.tvar X => by simp [SType.trename, FinFun.id]\n  | SType.forall E1 E2 => by\n    have ih1 := CType.trename_id (T := E1)\n    have ih2 := EType.trename_id (E := E2)\n    simp [SType.trename, FinFun.id_ext, ih1, ih2]\n  | SType.tforall S E => by\n    have ih1 := SType.trename_id (S := S)\n    have ih2 := EType.trename_id (E := E)\n    simp [SType.trename, FinFun.id_ext, ih1, ih2]\n  | SType.cforall B E => by\n    have ih := EType.trename_id (E := E)\n    simp [SType.trename, ih]\n  | SType.box T => by\n    have ih := CType.trename_id (T := T)\n    simp [SType.trename, ih]\n  | SType.label S => by\n    have ih := SType.trename_id (S := S)\n    simp [SType.trename, ih]", "nesting_depth": 4, "transitive_dep_count": 16, "subset_aristotle": false, "category": "Type systems"}
{"id": 102, "thm_name": "Capless.WellScoped.weaken", "thm_stmt": "theorem WellScoped.weaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.var T) cont.weaken Ct.weaken", "lean_root": "capless-lean", "rel_path": "Capless/Weakening/TypedCont/Term.lean", "imports": ["import Capless.Type.Basic", "import Capless.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.CaptureSet", "import Capless.Weakening.Subcapturing", "import Capless.Store"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "VarMap.weaken_ext", "content": "def VarMap.weaken_ext {Γ : Context n m k} :\n  VarMap\n    (Γ.var T)\n    FinFun.weaken.ext\n    ((Γ.var P).var T.weaken) :="}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CVarMap.weaken_ext", "content": "def CVarMap.weaken_ext {Γ : Context n m k} :\n  CVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.cvar b).var T.cweaken) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken_cext_ext", "content": "def CVarMap.weaken_cext_ext {Γ : Context n m k} :\n  CVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken.ext\n    (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarMap.weaken_cext_ext", "content": "def VarMap.weaken_cext_ext {Γ : Context n m k} :\n  VarMap\n    ((Γ.cvar (CBinding.bound b)).var T)\n    FinFun.weaken.ext\n    (((Γ.var P).cvar (CBinding.bound b.weaken)).var T.weaken) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.weaken_ext", "content": "def TVarMap.weaken_ext {Γ : Context n m k} :\n  TVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.tvar b).var T.tweaken) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TVarMap.weaken_cext_ext", "content": "def TVarMap.weaken_cext_ext {Γ : Context n m k} :\n  TVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken\n    (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "FinFun.comp_weaken", "content": "theorem FinFun.comp_weaken {f : FinFun n n'} :\n  weaken ∘ f = f.ext ∘ weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "CaptureSet.rename_rename", "content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n  (C.rename f).rename g = C.rename (g ∘ f)"}, {"name": "EType.rename_rename", "content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (E.rename f).rename g = E.rename (g ∘ f)"}, {"name": "CType.rename_rename", "content": "theorem CType.rename_rename (T : CType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (T.rename f).rename g = T.rename (g ∘ f)"}, {"name": "SType.rename_rename", "content": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f)"}, {"name": "CBound.rename_rename", "content": "theorem CBound.rename_rename {b : CBound n k} :\n  (b.rename f).rename g = b.rename (g ∘ f)"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "Typed.weaken_ext", "content": "theorem Typed.weaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.var P).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "Typed.weaken_cext_ext", "content": "theorem Typed.weaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.var P).cvar (CBinding.bound B.weaken)).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.weaken1_weaken", "content": "theorem EType.weaken1_weaken (E : EType n m k) :\n  E.weaken.weaken1 = E.weaken.weaken"}, {"name": "Capless.CaptureSet.weaken1_weaken", "content": "theorem CaptureSet.weaken1_weaken (C : CaptureSet n k) :\n  C.weaken.weaken1 = C.weaken.weaken"}, {"name": "Capless.EType.weaken_ex", "content": "theorem EType.weaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).weaken = EType.ex T.weaken"}, {"name": "Capless.EType.weaken_cweaken", "content": "theorem EType.weaken_cweaken (E : EType n m k) :\n  E.cweaken.weaken = E.weaken.cweaken"}, {"name": "Capless.CaptureSet.weaken_cweaken", "content": "theorem CaptureSet.weaken_cweaken (C : CaptureSet n k) :\n  C.cweaken.weaken = C.weaken.cweaken"}, {"name": "Capless.Cont.HasLabel.weaken", "content": "theorem Cont.HasLabel.weaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.weaken x.succ tail.weaken"}], "local_ctx": "import Capless.Store\n\nimport Capless.Weakening.Typing\n\nimport Capless.Weakening.Subtyping\n\nimport Capless.Weakening.Subcapturing\n\nnamespace Capless", "target_theorem": "theorem WellScoped.weaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.var T) cont.weaken Ct.weaken :=", "ground_truth_proof": ":= by\n  induction h\n  case empty => simp [CaptureSet.weaken]; constructor\n  case union ih1 ih2 =>\n    simp [CaptureSet.weaken] at *\n    apply union <;> aesop\n  case singleton hb _ ih =>\n    apply singleton\n    { simp [FinFun.weaken]\n      have hb1 := Context.Bound.there_var (E':=T) hb\n      simp [CType.weaken, CType.rename] at hb1\n      exact hb1 }\n    { exact ih }\n  case csingleton hb _ ih =>\n    apply csingleton\n    { have hb1 := Context.CBound.there_var (E:=T) hb\n      exact hb1 }\n    { exact ih }\n  case cbound hb _ ih =>\n    apply cbound\n    { have hb1 := Context.CBound.there_var (E:=T) hb\n      exact hb1 }\n    { exact ih }\n  case label hb hs =>\n    apply label\n    { have hb1 := Context.LBound.there_var (E:=T) hb\n      exact hb1 }\n    { apply hs.weaken }", "nesting_depth": 6, "transitive_dep_count": 128, "subset_aristotle": false, "category": "Type systems"}
{"id": 103, "thm_name": "Capless.WellScoped.lweaken", "thm_stmt": "theorem WellScoped.lweaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.label S) cont.weaken Ct.weaken", "lean_root": "capless-lean", "rel_path": "Capless/Weakening/TypedCont/Term.lean", "imports": ["import Capless.Type.Basic", "import Capless.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.CaptureSet", "import Capless.Weakening.Subcapturing", "import Capless.Store"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "VarMap.lweaken", "content": "def VarMap.lweaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.label S) :="}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "VarMap.lweaken_ext", "content": "def VarMap.lweaken_ext {Γ : Context n m k} :\n  VarMap\n    (Γ.var T)\n    FinFun.weaken.ext\n    ((Γ.label P).var T.weaken) :="}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "VarMap.lweaken_cext_ext", "content": "def VarMap.lweaken_cext_ext {Γ : Context n m k} :\n  VarMap\n    ((Γ.cvar (CBinding.bound b)).var T)\n    FinFun.weaken.ext\n    (((Γ.label P).cvar (CBinding.bound b.weaken)).var T.weaken) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "VarMap.weaken_ext", "content": "def VarMap.weaken_ext {Γ : Context n m k} :\n  VarMap\n    (Γ.var T)\n    FinFun.weaken.ext\n    ((Γ.var P).var T.weaken) :="}, {"name": "CVarMap.weaken_ext", "content": "def CVarMap.weaken_ext {Γ : Context n m k} :\n  CVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.cvar b).var T.cweaken) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken_cext_ext", "content": "def CVarMap.weaken_cext_ext {Γ : Context n m k} :\n  CVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken.ext\n    (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarMap.weaken_cext_ext", "content": "def VarMap.weaken_cext_ext {Γ : Context n m k} :\n  VarMap\n    ((Γ.cvar (CBinding.bound b)).var T)\n    FinFun.weaken.ext\n    (((Γ.var P).cvar (CBinding.bound b.weaken)).var T.weaken) :="}, {"name": "TVarMap.weaken_ext", "content": "def TVarMap.weaken_ext {Γ : Context n m k} :\n  TVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.tvar b).var T.tweaken) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TVarMap.weaken_cext_ext", "content": "def TVarMap.weaken_cext_ext {Γ : Context n m k} :\n  TVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken\n    (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "FinFun.comp_weaken", "content": "theorem FinFun.comp_weaken {f : FinFun n n'} :\n  weaken ∘ f = f.ext ∘ weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "CaptureSet.rename_rename", "content": "theorem CaptureSet.rename_rename {C : CaptureSet n k} :\n  (C.rename f).rename g = C.rename (g ∘ f)"}, {"name": "EType.rename_rename", "content": "theorem EType.rename_rename (E : EType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (E.rename f).rename g = E.rename (g ∘ f)"}, {"name": "CType.rename_rename", "content": "theorem CType.rename_rename (T : CType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (T.rename f).rename g = T.rename (g ∘ f)"}, {"name": "SType.rename_rename", "content": "theorem SType.rename_rename (S : SType n m k) (f : FinFun n n') (g : FinFun n' n'') :\n  (S.rename f).rename g = S.rename (g ∘ f)"}, {"name": "CBound.rename_rename", "content": "theorem CBound.rename_rename {b : CBound n k} :\n  (b.rename f).rename g = b.rename (g ∘ f)"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "Typed.weaken_ext", "content": "theorem Typed.weaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.var P).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "Typed.weaken_cext_ext", "content": "theorem Typed.weaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.var P).cvar (CBinding.bound B.weaken)).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "ESubtyp.lweaken", "content": "theorem ESubtyp.lweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.label S) E1.weaken E2.weaken"}, {"name": "CSubtyp.lweaken", "content": "theorem CSubtyp.lweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.label S) E1.weaken E2.weaken"}, {"name": "Typed.lweaken", "content": "theorem Typed.lweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.label S) t.weaken E.weaken Ct.weaken"}, {"name": "SSubtyp.lweaken", "content": "theorem SSubtyp.lweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ S, SSubtyp (Γ.label S) S1.weaken S2.weaken"}, {"name": "Typed.lweaken_cext_ext", "content": "theorem Typed.lweaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.label P).cvar (CBinding.bound B.weaken)).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}, {"name": "Typed.lweaken_ext", "content": "theorem Typed.lweaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.label P).var T.weaken) t.weaken1 E.weaken1 Ct.weaken1"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.weaken1_weaken", "content": "theorem EType.weaken1_weaken (E : EType n m k) :\n  E.weaken.weaken1 = E.weaken.weaken"}, {"name": "Capless.CaptureSet.weaken1_weaken", "content": "theorem CaptureSet.weaken1_weaken (C : CaptureSet n k) :\n  C.weaken.weaken1 = C.weaken.weaken"}, {"name": "Capless.EType.weaken_ex", "content": "theorem EType.weaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).weaken = EType.ex T.weaken"}, {"name": "Capless.EType.weaken_cweaken", "content": "theorem EType.weaken_cweaken (E : EType n m k) :\n  E.cweaken.weaken = E.weaken.cweaken"}, {"name": "Capless.CaptureSet.weaken_cweaken", "content": "theorem CaptureSet.weaken_cweaken (C : CaptureSet n k) :\n  C.cweaken.weaken = C.weaken.cweaken"}, {"name": "Capless.Cont.HasLabel.weaken", "content": "theorem Cont.HasLabel.weaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.weaken x.succ tail.weaken"}, {"name": "Capless.WellScoped.weaken", "content": "theorem WellScoped.weaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.var T) cont.weaken Ct.weaken"}, {"name": "Capless.TypedCont.weaken", "content": "theorem TypedCont.weaken\n  (h : TypedCont Γ E t E' C0) :\n  TypedCont (Γ.var T) E.weaken t.weaken E'.weaken C0.weaken"}, {"name": "Capless.Cont.HasLabel.lweaken", "content": "theorem Cont.HasLabel.lweaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.weaken x.succ tail.weaken"}], "local_ctx": "import Capless.Store\n\nimport Capless.Weakening.Typing\n\nimport Capless.Weakening.Subtyping\n\nimport Capless.Weakening.Subcapturing\n\nnamespace Capless", "target_theorem": "theorem WellScoped.lweaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.label S) cont.weaken Ct.weaken :=", "ground_truth_proof": ":= by\n  induction h\n  case empty => simp [CaptureSet.weaken]; constructor\n  case union ih1 ih2 =>\n    simp [CaptureSet.weaken] at *\n    apply union <;> aesop\n  case singleton hb _ ih =>\n    apply singleton\n    { simp [FinFun.weaken]\n      have hb1 := Context.Bound.there_label (S:=S) hb\n      simp [CaptureSet.weaken, CaptureSet.rename] at hb1\n      exact hb1 }\n    { exact ih }\n  case csingleton hb _ ih =>\n    apply csingleton\n    { have hb1 := Context.CBound.there_label (S:=S) hb\n      exact hb1 }\n    { exact ih }\n  case cbound hb _ ih =>\n    apply cbound\n    { have hb1 := Context.CBound.there_label (S:=S) hb\n      exact hb1 }\n    { exact ih }\n  case label hb hs =>\n    apply label\n    { have hb1 := Context.LBound.there_label (S':=S) hb\n      exact hb1 }\n    { apply hs.lweaken }", "nesting_depth": 7, "transitive_dep_count": 140, "subset_aristotle": false, "category": "Type systems"}
{"id": 104, "thm_name": "Capless.Typed.var_inv'", "thm_stmt": "theorem Typed.var_inv'\n  (he1 : t0 = Term.var x)\n  (he2 : E0 = EType.type T)\n  (h : Typed Γ t0 E0 Ct0) (hb : Γ.Bound x T0) :\n  ∃ C0 S0, Γ.Bound x (S0^C0) ∧ (Γ ⊢ (S0^{x=x}) <: T)", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Exists", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "ESubtyp.sub_type_inv'", "content": "theorem ESubtyp.sub_type_inv'\n  (he : E2 = EType.type T2)\n  (h : ESubtyp Γ E1 E2) :\n  ∃ T1, E1 = EType.type T1 ∧ CSubtyp Γ T1 T2"}, {"name": "Context.bound_lbound_absurd", "content": "theorem Context.bound_lbound_absurd\n  (hb1 : Context.Bound Γ x T)\n  (hb2 : Context.LBound Γ x S) : False"}, {"name": "Context.label_lbound_succ_inv", "content": "theorem Context.label_lbound_succ_inv\n  (hb : Context.LBound (Γ.label l) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.label_lbound_succ_inv'", "content": "theorem Context.label_lbound_succ_inv'\n  (he1 : Γ0 = Γ.label l) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.var_lbound_succ_inv", "content": "theorem Context.var_lbound_succ_inv\n  (hb : Context.LBound (Γ.var T) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.var_lbound_succ_inv'", "content": "theorem Context.var_lbound_succ_inv'\n  (he1 : Γ0 = Γ.var T) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.var_inv'\n  (he1 : t0 = Term.var x)\n  (he2 : E0 = EType.type T)\n  (h : Typed Γ t0 E0 Ct0) (hb : Γ.Bound x T0) :\n  ∃ C0 S0, Γ.Bound x (S0^C0) ∧ (Γ ⊢ (S0^{x=x}) <: T) :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he1 | cases he2)\n  case var C0 S0 hb =>\n    cases he1; cases he2\n    apply Exists.intro C0\n    apply Exists.intro S0\n    constructor\n    { trivial }\n    { apply CSubtyp.refl }\n  case label hbl =>\n    exfalso\n    cases he1\n    apply Context.bound_lbound_absurd hb hbl\n  case sub hs ih =>\n    have h := ESubtyp.sub_type_inv' he2 hs\n    have ⟨T1, he, hs1⟩ := h\n    have ih := ih he1 he hb\n    have ⟨C0, S0, hb, hs0⟩ := ih\n    apply Exists.intro C0\n    apply Exists.intro S0\n    constructor\n    { assumption }\n    { apply CSubtyp.trans <;> assumption }", "nesting_depth": 7, "transitive_dep_count": 58, "subset_aristotle": false, "category": "Type systems"}
{"id": 105, "thm_name": "Capless.Context.bound_lbound_absurd", "thm_stmt": "theorem Context.bound_lbound_absurd\n  (hb1 : Context.Bound Γ x T)\n  (hb2 : Context.LBound Γ x S) : False", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Context.lean", "imports": ["import Capless.Context", "import Capless.Store"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.Context.var_lbound_succ_inv'", "content": "theorem Context.var_lbound_succ_inv'\n  (he1 : Γ0 = Γ.var T) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Capless.Context.var_lbound_succ_inv", "content": "theorem Context.var_lbound_succ_inv\n  (hb : Context.LBound (Γ.var T) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Capless.Context.label_lbound_succ_inv'", "content": "theorem Context.label_lbound_succ_inv'\n  (he1 : Γ0 = Γ.label l) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Capless.Context.label_lbound_succ_inv", "content": "theorem Context.label_lbound_succ_inv\n  (hb : Context.LBound (Γ.label l) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}], "local_ctx": "import Capless.Context\n\nimport Capless.Store\n\nnamespace Capless", "target_theorem": "theorem Context.bound_lbound_absurd\n  (hb1 : Context.Bound Γ x T)\n  (hb2 : Context.LBound Γ x S) : False :=", "ground_truth_proof": ":= by\n  induction Γ\n  case empty => cases hb1\n  case var ih =>\n    cases hb1\n    case here =>\n      cases hb2\n    case there_var =>\n      have ⟨_, _, _⟩ := Context.var_lbound_succ_inv hb2\n      apply! ih\n  case tvar ih =>\n    cases hb1; cases hb2\n    apply ih <;> assumption\n  case cvar ih =>\n    cases hb1; cases hb2\n    apply ih <;> assumption\n  case label ih =>\n    cases hb1\n    case there_label =>\n      have ⟨_, _, _⟩ := Context.label_lbound_succ_inv hb2\n      apply ih <;> assumption", "nesting_depth": 6, "transitive_dep_count": 44, "subset_aristotle": false, "category": "Type systems"}
{"id": 106, "thm_name": "Capless.Typing.inv_subcapt'", "thm_stmt": "theorem Typing.inv_subcapt'\n  (he1 : t0 = Term.var x) (he2 : E0 = EType.type (CType.capt C S))\n  (h : Typed Γ t0 E0 C0) :\n  Subcapt Γ {x=x} C", "lean_root": "capless-lean", "rel_path": "Capless/Typing/Basic.lean", "imports": ["import Capless.Subcapturing", "import Capless.Subcapturing.Basic", "import Capless.Typing", "import Capless.Subtyping.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "abs", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "ESubtyp.type_inv_subcapt", "content": "theorem ESubtyp.type_inv_subcapt\n  (h : ESubtyp Γ E (EType.type (CType.capt C S))) :\n  ∃ C0 S0, E = EType.type (CType.capt C0 S0) ∧ Subcapt Γ C0 C"}, {"name": "ESubtyp.type_inv_subcapt'", "content": "theorem ESubtyp.type_inv_subcapt'\n  (heq : E1 = EType.type (CType.capt C S))\n  (h : ESubtyp Γ E E1) :\n  ∃ C0 S0, E = EType.type (CType.capt C0 S0) ∧ Subcapt Γ C0 C"}, {"name": "Subcapt.refl", "content": "theorem Subcapt.refl :\n  Subcapt Γ C C"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Typing\n\nimport Capless.Subcapturing\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Subtyping.Basic\n\nnamespace Capless", "target_theorem": "theorem Typing.inv_subcapt'\n  (he1 : t0 = Term.var x) (he2 : E0 = EType.type (CType.capt C S))\n  (h : Typed Γ t0 E0 C0) :\n  Subcapt Γ {x=x} C :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he1 | cases he2)\n  case var =>\n    cases he1; cases he2\n    apply Subcapt.refl\n  case label =>\n    cases he1; cases he2\n    apply Subcapt.refl\n  case sub hsub ih =>\n    subst he1 he2\n    have h := ESubtyp.type_inv_subcapt hsub\n    let ⟨C0, S0, he, hs⟩ := h\n    subst he\n    have ih := ih rfl rfl\n    apply Subcapt.trans; trivial; trivial", "nesting_depth": 5, "transitive_dep_count": 24, "subset_aristotle": false, "category": "Type systems"}
{"id": 107, "thm_name": "Capless.Typed.tapp_inv'", "thm_stmt": "theorem Typed.tapp_inv'\n  (he : t0 = Term.tapp x X)\n  (h : Typed Γ t0 E Ct) :\n  ∃ Cf F E0,\n    Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.tforall (SType.tvar X) F))) {x=x}\n    ∧ E0 = F.topen X\n    ∧ ESubtyp Γ E0 E", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Renaming.Capture.Typing", "import Capless.Renaming.Term.Typing", "import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Type.Subcapturing", "import Capless.Inversion.Subtyping", "import Capless.Renaming.Term.Subcapturing", "import Capless.Inversion.Context", "import Capless.Renaming.Type.Subtyping", "import Capless.Tactics", "import Capless.Typing", "import Capless.Renaming.Type.Typing", "import Capless.Subtyping.Basic", "import Capless.Renaming.Capture.Subtyping", "import Capless.Narrowing", "import Capless.Subst.Type.Typing", "import Capless.Weakening.Subcapturing", "import Capless.Renaming.Capture.Subcapturing"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Exists", "module": "Init.Core"}], "used_repo_defs": [{"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "SType.topen", "content": "def SType.topen (S : SType n (m+1) k) (X : Fin m) : SType n m k :=\n  S.trename (FinFun.open X)"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "CType.topen", "content": "def CType.topen (C : CType n (m+1) k) (X : Fin m) : CType n m k :=\n  C.trename (FinFun.open X)"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Term.topen", "content": "def Term.topen (t : Term n (m+1) k) (X : Fin m) : Term n m k :=\n  t.trename (FinFun.open X)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "EType.topen", "content": "def EType.topen (E : EType n (m+1) k) (X : Fin m) : EType n m k :=\n  E.trename (FinFun.open X)"}, {"name": "TVarSubst.open", "content": "def TVarSubst.open :\n  TVarSubst\n    (Γ.tvar (TBinding.bound (SType.tvar X)))\n    (FinFun.open X)\n    Γ :=\n  { map := fun x E hb => by admit /- proof elided -/"}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Typed.topen", "content": "theorem Typed.topen\n  (h : Typed (Γ,X<: (SType.tvar X)) t E Ct) :\n  Typed Γ (t.topen X) (E.topen X) Ct"}, {"name": "Typed.tsubst", "content": "theorem Typed.tsubst\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (σ : TVarSubst Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "ESubtyp.trans", "content": "theorem ESubtyp.trans\n  (h1 : ESubtyp Γ E1 E2)\n  (h2 : ESubtyp Γ E2 E3) :\n  ESubtyp Γ E1 E3"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.tapp_inv'\n  (he : t0 = Term.tapp x X)\n  (h : Typed Γ t0 E Ct) :\n  ∃ Cf F E0,\n    Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.tforall (SType.tvar X) F))) {x=x}\n    ∧ E0 = F.topen X\n    ∧ ESubtyp Γ E0 E :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he)\n  case tapp =>\n    cases he\n    repeat (apply Exists.intro)\n    apply And.intro\n    { trivial }\n    apply And.intro\n    { trivial }\n    { apply ESubtyp.refl }\n  case sub hs ih =>\n    have ih := ih he\n    have ⟨Cf, F, E0, hx, he0, hs0⟩ := ih\n    have h := ESubtyp.trans hs0 hs\n    aesop", "nesting_depth": 6, "transitive_dep_count": 166, "subset_aristotle": false, "category": "Type systems"}
{"id": 108, "thm_name": "Capless.Typed.capp_inv'", "thm_stmt": "theorem Typed.capp_inv'\n  (he : t0 = Term.capp x c)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ Cf F E0,\n    Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.cforall (CBound.upper {c=c}) F))) {x=x} ∧\n    E0 = F.copen c ∧\n    ESubtyp Γ E0 E", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Renaming.Capture.Typing", "import Capless.Renaming.Term.Typing", "import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Type.Subcapturing", "import Capless.Inversion.Subtyping", "import Capless.Renaming.Term.Subcapturing", "import Capless.Inversion.Context", "import Capless.Renaming.Type.Subtyping", "import Capless.Tactics", "import Capless.Typing", "import Capless.Renaming.Type.Typing", "import Capless.Subtyping.Basic", "import Capless.Renaming.Capture.Subtyping", "import Capless.Narrowing", "import Capless.Subst.Capture.Typing", "import Capless.Weakening.Subcapturing", "import Capless.Renaming.Capture.Subcapturing"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Exists", "module": "Init.Core"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "CType.copen", "content": "def CType.copen (C : CType n m (k+1)) (x : Fin k) : CType n m k :=\n  C.crename (FinFun.open x)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "CaptureSet.copen", "content": "def CaptureSet.copen (C : CaptureSet n (k+1)) (x : Fin k) : CaptureSet n k :=\n  C.crename (FinFun.open x)"}, {"name": "SType.copen", "content": "def SType.copen (S : SType n m (k+1)) (x : Fin k) : SType n m k :=\n  S.crename (FinFun.open x)"}, {"name": "Term.copen", "content": "def Term.copen (t : Term n m (k+1)) (c : Fin k) : Term n m k :=\n  t.crename (FinFun.open c)"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "EType.copen", "content": "def EType.copen (E : EType n m (k+1)) (x : Fin k) : EType n m k :=\n  E.crename (FinFun.open x)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CVarSubst.open", "content": "def CVarSubst.open :\n  CVarSubst\n    (Γ.cvar (CBinding.bound (CBound.upper {c=c})))\n    (FinFun.open c)\n    Γ :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Typed.copen", "content": "theorem Typed.copen\n  (h : Typed (Γ,c<:CBound.upper {c=c}) t E Ct) :\n  Typed Γ (t.copen c) (E.copen c) (Ct.copen c)"}, {"name": "Typed.csubst", "content": "theorem Typed.csubst\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (σ : CVarSubst Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "ESubtyp.trans", "content": "theorem ESubtyp.trans\n  (h1 : ESubtyp Γ E1 E2)\n  (h2 : ESubtyp Γ E2 E3) :\n  ESubtyp Γ E1 E3"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.capp_inv'\n  (he : t0 = Term.capp x c)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ Cf F E0,\n    Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.cforall (CBound.upper {c=c}) F))) {x=x} ∧\n    E0 = F.copen c ∧\n    ESubtyp Γ E0 E :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he)\n  case capp =>\n    cases he\n    repeat (apply Exists.intro)\n    apply And.intro\n    { trivial }\n    apply And.intro\n    { trivial }\n    { apply ESubtyp.refl }\n  case sub hs ih =>\n    have ih := ih he\n    have ⟨Cf, F, E0, hx, he0, hs0⟩ := ih\n    have h := ESubtyp.trans hs0 hs\n    aesop", "nesting_depth": 6, "transitive_dep_count": 167, "subset_aristotle": false, "category": "Type systems"}
{"id": 109, "thm_name": "Capless.Typed.boundary_inv'", "thm_stmt": "theorem Typed.boundary_inv' {Γ : Context n m k} {S : SType n m k}\n  (he : t0 = (boundary:S in t))\n  (ht : Typed Γ t0 E Ct) :\n  Typed\n    ((Γ,c<:*),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{})\n    (Ct.cweaken.weaken ∪ {c=0} ∪ {x=0}) ∧\n    (Γ ⊢ (S^{}) <:e E)", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "macro \"split_and\" : tactic => `(tactic| repeat any_goals app", "content": "macro \"split_and\" : tactic => `(tactic| repeat any_goals apply And.intro)"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "ESubtyp.trans", "content": "theorem ESubtyp.trans\n  (h1 : ESubtyp Γ E1 E2)\n  (h2 : ESubtyp Γ E2 E3) :\n  ESubtyp Γ E1 E3"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}, {"name": "Subcapt.refl", "content": "theorem Subcapt.refl :\n  Subcapt Γ C C"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "Subcapt.join", "content": "theorem Subcapt.join\n  (h1 : Γ ⊢ C1 <:c D1)\n  (h2 : Γ ⊢ C2 <:c D2) :\n  Γ ⊢ C1 ∪ C2 <:c D1 ∪ D2"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.boundary_inv' {Γ : Context n m k} {S : SType n m k}\n  (he : t0 = (boundary:S in t))\n  (ht : Typed Γ t0 E Ct) :\n  Typed\n    ((Γ,c<:*),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{})\n    (Ct.cweaken.weaken ∪ {c=0} ∪ {x=0}) ∧\n    (Γ ⊢ (S^{}) <:e E) :=", "ground_truth_proof": ":= by\n  induction ht <;> try (solve | cases he)\n  case boundary =>\n    cases he\n    split_and\n    { easy }\n    { apply ESubtyp.refl }\n  case sub hsc hsub ih =>\n    have ⟨ih, hsub0⟩ := ih he\n    split_and\n    { apply Typed.sub\n      { exact ih }\n      { apply Subcapt.join; apply Subcapt.join\n        all_goals try apply Subcapt.refl\n        apply hsc.cweaken.weaken }\n      apply ESubtyp.refl }\n    { apply ESubtyp.trans <;> easy }", "nesting_depth": 3, "transitive_dep_count": 89, "subset_aristotle": false, "category": "Type systems"}
{"id": 110, "thm_name": "Capless.WellScoped.subcapt", "thm_stmt": "theorem WellScoped.subcapt\n  (hsc : WellScoped Γ cont C)\n  (hs : Γ ⊢ C' <:c C) :\n  WellScoped Γ cont C'", "lean_root": "capless-lean", "rel_path": "Capless/WellScoped/Basic.lean", "imports": ["import Capless.Subcapturing", "import Capless.Store", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "Context.cbound_injective", "content": "theorem Context.cbound_injective\n  (hb1 : Context.CBound Γ c b1)\n  (hb2 : Context.CBound Γ c b2) : b1 = b2"}, {"name": "Context.cvar_cbound_succ_inv", "content": "theorem Context.cvar_cbound_succ_inv\n  (hb : Context.CBound (Γ.cvar p) (Fin.succ X) b) :\n  ∃ b0, Context.CBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Context.cvar_cbound_succ_inv'", "content": "theorem Context.cvar_cbound_succ_inv'\n  (he1 : Γ0 = Γ.cvar p) (he2 : X0 = Fin.succ X)\n  (hb : Context.CBound Γ0 X0 b) :\n  ∃ b0, Context.CBound Γ X b0 ∧ b = b0.cweaken"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.WellScoped.subset", "content": "theorem WellScoped.subset\n  (hsc : WellScoped Γ cont C)\n  (hs : C' ⊆ C) :\n  WellScoped Γ cont C'"}], "local_ctx": "import Capless.Store\n\nimport Capless.Subcapturing\n\nimport Capless.Inversion.Context\n\nnamespace Capless", "target_theorem": "theorem WellScoped.subcapt\n  (hsc : WellScoped Γ cont C)\n  (hs : Γ ⊢ C' <:c C) :\n  WellScoped Γ cont C' :=", "ground_truth_proof": ":= by\n  induction hs generalizing cont\n  case trans => aesop\n  case subset => apply WellScoped.subset <;> easy\n  case union => apply union <;> aesop\n  case var => apply WellScoped.singleton <;> aesop\n  case cinstl =>\n    cases hsc\n    rename_i hb1 _ _ hb2\n    have h := Context.cbound_injective hb1 hb2\n    cases h\n    rename_i h\n    exact h\n    rename_i hb1 _ _ hb2\n    have h := Context.cbound_injective hb1 hb2\n    cases h\n  case cinstr => apply WellScoped.csingleton <;> aesop\n  case cbound => apply WellScoped.cbound <;> aesop", "nesting_depth": 6, "transitive_dep_count": 50, "subset_aristotle": false, "category": "Type systems"}
{"id": 111, "thm_name": "Capless.Typed.letin_inv'", "thm_stmt": "theorem Typed.letin_inv' {Γ : Context n m k}\n  (he : t0 = Term.letin t u)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ T E0,\n    Typed Γ t (EType.type T) Ct0 ∧\n    Typed (Γ.var T) u E0.weaken Ct0.weaken ∧\n    ESubtyp Γ E0 E", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Exists", "module": "Init.Core"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.trans", "content": "theorem ESubtyp.trans\n  (h1 : ESubtyp Γ E1 E2)\n  (h2 : ESubtyp Γ E2 E3) :\n  ESubtyp Γ E1 E3"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.letin_inv' {Γ : Context n m k}\n  (he : t0 = Term.letin t u)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ T E0,\n    Typed Γ t (EType.type T) Ct0 ∧\n    Typed (Γ.var T) u E0.weaken Ct0.weaken ∧\n    ESubtyp Γ E0 E :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he)\n  case letin =>\n    cases he\n    repeat apply Exists.intro\n    constructor; trivial\n    constructor; trivial\n    apply ESubtyp.refl\n  case sub hs ih =>\n    have ih := ih he\n    obtain ⟨T, E0, ht, hu, hs0⟩ := ih\n    have hs1 := ESubtyp.trans hs0 hs\n    repeat apply Exists.intro\n    repeat any_goals apply And.intro\n    { apply Typed.sub\n      easy\n      easy\n      apply ESubtyp.refl }\n    { apply Typed.sub\n      easy\n      apply Subcapt.weaken; easy\n      apply ESubtyp.refl }\n    { easy }", "nesting_depth": 3, "transitive_dep_count": 64, "subset_aristotle": false, "category": "Type systems"}
{"id": 112, "thm_name": "Capless.Typed.label_inv'", "thm_stmt": "theorem Typed.label_inv'\n  (he1 : t0 = Term.var x)\n  (he2 : E0 = EType.type T)\n  (ht : Typed Γ t0 E0 Ct) (hb : Γ.LBound x S1) :\n  ∃ S0, Γ.LBound x S0 ∧ (Γ ⊢ (Label[S0]^{x=x}) <: T)", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Exists", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Context.bound_lbound_absurd", "content": "theorem Context.bound_lbound_absurd\n  (hb1 : Context.Bound Γ x T)\n  (hb2 : Context.LBound Γ x S) : False"}, {"name": "Context.label_lbound_succ_inv", "content": "theorem Context.label_lbound_succ_inv\n  (hb : Context.LBound (Γ.label l) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.label_lbound_succ_inv'", "content": "theorem Context.label_lbound_succ_inv'\n  (he1 : Γ0 = Γ.label l) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.var_lbound_succ_inv", "content": "theorem Context.var_lbound_succ_inv\n  (hb : Context.LBound (Γ.var T) x.succ S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "Context.var_lbound_succ_inv'", "content": "theorem Context.var_lbound_succ_inv'\n  (he1 : Γ0 = Γ.var T) (he2 : x0 = x.succ)\n  (hb : Context.LBound Γ0 x0 S) :\n  ∃ S0, Context.LBound Γ x S0 ∧ S = S0.weaken"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.label_inv'\n  (he1 : t0 = Term.var x)\n  (he2 : E0 = EType.type T)\n  (ht : Typed Γ t0 E0 Ct) (hb : Γ.LBound x S1) :\n  ∃ S0, Γ.LBound x S0 ∧ (Γ ⊢ (Label[S0]^{x=x}) <: T) :=", "ground_truth_proof": ":= by\n  induction ht <;> try (solve | cases he1 | cases he2)\n  case var hb0 =>\n    cases he1; cases he2\n    exfalso\n    apply! Context.bound_lbound_absurd\n  case label hb0 =>\n    cases he1; cases he2\n    apply Exists.intro; apply And.intro\n    { exact hb0 }\n    { apply CSubtyp.refl }\n  case sub hsub ih =>\n    cases he1; cases he2\n    cases hsub\n    have ⟨S0, hb0, hs0⟩ := ih rfl rfl hb\n    apply Exists.intro\n    apply And.intro\n    { easy }\n    { apply CSubtyp.trans <;> easy }", "nesting_depth": 3, "transitive_dep_count": 52, "subset_aristotle": false, "category": "Type systems"}
{"id": 113, "thm_name": "Capless.WellScoped.cweaken", "thm_stmt": "theorem WellScoped.cweaken\n  (h : WellScoped Γ E Ct) :\n  WellScoped (Γ.cvar b) E.cweaken Ct.cweaken", "lean_root": "capless-lean", "rel_path": "Capless/Weakening/TypedCont/Capture.lean", "imports": ["import Capless.Type.Basic", "import Capless.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subtyping", "import Capless.CaptureSet", "import Capless.Weakening.Subcapturing", "import Capless.Store"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "EType.cweaken_type", "content": "@[simp]\ndef EType.cweaken_type :\n  (EType.type T).cweaken = EType.type (T.cweaken) :="}, {"name": "Typed.cweaken_cext_ext", "content": "def Typed.cweaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) t.cweaken1 E.cweaken1 Ct.cweaken1 :="}, {"name": "CVarMap.weaken_cext_ext", "content": "def CVarMap.weaken_cext_ext {Γ : Context n m k} :\n  CVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken.ext\n    (((Γ.cvar b).cvar (CBinding.bound B.cweaken)).var T.cweaken1) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Typed.cweaken_ext", "content": "def Typed.cweaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.cvar b).var T.cweaken) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "CVarMap.weaken_ext", "content": "def CVarMap.weaken_ext {Γ : Context n m k} :\n  CVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.cvar b).var T.cweaken) :="}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "EType.crename_rename_comm", "content": "theorem EType.crename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (E.rename f).crename g = (E.crename g).rename f"}, {"name": "CType.crename_rename_comm", "content": "theorem CType.crename_rename_comm (C : CType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "SType.crename_rename_comm", "content": "theorem SType.crename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun k k') :\n  (S.rename f).crename g = (S.crename g).rename f"}, {"name": "CBound.crename_rename_comm", "content": "theorem CBound.crename_rename_comm {b : CBound n k} :\n  (b.crename f).rename g = (b.rename g).crename f"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "CaptureSet.crename_rename_comm", "content": "theorem CaptureSet.crename_rename_comm {C : CaptureSet n k} {f : FinFun n n'} {g : FinFun k k'} :\n  (C.rename f).crename g = (C.crename g).rename f"}, {"name": "EType.crename_crename", "content": "theorem EType.crename_crename (E : EType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (E.crename f).crename g = E.crename (g ∘ f)"}, {"name": "CType.crename_crename", "content": "theorem CType.crename_crename (T : CType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (T.crename f).crename g = T.crename (g ∘ f)"}, {"name": "SType.crename_crename", "content": "theorem SType.crename_crename (S : SType n m k) (f : FinFun k k') (g : FinFun k' k'') :\n  (S.crename f).crename g = S.crename (g ∘ f)"}, {"name": "CBound.crename_crename", "content": "theorem CBound.crename_crename {b : CBound n k} :\n  (b.crename f).crename g = b.crename (g ∘ f)"}, {"name": "FinFun.comp_weaken", "content": "theorem FinFun.comp_weaken {f : FinFun n n'} :\n  weaken ∘ f = f.ext ∘ weaken"}, {"name": "CaptureSet.weaken_crename", "content": "theorem CaptureSet.weaken_crename {C : CaptureSet n k} :\n  (C.crename f).weaken = C.weaken.crename f"}, {"name": "CaptureSet.crename_crename", "content": "theorem CaptureSet.crename_crename {C : CaptureSet n k} :\n  (C.crename f).crename g = C.crename (g ∘ f)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.cweaken_ex", "content": "theorem EType.cweaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).cweaken = EType.ex T.cweaken1"}, {"name": "Capless.EType.cweaken_weaken", "content": "theorem EType.cweaken_weaken (E : EType n m k) :\n  E.weaken.cweaken = E.cweaken.weaken"}, {"name": "Capless.EType.cweaken1_weaken", "content": "theorem EType.cweaken1_weaken (E : EType n m (k+1)) :\n  E.weaken.cweaken1 = E.cweaken1.weaken"}, {"name": "Capless.CaptureSet.cweaken1_weaken", "content": "theorem CaptureSet.cweaken1_weaken (C : CaptureSet n (k+1)) :\n  C.weaken.cweaken1 = C.cweaken1.weaken"}, {"name": "Capless.EType.cweaken1_cweaken", "content": "theorem EType.cweaken1_cweaken (E : EType n m k) :\n  E.cweaken.cweaken1 = E.cweaken.cweaken"}, {"name": "Capless.CaptureSet.cweaken1_cweaken", "content": "theorem CaptureSet.cweaken1_cweaken (C : CaptureSet n k) :\n  C.cweaken.cweaken1 = C.cweaken.cweaken"}, {"name": "Capless.Cont.HasLabel.cweaken", "content": "theorem Cont.HasLabel.cweaken\n  (h : Cont.HasLabel cont l tail) :\n  Cont.HasLabel (cont.cweaken) l tail.cweaken"}], "local_ctx": "import Capless.Store\n\nimport Capless.Weakening.Typing\n\nimport Capless.Weakening.Subtyping\n\nimport Capless.Weakening.Subcapturing\n\nnamespace Capless", "target_theorem": "theorem WellScoped.cweaken\n  (h : WellScoped Γ E Ct) :\n  WellScoped (Γ.cvar b) E.cweaken Ct.cweaken :=", "ground_truth_proof": ":= by\n  induction h\n  case empty => constructor\n  case union ih1 ih2 => apply union <;> aesop\n  case singleton hb _ ih =>\n    apply singleton\n    { have hb1 := Context.Bound.there_cvar (b := b) hb\n      simp [CType.cweaken, CType.crename] at hb1\n      exact hb1 }\n    { exact ih }\n  case csingleton hb _ ih =>\n    apply csingleton\n    { have hb1 := Context.CBound.there_cvar (b' := b) hb\n      simp [CType.cweaken, CType.crename] at hb1\n      exact hb1 }\n    { exact ih }\n  case cbound hb _ ih =>\n    apply cbound\n    { have hb1 := Context.CBound.there_cvar (b' := b) hb\n      simp [CType.cweaken, CType.crename] at hb1\n      exact hb1 }\n    { exact ih }\n  case label hb hs =>\n    apply label\n    { have hb1 := Context.LBound.there_cvar (b := b) hb\n      simp [CType.cweaken, CType.crename] at hb1\n      exact hb1 }\n    { apply hs.cweaken }", "nesting_depth": 5, "transitive_dep_count": 118, "subset_aristotle": false, "category": "Type systems"}
{"id": 114, "thm_name": "Capless.WellScoped.tweaken", "thm_stmt": "theorem WellScoped.tweaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.tvar b) cont.tweaken Ct", "lean_root": "capless-lean", "rel_path": "Capless/Weakening/TypedCont/Type.lean", "imports": ["import Capless.Type.Basic", "import Capless.Weakening.Typing", "import Capless.Weakening.Subcapturing", "import Capless.Store", "import Capless.Weakening.Subtyping"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "WellScoped", "content": "inductive WellScoped : Context n m k -> Cont n m k -> CaptureSet n k -> Prop where\n| empty :\n  WellScoped Γ cont {}\n| union :\n  WellScoped Γ cont C1 ->\n  WellScoped Γ cont C2 ->\n  WellScoped Γ cont (C1 ∪ C2)\n| singleton :\n  Context.Bound Γ x (S^C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {x=x}\n| csingleton :\n  Context.CBound Γ c (CBinding.inst C) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  WellScoped Γ cont C ->\n  WellScoped Γ cont {c=c}\n| label :\n  Context.LBound Γ x S ->\n  Cont.HasLabel cont x tail ->\n  WellScoped Γ cont {x=x}"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Subcapt.tweaken", "content": "def Subcapt.tweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.tvar b) ⊢ C1 <:c C2 :="}, {"name": "Cont.HasLabel", "content": "inductive Cont.HasLabel : Cont n m k -> Fin n -> Cont n m k -> Prop where\n| here :\n  Cont.HasLabel (Cont.scope l tail) l tail\n| there_val :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.cons t cont) l tail\n| there_tval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.conse t cont) l tail\n| there_cval :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail\n| there_label :\n  Cont.HasLabel cont l tail ->\n  Cont.HasLabel (Cont.scope l' cont) l tail"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Subcapt.weaken", "content": "def Subcapt.weaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ,x: T) ⊢ C1.weaken <:c C2.weaken :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "Subcapt.cweaken", "content": "def Subcapt.cweaken\n  (h : Γ ⊢ C1 <:c C2) :\n  (Γ.cvar b) ⊢ C1.cweaken <:c C2.cweaken :="}, {"name": "TVarMap.weaken_ext", "content": "def TVarMap.weaken_ext {Γ : Context n m k} :\n  TVarMap\n    (Γ.var T)\n    FinFun.weaken\n    ((Γ.tvar b).var T.tweaken) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "TVarMap.weaken_cext_ext", "content": "def TVarMap.weaken_cext_ext {Γ : Context n m k} :\n  TVarMap\n    ((Γ.cvar (CBinding.bound B)).var T)\n    FinFun.weaken\n    (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.weaken", "content": "theorem CSubtyp.weaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "Subbound.tweaken", "content": "theorem Subbound.tweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.tvar b) B1 B2"}, {"name": "Subbound.weaken", "content": "theorem Subbound.weaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.var b) B1.weaken B2.weaken"}, {"name": "ESubtyp.weaken", "content": "theorem ESubtyp.weaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.var T) E1.weaken E2.weaken"}, {"name": "SSubtyp.tweaken", "content": "theorem SSubtyp.tweaken\n  (h : SSubtyp Γ S1 S2) :\n  SSubtyp (Γ.tvar b) S1.tweaken S2.tweaken"}, {"name": "CSubtyp.tweaken", "content": "theorem CSubtyp.tweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.tvar b) E1.tweaken E2.tweaken"}, {"name": "Typed.weaken", "content": "theorem Typed.weaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.var T) t.weaken E.weaken Ct.weaken"}, {"name": "SSubtyp.weaken", "content": "theorem SSubtyp.weaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.var b) S1.weaken S2.weaken"}, {"name": "ESubtyp.tweaken", "content": "theorem ESubtyp.tweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.tvar b) E1.tweaken E2.tweaken"}, {"name": "EType.trename_rename_comm", "content": "theorem EType.trename_rename_comm (E : EType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (E.trename g).rename f = (E.rename f).trename g"}, {"name": "CType.trename_rename_comm", "content": "theorem CType.trename_rename_comm (T : CType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (T.trename g).rename f = (T.rename f).trename g"}, {"name": "SType.trename_rename_comm", "content": "theorem SType.trename_rename_comm (S : SType n m k) (f : FinFun n n') (g : FinFun m m') :\n  (S.trename g).rename f = (S.rename f).trename g"}, {"name": "CSubtyp.cweaken", "content": "theorem CSubtyp.cweaken\n  (h : CSubtyp Γ E1 E2) :\n  CSubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "Subbound.cweaken", "content": "theorem Subbound.cweaken\n  (h : Subbound Γ B1 B2) :\n  Subbound (Γ.cvar b) B1.cweaken B2.cweaken"}, {"name": "ESubtyp.cweaken", "content": "theorem ESubtyp.cweaken\n  (h : ESubtyp Γ E1 E2) :\n  ESubtyp (Γ.cvar b) E1.cweaken E2.cweaken"}, {"name": "EType.crename_trename_comm", "content": "theorem EType.crename_trename_comm (E : EType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (E.crename f).trename g = (E.trename g).crename f"}, {"name": "CType.crename_trename_comm", "content": "theorem CType.crename_trename_comm (T : CType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (T.crename f).trename g = (T.trename g).crename f"}, {"name": "SType.crename_trename_comm", "content": "theorem SType.crename_trename_comm (S : SType n m k) (f : FinFun k k') (g : FinFun m m') :\n  (S.crename f).trename g = (S.trename g).crename f"}, {"name": "SSubtyp.cweaken", "content": "theorem SSubtyp.cweaken\n  (h : SSubtyp Γ S1 S2) :\n  ∀ b, SSubtyp (Γ.cvar b) S1.cweaken S2.cweaken"}, {"name": "Typed.tweaken_ext", "content": "theorem Typed.tweaken_ext {Γ : Context n m k}\n  (h : Typed (Γ.var T) t E Ct) :\n  Typed ((Γ.tvar b).var T.tweaken) t.tweaken E.tweaken Ct"}, {"name": "Typed.tweaken_cext_ext", "content": "theorem Typed.tweaken_cext_ext {Γ : Context n m k}\n  (h : Typed ((Γ.cvar (CBinding.bound B)).var T) t E Ct) :\n  Typed (((Γ.tvar b).cvar (CBinding.bound B)).var T.tweaken) t.tweaken E.tweaken Ct"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.EType.tweaken_ex", "content": "theorem EType.tweaken_ex (T : CType n m (k+1)) :\n  (EType.ex T).tweaken = EType.ex T.tweaken"}, {"name": "Capless.EType.tweaken_weaken", "content": "theorem EType.tweaken_weaken (E : EType n m k) :\n  E.weaken.tweaken = E.tweaken.weaken"}, {"name": "Capless.EType.tweaken_cweaken", "content": "theorem EType.tweaken_cweaken (E : EType n m k) :\n  E.cweaken.tweaken = E.tweaken.cweaken"}, {"name": "Capless.Cont.HasLabel.tweaken", "content": "theorem Cont.HasLabel.tweaken\n  (h : Cont.HasLabel cont x tail) :\n  Cont.HasLabel cont.tweaken x tail.tweaken"}], "local_ctx": "import Capless.Store\n\nimport Capless.Weakening.Typing\n\nimport Capless.Weakening.Subtyping\n\nimport Capless.Weakening.Subcapturing\n\nnamespace Capless", "target_theorem": "theorem WellScoped.tweaken\n  (h : WellScoped Γ cont Ct) :\n  WellScoped (Γ.tvar b) cont.tweaken Ct :=", "ground_truth_proof": ":= by\n  induction h\n  case empty => constructor\n  case union ih1 ih2 => apply union <;> aesop\n  case singleton hb _ ih =>\n    apply singleton\n    { have hb1 := Context.Bound.there_tvar (b := b) hb\n      simp [CType.tweaken, CType.trename] at hb1\n      exact hb1 }\n    { exact ih }\n  case csingleton hb _ ih =>\n    apply csingleton\n    { have hb1 := Context.CBound.there_tvar (b' := b) hb\n      simp [CType.tweaken, CType.trename] at hb1\n      exact hb1 }\n    { exact ih }\n  case cbound hb _ ih =>\n    apply cbound\n    { have hb1 := Context.CBound.there_tvar (b' := b) hb\n      simp [CType.tweaken, CType.trename] at hb1\n      exact hb1 }\n    { exact ih }\n  case label hb hs =>\n    apply label\n    { have hb1 := Context.LBound.there_tvar (b := b) hb\n      simp [CType.tweaken, CType.trename] at hb1\n      exact hb1 }\n    { apply hs.tweaken }", "nesting_depth": 5, "transitive_dep_count": 125, "subset_aristotle": false, "category": "Type systems"}
{"id": 115, "thm_name": "Capless.Typed.forall_inv'", "thm_stmt": "theorem Typed.forall_inv' {v : Term n m k}\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.forall T E))\n  (he : E0 = EType.type (CType.capt Cv S0))\n  (hv : v.IsValue)\n  (ht : Typed Γ v E0 Ct) :\n  ∃ T0 t, v = Term.lam T0 t", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "CType.IsValue", "content": "inductive CType.IsValue : CType n m k -> Prop where\n| capt :\n  S.IsValue ->\n  CType.IsValue (S^C)"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "SSubtyp.dealias_right_forall.emotive", "content": "def SSubtyp.dealias_right_forall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.cmotive", "content": "def SSubtyp.dealias_right_forall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_forall.smotive", "content": "def SSubtyp.dealias_right_forall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.forall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "SSubtyp.dealias_right_forall", "content": "theorem SSubtyp.dealias_right_forall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.forall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.forall T1 E1)"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.forall_inv' {v : Term n m k}\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.forall T E))\n  (he : E0 = EType.type (CType.capt Cv S0))\n  (hv : v.IsValue)\n  (ht : Typed Γ v E0 Ct) :\n  ∃ T0 t, v = Term.lam T0 t :=", "ground_truth_proof": ":= by\n  induction ht <;> try (solve | cases hv | cases he | cases hv; cases he; cases hd)\n  case sub hsub ih =>\n    subst he\n    cases hsub\n    rename_i hsub\n    cases hsub\n    rename_i hsc hss\n    have ⟨T1, E1, hd1⟩ := SSubtyp.dealias_right_forall hss ht hd\n    aesop\n  case abs => aesop", "nesting_depth": 4, "transitive_dep_count": 28, "subset_aristotle": false, "category": "Type systems"}
{"id": 116, "thm_name": "Capless.Typed.tforall_inv'", "thm_stmt": "theorem Typed.tforall_inv' {v : Term n m k}\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.tforall X E))\n  (he : E0 = EType.type (CType.capt Cv S0))\n  (hv : v.IsValue)\n  (ht : Typed Γ v E0 Ct) :\n  ∃ X t, v = Term.tlam X t", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "CType.IsValue", "content": "inductive CType.IsValue : CType n m k -> Prop where\n| capt :\n  S.IsValue ->\n  CType.IsValue (S^C)"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "SSubtyp.dealias_right_tforall.smotive", "content": "def SSubtyp.dealias_right_tforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {T2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)),\n    ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "SSubtyp.dealias_right_tforall.emotive", "content": "def SSubtyp.dealias_right_tforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_tforall.cmotive", "content": "def SSubtyp.dealias_right_tforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "SSubtyp.dealias_right_tforall", "content": "theorem SSubtyp.dealias_right_tforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.tforall T2 E2)) :\n  ∃ T1 E1, SType.Dealias Γ S1 (SType.tforall T1 E1)"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.tforall_inv' {v : Term n m k}\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.tforall X E))\n  (he : E0 = EType.type (CType.capt Cv S0))\n  (hv : v.IsValue)\n  (ht : Typed Γ v E0 Ct) :\n  ∃ X t, v = Term.tlam X t :=", "ground_truth_proof": ":= by\n  induction ht <;> try (solve | cases hv | cases he | cases hv; cases he; cases hd)\n  case sub hsub ih =>\n    subst he\n    cases hsub\n    rename_i hsub\n    cases hsub\n    rename_i hsc hss\n    have ⟨T1, E1, hd1⟩ := SSubtyp.dealias_right_tforall hss ht hd\n    aesop\n  case tabs => aesop", "nesting_depth": 4, "transitive_dep_count": 28, "subset_aristotle": false, "category": "Type systems"}
{"id": 117, "thm_name": "Capless.Typed.cforall_inv'", "thm_stmt": "theorem Typed.cforall_inv' {v : Term n m k}\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.cforall B E))\n  (he : E0 = EType.type (CType.capt Cv S0))\n  (hv : v.IsValue)\n  (ht : Typed Γ v E0 Ct) :\n  ∃ B0 t, v = Term.clam B0 t", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Tactics", "import Capless.Typing", "import Capless.Subtyping.Basic", "import Capless.Inversion.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Inversion.Context"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "SType.Dealias", "content": "inductive SType.Dealias : Context n m k -> SType n m k -> SType n m k -> Prop where\n| refl :\n  Dealias Γ S S\n| step :\n  Context.TBound Γ X (TBinding.inst S) ->\n  Dealias Γ S S' ->\n  Dealias Γ (SType.tvar X) S'"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "CType.IsValue", "content": "inductive CType.IsValue : CType n m k -> Prop where\n| capt :\n  S.IsValue ->\n  CType.IsValue (S^C)"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "SSubtyp.dealias_right_cforall.cmotive", "content": "def SSubtyp.dealias_right_cforall.cmotive\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop := True"}, {"name": "SSubtyp.dealias_right_cforall.smotive", "content": "def SSubtyp.dealias_right_cforall.smotive\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {B2 E2} (ht : Γ.IsTight) (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)),\n    ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "SSubtyp.dealias_right_cforall.emotive", "content": "def SSubtyp.dealias_right_cforall.emotive\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop := True"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "SSubtyp.dealias_right_cforall", "content": "theorem SSubtyp.dealias_right_cforall\n  (h : SSubtyp Γ S1 S2) (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S2 (SType.cforall B2 E2)) :\n  ∃ B1 E1, SType.Dealias Γ S1 (SType.cforall B1 E1)"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.cforall_inv' {v : Term n m k}\n  (ht : Γ.IsTight)\n  (hd : SType.Dealias Γ S0 (SType.cforall B E))\n  (he : E0 = EType.type (CType.capt Cv S0))\n  (hv : v.IsValue)\n  (ht : Typed Γ v E0 Ct) :\n  ∃ B0 t, v = Term.clam B0 t :=", "ground_truth_proof": ":= by\n  induction ht <;> try (solve | cases hv | cases he | cases hv; cases he; cases hd)\n  case sub hsub ih =>\n    subst he\n    cases hsub\n    rename_i hsub\n    cases hsub\n    rename_i hsc hss\n    have ⟨E1, hd1⟩ := SSubtyp.dealias_right_cforall hss ht hd\n    aesop\n  case cabs => aesop", "nesting_depth": 4, "transitive_dep_count": 28, "subset_aristotle": false, "category": "Type systems"}
{"id": 118, "thm_name": "Capless.Context.tight_bound_tvar_absurd", "thm_stmt": "theorem Context.tight_bound_tvar_absurd\n  (ht : Context.IsTight Γ)\n  (hb : Context.TBound Γ X (TBinding.bound S)) : False", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Context.lean", "imports": ["import Capless.Context", "import Capless.Store"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "Exists", "module": "Init.Core"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Capless.Context.var_tbound_inv'", "content": "theorem Context.var_tbound_inv'\n  (he : Γ0 = Γ.var P)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Capless.Context.var_tbound_inv", "content": "theorem Context.var_tbound_inv\n  (hb : Context.TBound (Γ.var P) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Capless.Context.var_tbound_inv_bound", "content": "theorem Context.var_tbound_inv_bound\n  (hb : Context.TBound (Γ.var P) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}, {"name": "Capless.Context.tinst_tbound_bound_inv'", "content": "theorem Context.tinst_tbound_bound_inv'\n  (he1 : Γ0 = Γ.tvar (TBinding.inst P))\n  (he2 : b0 = TBinding.bound S)\n  (hb : Context.TBound Γ0 X b0) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Capless.Context.tinst_tbound_bound_inv", "content": "theorem Context.tinst_tbound_bound_inv\n  (hb : Context.TBound (Γ.tvar (TBinding.inst P)) X (TBinding.bound S)) :\n  ∃ X0 S0, Context.TBound Γ X0 (TBinding.bound S0)\n    ∧ S = SType.tweaken S0\n    ∧ X = X0.succ"}, {"name": "Capless.Context.cvar_tbound_inv'", "content": "theorem Context.cvar_tbound_inv'\n  (he : Γ0 = Γ.cvar p)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Capless.Context.cvar_tbound_inv", "content": "theorem Context.cvar_tbound_inv\n  (hb : Context.TBound (Γ.cvar p) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.cweaken"}, {"name": "Capless.Context.cvar_tbound_inv_bound", "content": "theorem Context.cvar_tbound_inv_bound\n  (hb : Context.TBound (Γ.cvar p) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = S0.cweaken"}, {"name": "Capless.Context.label_tbound_inv'", "content": "theorem Context.label_tbound_inv'\n  (he : Γ0 = Γ.label l)\n  (hb : Context.TBound Γ0 X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Capless.Context.label_tbound_inv", "content": "theorem Context.label_tbound_inv\n  (hb : Context.TBound (Γ.label l) X b) :\n  ∃ b0, Context.TBound Γ X b0 ∧ b = b0.weaken"}, {"name": "Capless.Context.label_tbound_inv_bound", "content": "theorem Context.label_tbound_inv_bound\n  (hb : Context.TBound (Γ.label l) X (TBinding.bound S)) :\n  ∃ S0, Context.TBound Γ X (TBinding.bound S0) ∧ S = SType.weaken S0"}], "local_ctx": "import Capless.Context\n\nimport Capless.Store\n\nnamespace Capless", "target_theorem": "theorem Context.tight_bound_tvar_absurd\n  (ht : Context.IsTight Γ)\n  (hb : Context.TBound Γ X (TBinding.bound S)) : False :=", "ground_truth_proof": ":= by\n  induction ht\n  case empty => cases hb\n  case var ih =>\n    have ⟨S0, hb0, he0⟩ := Context.var_tbound_inv_bound hb\n    aesop\n  case tvar =>\n    have ⟨X0, S0, hb0, hs0, hx0⟩ := Context.tinst_tbound_bound_inv hb\n    aesop\n  case cvar =>\n    have ⟨S0, hb0, he0⟩ := Context.cvar_tbound_inv_bound hb\n    aesop\n  case label =>\n    have ⟨S0, hb0, he0⟩ := Context.label_tbound_inv_bound hb\n    aesop", "nesting_depth": 5, "transitive_dep_count": 83, "subset_aristotle": false, "category": "Type systems"}
{"id": 119, "thm_name": "Capless.Typed.canonical_form_pack'", "thm_stmt": "theorem Typed.canonical_form_pack'\n  (ht : Γ.IsTight)\n  (he1 : t0 = Term.pack C x)\n  (he2 : E0 = EType.ex T)\n  (h : Typed Γ t0 E0 Ct) :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x}", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Renaming.Capture.Typing", "import Capless.Renaming.Term.Typing", "import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Type.Subcapturing", "import Capless.Inversion.Subtyping", "import Capless.Renaming.Term.Subcapturing", "import Capless.Inversion.Context", "import Capless.Subst.Capture.Subtyping", "import Capless.Renaming.Type.Subtyping", "import Capless.Tactics", "import Capless.Typing", "import Capless.Renaming.Type.Typing", "import Capless.Subtyping.Basic", "import Capless.Renaming.Capture.Subtyping", "import Capless.Narrowing", "import Capless.Subst.Capture.Typing", "import Capless.Weakening.Subcapturing", "import Capless.Renaming.Capture.Subcapturing"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "CVarSubst.instantiate", "content": "def CVarSubst.instantiate {Γ : Context n m k} :\n  CVarSubst\n    (Γ.cvar (CBinding.bound CBound.star))\n    FinFun.id\n    (Γ.cvar (CBinding.inst C)) :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "CBound.cweaken_upper", "content": "@[simp]\ndef CBound.cweaken_upper :\n  (CBound.upper C).cweaken = CBound.upper C.cweaken :="}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "FinFun.id", "content": "def FinFun.id : FinFun n n :=\n  fun i => i"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "SSubtyp.csubst_motive3", "content": "def SSubtyp.csubst_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "SSubtyp.csubst_motive1", "content": "def SSubtyp.csubst_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SSubtyp.csubst_motive2", "content": "def SSubtyp.csubst_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarSubst Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "CSubtyp.cinstantiate", "content": "theorem CSubtyp.cinstantiate {Γ : Context n m k}\n  (h : CSubtyp (Γ.cvar (CBinding.bound CBound.star)) T1 T2) :\n  CSubtyp (Γ.cvar (CBinding.inst C)) T1 T2"}, {"name": "CSubtyp.csubst", "content": "theorem CSubtyp.csubst\n  (h : CSubtyp Γ T1 T2)\n  (σ : CVarSubst Γ f Δ) :\n  CSubtyp Δ (T1.crename f) (T2.crename f)"}, {"name": "ESubtyp.csubst", "content": "theorem ESubtyp.csubst\n  (h : ESubtyp Γ E1 E2)\n  (σ : CVarSubst Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "Subbound.csubst", "content": "theorem Subbound.csubst\n  (h : Subbound Γ B1 B2)\n  (σ : CVarSubst Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "SSubtyp.csubst", "content": "theorem SSubtyp.csubst\n  (h : SSubtyp Γ S1 S2)\n  (σ : CVarSubst Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "Subcapt.refl", "content": "theorem Subcapt.refl :\n  Subcapt Γ C C"}, {"name": "Typed.cinstantiate", "content": "theorem Typed.cinstantiate {Γ : Context n m k}\n  (h : Typed (Γ,c<:CBound.star) t E Ct) :\n  Typed (Γ,c:= C) t E Ct"}, {"name": "Typed.csubst", "content": "theorem Typed.csubst\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (σ : CVarSubst Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.canonical_form_pack'\n  (ht : Γ.IsTight)\n  (he1 : t0 = Term.pack C x)\n  (he2 : E0 = EType.ex T)\n  (h : Typed Γ t0 E0 Ct) :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} :=", "ground_truth_proof": ":= by\n  induction h <;> try (solve | cases he1 | cases he2)\n  case pack =>\n    cases he1; cases he2\n    trivial\n  case sub hs ih =>\n    subst he2\n    cases hs\n    rename_i hs\n    have ih := ih ht he1 rfl\n    apply Typed.sub\n    exact ih\n    apply Subcapt.refl\n    constructor\n    apply hs.cinstantiate", "nesting_depth": 6, "transitive_dep_count": 167, "subset_aristotle": false, "category": "Type systems"}
{"id": 120, "thm_name": "Capless.Typed.app_inv'", "thm_stmt": "theorem Typed.app_inv'\n  (he : t0 = Term.app x y)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ T Cf F E0, Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.forall T F))) {x=x}\n    ∧ Typed Γ (Term.var y) (EType.type T) {x=y}\n    ∧ E0 = F.open y\n    ∧ ESubtyp Γ E0 E", "lean_root": "capless-lean", "rel_path": "Capless/Inversion/Typing.lean", "imports": ["import Capless.Subcapturing.Basic", "import Capless.Renaming.Capture.Typing", "import Capless.Renaming.Term.Typing", "import Capless.Renaming.Term.Subtyping", "import Capless.Renaming.Type.Subcapturing", "import Capless.Inversion.Subtyping", "import Capless.Renaming.Term.Subcapturing", "import Capless.Inversion.Context", "import Capless.Renaming.Type.Subtyping", "import Capless.Tactics", "import Capless.Typing", "import Capless.Renaming.Type.Typing", "import Capless.Subtyping.Basic", "import Capless.Subst.Term.Typing", "import Capless.Renaming.Capture.Subtyping", "import Capless.Narrowing", "import Capless.Weakening.Subcapturing", "import Capless.Renaming.Capture.Subcapturing"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.succ", "module": "Init.Data.Fin.Basic"}, {"name": "abs", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> eas", "content": "macro \"apply!\" e:term : tactic => `(tactic| apply $e <;> easy)"}, {"name": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont", "content": "notation:max \"⟨\" σ \" | \" cont \" | \" t \"⟩\" => State.mk σ cont t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "macro \"easy\" : tactic => `(tactic| assumption)", "content": "macro \"easy\" : tactic => `(tactic| assumption)"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try", "content": "macro \"apply?\" e:term : tactic => `(tactic| apply $e <;> try easy)"}, {"name": "CaptureSet", "content": "inductive CaptureSet : Nat -> Nat -> Type where\n| empty : CaptureSet n k\n| union : CaptureSet n k -> CaptureSet n k -> CaptureSet n k\n| singleton : Fin n -> CaptureSet n k\n| csingleton : Fin k -> CaptureSet n k"}, {"name": "Term.open", "content": "def Term.open (t : Term (n+1) m k) (x : Fin n) : Term n m k :=\n  t.rename (FinFun.open x)"}, {"name": "Term.rename", "content": "def Term.rename (t : Term n m k) (f : FinFun n n') : Term n' m k :=\n  match t with\n  | Term.var x => Term.var (f x)\n  | Term.lam E t => Term.lam (E.rename f) (t.rename f.ext)\n  | Term.tlam S t => Term.tlam (S.rename f) (t.rename f)\n  | Term.clam B t => Term.clam (B.rename f) (t.rename f)\n  | Term.pack C x => Term.pack (C.rename f) (f x)\n  | Term.app x y => Term.app (f x) (f y)\n  | Term.invoke x y => Term.invoke (f x) (f y)\n  | Term.tapp x X => Term.tapp (f x) X\n  | Term.capp x c => Term.capp (f x) c\n  | Term.letin t u => Term.letin (t.rename f) (u.rename f.ext)\n  | Term.letex t u => Term.letex (t.rename f) (u.rename f.ext)\n  | Term.bindt S t => Term.bindt (S.rename f) (t.rename f)\n  | Term.bindc c t => Term.bindc (c.rename f) (t.rename f)\n  | Term.boundary S t => Term.boundary (S.rename f) (t.rename f.ext)"}, {"name": "Term", "content": "inductive Term : Nat -> Nat -> Nat -> Type where\n \n| var : Fin n -> Term n m k\n \n| lam : CType n m k -> Term (n+1) m k -> Term n m k\n \n| tlam : SType n m k -> Term n (m+1) k -> Term n m k\n \n| clam : CBound n k -> Term n m (k+1) -> Term n m k\n \n| pack : CaptureSet n k -> Fin n -> Term n m k\n \n| app : Fin n -> Fin n -> Term n m k\n \n| invoke : Fin n -> Fin n -> Term n m k\n \n| tapp : Fin n -> Fin m -> Term n m k\n \n| capp : Fin n -> Fin k -> Term n m k\n \n| letin : Term n m k -> Term (n+1) m k -> Term n m k\n \n| letex : Term n m k -> Term (n+1) m (k+1) -> Term n m k\n \n| bindt : SType n m k -> Term n (m+1) k -> Term n m k\n \n| bindc : CaptureSet n k -> Term n m (k+1) -> Term n m k\n \n| boundary : SType n m k -> Term (n+1) m (k+1) -> Term n m k"}, {"name": "SType.rename", "content": "def SType.rename : SType n m k -> FinFun n n' -> SType n' m k\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.rename f) (E2.rename f.ext)\n| SType.tforall S E, f => SType.tforall (S.rename f) (E.rename f)\n| SType.cforall B E, f => SType.cforall (B.rename f) (E.rename f)\n| SType.box T, f => SType.box (T.rename f)\n| SType.label S, f => SType.label (S.rename f)"}, {"name": "CType.rename", "content": "def CType.rename : CType n m k -> FinFun n n' -> CType n' m k\n| CType.capt C S, f => CType.capt (C.rename f) (S.rename f)"}, {"name": "CType", "content": "inductive CType : Nat -> Nat -> Nat -> Type where\n| capt : CaptureSet n k -> SType n m k -> CType n m k"}, {"name": "FinFun", "content": "def FinFun (n n' : Nat) : Type :=\n  Fin n -> Fin n'"}, {"name": "CBound.rename", "content": "def CBound.rename (b : CBound n k) (f : FinFun n n') : CBound n' k :=\n  match b with\n  | upper C => upper (C.rename f)\n  | star => star"}, {"name": "EType.rename", "content": "def EType.rename : EType n m k -> FinFun n n' -> EType n' m k\n| EType.ex T, f => EType.ex (T.rename f)\n| EType.type T, f => EType.type (T.rename f)"}, {"name": "EType", "content": "inductive EType : Nat -> Nat -> Nat -> Type where\n| ex : CType n m (k+1) -> EType n m k\n| type : CType n m k -> EType n m k"}, {"name": "CaptureSet.rename", "content": "@[simp]\ndef CaptureSet.rename (C : CaptureSet n k) (f : FinFun n n') : CaptureSet n' k :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.rename f) ∪ (C2.rename f)\n  | singleton x => {x=f x}\n  | csingleton c => {c=c}"}, {"name": "SType", "content": "inductive SType : Nat -> Nat -> Nat -> Type where\n| top : SType n m k\n| tvar : Fin m -> SType n m k\n| forall : CType n m k -> EType (n+1) m k -> SType n m k\n| tforall : SType n m k -> EType n (m+1) k -> SType n m k\n| cforall : CBound n k -> EType n m (k+1) -> SType n m k\n| box : CType n m k -> SType n m k\n| label : SType n m k -> SType n m k"}, {"name": "FinFun.ext", "content": "def FinFun.ext (f : FinFun n n') : FinFun (n+1) (n'+1) :="}, {"name": "FinFun.open", "content": "def FinFun.open (x : Fin n) : FinFun (n+1) n :="}, {"name": "EType.open", "content": "def EType.open (E : EType (n+1) m k) (x : Fin n) : EType n m k :=\n  E.rename (FinFun.open x)"}, {"name": "Typed", "content": "inductive Typed : Context n m k -> Term n m k -> EType n m k -> CaptureSet n k -> Prop where\n| var :\n  Context.Bound Γ x (S^C) ->\n  Typed Γ (Term.var x) (S^{x=x}) {x=x}\n| label :\n  Context.LBound Γ x S ->\n  Typed Γ (Term.var x) (Label[S]^{x=x}) {x=x}\n| pack :\n  Typed (Γ.cvar (CBinding.inst C)) (Term.var x) (EType.type T) {x=x} ->\n  Typed Γ (Term.pack C x) (∃c.T) {}\n| sub :\n  Typed Γ t E1 C1 ->\n  (Γ ⊢ C1 <:c C2) ->\n  (Γ ⊢ E1 <:e E2) ->\n  Typed Γ t E2 C2\n| abs {C : CaptureSet n k} :\n  Typed (Γ,x:T) t E (C.weaken ∪ {x=0}) ->\n  Typed Γ (λ(x:T)t) ((∀(x:T)E)^C) {}\n| tabs {C : CaptureSet n k} :\n  Typed (Γ,X<:S) t E C ->\n  Typed Γ (λ[X<:S]t) ((∀[X<:S]E)^C) {}\n| cabs {C : CaptureSet n k} :\n  Typed (Γ,c<:B) t E C.cweaken ->\n  Typed Γ (λ[c<:B]t) ((∀[c<:B]E)^C) {}\n| app :\n  Typed Γ (Term.var x) (EType.type (∀(x:T)E)^C) {x=x} ->\n  Typed Γ (Term.var y) T {x=y} ->\n  Typed Γ (Term.app x y) (E.open y) ({x=x} ∪ {x=y})\n| invoke :\n  Typed Γ (Term.var x) (EType.type (Label[S])^C) {x=x} ->\n  Typed Γ (Term.var y) (S^{}) {x=y} ->\n  Typed Γ (Term.invoke x y) E ({x=x} ∪ {x=y})\n| tapp :\n  Typed Γ (Term.var x) (EType.type (∀[X<:SType.tvar X]E)^C) {x=x} ->\n  Typed Γ (Term.tapp x X) (E.topen X) {x=x}\n| capp :\n  Typed Γ (Term.var x) (EType.type (∀[c<:CBound.upper {c=c}]E)^C) {x=x} ->\n  Typed Γ (Term.capp x c) (E.copen c) {x=x}\n| letin :\n  Typed Γ t (EType.type T) C ->\n  Typed (Γ,x: T) u E.weaken C.weaken ->  \n  Typed Γ (let x=t in u) E C\n| letex :\n  Typed Γ t (EType.ex T) C ->\n  Typed ((Γ,c<:*),x: T) u E.cweaken.weaken C.cweaken.weaken ->\n  Typed Γ (let (c,x)=t in u) E C\n| bindt :\n  Typed (Γ,X:=S) t E.tweaken C ->\n  Typed Γ (let X=S in t) E C\n| bindc :\n  Typed (Γ,c:=C) t E.cweaken C0.cweaken ->\n  Typed Γ (let c=C in t) E C0\n| boundary {Γ : Context n m k} {S : SType n m k} :\n  Typed\n    ((Γ,c<:CBound.star),x: Label[S.cweaken]^{c=0})\n    t\n    (S.cweaken.weaken^{}) (C.cweaken.weaken ∪ {c=0} ∪ {x=0}) ->\n  Typed Γ (boundary: S in t) (S^CaptureSet.empty) C"}, {"name": "TVarSubst.open", "content": "def TVarSubst.open :\n  TVarSubst\n    (Γ.tvar (TBinding.bound (SType.tvar X)))\n    (FinFun.open X)\n    Γ :=\n  { map := fun x E hb => by admit /- proof elided -/"}, {"name": "TVarSubst", "content": "structure TVarSubst (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X S, Γ.TBound X (TBinding.bound S) ->\n    SSubtyp Δ (SType.tvar (f X)) (S.trename f)\n  tmap_inst : ∀ X S, Γ.TBound X (TBinding.inst S) ->\n    Δ.TBound (f X) (TBinding.inst (S.trename f))\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.trename f)"}, {"name": "Context.LBound", "content": "inductive Context.LBound : Context n m k -> Fin n -> SType n m k -> Prop where\n| here : LBound (label Γ0 S) 0 S.weaken\n| there_var :\n  LBound Γ x S ->\n  LBound (var Γ E) x.succ S.weaken\n| there_tvar :\n  LBound Γ x S ->\n  LBound (tvar Γ b) x S.tweaken\n| there_cvar :\n  LBound Γ x S ->\n  LBound (cvar Γ b) x S.cweaken\n| there_label :\n  LBound Γ x S ->\n  LBound (label Γ S') x.succ S.weaken"}, {"name": "CBinding", "content": "inductive CBinding : Nat -> Nat -> Type where\n| bound : CBound n k -> CBinding n k\n| inst : CaptureSet n k -> CBinding n k"}, {"name": "SType.tweaken", "content": "def SType.tweaken (S : SType n m k) : SType n (m+1) k :=\n  S.trename FinFun.weaken"}, {"name": "SType.trename", "content": "def SType.trename : SType n m k -> FinFun m m' -> SType n m' k\n| SType.top, _ => SType.top\n| SType.tvar X, f => SType.tvar (f X)\n| SType.forall E1 E2, f => SType.forall (E1.trename f) (E2.trename f)\n| SType.tforall S E, f => SType.tforall (S.trename f) (E.trename f.ext)\n| SType.cforall B E, f => SType.cforall B (E.trename f)\n| SType.box T, f => SType.box (T.trename f)\n| SType.label S, f => SType.label (S.trename f)"}, {"name": "CType.trename", "content": "def CType.trename : CType n m k -> FinFun m m' -> CType n m' k\n| CType.capt C S, f => CType.capt C (S.trename f)"}, {"name": "EType.trename", "content": "def EType.trename : EType n m k -> FinFun m m' -> EType n m' k\n| EType.ex T, f => EType.ex (T.trename f)\n| EType.type T, f => EType.type (T.trename f)"}, {"name": "FinFun.weaken", "content": "def FinFun.weaken : FinFun n (n+1) :=\n  Fin.succ"}, {"name": "TBinding.tweaken", "content": "def TBinding.tweaken (b : TBinding n m k) : TBinding n (m+1) k :=\n  b.trename FinFun.weaken"}, {"name": "TBinding.trename", "content": "def TBinding.trename (b : TBinding n m k) (f : FinFun m m') : TBinding n m' k :=\n  match b with\n  | bound S => bound (S.trename f)\n  | inst S => inst (S.trename f)"}, {"name": "SSubtyp", "content": "inductive SSubtyp : Context n m k -> SType n m k -> SType n m k -> Prop where\n| top :\n  SSubtyp Γ S SType.top\n| refl :\n  SSubtyp Γ S S\n| trans :\n  SSubtyp Γ S1 S2 ->\n  SSubtyp Γ S2 S3 ->\n  SSubtyp Γ S1 S3\n| tvar :\n  Context.TBound Γ X (TBinding.bound S) ->\n  SSubtyp Γ (SType.tvar X) S\n| tinstl :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ S (SType.tvar X)\n| tinstr :\n  Context.TBound Γ X (TBinding.inst S) ->\n  SSubtyp Γ (SType.tvar X) S\n| boxed :\n  CSubtyp Γ T1 T2 ->\n  SSubtyp Γ (□ T1) (□ T2)\n| label :\n  SSubtyp Γ S2 S1 ->\n  SSubtyp Γ (Label[S1]) (Label[S2])\n| xforall :\n  CSubtyp Γ E2 E1 ->\n  ESubtyp (Context.var Γ E2) F1 F2 ->\n  SSubtyp Γ (SType.forall E1 F1) (SType.forall E2 F2)\n| tforall :\n  SSubtyp Γ S2 S1 ->\n  ESubtyp (Context.tvar Γ (TBinding.bound S2)) E1 E2 ->\n  SSubtyp Γ (SType.tforall S1 E1) (SType.tforall S2 E2)\n| cforall :\n  Subbound Γ B2 B1 ->\n  ESubtyp (Context.cvar Γ (CBinding.bound B2)) E1 E2 ->\n  SSubtyp Γ (SType.cforall B1 E1) (SType.cforall B2 E2)"}, {"name": "TBinding", "content": "inductive TBinding : Nat -> Nat -> Nat -> Type where\n| bound : SType n m k -> TBinding n m k\n| inst : SType n m k -> TBinding n m k"}, {"name": "CType.tweaken", "content": "def CType.tweaken (C : CType n m k) : CType n (m+1) k :=\n  C.trename FinFun.weaken"}, {"name": "SType.open", "content": "def SType.open (S : SType (n+1) m k) (x : Fin n) : SType n m k :=\n  S.rename (FinFun.open x)"}, {"name": "VarSubst.open", "content": "def VarSubst.open\n  (hx : Typed Γ (Term.var x) (EType.type T) Cx) :\n  VarSubst (Γ.var T) (FinFun.open x) Γ :="}, {"name": "VarSubst", "content": "structure VarSubst (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Typed Δ (Term.var (f x)) (EType.type (E.rename f)) {x=f x}\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound (f l) (S.rename f)"}, {"name": "CType.weaken", "content": "def CType.weaken (C : CType n m k) : CType (n+1) m k :=\n  C.rename FinFun.weaken"}, {"name": "TBinding.weaken", "content": "def TBinding.weaken (b : TBinding n m k) : TBinding (n+1) m k :=\n  b.rename FinFun.weaken"}, {"name": "CBinding.rename", "content": "def CBinding.rename (b : CBinding n k) (f : FinFun n n') : CBinding n' k :=\n  match b with\n  | bound b0 => bound (b0.rename f)\n  | inst C => inst (C.rename f)"}, {"name": "TBinding.rename", "content": "def TBinding.rename (b : TBinding n m k) (f : FinFun n n') : TBinding n' m k :=\n  match b with\n  | bound S => bound (S.rename f)\n  | inst S => inst (S.rename f)"}, {"name": "CBinding.weaken", "content": "def CBinding.weaken (b : CBinding n k) : CBinding (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "SType.weaken", "content": "def SType.weaken (S : SType n m k) : SType (n+1) m k :=\n  S.rename FinFun.weaken"}, {"name": "Context", "content": "inductive Context : Nat -> Nat -> Nat -> Type where\n| empty : Context 0 0 0\n| var : Context n m k -> CType n m k -> Context (n+1) m k\n| label : Context n m k -> SType n m k -> Context (n+1) m k\n| tvar : Context n m k -> TBinding n m k -> Context n (m+1) k\n| cvar : Context n m k -> CBinding n k -> Context n m (k+1)"}, {"name": "CBound", "content": "inductive CBound : Nat -> Nat -> Type where\n| upper : CaptureSet n k -> CBound n k\n| star : CBound n k"}, {"name": "CaptureSet.Subset", "content": "inductive CaptureSet.Subset : CaptureSet n k → CaptureSet n k → Prop where\n| empty : Subset {} C\n| rfl : Subset C C\n| union_l :\n  Subset C1 C ->\n  Subset C2 C ->\n  Subset (C1 ∪ C2) C\n| union_rl :\n  Subset C C1 ->\n  Subset C (C1 ∪ C2)\n| union_rr :\n  Subset C C2 ->\n  Subset C (C1 ∪ C2)"}, {"name": "Context.Bound", "content": "inductive Context.Bound : Context n m k -> Fin n -> CType n m k -> Prop where\n| here : Bound (var Γ0 E) 0 E.weaken\n| there_var :\n  Bound Γ x E ->\n  Bound (var Γ E') (Fin.succ x) E.weaken\n| there_tvar :\n  Bound Γ x E ->\n  Bound (tvar Γ b) x E.tweaken\n| there_cvar :\n  Bound Γ x E ->\n  Bound (cvar Γ b) x E.cweaken\n| there_label :\n  Bound Γ x E ->\n  Bound (label Γ S) (Fin.succ x) E.weaken"}, {"name": "Term.IsValue", "content": "@[aesop safe constructors]\ninductive Term.IsValue : Term n m k -> Prop where\n| lam : Term.IsValue (lam E t)\n| tlam : Term.IsValue (tlam S t)\n| clam : Term.IsValue (clam B t)\n| pack : Term.IsValue (pack c x)"}, {"name": "CType.open", "content": "def CType.open (C : CType (n+1) m k) (x : Fin n) : CType n m k :=\n  C.rename (FinFun.open x)"}, {"name": "CVarSubst.open", "content": "def CVarSubst.open :\n  CVarSubst\n    (Γ.cvar (CBinding.bound (CBound.upper {c=c})))\n    (FinFun.open c)\n    Γ :="}, {"name": "CVarSubst", "content": "structure CVarSubst (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c C, Γ.CBound c (CBinding.inst C) ->\n    Δ.CBound (f c) (CBinding.inst (C.crename f))\n  cmap_bound : ∀ c B, Γ.CBound c (CBinding.bound B) ->\n    Subbound Δ (CBound.upper {c=f c}) (B.crename f)\n  lmap : ∀ l S, Γ.LBound l S -> Δ.LBound l (S.crename f)"}, {"name": "CType.cweaken", "content": "def CType.cweaken (C : CType n m k) : CType n m (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "CBound.crename", "content": "def CBound.crename (b : CBound n k) (f : FinFun k k') : CBound n k' :=\n  match b with\n  | upper C => upper (C.crename f)\n  | star => star"}, {"name": "EType.crename", "content": "def EType.crename : EType n m k -> FinFun k k' -> EType n m k'\n| EType.ex T, f => EType.ex (T.crename f.ext)\n| EType.type T, f => EType.type (T.crename f)"}, {"name": "CType.crename", "content": "def CType.crename : CType n m k -> FinFun k k' -> CType n m k'\n| CType.capt C S, f => CType.capt (C.crename f) (S.crename f)"}, {"name": "SType.crename", "content": "def SType.crename : SType n m k -> FinFun k k' -> SType n m k'\n| SType.top, _ => SType.top\n| SType.tvar X, _ => SType.tvar X\n| SType.forall E1 E2, f => SType.forall (E1.crename f) (E2.crename f)\n| SType.tforall S E, f => SType.tforall (S.crename f) (E.crename f)\n| SType.cforall B E, f => SType.cforall (B.crename f) (E.crename f.ext)\n| SType.box T, f => SType.box (T.crename f)\n| SType.label S, f => SType.label (S.crename f)"}, {"name": "CaptureSet.crename", "content": "@[simp]\ndef CaptureSet.crename (C : CaptureSet n k) (f : FinFun k k') : CaptureSet n k' :=\n  match C with\n  | empty => empty\n  | union C1 C2 => (C1.crename f) ∪ (C2.crename f)\n  | singleton x => {x=x}\n  | csingleton c => {c=f c}"}, {"name": "Subcapt", "content": "inductive Subcapt : Context n m k -> CaptureSet n k -> CaptureSet n k -> Prop where\n| trans :\n  Subcapt Γ C1 C2 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ C1 C3\n| subset :\n  C1 ⊆ C2 ->\n  Subcapt Γ C1 C2\n| union :\n  Subcapt Γ C1 C3 ->\n  Subcapt Γ C2 C3 ->\n  Subcapt Γ (C1 ∪ C2) C3\n| var :\n  Context.Bound Γ x (CType.capt C S) ->\n  Subcapt Γ {x=x} C\n| cinstl :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ C {c=c}\n| cinstr :\n  Context.CBound Γ c (CBinding.inst C) ->\n  Subcapt Γ {c=c} C\n| cbound :\n  Context.CBound Γ c (CBinding.bound (CBound.upper C)) ->\n  Subcapt Γ {c=c} C"}, {"name": "TBinding.cweaken", "content": "def TBinding.cweaken (b : TBinding n m k) : TBinding n m (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "CBinding.crename", "content": "def CBinding.crename (b : CBinding n k) (f : FinFun k k') : CBinding n k' :=\n  match b with\n  | bound b0 => bound (b0.crename f)\n  | inst C => inst (C.crename f)"}, {"name": "TBinding.crename", "content": "def TBinding.crename (b : TBinding n m k) (f : FinFun k k') : TBinding n m k' :=\n  match b with\n  | bound S => bound (S.crename f)\n  | inst S => inst (S.crename f)"}, {"name": "CaptureSet.cweaken", "content": "def CaptureSet.cweaken (C : CaptureSet n k) : CaptureSet n (k+1) :=\n  C.crename FinFun.weaken"}, {"name": "SType.cweaken", "content": "def SType.cweaken (S : SType n m k) : SType n m (k+1) :=\n  S.crename FinFun.weaken"}, {"name": "CaptureSet.open", "content": "def CaptureSet.open (C : CaptureSet (n+1) k) (x : Fin n) : CaptureSet n k :=\n  C.rename (FinFun.open x)"}, {"name": "ESubtyp", "content": "inductive ESubtyp : Context n m k -> EType n m k -> EType n m k -> Prop where\n| exist :\n  CSubtyp (Context.cvar Γ (CBinding.bound CBound.star)) T1 T2 ->\n  ESubtyp Γ (EType.ex T1) (EType.ex T2)\n| type :\n  CSubtyp Γ T1 T2 ->\n  ESubtyp Γ (EType.type T1) (EType.type T2)"}, {"name": "CSubtyp", "content": "inductive CSubtyp : Context n m k -> CType n m k -> CType n m k -> Prop where\n| capt :\n  (Γ ⊢ C1 <:c C2) ->\n  SSubtyp Γ S1 S2 ->\n  CSubtyp Γ (CType.capt C1 S1) (CType.capt C2 S2)"}, {"name": "Subbound", "content": "inductive Subbound : Context n m k -> CBound n k -> CBound n k -> Prop where\n| set :\n  (Γ ⊢ C1 <:c C2) ->\n  Subbound Γ (CBound.upper C1) (CBound.upper C2)\n| star :\n  Subbound Γ B CBound.star"}, {"name": "Context.TBound", "content": "inductive Context.TBound : Context n m k -> Fin m -> TBinding n m k -> Prop where\n| here : TBound (tvar Γ0 b) 0 b.tweaken\n| there_var :\n  TBound Γ x b ->\n  TBound (var Γ E) x b.weaken\n| there_tvar :\n  TBound Γ x b ->\n  TBound (tvar Γ b') (Fin.succ x) b.tweaken\n| there_cvar :\n  TBound Γ x b ->\n  TBound (cvar Γ b') x b.cweaken\n| there_label :\n  TBound Γ x b ->\n  TBound (label Γ S) x b.weaken"}, {"name": "TVarMap.cext", "content": "def TVarMap.cext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : CBinding n k) :\n  TVarMap (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CVarMap", "content": "structure CVarMap (Γ : Context n m k) (f : FinFun k k') (Δ : Context n m k') where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.crename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.crename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound (f c) (b.crename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.crename f)"}, {"name": "VarMap", "content": "structure VarMap (Γ : Context n m k) (f : FinFun n n') (Δ : Context n' m k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound (f x) (E.rename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound X (b.rename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c (b.rename f)\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound (f x) (S.rename f)"}, {"name": "TVarMap", "content": "structure TVarMap (Γ : Context n m k) (f : FinFun m m') (Δ : Context n m' k) where\n  map : ∀ x E, Γ.Bound x E -> Δ.Bound x (E.trename f)\n  tmap : ∀ X b, Γ.TBound X b -> Δ.TBound (f X) (b.trename f)\n  cmap : ∀ c b, Γ.CBound c b -> Δ.CBound c b\n  lmap : ∀ x S, Γ.LBound x S -> Δ.LBound x (S.trename f)"}, {"name": "CVarSubst.text", "content": "def CVarSubst.text {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.tvar T) f (Δ.tvar (T.crename f)) :="}, {"name": "SType.IsVar", "content": "inductive SType.IsVar : SType n m k -> Prop where\n| tvar : SType.IsVar (SType.tvar X)"}, {"name": "Context.IsTight", "content": "inductive Context.IsTight : Context n m k -> Prop where\n| empty : Context.IsTight Context.empty\n| var :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.var T)\n| tvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.tvar (TBinding.inst S))\n| cvar :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.cvar (CBinding.inst C))\n| label :\n  Context.IsTight Γ ->\n  Context.IsTight (Γ.label S)"}, {"name": "VarMap.ext", "content": "def VarMap.ext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (E : CType n m k) :\n  VarMap (Γ.var E) f.ext (Δ.var (E.rename f)) :="}, {"name": "EType.rename_open", "content": "def EType.rename_open :\n  (EType.open E x).rename f = (E.rename f.ext).open (f x) :="}, {"name": "CVarSubst.ext", "content": "def CVarSubst.ext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ)\n  (T : CType n m k) :\n  CVarSubst (Γ.var T) f (Δ.var (T.crename f)) :="}, {"name": "CBound.weaken_upper", "content": "@[simp]\ndef CBound.weaken_upper :\n  (CBound.upper C).weaken = CBound.upper C.weaken :="}, {"name": "CaptureSet.weaken", "content": "def CaptureSet.weaken (C : CaptureSet n k) : CaptureSet (n+1) k :=\n  C.rename FinFun.weaken"}, {"name": "CBound.weaken", "content": "def CBound.weaken (b : CBound n k) : CBound (n+1) k :=\n  b.rename FinFun.weaken"}, {"name": "Term.crename", "content": "def Term.crename (t : Term n m k) (f : FinFun k k') : Term n m k' :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.crename f) (t.crename f)\n  | Term.tlam S t => Term.tlam (S.crename f) (t.crename f)\n  | Term.clam B t => Term.clam (B.crename f) (t.crename f.ext)\n  | Term.pack C x => Term.pack (C.crename f) x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x X\n  | Term.capp x c => Term.capp x (f c)\n  | Term.letin t u => Term.letin (t.crename f) (u.crename f)\n  | Term.letex t u => Term.letex (t.crename f) (u.crename f.ext)\n  | Term.bindt S t => Term.bindt (S.crename f) (t.crename f)\n  | Term.bindc c t => Term.bindc (c.crename f) (t.crename f.ext)\n  | Term.boundary S t => Term.boundary (S.crename f) (t.crename f.ext)"}, {"name": "Context.CBound", "content": "inductive Context.CBound : Context n m k -> Fin k -> CBinding n k -> Prop where\n| here : CBound (cvar Γ0 b) 0 b.cweaken\n| there_var :\n  CBound Γ x b ->\n  CBound (var Γ E) x b.weaken\n| there_tvar :\n  CBound Γ x b ->\n  CBound (tvar Γ b') x b\n| there_cvar :\n  CBound Γ x b ->\n  CBound (cvar Γ b') (Fin.succ x) b.cweaken\n| there_label :\n  CBound Γ x b ->\n  CBound (label Γ S) x b.weaken"}, {"name": "SSubtyp.crename_motive2", "content": "def SSubtyp.crename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename_motive3", "content": "def SSubtyp.crename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "SSubtyp.crename_motive1", "content": "def SSubtyp.crename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {k'} (f : FinFun k k') (Δ : Context n m k') (ρ : CVarMap Γ f Δ),\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "SType.IsValue", "content": "inductive SType.IsValue : SType n m k -> Prop where\n| xforall : SType.IsValue (∀(x:T)U)\n| tforall : SType.IsValue (∀[X<:S]T)\n| cforall : SType.IsValue (∀[c<:B]T)\n| box : SType.IsValue (□ T)\n\n@[aesop safe [constructors, cases]]"}, {"name": "CVarMap.text", "content": "def CVarMap.text {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : TBinding n m k) :\n  CVarMap (Γ.tvar b) f (Δ.tvar (b.crename f)) :="}, {"name": "CVarMap.cext", "content": "def CVarMap.cext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (b : CBinding n k) :\n  CVarMap (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "CVarMap.ext", "content": "def CVarMap.ext {Γ : Context n m k} {Δ : Context n m k'}\n  (ρ : CVarMap Γ f Δ) (E : CType n m k) :\n  CVarMap (Γ.var E) f (Δ.var (E.crename f)) :="}, {"name": "VarMap.cext", "content": "def VarMap.cext {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : CBinding n k) :\n  VarMap (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "TVarMap.ext", "content": "def TVarMap.ext {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (E : CType n m k) :\n  TVarMap (Γ.var E) f (Δ.var (E.trename f)) :="}, {"name": "Term.trename", "content": "def Term.trename (t : Term n m k) (f : FinFun m m') : Term n m' k :=\n  match t with\n  | Term.var x => Term.var x\n  | Term.lam E t => Term.lam (E.trename f) (t.trename f)\n  | Term.tlam S t => Term.tlam (S.trename f) (t.trename f.ext)\n  | Term.clam B t => Term.clam B (t.trename f)\n  | Term.pack c x => Term.pack c x\n  | Term.app x y => Term.app x y\n  | Term.invoke x y => Term.invoke x y\n  | Term.tapp x X => Term.tapp x (f X)\n  | Term.capp x c => Term.capp x c\n  | Term.letin t u => Term.letin (t.trename f) (u.trename f)\n  | Term.letex t u => Term.letex (t.trename f) (u.trename f)\n  | Term.bindt S t => Term.bindt (S.trename f) (t.trename f.ext)\n  | Term.bindc c t => Term.bindc c (t.trename f)\n  | Term.boundary S t => Term.boundary (S.trename f) (t.trename f)"}, {"name": "VarMap.text", "content": "def VarMap.text {Γ : Context n m k} {Δ : Context n' m k}\n  (ρ : VarMap Γ f Δ) (b : TBinding n m k) :\n  VarMap (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "TVarMap.text", "content": "def TVarMap.text {Γ : Context n m k} {Δ : Context n m' k}\n  (ρ : TVarMap Γ f Δ) (b : TBinding n m k) :\n  TVarMap (Γ.tvar b) f.ext (Δ.tvar (b.trename f)) :="}, {"name": "TVarSubst.ext", "content": "def TVarSubst.ext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : CType n m k) :\n  TVarSubst (Γ.var T) f (Δ.var (T.trename f)) :="}, {"name": "EType.weaken", "content": "def EType.weaken (E : EType n m k) : EType (n+1) m k :=\n  E.rename FinFun.weaken"}, {"name": "CVarMap.weaken", "content": "def CVarMap.weaken {Γ : Context n m k} :\n  CVarMap Γ FinFun.weaken (Γ.cvar b) :="}, {"name": "Cont.weaken", "content": "def Cont.weaken : Cont n m k -> Cont (n+1) m k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.weaken1 cont.weaken\n| Cont.conse t cont => Cont.conse t.weaken1 cont.weaken\n| Cont.scope x cont => Cont.scope x.succ cont.weaken"}, {"name": "Cont", "content": "inductive Cont : Nat -> Nat -> Nat -> Type where\n| none : Cont n m k\n| cons :\n  (t : Term (n+1) m k) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| conse :\n  (t : Term (n+1) m (k+1)) ->\n  (cont : Cont n m k) ->\n  Cont n m k\n| scope :\n  (l : Fin n) ->\n  Cont n m k ->\n  Cont n m k"}, {"name": "TypedCont", "content": "inductive TypedCont : Context n m k -> EType n m k -> Cont n m k -> EType n m k -> CaptureSet n k -> Prop where\n| none :\n  ESubtyp Γ E E' ->\n  TypedCont Γ E Cont.none E' {}\n| cons {Ct : CaptureSet n k} :\n  Typed (Γ,x: T) t (EType.weaken E) Ct.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.type T) (Cont.cons t cont) E' (C ∪ Ct)\n| conse {Ct : CaptureSet n k} :\n  Typed ((Γ.cvar (CBinding.bound CBound.star)).var T) t (EType.weaken (EType.cweaken E)) Ct.cweaken.weaken ->\n  WellScoped Γ cont Ct ->\n  TypedCont Γ E cont E' C ->\n  TypedCont Γ (EType.ex T) (Cont.conse t cont) E' (C ∪ Ct)\n| scope :\n  Context.LBound Γ x S ->\n  TypedCont Γ (S^{}) cont E' C ->\n  (Γ ⊢ T0 <: S^{}) ->\n  TypedCont Γ (EType.type T0) (Cont.scope x cont) E' C"}, {"name": "State", "content": "structure State (n : Nat) (m : Nat) (k : Nat) where\n  σ : Store n m k\n  cont : Cont n m k\n  t : Term n m k"}, {"name": "Term.weaken", "content": "def Term.weaken (t : Term n m k) : Term (n+1) m k := t.rename FinFun.weaken"}, {"name": "Term.weaken1", "content": "def Term.weaken1 (t : Term (n+1) m k) : Term (n+2) m k :=\n  t.rename FinFun.weaken.ext"}, {"name": "CaptureSet.weaken1", "content": "def CaptureSet.weaken1 (C : CaptureSet (n+1) k) : CaptureSet (n+2) k :=\n  C.rename FinFun.weaken.ext"}, {"name": "EType.weaken1", "content": "def EType.weaken1 (E : EType (n+1) m k) : EType (n+2) m k :=\n  E.rename FinFun.weaken.ext"}, {"name": "TVarMap.weaken", "content": "def TVarMap.weaken {Γ : Context n m k} :\n  TVarMap Γ FinFun.weaken (Γ.tvar b) :="}, {"name": "VarMap.weaken", "content": "def VarMap.weaken {Γ : Context n m k} :\n  VarMap Γ FinFun.weaken (Γ.var T) :="}, {"name": "SSubtyp.trename_motive1", "content": "def SSubtyp.trename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "SSubtyp.trename_motive3", "content": "def SSubtyp.trename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "SSubtyp.trename_motive2", "content": "def SSubtyp.trename_motive2\n  (Γ : Context n m k)\n  (T1 : CType n m k)\n  (T2 : CType n m k)\n  : Prop :=\n  ∀ {m'} (f : FinFun m m') (Δ : Context n m' k) (ρ : TVarMap Γ f Δ),\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "TVarSubst.cext", "content": "def TVarSubst.cext {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ) :\n  TVarSubst (Γ.cvar b) f (Δ.cvar b) :="}, {"name": "CBinding.cweaken", "content": "def CBinding.cweaken (b : CBinding n k) : CBinding n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Typed.cweaken", "content": "def Typed.cweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.cvar b) t.cweaken E.cweaken Ct.cweaken :="}, {"name": "Term.cweaken", "content": "def Term.cweaken (t : Term n m k) : Term n m (k+1) := t.crename FinFun.weaken"}, {"name": "EType.cweaken", "content": "def EType.cweaken (E : EType n m k) : EType n m (k+1) :=\n  E.crename FinFun.weaken"}, {"name": "CBound.cweaken", "content": "def CBound.cweaken (b : CBound n k) : CBound n (k+1) :=\n  b.crename FinFun.weaken"}, {"name": "Cont.cweaken", "content": "def Cont.cweaken : Cont n m k -> Cont n m (k+1)\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.cweaken cont.cweaken\n| Cont.conse t cont => Cont.conse t.cweaken1 cont.cweaken\n| Cont.scope x cont => Cont.scope x cont.cweaken"}, {"name": "Term.cweaken1", "content": "def Term.cweaken1 (t : Term n m (k+1)) : Term n m (k+2) :=\n  t.crename FinFun.weaken.ext"}, {"name": "CType.cweaken1", "content": "def CType.cweaken1 (T : CType n m (k+1)) : CType n m (k+2) :=\n  T.crename FinFun.weaken.ext"}, {"name": "SType.cweaken1", "content": "def SType.cweaken1 (S : SType n m (k+1)) : SType n m (k+2) :=\n  S.crename FinFun.weaken.ext"}, {"name": "CaptureSet.cweaken1", "content": "def CaptureSet.cweaken1 (C : CaptureSet n (k+1)) : CaptureSet n (k+2) :=\n  C.crename FinFun.weaken.ext"}, {"name": "EType.cweaken1", "content": "def EType.cweaken1 (E : EType n m (k+1)) : EType n m (k+2) :=\n  E.crename FinFun.weaken.ext"}, {"name": "VarSubst.ext", "content": "def VarSubst.ext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ)\n  (T : CType n m k) :\n  VarSubst (Γ.var T) f.ext (Δ.var (T.rename f)) :="}, {"name": "SSubtyp.rename_motive3", "content": "def SSubtyp.rename_motive3\n  (Γ : Context n m k)\n  (S1 : SType n m k)\n  (S2 : SType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "SSubtyp.rename_motive1", "content": "def SSubtyp.rename_motive1\n  (Γ : Context n m k)\n  (E1 : EType n m k)\n  (E2 : EType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "SSubtyp.rename_motive2", "content": "def SSubtyp.rename_motive2\n  (Γ : Context n m k)\n  (C1 : CType n m k)\n  (C2 : CType n m k)\n  : Prop :=\n  ∀ {n'} (f : FinFun n n') (Δ : Context n' m k) (ρ : VarMap Γ f Δ),\n  CSubtyp Δ (C1.rename f) (C2.rename f)"}, {"name": "VarSubst.cext", "content": "def VarSubst.cext {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.cvar b) f (Δ.cvar (b.rename f)) :="}, {"name": "CVarSubst.cext", "content": "def CVarSubst.cext {Γ : Context n m k}\n  (σ : CVarSubst Γ f Δ) :\n  CVarSubst (Γ.cvar b) f.ext (Δ.cvar (b.crename f)) :="}, {"name": "VarSubst.text", "content": "def VarSubst.text {Γ : Context n m k}\n  (σ : VarSubst Γ f Δ) :\n  VarSubst (Γ.tvar b) f (Δ.tvar (b.rename f)) :="}, {"name": "Cont.tweaken", "content": "def Cont.tweaken : Cont n m k -> Cont n (m+1) k\n| Cont.none => Cont.none\n| Cont.cons t cont => Cont.cons t.tweaken cont.tweaken\n| Cont.conse t cont => Cont.conse t.tweaken cont.tweaken\n| Cont.scope x cont => Cont.scope x cont.tweaken"}, {"name": "Term.tweaken", "content": "def Term.tweaken (t : Term n m k) : Term n (m+1) k := t.trename FinFun.weaken"}, {"name": "EType.tweaken", "content": "def EType.tweaken (E : EType n m k) : EType n (m+1) k :=\n  E.trename FinFun.weaken"}, {"name": "Typed.tweaken", "content": "def Typed.tweaken\n  (h : Typed Γ t E Ct) :\n  Typed (Γ.tvar b) t.tweaken E.tweaken Ct :="}, {"name": "Store.LBound", "content": "inductive Store.LBound : Store n m k -> (Fin n) -> SType n m k -> Prop where\n| here :\n  Store.LBound (Store.label σ S) 0 S.weaken\n| there_val :\n  Store.LBound σ x S ->\n  Store.LBound (Store.val σ t hv) x.succ S.weaken\n| there_tval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.tval σ S') x S.tweaken\n| there_cval :\n  Store.LBound σ x S ->\n  Store.LBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.LBound σ x S ->\n  Store.LBound (Store.label σ S') x.succ S.weaken"}, {"name": "Store", "content": "inductive Store : Nat -> Nat -> Nat -> Type where\n| empty : Store 0 0 0\n| val :\n  Store n m k ->\n  (t : Term n m k) ->\n  t.IsValue ->\n  Store (n+1) m k\n| tval :\n  Store n m k ->\n  SType n m k ->\n  Store n (m+1) k\n| cval :\n  Store n m k ->\n  CaptureSet n k ->\n  Store n m (k+1)\n| label :\n  Store n m k ->\n  SType n m k ->\n  Store (n+1) m k"}, {"name": "Store.CBound", "content": "inductive Store.CBound : Store n m k -> (Fin k) -> CaptureSet n k -> Prop where\n| here :\n  Store.CBound (Store.cval σ C) 0 C.cweaken\n| there_val :\n  Store.CBound σ x C ->\n  Store.CBound (Store.val σ t hv) x C.weaken\n| there_tval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.tval σ S) x C\n| there_cval :\n  Store.CBound σ x C ->\n  Store.CBound (Store.cval σ C') (Fin.succ x) C.cweaken\n| there_label :\n  Store.CBound σ x C ->\n  Store.CBound (Store.label σ S) x C.weaken"}, {"name": "Store.TBound", "content": "inductive Store.TBound : Store n m k -> (Fin m) -> SType n m k -> Prop where\n| here :\n  Store.TBound (Store.tval σ S) 0 S.tweaken\n| there_val :\n  Store.TBound σ x S ->\n  Store.TBound (Store.val σ t hv) x S.weaken\n| there_tval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.tval σ S') (Fin.succ x) S.tweaken\n| there_cval :\n  Store.TBound σ x S ->\n  Store.TBound (Store.cval σ C) x S.cweaken\n| there_label :\n  Store.TBound σ x S ->\n  Store.TBound (Store.label σ S') x S.weaken"}, {"name": "Store.Bound", "content": "inductive Store.Bound : Store n m k -> (Fin n) -> Term n m k -> Prop where\n| here :\n  Store.Bound (Store.val σ t hv) 0 t.weaken\n| there_val :\n  Store.Bound σ x t ->\n  Store.Bound (Store.val σ t' hv) (Fin.succ x) t.weaken\n| there_tval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.tval σ S) x t.tweaken\n| there_cval :\n  Store.Bound σ x t ->\n  Store.Bound (Store.cval σ C) x t.cweaken\n| there_label :\n  Store.Bound σ x t ->\n  Store.Bound (Store.label σ S) (Fin.succ x) t.weaken"}, {"name": "TVarSubst.text", "content": "def TVarSubst.text {Γ : Context n m k}\n  (σ : TVarSubst Γ f Δ)\n  (T : TBinding n m k) :\n  TVarSubst (Γ.tvar T) f.ext (Δ.tvar (T.trename f)) :="}, {"name": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t", "content": "notation:50 \"λ(x:\" T \")\" t => Term.lam T t"}, {"name": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t", "content": "notation:50 \"λ[X<:\" S \"]\" t => Term.tlam S t"}, {"name": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t", "content": "notation:50 \"λ[c<:\" B \"]\" t => Term.clam B t"}, {"name": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u", "content": "notation:40 \"let\" \"x=\" t \" in \" u => Term.letin t u"}, {"name": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u", "content": "notation:40 \"let\" \"(c,x)=\" t \" in \" u => Term.letex t u"}, {"name": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t", "content": "notation:40 \"let\" \"X=\" S \" in \" t => Term.bindt S t"}, {"name": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t", "content": "notation:40 \"let\" \"c=\" C \" in \" t => Term.bindc C t"}, {"name": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t", "content": "notation:40 \"boundary:\" S \" in \" t => Term.boundary S t"}, {"name": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x", "content": "notation:max \"{x=\" x \"}\" => CaptureSet.singleton x"}, {"name": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c", "content": "notation:max \"{c=\" c \"}\" => CaptureSet.csingleton c"}, {"name": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2", "content": "notation:50 Γ \" ⊢ \" E1 \" <:e \" E2 => ESubtyp Γ E1 E2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <:s \" T2 => SSubtyp Γ T1 T2"}, {"name": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2", "content": "notation:50 Γ \" ⊢ \" T1 \" <: \" T2 => CSubtyp Γ T1 T2"}, {"name": "notation:30 Γ \",x:\" T => Context.var Γ T", "content": "notation:30 Γ \",x:\" T => Context.var Γ T"}, {"name": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)", "content": "notation:30 Γ \",X<:\" T => Context.tvar Γ (TBinding.bound T)"}, {"name": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)", "content": "notation:30 Γ \",X:=\" T => Context.tvar Γ (TBinding.inst T)"}, {"name": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)", "content": "notation:30 Γ \",c<:\" B => Context.cvar Γ (CBinding.bound B)"}, {"name": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBou", "content": "notation:30 Γ \",c<:*\" => Context.cvar Γ (CBinding.bound CBound.star)"}, {"name": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)", "content": "notation:30 Γ \",c:=\" C => Context.cvar Γ (CBinding.inst C)"}, {"name": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C", "content": "notation:40 Γ \" ⊢ \" t:80 \" : \" E \" @ \" C => Typed Γ t E C"}, {"name": "SType.top", "content": "notation \"⊤\" => SType.top"}, {"name": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U", "content": "notation:50 \"∀(x:\" T \")\" U => SType.forall T U"}, {"name": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T", "content": "notation:50 \"∀[X<:\" S \"]\" T => SType.tforall S T"}, {"name": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T", "content": "notation:50 \"∀[c<:\" B \"]\" T => SType.cforall B T"}, {"name": "notation:max S \" ^ \" C => CType.capt C S", "content": "notation:max S \" ^ \" C => CType.capt C S"}, {"name": "notation:40 \"∃c.\" T => EType.ex T", "content": "notation:40 \"∃c.\" T => EType.ex T"}, {"name": "notation:40 \"Label[\" S \"]\" => SType.label S", "content": "notation:40 \"Label[\" S \"]\" => SType.label S"}, {"name": "notation:60 \"□\" T => SType.box T", "content": "notation:60 \"□\" T => SType.box T"}, {"name": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2", "content": "notation:50 Γ \" ⊢ \" C1 \" <:c \" C2 => Subcapt Γ C1 C2"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "ESubtyp.trans", "content": "theorem ESubtyp.trans\n  (h1 : ESubtyp Γ E1 E2)\n  (h2 : ESubtyp Γ E2 E3) :\n  ESubtyp Γ E1 E3"}, {"name": "CSubtyp.trans", "content": "theorem CSubtyp.trans\n  (h1 : CSubtyp Γ T1 T2)\n  (h2 : CSubtyp Γ T2 T3) :\n  CSubtyp Γ T1 T3"}, {"name": "Typed.open", "content": "theorem Typed.open\n  (h : Typed (Γ,x: P) t E Ct)\n  (hx : Typed Γ (Term.var x) (EType.type P) Cx) :\n  Typed Γ (t.open x) (E.open x) (Ct.open x)"}, {"name": "Typed.subst", "content": "theorem Typed.subst\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (σ : VarSubst Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "CSubtyp.crename", "content": "theorem CSubtyp.crename\n  (h : CSubtyp Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  CSubtyp Δ (C1.crename f) (C2.crename f)"}, {"name": "SSubtyp.crename", "content": "theorem SSubtyp.crename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : CVarMap Γ f Δ) :\n  SSubtyp Δ (S1.crename f) (S2.crename f)"}, {"name": "Subbound.crename", "content": "theorem Subbound.crename\n  (h : Subbound Γ B1 B2)\n  (ρ : CVarMap Γ f Δ) :\n  Subbound Δ (B1.crename f) (B2.crename f)"}, {"name": "ESubtyp.crename", "content": "theorem ESubtyp.crename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : CVarMap Γ f Δ) :\n  ESubtyp Δ (E1.crename f) (E2.crename f)"}, {"name": "CaptureSet.Subset.crename", "content": "theorem CaptureSet.Subset.crename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.crename f ⊆ C2.crename f"}, {"name": "Subcapt.crename", "content": "theorem Subcapt.crename\n  (h : Subcapt Γ C1 C2)\n  (ρ : CVarMap Γ f Δ) :\n  Subcapt Δ (C1.crename f) (C2.crename f)"}, {"name": "Typed.crename", "content": "theorem Typed.crename\n  {Γ : Context n m k} {Δ : Context n m k'}\n  (h : Typed Γ t E Ct)\n  (ρ : CVarMap Γ f Δ) :\n  Typed Δ (t.crename f) (E.crename f) (Ct.crename f)"}, {"name": "CSubtyp.trename", "content": "theorem CSubtyp.trename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  CSubtyp Δ (T1.trename f) (T2.trename f)"}, {"name": "SSubtyp.trename", "content": "theorem SSubtyp.trename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : TVarMap Γ f Δ) :\n  SSubtyp Δ (S1.trename f) (S2.trename f)"}, {"name": "ESubtyp.trename", "content": "theorem ESubtyp.trename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : TVarMap Γ f Δ) :\n  ESubtyp Δ (E1.trename f) (E2.trename f)"}, {"name": "Subbound.trename", "content": "theorem Subbound.trename\n  (h : Subbound Γ T1 T2)\n  (ρ : TVarMap Γ f Δ) :\n  Subbound Δ T1 T2"}, {"name": "Subcapt.trename", "content": "theorem Subcapt.trename\n  (h : Subcapt Γ C1 C2)\n  (ρ : TVarMap Γ f Δ) :\n  Subcapt Δ C1 C2"}, {"name": "Typed.trename", "content": "theorem Typed.trename\n  {Γ : Context n m k} {Δ : Context n m' k}\n  (h : Typed Γ t E Ct)\n  (ρ : TVarMap Γ f Δ) :\n  Typed Δ (t.trename f) (E.trename f) Ct"}, {"name": "CSubtyp.rename", "content": "theorem CSubtyp.rename\n  (h : CSubtyp Γ T1 T2)\n  (ρ : VarMap Γ f Δ) :\n  CSubtyp Δ (T1.rename f) (T2.rename f)"}, {"name": "SSubtyp.rename", "content": "theorem SSubtyp.rename\n  (h : SSubtyp Γ S1 S2)\n  (ρ : VarMap Γ f Δ) :\n  SSubtyp Δ (S1.rename f) (S2.rename f)"}, {"name": "Subbound.rename", "content": "theorem Subbound.rename\n  (h : Subbound Γ B1 B2)\n  (ρ : VarMap Γ f Δ) :\n  Subbound Δ (B1.rename f) (B2.rename f)"}, {"name": "ESubtyp.rename", "content": "theorem ESubtyp.rename\n  (h : ESubtyp Γ E1 E2)\n  (ρ : VarMap Γ f Δ) :\n  ESubtyp Δ (E1.rename f) (E2.rename f)"}, {"name": "CaptureSet.Subset.rename", "content": "theorem CaptureSet.Subset.rename {C1 C2 : CaptureSet n k}\n  (h : C1 ⊆ C2) :\n  C1.rename f ⊆ C2.rename f"}, {"name": "Subcapt.rename", "content": "theorem Subcapt.rename\n  (h : Subcapt Γ C1 C2)\n  (ρ : VarMap Γ f Δ) :\n  Subcapt Δ (C1.rename f) (C2.rename f)"}, {"name": "Typed.rename", "content": "theorem Typed.rename\n  {Γ : Context n m k} {Δ : Context n' m k}\n  (h : Typed Γ t E Ct)\n  (ρ : VarMap Γ f Δ) :\n  Typed Δ (t.rename f) (E.rename f) (Ct.rename f)"}, {"name": "ESubtyp.refl", "content": "theorem ESubtyp.refl :\n  ESubtyp Γ E E"}, {"name": "CSubtyp.refl", "content": "theorem CSubtyp.refl :\n  CSubtyp Γ T T"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Capless.Tactics\n\nimport Capless.Typing\n\nimport Capless.Subtyping.Basic\n\nimport Capless.Subcapturing.Basic\n\nimport Capless.Narrowing\n\nimport Capless.Weakening.Subcapturing\n\nimport Capless.Inversion.Context\n\nimport Capless.Inversion.Subtyping\n\nnamespace Capless", "target_theorem": "theorem Typed.app_inv'\n  (he : t0 = Term.app x y)\n  (h : Typed Γ t0 E Ct0) :\n  ∃ T Cf F E0, Typed Γ (Term.var x) (EType.type (CType.capt Cf (SType.forall T F))) {x=x}\n    ∧ Typed Γ (Term.var y) (EType.type T) {x=y}\n    ∧ E0 = F.open y\n    ∧ ESubtyp Γ E0 E :=", "ground_truth_proof": ":= by\n    induction h <;> try (solve | cases he)\n    case app x C T F y h1 h2 _ _ =>\n      cases he\n      refine ⟨T, C, F, (F.open y), ?_⟩\n      repeat (constructor; trivial)\n      apply ESubtyp.refl\n    case sub hsub ih =>\n      have ih := ih he\n      have ⟨T0, Cf0, E0, F0, hx0, hy0, he0, hs0⟩ := ih\n      refine ⟨T0, Cf0, E0, F0, ?_⟩\n      repeat (any_goals apply And.intro)\n      all_goals try assumption\n      { apply! ESubtyp.trans }", "nesting_depth": 4, "transitive_dep_count": 168, "subset_aristotle": false, "category": "Type systems"}
{"id": 121, "thm_name": "InductiveTable.table_soundness_aux", "thm_stmt": "lemma table_soundness_aux (table : InductiveTable F State Input) (input output : State F)\n  (N : ℕ+) (trace : TraceOfLength F (ProvablePair State Input) N) (env : ℕ → ℕ → Environment F) :\n  table.Spec input [] 0 rfl input →\n  TableConstraintsHold (table.tableConstraints input output) trace env →\n    trace.ForAllRowsWithPrevious (fun row i rest => table.Spec input (traceInputs rest) i (traceInputs_length rest) row.1)\n    ∧ trace.lastRow.1 = output", "lean_root": "clean", "rel_path": "Clean/Table/Inductive.lean", "imports": ["import Clean.Table.Basic", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Table.Theorems", "import Clean.Circuit.Loops", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.snd", "module": "Init.Prelude"}, {"name": "StateM", "module": "Init.Control.State"}, {"name": "Vector.finRange", "module": "Init.Data.Vector.FinRange"}, {"name": "Vector.map", "module": "Init.Data.Vector.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Vector.push", "module": "Init.Data.Vector.Basic"}, {"name": "Fin.isValue", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "Nat.reduceAdd", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "PNat", "module": "Mathlib.Data.PNat.Notation"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "One", "module": "Init.Prelude"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "IO", "module": "Init.System.IO"}, {"name": "UInt32", "module": "Init.Prelude"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "Vector.foldlM", "module": "Init.Data.Vector.Basic"}, {"name": "List.ofFn", "module": "Init.Data.List.OfFn"}, {"name": "Vector.mk", "module": "Init.Data.Vector.Basic"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "Vector.toList", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.forM", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty", "content": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty\n\nsyntax \"infer_constant_length\" : tactic\n\nsyntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|simp_assign_row) =>\n    `(tactic|(\n    simp only [assignCurrRow, assignNextRow, size]\n    rw [List.finRange, List.ofFn]\n    repeat rw [Fin.foldr_succ]\n    rw [Fin.foldr_zero]\n    repeat rw [List.forM_cons]\n    rw [List.forM_nil, bind_pure_unit]\n    simp only [seval, toVars, toElements, Fin.cast_eq_self, Fin.val_zero, Fin.val_one, Fin.isValue,\n      List.getElem_toArray, List.getElem_cons_zero, List.getElem_cons_succ, Fin.succ_zero_eq_one]))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "toList", "content": "def toList : Trace F S → List (Row F S)\n  | <+> => []\n  | rest +> row => rest.toList.concat row"}, {"name": "Trace", "content": "inductive Trace (F : Type) (S : Type → Type) [ProvableType S] where\n   \n  | empty : Trace F S\n   \n  | cons (rest : Trace F S) (row : Row F S) : Trace F S"}, {"name": "CellOffset", "content": "structure CellOffset (W : ℕ+) (S : Type → Type) [ProvableType S]  where\n  row: Fin W\n  column: Fin (size S)"}, {"name": "empty", "content": "@[reducible, table_norm, table_assignment_norm]\ndef empty : TableContext W S F where\n  circuit := []\n  assignment := .empty W"}, {"name": "TableContext", "content": "structure TableContext (W : ℕ+) (S : Type → Type) (F : Type) [Field F] [ProvableType S] where\n  circuit : Operations F\n  assignment : CellAssignment W S\nderiving Repr"}, {"name": "CellAssignment", "content": "structure CellAssignment (W : ℕ+) (S : Type → Type) [ProvableType S] where\n  offset : ℕ \n  aux_length : ℕ \n\n   \n  vars : Vector (Cell W S) offset"}, {"name": "offset", "content": "@[reducible, table_norm, table_assignment_norm]\ndef offset (table : TableContext W S F) : ℕ := table.circuit.localLength"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "Cell", "content": "inductive Cell (W : ℕ+) (S : Type → Type) [ProvableType S] where\n  | input : CellOffset W S → Cell W S\n  | aux : ℕ → Cell W S"}, {"name": "Row", "content": "@[reducible]\ndef Row (F : Type) (S : Type → Type) [ProvableType S] := S F"}, {"name": "ProvablePair.instance", "content": "instance ProvablePair.instance {α β: TypeMap} [ProvableType α] [ProvableType β] : ProvableType (ProvablePair α β) where\n  size := size α + size β\n  toElements := fun (a, b) => toElements a ++ toElements b\n  fromElements {F} v :=\n    let a : α F := v.take (size α) |>.cast Nat.min_add_right_self |> fromElements\n    let b : β F := v.drop (size α) |>.cast (Nat.add_sub_self_left _ _) |> fromElements\n    (a, b)\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "TraceOfLength", "content": "def TraceOfLength (F : Type) (S : Type → Type) [ProvableType S] (N : ℕ) : Type :=\n  { env : Trace F S // env.len = N }"}, {"name": "toList", "content": "def toList {N : ℕ} (trace : TraceOfLength F S N) : List.Vector (Row F S) N :=\n  ⟨ trace.val.toList, by admit /- proof elided -/\n  ⟩"}, {"name": "ConstraintsHold", "content": "@[circuit_norm]\ndef ConstraintsHold (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold eval ops\n  | .lookup { table, entry, .. } :: ops =>\n    table.Contains (entry.map eval) ∧ ConstraintsHold eval ops\n  | .subcircuit s :: ops =>\n    ConstraintsHoldFlat eval s.ops ∧ ConstraintsHold eval ops"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "getCurrRow", "content": "@[table_norm, table_assignment_norm]\ndef getCurrRow : TableConstraint W S F (Var S F) := getRow 0"}, {"name": "TableConstraint", "content": "@[reducible, table_norm, table_assignment_norm]\ndef TableConstraint (W : ℕ+) (S : Type → Type) (F : Type) [Field F] [ProvableType S] :=\n  StateM (TableContext W S F)"}, {"name": "getRow", "content": "@[table_norm, table_assignment_norm]\ndef getRow (row : Fin W) : TableConstraint W S F (Var S F) :=\n  modifyGet fun ctx =>\n    let ctx' : TableContext W S F := {\n      circuit := ctx.circuit ++ [.witness (size S) fun env => .mapRange _ fun i => env.get (ctx.offset + i)],\n      assignment := ctx.assignment.pushRow row\n    }\n    (varFromOffset S ctx.offset, ctx')"}, {"name": "Row.get", "content": "@[table_norm, table_assignment_norm]\ndef Row.get (row : Row F S) (i : Fin (size S)) : F :=\n  (toElements row)[i.val]"}, {"name": "pushRow", "content": "@[table_assignment_norm]\ndef pushRow (assignment : CellAssignment W S) (row : Fin W) : CellAssignment W S :=\n  let row_vars : Vector (Cell W S) (size S) := .mapFinRange _ fun col => .input ⟨ row, col ⟩\n  {\n    offset := assignment.offset + size S\n    aux_length := assignment.aux_length\n    vars := assignment.vars ++ row_vars\n  }"}, {"name": "mapFinRange", "content": "def mapFinRange (n : ℕ) (create : Fin n → α) : Vector α n := finRange n |>.map create"}, {"name": "get", "content": "@[table_assignment_norm]\ndef get {M : ℕ} :\n    (env : TraceOfLength F S M) → (i : Fin M) → (j : Fin (size S)) → F\n  | ⟨env, h⟩, i, j => env.getLeFromBottom ⟨\n      M - 1 - i,\n      by admit /- proof elided -/\n    ⟩ j"}, {"name": "mapRange", "content": "def mapRange (n : ℕ) (create : ℕ → α) : Vector α n :=\n  match n with\n  | 0 => #v[]\n  | k + 1 => mapRange k create |>.push (create k)"}, {"name": "varFromOffset", "content": "@[explicit_provable_type]\ndef varFromOffset (α : TypeMap) [ProvableType α] (offset : ℕ) : Var α F :=\n  let vars := Vector.mapRange (size α) fun i => var ⟨offset + i⟩\n  fromVars vars"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "getNextRow", "content": "@[table_norm, table_assignment_norm]\ndef getNextRow : TableConstraint W S F (Var S F) := getRow 1"}, {"name": "HasAssertEq", "content": "class HasAssertEq (β : Type) (F : outParam Type) [Field F] where\n  assert_eq : β → β → Circuit F Unit"}, {"name": "SingleRowConstraint", "content": "@[reducible]\ndef SingleRowConstraint (S : Type → Type) (F : Type) [Field F] [ProvableType S] := TableConstraint 1 S F Unit"}, {"name": "TableOperation", "content": "inductive TableOperation (S : Type → Type) (F : Type) [Field F] [ProvableType S] where\n   \n  | boundary: RowIndex → SingleRowConstraint S F → TableOperation S F\n\n   \n  | everyRow: SingleRowConstraint S F → TableOperation S F\n\n   \n  | everyRowExceptLast: TwoRowsConstraint S F → TableOperation S F"}, {"name": "RowIndex", "content": "inductive RowIndex where\n  | fromStart : ℕ → RowIndex\n  | fromEnd : ℕ → RowIndex"}, {"name": "TwoRowsConstraint", "content": "@[reducible]\ndef TwoRowsConstraint (S : Type → Type) (F : Type) [Field F] [ProvableType S] := TableConstraint 2 S F Unit"}, {"name": "windowEnv", "content": "def windowEnv (table : TableConstraint W S F Unit)\n  (window : TraceOfLength F S W) (aux_env : Environment F) : Environment F :=\n  let assignment := table.finalAssignment\n  .mk fun i =>\n    if hi : i < assignment.offset then\n      match assignment.vars[i] with\n      | .input ⟨i, j⟩ => window.get i j\n      | .aux k => aux_env.get k\n    else aux_env.get (i + assignment.aux_length)"}, {"name": "finalAssignment", "content": "@[table_assignment_norm]\ndef finalAssignment (table : TableConstraint W S F α) : CellAssignment W S :=\n  table .empty |>.snd.assignment"}, {"name": "lastRow", "content": "@[table_norm]\ndef lastRow : (trace : Trace F S) → (hlen : trace.len > 0) → S F\n  | <+>, h  => nomatch h\n  | _ +> row, _ => row"}, {"name": "FormalTable", "content": "structure FormalTable (F : Type) [Field F] (S : Type → Type) [ProvableType S] where\n   \n  constraints : List (TableOperation S F)\n\n   \n  Assumption : ℕ → Prop := fun _ => True\n\n   \n  Spec {N : ℕ} : TraceOfLength F S N → Prop\n\n   \n  soundness :\n    ∀ (N : ℕ) (trace : TraceOfLength F S N) (env : ℕ → ℕ → Environment F),\n    Assumption N →\n    TableConstraintsHold constraints trace env →\n    Spec trace\n\n   \n  offset_consistent :\n    constraints.Forall fun cs =>\n      match cs with\n      | .boundary _ constraint => constraint.OffsetConsistent\n      | .everyRow constraint => constraint.OffsetConsistent\n      | .everyRowExceptLast constraint => constraint.OffsetConsistent\n    := by admit /- proof elided -/"}, {"name": "TableConstraintsHold", "content": "@[table_norm]\ndef TableConstraintsHold {N : ℕ} (constraints : List (TableOperation S F))\n  (trace : TraceOfLength F S N) (env : ℕ → ℕ → Environment F) : Prop :=\n  let constraints_and_envs := constraints.mapIdx (fun i cs => (cs, env i))\n  foldl N constraints_and_envs trace.val constraints_and_envs\n  where"}, {"name": "ForAllRowsWithPrevious", "content": "def ForAllRowsWithPrevious  : Trace F S → (Row F S → Trace F S → Prop) → Prop\n  | <+>, _ => true\n  | rest +> row, prop => (prop row rest) ∧ ForAllRowsWithPrevious rest prop"}, {"name": "ForAllRowsWithPrevious", "content": "def ForAllRowsWithPrevious {N : ℕ}\n    (trace : TraceOfLength F S N) (prop : Row F S → (i : ℕ) → TraceOfLength F S i → Prop) : Prop :=\n  trace.val.ForAllRowsWithPrevious fun row rest => prop row rest.len ⟨ rest, rfl ⟩"}, {"name": "len", "content": "@[table_norm, table_assignment_norm]\ndef len : Trace F S → ℕ\n  | <+> => 0\n  | rest +> _ => rest.len + 1"}, {"name": "len", "content": "def len (_ : Vector α n) : ℕ := n"}, {"name": "every_row_two_rows_induction", "content": "def every_row_two_rows_induction {P : Trace F S → Sort*}\n    (zero : P (<+>))\n    (one : ∀ row : Row F S, P (empty +> row))\n    (more : ∀ curr next : Row F S,\n      ∀ rest : Trace F S, P (rest) → P (rest +> curr) → P (rest +> curr +> next))\n    : ∀ trace, P trace\n  | <+> => zero\n  | <+> +> first => one first\n  | rest +> curr +> _ => more _ _ _\n    (every_row_two_rows_induction zero one more (rest))\n    (every_row_two_rows_induction zero one more (rest +> curr))"}, {"name": "curr", "content": "@[table_assignment_norm]\ndef curr {W : ℕ+} (j : Fin (size S)) : CellOffset W S := ⟨0, j⟩"}, {"name": "next", "content": "@[table_assignment_norm]\ndef next {W : ℕ+} (j : Fin (size S)) : CellOffset W S := ⟨1, j⟩"}, {"name": "lastRow", "content": "@[table_norm]\ndef lastRow {M : ℕ+} (trace : TraceOfLength F S M) : S F :=\n  trace.val.lastRow (by admit /- proof elided -/\n  )"}, {"name": "circuit", "content": "def circuit := do\n  let x ← witnessField (F := F pBabybear) fun _ => 246\n  let y ← witnessField fun _ => 20\n  let z ← Gadgets.Addition8.circuit.main { x, y }\n  Gadgets.Addition8.circuit.main { x, y := z }"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  x: U32 F\n  y: U32 F\n  carryIn: F"}, {"name": "Outputs", "content": "structure Outputs (F : Type) where\n  z: U32 F\n  carryOut: F\nderiving Repr"}, {"name": "witnessField", "content": "@[circuit_norm]\ndef witnessField (compute : Environment F → F) := do\n  let v ← witnessVar compute\n  return var v"}, {"name": "witnessVar", "content": "@[circuit_norm]\ndef witnessVar (compute : Environment F → F) : Circuit F (Variable F) :=\n  fun (offset : ℕ) =>\n    let var : Variable F := ⟨ offset ⟩\n    (var, [.witness 1 fun env => #v[compute env]])"}, {"name": "pBabybear", "content": "def pBabybear := 15 * 2^27 + 1"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) Inputs field where\n  main := fun { x, y } =>\n    Addition8Full.circuit { x, y, carryIn := 0 }\n\n  localLength _ := 2\n  output _ i0 := var ⟨i0⟩\n\n  Assumptions | { x, y } => x.val < 256 ∧ y.val < 256\n\n  Spec | { x, y }, z => z.val = (x.val + y.val) % 256\n\n  \n  soundness := by admit /- proof elided -/"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  x: F\n  y: F"}, {"name": "Addition8Full.circuit", "content": "def Addition8Full.circuit : FormalCircuit (F p) Addition8FullCarry.Inputs field where\n  main := fun inputs => do\n    let { z, .. } ← Addition8FullCarry.circuit inputs\n    return z\n\n  localLength _ := 2\n  output _ i0 := var ⟨i0⟩\n\n  Assumptions := fun { x, y, carryIn } =>\n    x.val < 256 ∧ y.val < 256 ∧ IsBool carryIn\n\n  Spec := fun { x, y, carryIn } z =>\n    z.val = (x.val + y.val + carryIn.val) % 256\n\n  \n  soundness := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "Spec", "content": "def Spec (input : Inputs (F p)) (out : Outputs (F p)) :=\n  let ⟨x, y, carryIn⟩ := input\n  out.z.val = (x.val + y.val + carryIn.val) % 256 ∧\n  out.carryOut.val = (x.val + y.val + carryIn.val) / 256"}, {"name": "Outputs", "content": "structure Outputs (F : Type) where\n  z: F\n  carryOut: F"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  x: F\n  y: F\n  carryIn: F"}, {"name": "Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y, carryIn⟩ := input\n  x.val < 256 ∧ y.val < 256 ∧ IsBool carryIn"}, {"name": "IsBool", "content": "def IsBool {α : Type*} [Zero α] [One α] (x : α) : Prop := x = 0 ∨ x = 1"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "main", "content": "def main (args : List String) : IO Unit := do\n  match args with\n  | [steps_str, output_path] =>\n    \n    match steps_str.toNat? with\n    | some steps => generateTrace steps output_path\n    | none => IO.println \"Error: Invalid number of steps\"\n  | _ =>\n    IO.println \"Usage: lake lean TraceGen.lean "}, {"name": "getLeFromBottom", "content": "@[table_assignment_norm]\ndef getLeFromBottom :\n    (trace : Trace F S) → (row : Fin trace.len) → (col : Fin (size S)) → F\n  | _ +> currRow, ⟨0, _⟩, j => currRow.get j\n  | rest +> _, ⟨i + 1, h⟩, j => getLeFromBottom rest ⟨i, Nat.le_of_succ_le_succ h⟩ j"}, {"name": "pushVarsAux", "content": "@[table_assignment_norm]\ndef pushVarsAux (assignment : CellAssignment W S) (n : ℕ) : CellAssignment W S where\n  offset := assignment.offset + n\n  aux_length := assignment.aux_length + n\n  vars := assignment.vars ++ (.mapRange n fun i => .aux (assignment.aux_length + i) : Vector (Cell W S) n)"}, {"name": "assignmentFromCircuit", "content": "@[table_assignment_norm]\ndef assignmentFromCircuit (as : CellAssignment W S) : Operations F → CellAssignment W S\n  | [] => as\n  | .witness m _ :: ops => assignmentFromCircuit (as.pushVarsAux m) ops\n  | .assert _ :: ops => assignmentFromCircuit as ops\n  | .lookup _ :: ops => assignmentFromCircuit as ops\n  | .subcircuit s :: ops => assignmentFromCircuit (as.pushVarsAux s.localLength) ops"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}, {"name": "foldlAcc", "content": "def foldlAcc (n : ℕ) (xs : Vector α m) (circuit : β → α → Circuit F β) (init : β) (j : Fin m) : β :=\n  Fin.foldl j (fun acc i => (circuit acc xs[i.val]).output (n + i*(circuit acc xs[i.val]).localLength)) init"}, {"name": "foldlRange", "content": "def foldlRange (m : ℕ) [Inhabited β] (init : β) (body : β → Fin m → Circuit F β)\n  (_constant : ConstantLength (fun (s, a) => body s a) := by admit /- proof elided -/\n    ) : Circuit F β :=\n  (Vector.finRange m).foldlM body init"}, {"name": "ConstantLength.fromConstantLength'", "content": "def ConstantLength.fromConstantLength' [Inhabited β] (body : β × Fin m → Circuit F β)\n    (h : ∀ (acc : β) (i i' : Fin m) n,\n      (body (acc, i)).localLength n = (body (default, i')).localLength 0) :\n    ConstantLength body where\n  localLength := match m with\n    | 0 => 0\n    | m + 1 => (body (default, 0)).localLength 0\n  localLength_eq := by admit /- proof elided -/"}, {"name": "prod", "content": "@[reducible]\ndef prod (circuit : β → α → Circuit F β) : β × α → Circuit F β := fun t => circuit t.1 t.2"}, {"name": "induct", "content": "def induct {motive : {n : ℕ} → Vector α n → Sort u}\n    (nil : motive #v[])\n    (cons: ∀ {n : ℕ} (a : α) (as : Vector α n), motive as → motive (cons a as))\n    {n : ℕ} (v : Vector α n) : motive v :=\n  match v with\n  | ⟨ .mk [], h ⟩ => by admit /- proof elided -/\n  | ⟨ .mk (a :: as), h ⟩ => by admit /- proof elided -/"}, {"name": "cons", "content": "def cons (a : α) (v : Vector α n) : Vector α (n + 1) :=\n  ⟨ .mk (a :: v.toList), by admit /- proof elided -/\n  ⟩"}, {"name": "circuit", "content": "@[simps! (attr := circuit_norm) (config := {isSimp := false})]\ndef circuit (α : TypeMap) [ProvableType α] : FormalAssertion F (ProvablePair α α) where\n  Assumptions _ := True\n\n  Spec : α F × α F → Prop\n  | (x, y) => x = y\n\n  soundness := by admit /- proof elided -/"}, {"name": "mapFinRange", "content": "def mapFinRange (m : ℕ) [NeZero m] (body : Fin m → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  Vector.mapFinRangeM m body"}, {"name": "mapFinRangeM", "content": "def mapFinRangeM (n : ℕ) {m : Type → Type} [Monad m] (f : Fin n → m β) : m (Vector β n) := (finRange n).mapM f"}, {"name": "allZero", "content": "def allZero {n} (xs : Vector (Expression F) n) : Circuit F Unit := .forEach xs assertZero"}, {"name": "forEach", "content": "def forEach {m : ℕ} (xs : Vector α m) [Inhabited α] (body : α → Circuit F Unit)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F Unit :=\n  xs.forM body"}, {"name": "assertZero", "content": "@[circuit_norm]\ndef assertZero (e : Expression F) : Circuit F Unit := fun _ =>\n  ((), [.assert e])"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) (var : α (Expression F)) : α F :=\n  toComponents var |> go (components α) |> fromComponents\nwhere"}, {"name": "ProvableStruct", "content": "class ProvableStruct (α : TypeMap) where\n  components : List WithProvableType\n  toComponents {F : Type} : α F → ProvableTypeList F components\n  fromComponents {F : Type} : ProvableTypeList F components → α F\n\n  combinedSize : ℕ := combinedSize' components\n  combinedSize_eq : combinedSize = combinedSize' components := by admit /- proof elided -/"}, {"name": "foldl", "content": "def foldl {m : ℕ} [Inhabited β] [Inhabited α] (xs : Vector α m) (init : β) (body : β → α → Circuit F β)\n  (_const_out : ConstantOutput (fun (s, a) => body s a) := by admit /- proof elided -/\n    )\n  (_constant : ConstantLength (fun (s, a) => body s a) := by admit /- proof elided -/\n  )\n   : Circuit F β :=\n  xs.foldlM body init"}, {"name": "ConstantOutput", "content": "@[circuit_norm]\ndef ConstantOutput (circuit : α → Circuit F β) [Inhabited α] :=\n  ∀ (x : α) (n : ℕ), (circuit x).output n = (circuit default).output n"}, {"name": "infix:50 \" === \" => HasAssertEq.assert_eq", "content": "infix:50 \" === \" => HasAssertEq.assert_eq"}, {"name": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty", "content": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty"}, {"name": "@[inherit_doc] infixl:67 \" +> \" => Trace.cons", "content": "@[inherit_doc] infixl:67 \" +> \" => Trace.cons"}], "lib_lemmas": [{"name": "List.length_map", "module": "Init.Data.List.Lemmas"}, {"name": "Vector.append_empty", "module": "Init.Data.Vector.Lemmas"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "NeZero.pos", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.length_cons", "module": "Init.Data.List.Basic"}, {"name": "List.length_nil", "module": "Init.Data.List.Basic"}, {"name": "List.map_concat", "module": "Init.Data.List.Lemmas"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "List.push_toArray", "module": "Init.Data.List.ToArray"}, {"name": "List.size_toArray", "module": "Init.Data.Array.Basic"}, {"name": "Nat.add_eq_zero", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.add_zero", "module": "Init.Core"}, {"name": "PNat.val_ofNat", "module": "Mathlib.Data.PNat.Basic"}, {"name": "Vector.empty_append", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_append", "module": "Init.Data.Vector.Lemmas"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_tsub_cancel_right", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "tsub_self", "module": "Mathlib.Algebra.Order.Sub.Basic"}, {"name": "tsub_zero", "module": "Mathlib.Algebra.Order.Sub.Defs"}], "repo_lemmas": [{"name": "toList_length", "content": "lemma toList_length : (trace : Trace F S) → trace.toList.length = trace.len\n  | <+> => rfl"}, {"name": "mapRange_zero", "content": "theorem mapRange_zero {create : ℕ → α} : mapRange 0 create = #v[]"}, {"name": "getElem_mapRange", "content": "theorem getElem_mapRange {n} {create : ℕ → α} :\n    ∀ (i : ℕ) (hi : i < n), (mapRange n create)[i] = create i"}, {"name": "mapRange_succ", "content": "theorem mapRange_succ {n} {create : ℕ → α} :\n    mapRange (n + 1) create = (mapRange n create).push (create n)"}, {"name": "ext_iff", "content": "theorem ext_iff {F : Type} {α : TypeMap} [ProvableType α] (x y : α F) :\n    x = y ↔ ∀ i (hi : i < size α), (toElements x)[i] = (toElements y)[i]"}, {"name": "eval_varFromOffset", "content": "theorem eval_varFromOffset {α : TypeMap} [ProvableType α] (env : Environment F) (offset : ℕ) :\n    eval env (varFromOffset α offset) = fromElements (.mapRange (size α) fun i => env.get (offset + i))"}, {"name": "getElem_mapFinRange", "content": "theorem getElem_mapFinRange {n} {create : Fin n → α} :\n    ∀ (i : ℕ) (hi : i < n), (mapFinRange n create)[i] = create ⟨ i, hi ⟩"}, {"name": "assignmentFromCircuit_offset", "content": "theorem assignmentFromCircuit_offset (as : CellAssignment W S) (ops : Operations F) :\n    (assignmentFromCircuit as ops).offset = as.offset + ops.localLength"}, {"name": "foldlRange.soundness", "content": "@[circuit_norm ↓]\nlemma foldlRange.soundness :\n  ConstraintsHold.Soundness env (foldlRange m init body constant |>.operations n) ↔\n    ∀ i : Fin m,\n    ConstraintsHold.Soundness env (body (FoldlM.foldlAcc n (Vector.finRange m) body init i) i\n    |>.operations (n + i * (body default i).localLength))"}, {"name": "foldlRange.forAll", "content": "@[circuit_norm ↓]\nlemma foldlRange.forAll :\n  Operations.forAll n prop (foldlRange m init body constant |>.operations n) ↔\n    ∀ i : Fin m,\n      body (FoldlM.foldlAcc n (Vector.finRange m) body init i) i\n      |>.forAll (n + i * (body default i).localLength) prop"}, {"name": "forAll_iff_finRange", "content": "theorem forAll_iff_finRange {constant : ConstantLength (prod circuit)} :\n  ((Vector.finRange m).foldlM circuit init).forAll n prop ↔\n    ∀ i : Fin m, (circuit (foldlAcc n (Vector.finRange m) circuit init i) i)\n    |>.forAll (n + i * (circuit init i).localLength) prop"}, {"name": "forAll_iff", "content": "theorem forAll_iff {constant : ConstantLength (prod circuit)} :\n  (xs.foldlM circuit init).forAll n prop ↔\n    ∀ i : Fin m, (circuit (foldlAcc n xs circuit init i) xs[i.val]).forAll (n + i * (circuit init xs[i.val]).localLength) prop"}, {"name": "forAll_flatten_foldl", "content": "lemma forAll_flatten_foldl :\n  Operations.forAll n prop (List.ofFn fun (i : Fin m) => (circuit (foldlAcc n xs circuit init i) xs[i.val]).operations (n + i * constant.localLength)).flatten\n    ↔ ∀ (i : Fin m), (circuit (foldlAcc n xs circuit init i) xs[i.val]).forAll (n + i * constant.localLength) prop"}, {"name": "forAll_flatten_abstract", "content": "lemma forAll_flatten_abstract (circuit : Fin m → Circuit F β) (constant : ConstantLength circuit) :\n  Operations.forAll n prop (List.ofFn fun i => (circuit i).operations (n + i * constant.localLength)).flatten\n    ↔ ∀ (i : Fin m), (circuit i).forAll (n + i * constant.localLength) prop"}, {"name": "localLength_eq", "content": "theorem localLength_eq :\n    (xs.foldlM circuit init).localLength n = m * constant.localLength"}, {"name": "foldlM_cons", "content": "lemma foldlM_cons (x : α) :\n  (Vector.cons x xs).foldlM circuit init = (do\n    let init' ← circuit init x\n    xs.foldlM circuit init')"}, {"name": "Vector.foldlM_toList", "content": "lemma Vector.foldlM_toList (xs : Vector α n) {m : Type → Type} [Monad m] (body : β → α → m β) (init : β) :\n    xs.foldlM body init = xs.toList.foldlM body init"}, {"name": "operations_eq", "content": "theorem operations_eq :\n  (Vector.foldlM circuit init xs).operations n =\n    (List.ofFn fun i => (circuit (foldlAcc n xs circuit init i) xs[i.val]).operations (n + i * constant.localLength)).flatten"}, {"name": "foldlAcc_zero", "content": "lemma foldlAcc_zero [NeZero m] : foldlAcc n xs circuit init 0 = init"}, {"name": "foldlAcc_cons_succ", "content": "lemma foldlAcc_cons_succ (i : Fin m) (x : α) [constant : ConstantLength (prod circuit)] :\n  foldlAcc n (Vector.cons x xs) circuit init i.succ =\n    foldlAcc (n + (circuit init x).localLength n) xs circuit ((circuit init x).output n) i"}, {"name": "soundness", "content": "@[circuit_norm]\ntheorem soundness (α : TypeMap) [ProvableType α] (n : ℕ) (env : Environment F) (x y : Var α F) :\n    ((circuit α).toSubcircuit n (x, y)).Soundness env = (eval env x = eval env y)"}, {"name": "mapFinRange.soundness", "content": "@[circuit_norm ↓]\nlemma mapFinRange.soundness :\n  ConstraintsHold.Soundness env (mapFinRange m body constant |>.operations n) ↔\n    ∀ i : Fin m, ConstraintsHold.Soundness env (body i |>.operations (n + i*(body 0).localLength))"}, {"name": "mapFinRangeM_forAll_iff", "content": "theorem mapFinRangeM_forAll_iff {circuit : Fin m → Circuit F β} [constant : ConstantLength circuit] :\n  (Vector.mapFinRangeM m circuit).forAll n prop ↔\n    ∀ i : Fin m, (circuit i).forAll (n + i*constant.localLength) prop"}, {"name": "forAll_iff", "content": "theorem forAll_iff :\n  (xs.mapM circuit).forAll n prop ↔\n    ∀ (i : Fin m), (circuit xs[i.val]).forAll (n + i * constant.localLength) prop"}, {"name": "operations_eq", "content": "theorem operations_eq : (xs.mapM circuit).operations n =\n    (List.ofFn fun (i : Fin m) => (circuit xs[i.val]).operations (n + i * constant.localLength)).flatten"}, {"name": "ofFn_flatten_cons", "content": "private lemma ofFn_flatten_cons {circuit : α → Circuit F β} (constant : ConstantLength circuit) (x : α) (xs : Vector α m) (n : ℕ) :\n  (List.ofFn fun i => (circuit (Vector.cons x xs)[i.val]).operations (n + i * constant.localLength)).flatten\n    = (circuit x).operations n ++ (List.ofFn fun i => (circuit xs[i.val]).operations (n + constant.localLength + i * constant.localLength)).flatten"}, {"name": "mapM_cons", "content": "lemma mapM_cons (xs : Vector α n) (body : α → Circuit F β) (x : α) :\n  (Vector.cons x xs).mapM body = do\n    let y ← body x\n    let ys ← xs.mapM body\n    return Vector.cons y ys"}, {"name": "Vector.toList_mapM", "content": "lemma Vector.toList_mapM (xs : Vector α n) {m : Type → Type} [monad: Monad m] [LawfulMonad m] (body : α → m β) :\n    Vector.toList <$> (xs.mapM body) = xs.toList.mapM body"}, {"name": "ext_map_toList", "content": "lemma ext_map_toList (f g : Circuit F (Vector α n)) :\n    (fun v => v.toList) <$> f = (fun v => v.toList) <$> g → f = g"}, {"name": "forAll_flatten", "content": "lemma forAll_flatten (xs : Vector α m) {circuit : α → Circuit F β} (constant : ConstantLength circuit) :\n  Operations.forAll n prop (List.ofFn fun (i : Fin m) => (circuit xs[i.val]).operations (n + i * constant.localLength)).flatten\n    ↔ ∀ (i : Fin m), (circuit xs[i.val]).forAll (n + i * constant.localLength) prop"}, {"name": "allZero.soundness", "content": "theorem allZero.soundness {offset : ℕ} {env : Environment F} {n} {xs : Vector (Expression F) n} :\n    ConstraintsHold.Soundness env ((allZero xs).operations offset) → ∀ x ∈ xs, x.eval env = 0"}, {"name": "foldl.soundness", "content": "@[circuit_norm ↓]\nlemma foldl.soundness [NeZero m] :\n  ConstraintsHold.Soundness env (foldl xs init body const_out constant |>.operations n) ↔\n    ConstraintsHold.Soundness env (body init (xs[0]'(NeZero.pos m)) |>.operations n) ∧\n    ∀ (i : ℕ) (hi : i + 1 < m),\n      let acc"}, {"name": "forAll_iff_const", "content": "theorem forAll_iff_const [NeZero m] (constant : ConstantLength (prod circuit))\n    (h_const_out : ConstantOutput (prod circuit)) :\n  (xs.foldlM circuit init).forAll n prop ↔\n  (circuit init (xs[0]'(NeZero.pos m))).forAll n prop ∧\n  ∀ (i : ℕ) (hi : i + 1 < m),\n    let acc"}, {"name": "ConstantLength.length_eq_default", "content": "lemma ConstantLength.length_eq_default {circuit : α → Circuit F β} (_ : ConstantLength circuit) [Inhabited α] (a : α) (n : ℕ) :\n   (circuit a).localLength n = (circuit default).localLength 0"}, {"name": "foldlAcc_const_succ", "content": "theorem foldlAcc_const_succ (constant : ConstantLength (prod circuit))\n  (h_const_out : ConstantOutput fun (t : β × α) => circuit t.1 t.2)\n  (i : ℕ) (hi : i + 1 < m) :\n  foldlAcc n xs circuit init ⟨ i + 1, hi ⟩ =\n    (circuit default xs[i]).output (n + i * (circuit default default).localLength)"}, {"name": "map.soundness", "content": "@[circuit_norm ↓]\nlemma map.soundness :\n  ConstraintsHold.Soundness env (map xs body constant |>.operations n) ↔\n    ∀ i : Fin m, ConstraintsHold.Soundness env (body xs[i.val] |>.operations (n + i*(body default).localLength))"}, {"name": "omit [Field F] in", "content": "omit [Field F] in\n@[circuit_norm ↓ high]\ntheorem varFromOffset_pair {α β: TypeMap} [ProvableType α] [ProvableType β] (offset : ℕ) :\n    varFromOffset (F:=F) (ProvablePair α β) offset\n    = (varFromOffset α offset, varFromOffset β (offset + size α))"}, {"name": "assignmentFromCircuit_vars", "content": "theorem assignmentFromCircuit_vars (as : CellAssignment W S) (ops : Operations F) :\n    (assignmentFromCircuit as ops).vars = (as.vars ++ (.mapRange ops.localLength fun i => .aux (as.aux_length + i) : Vector (Cell W S) _)\n      ).cast (assignmentFromCircuit_offset ..).symm"}, {"name": "forEach.soundness", "content": "@[circuit_norm ↓]\nlemma forEach.soundness :\n  ConstraintsHold.Soundness env ((forEach xs body constant).operations n) ↔\n    ∀ i : Fin m, ConstraintsHold.Soundness env (body xs[i.val] |>.operations (n + i*(body default).localLength))"}, {"name": "forAll_iff", "content": "theorem forAll_iff {prop : Condition F} :\n  (xs.forM circuit).forAll n prop ↔\n    ∀ (i : Fin m), (circuit xs[i.val]).forAll (n + i * constant.localLength) prop"}, {"name": "operations_eq", "content": "theorem operations_eq :\n  (xs.forM circuit).operations n =\n    (List.ofFn fun (i : Fin m) => (circuit xs[i.val]).operations (n + i * constant.localLength)).flatten"}, {"name": "Vector.forM_toList", "content": "lemma Vector.forM_toList (xs : Vector α n) {m : Type → Type} [Monad m] (body : α → m Unit) :\n    xs.forM body = forM xs.toList body"}], "used_local_defs": [{"name": "InductiveTable.Soundness", "content": "def InductiveTable.Soundness (F : Type) [Field F] (State Input : Type → Type) [ProvableType State] [ProvableType Input]\n    (Spec : (initialState : State F) → (xs : List (Input F)) → (i : ℕ) → (xs.length = i) → (currentState : State F) → Prop)\n    (step : Var State F → Var Input F → Circuit F (Var State F)) :=\n  ∀ (initialState : State F) (row_index : ℕ) (env : Environment F),\n  \n  ∀ (acc_var : Var State F) (x_var : Var Input F)\n    (acc : State F) (x : Input F) (xs : List (Input F)) (xs_len : xs.length = row_index),\n      (eval env acc_var = acc) ∧ (eval env x_var = x) →\n    \n    Circuit.ConstraintsHold.Soundness env (step acc_var x_var |>.operations ((size State) + (size Input))) →\n    \n    Spec initialState xs row_index xs_len acc →\n    \n    Spec initialState (xs.concat x) (row_index + 1) (xs_len ▸ List.length_concat) (eval env (step acc_var x_var |>.output ((size State) + (size Input))))"}, {"name": "InductiveTable.Completeness", "content": "def InductiveTable.Completeness (F : Type) [Field F] (State Input : Type → Type) [ProvableType State] [ProvableType Input]\n    (InputAssumptions : ℕ → Input F → Prop) (InitialStateAssumptions : State F → Prop)\n    (Spec : (initialState : State F) → (xs : List (Input F)) → (i : ℕ) → (xs.length = i) → (currentState : State F) → Prop)\n    (step : Var State F → Var Input F → Circuit F (Var State F)) :=\n  ∀ (initialState : State F) (row_index : ℕ) (env : Environment F),\n  \n  ∀ (acc_var : Var State F) (x_var : Var Input F)\n    (acc : State F) (x : Input F) (xs : List (Input F)) (xs_len : xs.length = row_index),\n    (eval env acc_var = acc) ∧ (eval env x_var = x) →\n  \n  env.UsesLocalWitnessesCompleteness ((size State) + (size Input)) (step acc_var x_var |>.operations ((size State) + (size Input))) →\n  \n  InitialStateAssumptions initialState ∧\n  Spec initialState xs row_index xs_len acc ∧ InputAssumptions row_index x →\n  \n  Circuit.ConstraintsHold.Completeness env (step acc_var x_var |>.operations ((size State) + (size Input)))"}, {"name": "InductiveTable", "content": "structure InductiveTable (F : Type) [Field F] (State Input : Type → Type) [ProvableType State] [ProvableType Input] where\n   \n  step : Var State F → Var Input F → Circuit F (Var State F)\n\n   \n  Spec : (initialState : State F) → (xs : List (Input F)) → (i : ℕ) → (xs.length = i) → (currentState : State F) → Prop\n\n   \n  InputAssumptions : ℕ → Input F → Prop := fun _ _ => True\n  InitialStateAssumptions : State F → Prop := fun _ => True\n\n  soundness : InductiveTable.Soundness F State Input Spec step\n\n  completeness : InductiveTable.Completeness F State Input InputAssumptions InitialStateAssumptions Spec step\n\n  subcircuitsConsistent : ∀ acc x, ((step acc x).operations ((size State) + (size Input))).SubcircuitsConsistent ((size State) + (size Input))\n    := by admit /- proof elided -/"}, {"name": "InductiveTable.inductiveConstraint", "content": "def inductiveConstraint (table : InductiveTable F State Input) : TableConstraint 2 (ProvablePair State Input) F Unit := do\n  let (acc, x) ← getCurrRow\n  let output ← table.step acc x\n  let (output', _) ← getNextRow\n  \n  output' === output"}, {"name": "InductiveTable.equalityConstraint", "content": "def equalityConstraint (Input : TypeMap) [ProvableType Input] (target : State F) : SingleRowConstraint (ProvablePair State Input) F := do\n  let (actual, _) ← getCurrRow\n  actual === (const target)"}, {"name": "InductiveTable.tableConstraints", "content": "def tableConstraints (table : InductiveTable F State Input) (input_state output_state : State F) :\n  List (TableOperation (ProvablePair State Input) F) := [\n    .everyRowExceptLast table.inductiveConstraint,\n    .boundary (.fromStart 0) (equalityConstraint Input input_state),\n    .boundary (.fromEnd 0) (equalityConstraint Input output_state),\n  ]"}, {"name": "InductiveTable.traceInputs", "content": "def traceInputs {N : ℕ} (trace : TraceOfLength F (ProvablePair State Input) N) : List (Input F) :=\n  trace.val.toList.map Prod.snd"}], "used_local_lemmas": [{"name": "InductiveTable.equalityConstraint.soundness", "content": "theorem equalityConstraint.soundness {row : State F × Input F} {input_state : State F} {env : Environment F} :\n  Circuit.ConstraintsHold.Soundness (windowEnv (equalityConstraint Input input_state) ⟨<+> +> row, rfl⟩ env)\n    (equalityConstraint Input input_state .empty).2.circuit\n    ↔ row.1 = input_state"}, {"name": "InductiveTable.traceInputs_length", "content": "omit [Field F] in\nlemma traceInputs_length {N : ℕ} (trace : TraceOfLength F (ProvablePair State Input) N) :\n    (traceInputs trace).length = N"}], "local_ctx": "import Clean.Table.Theorems\n\nimport Clean.Gadgets.Equality\n\ndef InductiveTable.Soundness (F : Type) [Field F] (State Input : Type → Type) [ProvableType State] [ProvableType Input]\n    (Spec : (initialState : State F) → (xs : List (Input F)) → (i : ℕ) → (xs.length = i) → (currentState : State F) → Prop)\n    (step : Var State F → Var Input F → Circuit F (Var State F)) :=\n  ∀ (initialState : State F) (row_index : ℕ) (env : Environment F),\n  \n  ∀ (acc_var : Var State F) (x_var : Var Input F)\n    (acc : State F) (x : Input F) (xs : List (Input F)) (xs_len : xs.length = row_index),\n      (eval env acc_var = acc) ∧ (eval env x_var = x) →\n    \n    Circuit.ConstraintsHold.Soundness env (step acc_var x_var |>.operations ((size State) + (size Input))) →\n    \n    Spec initialState xs row_index xs_len acc →\n    \n    Spec initialState (xs.concat x) (row_index + 1) (xs_len ▸ List.length_concat) (eval env (step acc_var x_var |>.output ((size State) + (size Input))))\n\ndef InductiveTable.Completeness (F : Type) [Field F] (State Input : Type → Type) [ProvableType State] [ProvableType Input]\n    (InputAssumptions : ℕ → Input F → Prop) (InitialStateAssumptions : State F → Prop)\n    (Spec : (initialState : State F) → (xs : List (Input F)) → (i : ℕ) → (xs.length = i) → (currentState : State F) → Prop)\n    (step : Var State F → Var Input F → Circuit F (Var State F)) :=\n  ∀ (initialState : State F) (row_index : ℕ) (env : Environment F),\n  \n  ∀ (acc_var : Var State F) (x_var : Var Input F)\n    (acc : State F) (x : Input F) (xs : List (Input F)) (xs_len : xs.length = row_index),\n    (eval env acc_var = acc) ∧ (eval env x_var = x) →\n  \n  env.UsesLocalWitnessesCompleteness ((size State) + (size Input)) (step acc_var x_var |>.operations ((size State) + (size Input))) →\n  \n  InitialStateAssumptions initialState ∧\n  Spec initialState xs row_index xs_len acc ∧ InputAssumptions row_index x →\n  \n  Circuit.ConstraintsHold.Completeness env (step acc_var x_var |>.operations ((size State) + (size Input)))\n\nstructure InductiveTable (F : Type) [Field F] (State Input : Type → Type) [ProvableType State] [ProvableType Input] where\n   \n  step : Var State F → Var Input F → Circuit F (Var State F)\n\n   \n  Spec : (initialState : State F) → (xs : List (Input F)) → (i : ℕ) → (xs.length = i) → (currentState : State F) → Prop\n\n   \n  InputAssumptions : ℕ → Input F → Prop := fun _ _ => True\n  InitialStateAssumptions : State F → Prop := fun _ => True\n\n  soundness : InductiveTable.Soundness F State Input Spec step\n\n  completeness : InductiveTable.Completeness F State Input InputAssumptions InitialStateAssumptions Spec step\n\n  subcircuitsConsistent : ∀ acc x, ((step acc x).operations ((size State) + (size Input))).SubcircuitsConsistent ((size State) + (size Input))\n    := by admit /- proof elided -/\n\nnamespace InductiveTable\n\nvariable {F : Type} [Field F] {State Input : TypeMap} [ProvableType State] [ProvableType Input]\n\ndef inductiveConstraint (table : InductiveTable F State Input) : TableConstraint 2 (ProvablePair State Input) F Unit := do\n  let (acc, x) ← getCurrRow\n  let output ← table.step acc x\n  let (output', _) ← getNextRow\n  \n  output' === output\n\ndef equalityConstraint (Input : TypeMap) [ProvableType Input] (target : State F) : SingleRowConstraint (ProvablePair State Input) F := do\n  let (actual, _) ← getCurrRow\n  actual === (const target)\n\ndef tableConstraints (table : InductiveTable F State Input) (input_state output_state : State F) :\n  List (TableOperation (ProvablePair State Input) F) := [\n    .everyRowExceptLast table.inductiveConstraint,\n    .boundary (.fromStart 0) (equalityConstraint Input input_state),\n    .boundary (.fromEnd 0) (equalityConstraint Input output_state),\n  ]\n\ndef traceInputs {N : ℕ} (trace : TraceOfLength F (ProvablePair State Input) N) : List (Input F) :=\n  trace.val.toList.map Prod.snd", "target_theorem": "lemma table_soundness_aux (table : InductiveTable F State Input) (input output : State F)\n  (N : ℕ+) (trace : TraceOfLength F (ProvablePair State Input) N) (env : ℕ → ℕ → Environment F) :\n  table.Spec input [] 0 rfl input →\n  TableConstraintsHold (table.tableConstraints input output) trace env →\n    trace.ForAllRowsWithPrevious (fun row i rest => table.Spec input (traceInputs rest) i (traceInputs_length rest) row.1)\n    ∧ trace.lastRow.1 = output :=", "ground_truth_proof": ":= by\n  intro input_spec\n\n  -- add a condition on the trace length to the goal,\n  -- so that we can change the induction to not depend on `N` (which would make it unprovable)\n  rcases trace with ⟨ trace, h_trace ⟩\n  suffices goal : TableConstraintsHold (table.tableConstraints input output) ⟨ trace, h_trace ⟩ env →\n    trace.ForAllRowsWithPrevious (fun row rest => table.Spec input (traceInputs ⟨ rest, rfl ⟩) rest.len (traceInputs_length ⟨ rest, rfl ⟩) row.1) ∧\n    (∀ (h_len : trace.len = N), (trace.lastRow (by rw [h_len]; exact N.pos)).1 = output) by\n      intro constraints\n      specialize goal constraints\n      exact ⟨ goal.left, goal.right h_trace ⟩\n\n  simp only [table_norm, tableConstraints]\n  clear h_trace\n  induction trace using Trace.every_row_two_rows_induction\n\n  case zero =>\n    intro constraints\n    simp only [Trace.ForAllRowsWithPrevious, true_and]\n    intros\n    nomatch N\n\n  case one first_row =>\n    intro constraints\n    simp only [table_norm,\n      List.size_toArray, List.length_nil, List.push_toArray, List.nil_append,\n      List.length_cons, zero_add, List.cons_append, reduceIte, and_true] at constraints\n    obtain ⟨ input_eq, output_eq ⟩ := constraints\n    rw [equalityConstraint.soundness] at input_eq output_eq\n    simp only [table_norm, and_true, Trace.ForAllRowsWithPrevious]\n    constructor\n    · rw [input_eq]\n      exact input_spec\n    intro h_len\n    rw [←h_len] at output_eq\n    simp only [zero_add, tsub_self, reduceIte] at output_eq\n    exact output_eq\n\n  case more curr next rest ih1 ih2 =>\n    intro constraints\n    simp only [table_norm, List.size_toArray, List.length_nil, List.push_toArray,\n      List.nil_append, List.length_cons, zero_add, List.cons_append, Nat.add_eq_zero, one_ne_zero,\n      and_false, reduceIte, tsub_zero,\n      Nat.reduceAdd, true_and, Trace.ForAllRowsWithPrevious] at constraints ih1 ih2 ⊢\n    rcases constraints with ⟨ constraints, output_eq, h_rest ⟩\n    specialize ih2 h_rest\n    have spec_previous : table.Spec input (traceInputs ⟨rest, rfl⟩) rest.len (traceInputs_length ⟨rest, rfl⟩) curr.1 := by\n      simp [ih2]\n    simp only [ih2, and_self, and_true]\n    clear ih1 ih2\n    set env' := windowEnv table.inductiveConstraint ⟨<+> +> curr +> next, _⟩ (env 0 (rest.len + 1))\n    simp only [table_norm, circuit_norm, inductiveConstraint] at constraints\n    obtain ⟨ main_constraints, return_eq ⟩ := constraints\n    have h_env' : env' = windowEnv table.inductiveConstraint ⟨<+> +> curr +> next, _⟩ (env 0 (rest.len + 1)) := rfl\n    simp only [windowEnv, table_assignment_norm, inductiveConstraint, circuit_norm] at h_env'\n    simp only [zero_add, Nat.add_zero, Fin.isValue, PNat.val_ofNat, Nat.reduceAdd, Nat.add_one_sub_one,\n      CellAssignment.assignmentFromCircuit_offset, CellAssignment.assignmentFromCircuit_vars] at h_env'\n    set curr_var : Var State F × Var Input F := varFromOffset (ProvablePair State Input) 0\n    set s := size State\n    set x := size Input\n    set main_ops : Operations F := (table.step (varFromOffset State 0) (varFromOffset Input s) (s + x)).2\n    set t := main_ops.localLength\n\n    have h_env_input_1 i (hi : i < s) : (toElements curr.1)[i] = env'.get i := by\n      have hi' : i < s + x + t + (s + x) := by linarith\n      have hi'' : i < 0 + (s + x) := by linarith\n      have hi''' : i < 0 + (s + x) + t := by linarith\n      rw [h_env']\n      simp +arith only [main_ops, s, t, x, hi, hi', hi'', hi''', table_assignment_norm, circuit_norm, reduceDIte,\n        CellAssignment.assignmentFromCircuit_offset,\n        Vector.mapRange_zero, Vector.empty_append, Vector.append_empty, Vector.getElem_append]\n\n    have h_env_input_2 i (hi : i < x) : (toElements curr.2)[i] = env'.get (i + s) := by\n      have hi' : i + s < s + x + t + (s + x) := by linarith\n      have hi'' : i + s < 0 + (s + x) := by linarith\n      have hi''' : i + s < 0 + (s + x) + t := by linarith\n      rw [h_env']\n      simp +arith only [main_ops, s, t, x, hi', hi'', hi''', table_assignment_norm, circuit_norm, reduceDIte,\n        CellAssignment.assignmentFromCircuit_offset,\n        Vector.mapRange_zero, Vector.empty_append, Vector.append_empty, Vector.getElem_append]\n      congr; omega\n\n    have h_env_output i (hi : i < s) : (toElements next.1)[i] = env'.get (i + (s + x) + t) := by\n      have hi' : i + (s + x) + t < s + x + t + (s + x) := by linarith\n      have hi'' : ¬(i + (s + x) + t < 0 + (s + x)) := by linarith\n      have hi''' : ¬(i + (s + x) + t < 0 + (s + x) + t) := by linarith\n      rw [h_env']\n      simp +arith only [main_ops, hi', s, t, x, table_assignment_norm, circuit_norm, reduceDIte,\n        CellAssignment.assignmentFromCircuit_offset,\n        Vector.mapRange_zero, Vector.empty_append, Vector.append_empty, Vector.getElem_append]\n      simp +arith [hi, s, add_assoc]\n    clear h_env'\n\n    have input_eq_1 : eval env' curr_var.1 = curr.1 := by\n      rw [ProvableType.ext_iff]\n      intro i hi\n      simp only [curr_var, varFromOffset_pair]\n      rw [h_env_input_1 i hi]\n      simp only [ProvableType.eval_varFromOffset,\n        ProvableType.toElements_fromElements, Vector.getElem_mapRange, zero_add]\n\n    have input_eq_2 : eval env' curr_var.2 = curr.2 := by\n      rw [ProvableType.ext_iff]\n      intro i hi\n      simp only [curr_var, varFromOffset_pair]\n      rw [h_env_input_2 i hi]\n      simp only [s, ProvableType.eval_varFromOffset,\n        ProvableType.toElements_fromElements, Vector.getElem_mapRange, zero_add]\n      ac_rfl\n\n    have next_eq : eval env' (varFromOffset State (size State + size Input + main_ops.localLength)) = next.1 := by\n      rw [ProvableType.ext_iff]\n      intro i hi\n      rw [h_env_output i hi, ProvableType.eval_varFromOffset,\n        ProvableType.toElements_fromElements, Vector.getElem_mapRange]\n      simp only [t, s, x]\n      ac_rfl\n\n    simp only [x] at main_constraints\n    have constraints : Circuit.ConstraintsHold.Soundness\n        env' ((table.step curr_var.1 curr_var.2).operations (size State + size Input)) := by\n      simp only [curr_var, varFromOffset_pair]\n      exact main_constraints\n\n    let xs := traceInputs ⟨ rest, rfl ⟩\n    have xs_len := traceInputs_length ⟨ rest, rfl ⟩\n    have xs_concat : traceInputs ⟨rest +> curr, rfl⟩ = xs.concat curr.2 := by\n      simp only [traceInputs, xs, Trace.toList, List.map_concat]\n\n    have h_soundness := table.soundness input rest.len env' curr_var.1 curr_var.2 curr.1 curr.2 xs xs_len\n      ⟨ input_eq_1, input_eq_2 ⟩ constraints spec_previous\n    simp only [curr_var, varFromOffset_pair] at h_soundness\n    simp only [s, x, t, main_ops] at *\n    simp +arith only at return_eq h_soundness\n    rw [←return_eq, next_eq] at h_soundness\n    simp only [xs_concat]\n    use h_soundness\n\n    intro h_len\n    rw [equalityConstraint.soundness] at output_eq\n    rw [←h_len] at output_eq\n    simp only [add_tsub_cancel_right, reduceIte] at output_eq\n    exact output_eq", "nesting_depth": 9, "transitive_dep_count": 251, "subset_aristotle": true, "category": "Applied verif."}
{"id": 122, "thm_name": "Circomlib.MultiAND.soundness", "thm_stmt": "theorem soundness {p : ℕ} [Fact p.Prime] (n : ℕ) :\n    ∀ (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields n) (F p))\n      (input : fields n (F p)),\n    input = eval env input_var →\n    Assumptions n input →\n    Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset) →\n    Spec n input (env ((main input_var).output offset))", "lean_root": "clean", "rel_path": "Clean/Circomlib/Gates.lean", "imports": ["import Clean.Circuit.Theorems", "import Clean.Circuit.Provable", "import Clean.Utils.Field", "import Clean.Circuit", "import Mathlib.Data.Nat.Bitwise", "import Clean.Gadgets.Boolean", "import Clean.Utils.Bitwise", "import Clean.Utils.BinaryOps", "import Clean.Circuit.Basic", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "One", "module": "Init.Prelude"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Vector.cast", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.foldl", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.map", "module": "Init.Data.Vector.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "List.foldl", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "fields", "content": "@[reducible]\ndef fields (n : ℕ) := fun F => Vector F n"}, {"name": "IsBool", "content": "def IsBool {α : Type*} [Zero α] [One α] (x : α) : Prop := x = 0 ∨ x = 1"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "fieldPair", "content": "@[reducible]\ndef fieldPair : TypeMap := fun F => F × F"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "ConstraintsHold", "content": "@[circuit_norm]\ndef ConstraintsHold (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold eval ops\n  | .lookup { table, entry, .. } :: ops =>\n    table.Contains (entry.map eval) ∧ ConstraintsHold eval ops\n  | .subcircuit s :: ops =>\n    ConstraintsHoldFlat eval s.ops ∧ ConstraintsHold eval ops"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}, {"name": "forAll", "content": "@[reducible, circuit_norm]\ndef forAll (circuit : Circuit F α) (n : ℕ) (prop : Condition F) :=\n  (circuit.operations n).forAll n prop"}], "lib_lemmas": [{"name": "Array.foldl_toList", "module": "Init.Data.Array.Bootstrap"}, {"name": "List.foldl_cons", "module": "Init.Data.List.Basic"}, {"name": "List.foldl_nil", "module": "Init.Data.List.Basic"}, {"name": "Vector.foldl_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_map", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.toList_toArray", "module": "Init.Data.Vector.Lemmas"}, {"name": "ZMod.val_one", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Vector.foldl_append", "module": "Init.Data.Vector.Lemmas"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.min_add_right_self", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.min_def", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.strong_induction_on", "module": "Mathlib.Data.Nat.Init"}, {"name": "Vector.cast_cast", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_cast", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_drop", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_take", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.map_append", "module": "Init.Data.Vector.Lemmas"}], "repo_lemmas": [{"name": "toList_length_one", "content": "theorem toList_length_one {α : Type} (v : Vector α 1) :\n    v.toList = [v[0]]"}, {"name": "one_land_of_IsBool", "content": "theorem one_land_of_IsBool (a : ℕ) (h : IsBool a) : 1 &&& a = a"}, {"name": "land_one_of_IsBool", "content": "theorem land_one_of_IsBool (a : ℕ) (h : IsBool a) : a &&& 1 = a"}, {"name": "pure_def", "content": "@[circuit_norm]\ntheorem pure_def {α} (a : α) : (pure a : Circuit F α) = fun _ => (a, [])"}, {"name": "val_of_IsBool", "content": "theorem val_of_IsBool {p : ℕ} [Fact p.Prime] {x : F p} (h : IsBool x) : IsBool x.val"}, {"name": "one", "content": "@[circuit_norm]\ntheorem one {α : Type*} [Zero α] [One α] : IsBool (1 : α)"}, {"name": "zero", "content": "@[circuit_norm]\ntheorem zero {α : Type*} [Zero α] [One α] : IsBool (0 : α)"}, {"name": "eval_fieldPair", "content": "@[circuit_norm ↓]\ntheorem eval_fieldPair {F : Type} [Field F] (env : Environment F) (t : Var fieldPair F) :\n  ProvableType.eval env t = (match t with | (x, y) => (Expression.eval env x, Expression.eval env y))"}, {"name": "toList_length_two", "content": "theorem toList_length_two {α : Type} (v : Vector α 2) :\n    v.toList = [v[0], v[1]]"}, {"name": "List.foldl_and_IsBool", "content": "theorem List.foldl_and_IsBool (l : List ℕ) :\n    IsBool (List.foldl (· &&& ·) 1 l : ℕ)"}, {"name": "List.and_foldl_eq_foldl", "content": "theorem List.and_foldl_eq_foldl (a : ℕ) (orig : ℕ) (l : List ℕ) :\n    a &&& List.foldl (· &&& ·) orig l = List.foldl (· &&& ·) (a &&& orig) l"}, {"name": "eval_fields", "content": "@[circuit_norm ↓]\ntheorem eval_fields {F : Type} [Field F] (env : Environment F) (x : Var (fields n) F) :\n  ProvableType.eval env x = x.map (Expression.eval env)"}, {"name": "append_take_drop", "content": "theorem append_take_drop {v : Vector α (n + m)} :\n    (v.take n |>.cast Nat.min_add_right_self) ++ (v.drop n |>.cast (Nat.add_sub_self_left n m)) = v"}, {"name": "ext", "content": "@[ext]\ntheorem ext {f g : Circuit F α}\n  (h_output : ∀ n, f.output n = g.output n)\n  (h_operations : ∀ n, f.operations n = g.operations n) :\n    f = g"}, {"name": "ext_iff", "content": "theorem ext_iff {f g : Circuit F α} :\n  (f = g) ↔ (∀ n, (f.output n = g.output n) ∧ (f.operations n = g.operations n))"}, {"name": "ConstraintsHold.bind_soundness", "content": "@[circuit_norm] theorem ConstraintsHold.bind_soundness {f : Circuit F α} {g : α → Circuit F β} (n : ℕ) :\n  ConstraintsHold.Soundness env ((f >>= g).operations n)\n  ↔ ConstraintsHold.Soundness env (f.operations n) ∧\n    ConstraintsHold.Soundness env ((g (f.output n)).operations (n + f.localLength n))"}, {"name": "bind_forAll", "content": "@[circuit_norm]\ntheorem bind_forAll {f : Circuit F α} {g : α → Circuit F β} :\n  ((f >>= g).operations n).forAll n prop ↔\n    (f.operations n).forAll n prop ∧ (((g (f.output n)).operations (n + f.localLength n)).forAll (n + f.localLength n)) prop"}, {"name": "forAll_append", "content": "@[circuit_norm]\ntheorem forAll_append {condition : Condition F} {offset : ℕ} {as bs: Operations F} :\n  forAll offset condition (as ++ bs) ↔\n    forAll offset condition as ∧ forAll (as.localLength + offset) condition bs"}, {"name": "forAll_empty", "content": "@[circuit_norm]\ntheorem forAll_empty {condition : Condition F} {n : ℕ} : forAll n condition [] = True"}, {"name": "ConstraintsHold.soundness_iff_forAll", "content": "theorem ConstraintsHold.soundness_iff_forAll (n : ℕ) (env : Environment F) (ops : Operations F) :\n  ConstraintsHold.Soundness env ops ↔ ops.forAll n {\n    assert _ e := env e = 0,\n    lookup _ l := l.table.Soundness (l.entry.map env),\n    subcircuit _ _ s := s.Soundness env\n  }"}], "used_local_defs": [{"name": "Circomlib.AND.main", "content": "def main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out"}, {"name": "Circomlib.MultiAND.main", "content": "def main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)"}, {"name": "Circomlib.MultiAND.Assumptions", "content": "def Assumptions (n : ℕ) (input : fields n (F p)) : Prop :=\n  ∀ (i : ℕ) (h : i < n), IsBool input[i]"}, {"name": "Circomlib.MultiAND.Spec", "content": "def Spec (n : ℕ) (input : fields n (F p)) (output : F p) : Prop :=\n  output.val = (input.map (·.val)).foldl (· &&& ·) 1 ∧ IsBool output"}], "used_local_lemmas": [{"name": "Circomlib.MultiAND.Vector.foldl_empty'", "content": "lemma Vector.foldl_empty' {α β : Type} (init : β) (f : β → α → β) (v : Vector α 0) :\n    Vector.foldl f init v = init"}, {"name": "Circomlib.MultiAND.Vector.foldl_and_split", "content": "lemma Vector.foldl_and_split {n1 n2 n3 : ℕ} (v : Vector ℕ n3)\n    (v1 : Vector ℕ n1) (v2 : Vector ℕ n2) (h_sum : n1 + n2 = n3)\n    (h_split : v = h_sum ▸ (v1 ++ v2)) :\n    Vector.foldl (· &&& ·) 1 v =\n    Vector.foldl (· &&& ·) 1 v1 &&& Vector.foldl (· &&& ·) 1 v2"}, {"name": "Circomlib.MultiAND.soundness_zero", "content": "lemma soundness_zero {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 0) (F p))\n    (input : fields 0 (F p)) (_h_env : input = eval env input_var)\n    (_h_assumptions : Assumptions 0 input)\n    (_h_hold : Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset)) :\n    Spec 0 input (env ((main input_var).output offset))"}, {"name": "Circomlib.MultiAND.soundness_one", "content": "lemma soundness_one {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 1) (F p))\n    (input : fields 1 (F p)) (h_env : input = eval env input_var)\n    (h_assumptions : Assumptions 1 input)\n    (_h_hold : Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset)) :\n    Spec 1 input (env ((main input_var).output offset))"}, {"name": "Circomlib.MultiAND.soundness_two", "content": "lemma soundness_two {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 2) (F p))\n    (input : fields 2 (F p)) (h_env : input = eval env input_var)\n    (h_assumptions : Assumptions 2 input)\n    (h_hold : Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset)) :\n    Spec 2 input (env ((main input_var).output offset))"}], "local_ctx": "import Clean.Circuit\n\nimport Clean.Utils.Field\n\nimport Clean.Gadgets.Boolean\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Utils.Vector\n\nimport Clean.Utils.BinaryOps\n\nimport Clean.Circuit.Theorems\n\nimport Mathlib.Data.Nat.Bitwise\n\nopen IsBool\n\nnamespace Circomlib\n\nvariable {p : ℕ} [Fact p.Prime]\n\nopen Circuit (bind_output_eq bind_localLength_eq bind_forAll)\n\nopen Operations (append_localLength)\n\nopen BinaryOps (List.foldl_and_IsBool List.and_foldl_eq_foldl)\n\nnamespace XOR\n\nend XOR\n\nnamespace AND\n\ndef main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out\n\nend AND\n\nnamespace OR\n\nend OR\n\nnamespace NOT\n\nend NOT\n\nnamespace NAND\n\nend NAND\n\nnamespace NOR\n\nend NOR\n\nnamespace MultiAND\n\ndef main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)\n\ndef Assumptions (n : ℕ) (input : fields n (F p)) : Prop :=\n  ∀ (i : ℕ) (h : i < n), IsBool input[i]\n\ndef Spec (n : ℕ) (input : fields n (F p)) (output : F p) : Prop :=\n  output.val = (input.map (·.val)).foldl (· &&& ·) 1 ∧ IsBool output", "target_theorem": "theorem soundness {p : ℕ} [Fact p.Prime] (n : ℕ) :\n    ∀ (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields n) (F p))\n      (input : fields n (F p)),\n    input = eval env input_var →\n    Assumptions n input →\n    Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset) →\n    Spec n input (env ((main input_var).output offset)) :=", "ground_truth_proof": ":= by\n  induction n using Nat.strong_induction_on with\n  | _ n IH =>\n    intro offset env input_var input h_env h_assumptions h_hold\n    match n with\n    | 0 => exact soundness_zero offset env input_var input h_env h_assumptions h_hold\n    | 1 => exact soundness_one offset env input_var input h_env h_assumptions h_hold\n    | 2 => exact soundness_two offset env input_var input h_env h_assumptions h_hold\n    | m + 3 =>\n      simp only [main] at h_hold ⊢\n      simp only [Spec]\n      let n1 := (m + 3) / 2\n      let n2 := (m + 3) - n1\n      have h_sum : n1 + n2 = m + 3 := by unfold n1 n2; omega\n      have h_n1_lt : n1 < m + 3 := by unfold n1; omega\n      have h_n2_lt : n2 < m + 3 := by unfold n2; omega\n      let input1 : fields n1 (F p) := input.take n1 |>.cast (by simp only [Nat.min_def, n1]; split <;> omega)\n      let input2 : fields n2 (F p) := input.drop n1 |>.cast (by omega)\n      let input_var1 : Var (fields n1) (F p) := input_var.take n1 |>.cast (by simp only [Nat.min_def, n1]; split <;> omega)\n      let input_var2 : Var (fields n2) (F p) := input_var.drop n1 |>.cast (by omega)\n      have h_eval1 : input1 = eval env input_var1 := by\n        simp only [input_var1, input1]\n        apply Vector.ext\n        intro i hi\n        simp only [h_env, ProvableType.eval_fields, Vector.getElem_map, Vector.getElem_cast, Vector.getElem_take]\n\n      have h_eval2 : input2 = eval env input_var2 := by\n        simp only [input_var2, input2]\n        apply Vector.ext\n        intro i hi\n        simp only [h_env, ProvableType.eval_fields, Vector.getElem_map, Vector.getElem_cast, Vector.getElem_drop]\n\n      have h_assumptions1 : Assumptions n1 input1 := by\n        intro i hi\n        -- input1[i] = input[i] since input1 is take of input\n        simp only [input1]\n        have : (input.take n1 |>.cast (by simp only [Nat.min_def, n1]; split <;> omega))[i]'hi = input[i]'(by omega) := by\n          rw [Vector.getElem_cast, Vector.getElem_take]\n        rw [this]\n        apply h_assumptions i (by omega)\n      have h_assumptions2 : Assumptions n2 input2 := by\n        intro i hi\n        -- input2[i] = input[n1 + i] since input2 is drop of input\n        simp only [input2]\n        have : (input.drop n1 |>.cast (by omega))[i]'hi = input[n1 + i]'(by omega) := by\n          rw [Vector.getElem_cast, Vector.getElem_drop]\n        rw [this]\n        apply h_assumptions (n1 + i) (by omega)\n      have h_spec1 : Spec n1 input1 (env ((main input_var1).output offset)) := by\n        apply IH n1 h_n1_lt offset env input_var1 input1 h_eval1 h_assumptions1\n        rw [Circuit.ConstraintsHold.bind_soundness] at h_hold\n        exact h_hold.1\n      have h_spec2 : Spec n2 input2 (env ((main input_var2).output (offset + (main input_var1).localLength offset))) := by\n        apply IH n2 h_n2_lt (offset + (main input_var1).localLength offset) env input_var2 input2 h_eval2 h_assumptions2\n        rw [Circuit.ConstraintsHold.bind_soundness] at h_hold\n        rw [Circuit.ConstraintsHold.bind_soundness] at h_hold\n        exact h_hold.2.1\n      have h_hold' := h_hold\n      rw [Circuit.ConstraintsHold.bind_soundness] at h_hold'\n      rw [Circuit.ConstraintsHold.bind_soundness] at h_hold'\n      let out1 := (main input_var1).output offset\n      let out2 := (main input_var2).output (offset + (main input_var1).localLength offset)\n      have h_and_spec := AND.circuit.soundness\n        (offset + (main input_var1).localLength offset + (main input_var2).localLength (offset + (main input_var1).localLength offset))\n        env\n        (out1, out2)\n        (env out1, env out2)\n        (by simp only [ProvableType.eval_fieldPair])\n        ⟨by rcases h_spec1 with ⟨_, h_binary1⟩; exact h_binary1,\n         by rcases h_spec2 with ⟨_, h_binary2⟩; exact h_binary2⟩\n        h_hold'.2.2\n\n      rcases h_spec1 with ⟨h_val1, h_binary1⟩\n      rcases h_spec2 with ⟨h_val2, h_binary2⟩\n      rcases h_and_spec with ⟨h_and_val, h_and_binary⟩\n      constructor\n      · trans (Vector.foldl (fun x1 x2 => x1 &&& x2) 1 (input1.map (·.val)) &&&\n               Vector.foldl (fun x1 x2 => x1 &&& x2) 1 (input2.map (·.val)))\n        · convert h_and_val using 1\n          simp only [out1, out2]\n          simp only [h_val1, h_val2]\n\n        have h_append : input1.cast (by omega : n1 = n1) ++ input2.cast (by omega : n2 = n2) =\n                          input.cast (by omega : m + 3 = n1 + n2) := by\n            simp only [input1, input2]\n            have h_eq : n1 + n2 = m + 3 := by omega\n            simp only [Vector.cast_cast]\n            rw [← Vector.append_take_drop (n := n1) (m := n2) (v := input.cast h_eq.symm)]\n            congr 1\n\n        symm\n        refine Vector.foldl_and_split (Vector.map (·.val) input) (Vector.map (·.val) input1) (Vector.map (·.val) input2) ?_ ?_\n        · exact h_sum\n        · have h_map_append : Vector.map (·.val) (input.cast (by omega : m + 3 = n1 + n2)) =\n                             Vector.map (·.val) (input1.cast (by omega : n1 = n1) ++ input2.cast (by omega : n2 = n2)) := by\n            congr 1\n            exact h_append.symm\n\n          simp only [Vector.map_append] at h_map_append\n\n          have h1 : Vector.map (·.val) input = (Vector.map (·.val) (input.cast (by omega : m + 3 = n1 + n2))).cast h_sum := by\n            ext i\n            simp only [Vector.getElem_map, Vector.getElem_cast]\n\n          have h2 : Vector.map (·.val) (input1.cast (by omega : n1 = n1)) = Vector.map (·.val) input1 := by\n            ext i\n            simp only [Vector.getElem_map, Vector.getElem_cast]\n\n          have h3 : Vector.map (·.val) (input2.cast (by omega : n2 = n2)) = Vector.map (·.val) input2 := by\n            ext i\n            simp only [Vector.getElem_map, Vector.getElem_cast]\n\n          rw [h1, h_map_append, h2, h3]\n\n          have h_cast_transport : ∀ {n m : ℕ} (h : n = m) (v : Vector ℕ n),\n                                  Vector.cast h v = h ▸ v := by\n            intros n m h v\n            subst h\n            rfl\n\n          rw [h_cast_transport]\n\n      · exact h_and_binary", "nesting_depth": 9, "transitive_dep_count": 134, "subset_aristotle": true, "category": "Applied verif."}
{"id": 123, "thm_name": "MemoryAccessList.isConsistent_iff_all_single_address", "thm_stmt": "theorem MemoryAccessList.isConsistent_iff_all_single_address (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    (∀ addr : ℕ, MemoryAccessList.isConsistentSingleAddress (MemoryAccessList.filterAddress accesses addr) (MemoryAccessList.filterAddress_sorted accesses h_sorted addr))", "lean_root": "clean", "rel_path": "Clean/Utils/OfflineMemory.lean", "imports": ["import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Mathlib.Data.List.Sort", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Sorted", "module": "Mathlib.Deprecated.Sort"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.Sorted.filter", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Sorted.of_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.filter_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.filter_nil", "module": "Init.Data.List.Basic"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "forall_true_left", "module": "Mathlib.Logic.Basic"}, {"name": "Bool.and_true", "module": "Init.SimpLemmas"}, {"name": "List.all_cons", "module": "Init.Data.List.Basic"}, {"name": "List.all_nil", "module": "Init.Data.List.Basic"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "decide_eq_true_eq", "module": "Init.SimpLemmas"}, {"name": "imp_self", "module": "Init.Core"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "true_iff", "module": "Init.SimpLemmas"}, {"name": "List.sorted_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.filter_cons_of_pos", "module": "Init.Data.List.Lemmas"}, {"name": "decide_true", "module": "Init.Core"}, {"name": "Bool.not_or_self", "module": "Init.Data.Bool"}, {"name": "Bool.true_and", "module": "Init.SimpLemmas"}, {"name": "List.all_eq_true", "module": "Init.Data.List.Lemmas"}, {"name": "List.all_filter", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_filter", "module": "Init.Data.List.Lemmas"}, {"name": "and_imp", "module": "Init.SimpLemmas"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MemoryAccess", "content": "def MemoryAccess := ℕ × ℕ × ℕ × ℕ"}, {"name": "MemoryAccessList", "content": "def MemoryAccessList := List MemoryAccess"}, {"name": "timestamp_ordering", "content": "abbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2"}, {"name": "MemoryAccessList.isTimestampSorted", "content": "def MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering"}, {"name": "MemoryAccessList.lastWriteValue", "content": "def MemoryAccessList.lastWriteValue (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) (addr : ℕ) : ℕ := match accesses with\n  \n  | [] => 0\n  | (_t, addr', _readValue, writeValue) :: rest =>\n    if addr' = addr then\n      \n      writeValue\n    else\n      MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr"}, {"name": "MemoryAccessList.isConsistentOnline", "content": "def MemoryAccessList.isConsistentOnline (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, addr, readValue, _writeValue) :: rest =>\n    \n    readValue = MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n    ∧ MemoryAccessList.isConsistentOnline rest (List.Sorted.of_cons h)\n\nexample : MemoryAccessList.isConsistentOnline [] (by admit /- proof elided -/\n) := by admit /- proof elided -/"}, {"name": "MemoryAccessList.filterAddress", "content": "def MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)"}, {"name": "MemoryAccessList.isConsistentSingleAddress", "content": "def MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}], "used_local_lemmas": [{"name": "MemoryAccessList.filterAddress_sorted", "content": "theorem MemoryAccessList.filterAddress_sorted (accesses : MemoryAccessList)\n    (h : accesses.isTimestampSorted) (addr : ℕ) :\n    (MemoryAccessList.filterAddress accesses addr).isTimestampSorted"}, {"name": "MemoryAccessList.filterAddress_cons", "content": "theorem MemoryAccessList.filterAddress_cons (head : MemoryAccess) (tail : MemoryAccessList) (addr : ℕ) :\n    MemoryAccessList.filterAddress (head :: tail) addr =\n    match head with\n    | ⟨_t, a, _r, _w⟩ => ((if a = addr then\n      (head :: (MemoryAccessList.filterAddress tail addr))\n        else (MemoryAccessList.filterAddress tail addr)))"}, {"name": "MemoryAccessList.isConsistentSingleAddress_iff", "content": "theorem MemoryAccessList.isConsistentSingleAddress_iff (accesses : MemoryAccessList) (addr : ℕ) (h_sorted : accesses.isTimestampSorted)\n    (h_eq : accesses.all (fun (_t, addr', _readValue, _writeValue) => addr' = addr)) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    MemoryAccessList.isConsistentSingleAddress accesses h_sorted"}, {"name": "MemoryAccessList.lastWriteValue_filter", "content": "theorem MemoryAccessList.lastWriteValue_filter (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted) (addr : ℕ) (h_sorted' : ((MemoryAccessList.filterAddress accesses addr).isTimestampSorted))  :\n    MemoryAccessList.lastWriteValue accesses h_sorted addr =\n    MemoryAccessList.lastWriteValue (MemoryAccessList.filterAddress accesses addr) h_sorted' addr"}, {"name": "MemoryAccessList.isConsistentOnline_filter_of_consistentOnline", "content": "theorem MemoryAccessList.isConsistentOnline_filter_of_consistentOnline (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted)\n    (h_consistent : MemoryAccessList.isConsistentOnline accesses h_sorted) (addr : ℕ) :\n    MemoryAccessList.isConsistentOnline (MemoryAccessList.filterAddress accesses addr) (MemoryAccessList.filterAddress_sorted accesses h_sorted addr)"}, {"name": "MemoryAccessList.isConsistentSingleAddress_cons", "content": "theorem MemoryAccessList.isConsistentSingleAddress_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail)) (h_sorted' : tail.isTimestampSorted)\n    (h : isConsistentSingleAddress (head :: tail) h_sorted) :\n    isConsistentSingleAddress tail h_sorted'"}, {"name": "MemoryAccessList.isConsistentSingleAddress_cons_forall", "content": "theorem MemoryAccessList.isConsistentSingleAddress_cons_forall (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail))\n    : (∀ addr : ℕ, (filterAddress (head :: tail) addr).isConsistentSingleAddress (MemoryAccessList.filterAddress_sorted (head :: tail) h_sorted addr)) →\n    (∀ addr : ℕ, isConsistentSingleAddress (filterAddress tail addr) (MemoryAccessList.filterAddress_sorted tail (by simp_all only [isTimestampSorted,\n      List.sorted_cons]) addr))"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Utils.Field\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Tactics\n\nimport Mathlib.Data.List.Sort\n\ndef MemoryAccess := ℕ × ℕ × ℕ × ℕ \n\ndef MemoryAccessList := List MemoryAccess\n\nabbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2\n\ndef MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering\n\ndef MemoryAccessList.lastWriteValue (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) (addr : ℕ) : ℕ := match accesses with\n  \n  | [] => 0\n  | (_t, addr', _readValue, writeValue) :: rest =>\n    if addr' = addr then\n      \n      writeValue\n    else\n      MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n\ndef MemoryAccessList.isConsistentOnline (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, addr, readValue, _writeValue) :: rest =>\n    \n    readValue = MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n    ∧ MemoryAccessList.isConsistentOnline rest (List.Sorted.of_cons h)\n\nexample : MemoryAccessList.isConsistentOnline [] (by admit /- proof elided -/\n) := by admit /- proof elided -/\n\ndef MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)\n\ndef MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)", "target_theorem": "theorem MemoryAccessList.isConsistent_iff_all_single_address (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    (∀ addr : ℕ, MemoryAccessList.isConsistentSingleAddress (MemoryAccessList.filterAddress accesses addr) (MemoryAccessList.filterAddress_sorted accesses h_sorted addr)) :=", "ground_truth_proof": ":= by\n  constructor\n  · intro h addr\n    have h' := MemoryAccessList.isConsistentSingleAddress_iff (accesses.filterAddress addr) addr (MemoryAccessList.filterAddress_sorted accesses h_sorted addr)\n        (by simp only [filterAddress, List.all_filter, Bool.not_or_self, List.all_eq_true,\n          implies_true])\n    exact h'.mp (MemoryAccessList.isConsistentOnline_filter_of_consistentOnline accesses h_sorted h addr)\n  · intro h\n    -- by induction on the list of accesses\n    induction accesses with\n    | nil =>\n      simp [isConsistentOnline]\n    | cons head tail ih =>\n      obtain ⟨t, a, r, w⟩ := head\n      have h_sorted' : isTimestampSorted tail := by\n        unfold isTimestampSorted at h_sorted\n        exact List.Sorted.of_cons h_sorted\n      specialize ih h_sorted'\n      have h_tail := MemoryAccessList.isConsistentSingleAddress_cons_forall (t, a, r, w) tail h_sorted h\n      specialize ih h_tail\n      simp only [isConsistentOnline, ih, and_true]\n\n      have h_tail_filter_sorted : (MemoryAccessList.filterAddress tail a).isTimestampSorted := by\n        simp only [filterAddress]\n        apply List.Sorted.filter\n        exact h_sorted'\n\n      have h_filtered_sorted : MemoryAccessList.isTimestampSorted ((t, a, r, w) :: (MemoryAccessList.filterAddress tail a)) := by\n        simp only [isTimestampSorted, List.sorted_cons, filterAddress, List.mem_filter,\n          and_imp] at ⊢ h_sorted\n        simp_all only [implies_true, true_and]\n        apply List.Sorted.filter\n        assumption\n\n      have h' := MemoryAccessList.lastWriteValue_filter tail h_sorted' a h_tail_filter_sorted\n\n      specialize h a\n      simp [filterAddress_cons] at h\n      -- rw [h']\n\n      have h_iff := MemoryAccessList.isConsistentSingleAddress_iff ((t, a, r, w) :: filterAddress tail a) a h_filtered_sorted\n      simp only [filterAddress, List.all_cons, decide_true, List.all_filter, Bool.not_or_self,\n        Bool.true_and, List.all_eq_true, implies_true, forall_const] at h_iff\n      have is_consistent_online_full := h_iff.mpr h\n      simp only [isConsistentOnline] at is_consistent_online_full\n      simp only [is_consistent_online_full.left]\n      rw [eq_comm]\n      apply MemoryAccessList.lastWriteValue_filter", "nesting_depth": 3, "transitive_dep_count": 44, "subset_aristotle": true, "category": "Applied verif."}
{"id": 124, "thm_name": "Gadgets.Addition32.Theorems.add32_soundness", "thm_stmt": "theorem add32_soundness {x0 x1 x2 x3 y0 y1 y2 y3 carry_in c0 c1 c2 c3 z0 z1 z2 z3 : F p}\n    (x0_byte : x0.val < 256) (x1_byte : x1.val < 256) (x2_byte : x2.val < 256) (x3_byte : x3.val < 256)\n    (y0_byte : y0.val < 256) (y1_byte : y1.val < 256) (y2_byte : y2.val < 256) (y3_byte : y3.val < 256)\n    (z0_byte : z0.val < 256) (z1_byte : z1.val < 256) (z2_byte : z2.val < 256) (z3_byte : z3.val < 256)\n    (carry_in_bool : IsBool carry_in) (c0_bool : IsBool c0)\n    (c1_bool : IsBool c1) (c2_bool : IsBool c2) (c3_bool : IsBool c3)\n    (h0 : x0 + y0 + carry_in = c0 * 256 + z0)\n    (h1 : x1 + y1 + c0 = c1 * 256 + z1)\n    (h2 : x2 + y2 + c1 = c2 * 256 + z2)\n    (h3 : x3 + y3 + c2 = c3 * 256 + z3) :\n    ZMod.val z0 + ZMod.val z1 * 256 + ZMod.val z2 * 256 ^ 2 + ZMod.val z3 * 256 ^ 3 =\n      (ZMod.val x0 + ZMod.val x1 * 256 + ZMod.val x2 * 256 ^ 2 + ZMod.val x3 * 256 ^ 3 +\n            (ZMod.val y0 + ZMod.val y1 * 256 + ZMod.val y2 * 256 ^ 2 + ZMod.val y3 * 256 ^ 3) +\n          ZMod.val carry_in) %\n        2 ^ 32 ∧\n    ZMod.val c3 =\n      (ZMod.val x0 + ZMod.val x1 * 256 + ZMod.val x2 * 256 ^ 2 + ZMod.val x3 * 256 ^ 3 +\n            (ZMod.val y0 + ZMod.val y1 * 256 + ZMod.val y2 * 256 ^ 2 + ZMod.val y3 * 256 ^ 3) +\n          ZMod.val carry_in) /\n        2 ^ 32", "lean_root": "clean", "rel_path": "Clean/Gadgets/Addition32/Theorems.lean", "imports": ["import Clean.Types.U32", "import Clean.Gadgets.Addition8.Addition8FullCarry", "import Clean.Gadgets.Boolean", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "One", "module": "Init.Prelude"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.reducePow", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "CharP", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}], "used_repo_defs": [{"name": "syntax \"field_to_nat\" : tactic", "content": "syntax \"field_to_nat\" : tactic\n\nsyntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|field_to_nat) =>\n    `(tactic|(\n      intros\n      repeat rw [ZMod.val_add] \n      repeat rw [ZMod.val_mul] \n      repeat rw [val_eq_256]\n      try simp only [Nat.add_mod_mod, Nat.mod_add_mod, Nat.mul_mod_mod, Nat.mod_mul_mod]\n      rw [Nat.mod_eq_of_lt _]\n      repeat linarith [‹Fact (_ > 512)›.elim]))\n\nexample [Fact (p > 512)] (x y : F p) (hx : x.val < 256) (hy : y.val < 2) :\n    (x + y * 256).val = x.val + y.val * 256 := by admit /- proof elided -/"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  x: F\n  y: F\n  carryIn: F"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "IsBool", "content": "def IsBool {α : Type*} [Zero α] [One α] (x : α) : Prop := x = 0 ∨ x = 1"}, {"name": "Outputs", "content": "structure Outputs (F : Type) where\n  z: F\n  carryOut: F"}], "lib_lemmas": [{"name": "CharP.cast_eq_zero", "module": "Mathlib.Algebra.CharP.Defs"}, {"name": "Nat.cast_one", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "ZMod.val_one", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_zero", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "or_true", "module": "Init.SimpLemmas"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}, {"name": "Int.add_emod_right", "module": "Init.Data.Int.DivMod.Lemmas"}, {"name": "Int.add_mul_ediv_left", "module": "Init.Data.Int.DivMod.Bootstrap"}, {"name": "Int.ediv_eq_zero_of_lt", "module": "Init.Data.Int.DivMod.Lemmas"}, {"name": "Int.emod_eq_of_lt", "module": "Init.Data.Int.DivMod.Lemmas"}, {"name": "Nat.zero_le", "module": "Init.Prelude"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "val_lt_two", "content": "theorem val_lt_two {p : ℕ} [Fact p.Prime] {x : F p} (h : IsBool x) : x.val < 2"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Gadgets.Addition32.Theorems.lift_val1", "content": "lemma lift_val1 {x y b : (F p)} (x_byte : x.val < 256) (y_byte : y.val < 256) (b_bool : IsBool b) :\n    (x + y + b).val = (x.val + y.val + b.val)"}, {"name": "Gadgets.Addition32.Theorems.lift_val2", "content": "lemma lift_val2 {x b : (F p)} (x_byte : x.val < 256) (b_bool : IsBool b) :\n    (b * 256 + x).val = (b.val * 256 + x.val)"}, {"name": "Gadgets.Addition32.Theorems.zify_bool", "content": "omit p_large_enough in\nlemma zify_bool {b : (F p)} (b_bool : IsBool b) : (↑(b.val) : ℤ) = 0 ∨ (↑(b.val) : ℤ) = 1"}], "local_ctx": "import Clean.Gadgets.Addition8.Addition8FullCarry\n\nimport Clean.Types.U32\n\nimport Clean.Utils.Field\n\nimport Clean.Gadgets.Boolean\n\nvariable {p : ℕ} [Fact p.Prime]\n\nvariable [p_large_enough: Fact (p > 512)]\n\nnamespace Gadgets.Addition32.Theorems", "target_theorem": "theorem add32_soundness {x0 x1 x2 x3 y0 y1 y2 y3 carry_in c0 c1 c2 c3 z0 z1 z2 z3 : F p}\n    (x0_byte : x0.val < 256) (x1_byte : x1.val < 256) (x2_byte : x2.val < 256) (x3_byte : x3.val < 256)\n    (y0_byte : y0.val < 256) (y1_byte : y1.val < 256) (y2_byte : y2.val < 256) (y3_byte : y3.val < 256)\n    (z0_byte : z0.val < 256) (z1_byte : z1.val < 256) (z2_byte : z2.val < 256) (z3_byte : z3.val < 256)\n    (carry_in_bool : IsBool carry_in) (c0_bool : IsBool c0)\n    (c1_bool : IsBool c1) (c2_bool : IsBool c2) (c3_bool : IsBool c3)\n    (h0 : x0 + y0 + carry_in = c0 * 256 + z0)\n    (h1 : x1 + y1 + c0 = c1 * 256 + z1)\n    (h2 : x2 + y2 + c1 = c2 * 256 + z2)\n    (h3 : x3 + y3 + c2 = c3 * 256 + z3) :\n    ZMod.val z0 + ZMod.val z1 * 256 + ZMod.val z2 * 256 ^ 2 + ZMod.val z3 * 256 ^ 3 =\n      (ZMod.val x0 + ZMod.val x1 * 256 + ZMod.val x2 * 256 ^ 2 + ZMod.val x3 * 256 ^ 3 +\n            (ZMod.val y0 + ZMod.val y1 * 256 + ZMod.val y2 * 256 ^ 2 + ZMod.val y3 * 256 ^ 3) +\n          ZMod.val carry_in) %\n        2 ^ 32 ∧\n    ZMod.val c3 =\n      (ZMod.val x0 + ZMod.val x1 * 256 + ZMod.val x2 * 256 ^ 2 + ZMod.val x3 * 256 ^ 3 +\n            (ZMod.val y0 + ZMod.val y1 * 256 + ZMod.val y2 * 256 ^ 2 + ZMod.val y3 * 256 ^ 3) +\n          ZMod.val carry_in) /\n        2 ^ 32 :=", "ground_truth_proof": ":= by\n\n  apply_fun ZMod.val at h0 h1 h2 h3\n  rw [lift_val1 x0_byte y0_byte carry_in_bool, lift_val2 z0_byte c0_bool] at h0\n  rw [lift_val1 x1_byte y1_byte c0_bool, lift_val2 z1_byte c1_bool] at h1\n  rw [lift_val1 x2_byte y2_byte c1_bool, lift_val2 z2_byte c2_bool] at h2\n  rw [lift_val1 x3_byte y3_byte c2_bool, lift_val2 z3_byte c3_bool] at h3\n\n  have c0_bool := zify_bool c0_bool\n  have c1_bool := zify_bool c1_bool\n  have c2_bool := zify_bool c2_bool\n  have c3_bool := zify_bool c3_bool\n  have carry_in_bool := zify_bool carry_in_bool\n\n  have x0_pos : 0 ≤ x0.val := by exact Nat.zero_le _\n  have y0_pos : 0 ≤ y0.val := by exact Nat.zero_le _\n  have x1_pos : 0 ≤ x1.val := by exact Nat.zero_le _\n  have y1_pos : 0 ≤ y1.val := by exact Nat.zero_le _\n  have x2_pos : 0 ≤ x2.val := by exact Nat.zero_le _\n  have y2_pos : 0 ≤ y2.val := by exact Nat.zero_le _\n  have x3_pos : 0 ≤ x3.val := by exact Nat.zero_le _\n  have y3_pos : 0 ≤ y3.val := by exact Nat.zero_le _\n  have z0_pos : 0 ≤ z0.val := by exact Nat.zero_le _\n  have z1_pos : 0 ≤ z1.val := by exact Nat.zero_le _\n  have z2_pos : 0 ≤ z2.val := by exact Nat.zero_le _\n  have z3_pos : 0 ≤ z3.val := by exact Nat.zero_le _\n\n  zify at *\n\n  set x0 : ℤ := ↑(ZMod.val x0)\n  set x1 : ℤ := ↑(ZMod.val x1)\n  set x2 : ℤ := ↑(ZMod.val x2)\n  set x3 : ℤ := ↑(ZMod.val x3)\n  set y0 : ℤ := ↑(ZMod.val y0)\n  set y1 : ℤ := ↑(ZMod.val y1)\n  set y2 : ℤ := ↑(ZMod.val y2)\n  set y3 : ℤ := ↑(ZMod.val y3)\n  set z0 : ℤ := ↑(ZMod.val z0)\n  set z1 : ℤ := ↑(ZMod.val z1)\n  set z2 : ℤ := ↑(ZMod.val z2)\n  set z3 : ℤ := ↑(ZMod.val z3)\n  set c0 : ℤ := ↑(ZMod.val c0)\n  set c1 : ℤ := ↑(ZMod.val c1)\n  set c2 : ℤ := ↑(ZMod.val c2)\n  set c3 : ℤ := ↑(ZMod.val c3)\n  set carry_in : ℤ := ↑(ZMod.val carry_in)\n\n  -- now everything, assumptions and goal, is over Z\n\n  -- add up all the equations\n  set z := z0 + z1*256 + z2*256^2 + z3*256^3\n  set x := x0 + x1*256 + x2*256^2 + x3*256^3\n  set y := y0 + y1*256 + y2*256^2 + y3*256^3\n  let lhs := z + c3*2^32\n  let rhs₀ := x0 + y0 + carry_in + -1 * z0 + -1 * (c0 * 256) -- h0 expression\n  let rhs₁ := x1 + y1 + c0 + -1 * z1 + -1 * (c1 * 256) -- h1 expression\n  let rhs₂ := x2 + y2 + c1 + -1 * z2 + -1 * (c2 * 256) -- h2 expression\n  let rhs₃ := x3 + y3 + c2 + -1 * z3 + -1 * (c3 * 256) -- h3 expression\n\n  have h_add := calc z + c3*2^32\n    -- substitute equations\n    _ = lhs + 0 + 256*0 + 256^2*0 + 256^3*0 := by ring\n    _ = lhs + rhs₀ + 256*rhs₁ + 256^2*rhs₂ + 256^3*rhs₃ := by\n      simp only [rhs₀, rhs₁, rhs₂, rhs₃]\n      simp only [h0, h1, h2, h3]\n      ring\n    -- simplify\n    _ = x + y + carry_in := by ring\n\n  have z_lt_2_32 : z < 2^32 := by dsimp only [z]; linarith\n  have z_pos : 0 ≤ z := by dsimp [z]; linarith\n\n  rcases c3_bool with c3_zero | c3_one\n  · rw [c3_zero] at h_add\n    simp only [Int.reducePow, zero_mul, add_zero] at h_add\n    have sum_pos : 0 ≤ x + y + carry_in := by\n      rcases carry_in_bool with c_zero | c_one\n      · rw [c_zero]; linarith\n      · rw [c_one]; linarith\n    rw [Int.emod_eq_of_lt sum_pos (by linarith)]\n    rw [Int.ediv_eq_zero_of_lt sum_pos (by linarith)]\n    simp only [h_add, c3_zero, and_self]\n  · simp only [c3_one, one_mul] at h_add\n    rw [←h_add, Int.add_emod_right, Int.emod_eq_of_lt z_pos (by linarith)]\n    rw [\n      show z + 2^32 = z + 2^32 * 1 from rfl,\n      Int.add_mul_ediv_left _ _ (by linarith),\n      Int.ediv_eq_zero_of_lt z_pos z_lt_2_32\n    ]\n    simp only [c3_one, zero_add, and_self]", "nesting_depth": 3, "transitive_dep_count": 34, "subset_aristotle": true, "category": "Applied verif."}
{"id": 125, "thm_name": "Gadgets.ByteDecomposition.Theorems.soundness", "thm_stmt": "theorem soundness (offset : Fin 8) (x low high : F p)\n  (x_lt : x.val < 2^8) (low_lt : low.val < 2^offset.val) (high_lt : high.val < 2^8)\n  (h_eq : x = low + high * 2^offset.val) :\n    low.val = x.val % 2^offset.val ∧ high.val = x.val / 2^offset.val", "lean_root": "clean", "rel_path": "Clean/Gadgets/ByteDecomposition/Theorems.lean", "imports": ["import Clean.Types.U32", "import Clean.Utils.Bitwise", "import Clean.Types.U64", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.reduceMod", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Nat.reducePow", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "UInt32", "module": "Init.Prelude"}, {"name": "UInt32.ofNat", "module": "Init.Data.UInt.BasicAux"}, {"name": "UInt32.reduceToNat", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.UInt"}, {"name": "UInt32.toNat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"field_to_nat\" : tactic", "content": "syntax \"field_to_nat\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|field_to_nat) =>\n    `(tactic|(\n      intros\n      repeat rw [ZMod.val_add] \n      repeat rw [ZMod.val_mul] \n      repeat rw [val_eq_256]\n      try simp only [Nat.add_mod_mod, Nat.mod_add_mod, Nat.mul_mod_mod, Nat.mod_mul_mod]\n      rw [Nat.mod_eq_of_lt _]\n      repeat linarith [‹Fact (_ > 512)›.elim]))\n\nexample [Fact (p > 512)] (x y : F p) (hx : x.val < 256) (hy : y.val < 2) :\n    (x + y * 256).val = x.val + y.val * 256 := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}], "lib_lemmas": [{"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.mul_lt_mul_of_lt_of_le", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.pow_pos", "module": "Init.Prelude"}, {"name": "Nat.add_mod_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_add_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_iff_lt", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_mul_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mul_mod_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.pow_lt_pow_of_lt", "module": "Init.Data.Nat.Lemmas"}, {"name": "OfNat.ofNat_ne_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "UInt32.lt_iff_toNat_lt", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt32.toNat_add", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt32.toNat_div", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt32.toNat_inj", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt32.toNat_mod", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt32.toNat_mul", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt32.toNat_ofNat", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt32.toNat_ofNat'", "module": "Init.Data.UInt.Lemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "two_pow_val", "content": "lemma two_pow_val (n : ℕ) (hn : n ≤ 8) : (2^n : F p).val = 2^n"}, {"name": "two_pow_lt", "content": "omit p_prime in\nlemma two_pow_lt (n : ℕ) (hn : n ≤ 8) : 2^n < p"}, {"name": "two_val", "content": "lemma two_val : (2 : F p).val = 2"}, {"name": "two_lt", "content": "omit p_prime in\nlemma two_lt : 2 < p"}, {"name": "val_lt_p", "content": "theorem val_lt_p {p : ℕ} (x : ℕ) : (x < p) → (x : F p).val = x"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Gadgets.ByteDecomposition.Theorems.byteDecomposition_lift", "content": "theorem byteDecomposition_lift {low high two_power : F p}\n  (h_low : low.val < 2^8) (h_high : high.val < 2^8) (h_two_power : two_power.val ≤ 2^8) :\n    (low + high * two_power).val = low.val + high.val * two_power.val"}], "local_ctx": "import Clean.Utils.Field\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Types.U64\n\nimport Clean.Types.U32\n\nnamespace Gadgets.ByteDecomposition.Theorems\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 2^16 + 2^8)]\n\nopen FieldUtils (two_val two_pow_val)", "target_theorem": "theorem soundness (offset : Fin 8) (x low high : F p)\n  (x_lt : x.val < 2^8) (low_lt : low.val < 2^offset.val) (high_lt : high.val < 2^8)\n  (h_eq : x = low + high * 2^offset.val) :\n    low.val = x.val % 2^offset.val ∧ high.val = x.val / 2^offset.val :=", "ground_truth_proof": ":= by\n\n  have two_power_lt : 2^offset.val < 2^8 := Nat.pow_lt_pow_of_lt (by linarith) offset.is_lt\n  have two_power_val : ((2 : F p)^offset.val).val = 2^offset.val := two_pow_val offset.val (by linarith [offset.is_lt])\n  have two_power_le : (2^offset.val : F p).val ≤ 2^8 := by rw [two_power_val]; linarith\n\n  have low_byte : low.val < 256 := by linarith\n  have h := byteDecomposition_lift low_byte high_lt two_power_le\n  rw [two_power_val, ←h_eq] at h\n\n  set low_b := UInt32.ofNat low.val\n  set high_b := UInt32.ofNat high.val\n  set x_b := UInt32.ofNat x.val\n\n  -- The heavy lifting is done by using a SAT solver\n  have h_decomposition_bv (base : UInt32) :\n      base < 256 → low_b < base → high_b < 256 → x_b < 256 →\n      x_b = low_b + high_b * base →\n      low_b = x_b % base ∧ high_b = x_b / base := by\n    bv_decide\n\n  -- now it is left to prove that the bv variant is equivalent\n  -- to the field variant of the theorem\n\n  -- TODO: when updating to new Mathlib, all the following lemmas should be much easier to prove\n  -- thanks new UInt32 lemmas\n  have two_power_mod : (2^offset.val % 2 ^ 32) = 2^offset.val := by\n    rw [Nat.mod_eq_iff_lt]\n    linarith\n    simp only [Nat.reducePow, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true]\n\n  have two_power_lt_bv : UInt32.ofNat (2^offset.val) < 256 := by\n    rw [UInt32.lt_iff_toNat_lt]\n    simp only [UInt32.toNat_ofNat', Nat.reducePow, UInt32.toNat_ofNat, Nat.reduceMod]\n    rw [Nat.mod_eq_of_lt (by linarith)]\n    linarith\n\n  have low_mod : low.val % 2^32 = low.val := by\n    rw [Nat.mod_eq_iff_lt]\n    linarith\n    simp only [Nat.reducePow, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true]\n\n  have high_mod : high.val % 2^32 = high.val := by\n    rw [Nat.mod_eq_iff_lt]\n    linarith\n    simp only [Nat.reducePow, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true]\n\n  have x_mod : x.val % 2^32 = x.val := by\n    rw [Nat.mod_eq_iff_lt]\n    linarith\n    simp only [Nat.reducePow, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true]\n\n  have low_b_lt : low_b < UInt32.ofNat (2^offset.val) := by\n    simp only [low_b]\n    rw [UInt32.lt_iff_toNat_lt]\n    simp only [UInt32.toNat_ofNat']\n    rw [low_mod, two_power_mod]\n    assumption\n\n  have high_b_lt : high_b < 256 := by\n    simp only [high_b]\n    rw [UInt32.lt_iff_toNat_lt]\n    simp only [UInt32.toNat_ofNat', UInt32.reduceToNat]\n    rw [high_mod]\n    assumption\n\n  have x_lt : x_b < 256 := by\n    simp only [x_b]\n    rw [UInt32.lt_iff_toNat_lt]\n    simp only [UInt32.toNat_ofNat', UInt32.reduceToNat]\n    rw [x_mod]\n    assumption\n\n  have eq_holds_bv : x_b = low_b + high_b * UInt32.ofNat (2^offset.val) := by\n    simp only [x_b, low_b, high_b]\n    rw [←UInt32.toNat_inj]\n    simp only [UInt32.toNat_ofNat', UInt32.toNat_add, UInt32.toNat_mul,\n      Nat.mul_mod_mod, Nat.mod_mul_mod, Nat.add_mod_mod, Nat.mod_add_mod]\n    rw [x_mod]\n    have h : (low.val + high.val * (2^offset.val)) % 2^32 = low.val + high.val * (2^offset.val) := by\n      apply Nat.mod_eq_of_lt\n      linarith [p_large_enough.elim]\n    rw [h]\n    assumption\n\n  specialize h_decomposition_bv (UInt32.ofNat (2^offset.val))\n    two_power_lt_bv low_b_lt high_b_lt x_lt eq_holds_bv\n\n  obtain ⟨h1, h2⟩ := h_decomposition_bv\n\n  constructor\n  · apply_fun UInt32.toNat at h1\n    simp only [UInt32.toNat_ofNat', UInt32.toNat_mod, low_b, x_b] at h1\n    rw [low_mod, x_mod, two_power_mod] at h1\n    assumption\n  · apply_fun UInt32.toNat at h2\n    simp only [UInt32.toNat_ofNat', UInt32.toNat_div, high_b, x_b] at h2\n    rw [high_mod, x_mod, two_power_mod] at h2\n    assumption", "nesting_depth": 3, "transitive_dep_count": 38, "subset_aristotle": true, "category": "Applied verif."}
{"id": 126, "thm_name": "MemoryAccessList.isConsistentOffline_implies_single_address", "thm_stmt": "theorem MemoryAccessList.isConsistentOffline_implies_single_address\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (h_offline : accesses.isConsistentOffline h_sorted)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isConsistentSingleAddress\n      (filterAddress_sorted_from_addressTimestampSorted accesses h_sorted h_nodup addr)", "lean_root": "clean", "rel_path": "Clean/Utils/OfflineMemory.lean", "imports": ["import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Mathlib.Data.List.Sort", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Sorted", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Pairwise", "module": "Init.Data.List.Basic"}, {"name": "List.filter", "module": "Init.Data.List.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.Sorted.of_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Pairwise.of_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.filter_eq_nil_iff", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_cons_self", "module": "Init.Data.List.Lemmas"}, {"name": "List.sorted_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "decide_eq_true_eq", "module": "Init.SimpLemmas"}, {"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "List.filter_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_filter", "module": "Init.Data.List.Lemmas"}, {"name": "decide_true", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MemoryAccess", "content": "def MemoryAccess := ℕ × ℕ × ℕ × ℕ"}, {"name": "MemoryAccessList", "content": "def MemoryAccessList := List MemoryAccess"}, {"name": "timestamp_ordering", "content": "abbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2"}, {"name": "MemoryAccessList.isTimestampSorted", "content": "def MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering"}, {"name": "MemoryAccessList.timestamps_neq", "content": "def MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y"}, {"name": "MemoryAccessList.Notimestampdup", "content": "def MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses"}, {"name": "address_timestamp_ordering", "content": "abbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2"}, {"name": "address_strict_timestamp_ordering", "content": "abbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2"}, {"name": "MemoryAccessList.isAddressTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering"}, {"name": "MemoryAccessList.isAddressStrictTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering"}, {"name": "MemoryAccessList.filterAddress", "content": "def MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)"}, {"name": "MemoryAccessList.isConsistentSingleAddress", "content": "def MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}, {"name": "MemoryAccessList.isConsistentOffline", "content": "def MemoryAccessList.isConsistentOffline (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  | (_t2, addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    (if addr1 = addr2 then readValue2 = writeValue1 else readValue2 = 0) ∧\n    MemoryAccessList.isConsistentOffline ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}], "used_local_lemmas": [{"name": "MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup", "content": "theorem MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup\n    (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted)\n    (h_no_timestamp_dup : accesses.Notimestampdup) :\n    accesses.isAddressStrictTimestampSorted"}, {"name": "MemoryAccessList.noTimestampDup_of_cons", "content": "theorem MemoryAccessList.noTimestampDup_of_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h : Notimestampdup (head :: tail)) :\n    Notimestampdup tail"}, {"name": "MemoryAccessList.isAddressTimestampSorted_of_cons", "content": "theorem MemoryAccessList.isAddressTimestampSorted_of_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h : isAddressTimestampSorted (head :: tail)) :\n    isAddressTimestampSorted tail"}, {"name": "MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted", "content": "theorem MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isTimestampSorted"}, {"name": "MemoryAccessList.filterAddress_empty_when_address_changes", "content": "theorem MemoryAccessList.filterAddress_empty_when_address_changes\n    (head : MemoryAccess) (second : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: second :: tail))\n    (h_addr_ne : head.2.1 ≠ second.2.1) :\n    filterAddress (second :: tail) head.2.1 = []"}, {"name": "MemoryAccessList.isConsistentOffline_of_cons", "content": "theorem MemoryAccessList.isConsistentOffline_of_cons\n    (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: tail))\n    (h_offline : isConsistentOffline (head :: tail) h_sorted) :\n    isConsistentOffline tail (isAddressTimestampSorted_of_cons head tail h_sorted)"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Utils.Field\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Tactics\n\nimport Mathlib.Data.List.Sort\n\ndef MemoryAccess := ℕ × ℕ × ℕ × ℕ \n\ndef MemoryAccessList := List MemoryAccess\n\nabbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2\n\ndef MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering\n\ndef MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y\n\ndef MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses\n\nabbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2\n\nabbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2\n\n@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering\n\n@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering\n\ndef MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)\n\ndef MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)\n\ndef MemoryAccessList.isConsistentOffline (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  | (_t2, addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    (if addr1 = addr2 then readValue2 = writeValue1 else readValue2 = 0) ∧\n    MemoryAccessList.isConsistentOffline ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)", "target_theorem": "theorem MemoryAccessList.isConsistentOffline_implies_single_address\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (h_offline : accesses.isConsistentOffline h_sorted)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isConsistentSingleAddress\n      (filterAddress_sorted_from_addressTimestampSorted accesses h_sorted h_nodup addr) :=", "ground_truth_proof": ":= by\n  induction accesses with\n  | nil =>\n    simp [filterAddress, isConsistentSingleAddress]\n  | cons head tail ih =>\n    obtain ⟨t_head, a_head, r_head, w_head⟩ := head\n    simp only [filterAddress, List.filter_cons]\n    split_ifs with h_addr\n    · -- head has the target address\n      rw [decide_eq_true_eq] at h_addr\n      subst h_addr\n      -- Now we need to show consistency for (head :: filterAddress tail addr)\n      cases tail with\n      | nil =>\n        simp [isConsistentSingleAddress]\n        -- For a singleton list, need to show r_head = 0\n        exact h_offline\n      | cons second tail_rest =>\n        obtain ⟨t_second, a_second, r_second, w_second⟩ := second\n        -- Filter the tail\n        simp only [List.filter_cons]\n        split_ifs with h_second_addr\n        · -- second also has address a_head\n          rw [decide_eq_true_eq] at h_second_addr\n          subst h_second_addr\n          -- Now we show consistency for (head :: second :: filterAddress tail_rest a_head)\n          simp only [isConsistentSingleAddress]\n          constructor\n          · -- Show r_second = w_head from offline consistency\n            simp only [isConsistentOffline] at h_offline\n            simp at h_offline\n            exact h_offline.1\n          · -- Apply IH to (second :: tail_rest)\n            have ih_applied := ih\n              (isAddressTimestampSorted_of_cons (t_head, a_second, r_head, w_head) ((t_second, a_second, r_second, w_second) :: tail_rest) h_sorted)\n              (noTimestampDup_of_cons (t_head, a_second, r_head, w_head) ((t_second, a_second, r_second, w_second) :: tail_rest) h_nodup)\n              (isConsistentOffline_of_cons (t_head, a_second, r_head, w_head) ((t_second, a_second, r_second, w_second) :: tail_rest) h_sorted h_offline)\n            simp only [filterAddress, List.filter_cons, decide_true] at ih_applied\n            exact ih_applied\n        · -- second has a different address\n          -- Need to show isConsistentSingleAddress for (head :: filterAddress tail_rest a_head)\n          -- Key insight: since a_head ≠ a_second and list is address-sorted,\n          -- all elements in (second :: tail_rest) have address ≠ a_head\n          have h_addr_ne : ¬(a_second = a_head) := by\n            simp only [decide_eq_true_eq] at h_second_addr\n            exact h_second_addr\n          -- Use filterAddress_empty_when_address_changes to show filterAddress (second :: tail_rest) a_head = []\n          have h_empty := filterAddress_empty_when_address_changes\n            (t_head, a_head, r_head, w_head) (t_second, a_second, r_second, w_second) tail_rest h_sorted\n            (by simp; intro h_eq; exact h_addr_ne h_eq.symm)\n          simp only [filterAddress, List.filter_cons] at h_empty\n          have h_second_ne : decide (a_second = a_head) = false := by\n            simp [h_addr_ne]\n          simp only [h_second_ne] at h_empty\n          have h_empty_simplified : List.filter (fun x => match x with | (_, addr', _, _) => decide (addr' = a_head)) tail_rest = [] := by\n            simp at h_empty\n            exact h_empty\n          simp only [h_empty_simplified, isConsistentSingleAddress]\n          simp only [isConsistentOffline] at h_offline\n          simp [h_addr_ne] at h_offline\n          exact h_offline.1\n    · -- head doesn't have the target address\n      -- Apply IH to tail\n      apply ih (isAddressTimestampSorted_of_cons (t_head, a_head, r_head, w_head) tail h_sorted)\n        (noTimestampDup_of_cons (t_head, a_head, r_head, w_head) tail h_nodup)\n      exact isConsistentOffline_of_cons (t_head, a_head, r_head, w_head) tail h_sorted h_offline", "nesting_depth": 4, "transitive_dep_count": 34, "subset_aristotle": true, "category": "Applied verif."}
{"id": 127, "thm_name": "Circomlib.MultiAND.soundness_two", "thm_stmt": "lemma soundness_two {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 2) (F p))\n    (input : fields 2 (F p)) (h_env : input = eval env input_var)\n    (h_assumptions : Assumptions 2 input)\n    (h_hold : Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset)) :\n    Spec 2 input (env ((main input_var).output offset))", "lean_root": "clean", "rel_path": "Clean/Circomlib/Gates.lean", "imports": ["import Clean.Circuit.Theorems", "import Clean.Circuit.Provable", "import Clean.Utils.Field", "import Clean.Circuit", "import Mathlib.Data.Nat.Bitwise", "import Clean.Gadgets.Boolean", "import Clean.Utils.Bitwise", "import Clean.Utils.BinaryOps", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "One", "module": "Init.Prelude"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "Vector.foldl", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "fields", "content": "@[reducible]\ndef fields (n : ℕ) := fun F => Vector F n"}, {"name": "IsBool", "content": "def IsBool {α : Type*} [Zero α] [One α] (x : α) : Prop := x = 0 ∨ x = 1"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "fieldPair", "content": "@[reducible]\ndef fieldPair : TypeMap := fun F => F × F"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "ConstraintsHold", "content": "@[circuit_norm]\ndef ConstraintsHold (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold eval ops\n  | .lookup { table, entry, .. } :: ops =>\n    table.Contains (entry.map eval) ∧ ConstraintsHold eval ops\n  | .subcircuit s :: ops =>\n    ConstraintsHoldFlat eval s.ops ∧ ConstraintsHold eval ops"}], "lib_lemmas": [{"name": "Array.foldl_toList", "module": "Init.Data.Array.Bootstrap"}, {"name": "List.foldl_cons", "module": "Init.Data.List.Basic"}, {"name": "List.foldl_nil", "module": "Init.Data.List.Basic"}, {"name": "Vector.foldl_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_map", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.toList_toArray", "module": "Init.Data.Vector.Lemmas"}], "repo_lemmas": [{"name": "eval_fieldPair", "content": "@[circuit_norm ↓]\ntheorem eval_fieldPair {F : Type} [Field F] (env : Environment F) (t : Var fieldPair F) :\n  ProvableType.eval env t = (match t with | (x, y) => (Expression.eval env x, Expression.eval env y))"}, {"name": "toList_length_two", "content": "theorem toList_length_two {α : Type} (v : Vector α 2) :\n    v.toList = [v[0], v[1]]"}, {"name": "one_land_of_IsBool", "content": "theorem one_land_of_IsBool (a : ℕ) (h : IsBool a) : 1 &&& a = a"}, {"name": "land_one_of_IsBool", "content": "theorem land_one_of_IsBool (a : ℕ) (h : IsBool a) : a &&& 1 = a"}, {"name": "val_of_IsBool", "content": "theorem val_of_IsBool {p : ℕ} [Fact p.Prime] {x : F p} (h : IsBool x) : IsBool x.val"}, {"name": "one", "content": "@[circuit_norm]\ntheorem one {α : Type*} [Zero α] [One α] : IsBool (1 : α)"}, {"name": "zero", "content": "@[circuit_norm]\ntheorem zero {α : Type*} [Zero α] [One α] : IsBool (0 : α)"}], "used_local_defs": [{"name": "Circomlib.AND.main", "content": "def main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out"}, {"name": "Circomlib.MultiAND.main", "content": "def main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)"}, {"name": "Circomlib.MultiAND.Assumptions", "content": "def Assumptions (n : ℕ) (input : fields n (F p)) : Prop :=\n  ∀ (i : ℕ) (h : i < n), IsBool input[i]"}, {"name": "Circomlib.MultiAND.Spec", "content": "def Spec (n : ℕ) (input : fields n (F p)) (output : F p) : Prop :=\n  output.val = (input.map (·.val)).foldl (· &&& ·) 1 ∧ IsBool output"}], "used_local_lemmas": [], "local_ctx": "import Clean.Circuit\n\nimport Clean.Utils.Field\n\nimport Clean.Gadgets.Boolean\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Utils.Vector\n\nimport Clean.Utils.BinaryOps\n\nimport Clean.Circuit.Theorems\n\nimport Mathlib.Data.Nat.Bitwise\n\nopen IsBool\n\nnamespace Circomlib\n\nvariable {p : ℕ} [Fact p.Prime]\n\nopen Circuit (bind_output_eq bind_localLength_eq bind_forAll)\n\nopen Operations (append_localLength)\n\nopen BinaryOps (List.foldl_and_IsBool List.and_foldl_eq_foldl)\n\nnamespace XOR\n\nend XOR\n\nnamespace AND\n\ndef main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out\n\nend AND\n\nnamespace OR\n\nend OR\n\nnamespace NOT\n\nend NOT\n\nnamespace NAND\n\nend NAND\n\nnamespace NOR\n\nend NOR\n\nnamespace MultiAND\n\ndef main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)\n\ndef Assumptions (n : ℕ) (input : fields n (F p)) : Prop :=\n  ∀ (i : ℕ) (h : i < n), IsBool input[i]\n\ndef Spec (n : ℕ) (input : fields n (F p)) (output : F p) : Prop :=\n  output.val = (input.map (·.val)).foldl (· &&& ·) 1 ∧ IsBool output", "target_theorem": "lemma soundness_two {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 2) (F p))\n    (input : fields 2 (F p)) (h_env : input = eval env input_var)\n    (h_assumptions : Assumptions 2 input)\n    (h_hold : Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset)) :\n    Spec 2 input (env ((main input_var).output offset)) :=", "ground_truth_proof": ":= by\n  simp only [main] at h_hold ⊢\n  simp only [Spec]\n  have h_input0 := h_assumptions 0 (by norm_num : 0 < 2)\n  have h_input1 := h_assumptions 1 (by norm_num : 1 < 2)\n  have h_eval0 : env input_var[0] = input[0] := by simp [h_env, circuit_norm]\n  have h_eval1 : env input_var[1] = input[1] := by simp [h_env, circuit_norm]\n  have h_and_spec := AND.circuit.soundness offset env (input_var[0], input_var[1])\n    (input[0], input[1])\n    (by simp only [ProvableType.eval_fieldPair, h_eval0, h_eval1])\n    ⟨h_input0, h_input1⟩ h_hold\n\n  rcases h_and_spec with ⟨h_val, h_binary⟩\n  constructor\n  · -- Prove output.val = fold\n    have h_fold_two : Vector.foldl (fun x1 x2 => x1 &&& x2) 1 (input.map (·.val)) = input[0].val &&& input[1].val := by\n      rw [Vector.foldl_mk, ← Array.foldl_toList]\n      have h_toList : (input.map (·.val)).toList = [input[0].val, input[1].val] := by\n        rw [Vector.toList_length_two]\n        simp only [Vector.getElem_map]\n      rw [Vector.toList_toArray, h_toList]\n      simp only [List.foldl_cons, List.foldl_nil]\n      rw [one_land_of_IsBool input[0].val (val_of_IsBool h_input0)]\n    rw [h_fold_two]\n    exact h_val\n  · exact h_binary", "nesting_depth": 9, "transitive_dep_count": 99, "subset_aristotle": true, "category": "Applied verif."}
{"id": 128, "thm_name": "Gadgets.ByteDecomposition.soundness", "thm_stmt": "theorem soundness (offset : Fin 8) : Soundness (F p) (circuit := elaborated offset) Assumptions (Spec offset)", "lean_root": "clean", "rel_path": "Clean/Gadgets/ByteDecomposition/ByteDecomposition.lean", "imports": ["import Clean.Utils.Primes", "import Init.Data.Nat.Div.Basic", "import Clean.Gadgets.ByteDecomposition.Theorems", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Vector.push", "module": "Init.Data.Vector.Basic"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}], "used_repo_defs": [{"name": "syntax \"let \" ident \" <== \" term : doElem", "content": "syntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem\n\nsyntax \"infer_constant_length\" : tactic\n\nsyntax \"field_to_nat\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|field_to_nat) =>\n    `(tactic|(\n      intros\n      repeat rw [ZMod.val_add] \n      repeat rw [ZMod.val_mul] \n      repeat rw [val_eq_256]\n      try simp only [Nat.add_mod_mod, Nat.mod_add_mod, Nat.mul_mod_mod, Nat.mod_mul_mod]\n      rw [Nat.mod_eq_of_lt _]\n      repeat linarith [‹Fact (_ > 512)›.elim]))\n\nexample [Fact (p > 512)] (x y : F p) (hx : x.val < 256) (hy : y.val < 2) :\n    (x + y * 256).val = x.val + y.val * 256 := by admit /- proof elided -/"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "floorDiv", "content": "def floorDiv (x : F p) (c : ℕ+) : F p :=\n  FieldUtils.natToField (x.val / c) (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "less_than_p", "content": "def less_than_p (x : F p) : x.val < p :="}, {"name": "ByteTable", "content": "def ByteTable : Table (F p) field := .fromStatic {\n  name := \"Bytes\"\n  length := 256\n\n  row i := fromByte i\n  index x := x.val\n\n  Spec x := x.val < 256\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "mod", "content": "def mod (x : F p) (c : ℕ+) (lt : c < p) : F p :=\n  FieldUtils.natToField (x.val % c) (by admit /- proof elided -/\n  )"}, {"name": "HasAssertEq", "content": "class HasAssertEq (β : Type) (F : outParam Type) [Field F] where\n  assert_eq : β → β → Circuit F Unit"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "varFromOffset", "content": "@[explicit_provable_type]\ndef varFromOffset (α : TypeMap) [ProvableType α] (offset : ℕ) : Var α F :=\n  let vars := Vector.mapRange (size α) fun i => var ⟨offset + i⟩\n  fromVars vars"}, {"name": "mapRange", "content": "def mapRange (n : ℕ) (create : ℕ → α) : Vector α n :=\n  match n with\n  | 0 => #v[]\n  | k + 1 => mapRange k create |>.push (create k)"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "infix:50 \" === \" => HasAssertEq.assert_eq", "content": "infix:50 \" === \" => HasAssertEq.assert_eq"}], "lib_lemmas": [{"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.add_mod_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.div_lt_iff_lt_mul", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mul_lt_mul_left", "module": "Init.Data.Nat.Mod"}, {"name": "Nat.mul_lt_mul_of_pos_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_sub", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pow_le_pow_of_le", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.sub_eq_of_eq_add", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.two_pow_pos", "module": "Init.Data.Nat.Basic"}, {"name": "ZMod.val_add", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_mul", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_mul_of_lt", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "mul_add", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pow_add", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "soundness", "content": "theorem soundness (offset : Fin 8) (x low high : F p)\n  (x_lt : x.val < 2^8) (low_lt : low.val < 2^offset.val) (high_lt : high.val < 2^8)\n  (h_eq : x = low + high * 2^offset.val) :\n    low.val = x.val % 2^offset.val ∧ high.val = x.val / 2^offset.val"}, {"name": "byteDecomposition_lift", "content": "theorem byteDecomposition_lift {low high two_power : F p}\n  (h_low : low.val < 2^8) (h_high : high.val < 2^8) (h_two_power : two_power.val ≤ 2^8) :\n    (low + high * two_power).val = low.val + high.val * two_power.val"}, {"name": "two_pow_lt", "content": "omit p_prime in\nlemma two_pow_lt (n : ℕ) (hn : n ≤ 8) : 2^n < p"}, {"name": "mul_nat_val_of_dvd", "content": "theorem mul_nat_val_of_dvd {x : F p} (c : ℕ) (c_lt : c < p) {z : ℕ} :\n    (c * x).val = c * z → (c * x).val = c * x.val"}, {"name": "mul_val_of_dvd", "content": "theorem mul_val_of_dvd {x c : F p} :\n    c.val ∣ (c * x).val → (c * x).val = c.val * x.val"}, {"name": "natToField_eq", "content": "theorem natToField_eq {n : ℕ} {lt : n < p} (x : F p) (hx : x = natToField n lt) : x.val = n"}, {"name": "ext", "content": "theorem ext {x y : F p} (h : x.val = y.val) : x = y"}, {"name": "p_ne_zero", "content": "theorem p_ne_zero : p ≠ 0"}, {"name": "two_pow_val", "content": "lemma two_pow_val (n : ℕ) (hn : n ≤ 8) : (2^n : F p).val = 2^n"}, {"name": "two_val", "content": "lemma two_val : (2 : F p).val = 2"}, {"name": "two_lt", "content": "omit p_prime in\nlemma two_lt : 2 < p"}, {"name": "val_lt_p", "content": "theorem val_lt_p {p : ℕ} (x : ℕ) : (x < p) → (x : F p).val = x"}], "used_local_defs": [{"name": "Gadgets.ByteDecomposition.Outputs", "content": "structure Outputs (F : Type) where\n  low : F\n  high : F"}, {"name": "Gadgets.ByteDecomposition.main", "content": "def main (offset : Fin 8) (x : Expression (F p)) : Circuit (F p) (Var Outputs (F p)) := do\n  let low ← witness fun env => mod (env x) (2^offset.val) (by admit /- proof elided -/\n  )\n  let high ← witness fun env => floorDiv (env x) (2^offset.val)\n\n  lookup ByteTable ((2^(8-offset.val) : F p) * low)\n  lookup ByteTable high\n\n  x === low + high * (2^offset.val : F p)\n\n  return { low, high }"}, {"name": "Gadgets.ByteDecomposition.Assumptions", "content": "def Assumptions (x : F p) := x.val < 256"}, {"name": "Gadgets.ByteDecomposition.Spec", "content": "def Spec (offset : Fin 8) (x : F p) (out : Outputs (F p)) :=\n  let ⟨low, high⟩ := out\n  (low.val = x.val % (2^offset.val) ∧ high.val = x.val / (2^offset.val))\n  ∧ (low.val < 2^offset.val ∧ high.val < 2^(8-offset.val))"}, {"name": "Gadgets.ByteDecomposition.elaborated", "content": "def elaborated (offset : Fin 8) : ElaboratedCircuit (F p) field Outputs where\n  main := main offset\n  localLength _ := 2\n  output _ i0 := varFromOffset Outputs i0"}], "used_local_lemmas": [], "local_ctx": "import Clean.Utils.Primes\n\nimport Clean.Utils.Field\n\nimport Clean.Gadgets.ByteDecomposition.Theorems\n\nimport Init.Data.Nat.Div.Basic\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 2^16 + 2^8)]\n\nnamespace Gadgets.ByteDecomposition\n\nopen FieldUtils (mod floorDiv two_lt two_pow_lt two_val two_pow_val)\n\nstructure Outputs (F : Type) where\n  low : F\n  high : F\n\ndef main (offset : Fin 8) (x : Expression (F p)) : Circuit (F p) (Var Outputs (F p)) := do\n  let low ← witness fun env => mod (env x) (2^offset.val) (by admit /- proof elided -/\n  )\n  let high ← witness fun env => floorDiv (env x) (2^offset.val)\n\n  lookup ByteTable ((2^(8-offset.val) : F p) * low)\n  lookup ByteTable high\n\n  x === low + high * (2^offset.val : F p)\n\n  return { low, high }\n\ndef Assumptions (x : F p) := x.val < 256\n\ndef Spec (offset : Fin 8) (x : F p) (out : Outputs (F p)) :=\n  let ⟨low, high⟩ := out\n  (low.val = x.val % (2^offset.val) ∧ high.val = x.val / (2^offset.val))\n  ∧ (low.val < 2^offset.val ∧ high.val < 2^(8-offset.val))\n\ndef elaborated (offset : Fin 8) : ElaboratedCircuit (F p) field Outputs where\n  main := main offset\n  localLength _ := 2\n  output _ i0 := varFromOffset Outputs i0", "target_theorem": "theorem soundness (offset : Fin 8) : Soundness (F p) (circuit := elaborated offset) Assumptions (Spec offset) :=", "ground_truth_proof": ":= by\n  intro i0 env x_var (x : F p) h_input (x_byte : x.val < 256) h_holds\n  simp only [id_eq, circuit_norm] at h_input\n  simp only [circuit_norm, elaborated, main, Spec, ByteTable, h_input] at h_holds ⊢\n  clear h_input\n\n  obtain ⟨low_lt, high_lt, h_eq⟩ := h_holds\n  set low := env.get i0\n  set high := env.get (i0 + 1)\n\n  have : 2^16 < p := by linarith [p_large_enough.elim]\n  let n : ℕ := 8 - offset.val\n  have neg_off_le : n ≤ 8 := by omega\n  have pow_8 : 2^n * 2^offset.val = (2^8 : F p) := by simp [n, ←pow_add]\n  have pow_8_nat : 2^n * 2^offset.val = 2^8 := by simp [n, ←pow_add]\n\n  -- we first work with the equation multiplied by `2^n`, where we can make use of the range check on `2^n * low`\n  -- the goal is to apply `FieldUtils.mul_nat_val_of_dvd` to get to the stronger inequality `low < 2^offset`\n  have h_eq_mul : 2^n * x = 2^n * low + 2^n * 2^offset.val * high := by rw [h_eq, mul_add, mul_comm high, mul_assoc]\n  replace h_eq_mul := congrArg ZMod.val h_eq_mul\n\n  have h_lt_mul {x n} (hn : n ≤ 8) (hx : x < 2^8) : 2^n * x < 2^16 := by\n    have : 2^(n+8) ≤ 2^16 := Nat.pow_le_pow_of_le (by norm_num) (by omega)\n    suffices 2^n * x < 2^(n+8) by linarith\n    rw [pow_add]\n    exact Nat.mul_lt_mul_of_pos_left hx (Nat.two_pow_pos n)\n\n  have h_lt_mul_x : 2^n * x.val < 2^16 := h_lt_mul neg_off_le x_byte\n  have h_pow8_val : (2^8 : F p).val = 2^8 := two_pow_val _ (by norm_num)\n  have h_lt_mul_low : (2 ^ n * low).val < 2^8 := low_lt\n\n  have h_mul_x : (2^n : F p).val * x.val = 2^n * ZMod.val x := by rw [two_pow_val _ neg_off_le]\n  have : (2 ^ n * x).val = 2^n * x.val := by rw [ZMod.val_mul_of_lt (by linarith), h_mul_x]\n  rw [this] at h_eq_mul\n\n  have : (2^n * low + 2^n * 2^offset.val * high).val = (2^n * low).val + 2^n * 2^offset.val * high.val := by\n    rw [ZMod.val_add, ZMod.val_mul _ high, Nat.add_mod_mod, pow_8_nat, pow_8, h_pow8_val, Nat.mod_eq_of_lt]\n    linarith\n  rw [this, mul_assoc (2^n)] at h_eq_mul\n  replace h_eq_mul := Nat.sub_eq_of_eq_add h_eq_mul |>.symm\n  have two_pow_cast : 2^n = ((2^n : ℕ) : F p) := by simp\n  rw [←Nat.mul_sub, two_pow_cast] at h_eq_mul\n  have h_eq_mul_low := FieldUtils.mul_nat_val_of_dvd (2^n) (two_pow_lt n ‹_›) h_eq_mul\n  rw [←two_pow_cast] at h_eq_mul_low\n  rw [h_eq_mul_low, ←pow_8_nat, Nat.mul_lt_mul_left (show 2^n > 0 by simp)] at h_lt_mul_low\n\n  -- finally we have the desired inequality on `low`\n  have h_lt_low : low.val < 2^offset.val := h_lt_mul_low\n  have ⟨ low_eq, high_eq ⟩ := Theorems.soundness offset x low high x_byte h_lt_low high_lt h_eq\n  use ⟨ low_eq, high_eq ⟩, h_lt_low\n  rwa [high_eq, Nat.div_lt_iff_lt_mul (by simp), pow_8_nat]", "nesting_depth": 7, "transitive_dep_count": 119, "subset_aristotle": true, "category": "Applied verif."}
{"id": 129, "thm_name": "Gadgets.Or.Or8.completeness", "thm_stmt": "theorem completeness : Completeness (F p) elaborated Assumptions", "lean_root": "clean", "rel_path": "Clean/Gadgets/Or/Or8.lean", "imports": ["import Clean.Gadgets.Xor.ByteXorTable", "import Clean.Utils.Field", "import Clean.Circuit.Basic"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Fin.xor", "module": "Init.Data.Fin.Basic"}, {"name": "HXor", "module": "Init.Prelude"}, {"name": "HXor.hXor", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "XorOp", "module": "Init.Prelude"}, {"name": "XorOp.xor", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat.reduceMod", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Nat.reducePow", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "UInt16", "module": "Init.Prelude"}, {"name": "UInt16.toNat", "module": "Init.Data.UInt.BasicAux"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"field_to_nat\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|field_to_nat) =>\n    `(tactic|(\n      intros\n      repeat rw [ZMod.val_add] \n      repeat rw [ZMod.val_mul] \n      repeat rw [val_eq_256]\n      try simp only [Nat.add_mod_mod, Nat.mod_add_mod, Nat.mul_mod_mod, Nat.mod_mul_mod]\n      rw [Nat.mod_eq_of_lt _]\n      repeat linarith [‹Fact (_ > 512)›.elim]))\n\nexample [Fact (p > 512)] (x y : F p) (hx : x.val < 256) (hy : y.val < 2) :\n    (x + y * 256).val = x.val + y.val * 256 := by admit /- proof elided -/"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "ByteXorTable", "content": "def ByteXorTable : Table (F p) fieldTriple := .fromStatic {\n  name := \"ByteXor\"\n  length := 256*256\n\n  row i :=\n    let (x, y) := splitTwoBytes i\n    (fromByte x, fromByte y, fromByte (x ^^^ y))\n\n  index := fun (x, y, _) => x.val * 256 + y.val\n\n  Spec := fun (x, y, z) =>\n    x.val < 256 ∧ y.val < 256 ∧ z.val = x.val ^^^ y.val\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "splitTwoBytes", "content": "def splitTwoBytes (i : Fin (256 * 256)) : Fin 256 × Fin 256 :=\n  let x := i.val / 256\n  let y := i.val % 256\n  have x_lt : x < 256 := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "fieldTriple", "content": "@[reducible]\ndef fieldTriple : TypeMap := fun F => F × F × F"}, {"name": "concatTwoBytes", "content": "def concatTwoBytes (x y : Fin 256) : Fin (256 * 256) :=\n  let i := x.val * 256 + y.val\n  have i_lt : i < 256 * 256 := by admit /- proof elided -/"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "fieldVar", "content": "@[reducible] def fieldVar (F : Type) := field (Expression F)"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}], "lib_lemmas": [{"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.or_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "UInt16.toBitVec_ofNat", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt16.toNat_ofNat_of_lt", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt16.toNat_or", "module": "Init.Data.UInt.Bitwise"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.le_sub_of_add_le", "module": "Init.Data.Nat.Basic"}, {"name": "ZMod.val_cast_of_lt", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_mul", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_mul_of_lt", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_sub", "module": "Mathlib.Data.ZMod.Basic"}], "repo_lemmas": [{"name": "val_lt_p", "content": "theorem val_lt_p {p : ℕ} (x : ℕ) : (x < p) → (x : F p).val = x"}, {"name": "natToField_eq", "content": "theorem natToField_eq {n : ℕ} {lt : n < p} (x : F p) (hx : x = natToField n lt) : x.val = n"}, {"name": "natToField_eq_natCast", "content": "theorem natToField_eq_natCast {n : ℕ} (lt : n < p) : ↑n = FieldUtils.natToField n lt"}], "used_local_defs": [{"name": "Gadgets.Or.Or8.Inputs", "content": "structure Inputs (F : Type) where\n  x: F\n  y: F"}, {"name": "Gadgets.Or.Or8.Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.val < 256 ∧ y.val < 256"}, {"name": "Gadgets.Or.Or8.main", "content": "def main (input : Var Inputs (F p)) : Circuit (F p) (fieldVar (F p)) := do\n  let ⟨x, y⟩ := input\n  let or ← witness fun eval => (eval x).val ||| (eval y).val\n  \n  let xor := 2*or - x - y\n  lookup ByteXorTable (x, y, xor)\n  return or"}, {"name": "Gadgets.Or.Or8.elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs field where\n  main\n  localLength _ := 1"}], "used_local_lemmas": [{"name": "Gadgets.Or.Or8.or_times_two_sub_xor", "content": "private theorem or_times_two_sub_xor {x y : ℕ} (hx : x < 256) (hy : y < 256) :\n    2 * (x ||| y) = x + y + (x ^^^ y)"}, {"name": "Gadgets.Or.Or8.or_times_two_sub_xor'", "content": "private theorem or_times_two_sub_xor' {x y : ℕ} (hx : x < 256) (hy : y < 256) :\n    2 * (x ||| y) - x - y = (x ^^^ y)"}, {"name": "Gadgets.Or.Or8.two_or_ge_add", "content": "private theorem two_or_ge_add {x y : ℕ} (hx : x < 256) (hy : y < 256) : 2 * (x ||| y) ≥ x + y"}, {"name": "Gadgets.Or.Or8.val_two", "content": "lemma val_two : (2 : F p).val = 2"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Gadgets.Xor.ByteXorTable\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nnamespace Gadgets.Or.Or8\n\nopen Xor (ByteXorTable)\n\nopen FieldUtils\n\nstructure Inputs (F : Type) where\n  x: F\n  y: F\n\ndef Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.val < 256 ∧ y.val < 256\n\ndef main (input : Var Inputs (F p)) : Circuit (F p) (fieldVar (F p)) := do\n  let ⟨x, y⟩ := input\n  let or ← witness fun eval => (eval x).val ||| (eval y).val\n  \n  let xor := 2*or - x - y\n  lookup ByteXorTable (x, y, xor)\n  return or\n\ninstance elaborated : ElaboratedCircuit (F p) Inputs field where\n  main\n  localLength _ := 1", "target_theorem": "theorem completeness : Completeness (F p) elaborated Assumptions :=", "ground_truth_proof": ":= by\n  intro i env ⟨ x_var, y_var ⟩ h_env ⟨ x, y ⟩ h_input h_assumptions\n  simp_all only [circuit_norm, main, Assumptions, ByteXorTable, Inputs.mk.injEq]\n  obtain ⟨ hx_byte, hy_byte ⟩ := h_assumptions\n  set w : F p := ZMod.val x ||| ZMod.val y\n  have hw : w = ZMod.val x ||| ZMod.val y := rfl\n  let z := 2*w - x - y\n\n  -- now it's pretty much the soundness proof in reverse\n  have or_byte : x.val ||| y.val < 256 := Nat.or_lt_two_pow (n:=8) hx_byte hy_byte\n  have p_large := p_large_enough.elim\n  have or_lt : x.val ||| y.val < p := by linarith\n  rw [natToField_eq_natCast or_lt] at hw\n  have h_or : w.val = x.val ||| y.val := natToField_eq w hw\n\n  have two_or_val : (2*w).val = 2*(x.val ||| y.val) := by\n    rw [ZMod.val_mul_of_lt, val_two, h_or]\n    rw [val_two]\n    linarith\n\n  have x_y_val : (x + y).val = x.val + y.val := by field_to_nat\n\n  have two_or_ge : (2*w).val ≥ (x + y).val := by\n    rw [two_or_val, x_y_val]\n    exact two_or_ge_add hx_byte hy_byte\n\n  have : 2 * w + -x + -y = 2*w - x - y := by ring\n  rw [this]\n\n  simp only [w]\n  rw [← or_times_two_sub_xor']\n  · rw [ZMod.val_sub]\n    · rw [ZMod.val_sub]\n      · rw [ZMod.val_mul]\n        simp only [val_two]\n        rw [ZMod.val_cast_of_lt]\n        · rw [Nat.mod_eq_of_lt]\n          omega\n        omega\n      · calc\n          _ ≤ ZMod.val (x + y) := by linarith\n          _ ≤ _ := by omega\n    · rw [ZMod.val_sub]\n      · apply Nat.le_sub_of_add_le\n        calc\n          _ ≤ ZMod.val (x + y) := by linarith\n          _ ≤ _ := by omega\n      · calc\n          _ ≤ ZMod.val (x + y) := by linarith\n          _ ≤ _ := by omega\n  · omega\n  · omega", "nesting_depth": 7, "transitive_dep_count": 117, "subset_aristotle": true, "category": "Applied verif."}
{"id": 130, "thm_name": "Gadgets.BLAKE3.ApplyRounds.soundness", "thm_stmt": "theorem soundness : Soundness (F p) elaborated Assumptions Spec", "lean_root": "clean", "rel_path": "Clean/Gadgets/BLAKE3/ApplyRounds.lean", "imports": ["import Clean.Specs.BLAKE3", "import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.BLAKE3.BLAKE3State", "import Clean.Gadgets.BLAKE3.Permute", "import Clean.Types.U32", "import Clean.Gadgets.BLAKE3.Round", "import Clean.Circuit.StructuralLemmas"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "UInt32", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Vector.push", "module": "Init.Data.Vector.Basic"}, {"name": "Fin.reduceEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "Fin.reduceFinMk", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "IO", "module": "Init.System.IO"}, {"name": "Nat.reduceAdd", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.Prime", "module": "Mathlib.Data.Nat.Prime.Defs"}], "used_repo_defs": [{"name": "syntax \"let \" ident \" <== \" term : doElem", "content": "syntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem\n\nsyntax \"infer_constant_length\" : tactic"}, {"name": "macro \"provable_struct_simp\" : tactic =>", "content": "macro \"provable_struct_simp\" : tactic =>\n  `(tactic|\n    repeat (\n      fail_if_no_progress (\n      try split_provable_struct_eq;\n      try decompose_provable_struct;\n      try simplify_provable_struct_eval;\n      try simp only at *\n      )\n    )\n  )\n\nsyntax \"state_vec_norm_simp\" : tactic\n\nsyntax \"state_vec_norm_simp_simple\" : tactic\n\nsyntax \"circuit_proof_start\" (\"[\" term,* \"]\")? : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic| state_vec_norm_simp) => `(tactic|\n      simp only [Vector.getElem_mk];\n      rw [Vector.getElem_map, getElem_eval_vector];\n      simp only [eval_vector, Vector.map_mk, List.map_toArray, List.map_cons, List.map_nil, Vector.getElem_mk,\n        List.getElem_toArray, List.getElem_cons_succ, List.getElem_cons_zero])"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic| state_vec_norm_simp_simple) => `(tactic|\n      simp only [Vector.getElem_mk, Vector.getElem_map, Vector.map_mk, List.map_toArray, List.map_cons, List.map_nil, Vector.getElem_mk,\n        List.getElem_toArray, List.getElem_cons_succ, List.getElem_cons_zero, circuit_norm, U32.fromUInt32_normalized])"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "permute", "content": "def permute (state : Vector ℕ 16) : Vector ℕ 16 :=\n  Vector.ofFn (fun i => state[msgPermutation[i]])"}, {"name": "msgPermutation", "content": "def msgPermutation : Vector (Fin 16) 16 :=\n\n  #v[2, 6, 3, 10, 7, 0, 4, 13, 1, 11, 12, 5, 9, 14, 15, 8]"}, {"name": "round", "content": "def round (state : Vector ℕ 16) (m : Vector ℕ 16) : Vector ℕ 16 :=\n  roundConstants.foldl (fun state (a, b, c, d, i, j) =>\n    g state a b c d m[i] m[j]\n  ) state"}, {"name": "roundConstants", "content": "def roundConstants : Vector (Fin 16 × Fin 16 × Fin 16 × Fin 16 × Fin 16 × Fin 16) 8 := #v[\n  (0, 4, 8, 12, 0, 1),\n  (1, 5, 9, 13, 2, 3),\n  (2, 6, 10, 14, 4, 5),\n  (3, 7, 11, 15, 6, 7),\n  (0, 5, 10, 15, 8, 9),\n  (1, 6, 11, 12, 10, 11),\n  (2, 7, 8, 13, 12, 13),\n  (3, 4, 9, 14, 14, 15)\n]"}, {"name": "g", "content": "def g (state : Vector ℕ 16) (a b c d : Fin 16) (mx my : ℕ) : Vector ℕ 16 :=\n  let state_a := add32 (state[a]) (add32 state[b] mx)\n  let state_d := rotRight32 (state[d] ^^^ state_a) 16\n  let state_c := add32 (state[c]) state_d\n  let state_b := rotRight32 (state[b] ^^^ state_c) 12\n\n  let state_a := add32 state_a (add32 state_b my)\n  let state_d := rotRight32 (state_d ^^^ state_a) 8\n  let state_c := add32 state_c state_d\n  let state_b := rotRight32 (state_b ^^^ state_c) 7\n\n  state.set a state_a\n        |>.set b state_b\n        |>.set c state_c\n        |>.set d state_d"}, {"name": "rotRight32", "content": "def rotRight32 (x : ℕ) (offset : ℕ) : ℕ :=\n  let offset := offset % 32\n  let low := x % (2^offset)\n  let high := x / (2^offset)\n  low * (2^(32-offset)) + high"}, {"name": "add32", "content": "def add32 (a b : ℕ) : ℕ := (a + b) % 2^32"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  state : BLAKE3State F\n  message : Vector (U32 F) 16"}, {"name": "value", "content": "def value (x : U64 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3 +\n  x.x4.val * 256^4 + x.x5.val * 256^5 + x.x6.val * 256^6 + x.x7.val * 256^7"}, {"name": "U64", "content": "structure U64 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\n  x4 : T\n  x5 : T\n  x6 : T\n  x7 : T\nderiving DecidableEq"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  state : BLAKE3State F\n  chaining_value : Vector (U32 F) 8"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  state : BLAKE3State F\n  x : U32 F\n  y : U32 F"}, {"name": "BLAKE3State.Normalized", "content": "def BLAKE3State.Normalized (state : BLAKE3State (F p)) :=\n  ∀ i : Fin 16, state[i.val].Normalized"}, {"name": "Normalized", "content": "def Normalized (x : U32 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256"}, {"name": "U32", "content": "structure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq"}, {"name": "BLAKE3State", "content": "@[reducible] def BLAKE3State := ProvableVector U32 16"}, {"name": "ProvableVector.instance", "content": "instance ProvableVector.instance : ProvableType (ProvableVector α n) where\n  size := n * size α\n  toElements x := x.map toElements |>.flatten\n  fromElements v := v.toChunks (psize α) |>.map fromElements\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "map", "content": "def map {α β : Type} (x : U64 α) (f : α → β) : U64 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3, f x.x4, f x.x5, f x.x6, f x.x7 ⟩"}, {"name": "BLAKE3State.value", "content": "def BLAKE3State.value (state : BLAKE3State (F p)) := state.map U32.value"}, {"name": "value", "content": "def value (x : U32 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3"}, {"name": "map", "content": "def map {α β : Type} (x : U32 α) (f : α → β) : U32 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3 ⟩"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "Normalized", "content": "def Normalized (x : U64 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256 ∧\n  x.x4.val < 256 ∧ x.x5.val < 256 ∧ x.x6.val < 256 ∧ x.x7.val < 256"}, {"name": "fromUInt32", "content": "def fromUInt32 (x : UInt32) : U32 (F p) :=\n  decomposeNat x.toFin"}, {"name": "decomposeNat", "content": "def decomposeNat (x : ℕ) : U32 (F p) :=\n  let x0 := x % 256\n  let x1 : ℕ := (x / 256) % 256\n  let x2 : ℕ := (x / 256^2) % 256\n  let x3 : ℕ := (x / 256^3) % 256\n  ⟨ x0, x1, x2, x3 ⟩"}, {"name": "iv", "content": "def iv : Vector UInt32 8 := #v[\n  0x6a09e667,\n  0xbb67ae85,\n  0x3c6ef372,\n  0xa54ff53a,\n  0x510e527f,\n  0x9b05688c,\n  0x1f83d9ab,\n  0x5be0cd19\n]"}, {"name": "applyRounds", "content": "def applyRounds (chaining_value : Vector ℕ 8) (block_words : Vector ℕ 16) (counter : ℕ) (block_len : ℕ) (flags : ℕ) : Vector ℕ 16 :=\n  \n  let counter_low := counter % 2^32\n  let counter_high := counter / 2^32\n\n  \n  let state := #v[\n    chaining_value[0], chaining_value[1], chaining_value[2], chaining_value[3],\n    chaining_value[4], chaining_value[5], chaining_value[6], chaining_value[7],\n    iv[0].toNat, iv[1].toNat, iv[2].toNat, iv[3].toNat,\n    counter_low, counter_high, block_len, flags\n  ]\n\n  let state := round state block_words\n  let block_words := permute block_words\n  let state := round state block_words\n  let block_words := permute block_words\n  let state := round state block_words\n  let block_words := permute block_words\n  let state := round state block_words\n  let block_words := permute block_words\n  let state := round state block_words\n  let block_words := permute block_words\n  let state := round state block_words\n  let block_words := permute block_words\n  let state := round state block_words\n\n  state"}, {"name": "Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let { state, message } := input\n  state.Normalized ∧ (∀ i : Fin 16, message[i].Normalized)"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) (var : α (Expression F)) : α F :=\n  toComponents var |> go (components α) |> fromComponents\nwhere"}, {"name": "ProvableStruct", "content": "class ProvableStruct (α : TypeMap) where\n  components : List WithProvableType\n  toComponents {F : Type} : α F → ProvableTypeList F components\n  fromComponents {F : Type} : ProvableTypeList F components → α F\n\n  combinedSize : ℕ := combinedSize' components\n  combinedSize_eq : combinedSize = combinedSize' components := by admit /- proof elided -/"}, {"name": "Spec", "content": "def Spec (input : Inputs (F p)) (out : BLAKE3State (F p)) :=\n  let { state, message } := input\n  out.value = round state.value (message.map U32.value) ∧ out.Normalized"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) BLAKE3State BLAKE3State :=\n  { elaborated with Assumptions, Spec, soundness, completeness }"}, {"name": "elaborated", "content": "instance elaborated: ElaboratedCircuit (F p) BLAKE3State BLAKE3State where\n  main := main\n  localLength _ := 0\n  output state i0 := Vector.ofFn (fun i => state[msgPermutation[i]])"}, {"name": "main", "content": "def main (state : Var BLAKE3State (F p)) : Circuit (F p) (Var BLAKE3State (F p)) := do\n  return Vector.ofFn (fun i => state[msgPermutation[i]])"}, {"name": "Spec", "content": "def Spec (state : BLAKE3State (F p)) (out : BLAKE3State (F p)) :=\n  out.value = permute state.value ∧ out.Normalized"}, {"name": "Assumptions", "content": "def Assumptions (state : BLAKE3State (F p)) := state.Normalized"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) Inputs BLAKE3State := {\n  elaborated with Assumptions, Spec, soundness, completeness\n}"}, {"name": "elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs BLAKE3State where\n  main := main\n  localLength _ := 768\n  localLength_eq input i0 := by admit /- proof elided -/"}, {"name": "main", "content": "def main (input : Var Inputs (F p)) : Circuit (F p) (Var BLAKE3State (F p)) := do\n  let { state, message } := input\n  \n  let state ← G.circuit 0 4 8 12 ⟨state, message[0], message[1]⟩\n  let state ← G.circuit 1 5 9 13 ⟨state, message[2], message[3]⟩\n  let state ← G.circuit 2 6 10 14 ⟨state, message[4], message[5]⟩\n  let state ← G.circuit 3 7 11 15 ⟨state, message[6], message[7]⟩\n  let state ← G.circuit 0 5 10 15 ⟨state, message[8], message[9]⟩\n  let state ← G.circuit 1 6 11 12 ⟨state, message[10], message[11]⟩\n  let state ← G.circuit 2 7 8 13 ⟨state, message[12], message[13]⟩\n  let state ← G.circuit 3 4 9 14 ⟨state, message[14], message[15]⟩\n  return state"}, {"name": "circuit", "content": "def circuit (a b c d : Fin 16) : FormalCircuit (F p) Inputs BLAKE3State := {\n  elaborated a b c d with\n  Assumptions\n  Spec := Spec a b c d\n  soundness := soundness a b c d\n  completeness := completeness a b c d\n}"}, {"name": "Spec", "content": "def Spec (a b c d : Fin 16) (input : Inputs (F p)) (out : BLAKE3State (F p)) :=\n  let { state, x, y } := input\n  out.value = g state.value a b c d x.value y.value ∧ out.Normalized"}, {"name": "Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let { state, x, y } := input\n  state.Normalized ∧ x.Normalized ∧ y.Normalized"}, {"name": "elaborated", "content": "instance elaborated (a b c d : Fin 16): ElaboratedCircuit (F p) Inputs BLAKE3State where\n  main := main a b c d\n  localLength _ := 96\n  output inputs i0 := (inputs.state : Vector (U32 (Expression (F p))) 16)\n    |>.set a (⟨var ⟨i0 + 56⟩, var ⟨i0 + 58⟩, var ⟨i0 + 60⟩, var ⟨i0 + 62⟩⟩) a.is_lt\n    |>.set b (Rotation32.output 7 (i0 + 88)) b.is_lt\n    |>.set c (⟨var ⟨i0 + 76⟩, var ⟨i0 + 78⟩, var ⟨i0 + 80⟩, var ⟨i0 + 82⟩⟩) c.is_lt\n    |>.set d (Rotation32.output 8 (i0 + 68)) d.is_lt\n\n  localLength_eq _ n := by admit /- proof elided -/"}, {"name": "main", "content": "def main (a b c d : Fin 16) (input : Var Inputs (F p)) : Circuit (F p) (Var BLAKE3State (F p)) := do\n  let { state, x, y } := input\n\n  let state_a ← Addition32.circuit ⟨state[a], ← Addition32.circuit ⟨state[b], x⟩⟩\n\n  let state_d ← Rotation32.circuit 16 <|\n    ← Xor32.circuit ⟨state[d], state_a⟩\n\n  let state_c ← Addition32.circuit ⟨state[c], state_d⟩\n\n  let state_b ← Rotation32.circuit 12 <|\n    ← Xor32.circuit ⟨state[b], state_c⟩\n\n  let state_a ← Addition32.circuit ⟨state_a, ← Addition32.circuit ⟨state_b, y⟩⟩\n\n  let state_d ← Rotation32.circuit 8 <|\n    ← Xor32.circuit ⟨state_d, state_a⟩\n\n  let state_c ← Addition32.circuit ⟨state_c, state_d⟩\n\n  let state_b ← Rotation32.circuit 7 <|\n    ← Xor32.circuit ⟨state_b, state_c⟩\n\n  return state\n    |>.set a state_a\n    |>.set b state_b\n    |>.set c state_c\n    |>.set d state_d"}, {"name": "circuit", "content": "def circuit (offset : Fin 32) : FormalCircuit (F p) U32 U32 := {\n  elaborated offset with\n  Assumptions\n  Spec := Spec offset\n  soundness := soundness offset\n  completeness := completeness offset\n}"}, {"name": "Spec", "content": "def Spec (offset : Fin 32) (x : U32 (F p)) (y : U32 (F p)) :=\n  y.value = rotRight32 x.value offset.val\n  ∧ y.Normalized"}, {"name": "elaborated", "content": "def elaborated (off : Fin 32) : ElaboratedCircuit (F p) U32 U32 where\n  main := main off\n  localLength _ := 8\n  output _inputs i0 := output off i0"}, {"name": "output", "content": "def output (offset : Fin 32) (i0 : ℕ) : U32 (Expression (F p)) :=\n  Rotation32Bits.output ⟨ offset.val % 8, by admit /- proof elided -/\n  ⟩ i0"}, {"name": "output", "content": "def output (offset : Fin 8) (i0 : ℕ) : U32 (Expression (F p)) :=\n  U32.fromLimbs (.ofFn fun ⟨i,_⟩ =>\n    (var ⟨i0 + i*2 + 1⟩) + var ⟨i0 + (i + 1) % 4 * 2⟩ * .const ((2^(8-offset.val) : ℕ) : F p))"}, {"name": "fromLimbs", "content": "def fromLimbs {F} (v : Vector F 4) : U32 F := fromElements v"}, {"name": "main", "content": "def main (offset : Fin 32) (x : Var U32 (F p)) : Circuit (F p) (Var U32 (F p)) := do\n  let byte_offset : Fin 4 := ⟨ offset.val / 8, by admit /- proof elided -/\n  ⟩\n  let bit_offset : Fin 8 := ⟨ offset.val % 8, by admit /- proof elided -/\n  ⟩\n\n  \n  let byte_rotated ← Rotation32Bytes.circuit byte_offset x\n  Rotation32Bits.circuit bit_offset byte_rotated"}, {"name": "circuit", "content": "def circuit (offset : Fin 8) : FormalCircuit (F p) U32 U32 := {\n  elaborated offset with\n  Assumptions\n  Spec := Spec offset\n  soundness := soundness offset\n  completeness := completeness offset\n}"}, {"name": "Assumptions", "content": "def Assumptions (input : U32 (F p)) := input.Normalized"}, {"name": "elaborated", "content": "def elaborated (off : Fin 8) : ElaboratedCircuit (F p) U32 U32 where\n  main := main off\n  localLength _ := 8\n  output _inputs i0 := output off i0\n  localLength_eq _ i0 := by admit /- proof elided -/"}, {"name": "main", "content": "def main (offset : Fin 8) (x : U32 (Expression (F p))) : Circuit (F p) (Var U32 (F p)) := do\n  let parts ← Circuit.map x.toLimbs (ByteDecomposition.circuit offset)\n  let lows := parts.map Outputs.low\n  let highs := parts.map Outputs.high\n\n  let rotated := highs.zip (lows.rotate 1) |>.map fun (high, low) =>\n    high + low * ((2^(8-offset.val) : ℕ) : F p)\n\n  return U32.fromLimbs rotated"}, {"name": "Outputs", "content": "structure Outputs (F : Type) where\n  low : F\n  high : F"}, {"name": "toLimbs", "content": "def toLimbs {F} (x : U64 F) : Vector F 8 := toElements x"}, {"name": "circuit", "content": "def circuit (offset : Fin 8) : FormalCircuit (F p) field Outputs := {\n  elaborated offset with\n  main := main offset\n  Assumptions\n  Spec := Spec offset\n  soundness := soundness offset\n  completeness := completeness offset\n}"}, {"name": "Spec", "content": "def Spec (offset : Fin 8) (x : F p) (out : Outputs (F p)) :=\n  let ⟨low, high⟩ := out\n  (low.val = x.val % (2^offset.val) ∧ high.val = x.val / (2^offset.val))\n  ∧ (low.val < 2^offset.val ∧ high.val < 2^(8-offset.val))"}, {"name": "main", "content": "def main (offset : Fin 8) (x : Expression (F p)) : Circuit (F p) (Var Outputs (F p)) := do\n  let low ← witness fun env => mod (env x) (2^offset.val) (by admit /- proof elided -/\n  )\n  let high ← witness fun env => floorDiv (env x) (2^offset.val)\n\n  lookup ByteTable ((2^(8-offset.val) : F p) * low)\n  lookup ByteTable high\n\n  x === low + high * (2^offset.val : F p)\n\n  return { low, high }"}, {"name": "floorDiv", "content": "def floorDiv (x : F p) (c : ℕ+) : F p :=\n  FieldUtils.natToField (x.val / c) (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "less_than_p", "content": "def less_than_p (x : F p) : x.val < p :="}, {"name": "ByteTable", "content": "def ByteTable : Table (F p) field := .fromStatic {\n  name := \"Bytes\"\n  length := 256\n\n  row i := fromByte i\n  index x := x.val\n\n  Spec x := x.val < 256\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "mod", "content": "def mod (x : F p) (c : ℕ+) (lt : c < p) : F p :=\n  FieldUtils.natToField (x.val % c) (by admit /- proof elided -/\n  )"}, {"name": "HasAssertEq", "content": "class HasAssertEq (β : Type) (F : outParam Type) [Field F] where\n  assert_eq : β → β → Circuit F Unit"}, {"name": "Assumptions", "content": "def Assumptions (x : F p) := x.val < 256"}, {"name": "elaborated", "content": "def elaborated (offset : Fin 8) : ElaboratedCircuit (F p) field Outputs where\n  main := main offset\n  localLength _ := 2\n  output _ i0 := varFromOffset Outputs i0"}, {"name": "varFromOffset", "content": "@[explicit_provable_type]\ndef varFromOffset (α : TypeMap) [ProvableType α] (offset : ℕ) : Var α F :=\n  let vars := Vector.mapRange (size α) fun i => var ⟨offset + i⟩\n  fromVars vars"}, {"name": "mapRange", "content": "def mapRange (n : ℕ) (create : ℕ → α) : Vector α n :=\n  match n with\n  | 0 => #v[]\n  | k + 1 => mapRange k create |>.push (create k)"}, {"name": "rotate", "content": "def rotate {α : Type} {n : ℕ} (v : Vector α n) (off : ℕ) : Vector α n :=\n  ⟨(v.toList.rotate off).toArray, by admit /- proof elided -/\n  ⟩"}, {"name": "toLimbs", "content": "def toLimbs {F} (x : U32 F) : Vector F 4 := toElements x"}, {"name": "Spec", "content": "def Spec (offset : Fin 8) (x : U32 (F p)) (y : U32 (F p)) :=\n  y.value = rotRight32 x.value offset.val\n  ∧ y.Normalized"}, {"name": "circuit", "content": "def circuit (off : Fin 4) : FormalCircuit (F p) U32 U32 := {\n  elaborated off with\n  main := main off\n  Assumptions\n  Spec := Spec off\n  soundness := soundness off\n  completeness := completeness off\n}"}, {"name": "elaborated", "content": "instance elaborated (off : Fin 4): ElaboratedCircuit (F p) U32 U32 where\n  main := main off\n  localLength _ := 0\n  output input i0 :=\n    let ⟨x0, x1, x2, x3⟩ := input\n    match off with\n    | 0 => ⟨ x0, x1, x2, x3 ⟩\n    | 1 => ⟨ x1, x2, x3, x0 ⟩\n    | 2 => ⟨ x2, x3, x0, x1 ⟩\n    | 3 => ⟨ x3, x0, x1, x2 ⟩\n\n  subcircuitsConsistent x i0 := by admit /- proof elided -/"}, {"name": "main", "content": "def main (offset : Fin 4) (input : Var U32 (F p)) : Circuit (F p) (Var U32 (F p)) := do\n  let ⟨x0, x1, x2, x3⟩ := input\n\n  if offset = 0 then\n    return ⟨ x0, x1, x2, x3 ⟩\n  else if offset = 1 then\n    return ⟨ x1, x2, x3, x0 ⟩\n  else if offset = 2 then\n    return ⟨ x2, x3, x0, x1 ⟩\n  else\n    return ⟨ x3, x0, x1, x2 ⟩"}, {"name": "Spec", "content": "def Spec (offset : Fin 4) (x : U32 (F p)) (y : U32 (F p)) :=\n  y.value = rotRight32 x.value (offset.val * 8) ∧ y.Normalized"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) Inputs U32 where\n  Assumptions\n  Spec\n  soundness\n  completeness"}, {"name": "Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.Normalized ∧ y.Normalized"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  x: U32 F\n  y: U32 F"}, {"name": "Spec", "content": "def Spec (input : Inputs (F p)) (z : U32 (F p)) :=\n  let ⟨x, y⟩ := input\n  z.value = (x.value + y.value) % 2^32 ∧ z.Normalized"}, {"name": "Spec", "content": "def Spec (input : Inputs (F p)) (z : U32 (F p)) :=\n  let ⟨x, y⟩ := input\n  z.value = x.value ^^^ y.value ∧ z.Normalized"}, {"name": "FormalCircuit.weakenSpec", "content": "def FormalCircuit.weakenSpec\n    {F : Type} [Field F]\n    {Input Output : TypeMap} [ProvableType Input] [ProvableType Output]\n    (circuit : FormalCircuit F Input Output)\n    (WeakerSpec : Input F → Output F → Prop)\n    (h_spec_implication : ∀ input output,\n      circuit.Assumptions input →\n      circuit.Spec input output →\n      WeakerSpec input output) :\n    FormalCircuit F Input Output := {\n  elaborated := circuit.elaborated\n  Assumptions := circuit.Assumptions\n  Spec := WeakerSpec\n  soundness := by admit /- proof elided -/"}, {"name": "FormalCircuit.concat", "content": "def FormalCircuit.concat\n    {F : Type} [Field F]\n    {Input Mid Output : TypeMap} [ProvableType Input] [ProvableType Mid] [ProvableType Output]\n    (circuit1 : FormalCircuit F Input Mid)\n    (circuit2 : FormalCircuit F Mid Output)\n    (h_compat : ∀ input mid, circuit1.Assumptions input → circuit1.Spec input mid → circuit2.Assumptions mid)\n    (h_localLength_stable : ∀ mid mid', circuit2.localLength mid = circuit2.localLength mid') :\n    FormalCircuit F Input Output := {\n  elaborated := {\n    main := (circuit1 · >>= circuit2)\n    localLength input := circuit1.localLength input + circuit2.localLength (circuit1.output input 0)\n    localLength_eq := by admit /- proof elided -/"}, {"name": "main", "content": "def main (args : List String) : IO Unit := do\n  match args with\n  | [steps_str, output_path] =>\n    \n    match steps_str.toNat? with\n    | some steps => generateTrace steps output_path\n    | none => IO.println \"Error: Invalid number of steps\"\n  | _ =>\n    IO.println \"Usage: lake lean TraceGen.lean "}, {"name": "NonEmptyProvableType", "content": "class NonEmptyProvableType (M : TypeMap) extends ProvableType M where\n  nonempty : size > 0 := by admit /- proof elided -/"}, {"name": "KeccakRow.value", "content": "def KeccakRow.value (row : KeccakRow (F p)) := row.map U64.value"}, {"name": "KeccakRow.Normalized", "content": "def KeccakRow.Normalized (row : KeccakRow (F p)) :=\n  ∀ i : Fin 5, row[i.val].Normalized"}, {"name": "KeccakBlock.value", "content": "def KeccakBlock.value (block : KeccakBlock (F p)) := block.map U64.value"}, {"name": "KeccakState.value", "content": "def KeccakState.value (state : KeccakState (F p)) := state.map U64.value"}, {"name": "KeccakBlock.Normalized", "content": "def KeccakBlock.Normalized (block : KeccakBlock (F p)) :=\n  ∀ i : Fin RATE, block[i.val].Normalized"}, {"name": "RATE", "content": "@[reducible] def RATE := 17\nexample : RATE + CAPACITY = 25 := rfl"}, {"name": "CAPACITY", "content": "@[reducible] def CAPACITY := 8"}, {"name": "KeccakState.Normalized", "content": "def KeccakState.Normalized (state : KeccakState (F p)) :=\n  ∀ i : Fin 25, state[i.val].Normalized"}, {"name": "infix:50 \" === \" => HasAssertEq.assert_eq", "content": "infix:50 \" === \" => HasAssertEq.assert_eq"}], "lib_lemmas": [{"name": "Nat.add_mul_div_left", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.div_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}], "repo_lemmas": [{"name": "eval_vector", "content": "theorem eval_vector (env : Environment F)\n  (x : Var (ProvableVector α n) F) :\n    eval env x = x.map (eval env)"}, {"name": "eval_vector_eq_get", "content": "lemma eval_vector_eq_get {M : TypeMap} [NonEmptyProvableType M] {n : ℕ} (env : Environment F)\n    (vars : Vector (Var M F) n)\n    (vals : Vector (M F) n)\n    (h : (eval env vars : ProvableVector _ _ _) = (vals : ProvableVector _ _ _))\n    (i : ℕ) (h_i : i < n) :\n    eval env vars[i] = vals[i]"}, {"name": "fromUInt32_normalized", "content": "lemma fromUInt32_normalized (x : UInt32) : (fromUInt32 (p:=p) x).Normalized"}, {"name": "value_fromUInt32", "content": "theorem value_fromUInt32 (x : UInt32) : value (fromUInt32 (p:=p) x) = x.toNat"}, {"name": "value_of_decomposedNat_of_small", "content": "lemma value_of_decomposedNat_of_small (x : ℕ) :\n    x < 256^4 ->\n    (decomposeNat (p:=p) x).value = x"}, {"name": "value_lt_of_normalized", "content": "omit [Fact (Nat.Prime p)] p_large_enough in\ntheorem value_lt_of_normalized {x : U32 (F p)} (hx : x.Normalized) : x.value < 2^32"}, {"name": "getElem_eval_vector", "content": "theorem getElem_eval_vector (env : Environment F) (x : Var (ProvableVector α n) F) (i : ℕ) (h : i < n) :\n    (eval env x[i]) = (eval env x)[i]"}], "used_local_defs": [{"name": "Gadgets.BLAKE3.ApplyRounds.roundWithPermute", "content": "def roundWithPermute : FormalCircuit (F p) Round.Inputs Round.Inputs where\n  main := fun input => do\n    let state ← subcircuit Round.circuit input\n    let permuted_message ← subcircuit Permute.circuit input.message\n    return ⟨state, permuted_message⟩\n  localLength := fun _ => Round.circuit.localLength _ + Permute.circuit.localLength _\n  localLength_eq := by admit /- proof elided -/"}, {"name": "Gadgets.BLAKE3.ApplyRounds.twoRoundsWithPermute", "content": "def twoRoundsWithPermute : FormalCircuit (F p) Round.Inputs Round.Inputs :=\n  roundWithPermute.concat roundWithPermute (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )"}, {"name": "Gadgets.BLAKE3.ApplyRounds.fourRoundsWithPermute", "content": "def fourRoundsWithPermute : FormalCircuit (F p) Round.Inputs Round.Inputs :=\n  twoRoundsWithPermute.concat twoRoundsWithPermute (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )"}, {"name": "Gadgets.BLAKE3.ApplyRounds.sixRoundsWithPermute", "content": "def sixRoundsWithPermute : FormalCircuit (F p) Round.Inputs Round.Inputs :=\n  fourRoundsWithPermute.concat twoRoundsWithPermute (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )"}, {"name": "Gadgets.BLAKE3.ApplyRounds.applySixRounds", "content": "def applySixRounds (state : Vector ℕ 16) (message : Vector ℕ 16) : Vector ℕ 16 × Vector ℕ 16 :=\n  let state1 := round state message\n  let msg1 := permute message\n  let state2 := round state1 msg1\n  let msg2 := permute msg1\n  let state3 := round state2 msg2\n  let msg3 := permute msg2\n  let state4 := round state3 msg3\n  let msg4 := permute msg3\n  let state5 := round state4 msg4\n  let msg5 := permute msg4\n  let state6 := round state5 msg5\n  let msg6 := permute msg5\n  (state6, msg6)"}, {"name": "Gadgets.BLAKE3.ApplyRounds.SixRoundsSpec", "content": "def SixRoundsSpec (input : Round.Inputs (F p)) (output : Round.Inputs (F p)) : Prop :=\n  let (final_state, final_message) := applySixRounds input.state.value (input.message.map U32.value)\n  output.state.value = final_state ∧\n  output.message.map U32.value = final_message ∧\n  output.state.Normalized ∧\n  (∀ i : Fin 16, output.message[i].Normalized)"}, {"name": "Gadgets.BLAKE3.ApplyRounds.sixRoundsApplyStyle", "content": "def sixRoundsApplyStyle : FormalCircuit (F p) Round.Inputs Round.Inputs :=\n  sixRoundsWithPermute.weakenSpec SixRoundsSpec (by admit /- proof elided -/\n  )"}, {"name": "Gadgets.BLAKE3.ApplyRounds.sevenRoundsFinal", "content": "def sevenRoundsFinal : FormalCircuit (F p) Round.Inputs BLAKE3State :=\n  sixRoundsApplyStyle.concat Round.circuit (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )"}, {"name": "Gadgets.BLAKE3.ApplyRounds.applySevenRounds", "content": "def applySevenRounds (state : Vector ℕ 16) (message : Vector ℕ 16) : Vector ℕ 16 :=\n  let state1 := round state message\n  let msg1 := permute message\n  let state2 := round state1 msg1\n  let msg2 := permute msg1\n  let state3 := round state2 msg2\n  let msg3 := permute msg2\n  let state4 := round state3 msg3\n  let msg4 := permute msg3\n  let state5 := round state4 msg4\n  let msg5 := permute msg4\n  let state6 := round state5 msg5\n  let msg6 := permute msg5\n  let state7 := round state6 msg6\n  state7"}, {"name": "Gadgets.BLAKE3.ApplyRounds.SevenRoundsSpec", "content": "def SevenRoundsSpec (input : Round.Inputs (F p)) (output : BLAKE3State (F p)) : Prop :=\n  let final_state := applySevenRounds input.state.value (input.message.map U32.value)\n  output.value = final_state ∧\n  output.Normalized"}, {"name": "Gadgets.BLAKE3.ApplyRounds.sevenRoundsApplyStyle", "content": "def sevenRoundsApplyStyle : FormalCircuit (F p) Round.Inputs BLAKE3State :=\n  sevenRoundsFinal.weakenSpec SevenRoundsSpec (by admit /- proof elided -/\n  )"}, {"name": "Gadgets.BLAKE3.ApplyRounds.Inputs", "content": "structure Inputs (F : Type) where\n  chaining_value : Vector (U32 F) 8\n  block_words : Vector (U32 F) 16\n  counter_high : U32 F\n  counter_low : U32 F\n  block_len : U32 F\n  flags : U32 F"}, {"name": "Gadgets.BLAKE3.ApplyRounds.initializeStateVector", "content": "def initializeStateVector (input_var : Var Inputs (F p)) : Var BLAKE3State (F p) :=\n  let { chaining_value, block_words, counter_high, counter_low, block_len, flags } := input_var\n  #v[\n    chaining_value[0], chaining_value[1], chaining_value[2], chaining_value[3],\n    chaining_value[4], chaining_value[5], chaining_value[6], chaining_value[7],\n    const (U32.fromUInt32 iv[0]), const (U32.fromUInt32 iv[1]),\n    const (U32.fromUInt32 iv[2]), const (U32.fromUInt32 iv[3]),\n    counter_low, counter_high, block_len, flags\n  ]"}, {"name": "Gadgets.BLAKE3.ApplyRounds.main", "content": "def main (input : Var Inputs (F p)) : Circuit (F p) (Var BLAKE3State (F p)) := do\n  let state := initializeStateVector input\n  \n  sevenRoundsApplyStyle ⟨state, input.block_words⟩"}, {"name": "Gadgets.BLAKE3.ApplyRounds.elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs BLAKE3State where\n  main := main\n  localLength _ := 5376\n  localLength_eq input i0 := by admit /- proof elided -/"}, {"name": "Gadgets.BLAKE3.ApplyRounds.Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let { chaining_value, block_words, counter_high, counter_low, block_len, flags } := input\n  (∀ i : Fin 8, chaining_value[i].Normalized) ∧\n  (∀ i : Fin 16, block_words[i].Normalized) ∧\n  counter_high.Normalized ∧ counter_low.Normalized ∧ block_len.Normalized ∧ flags.Normalized"}, {"name": "Gadgets.BLAKE3.ApplyRounds.Spec", "content": "def Spec (input : Inputs (F p)) (out : BLAKE3State (F p)) :=\n  let { chaining_value, block_words, counter_high, counter_low, block_len, flags } := input\n  out.value = applyRounds\n    (chaining_value.map U32.value)\n    (block_words.map U32.value)\n    (counter_low.value + 2^32 * counter_high.value)\n    block_len.value\n    flags.value ∧\n  out.Normalized"}], "used_local_lemmas": [{"name": "Gadgets.BLAKE3.ApplyRounds.applyRounds_eq_applySevenRounds", "content": "lemma applyRounds_eq_applySevenRounds\n    (chaining_value : Vector ℕ 8)\n    (block_words : Vector ℕ 16)\n    (counter : ℕ)\n    (block_len : ℕ)\n    (flags : ℕ) :\n    applyRounds chaining_value block_words counter block_len flags =\n    applySevenRounds\n      (#v[\n        chaining_value[0], chaining_value[1], chaining_value[2], chaining_value[3],\n        chaining_value[4], chaining_value[5], chaining_value[6], chaining_value[7],\n        iv[0].toNat, iv[1].toNat, iv[2].toNat, iv[3].toNat,\n        counter % 2^32, counter / 2^32, block_len, flags\n      ])\n      block_words"}, {"name": "Gadgets.BLAKE3.ApplyRounds.initial_state_and_messages_are_normalized", "content": "lemma initial_state_and_messages_are_normalized\n    (env : Environment (F p))\n    (input_var : Var Inputs (F p))\n    (block_words : BLAKE3State (F p))\n    (chaining_value counter_high counter_low block_len flags)\n    (h_input : eval env input_var = { chaining_value, block_words, counter_high, counter_low, block_len, flags })\n    (h_normalized : Assumptions { chaining_value, block_words, counter_high, counter_low, block_len, flags }) :\n    (eval env (initializeStateVector input_var)).Normalized ∧ ∀ (i : Fin 16), block_words[i].Normalized"}], "local_ctx": "import Clean.Gadgets.BLAKE3.BLAKE3State\n\nimport Clean.Gadgets.BLAKE3.Round\n\nimport Clean.Gadgets.BLAKE3.Permute\n\nimport Clean.Types.U32\n\nimport Clean.Circuit.Provable\n\nimport Clean.Specs.BLAKE3\n\nimport Clean.Circuit.StructuralLemmas\n\nimport Clean.Utils.Tactics\n\nnamespace Gadgets.BLAKE3.ApplyRounds\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 2^16 + 2^8)]\n\nopen Specs.BLAKE3 (applyRounds iv round permute)\n\ndef roundWithPermute : FormalCircuit (F p) Round.Inputs Round.Inputs where\n  main := fun input => do\n    let state ← subcircuit Round.circuit input\n    let permuted_message ← subcircuit Permute.circuit input.message\n    return ⟨state, permuted_message⟩\n  localLength := fun _ => Round.circuit.localLength _ + Permute.circuit.localLength _\n  localLength_eq := by admit /- proof elided -/\n\ndef twoRoundsWithPermute : FormalCircuit (F p) Round.Inputs Round.Inputs :=\n  roundWithPermute.concat roundWithPermute (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n\ndef fourRoundsWithPermute : FormalCircuit (F p) Round.Inputs Round.Inputs :=\n  twoRoundsWithPermute.concat twoRoundsWithPermute (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n\ndef sixRoundsWithPermute : FormalCircuit (F p) Round.Inputs Round.Inputs :=\n  fourRoundsWithPermute.concat twoRoundsWithPermute (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n\ndef applySixRounds (state : Vector ℕ 16) (message : Vector ℕ 16) : Vector ℕ 16 × Vector ℕ 16 :=\n  let state1 := round state message\n  let msg1 := permute message\n  let state2 := round state1 msg1\n  let msg2 := permute msg1\n  let state3 := round state2 msg2\n  let msg3 := permute msg2\n  let state4 := round state3 msg3\n  let msg4 := permute msg3\n  let state5 := round state4 msg4\n  let msg5 := permute msg4\n  let state6 := round state5 msg5\n  let msg6 := permute msg5\n  (state6, msg6)\n\ndef SixRoundsSpec (input : Round.Inputs (F p)) (output : Round.Inputs (F p)) : Prop :=\n  let (final_state, final_message) := applySixRounds input.state.value (input.message.map U32.value)\n  output.state.value = final_state ∧\n  output.message.map U32.value = final_message ∧\n  output.state.Normalized ∧\n  (∀ i : Fin 16, output.message[i].Normalized)\n\ndef sixRoundsApplyStyle : FormalCircuit (F p) Round.Inputs Round.Inputs :=\n  sixRoundsWithPermute.weakenSpec SixRoundsSpec (by admit /- proof elided -/\n  )\n\ndef sevenRoundsFinal : FormalCircuit (F p) Round.Inputs BLAKE3State :=\n  sixRoundsApplyStyle.concat Round.circuit (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n\ndef applySevenRounds (state : Vector ℕ 16) (message : Vector ℕ 16) : Vector ℕ 16 :=\n  let state1 := round state message\n  let msg1 := permute message\n  let state2 := round state1 msg1\n  let msg2 := permute msg1\n  let state3 := round state2 msg2\n  let msg3 := permute msg2\n  let state4 := round state3 msg3\n  let msg4 := permute msg3\n  let state5 := round state4 msg4\n  let msg5 := permute msg4\n  let state6 := round state5 msg5\n  let msg6 := permute msg5\n  let state7 := round state6 msg6\n  state7\n\ndef SevenRoundsSpec (input : Round.Inputs (F p)) (output : BLAKE3State (F p)) : Prop :=\n  let final_state := applySevenRounds input.state.value (input.message.map U32.value)\n  output.value = final_state ∧\n  output.Normalized\n\ndef sevenRoundsApplyStyle : FormalCircuit (F p) Round.Inputs BLAKE3State :=\n  sevenRoundsFinal.weakenSpec SevenRoundsSpec (by admit /- proof elided -/\n  )\n\nstructure Inputs (F : Type) where\n  chaining_value : Vector (U32 F) 8\n  block_words : Vector (U32 F) 16\n  counter_high : U32 F\n  counter_low : U32 F\n  block_len : U32 F\n  flags : U32 F\n\ndef initializeStateVector (input_var : Var Inputs (F p)) : Var BLAKE3State (F p) :=\n  let { chaining_value, block_words, counter_high, counter_low, block_len, flags } := input_var\n  #v[\n    chaining_value[0], chaining_value[1], chaining_value[2], chaining_value[3],\n    chaining_value[4], chaining_value[5], chaining_value[6], chaining_value[7],\n    const (U32.fromUInt32 iv[0]), const (U32.fromUInt32 iv[1]),\n    const (U32.fromUInt32 iv[2]), const (U32.fromUInt32 iv[3]),\n    counter_low, counter_high, block_len, flags\n  ]\n\ndef main (input : Var Inputs (F p)) : Circuit (F p) (Var BLAKE3State (F p)) := do\n  let state := initializeStateVector input\n  \n  sevenRoundsApplyStyle ⟨state, input.block_words⟩\n\ninstance elaborated : ElaboratedCircuit (F p) Inputs BLAKE3State where\n  main := main\n  localLength _ := 5376\n  localLength_eq input i0 := by admit /- proof elided -/\n\ndef Assumptions (input : Inputs (F p)) :=\n  let { chaining_value, block_words, counter_high, counter_low, block_len, flags } := input\n  (∀ i : Fin 8, chaining_value[i].Normalized) ∧\n  (∀ i : Fin 16, block_words[i].Normalized) ∧\n  counter_high.Normalized ∧ counter_low.Normalized ∧ block_len.Normalized ∧ flags.Normalized\n\ndef Spec (input : Inputs (F p)) (out : BLAKE3State (F p)) :=\n  let { chaining_value, block_words, counter_high, counter_low, block_len, flags } := input\n  out.value = applyRounds\n    (chaining_value.map U32.value)\n    (block_words.map U32.value)\n    (counter_low.value + 2^32 * counter_high.value)\n    block_len.value\n    flags.value ∧\n  out.Normalized", "target_theorem": "theorem soundness : Soundness (F p) elaborated Assumptions Spec :=", "ground_truth_proof": ":= by\n  circuit_proof_start\n\n  -- Equations for counter values\n  have h_counter_low_eq : input_counter_low.value % 4294967296 = input_counter_low.value := by\n    apply Nat.mod_eq_of_lt\n\n    exact U32.value_lt_of_normalized h_assumptions.2.2.2.1\n  have h_counter_high_eq : (input_counter_low.value + 4294967296 * input_counter_high.value) / 4294967296 = input_counter_high.value := by\n    -- We want to show (input_counter_low.value + 2^32 * input_counter_high.value) / 2^32 = input_counter_high.value\n    -- Since input_counter_low.value < 2^32, this follows from properties of division\n    have h1 : input_counter_low.value < 4294967296 := U32.value_lt_of_normalized h_assumptions.2.2.2.1\n    have h2 : 4294967296 > 0 := by norm_num\n    -- Now we have (2^32 * input_counter_high.value + input_counter_low.value) / 2^32\n    -- This equals input_counter_high.value + input_counter_low.value / 2^32\n    rw [Nat.add_mul_div_left _ _ h2]\n    rw [Nat.div_eq_of_lt h1]\n    simp\n\n  -- Apply h_holds with the proven assumptions\n  have h_spec := h_holds (by\n    apply initial_state_and_messages_are_normalized\n    · simp only [circuit_norm, h_input]\n      rfl\n    · simp only [Assumptions]\n      aesop\n  )\n  clear h_holds\n\n  -- Now we need to show that the spec holds\n  -- h_spec tells us that sevenRoundsApplyStyle.Spec holds for the inputs and output\n  -- We need to unpack what this means and relate it to our Spec\n\n  simp only [sevenRoundsApplyStyle, FormalCircuit.weakenSpec, sevenRoundsFinal,\n             FormalCircuit.concat] at h_spec\n\n  -- The spec for sevenRoundsApplyStyle says the output equals applySevenRounds\n  simp only [SevenRoundsSpec] at h_spec\n\n  obtain ⟨h_value, h_normalized⟩ := h_spec\n\n  constructor\n  · -- Show out.value = applyRounds ...\n    -- Use our lemma to express applyRounds in terms of applySevenRounds\n    rw [applyRounds_eq_applySevenRounds]\n\n    -- h_value tells us the output equals applySevenRounds on our constructed state\n    simp only [BLAKE3State.value] at h_value ⊢\n    calc\n      _ = _ := h_value\n      _ = _ := by\n        clear h_value\n        simp only [initializeStateVector, h_input, eval_vector, circuit_norm, getElem_eval_vector]\n        simp [circuit_norm, U32.value_fromUInt32, h_counter_low_eq, h_counter_high_eq]\n\n  · -- Show out.Normalized\n    exact h_normalized", "nesting_depth": 21, "transitive_dep_count": 204, "subset_aristotle": false, "category": "Applied verif."}
{"id": 131, "thm_name": "Utils.StateTransition.size_removeCycle_lt", "thm_stmt": "lemma size_removeCycle_lt (R : Run S) (cycle : List S)\n    (h_len : cycle.length ≥ 2)\n    (h_contains : R.containsPath cycle)\n    (_h_cycle : cycle.head? = cycle.getLast?) :\n    (R.removeCycle cycle).size < R.size", "lean_root": "clean", "rel_path": "Clean/Utils/SourceSinkPath.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import Mathlib.Data.Fintype.Prod", "import Mathlib.Data.List.Basic", "import Mathlib.Algebra.BigOperators.Group.Finset.Basic", "import Mathlib.Data.Finset.Basic", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise", "import Mathlib.Data.Fintype.Basic"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "List.tail", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}, {"name": "Finset.erase", "module": "Mathlib.Data.Finset.Erase"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_erase_add", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_le_sum", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "List.count_pos_iff", "module": "Init.Data.List.Count"}, {"name": "Finset.mem_toList", "module": "Mathlib.Data.Finset.Dedup"}, {"name": "Nat.sub_le", "module": "Init.Prelude"}, {"name": "Nat.sub_lt", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Utils.StateTransition.Transition", "content": "def Transition (S : Type*) := S × S"}, {"name": "Utils.StateTransition.Run", "content": "def Run (S : Type*) := Transition S → ℕ"}, {"name": "Utils.StateTransition.Run.size", "content": "noncomputable def Run.size {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) : ℕ :=\n  ∑ t : Transition S, R t"}, {"name": "Utils.StateTransition.countTransitionInPath", "content": "def countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t"}, {"name": "Utils.StateTransition.Run.containsPath", "content": "def Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t"}, {"name": "Utils.StateTransition.Run.removeCycle", "content": "def Run.removeCycle (R : Run S) (cycle : List S) : Run S :=\n  fun t => R t - countTransitionInPath t cycle"}], "used_local_lemmas": [{"name": "Utils.StateTransition.sum_decrease", "content": "lemma sum_decrease {α : Type*} [Fintype α] [DecidableEq α] (f g : α → ℕ) (a : α)\n    (h_a_decrease : g a < f a)\n    (h_others_le : ∀ x, g x ≤ f x) :\n    ∑ x : α, g x < ∑ x : α, f x"}, {"name": "Utils.StateTransition.path_has_transition", "content": "lemma path_has_transition {S : Type*} [DecidableEq S] (path : List S)\n    (h_len : path.length ≥ 2) :\n    ∃ (t : Transition S), t ∈ path.zip path.tail"}, {"name": "Utils.StateTransition.containsPath_has_positive_transition", "content": "lemma containsPath_has_positive_transition (R : Run S) (path : List S)\n    (h_contains : R.containsPath path) (t : Transition S)\n    (h_in : t ∈ path.zip path.tail) :\n    R t > 0"}], "local_ctx": "import Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Fintype.Basic\n\nimport Mathlib.Data.Fintype.Prod\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Piecewise\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nnamespace Utils.StateTransition\n\nvariable {S : Type*} [DecidableEq S] [Fintype S]\n\ndef Transition (S : Type*) := S × S\n\ndef Run (S : Type*) := Transition S → ℕ\n\nnoncomputable def Run.size {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) : ℕ :=\n  ∑ t : Transition S, R t\n\ndef countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t\n\ndef Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t\n\ndef Run.removeCycle (R : Run S) (cycle : List S) : Run S :=\n  fun t => R t - countTransitionInPath t cycle", "target_theorem": "lemma size_removeCycle_lt (R : Run S) (cycle : List S)\n    (h_len : cycle.length ≥ 2)\n    (h_contains : R.containsPath cycle)\n    (_h_cycle : cycle.head? = cycle.getLast?) :\n    (R.removeCycle cycle).size < R.size :=", "ground_truth_proof": ":= by\n  -- Get a transition in the path\n  obtain ⟨t, h_in_zip⟩ := path_has_transition cycle h_len\n  -- This transition has positive capacity\n  have h_pos := containsPath_has_positive_transition R cycle h_contains t h_in_zip\n  let (x, y) := t\n  -- Show that this transition appears in the cycle, so countTransitionInPath > 0\n  have h_count_pos : countTransitionInPath (x, y) cycle > 0 := by\n    unfold countTransitionInPath\n    exact List.count_pos_iff.mpr h_in_zip\n  -- The size decreases because we subtract countTransitionInPath (x,y) cycle from R(x,y)\n  -- Since R(x,y) > 0 and countTransitionInPath (x,y) cycle > 0, the total decreases.\n  unfold Run.size Run.removeCycle\n  -- We'll prove that the sum of (R t - countTransitionInPath t cycle) is less than sum of R t\n  have h_decrease : (fun t => R t - countTransitionInPath t cycle) (x, y) < R (x, y) := by\n    -- R(x,y) > 0 and countTransitionInPath (x,y) cycle > 0\n    -- So R(x,y) - count < R(x,y) by Nat.sub_lt\n    exact Nat.sub_lt h_pos h_count_pos\n  have h_xy_in_univ : (x, y) ∈ (Finset.univ : Finset (Transition S)).toList := by\n    simp [Finset.mem_toList]\n  have h_others_le : ∀ t, (fun t => R t - countTransitionInPath t cycle) t ≤ R t := fun t =>\n    Nat.sub_le (R t) (countTransitionInPath t cycle)\n  -- Apply the sum_decrease lemma\n  exact sum_decrease R (fun t => R t - countTransitionInPath t cycle) (x, y) h_decrease h_others_le", "nesting_depth": 2, "transitive_dep_count": 25, "subset_aristotle": true, "category": "Applied verif."}
{"id": 132, "thm_name": "U32.ByteVector.bitwise_componentwise", "thm_stmt": "omit [Fact (Nat.Prime p)] p_large_enough in\nlemma bitwise_componentwise (f : Bool → Bool → Bool)\n    {x y : U32 (F p)} (x_norm : x.Normalized) (y_norm : y.Normalized) :\n    f false false = false →\n    Nat.bitwise f x.value y.value =\n      Nat.bitwise f x.x0.val y.x0.val + 256 *\n        (Nat.bitwise f x.x1.val y.x1.val + 256 *\n          (Nat.bitwise f x.x2.val y.x2.val + 256 * Nat.bitwise f x.x3.val y.x3.val))", "lean_root": "clean", "rel_path": "Clean/Types/U32.lean", "imports": ["import Clean.Circuit.Provable", "import Clean.Circuit.Subcircuit", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Clean.Utils.Bitwise", "import Clean.Circuit.Extensions", "import Clean.Gadgets.ByteLookup"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.Prime", "module": "Mathlib.Data.Nat.Prime.Defs"}, {"name": "Nat.bitwise", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "KeccakRow.value", "content": "def KeccakRow.value (row : KeccakRow (F p)) := row.map U64.value"}, {"name": "map", "content": "def map {α β : Type} (x : U64 α) (f : α → β) : U64 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3, f x.x4, f x.x5, f x.x6, f x.x7 ⟩"}, {"name": "U64", "content": "structure U64 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\n  x4 : T\n  x5 : T\n  x6 : T\n  x7 : T\nderiving DecidableEq"}, {"name": "value", "content": "def value (x : U64 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3 +\n  x.x4.val * 256^4 + x.x5.val * 256^5 + x.x6.val * 256^6 + x.x7.val * 256^7"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "KeccakRow.Normalized", "content": "def KeccakRow.Normalized (row : KeccakRow (F p)) :=\n  ∀ i : Fin 5, row[i.val].Normalized"}, {"name": "BLAKE3State.Normalized", "content": "def BLAKE3State.Normalized (state : BLAKE3State (F p)) :=\n  ∀ i : Fin 16, state[i.val].Normalized"}, {"name": "KeccakBlock.value", "content": "def KeccakBlock.value (block : KeccakBlock (F p)) := block.map U64.value"}, {"name": "BLAKE3State.value", "content": "def BLAKE3State.value (state : BLAKE3State (F p)) := state.map U32.value"}, {"name": "KeccakState.value", "content": "def KeccakState.value (state : KeccakState (F p)) := state.map U64.value"}, {"name": "KeccakBlock.Normalized", "content": "def KeccakBlock.Normalized (block : KeccakBlock (F p)) :=\n  ∀ i : Fin RATE, block[i.val].Normalized"}, {"name": "RATE", "content": "@[reducible] def RATE := 17\nexample : RATE + CAPACITY = 25 := rfl"}, {"name": "CAPACITY", "content": "@[reducible] def CAPACITY := 8"}, {"name": "KeccakState.Normalized", "content": "def KeccakState.Normalized (state : KeccakState (F p)) :=\n  ∀ i : Fin 25, state[i.val].Normalized"}], "lib_lemmas": [{"name": "Nat.bitwise_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.eq_of_testBit_eq", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.testBit_bitwise", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.testBit_two_pow_mul_add", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "U32", "content": "structure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq"}, {"name": "U32.map", "content": "def map {α β : Type} (x : U32 α) (f : α → β) : U32 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3 ⟩"}, {"name": "U32.Normalized", "content": "def Normalized (x : U32 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256"}, {"name": "U32.value", "content": "def value (x : U32 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3"}], "used_local_lemmas": [{"name": "U32.ByteVector.reorganize_value", "content": "private lemma reorganize_value (a b c d : ℕ) :\n  a + 256 * (b + 256 * (c + 256 * d)) =\n  2^8 * (2^8 * (2^8 * d + c) + b) + a"}, {"name": "U32.ByteVector.reorganize_value'", "content": "private lemma reorganize_value' (a b c d : ℕ) :\n  a + b * 256 + c * 256 ^ 2 + d * 256 ^ 3 =\n  2^8 * (2^8 * (2^8 * d + c) + b) + a"}], "local_ctx": "import Clean.Gadgets.ByteLookup\n\nimport Clean.Circuit.Extensions\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Circuit.Provable\n\nimport Clean.Utils.Primes\n\nimport Clean.Circuit.Subcircuit\n\nimport Clean.Gadgets.Equality\n\nsection\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nstructure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq\n\nnamespace U32\n\ndef map {α β : Type} (x : U32 α) (f : α → β) : U32 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3 ⟩\n\ndef Normalized (x : U32 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256\n\ndef value (x : U32 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3\n\nend U32\n\nnamespace U32.AssertNormalized\n\nopen Gadgets (ByteTable)\n\nend U32.AssertNormalized\n\nnamespace U32.ByteVector\n\nend ByteVector\n\nsection Bitwise", "target_theorem": "omit [Fact (Nat.Prime p)] p_large_enough in\nlemma bitwise_componentwise (f : Bool → Bool → Bool)\n    {x y : U32 (F p)} (x_norm : x.Normalized) (y_norm : y.Normalized) :\n    f false false = false →\n    Nat.bitwise f x.value y.value =\n      Nat.bitwise f x.x0.val y.x0.val + 256 *\n        (Nat.bitwise f x.x1.val y.x1.val + 256 *\n          (Nat.bitwise f x.x2.val y.x2.val + 256 * Nat.bitwise f x.x3.val y.x3.val)) :=", "ground_truth_proof": ":= by\n  intro _\n  simp only [value]\n\n  have ⟨_, _, _, _⟩ := x_norm\n  have ⟨_, _, _, _⟩ := y_norm\n  apply Nat.eq_of_testBit_eq\n  intro i\n  simp only [reorganize_value, reorganize_value']\n  rw [Nat.testBit_bitwise] <;> try assumption\n  rw [Nat.testBit_two_pow_mul_add (i:=8) (b:=ZMod.val x.x0)] <;> try assumption\n  rw [Nat.testBit_two_pow_mul_add (i:=8) (b:=ZMod.val y.x0)] <;> try assumption\n  rw [Nat.testBit_two_pow_mul_add (i:=8) (b:=Nat.bitwise f (ZMod.val x.x0) (ZMod.val y.x0))] <;> try (apply Nat.bitwise_lt_two_pow <;> assumption)\n  split\n  · simp_all only [Nat.testBit_bitwise]\n  rw [Nat.testBit_two_pow_mul_add (i:=8) (b:=ZMod.val x.x1)] <;> try assumption\n  rw [Nat.testBit_two_pow_mul_add (i:=8) (b:=ZMod.val y.x1)] <;> try assumption\n  rw [Nat.testBit_two_pow_mul_add (i:=8) (b:=Nat.bitwise f (ZMod.val x.x1) (ZMod.val y.x1))] <;> try (apply Nat.bitwise_lt_two_pow <;> assumption)\n  split\n  · simp_all only [not_lt, Nat.testBit_bitwise]\n  rw [Nat.testBit_two_pow_mul_add (i:=8) (b:=ZMod.val x.x2)] <;> try assumption\n  rw [Nat.testBit_two_pow_mul_add (i:=8) (b:=ZMod.val y.x2)] <;> try assumption\n  rw [Nat.testBit_two_pow_mul_add (i:=8) (b:=Nat.bitwise f (ZMod.val x.x2) (ZMod.val y.x2))] <;> try (apply Nat.bitwise_lt_two_pow <;> assumption)\n  aesop", "nesting_depth": 7, "transitive_dep_count": 66, "subset_aristotle": true, "category": "Applied verif."}
{"id": 133, "thm_name": "MemoryAccessList.isConsistentOffline_iff_all_single_addresses", "thm_stmt": "theorem MemoryAccessList.isConsistentOffline_iff_all_single_addresses (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) (h_nodup : accesses.Notimestampdup) :\n    MemoryAccessList.isConsistentOffline accesses h_sorted ↔\n    ∀ addr, MemoryAccessList.isConsistentSingleAddress (MemoryAccessList.filterAddress accesses addr) (filterAddress_sorted_from_addressTimestampSorted accesses h_sorted h_nodup addr)", "lean_root": "clean", "rel_path": "Clean/Utils/OfflineMemory.lean", "imports": ["import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Mathlib.Data.List.Sort", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Sorted", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Pairwise", "module": "Init.Data.List.Basic"}, {"name": "List.filter", "module": "Init.Data.List.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.Sorted.of_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Pairwise.of_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.filter_eq_nil_iff", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_cons_self", "module": "Init.Data.List.Lemmas"}, {"name": "List.sorted_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "decide_eq_true_eq", "module": "Init.SimpLemmas"}, {"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "List.filter_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_filter", "module": "Init.Data.List.Lemmas"}, {"name": "decide_true", "module": "Init.Core"}, {"name": "decide_eq_false_iff_not", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MemoryAccess", "content": "def MemoryAccess := ℕ × ℕ × ℕ × ℕ"}, {"name": "MemoryAccessList", "content": "def MemoryAccessList := List MemoryAccess"}, {"name": "timestamp_ordering", "content": "abbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2"}, {"name": "MemoryAccessList.isTimestampSorted", "content": "def MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering"}, {"name": "MemoryAccessList.timestamps_neq", "content": "def MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y"}, {"name": "MemoryAccessList.Notimestampdup", "content": "def MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses"}, {"name": "address_timestamp_ordering", "content": "abbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2"}, {"name": "address_strict_timestamp_ordering", "content": "abbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2"}, {"name": "MemoryAccessList.isAddressTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering"}, {"name": "MemoryAccessList.isAddressStrictTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering"}, {"name": "MemoryAccessList.filterAddress", "content": "def MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)"}, {"name": "MemoryAccessList.isConsistentSingleAddress", "content": "def MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}, {"name": "MemoryAccessList.isConsistentOffline", "content": "def MemoryAccessList.isConsistentOffline (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  | (_t2, addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    (if addr1 = addr2 then readValue2 = writeValue1 else readValue2 = 0) ∧\n    MemoryAccessList.isConsistentOffline ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}], "used_local_lemmas": [{"name": "MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup", "content": "theorem MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup\n    (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted)\n    (h_no_timestamp_dup : accesses.Notimestampdup) :\n    accesses.isAddressStrictTimestampSorted"}, {"name": "MemoryAccessList.noTimestampDup_of_cons", "content": "theorem MemoryAccessList.noTimestampDup_of_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h : Notimestampdup (head :: tail)) :\n    Notimestampdup tail"}, {"name": "MemoryAccessList.isAddressTimestampSorted_of_cons", "content": "theorem MemoryAccessList.isAddressTimestampSorted_of_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h : isAddressTimestampSorted (head :: tail)) :\n    isAddressTimestampSorted tail"}, {"name": "MemoryAccessList.isConsistentSingleAddress_cons", "content": "theorem MemoryAccessList.isConsistentSingleAddress_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail)) (h_sorted' : tail.isTimestampSorted)\n    (h : isConsistentSingleAddress (head :: tail) h_sorted) :\n    isConsistentSingleAddress tail h_sorted'"}, {"name": "MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted", "content": "theorem MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isTimestampSorted"}, {"name": "MemoryAccessList.isConsistentSingleAddress_filterAddress_forall_of_cons", "content": "theorem MemoryAccessList.isConsistentSingleAddress_filterAddress_forall_of_cons\n    (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: tail))\n    (h_nodup : Notimestampdup (head :: tail))\n    (h : ∀ addr, isConsistentSingleAddress (filterAddress (head :: tail) addr)\n         (filterAddress_sorted_from_addressTimestampSorted (head :: tail) h_sorted h_nodup addr)) :\n    ∀ addr, isConsistentSingleAddress (filterAddress tail addr)\n      (filterAddress_sorted_from_addressTimestampSorted tail\n        (isAddressTimestampSorted_of_cons head tail h_sorted)\n        (noTimestampDup_of_cons head tail h_nodup) addr)"}, {"name": "MemoryAccessList.filterAddress_empty_when_address_changes", "content": "theorem MemoryAccessList.filterAddress_empty_when_address_changes\n    (head : MemoryAccess) (second : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: second :: tail))\n    (h_addr_ne : head.2.1 ≠ second.2.1) :\n    filterAddress (second :: tail) head.2.1 = []"}, {"name": "MemoryAccessList.isConsistentOffline_of_cons", "content": "theorem MemoryAccessList.isConsistentOffline_of_cons\n    (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: tail))\n    (h_offline : isConsistentOffline (head :: tail) h_sorted) :\n    isConsistentOffline tail (isAddressTimestampSorted_of_cons head tail h_sorted)"}, {"name": "MemoryAccessList.isConsistentOffline_implies_single_address", "content": "theorem MemoryAccessList.isConsistentOffline_implies_single_address\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (h_offline : accesses.isConsistentOffline h_sorted)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isConsistentSingleAddress\n      (filterAddress_sorted_from_addressTimestampSorted accesses h_sorted h_nodup addr)"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Utils.Field\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Tactics\n\nimport Mathlib.Data.List.Sort\n\ndef MemoryAccess := ℕ × ℕ × ℕ × ℕ \n\ndef MemoryAccessList := List MemoryAccess\n\nabbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2\n\ndef MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering\n\ndef MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y\n\ndef MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses\n\nabbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2\n\nabbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2\n\n@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering\n\n@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering\n\ndef MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)\n\ndef MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)\n\ndef MemoryAccessList.isConsistentOffline (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  | (_t2, addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    (if addr1 = addr2 then readValue2 = writeValue1 else readValue2 = 0) ∧\n    MemoryAccessList.isConsistentOffline ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)", "target_theorem": "theorem MemoryAccessList.isConsistentOffline_iff_all_single_addresses (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) (h_nodup : accesses.Notimestampdup) :\n    MemoryAccessList.isConsistentOffline accesses h_sorted ↔\n    ∀ addr, MemoryAccessList.isConsistentSingleAddress (MemoryAccessList.filterAddress accesses addr) (filterAddress_sorted_from_addressTimestampSorted accesses h_sorted h_nodup addr) :=", "ground_truth_proof": ":= by\n  constructor\n  · intro h_offline addr\n    exact isConsistentOffline_implies_single_address accesses h_sorted h_nodup h_offline addr\n  · induction accesses\n    · simp [isConsistentOffline]\n    rename_i hd tl h_ih\n    intro h\n    rcases tl\n    · rcases hd with ⟨ hd_t, hd_a, hd_r, hd_w ⟩\n      simp only [isConsistentOffline]\n      specialize h hd_a\n      simp only [filterAddress, List.filter, decide_true, isConsistentSingleAddress] at h\n      assumption\n    rename_i snd tl\n    rcases hd with ⟨ hd_t, hd_a, hd_r, hd_w ⟩\n    rcases snd with ⟨ snd_t, snd_a, snd_r, snd_w ⟩\n    simp only [isConsistentOffline]\n    and_intros\n    · split\n      · rename_i addr_eq\n        subst addr_eq\n        specialize h snd_a\n        simp only [filterAddress, List.filter, decide_true, isConsistentSingleAddress] at h\n        aesop\n      · -- addresstimestampsorted, and seeing two different addresses, then the first address will never appear again\n        -- Since hd_a ≠ snd_a and the list is address-timestamp sorted, hd_a won't appear in (snd :: tl)\n        -- Therefore filterAddress ((hd :: snd :: tl)) hd_a = [hd]\n        -- And since isConsistentSingleAddress [hd] must hold, we get hd_r = 0\n        rename_i h_addr_ne\n        -- Use the lemma to show filterAddress (snd :: tl) hd_a is empty\n        have h_empty := filterAddress_empty_when_address_changes (hd_t, hd_a, hd_r, hd_w) (snd_t, snd_a, snd_r, snd_w) tl h_sorted (by simp; intro h_eq; exact h_addr_ne h_eq.symm)\n        specialize h hd_a\n        simp only [filterAddress, List.filter_cons, decide_true] at h\n        have h_snd_ne : decide (snd_a = hd_a) = false := by\n          simp only [decide_eq_false_iff_not]\n          exact h_addr_ne\n        simp only [h_snd_ne, ↓reduceIte] at h\n        simp only [filterAddress, List.filter_cons, h_snd_ne] at h_empty\n        simp only [h_empty, isConsistentSingleAddress] at h\n        exact h\n    apply h_ih\n    · exact isConsistentSingleAddress_filterAddress_forall_of_cons (hd_t, hd_a, hd_r, hd_w) ((snd_t, snd_a, snd_r, snd_w) :: tl) h_sorted h_nodup h\n    · exact noTimestampDup_of_cons (hd_t, hd_a, hd_r, hd_w) ((snd_t, snd_a, snd_r, snd_w) :: tl) h_nodup", "nesting_depth": 4, "transitive_dep_count": 38, "subset_aristotle": true, "category": "Applied verif."}
{"id": 134, "thm_name": "FlatOperation.proverEnvironment_usesLocalWitnesses", "thm_stmt": "theorem proverEnvironment_usesLocalWitnesses {ops : List (FlatOperation F)} (init : List F) :\n  (∀ (env env' : Environment F),\n    forAll init.length { witness n _ c := env.AgreesBelow n env' → c env = c env' } ops) →\n    (proverEnvironment ops init).UsesLocalWitnessesFlat init.length ops", "lean_root": "clean", "rel_path": "Clean/Circuit/Theorems.lean", "imports": ["import Clean.Circuit.Provable", "import Clean.Circuit.Basic"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.foldlM", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "FlatOperation.dynamicWitness", "content": "def FlatOperation.dynamicWitness (op : FlatOperation F) (acc : List F) : List F := match op with\n  | .witness _ compute => (compute (.fromList acc)).toList\n  | .assert _ => []\n  | .lookup _ => []"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Environment.AgreesBelow", "content": "def Environment.AgreesBelow (n : ℕ) (env env' : Environment F) :=\n  ∀ i < n, env.get i = env'.get i"}, {"name": "Environment.UsesLocalWitnessesFlat", "content": "def Environment.UsesLocalWitnessesFlat (env : Environment F) (n : ℕ) (ops : List (FlatOperation F)) : Prop :=\n  FlatOperation.forAll n { witness n _ compute := env.ExtendsVector (compute env) n } ops"}, {"name": "Condition.applyFlat", "content": "def Condition.applyFlat (condition : Condition F) (offset : ℕ) : FlatOperation F → Prop\n  | .witness m c => condition.witness offset m c\n  | .assert e => condition.assert offset e\n  | .lookup l => condition.lookup offset l"}, {"name": "Environment.fromList", "content": "def Environment.fromList (witnesses : List F) : Environment F :=\n  .mk fun i => witnesses[i]?.getD 0"}, {"name": "FlatOperation.proverEnvironment", "content": "def FlatOperation.proverEnvironment (ops : List (FlatOperation F)) (init : List F) : Environment F :=\n  .fromList (FlatOperation.dynamicWitnesses ops init)"}, {"name": "FlatOperation.dynamicWitnesses", "content": "def FlatOperation.dynamicWitnesses (ops : List (FlatOperation F)) (init : List F) : List F :=\n  ops.foldl (fun (acc : List F) (op : FlatOperation F) =>\n    acc ++ op.dynamicWitness acc\n  ) init"}, {"name": "foldl", "content": "def foldl {m : ℕ} [Inhabited β] [Inhabited α] (xs : Vector α m) (init : β) (body : β → α → Circuit F β)\n  (_const_out : ConstantOutput (fun (s, a) => body s a) := by admit /- proof elided -/\n    )\n  (_constant : ConstantLength (fun (s, a) => body s a) := by admit /- proof elided -/\n  )\n   : Circuit F β :=\n  xs.foldlM body init"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantOutput", "content": "@[circuit_norm]\ndef ConstantOutput (circuit : α → Circuit F β) [Inhabited α] :=\n  ∀ (x : α) (n : ℕ), (circuit x).output n = (circuit default).output n"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "FlatOperation.singleLocalLength", "content": "def FlatOperation.singleLocalLength : FlatOperation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0"}], "lib_lemmas": [{"name": "List.foldl_cons", "module": "Init.Data.List.Basic"}, {"name": "List.getElem_append_left", "module": "Init.Data.List.BasicAux"}, {"name": "List.length_append", "module": "Init.Data.List.Basic"}, {"name": "List.getElem_append_right", "module": "Init.Data.List.BasicAux"}, {"name": "add_tsub_cancel_left", "module": "Mathlib.Algebra.Order.Sub.Defs"}, {"name": "Vector.getElem_toList", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.length_toList", "module": "Init.Data.Vector.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "FlatOperation.forAll_cons", "content": "theorem forAll_cons {condition : Condition F} {offset : ℕ} {op : FlatOperation F} {ops : List (FlatOperation F)} :\n  forAll offset condition (op :: ops) ↔\n    condition.applyFlat offset op ∧ forAll (op.singleLocalLength + offset) condition ops"}, {"name": "FlatOperation.dynamicWitness_length", "content": "lemma dynamicWitness_length {op : FlatOperation F} {init : List F} :\n    (op.dynamicWitness init).length = op.singleLocalLength"}, {"name": "FlatOperation.dynamicWitnesses_cons", "content": "lemma dynamicWitnesses_cons {op : FlatOperation F} {ops : List (FlatOperation F)} {acc : List F} :\n    dynamicWitnesses (op :: ops) acc = dynamicWitnesses ops (acc ++ op.dynamicWitness acc)"}, {"name": "FlatOperation.getElem?_dynamicWitnesses_of_lt", "content": "lemma getElem?_dynamicWitnesses_of_lt {ops : List (FlatOperation F)} {acc : List F} {i : ℕ} (hi : i < acc.length) :\n    (dynamicWitnesses ops acc)[i]?.getD 0 = acc[i]"}, {"name": "FlatOperation.getElem?_dynamicWitnesses_cons_right", "content": "lemma getElem?_dynamicWitnesses_cons_right {op : FlatOperation F} {ops : List (FlatOperation F)} {init : List F} {i : ℕ} (hi : i < op.singleLocalLength) :\n    (dynamicWitnesses (op :: ops) init)[init.length + i]?.getD 0 = (op.dynamicWitness init)[i]'(dynamicWitness_length (F:=F) ▸ hi)"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nvariable {F : Type} [Field F] {α β : Type}\n\nnamespace Operations\n\nend Operations\n\nnamespace Circuit\n\nend Circuit\n\nnamespace FlatOperation\n\nend FlatOperation\n\nnamespace Environment\n\nopen FlatOperation (localLength localWitnesses)\n\nend Environment\n\nnamespace Circuit\n\nend Circuit\n\nnamespace Circuit\n\nvariable {α β : Type} {n : ℕ} {prop : Condition F} {env : Environment F}\n\nend Circuit\n\nnamespace FlatOperation\n\nend FlatOperation\n\nnamespace Operations\n\nend Operations\n\nnamespace FlatOperation", "target_theorem": "theorem proverEnvironment_usesLocalWitnesses {ops : List (FlatOperation F)} (init : List F) :\n  (∀ (env env' : Environment F),\n    forAll init.length { witness n _ c := env.AgreesBelow n env' → c env = c env' } ops) →\n    (proverEnvironment ops init).UsesLocalWitnessesFlat init.length ops :=", "ground_truth_proof": ":= by\n  simp only [proverEnvironment, Environment.UsesLocalWitnessesFlat, Environment.ExtendsVector]\n  intro h_computable\n  induction ops generalizing init with\n  | nil => trivial\n  | cons op ops ih =>\n    simp only [forAll_cons] at h_computable ⊢\n    cases op with\n    | assert | lookup  =>\n      simp_all [dynamicWitnesses_cons, Condition.applyFlat, singleLocalLength, dynamicWitness]\n    | witness m compute =>\n      simp_all only [Condition.applyFlat, singleLocalLength, Environment.AgreesBelow]\n      -- get rid of ih first\n      constructor; case right =>\n        specialize ih (init ++ (compute (.fromList init)).toList)\n        simp only [List.length_append, Vector.length_toList] at ih\n        ring_nf at *\n        exact ih fun _ _ => (h_computable ..).right\n      clear ih\n      replace h_computable := fun env env' => (h_computable env env').left\n      intro i\n      simp only [Environment.fromList]\n      rw [getElem?_dynamicWitnesses_cons_right i.is_lt]\n      simp only [dynamicWitness, Vector.getElem_toList]\n      congr 1\n      apply h_computable\n      intro j hj\n      simp [Environment.fromList, hj, getElem?_dynamicWitnesses_of_lt]", "nesting_depth": 7, "transitive_dep_count": 57, "subset_aristotle": true, "category": "Applied verif."}
{"id": 135, "thm_name": "Utils.StateTransition.acyclic_has_leaf_aux", "thm_stmt": "lemma acyclic_has_leaf_aux (R : Run S) (root current : S)\n    (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_start : path.head? = some root)\n    (h_end : path.getLast? = some current)\n    (h_nonempty : path ≠ [])\n    (h_contains : R.containsPath path)\n    (h_has_out : ∃ y, y ∉ path ∧ R (current, y) > 0) :\n    ∃ leaf, R.isLeaf root leaf", "lean_root": "clean", "rel_path": "Clean/Utils/SourceSinkPath.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import Mathlib.Data.Fintype.Prod", "import Mathlib.Data.List.Basic", "import Mathlib.Algebra.BigOperators.Group.Finset.Basic", "import Mathlib.Data.Finset.Basic", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise", "import Mathlib.Data.Fintype.Basic"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List.count", "module": "Init.Data.List.Basic"}, {"name": "List.tail", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "List.Sublist", "module": "Init.Data.List.Basic"}, {"name": "List.drop", "module": "Init.Data.List.Basic"}, {"name": "List.take", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.cons₂", "module": "Init.Data.List.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat.reduceLeDiff", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.countP_cons_of_pos", "module": "Init.Data.List.Count"}, {"name": "List.countP_nil", "module": "Init.Data.List.Count"}, {"name": "List.count_cons", "module": "Init.Data.List.Count"}, {"name": "List.count_nil", "module": "Init.Data.List.Count"}, {"name": "List.zipWith_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zipWith_nil_right", "module": "Init.Data.List.Basic"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.exists_cons_of_ne_nil", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast?_cons_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.tail_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zip_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.countP_singleton", "module": "Init.Data.List.Count"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "List.zip_nil_right", "module": "Init.Data.List.Basic"}, {"name": "Nat.add_right_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "List.mem_of_mem_tail", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "List.drop_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.count_le", "module": "Init.Data.List.Count"}, {"name": "List.tail_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.take_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.length_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.length_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "Nat.min_eq_left", "module": "Init.Data.Nat.MinMax"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "List.getElem?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getElem?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getLast?_eq_getElem?", "module": "Init.Data.List.Lemmas"}, {"name": "List.head?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.head?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "List.duplicate_iff_exists_distinct_get", "module": "Mathlib.Data.List.NodupEquivFin"}, {"name": "List.exists_duplicate_iff_not_nodup", "module": "Mathlib.Data.List.Duplicate"}, {"name": "List.getLast?_eq_getLast", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast_mem", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.nodup_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.mem_iff_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.ssubset_univ_iff", "module": "Mathlib.Data.Finset.BooleanAlgebra"}, {"name": "Finset.card_insert_of_notMem", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.card_lt_card", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.card_univ", "module": "Mathlib.Data.Fintype.Card"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Utils.StateTransition.Transition", "content": "def Transition (S : Type*) := S × S"}, {"name": "Utils.StateTransition.Run", "content": "def Run (S : Type*) := Transition S → ℕ"}, {"name": "Utils.StateTransition.countTransitionInPath", "content": "def countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t"}, {"name": "Utils.StateTransition.Run.containsPath", "content": "def Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t"}, {"name": "Utils.StateTransition.Run.hasCycle", "content": "def Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle"}, {"name": "Utils.StateTransition.Run.isAcyclic", "content": "def Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle"}, {"name": "Utils.StateTransition.Run.reachable", "content": "def Run.reachable [DecidableEq S] (R : Run S) (start finish : S) : Prop :=\n  ∃ (path : List S), path.head? = some start ∧ path.getLast? = some finish ∧\n    path ≠ [] ∧ R.containsPath path"}, {"name": "Utils.StateTransition.Run.isLeaf", "content": "def Run.isLeaf (R : Run S) (root leaf : S) : Prop :=\n  R.reachable root leaf ∧ ∀ y, R (leaf, y) = 0"}], "used_local_lemmas": [{"name": "Utils.StateTransition.finset_ssubset_univ_of_not_mem", "content": "lemma finset_ssubset_univ_of_not_mem {α : Type*} [Fintype α] (s : Finset α) (x : α)\n    (h : x ∉ s) :\n    s ⊂ Finset.univ"}, {"name": "Utils.StateTransition.zip_drop_sublist", "content": "lemma zip_drop_sublist (l : List S) (n : ℕ) :\n    ((l.drop n).zip (l.drop (n + 1))).Sublist (l.zip l.tail)"}, {"name": "Utils.StateTransition.containsPath_drop", "content": "lemma containsPath_drop (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.drop n)"}, {"name": "Utils.StateTransition.zip_take_sublist", "content": "lemma zip_take_sublist (l1 l2 : List S) (n m : ℕ) :\n    ((l1.take n).zip (l2.take m)).Sublist (l1.zip l2)"}, {"name": "Utils.StateTransition.tail_take", "content": "lemma tail_take {α : Type*} (l : List α) (n : ℕ) :\n    (l.take n).tail = (l.tail).take (n - 1)"}, {"name": "Utils.StateTransition.containsPath_take", "content": "lemma containsPath_take (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.take n)"}, {"name": "Utils.StateTransition.drop_take_length_ge_two", "content": "lemma drop_take_length_ge_two {α : Type*} (path : List α) (n m : Fin path.length)\n    (h_n_lt_m : n < m) :\n    ((path.drop n.val).take (m.val - n.val + 1)).length ≥ 2"}, {"name": "Utils.StateTransition.getLast_drop_take", "content": "lemma getLast_drop_take {α : Type*} (path : List α) (n k : ℕ)\n    (h_n_lt : n < path.length)\n    (h_bound : n + k ≤ path.length)\n    (h_k_pos : k > 0) :\n    ((path.drop n).take k).getLast? = path[n + k - 1]?"}, {"name": "Utils.StateTransition.drop_take_cycle_same_endpoints", "content": "lemma drop_take_cycle_same_endpoints (path : List S) (x : S) (n m : Fin path.length)\n    (h_n_lt_m : n < m)\n    (h_x_at_n : path[n] = x)\n    (h_x_at_m : path[m] = x) :\n    ((path.drop n.val).take (m.val - n.val + 1)).head? =\n    ((path.drop n.val).take (m.val - n.val + 1)).getLast?"}, {"name": "Utils.StateTransition.containsPath_drop_take", "content": "lemma containsPath_drop_take (R : Run S) (path : List S) (n m : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath ((path.drop n).take m)"}, {"name": "Utils.StateTransition.acyclic_containsPath_nodup", "content": "lemma acyclic_containsPath_nodup (R : Run S) (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_contains : R.containsPath path) :\n    path.Nodup"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton", "content": "lemma countTransitionInPath_append_singleton (path : List S) (x y : S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_not_in : (x, y) ∉ path.zip path.tail) :\n    countTransitionInPath (x, y) (path ++ [y]) = 1"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton_other", "content": "lemma countTransitionInPath_append_singleton_other (path : List S) (x y : S) (t : Transition S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_ne : t ≠ (x, y)) :\n    countTransitionInPath t (path ++ [y]) = countTransitionInPath t path"}, {"name": "Utils.StateTransition.acyclic_no_self_loop", "content": "lemma acyclic_no_self_loop (R : Run S) (s : S) (h_acyclic : R.isAcyclic) (h_edge : R (s, s) > 0) : False"}, {"name": "Utils.StateTransition.getLast_mem", "content": "lemma getLast_mem {α : Type*} (l : List α) (x : α) (h_last : l.getLast? = some x) :\n    x ∈ l"}, {"name": "Utils.StateTransition.last_not_in_zip_tail", "content": "lemma last_not_in_zip_tail {α : Type*} [DecidableEq α] (l : List α) (x : α)\n    (h_nodup : l.Nodup)\n    (h_last : l.getLast? = some x) :\n    ∀ y : α, (x, y) ∉ l.zip l.tail"}, {"name": "Utils.StateTransition.drop_of_lt_length_nonempty", "content": "lemma drop_of_lt_length_nonempty {α : Type*} (path : List α) (i : ℕ)\n    (h_i_lt : i < path.length) :\n    path.drop i ≠ []"}, {"name": "Utils.StateTransition.cycle_from_suffix_contains", "content": "lemma cycle_from_suffix_contains (R : Run S) (suffix : List S) (current y : S)\n    (h_suffix_nodup : suffix.Nodup)\n    (h_contains_suffix : R.containsPath suffix)\n    (h_suffix_nonempty : suffix ≠ [])\n    (h_suffix_last : suffix.getLast? = some current)\n    (h_edge : R (current, y) > 0) :\n    ∀ t : Transition S, countTransitionInPath t (suffix ++ [y]) ≤ R t"}, {"name": "Utils.StateTransition.path_with_back_edge_creates_cycle", "content": "lemma path_with_back_edge_creates_cycle (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_y_in_path : y ∈ path)\n    (h_edge : R (current, y) > 0) :\n    R.hasCycle"}, {"name": "Utils.StateTransition.acyclic_edge_not_in_path", "content": "lemma acyclic_edge_not_in_path (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_edge : R (current, y) > 0)\n    (h_y_in_path : y ∈ path) :\n    False"}, {"name": "Utils.StateTransition.not_mem_implies_transition_not_in_zip_tail", "content": "lemma not_mem_implies_transition_not_in_zip_tail {α : Type*} (path : List α) (x y : α)\n    (h_y_not_in : y ∉ path) :\n    (x, y) ∉ path.zip path.tail"}, {"name": "Utils.StateTransition.containsPath_append_singleton", "content": "lemma containsPath_append_singleton (R : Run S) (path : List S) (x y : S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_contains : R.containsPath path)\n    (h_y_not_in_path : y ∉ path)\n    (h_edge : R (x, y) > 0) :\n    R.containsPath (path ++ [y])"}], "local_ctx": "import Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Fintype.Basic\n\nimport Mathlib.Data.Fintype.Prod\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Piecewise\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nnamespace Utils.StateTransition\n\nvariable {S : Type*} [DecidableEq S] [Fintype S]\n\ndef Transition (S : Type*) := S × S\n\ndef Run (S : Type*) := Transition S → ℕ\n\ndef countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t\n\ndef Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t\n\ndef Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle\n\ndef Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle\n\ndef Run.reachable [DecidableEq S] (R : Run S) (start finish : S) : Prop :=\n  ∃ (path : List S), path.head? = some start ∧ path.getLast? = some finish ∧\n    path ≠ [] ∧ R.containsPath path\n\ndef Run.isLeaf (R : Run S) (root leaf : S) : Prop :=\n  R.reachable root leaf ∧ ∀ y, R (leaf, y) = 0", "target_theorem": "lemma acyclic_has_leaf_aux (R : Run S) (root current : S)\n    (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_start : path.head? = some root)\n    (h_end : path.getLast? = some current)\n    (h_nonempty : path ≠ [])\n    (h_contains : R.containsPath path)\n    (h_has_out : ∃ y, y ∉ path ∧ R (current, y) > 0) :\n    ∃ leaf, R.isLeaf root leaf :=", "ground_truth_proof": ":= by\n  -- Get a successor not in path\n  obtain ⟨y, h_y_not_in_path, h_edge⟩ := h_has_out\n\n  -- Check if y has any outgoing edges to states not in path ++ [y]\n  by_cases h_y_has_out : ∃ z, z ∉ path ∧ z ≠ y ∧ R (y, z) > 0\n  case neg =>\n    -- y has no outgoing edges to states not in path ++ [y]\n    -- We'll show y is actually a leaf (has NO outgoing edges at all)\n    use y\n    constructor\n    · -- Show y is reachable from root\n      -- Extend the path by adding y\n      use path ++ [y]\n      constructor\n      · aesop\n      constructor\n      · simp\n      constructor\n      · aesop\n      · exact containsPath_append_singleton R path current y h_nonempty h_end h_contains h_y_not_in_path h_edge\n    · -- Show y has no outgoing edges\n      intro z\n      by_contra h_pos\n      push_neg at h_y_has_out\n      -- If R(y,z) > 0, then by h_y_has_out, either z ∈ path or z = y\n      have h_z_pos : R (y, z) > 0 := by omega\n      have h_z_in_path_or_y : z ∈ path ∨ z = y := by grind\n\n      -- Derive contradiction from cycle\n      cases h_z_in_path_or_y with\n      | inl h_z_in_path =>\n        -- z ∈ path, so we can construct a cycle\n        -- path ends with current, current → y, y → z, and z ∈ path\n        -- This creates a cycle: (suffix of path from z to current) → y → z\n        have h_z_in_extended : z ∈ path ++ [y] := by aesop\n        exact acyclic_edge_not_in_path R (path ++ [y]) y z h_acyclic (by simp) (containsPath_append_singleton R path current y h_nonempty h_end h_contains h_y_not_in_path h_edge) h_z_pos h_z_in_extended\n      | inr h_z_eq_y =>\n        -- z = y, so we have a self-loop y → y\n        rw [h_z_eq_y] at h_z_pos\n        exact acyclic_no_self_loop R y h_acyclic h_z_pos\n\n  case pos =>\n    -- y has an outgoing edge to some z not in path and z ≠ y\n    -- Recurse with path ++ [y]\n    obtain ⟨z, h_z_not_in_path, h_z_ne_y, h_y_z_edge⟩ := h_y_has_out\n\n    -- Construct new path\n    let new_path := path ++ [y]\n\n    -- Show properties of new_path\n    have h_new_start : new_path.head? = some root := by aesop\n    have h_new_end : new_path.getLast? = some y := by aesop\n    have h_new_nonempty : new_path ≠ [] := by aesop\n    have h_new_contains : R.containsPath new_path :=\n      containsPath_append_singleton R path current y h_nonempty h_end h_contains h_y_not_in_path h_edge\n\n    have h_new_has_out : ∃ w, w ∉ new_path ∧ R (y, w) > 0 := by grind\n    exact acyclic_has_leaf_aux R root y new_path h_acyclic h_new_start h_new_end h_new_nonempty h_new_contains h_new_has_out\ntermination_by Fintype.card S - path.toFinset.card\ndecreasing_by\n  simp_wf\n  -- new_path = path ++ [y], and y ∉ path, so toFinset increases by 1\n  have h_y_not_mem : y ∉ path.toFinset := by\n    simp\n    exact h_y_not_in_path\n  have h_card_increase : (insert y path.toFinset).card = path.toFinset.card + 1 := by\n    rw [Finset.card_insert_of_notMem h_y_not_mem]\n  -- path.toFinset is a strict subset of univ\n  have h_path_subset := finset_ssubset_univ_of_not_mem path.toFinset y h_y_not_mem\n  have h_card_bound : path.toFinset.card < Fintype.card S := by\n    rw [← Finset.card_univ]\n    exact Finset.card_lt_card h_path_subset\n  rw [h_card_increase]\n  omega", "nesting_depth": 7, "transitive_dep_count": 97, "subset_aristotle": true, "category": "Applied verif."}
{"id": 136, "thm_name": "MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted", "thm_stmt": "theorem MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isTimestampSorted", "lean_root": "clean", "rel_path": "Clean/Utils/OfflineMemory.lean", "imports": ["import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Mathlib.Data.List.Sort", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Pairwise", "module": "Init.Data.List.Basic"}, {"name": "List.Sorted", "module": "Mathlib.Deprecated.Sort"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "List.Pairwise.of_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.Sorted.of_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.filter_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_filter", "module": "Init.Data.List.Lemmas"}, {"name": "List.sorted_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "decide_eq_true_eq", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MemoryAccess", "content": "def MemoryAccess := ℕ × ℕ × ℕ × ℕ"}, {"name": "MemoryAccessList", "content": "def MemoryAccessList := List MemoryAccess"}, {"name": "timestamp_ordering", "content": "abbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2"}, {"name": "MemoryAccessList.isTimestampSorted", "content": "def MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering"}, {"name": "MemoryAccessList.timestamps_neq", "content": "def MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y"}, {"name": "MemoryAccessList.Notimestampdup", "content": "def MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses"}, {"name": "address_timestamp_ordering", "content": "abbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2"}, {"name": "address_strict_timestamp_ordering", "content": "abbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2"}, {"name": "MemoryAccessList.isAddressTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering"}, {"name": "MemoryAccessList.isAddressStrictTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering"}, {"name": "MemoryAccessList.filterAddress", "content": "def MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)"}], "used_local_lemmas": [{"name": "MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup", "content": "theorem MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup\n    (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted)\n    (h_no_timestamp_dup : accesses.Notimestampdup) :\n    accesses.isAddressStrictTimestampSorted"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Utils.Field\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Tactics\n\nimport Mathlib.Data.List.Sort\n\ndef MemoryAccess := ℕ × ℕ × ℕ × ℕ \n\ndef MemoryAccessList := List MemoryAccess\n\nabbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2\n\ndef MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering\n\ndef MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y\n\ndef MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses\n\nabbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2\n\nabbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2\n\n@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering\n\n@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering\n\ndef MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)", "target_theorem": "theorem MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isTimestampSorted :=", "ground_truth_proof": ":= by\n  have h_strict := addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup accesses h_sorted h_nodup\n  simp only [isAddressStrictTimestampSorted, filterAddress, isTimestampSorted] at h_strict ⊢\n  induction accesses with\n  | nil => simp\n  | cons head tail ih =>\n    obtain ⟨t, a, r, w⟩ := head\n    simp only [List.filter_cons]\n    split_ifs with h_addr\n    · simp only [List.sorted_cons]\n      constructor\n      · intro z hz\n        simp only [List.mem_filter] at hz\n        obtain ⟨hz_mem, hz_addr⟩ := hz\n        obtain ⟨t_z, a_z, r_z, w_z⟩ := z\n        simp only [decide_eq_true_eq] at hz_addr\n        simp only [List.sorted_cons] at h_strict\n        have h_ord := h_strict.1 (t_z, a_z, r_z, w_z) hz_mem\n        simp only [address_strict_timestamp_ordering] at h_ord\n        simp only [decide_eq_true_eq] at h_addr\n        rw [hz_addr, h_addr] at h_ord\n        simp only [↓reduceIte] at h_ord\n        simp only [timestamp_ordering]\n        exact h_ord\n      · apply ih\n        · simp only [isAddressTimestampSorted] at h_sorted ⊢\n          exact List.Sorted.of_cons h_sorted\n        · simp only [Notimestampdup] at h_nodup ⊢\n          exact List.Pairwise.of_cons h_nodup\n        · exact List.Sorted.of_cons h_strict\n    · apply ih\n      · simp only [isAddressTimestampSorted] at h_sorted ⊢\n        exact List.Sorted.of_cons h_sorted\n      · simp only [Notimestampdup] at h_nodup ⊢\n        exact List.Pairwise.of_cons h_nodup\n      · exact List.Sorted.of_cons h_strict", "nesting_depth": 2, "transitive_dep_count": 23, "subset_aristotle": true, "category": "Applied verif."}
{"id": 137, "thm_name": "MemoryAccessList.isConsistentOnline_iff_isConsistentOffline", "thm_stmt": "theorem MemoryAccessList.isConsistentOnline_iff_isConsistentOffline\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted)\n    (h_nodup : accesses.Notimestampdup) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    ∃ permuted : AddressSortedMemoryAccessList,\n      permuted.val.Perm accesses ∧\n      MemoryAccessList.isConsistentOffline permuted.val permuted.property", "lean_root": "clean", "rel_path": "Clean/Utils/OfflineMemory.lean", "imports": ["import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Mathlib.Data.List.Sort", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Sorted", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.insertionSort", "module": "Mathlib.Data.List.Sort"}, {"name": "List.Pairwise", "module": "Init.Data.List.Basic"}, {"name": "List.filter", "module": "Init.Data.List.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "List.Perm", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.Sorted.of_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Pairwise.of_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.filter_eq_nil_iff", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_cons_self", "module": "Init.Data.List.Lemmas"}, {"name": "List.sorted_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "decide_eq_true_eq", "module": "Init.SimpLemmas"}, {"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "List.filter_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_filter", "module": "Init.Data.List.Lemmas"}, {"name": "decide_true", "module": "Init.Core"}, {"name": "decide_eq_false_iff_not", "module": "Init.SimpLemmas"}, {"name": "List.Sorted.filter", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.filter_nil", "module": "Init.Data.List.Basic"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "forall_true_left", "module": "Mathlib.Logic.Basic"}, {"name": "Bool.and_true", "module": "Init.SimpLemmas"}, {"name": "List.all_cons", "module": "Init.Data.List.Basic"}, {"name": "List.all_nil", "module": "Init.Data.List.Basic"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "imp_self", "module": "Init.Core"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "true_iff", "module": "Init.SimpLemmas"}, {"name": "List.filter_cons_of_pos", "module": "Init.Data.List.Lemmas"}, {"name": "Bool.not_or_self", "module": "Init.Data.Bool"}, {"name": "Bool.true_and", "module": "Init.SimpLemmas"}, {"name": "List.all_eq_true", "module": "Init.Data.List.Lemmas"}, {"name": "List.all_filter", "module": "Init.Data.List.Lemmas"}, {"name": "and_imp", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "List.Pairwise.perm", "module": "Init.Data.List.Perm"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "List.perm_insertionSort", "module": "Mathlib.Data.List.Sort"}, {"name": "List.sorted_insertionSort", "module": "Mathlib.Data.List.Sort"}, {"name": "List.Perm.filter", "module": "Init.Data.List.Perm"}, {"name": "List.eq_of_perm_of_sorted", "module": "Mathlib.Data.List.Sort"}, {"name": "List.Pairwise.imp", "module": "Init.Data.List.Pairwise"}, {"name": "List.Perm.nodup_iff", "module": "Init.Data.List.Perm"}, {"name": "List.Sorted.insertionSort_eq", "module": "Mathlib.Data.List.Sort"}, {"name": "List.Sorted.nodup", "module": "Mathlib.Data.List.Nodup"}, {"name": "List.perm_comm", "module": "Init.Data.List.Perm"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MemoryAccess", "content": "def MemoryAccess := ℕ × ℕ × ℕ × ℕ"}, {"name": "MemoryAccessList", "content": "def MemoryAccessList := List MemoryAccess"}, {"name": "timestamp_ordering", "content": "abbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2"}, {"name": "MemoryAccessList.isTimestampSorted", "content": "def MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering"}, {"name": "MemoryAccessList.timestamps_neq", "content": "def MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y"}, {"name": "MemoryAccessList.Notimestampdup", "content": "def MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses"}, {"name": "address_timestamp_ordering", "content": "abbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2"}, {"name": "address_strict_timestamp_ordering", "content": "abbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2"}, {"name": "MemoryAccessList.isAddressTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering"}, {"name": "MemoryAccessList.isAddressStrictTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering"}, {"name": "AddressSortedMemoryAccessList", "content": "def AddressSortedMemoryAccessList := {accesses : MemoryAccessList // accesses.isAddressTimestampSorted}"}, {"name": "MemoryAccessList.addressTimestampSort", "content": "def MemoryAccessList.addressTimestampSort (accesses : MemoryAccessList) : MemoryAccessList :=\n  List.insertionSort address_timestamp_ordering accesses"}, {"name": "MemoryAccessList.lastWriteValue", "content": "def MemoryAccessList.lastWriteValue (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) (addr : ℕ) : ℕ := match accesses with\n  \n  | [] => 0\n  | (_t, addr', _readValue, writeValue) :: rest =>\n    if addr' = addr then\n      \n      writeValue\n    else\n      MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr"}, {"name": "MemoryAccessList.isConsistentOnline", "content": "def MemoryAccessList.isConsistentOnline (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, addr, readValue, _writeValue) :: rest =>\n    \n    readValue = MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n    ∧ MemoryAccessList.isConsistentOnline rest (List.Sorted.of_cons h)\n\nexample : MemoryAccessList.isConsistentOnline [] (by admit /- proof elided -/\n) := by admit /- proof elided -/"}, {"name": "MemoryAccessList.filterAddress", "content": "def MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)"}, {"name": "MemoryAccessList.isConsistentSingleAddress", "content": "def MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}, {"name": "MemoryAccessList.isConsistentOffline", "content": "def MemoryAccessList.isConsistentOffline (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  | (_t2, addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    (if addr1 = addr2 then readValue2 = writeValue1 else readValue2 = 0) ∧\n    MemoryAccessList.isConsistentOffline ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}], "used_local_lemmas": [{"name": "MemoryAccessList.addressTimestampSort_sorted", "content": "theorem MemoryAccessList.addressTimestampSort_sorted (accesses : MemoryAccessList) :\n    (MemoryAccessList.addressTimestampSort accesses).Sorted address_timestamp_ordering"}, {"name": "MemoryAccessList.addressTimestampSort_perm", "content": "theorem MemoryAccessList.addressTimestampSort_perm (accesses : MemoryAccessList) :\n    (MemoryAccessList.addressTimestampSort accesses).Perm accesses"}, {"name": "MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup", "content": "theorem MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup\n    (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted)\n    (h_no_timestamp_dup : accesses.Notimestampdup) :\n    accesses.isAddressStrictTimestampSorted"}, {"name": "MemoryAccessList.noTimestampDup_perm", "content": "theorem MemoryAccessList.noTimestampDup_perm (l1 l2 : MemoryAccessList)\n    (h_l1_nodup : l1.Notimestampdup) (h_perm : l1.Perm l2) :\n    l2.Notimestampdup"}, {"name": "MemoryAccessList.noTimestampDup_of_cons", "content": "theorem MemoryAccessList.noTimestampDup_of_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h : Notimestampdup (head :: tail)) :\n    Notimestampdup tail"}, {"name": "MemoryAccessList.isAddressTimestampSorted_of_cons", "content": "theorem MemoryAccessList.isAddressTimestampSorted_of_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h : isAddressTimestampSorted (head :: tail)) :\n    isAddressTimestampSorted tail"}, {"name": "MemoryAccessList.noTimestampDup_of_TimestampSorted", "content": "theorem MemoryAccessList.noTimestampDup_of_TimestampSorted\n    (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) :\n    accesses.Notimestampdup"}, {"name": "MemoryAccessList.filterAddress_sorted", "content": "theorem MemoryAccessList.filterAddress_sorted (accesses : MemoryAccessList)\n    (h : accesses.isTimestampSorted) (addr : ℕ) :\n    (MemoryAccessList.filterAddress accesses addr).isTimestampSorted"}, {"name": "MemoryAccessList.filterAddress_cons", "content": "theorem MemoryAccessList.filterAddress_cons (head : MemoryAccess) (tail : MemoryAccessList) (addr : ℕ) :\n    MemoryAccessList.filterAddress (head :: tail) addr =\n    match head with\n    | ⟨_t, a, _r, _w⟩ => ((if a = addr then\n      (head :: (MemoryAccessList.filterAddress tail addr))\n        else (MemoryAccessList.filterAddress tail addr)))"}, {"name": "MemoryAccessList.isConsistentSingleAddress_iff", "content": "theorem MemoryAccessList.isConsistentSingleAddress_iff (accesses : MemoryAccessList) (addr : ℕ) (h_sorted : accesses.isTimestampSorted)\n    (h_eq : accesses.all (fun (_t, addr', _readValue, _writeValue) => addr' = addr)) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    MemoryAccessList.isConsistentSingleAddress accesses h_sorted"}, {"name": "MemoryAccessList.lastWriteValue_filter", "content": "theorem MemoryAccessList.lastWriteValue_filter (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted) (addr : ℕ) (h_sorted' : ((MemoryAccessList.filterAddress accesses addr).isTimestampSorted))  :\n    MemoryAccessList.lastWriteValue accesses h_sorted addr =\n    MemoryAccessList.lastWriteValue (MemoryAccessList.filterAddress accesses addr) h_sorted' addr"}, {"name": "MemoryAccessList.isConsistentOnline_filter_of_consistentOnline", "content": "theorem MemoryAccessList.isConsistentOnline_filter_of_consistentOnline (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted)\n    (h_consistent : MemoryAccessList.isConsistentOnline accesses h_sorted) (addr : ℕ) :\n    MemoryAccessList.isConsistentOnline (MemoryAccessList.filterAddress accesses addr) (MemoryAccessList.filterAddress_sorted accesses h_sorted addr)"}, {"name": "MemoryAccessList.isConsistentSingleAddress_cons", "content": "theorem MemoryAccessList.isConsistentSingleAddress_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail)) (h_sorted' : tail.isTimestampSorted)\n    (h : isConsistentSingleAddress (head :: tail) h_sorted) :\n    isConsistentSingleAddress tail h_sorted'"}, {"name": "MemoryAccessList.isConsistentSingleAddress_cons_forall", "content": "theorem MemoryAccessList.isConsistentSingleAddress_cons_forall (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail))\n    : (∀ addr : ℕ, (filterAddress (head :: tail) addr).isConsistentSingleAddress (MemoryAccessList.filterAddress_sorted (head :: tail) h_sorted addr)) →\n    (∀ addr : ℕ, isConsistentSingleAddress (filterAddress tail addr) (MemoryAccessList.filterAddress_sorted tail (by simp_all only [isTimestampSorted,\n      List.sorted_cons]) addr))"}, {"name": "MemoryAccessList.isConsistent_iff_all_single_address", "content": "theorem MemoryAccessList.isConsistent_iff_all_single_address (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    (∀ addr : ℕ, MemoryAccessList.isConsistentSingleAddress (MemoryAccessList.filterAddress accesses addr) (MemoryAccessList.filterAddress_sorted accesses h_sorted addr))"}, {"name": "MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted", "content": "theorem MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isTimestampSorted"}, {"name": "MemoryAccessList.isConsistentSingleAddress_filterAddress_forall_of_cons", "content": "theorem MemoryAccessList.isConsistentSingleAddress_filterAddress_forall_of_cons\n    (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: tail))\n    (h_nodup : Notimestampdup (head :: tail))\n    (h : ∀ addr, isConsistentSingleAddress (filterAddress (head :: tail) addr)\n         (filterAddress_sorted_from_addressTimestampSorted (head :: tail) h_sorted h_nodup addr)) :\n    ∀ addr, isConsistentSingleAddress (filterAddress tail addr)\n      (filterAddress_sorted_from_addressTimestampSorted tail\n        (isAddressTimestampSorted_of_cons head tail h_sorted)\n        (noTimestampDup_of_cons head tail h_nodup) addr)"}, {"name": "MemoryAccessList.filterAddress_empty_when_address_changes", "content": "theorem MemoryAccessList.filterAddress_empty_when_address_changes\n    (head : MemoryAccess) (second : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: second :: tail))\n    (h_addr_ne : head.2.1 ≠ second.2.1) :\n    filterAddress (second :: tail) head.2.1 = []"}, {"name": "MemoryAccessList.isConsistentOffline_of_cons", "content": "theorem MemoryAccessList.isConsistentOffline_of_cons\n    (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: tail))\n    (h_offline : isConsistentOffline (head :: tail) h_sorted) :\n    isConsistentOffline tail (isAddressTimestampSorted_of_cons head tail h_sorted)"}, {"name": "MemoryAccessList.isConsistentOffline_implies_single_address", "content": "theorem MemoryAccessList.isConsistentOffline_implies_single_address\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (h_offline : accesses.isConsistentOffline h_sorted)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isConsistentSingleAddress\n      (filterAddress_sorted_from_addressTimestampSorted accesses h_sorted h_nodup addr)"}, {"name": "MemoryAccessList.isConsistentOffline_iff_all_single_addresses", "content": "theorem MemoryAccessList.isConsistentOffline_iff_all_single_addresses (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) (h_nodup : accesses.Notimestampdup) :\n    MemoryAccessList.isConsistentOffline accesses h_sorted ↔\n    ∀ addr, MemoryAccessList.isConsistentSingleAddress (MemoryAccessList.filterAddress accesses addr) (filterAddress_sorted_from_addressTimestampSorted accesses h_sorted h_nodup addr)"}, {"name": "MemoryAccessList.addressTimestampSort_noTimestampDup", "content": "theorem MemoryAccessList.addressTimestampSort_noTimestampDup\n    (accesses : MemoryAccessList)\n    (h_nodup : accesses.Notimestampdup) :\n    accesses.addressTimestampSort.Notimestampdup"}, {"name": "MemoryAccessList.filterAddress_addressTimestampSort_eq", "content": "theorem MemoryAccessList.filterAddress_addressTimestampSort_eq\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isConsistentSingleAddress\n      (filterAddress_sorted accesses h_sorted addr) ↔\n    (accesses.addressTimestampSort.filterAddress addr).isConsistentSingleAddress\n      (filterAddress_sorted_from_addressTimestampSorted accesses.addressTimestampSort\n        (addressTimestampSort_sorted accesses)\n        (addressTimestampSort_noTimestampDup accesses h_nodup) addr)"}, {"name": "MemoryAccessList.isConsistentOnline_iff_sorted_isConsistentOffline", "content": "theorem MemoryAccessList.isConsistentOnline_iff_sorted_isConsistentOffline\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted)\n    (h_nodup : accesses.Notimestampdup) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    MemoryAccessList.isConsistentOffline (MemoryAccessList.addressTimestampSort accesses) (MemoryAccessList.addressTimestampSort_sorted accesses)"}, {"name": "MemoryAccessList.eq_of_perm_of_sorted", "content": "lemma MemoryAccessList.eq_of_perm_of_sorted {l1 l2 l3 : MemoryAccessList} (h_l1_sorted: l1.isTimestampSorted)\n    (h_l2_sorted : l2.isAddressTimestampSorted) (h_l3_sorted : l3.isAddressTimestampSorted)\n    (h_perm1 : l1.Perm l2) (h_perm2 : l1.Perm l3) : l2 = l3"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Utils.Field\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Tactics\n\nimport Mathlib.Data.List.Sort\n\ndef MemoryAccess := ℕ × ℕ × ℕ × ℕ \n\ndef MemoryAccessList := List MemoryAccess\n\nabbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2\n\ndef MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering\n\ndef MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y\n\ndef MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses\n\nabbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2\n\nabbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2\n\n@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering\n\n@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering\n\ndef AddressSortedMemoryAccessList := {accesses : MemoryAccessList // accesses.isAddressTimestampSorted}\n\ndef MemoryAccessList.addressTimestampSort (accesses : MemoryAccessList) : MemoryAccessList :=\n  List.insertionSort address_timestamp_ordering accesses\n\ndef MemoryAccessList.lastWriteValue (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) (addr : ℕ) : ℕ := match accesses with\n  \n  | [] => 0\n  | (_t, addr', _readValue, writeValue) :: rest =>\n    if addr' = addr then\n      \n      writeValue\n    else\n      MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n\ndef MemoryAccessList.isConsistentOnline (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, addr, readValue, _writeValue) :: rest =>\n    \n    readValue = MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n    ∧ MemoryAccessList.isConsistentOnline rest (List.Sorted.of_cons h)\n\nexample : MemoryAccessList.isConsistentOnline [] (by admit /- proof elided -/\n) := by admit /- proof elided -/\n\ndef MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)\n\ndef MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)\n\ndef MemoryAccessList.isConsistentOffline (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  | (_t2, addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    (if addr1 = addr2 then readValue2 = writeValue1 else readValue2 = 0) ∧\n    MemoryAccessList.isConsistentOffline ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)", "target_theorem": "theorem MemoryAccessList.isConsistentOnline_iff_isConsistentOffline\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted)\n    (h_nodup : accesses.Notimestampdup) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    ∃ permuted : AddressSortedMemoryAccessList,\n      permuted.val.Perm accesses ∧\n      MemoryAccessList.isConsistentOffline permuted.val permuted.property :=", "ground_truth_proof": ":= by\n  constructor\n  · intro h\n    use ⟨MemoryAccessList.addressTimestampSort accesses, MemoryAccessList.addressTimestampSort_sorted accesses⟩\n    constructor\n    · simp only\n      apply MemoryAccessList.addressTimestampSort_perm\n    · simp only\n      have h' := MemoryAccessList.isConsistentOnline_iff_sorted_isConsistentOffline accesses h_sorted h_nodup\n      rw [←h']\n      assumption\n  · rintro ⟨⟨permuted, h_permuted_sorted⟩, h_perm, h_offline⟩\n    simp_all only\n    rw [List.perm_comm] at h_perm\n    have h_eq := MemoryAccessList.eq_of_perm_of_sorted h_sorted h_permuted_sorted (MemoryAccessList.addressTimestampSort_sorted accesses)\n      h_perm (by rw [List.perm_comm]; apply MemoryAccessList.addressTimestampSort_perm)\n    simp only [h_eq] at h_offline\n    rw [←MemoryAccessList.isConsistentOnline_iff_sorted_isConsistentOffline accesses h_sorted h_nodup] at h_offline\n    assumption", "nesting_depth": 6, "transitive_dep_count": 94, "subset_aristotle": true, "category": "Applied verif."}
{"id": 138, "thm_name": "Utils.StateTransition.path_with_back_edge_creates_cycle", "thm_stmt": "lemma path_with_back_edge_creates_cycle (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_y_in_path : y ∈ path)\n    (h_edge : R (current, y) > 0) :\n    R.hasCycle", "lean_root": "clean", "rel_path": "Clean/Utils/SourceSinkPath.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import Mathlib.Data.Fintype.Prod", "import Mathlib.Data.List.Basic", "import Mathlib.Algebra.BigOperators.Group.Finset.Basic", "import Mathlib.Data.Finset.Basic", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise", "import Mathlib.Data.Fintype.Basic"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Sublist", "module": "Init.Data.List.Basic"}, {"name": "List.count", "module": "Init.Data.List.Basic"}, {"name": "List.drop", "module": "Init.Data.List.Basic"}, {"name": "List.tail", "module": "Init.Data.List.Basic"}, {"name": "List.take", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.cons₂", "module": "Init.Data.List.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.reduceLeDiff", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.drop_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.tail_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zip_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.count_le", "module": "Init.Data.List.Count"}, {"name": "List.tail_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.take_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.length_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.length_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "Nat.min_eq_left", "module": "Init.Data.Nat.MinMax"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "List.getElem?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getElem?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getLast?_eq_getElem?", "module": "Init.Data.List.Lemmas"}, {"name": "List.head?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.head?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "List.duplicate_iff_exists_distinct_get", "module": "Mathlib.Data.List.NodupEquivFin"}, {"name": "List.exists_duplicate_iff_not_nodup", "module": "Mathlib.Data.List.Duplicate"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.count_cons", "module": "Init.Data.List.Count"}, {"name": "List.exists_cons_of_ne_nil", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast?_cons_cons", "module": "Init.Data.List.Lemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "List.countP_nil", "module": "Init.Data.List.Count"}, {"name": "List.countP_singleton", "module": "Init.Data.List.Count"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "List.zip_nil_right", "module": "Init.Data.List.Basic"}, {"name": "Nat.add_right_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "List.getLast?_eq_getLast", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast_mem", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.nodup_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.mem_iff_getElem", "module": "Init.Data.List.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Utils.StateTransition.Transition", "content": "def Transition (S : Type*) := S × S"}, {"name": "Utils.StateTransition.Run", "content": "def Run (S : Type*) := Transition S → ℕ"}, {"name": "Utils.StateTransition.countTransitionInPath", "content": "def countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t"}, {"name": "Utils.StateTransition.Run.containsPath", "content": "def Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t"}, {"name": "Utils.StateTransition.Run.hasCycle", "content": "def Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle"}, {"name": "Utils.StateTransition.Run.isAcyclic", "content": "def Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle"}], "used_local_lemmas": [{"name": "Utils.StateTransition.zip_drop_sublist", "content": "lemma zip_drop_sublist (l : List S) (n : ℕ) :\n    ((l.drop n).zip (l.drop (n + 1))).Sublist (l.zip l.tail)"}, {"name": "Utils.StateTransition.containsPath_drop", "content": "lemma containsPath_drop (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.drop n)"}, {"name": "Utils.StateTransition.zip_take_sublist", "content": "lemma zip_take_sublist (l1 l2 : List S) (n m : ℕ) :\n    ((l1.take n).zip (l2.take m)).Sublist (l1.zip l2)"}, {"name": "Utils.StateTransition.tail_take", "content": "lemma tail_take {α : Type*} (l : List α) (n : ℕ) :\n    (l.take n).tail = (l.tail).take (n - 1)"}, {"name": "Utils.StateTransition.containsPath_take", "content": "lemma containsPath_take (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.take n)"}, {"name": "Utils.StateTransition.drop_take_length_ge_two", "content": "lemma drop_take_length_ge_two {α : Type*} (path : List α) (n m : Fin path.length)\n    (h_n_lt_m : n < m) :\n    ((path.drop n.val).take (m.val - n.val + 1)).length ≥ 2"}, {"name": "Utils.StateTransition.getLast_drop_take", "content": "lemma getLast_drop_take {α : Type*} (path : List α) (n k : ℕ)\n    (h_n_lt : n < path.length)\n    (h_bound : n + k ≤ path.length)\n    (h_k_pos : k > 0) :\n    ((path.drop n).take k).getLast? = path[n + k - 1]?"}, {"name": "Utils.StateTransition.drop_take_cycle_same_endpoints", "content": "lemma drop_take_cycle_same_endpoints (path : List S) (x : S) (n m : Fin path.length)\n    (h_n_lt_m : n < m)\n    (h_x_at_n : path[n] = x)\n    (h_x_at_m : path[m] = x) :\n    ((path.drop n.val).take (m.val - n.val + 1)).head? =\n    ((path.drop n.val).take (m.val - n.val + 1)).getLast?"}, {"name": "Utils.StateTransition.containsPath_drop_take", "content": "lemma containsPath_drop_take (R : Run S) (path : List S) (n m : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath ((path.drop n).take m)"}, {"name": "Utils.StateTransition.acyclic_containsPath_nodup", "content": "lemma acyclic_containsPath_nodup (R : Run S) (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_contains : R.containsPath path) :\n    path.Nodup"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton", "content": "lemma countTransitionInPath_append_singleton (path : List S) (x y : S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_not_in : (x, y) ∉ path.zip path.tail) :\n    countTransitionInPath (x, y) (path ++ [y]) = 1"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton_other", "content": "lemma countTransitionInPath_append_singleton_other (path : List S) (x y : S) (t : Transition S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_ne : t ≠ (x, y)) :\n    countTransitionInPath t (path ++ [y]) = countTransitionInPath t path"}, {"name": "Utils.StateTransition.getLast_mem", "content": "lemma getLast_mem {α : Type*} (l : List α) (x : α) (h_last : l.getLast? = some x) :\n    x ∈ l"}, {"name": "Utils.StateTransition.last_not_in_zip_tail", "content": "lemma last_not_in_zip_tail {α : Type*} [DecidableEq α] (l : List α) (x : α)\n    (h_nodup : l.Nodup)\n    (h_last : l.getLast? = some x) :\n    ∀ y : α, (x, y) ∉ l.zip l.tail"}, {"name": "Utils.StateTransition.drop_of_lt_length_nonempty", "content": "lemma drop_of_lt_length_nonempty {α : Type*} (path : List α) (i : ℕ)\n    (h_i_lt : i < path.length) :\n    path.drop i ≠ []"}, {"name": "Utils.StateTransition.cycle_from_suffix_contains", "content": "lemma cycle_from_suffix_contains (R : Run S) (suffix : List S) (current y : S)\n    (h_suffix_nodup : suffix.Nodup)\n    (h_contains_suffix : R.containsPath suffix)\n    (h_suffix_nonempty : suffix ≠ [])\n    (h_suffix_last : suffix.getLast? = some current)\n    (h_edge : R (current, y) > 0) :\n    ∀ t : Transition S, countTransitionInPath t (suffix ++ [y]) ≤ R t"}], "local_ctx": "import Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Fintype.Basic\n\nimport Mathlib.Data.Fintype.Prod\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Piecewise\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nnamespace Utils.StateTransition\n\nvariable {S : Type*} [DecidableEq S] [Fintype S]\n\ndef Transition (S : Type*) := S × S\n\ndef Run (S : Type*) := Transition S → ℕ\n\ndef countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t\n\ndef Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t\n\ndef Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle\n\ndef Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle", "target_theorem": "lemma path_with_back_edge_creates_cycle (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_y_in_path : y ∈ path)\n    (h_edge : R (current, y) > 0) :\n    R.hasCycle :=", "ground_truth_proof": ":= by\n  -- Find y's position in path\n  rw [List.mem_iff_getElem] at h_y_in_path\n  obtain ⟨i, h_i_lt, h_y_eq⟩ := h_y_in_path\n  let suffix := path.drop i\n  have h_suffix_nonempty : suffix ≠ [] := drop_of_lt_length_nonempty path i h_i_lt\n  have h_suffix_head : suffix.head? = some y := by aesop\n  have h_suffix_last : suffix.getLast? = some current := by grind\n  let cycle := suffix ++ [y]\n  have h_cycle_head : cycle.head? = some y := by grind\n  have h_cycle_last : cycle.getLast? = some y := by grind\n  have h_cycle_len : cycle.length ≥ 2 := by grind\n  use cycle\n  constructor\n  · exact h_cycle_len\n  constructor\n  · rw [h_cycle_head, h_cycle_last]\n  · unfold cycle\n    have h_suffix_contains : R.containsPath suffix := by\n      unfold suffix\n      exact containsPath_drop R path i h_contains\n    have h_path_nodup := acyclic_containsPath_nodup R path h_acyclic h_contains\n    have h_suffix_nodup : suffix.Nodup := by grind\n    exact cycle_from_suffix_contains R suffix current y h_suffix_nodup h_suffix_contains h_suffix_nonempty h_suffix_last h_edge", "nesting_depth": 5, "transitive_dep_count": 71, "subset_aristotle": true, "category": "Applied verif."}
{"id": 139, "thm_name": "Vector.flatten_toChunks", "thm_stmt": "theorem flatten_toChunks {α : Type} (m : ℕ+) (v : Vector (Vector α m) n) :\n    v.flatten.toChunks m = v", "lean_root": "clean", "rel_path": "Clean/Utils/Vector.lean", "imports": ["import Mathlib.Combinatorics.Enumerative.Composition", "import Mathlib.Analysis.Normed.Ring.Lemmas", "import Init.Data.List.Find"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector.mk", "module": "Init.Data.Vector.Basic"}, {"name": "Composition", "module": "Mathlib.Combinatorics.Enumerative.Composition"}, {"name": "Composition.blocks", "module": "Mathlib.Combinatorics.Enumerative.Composition"}, {"name": "List.replicate", "module": "Init.Data.List.Basic"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "Vector.attachWith", "module": "Init.Data.Vector.Attach"}, {"name": "Vector.map", "module": "Init.Data.Vector.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "Array.toList", "module": "Init.Prelude"}, {"name": "Vector.toArray", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Array.length_toList", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.toArray_toList", "module": "Init.Data.Array.Basic"}, {"name": "Array.toList_flatten", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.toList_inj", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.toList_map", "module": "Init.Data.Array.Lemmas"}, {"name": "Function.comp_apply", "module": "Init.Core"}, {"name": "List.length_flatten", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_attachWith", "module": "Init.Data.List.Attach"}, {"name": "List.map_attach_eq_pmap", "module": "Init.Data.List.Attach"}, {"name": "List.map_cons", "module": "Init.Data.List.Basic"}, {"name": "List.map_id_fun'", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_map", "module": "Init.Data.List.Lemmas"}, {"name": "List.pmap_eq_map", "module": "Init.Data.List.Attach"}, {"name": "List.pmap_map", "module": "Init.Data.List.Attach"}, {"name": "List.replicate_succ", "module": "Init.Data.List.Basic"}, {"name": "Vector.flatten_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.length_toList", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.mk_toArray", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.size_toArray", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.toArray_inj", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.toList_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "id_eq", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Vector.cons", "content": "def cons (a : α) (v : Vector α n) : Vector α (n + 1) :=\n  ⟨ .mk (a :: v.toList), by admit /- proof elided -/\n  ⟩"}, {"name": "Vector.induct", "content": "def induct {motive : {n : ℕ} → Vector α n → Sort u}\n    (nil : motive #v[])\n    (cons: ∀ {n : ℕ} (a : α) (as : Vector α n), motive as → motive (cons a as))\n    {n : ℕ} (v : Vector α n) : motive v :=\n  match v with\n  | ⟨ .mk [], h ⟩ => by admit /- proof elided -/\n  | ⟨ .mk (a :: as), h ⟩ => by admit /- proof elided -/"}, {"name": "Composition.ofProductLength", "content": "def Composition.ofProductLength (m : ℕ+) {α : Type} {l : List α} (hl : l.length = n * m.val) : Composition l.length := {\n  blocks := List.replicate n m.val\n  blocks_pos hi := (List.mem_replicate.mp hi).right ▸ m.pos\n  blocks_sum := hl ▸ List.sum_replicate_nat\n}"}, {"name": "Vector.toChunks", "content": "def toChunks (m : ℕ+) {α : Type} (v : Vector α (n*m)) : Vector (Vector α m) n :=\n  let comp := Composition.ofProductLength m v.length_toList\n  let list : List (Vector α m) := v.toList.splitWrtComposition comp\n    |>.attachWith (List.length · = m) (comp.ofProductLength_mem_length rfl)\n    |>.map fun ⟨ l, hl ⟩ => .mk (.mk l) hl\n  .mk (.mk list) (by admit /- proof elided -/\n  )"}], "used_local_lemmas": [{"name": "Vector.toList_cons", "content": "theorem toList_cons {a : α} {v : Vector α n} : (cons a v).toList = a :: v.toList"}], "local_ctx": "import Mathlib.Analysis.Normed.Ring.Lemmas\n\nimport Mathlib.Combinatorics.Enumerative.Composition\n\nimport Init.Data.List.Find\n\nvariable {α β : Type} {n m : ℕ}\n\nopen Vector (finRange)\n\nnamespace Vector\n\ndef cons (a : α) (v : Vector α n) : Vector α (n + 1) :=\n  ⟨ .mk (a :: v.toList), by admit /- proof elided -/\n  ⟩\n\ndef induct {motive : {n : ℕ} → Vector α n → Sort u}\n    (nil : motive #v[])\n    (cons: ∀ {n : ℕ} (a : α) (as : Vector α n), motive as → motive (cons a as))\n    {n : ℕ} (v : Vector α n) : motive v :=\n  match v with\n  | ⟨ .mk [], h ⟩ => by admit /- proof elided -/\n  | ⟨ .mk (a :: as), h ⟩ => by admit /- proof elided -/\n\nend Vector\n\ndef Composition.ofProductLength (m : ℕ+) {α : Type} {l : List α} (hl : l.length = n * m.val) : Composition l.length := {\n  blocks := List.replicate n m.val\n  blocks_pos hi := (List.mem_replicate.mp hi).right ▸ m.pos\n  blocks_sum := hl ▸ List.sum_replicate_nat\n}\n\nnamespace Vector\n\ndef toChunks (m : ℕ+) {α : Type} (v : Vector α (n*m)) : Vector (Vector α m) n :=\n  let comp := Composition.ofProductLength m v.length_toList\n  let list : List (Vector α m) := v.toList.splitWrtComposition comp\n    |>.attachWith (List.length · = m) (comp.ofProductLength_mem_length rfl)\n    |>.map fun ⟨ l, hl ⟩ => .mk (.mk l) hl\n  .mk (.mk list) (by admit /- proof elided -/\n  )", "target_theorem": "theorem flatten_toChunks {α : Type} (m : ℕ+) (v : Vector (Vector α m) n) :\n    v.flatten.toChunks m = v :=", "ground_truth_proof": ":= by\n  simp only [toChunks]\n  rw [←Vector.toArray_inj,←Array.toList_inj]\n  simp only\n  let v_list_list := v.toList.map (Array.toList ∘ toArray)\n  have h_flatten : v.flatten.toList = v_list_list.flatten := by\n    rw [Vector.flatten_mk, Vector.toList_mk, Array.toList_flatten, Array.toList_map, List.map_map]\n    congr\n  have h_length : v.flatten.toList.length = n * ↑m := by rw [length_toList]\n  have h_flatten_length : v_list_list.flatten.length = n * ↑m := by rw [←h_flatten, h_length]\n  have h' : (v.flatten.toList.splitWrtComposition (Composition.ofProductLength m h_length)) = v_list_list := by\n    rw [← v_list_list.splitWrtComposition_flatten (Composition.ofProductLength m h_flatten_length)]\n    congr 1\n    · rw [h_length, h_flatten_length]\n    congr\n    · simp [h_flatten]\n    simp only [List.map_map, Composition.ofProductLength, v_list_list]\n    clear *-\n    induction v using Vector.induct\n    case nil => rfl\n    case cons xs x hi => rw [List.replicate_succ, Vector.toList_cons, List.map_cons, hi,\n      Function.comp_apply, Function.comp_apply, Array.length_toList, size_toArray]\n  simp_all only [List.length_flatten, List.map_map, List.map_attachWith, v_list_list]\n  rw [List.map_attach_eq_pmap, List.pmap_map]\n  simp only [Function.comp_apply, Array.toArray_toList, mk_toArray, List.pmap_eq_map,\n    List.map_id_fun', id_eq]\n  congr", "nesting_depth": 3, "transitive_dep_count": 39, "subset_aristotle": true, "category": "Applied verif."}
{"id": 140, "thm_name": "Tables.Fibonacci32.fib_constraints", "thm_stmt": "lemma fib_constraints (curr next : Row (F p) RowType) (aux_env : Environment (F p))\n  : recursiveRelation.ConstraintsHoldOnWindow ⟨<+> +> curr +> next, rfl⟩ aux_env →\n  curr.y = next.x ∧\n  (curr.x.Normalized → curr.y.Normalized → next.y.value = (curr.x.value + curr.y.value) % 2^32 ∧ next.y.Normalized)", "lean_root": "clean", "rel_path": "Clean/Tables/Fibonacci32.lean", "imports": ["import Clean.Types.U32", "import Clean.Utils.Vector", "import Clean.Gadgets.Equality", "import Clean.Gadgets.Addition32.Addition32", "import Clean.Circuit.Basic", "import Clean.Table.Theorems"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "StateM", "module": "Init.Control.State"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "Vector.mk", "module": "Init.Data.Vector.Basic"}, {"name": "modify", "module": "Init.Prelude"}, {"name": "Vector.finRange", "module": "Init.Data.Vector.FinRange"}, {"name": "Vector.map", "module": "Init.Data.Vector.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Vector.push", "module": "Init.Data.Vector.Basic"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "Nat.reduceAdd", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Fin.isValue", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "Nat.reduceLT", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "PNat", "module": "Mathlib.Data.PNat.Notation"}, {"name": "Vector.get", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.instGetElemNatLt", "module": "Init.Data.Vector.Basic"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}], "used_repo_defs": [{"name": "syntax \"let \" ident \" <== \" term : doElem", "content": "syntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem\n\nsyntax \"infer_constant_length\" : tactic"}, {"name": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty", "content": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|simp_assign_row) =>\n    `(tactic|(\n    simp only [assignCurrRow, assignNextRow, size]\n    rw [List.finRange, List.ofFn]\n    repeat rw [Fin.foldr_succ]\n    rw [Fin.foldr_zero]\n    repeat rw [List.forM_cons]\n    rw [List.forM_nil, bind_pure_unit]\n    simp only [seval, toVars, toElements, Fin.cast_eq_self, Fin.val_zero, Fin.val_one, Fin.isValue,\n      List.getElem_toArray, List.getElem_cons_zero, List.getElem_cons_succ, Fin.succ_zero_eq_one]))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "U32", "content": "structure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq"}, {"name": "CellOffset", "content": "structure CellOffset (W : ℕ+) (S : Type → Type) [ProvableType S]  where\n  row: Fin W\n  column: Fin (size S)"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "TwoRowsConstraint", "content": "@[reducible]\ndef TwoRowsConstraint (S : Type → Type) (F : Type) [Field F] [ProvableType S] := TableConstraint 2 S F Unit"}, {"name": "TableConstraint", "content": "@[reducible, table_norm, table_assignment_norm]\ndef TableConstraint (W : ℕ+) (S : Type → Type) (F : Type) [Field F] [ProvableType S] :=\n  StateM (TableContext W S F)"}, {"name": "TableContext", "content": "structure TableContext (W : ℕ+) (S : Type → Type) (F : Type) [Field F] [ProvableType S] where\n  circuit : Operations F\n  assignment : CellAssignment W S\nderiving Repr"}, {"name": "CellAssignment", "content": "structure CellAssignment (W : ℕ+) (S : Type → Type) [ProvableType S] where\n  offset : ℕ \n  aux_length : ℕ \n\n   \n  vars : Vector (Cell W S) offset"}, {"name": "offset", "content": "@[reducible, table_norm, table_assignment_norm]\ndef offset (table : TableContext W S F) : ℕ := table.circuit.localLength"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "Cell", "content": "inductive Cell (W : ℕ+) (S : Type → Type) [ProvableType S] where\n  | input : CellOffset W S → Cell W S\n  | aux : ℕ → Cell W S"}, {"name": "assign", "content": "@[table_norm, table_assignment_norm]\ndef assign (off : CellOffset W S) : Expression F → TableConstraint W S F Unit\n  \n  | .var v => assignVar off v\n  \n  | x => do\n    let new_var ← witnessVar x.eval\n    assertZero (x - var new_var)\n    assignVar off new_var"}, {"name": "assignVar", "content": "@[table_norm, table_assignment_norm]\ndef assignVar (off : CellOffset W S) (v : Variable F) : TableConstraint W S F Unit :=\n  modify fun ctx =>\n    let assignment := ctx.assignment.setVarInput off v.index\n    { ctx with assignment }"}, {"name": "setVarInput", "content": "@[table_assignment_norm]\ndef setVarInput (assignment : CellAssignment W S) (off : CellOffset W S) (var : ℕ) : CellAssignment W S :=\n  let vars := assignment.vars.set? var (.input off)\n  \n  \n  { assignment with vars }"}, {"name": "set?", "content": "def set? (v : Vector α n) (i : ℕ) (a : α) : Vector α n :=\n  ⟨ .mk <| v.toList.set i a, by admit /- proof elided -/\n  ⟩"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "witnessVar", "content": "@[circuit_norm]\ndef witnessVar (compute : Environment F → F) : Circuit F (Variable F) :=\n  fun (offset : ℕ) =>\n    let var : Variable F := ⟨ offset ⟩\n    (var, [.witness 1 fun env => #v[compute env]])"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) (var : α (Expression F)) : α F :=\n  toComponents var |> go (components α) |> fromComponents\nwhere"}, {"name": "ProvableStruct", "content": "class ProvableStruct (α : TypeMap) where\n  components : List WithProvableType\n  toComponents {F : Type} : α F → ProvableTypeList F components\n  fromComponents {F : Type} : ProvableTypeList F components → α F\n\n  combinedSize : ℕ := combinedSize' components\n  combinedSize_eq : combinedSize = combinedSize' components := by admit /- proof elided -/"}, {"name": "assertZero", "content": "@[circuit_norm]\ndef assertZero (e : Expression F) : Circuit F Unit := fun _ =>\n  ((), [.assert e])"}, {"name": "getCurrRow", "content": "@[table_norm, table_assignment_norm]\ndef getCurrRow : TableConstraint W S F (Var S F) := getRow 0"}, {"name": "getRow", "content": "@[table_norm, table_assignment_norm]\ndef getRow (row : Fin W) : TableConstraint W S F (Var S F) :=\n  modifyGet fun ctx =>\n    let ctx' : TableContext W S F := {\n      circuit := ctx.circuit ++ [.witness (size S) fun env => .mapRange _ fun i => env.get (ctx.offset + i)],\n      assignment := ctx.assignment.pushRow row\n    }\n    (varFromOffset S ctx.offset, ctx')"}, {"name": "Row.get", "content": "@[table_norm, table_assignment_norm]\ndef Row.get (row : Row F S) (i : Fin (size S)) : F :=\n  (toElements row)[i.val]"}, {"name": "pushRow", "content": "@[table_assignment_norm]\ndef pushRow (assignment : CellAssignment W S) (row : Fin W) : CellAssignment W S :=\n  let row_vars : Vector (Cell W S) (size S) := .mapFinRange _ fun col => .input ⟨ row, col ⟩\n  {\n    offset := assignment.offset + size S\n    aux_length := assignment.aux_length\n    vars := assignment.vars ++ row_vars\n  }"}, {"name": "mapFinRange", "content": "def mapFinRange (n : ℕ) (create : Fin n → α) : Vector α n := finRange n |>.map create"}, {"name": "get", "content": "@[table_assignment_norm]\ndef get {M : ℕ} :\n    (env : TraceOfLength F S M) → (i : Fin M) → (j : Fin (size S)) → F\n  | ⟨env, h⟩, i, j => env.getLeFromBottom ⟨\n      M - 1 - i,\n      by admit /- proof elided -/\n    ⟩ j"}, {"name": "TraceOfLength", "content": "def TraceOfLength (F : Type) (S : Type → Type) [ProvableType S] (N : ℕ) : Type :=\n  { env : Trace F S // env.len = N }"}, {"name": "Trace", "content": "inductive Trace (F : Type) (S : Type → Type) [ProvableType S] where\n   \n  | empty : Trace F S\n   \n  | cons (rest : Trace F S) (row : Row F S) : Trace F S"}, {"name": "empty", "content": "@[reducible, table_norm, table_assignment_norm]\ndef empty : TableContext W S F where\n  circuit := []\n  assignment := .empty W"}, {"name": "Row", "content": "@[reducible]\ndef Row (F : Type) (S : Type → Type) [ProvableType S] := S F"}, {"name": "mapRange", "content": "def mapRange (n : ℕ) (create : ℕ → α) : Vector α n :=\n  match n with\n  | 0 => #v[]\n  | k + 1 => mapRange k create |>.push (create k)"}, {"name": "varFromOffset", "content": "@[explicit_provable_type]\ndef varFromOffset (α : TypeMap) [ProvableType α] (offset : ℕ) : Var α F :=\n  let vars := Vector.mapRange (size α) fun i => var ⟨offset + i⟩\n  fromVars vars"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "getNextRow", "content": "@[table_norm, table_assignment_norm]\ndef getNextRow : TableConstraint W S F (Var S F) := getRow 1"}, {"name": "HasAssertEq", "content": "class HasAssertEq (β : Type) (F : outParam Type) [Field F] where\n  assert_eq : β → β → Circuit F Unit"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) Inputs U32 where\n  Assumptions\n  Spec\n  soundness\n  completeness"}, {"name": "Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.Normalized ∧ y.Normalized"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  x: U32 F\n  y: U32 F"}, {"name": "Normalized", "content": "def Normalized (x : U32 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256"}, {"name": "Spec", "content": "def Spec (input : Inputs (F p)) (z : U32 (F p)) :=\n  let ⟨x, y⟩ := input\n  z.value = (x.value + y.value) % 2^32 ∧ z.Normalized"}, {"name": "value", "content": "def value (x : U32 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "RowType", "content": "structure RowType (F : Type) where\n  x: F\n  y: F"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  x: F\n  y: F\n  carryIn: F"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  x: U32 F\n  y: U32 F\n  carryIn: F"}, {"name": "windowEnv", "content": "def windowEnv (table : TableConstraint W S F Unit)\n  (window : TraceOfLength F S W) (aux_env : Environment F) : Environment F :=\n  let assignment := table.finalAssignment\n  .mk fun i =>\n    if hi : i < assignment.offset then\n      match assignment.vars[i] with\n      | .input ⟨i, j⟩ => window.get i j\n      | .aux k => aux_env.get k\n    else aux_env.get (i + assignment.aux_length)"}, {"name": "finalAssignment", "content": "@[table_assignment_norm]\ndef finalAssignment (table : TableConstraint W S F α) : CellAssignment W S :=\n  table .empty |>.snd.assignment"}, {"name": "infix:50 \" === \" => HasAssertEq.assert_eq", "content": "infix:50 \" === \" => HasAssertEq.assert_eq"}, {"name": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty", "content": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty"}, {"name": "@[inherit_doc] infixl:67 \" +> \" => Trace.cons", "content": "@[inherit_doc] infixl:67 \" +> \" => Trace.cons"}], "lib_lemmas": [{"name": "Fin.cast_mk", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.getElem_cons_succ", "module": "Init.GetElem"}, {"name": "List.getElem_cons_zero", "module": "Init.GetElem"}, {"name": "List.getElem_toArray", "module": "Init.Data.Array.Basic"}, {"name": "PNat.val_ofNat", "module": "Mathlib.Data.PNat.Basic"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "mapRange_zero", "content": "theorem mapRange_zero {create : ℕ → α} : mapRange 0 create = #v[]"}, {"name": "mapFinRange_succ", "content": "theorem mapFinRange_succ {n : ℕ} {create : Fin (n + 1) → α} :\n    mapFinRange (n + 1) create = (mapFinRange n (fun i => create i.castSucc)).push (create (.last n))"}, {"name": "mapRange_succ", "content": "theorem mapRange_succ {n} {create : ℕ → α} :\n    mapRange (n + 1) create = (mapRange n create).push (create n)"}, {"name": "mapFinRange_zero", "content": "theorem mapFinRange_zero {create : Fin 0 → α} : mapFinRange 0 create = #v[]"}], "used_local_defs": [{"name": "Tables.Fibonacci32.RowType", "content": "structure RowType (F : Type) where\n  x: U32 F\n  y: U32 F"}, {"name": "Tables.Fibonacci32.nextRowOff", "content": "@[reducible]\ndef nextRowOff : RowType (CellOffset 2 RowType) := {\n  x := ⟨.next 0, .next 1, .next 2, .next 3⟩,\n  y := ⟨.next 4, .next 5, .next 6, .next 7⟩\n}"}, {"name": "Tables.Fibonacci32.assignU32", "content": "def assignU32 (offs : U32 (CellOffset 2 RowType)) (x : Var U32 (F p)) : TwoRowsConstraint RowType (F p) := do\n  assign offs.x0 x.x0\n  assign offs.x1 x.x1\n  assign offs.x2 x.x2\n  assign offs.x3 x.x3"}, {"name": "Tables.Fibonacci32.recursiveRelation", "content": "def recursiveRelation : TwoRowsConstraint RowType (F p) := do\n  let curr ← TableConstraint.getCurrRow\n  let next ← TableConstraint.getNextRow\n\n  let z ← Gadgets.Addition32.circuit { x := curr.x, y := curr.y }\n\n  assignU32 nextRowOff.y z\n  curr.y === next.x"}], "used_local_lemmas": [{"name": "Tables.Fibonacci32.fib_assignment", "content": "lemma fib_assignment : (recursiveRelation (p:=p)).finalAssignment.vars =\n   #v[.input ⟨0, 0⟩, .input ⟨0, 1⟩, .input ⟨0, 2⟩, .input ⟨0, 3⟩, .input ⟨0, 4⟩, .input ⟨0, 5⟩, .input ⟨0, 6⟩,\n      .input ⟨0, 7⟩, .input ⟨1, 0⟩, .input ⟨1, 1⟩, .input ⟨1, 2⟩, .input ⟨1, 3⟩, .input ⟨1, 4⟩, .input ⟨1, 5⟩,\n      .input ⟨1, 6⟩, .input ⟨1, 7⟩, .input ⟨1, 4⟩, .aux 1, .input ⟨1, 5⟩, .aux 3, .input ⟨1, 6⟩, .aux 5,\n      .input ⟨1, 7⟩, .aux 7]"}, {"name": "Tables.Fibonacci32.fib_vars", "content": "lemma fib_vars (curr next : Row (F p) RowType) (aux_env : Environment (F p)) :\n    let env := recursiveRelation.windowEnv ⟨<+> +> curr +> next, rfl⟩ aux_env;\n    eval env (varFromOffset U32 0) = curr.x ∧\n    eval env (varFromOffset U32 4) = curr.y ∧\n    eval env (varFromOffset U32 8) = next.x ∧\n    eval env (U32.mk (var ⟨16⟩) (var ⟨18⟩) (var ⟨20⟩) (var ⟨22⟩)) = next.y"}], "local_ctx": "import Clean.Utils.Vector\n\nimport Clean.Circuit.Basic\n\nimport Clean.Table.Theorems\n\nimport Clean.Gadgets.Addition32.Addition32\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Types.U32\n\nnamespace Tables.Fibonacci32\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nstructure RowType (F : Type) where\n  x: U32 F\n  y: U32 F\n\n@[reducible]\ndef nextRowOff : RowType (CellOffset 2 RowType) := {\n  x := ⟨.next 0, .next 1, .next 2, .next 3⟩,\n  y := ⟨.next 4, .next 5, .next 6, .next 7⟩\n}\n\ndef assignU32 (offs : U32 (CellOffset 2 RowType)) (x : Var U32 (F p)) : TwoRowsConstraint RowType (F p) := do\n  assign offs.x0 x.x0\n  assign offs.x1 x.x1\n  assign offs.x2 x.x2\n  assign offs.x3 x.x3\n\ndef recursiveRelation : TwoRowsConstraint RowType (F p) := do\n  let curr ← TableConstraint.getCurrRow\n  let next ← TableConstraint.getNextRow\n\n  let z ← Gadgets.Addition32.circuit { x := curr.x, y := curr.y }\n\n  assignU32 nextRowOff.y z\n  curr.y === next.x\n\nvariable {α : Type}", "target_theorem": "lemma fib_constraints (curr next : Row (F p) RowType) (aux_env : Environment (F p))\n  : recursiveRelation.ConstraintsHoldOnWindow ⟨<+> +> curr +> next, rfl⟩ aux_env →\n  curr.y = next.x ∧\n  (curr.x.Normalized → curr.y.Normalized → next.y.value = (curr.x.value + curr.y.value) % 2^32 ∧ next.y.Normalized) :=", "ground_truth_proof": ":= by\n  simp only [table_norm]\n  obtain ⟨ hcurr_x, hcurr_y, hnext_x, hnext_y ⟩ := fib_vars curr next aux_env\n  set env := recursiveRelation.windowEnv  ⟨<+> +> curr +> next, rfl⟩ aux_env\n  simp only [table_norm, circuit_norm, recursiveRelation,\n    assignU32, Gadgets.Addition32.circuit]\n  rintro ⟨ h_add, h_eq ⟩\n  simp only [table_norm, circuit_norm, Nat.reduceAdd, zero_add] at h_add\n  simp only [circuit_norm] at hnext_y\n  rw [hcurr_x, hcurr_y, hnext_y] at h_add\n  rw [hcurr_y, hnext_x] at h_eq\n  clear hcurr_x hcurr_y hnext_x hnext_y\n  constructor\n  · exact h_eq\n  rw [Gadgets.Addition32.Assumptions, Gadgets.Addition32.Spec] at h_add\n  intro h_norm_x h_norm_y\n  specialize h_add ⟨ h_norm_x, h_norm_y ⟩\n  obtain ⟨ h_add_mod, h_norm_next_y ⟩ := h_add\n  exact ⟨h_add_mod, h_norm_next_y⟩", "nesting_depth": 17, "transitive_dep_count": 142, "subset_aristotle": true, "category": "Applied verif."}
{"id": 141, "thm_name": "MemoryAccessList.isConsistentOnline_iff_sorted_isConsistentOffline", "thm_stmt": "theorem MemoryAccessList.isConsistentOnline_iff_sorted_isConsistentOffline\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted)\n    (h_nodup : accesses.Notimestampdup) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    MemoryAccessList.isConsistentOffline (MemoryAccessList.addressTimestampSort accesses) (MemoryAccessList.addressTimestampSort_sorted accesses)", "lean_root": "clean", "rel_path": "Clean/Utils/OfflineMemory.lean", "imports": ["import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Mathlib.Data.List.Sort", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Sorted", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.insertionSort", "module": "Mathlib.Data.List.Sort"}, {"name": "List.Pairwise", "module": "Init.Data.List.Basic"}, {"name": "List.filter", "module": "Init.Data.List.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "List.Perm", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.Sorted.of_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Pairwise.of_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.filter_eq_nil_iff", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_cons_self", "module": "Init.Data.List.Lemmas"}, {"name": "List.sorted_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "decide_eq_true_eq", "module": "Init.SimpLemmas"}, {"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "List.filter_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_filter", "module": "Init.Data.List.Lemmas"}, {"name": "decide_true", "module": "Init.Core"}, {"name": "decide_eq_false_iff_not", "module": "Init.SimpLemmas"}, {"name": "List.Sorted.filter", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.filter_nil", "module": "Init.Data.List.Basic"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "forall_true_left", "module": "Mathlib.Logic.Basic"}, {"name": "Bool.and_true", "module": "Init.SimpLemmas"}, {"name": "List.all_cons", "module": "Init.Data.List.Basic"}, {"name": "List.all_nil", "module": "Init.Data.List.Basic"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "imp_self", "module": "Init.Core"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "true_iff", "module": "Init.SimpLemmas"}, {"name": "List.filter_cons_of_pos", "module": "Init.Data.List.Lemmas"}, {"name": "Bool.not_or_self", "module": "Init.Data.Bool"}, {"name": "Bool.true_and", "module": "Init.SimpLemmas"}, {"name": "List.all_eq_true", "module": "Init.Data.List.Lemmas"}, {"name": "List.all_filter", "module": "Init.Data.List.Lemmas"}, {"name": "and_imp", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "List.Pairwise.perm", "module": "Init.Data.List.Perm"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "List.perm_insertionSort", "module": "Mathlib.Data.List.Sort"}, {"name": "List.sorted_insertionSort", "module": "Mathlib.Data.List.Sort"}, {"name": "List.Perm.filter", "module": "Init.Data.List.Perm"}, {"name": "List.eq_of_perm_of_sorted", "module": "Mathlib.Data.List.Sort"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MemoryAccess", "content": "def MemoryAccess := ℕ × ℕ × ℕ × ℕ"}, {"name": "MemoryAccessList", "content": "def MemoryAccessList := List MemoryAccess"}, {"name": "timestamp_ordering", "content": "abbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2"}, {"name": "MemoryAccessList.isTimestampSorted", "content": "def MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering"}, {"name": "MemoryAccessList.timestamps_neq", "content": "def MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y"}, {"name": "MemoryAccessList.Notimestampdup", "content": "def MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses"}, {"name": "address_timestamp_ordering", "content": "abbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2"}, {"name": "address_strict_timestamp_ordering", "content": "abbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2"}, {"name": "MemoryAccessList.isAddressTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering"}, {"name": "MemoryAccessList.isAddressStrictTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering"}, {"name": "MemoryAccessList.addressTimestampSort", "content": "def MemoryAccessList.addressTimestampSort (accesses : MemoryAccessList) : MemoryAccessList :=\n  List.insertionSort address_timestamp_ordering accesses"}, {"name": "MemoryAccessList.lastWriteValue", "content": "def MemoryAccessList.lastWriteValue (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) (addr : ℕ) : ℕ := match accesses with\n  \n  | [] => 0\n  | (_t, addr', _readValue, writeValue) :: rest =>\n    if addr' = addr then\n      \n      writeValue\n    else\n      MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr"}, {"name": "MemoryAccessList.isConsistentOnline", "content": "def MemoryAccessList.isConsistentOnline (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, addr, readValue, _writeValue) :: rest =>\n    \n    readValue = MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n    ∧ MemoryAccessList.isConsistentOnline rest (List.Sorted.of_cons h)\n\nexample : MemoryAccessList.isConsistentOnline [] (by admit /- proof elided -/\n) := by admit /- proof elided -/"}, {"name": "MemoryAccessList.filterAddress", "content": "def MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)"}, {"name": "MemoryAccessList.isConsistentSingleAddress", "content": "def MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}, {"name": "MemoryAccessList.isConsistentOffline", "content": "def MemoryAccessList.isConsistentOffline (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  | (_t2, addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    (if addr1 = addr2 then readValue2 = writeValue1 else readValue2 = 0) ∧\n    MemoryAccessList.isConsistentOffline ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}], "used_local_lemmas": [{"name": "MemoryAccessList.addressTimestampSort_sorted", "content": "theorem MemoryAccessList.addressTimestampSort_sorted (accesses : MemoryAccessList) :\n    (MemoryAccessList.addressTimestampSort accesses).Sorted address_timestamp_ordering"}, {"name": "MemoryAccessList.addressTimestampSort_perm", "content": "theorem MemoryAccessList.addressTimestampSort_perm (accesses : MemoryAccessList) :\n    (MemoryAccessList.addressTimestampSort accesses).Perm accesses"}, {"name": "MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup", "content": "theorem MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup\n    (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted)\n    (h_no_timestamp_dup : accesses.Notimestampdup) :\n    accesses.isAddressStrictTimestampSorted"}, {"name": "MemoryAccessList.noTimestampDup_perm", "content": "theorem MemoryAccessList.noTimestampDup_perm (l1 l2 : MemoryAccessList)\n    (h_l1_nodup : l1.Notimestampdup) (h_perm : l1.Perm l2) :\n    l2.Notimestampdup"}, {"name": "MemoryAccessList.noTimestampDup_of_cons", "content": "theorem MemoryAccessList.noTimestampDup_of_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h : Notimestampdup (head :: tail)) :\n    Notimestampdup tail"}, {"name": "MemoryAccessList.isAddressTimestampSorted_of_cons", "content": "theorem MemoryAccessList.isAddressTimestampSorted_of_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h : isAddressTimestampSorted (head :: tail)) :\n    isAddressTimestampSorted tail"}, {"name": "MemoryAccessList.filterAddress_sorted", "content": "theorem MemoryAccessList.filterAddress_sorted (accesses : MemoryAccessList)\n    (h : accesses.isTimestampSorted) (addr : ℕ) :\n    (MemoryAccessList.filterAddress accesses addr).isTimestampSorted"}, {"name": "MemoryAccessList.filterAddress_cons", "content": "theorem MemoryAccessList.filterAddress_cons (head : MemoryAccess) (tail : MemoryAccessList) (addr : ℕ) :\n    MemoryAccessList.filterAddress (head :: tail) addr =\n    match head with\n    | ⟨_t, a, _r, _w⟩ => ((if a = addr then\n      (head :: (MemoryAccessList.filterAddress tail addr))\n        else (MemoryAccessList.filterAddress tail addr)))"}, {"name": "MemoryAccessList.isConsistentSingleAddress_iff", "content": "theorem MemoryAccessList.isConsistentSingleAddress_iff (accesses : MemoryAccessList) (addr : ℕ) (h_sorted : accesses.isTimestampSorted)\n    (h_eq : accesses.all (fun (_t, addr', _readValue, _writeValue) => addr' = addr)) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    MemoryAccessList.isConsistentSingleAddress accesses h_sorted"}, {"name": "MemoryAccessList.lastWriteValue_filter", "content": "theorem MemoryAccessList.lastWriteValue_filter (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted) (addr : ℕ) (h_sorted' : ((MemoryAccessList.filterAddress accesses addr).isTimestampSorted))  :\n    MemoryAccessList.lastWriteValue accesses h_sorted addr =\n    MemoryAccessList.lastWriteValue (MemoryAccessList.filterAddress accesses addr) h_sorted' addr"}, {"name": "MemoryAccessList.isConsistentOnline_filter_of_consistentOnline", "content": "theorem MemoryAccessList.isConsistentOnline_filter_of_consistentOnline (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted)\n    (h_consistent : MemoryAccessList.isConsistentOnline accesses h_sorted) (addr : ℕ) :\n    MemoryAccessList.isConsistentOnline (MemoryAccessList.filterAddress accesses addr) (MemoryAccessList.filterAddress_sorted accesses h_sorted addr)"}, {"name": "MemoryAccessList.isConsistentSingleAddress_cons", "content": "theorem MemoryAccessList.isConsistentSingleAddress_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail)) (h_sorted' : tail.isTimestampSorted)\n    (h : isConsistentSingleAddress (head :: tail) h_sorted) :\n    isConsistentSingleAddress tail h_sorted'"}, {"name": "MemoryAccessList.isConsistentSingleAddress_cons_forall", "content": "theorem MemoryAccessList.isConsistentSingleAddress_cons_forall (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail))\n    : (∀ addr : ℕ, (filterAddress (head :: tail) addr).isConsistentSingleAddress (MemoryAccessList.filterAddress_sorted (head :: tail) h_sorted addr)) →\n    (∀ addr : ℕ, isConsistentSingleAddress (filterAddress tail addr) (MemoryAccessList.filterAddress_sorted tail (by simp_all only [isTimestampSorted,\n      List.sorted_cons]) addr))"}, {"name": "MemoryAccessList.isConsistent_iff_all_single_address", "content": "theorem MemoryAccessList.isConsistent_iff_all_single_address (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    (∀ addr : ℕ, MemoryAccessList.isConsistentSingleAddress (MemoryAccessList.filterAddress accesses addr) (MemoryAccessList.filterAddress_sorted accesses h_sorted addr))"}, {"name": "MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted", "content": "theorem MemoryAccessList.filterAddress_sorted_from_addressTimestampSorted\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isTimestampSorted"}, {"name": "MemoryAccessList.isConsistentSingleAddress_filterAddress_forall_of_cons", "content": "theorem MemoryAccessList.isConsistentSingleAddress_filterAddress_forall_of_cons\n    (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: tail))\n    (h_nodup : Notimestampdup (head :: tail))\n    (h : ∀ addr, isConsistentSingleAddress (filterAddress (head :: tail) addr)\n         (filterAddress_sorted_from_addressTimestampSorted (head :: tail) h_sorted h_nodup addr)) :\n    ∀ addr, isConsistentSingleAddress (filterAddress tail addr)\n      (filterAddress_sorted_from_addressTimestampSorted tail\n        (isAddressTimestampSorted_of_cons head tail h_sorted)\n        (noTimestampDup_of_cons head tail h_nodup) addr)"}, {"name": "MemoryAccessList.filterAddress_empty_when_address_changes", "content": "theorem MemoryAccessList.filterAddress_empty_when_address_changes\n    (head : MemoryAccess) (second : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: second :: tail))\n    (h_addr_ne : head.2.1 ≠ second.2.1) :\n    filterAddress (second :: tail) head.2.1 = []"}, {"name": "MemoryAccessList.isConsistentOffline_of_cons", "content": "theorem MemoryAccessList.isConsistentOffline_of_cons\n    (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isAddressTimestampSorted (head :: tail))\n    (h_offline : isConsistentOffline (head :: tail) h_sorted) :\n    isConsistentOffline tail (isAddressTimestampSorted_of_cons head tail h_sorted)"}, {"name": "MemoryAccessList.isConsistentOffline_implies_single_address", "content": "theorem MemoryAccessList.isConsistentOffline_implies_single_address\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isAddressTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (h_offline : accesses.isConsistentOffline h_sorted)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isConsistentSingleAddress\n      (filterAddress_sorted_from_addressTimestampSorted accesses h_sorted h_nodup addr)"}, {"name": "MemoryAccessList.isConsistentOffline_iff_all_single_addresses", "content": "theorem MemoryAccessList.isConsistentOffline_iff_all_single_addresses (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) (h_nodup : accesses.Notimestampdup) :\n    MemoryAccessList.isConsistentOffline accesses h_sorted ↔\n    ∀ addr, MemoryAccessList.isConsistentSingleAddress (MemoryAccessList.filterAddress accesses addr) (filterAddress_sorted_from_addressTimestampSorted accesses h_sorted h_nodup addr)"}, {"name": "MemoryAccessList.addressTimestampSort_noTimestampDup", "content": "theorem MemoryAccessList.addressTimestampSort_noTimestampDup\n    (accesses : MemoryAccessList)\n    (h_nodup : accesses.Notimestampdup) :\n    accesses.addressTimestampSort.Notimestampdup"}, {"name": "MemoryAccessList.filterAddress_addressTimestampSort_eq", "content": "theorem MemoryAccessList.filterAddress_addressTimestampSort_eq\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted)\n    (h_nodup : accesses.Notimestampdup)\n    (addr : ℕ) :\n    (accesses.filterAddress addr).isConsistentSingleAddress\n      (filterAddress_sorted accesses h_sorted addr) ↔\n    (accesses.addressTimestampSort.filterAddress addr).isConsistentSingleAddress\n      (filterAddress_sorted_from_addressTimestampSorted accesses.addressTimestampSort\n        (addressTimestampSort_sorted accesses)\n        (addressTimestampSort_noTimestampDup accesses h_nodup) addr)"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Utils.Field\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Tactics\n\nimport Mathlib.Data.List.Sort\n\ndef MemoryAccess := ℕ × ℕ × ℕ × ℕ \n\ndef MemoryAccessList := List MemoryAccess\n\nabbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2\n\ndef MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering\n\ndef MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y\n\ndef MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses\n\nabbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2\n\nabbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2\n\n@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering\n\n@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering\n\ndef MemoryAccessList.addressTimestampSort (accesses : MemoryAccessList) : MemoryAccessList :=\n  List.insertionSort address_timestamp_ordering accesses\n\ndef MemoryAccessList.lastWriteValue (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) (addr : ℕ) : ℕ := match accesses with\n  \n  | [] => 0\n  | (_t, addr', _readValue, writeValue) :: rest =>\n    if addr' = addr then\n      \n      writeValue\n    else\n      MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n\ndef MemoryAccessList.isConsistentOnline (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, addr, readValue, _writeValue) :: rest =>\n    \n    readValue = MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n    ∧ MemoryAccessList.isConsistentOnline rest (List.Sorted.of_cons h)\n\nexample : MemoryAccessList.isConsistentOnline [] (by admit /- proof elided -/\n) := by admit /- proof elided -/\n\ndef MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)\n\ndef MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)\n\ndef MemoryAccessList.isConsistentOffline (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  | (_t2, addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    (if addr1 = addr2 then readValue2 = writeValue1 else readValue2 = 0) ∧\n    MemoryAccessList.isConsistentOffline ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)", "target_theorem": "theorem MemoryAccessList.isConsistentOnline_iff_sorted_isConsistentOffline\n    (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted)\n    (h_nodup : accesses.Notimestampdup) :\n    MemoryAccessList.isConsistentOnline accesses h_sorted ↔\n    MemoryAccessList.isConsistentOffline (MemoryAccessList.addressTimestampSort accesses) (MemoryAccessList.addressTimestampSort_sorted accesses) :=", "ground_truth_proof": ":= by\n  rw [isConsistent_iff_all_single_address]\n  -- Use isConsistentOffline_iff_all_single_addresses\n  rw [isConsistentOffline_iff_all_single_addresses (addressTimestampSort accesses)\n    (addressTimestampSort_sorted accesses)\n    (addressTimestampSort_noTimestampDup accesses h_nodup)]\n  -- Now use filterAddress_addressTimestampSort_eq to relate the two sides\n  constructor\n  · intro h addr\n    rw [← filterAddress_addressTimestampSort_eq accesses h_sorted h_nodup addr]\n    exact h addr\n  · intro h addr\n    rw [filterAddress_addressTimestampSort_eq accesses h_sorted h_nodup addr]\n    exact h addr", "nesting_depth": 5, "transitive_dep_count": 84, "subset_aristotle": true, "category": "Applied verif."}
{"id": 142, "thm_name": "U32.value_injective_on_normalized", "thm_stmt": "omit p_large_enough in\nlemma value_injective_on_normalized (x y : U32 (F p))\n    (hx : x.Normalized) (hy : y.Normalized) :\n    x.value = y.value → x = y", "lean_root": "clean", "rel_path": "Clean/Types/U32.lean", "imports": ["import Clean.Circuit.Provable", "import Clean.Circuit.Subcircuit", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Clean.Utils.Bitwise", "import Clean.Circuit.Extensions", "import Clean.Gadgets.ByteLookup"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.Prime", "module": "Mathlib.Data.Nat.Prime.Defs"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "KeccakRow.value", "content": "def KeccakRow.value (row : KeccakRow (F p)) := row.map U64.value"}, {"name": "map", "content": "def map {α β : Type} (x : U64 α) (f : α → β) : U64 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3, f x.x4, f x.x5, f x.x6, f x.x7 ⟩"}, {"name": "U64", "content": "structure U64 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\n  x4 : T\n  x5 : T\n  x6 : T\n  x7 : T\nderiving DecidableEq"}, {"name": "value", "content": "def value (x : U64 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3 +\n  x.x4.val * 256^4 + x.x5.val * 256^5 + x.x6.val * 256^6 + x.x7.val * 256^7"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "KeccakRow.Normalized", "content": "def KeccakRow.Normalized (row : KeccakRow (F p)) :=\n  ∀ i : Fin 5, row[i.val].Normalized"}, {"name": "BLAKE3State.Normalized", "content": "def BLAKE3State.Normalized (state : BLAKE3State (F p)) :=\n  ∀ i : Fin 16, state[i.val].Normalized"}, {"name": "KeccakBlock.value", "content": "def KeccakBlock.value (block : KeccakBlock (F p)) := block.map U64.value"}, {"name": "BLAKE3State.value", "content": "def BLAKE3State.value (state : BLAKE3State (F p)) := state.map U32.value"}, {"name": "KeccakState.value", "content": "def KeccakState.value (state : KeccakState (F p)) := state.map U64.value"}, {"name": "KeccakBlock.Normalized", "content": "def KeccakBlock.Normalized (block : KeccakBlock (F p)) :=\n  ∀ i : Fin RATE, block[i.val].Normalized"}, {"name": "RATE", "content": "@[reducible] def RATE := 17\nexample : RATE + CAPACITY = 25 := rfl"}, {"name": "CAPACITY", "content": "@[reducible] def CAPACITY := 8"}, {"name": "KeccakState.Normalized", "content": "def KeccakState.Normalized (state : KeccakState (F p)) :=\n  ∀ i : Fin 25, state[i.val].Normalized"}], "lib_lemmas": [{"name": "ZMod.val_injective", "module": "Mathlib.Data.ZMod.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "U32", "content": "structure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq"}, {"name": "U32.map", "content": "def map {α β : Type} (x : U32 α) (f : α → β) : U32 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3 ⟩"}, {"name": "U32.Normalized", "content": "def Normalized (x : U32 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256"}, {"name": "U32.value", "content": "def value (x : U32 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3"}], "used_local_lemmas": [{"name": "U32.value_horner", "content": "omit [Fact (Nat.Prime p)] p_large_enough in\ntheorem value_horner (x : U32 (F p)) : x.value =\n    x.x0.val + 2^8 * (x.x1.val + 2^8 * (x.x2.val + 2^8 * x.x3.val))"}], "local_ctx": "import Clean.Gadgets.ByteLookup\n\nimport Clean.Circuit.Extensions\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Circuit.Provable\n\nimport Clean.Utils.Primes\n\nimport Clean.Circuit.Subcircuit\n\nimport Clean.Gadgets.Equality\n\nsection\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nstructure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq\n\nnamespace U32\n\ndef map {α β : Type} (x : U32 α) (f : α → β) : U32 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3 ⟩\n\ndef Normalized (x : U32 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256\n\ndef value (x : U32 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3", "target_theorem": "omit p_large_enough in\nlemma value_injective_on_normalized (x y : U32 (F p))\n    (hx : x.Normalized) (hy : y.Normalized) :\n    x.value = y.value → x = y :=", "ground_truth_proof": ":= by\n  intro h_eq\n  -- Use horner form of value\n  have hx_value := U32.value_horner x\n  have hy_value := U32.value_horner y\n\n  simp only [U32.Normalized] at hx hy\n\n  have : x.x0 = y.x0 := by apply ZMod.val_injective; omega\n  have : x.x1 = y.x1 := by apply ZMod.val_injective; omega\n  have : x.x2 = y.x2 := by apply ZMod.val_injective; omega\n  have : x.x3 = y.x3 := by apply ZMod.val_injective; omega\n\n  cases x; cases y\n  simp_all", "nesting_depth": 7, "transitive_dep_count": 58, "subset_aristotle": true, "category": "Applied verif."}
{"id": 143, "thm_name": "MemoryAccessList.isConsistentSingleAddress_cons_forall", "thm_stmt": "theorem MemoryAccessList.isConsistentSingleAddress_cons_forall (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail))\n    : (∀ addr : ℕ, (filterAddress (head :: tail) addr).isConsistentSingleAddress (MemoryAccessList.filterAddress_sorted (head :: tail) h_sorted addr)) →\n    (∀ addr : ℕ, isConsistentSingleAddress (filterAddress tail addr) (MemoryAccessList.filterAddress_sorted tail (by simp_all only [isTimestampSorted,\n      List.sorted_cons]) addr))", "lean_root": "clean", "rel_path": "Clean/Utils/OfflineMemory.lean", "imports": ["import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Mathlib.Data.List.Sort", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Sorted", "module": "Mathlib.Deprecated.Sort"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.Sorted.filter", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.filter_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.sorted_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Sorted.of_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.filter_cons_of_pos", "module": "Init.Data.List.Lemmas"}, {"name": "decide_true", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MemoryAccess", "content": "def MemoryAccess := ℕ × ℕ × ℕ × ℕ"}, {"name": "MemoryAccessList", "content": "def MemoryAccessList := List MemoryAccess"}, {"name": "timestamp_ordering", "content": "abbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2"}, {"name": "MemoryAccessList.isTimestampSorted", "content": "def MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering"}, {"name": "MemoryAccessList.filterAddress", "content": "def MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)"}, {"name": "MemoryAccessList.isConsistentSingleAddress", "content": "def MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)"}], "used_local_lemmas": [{"name": "MemoryAccessList.filterAddress_sorted", "content": "theorem MemoryAccessList.filterAddress_sorted (accesses : MemoryAccessList)\n    (h : accesses.isTimestampSorted) (addr : ℕ) :\n    (MemoryAccessList.filterAddress accesses addr).isTimestampSorted"}, {"name": "MemoryAccessList.filterAddress_cons", "content": "theorem MemoryAccessList.filterAddress_cons (head : MemoryAccess) (tail : MemoryAccessList) (addr : ℕ) :\n    MemoryAccessList.filterAddress (head :: tail) addr =\n    match head with\n    | ⟨_t, a, _r, _w⟩ => ((if a = addr then\n      (head :: (MemoryAccessList.filterAddress tail addr))\n        else (MemoryAccessList.filterAddress tail addr)))"}, {"name": "MemoryAccessList.isConsistentSingleAddress_cons", "content": "theorem MemoryAccessList.isConsistentSingleAddress_cons (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail)) (h_sorted' : tail.isTimestampSorted)\n    (h : isConsistentSingleAddress (head :: tail) h_sorted) :\n    isConsistentSingleAddress tail h_sorted'"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Utils.Field\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Tactics\n\nimport Mathlib.Data.List.Sort\n\ndef MemoryAccess := ℕ × ℕ × ℕ × ℕ \n\ndef MemoryAccessList := List MemoryAccess\n\nabbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2\n\ndef MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering\n\ndef MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)\n\ndef MemoryAccessList.isConsistentSingleAddress (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) : Prop := match accesses with\n  \n  | [] => True\n  \n  | (_timestamp, _addr, readValue, _writeValue) :: [] => readValue = 0\n  \n  | (_t2, _addr2, readValue2, _writeValue2) :: (t1, addr1, readValue1, writeValue1) :: rest =>\n    readValue2 = writeValue1 ∧\n    MemoryAccessList.isConsistentSingleAddress ((t1, addr1, readValue1, writeValue1) :: rest) (List.Sorted.of_cons h_sorted)", "target_theorem": "theorem MemoryAccessList.isConsistentSingleAddress_cons_forall (head : MemoryAccess) (tail : MemoryAccessList)\n    (h_sorted : isTimestampSorted (head :: tail))\n    : (∀ addr : ℕ, (filterAddress (head :: tail) addr).isConsistentSingleAddress (MemoryAccessList.filterAddress_sorted (head :: tail) h_sorted addr)) →\n    (∀ addr : ℕ, isConsistentSingleAddress (filterAddress tail addr) (MemoryAccessList.filterAddress_sorted tail (by simp_all only [isTimestampSorted,\n      List.sorted_cons]) addr)) :=", "ground_truth_proof": ":= by\n  intro h addr'\n  obtain ⟨t_head, a_head, r_head, w_head⟩ := head\n  simp_all [MemoryAccessList.filterAddress_cons]\n  specialize h addr'\n  by_cases h_addr : a_head = addr'\n  · simp_all only [↓reduceIte]\n    rw [h_addr] at h_sorted\n    have tail_sorted : tail.isTimestampSorted := by\n      unfold isTimestampSorted at h_sorted\n      exact List.Sorted.of_cons h_sorted\n    have filtered_tail_sorted : (MemoryAccessList.filterAddress tail addr').isTimestampSorted := by\n      simp only [filterAddress]\n      apply List.Sorted.filter\n      exact tail_sorted\n\n    have filter_eq_head : MemoryAccessList.filterAddress (⟨t_head, addr', r_head, w_head⟩ :: tail) addr' =\n      (⟨t_head, addr', r_head, w_head⟩ :: (MemoryAccessList.filterAddress tail addr')) := by\n      simp only [filterAddress, decide_true, List.filter_cons_of_pos]\n\n    have h_filtered_sorted : MemoryAccessList.isTimestampSorted (⟨t_head, addr', r_head, w_head⟩ :: (MemoryAccessList.filterAddress tail addr')) := by\n      rw [←filter_eq_head]\n      apply MemoryAccessList.filterAddress_sorted\n      assumption\n\n    have h' := MemoryAccessList.isConsistentSingleAddress_cons ⟨t_head, addr', r_head, w_head⟩ (tail.filterAddress addr') h_filtered_sorted filtered_tail_sorted\n    specialize h' h\n    simp_all only\n  · simp_all only [↓reduceIte]", "nesting_depth": 2, "transitive_dep_count": 16, "subset_aristotle": true, "category": "Applied verif."}
{"id": 144, "thm_name": "Utils.StateTransition.acyclic_containsPath_nodup", "thm_stmt": "lemma acyclic_containsPath_nodup (R : Run S) (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_contains : R.containsPath path) :\n    path.Nodup", "lean_root": "clean", "rel_path": "Clean/Utils/SourceSinkPath.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import Mathlib.Data.Fintype.Prod", "import Mathlib.Data.List.Basic", "import Mathlib.Algebra.BigOperators.Group.Finset.Basic", "import Mathlib.Data.Finset.Basic", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise", "import Mathlib.Data.Fintype.Basic"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Sublist", "module": "Init.Data.List.Basic"}, {"name": "List.count", "module": "Init.Data.List.Basic"}, {"name": "List.drop", "module": "Init.Data.List.Basic"}, {"name": "List.tail", "module": "Init.Data.List.Basic"}, {"name": "List.take", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.cons₂", "module": "Init.Data.List.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.reduceLeDiff", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.drop_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.tail_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zip_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.count_le", "module": "Init.Data.List.Count"}, {"name": "List.tail_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.take_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.length_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.length_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "Nat.min_eq_left", "module": "Init.Data.Nat.MinMax"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "List.getElem?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getElem?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getLast?_eq_getElem?", "module": "Init.Data.List.Lemmas"}, {"name": "List.head?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.head?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "List.duplicate_iff_exists_distinct_get", "module": "Mathlib.Data.List.NodupEquivFin"}, {"name": "List.exists_duplicate_iff_not_nodup", "module": "Mathlib.Data.List.Duplicate"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Utils.StateTransition.Transition", "content": "def Transition (S : Type*) := S × S"}, {"name": "Utils.StateTransition.Run", "content": "def Run (S : Type*) := Transition S → ℕ"}, {"name": "Utils.StateTransition.countTransitionInPath", "content": "def countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t"}, {"name": "Utils.StateTransition.Run.containsPath", "content": "def Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t"}, {"name": "Utils.StateTransition.Run.hasCycle", "content": "def Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle"}, {"name": "Utils.StateTransition.Run.isAcyclic", "content": "def Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle"}], "used_local_lemmas": [{"name": "Utils.StateTransition.zip_drop_sublist", "content": "lemma zip_drop_sublist (l : List S) (n : ℕ) :\n    ((l.drop n).zip (l.drop (n + 1))).Sublist (l.zip l.tail)"}, {"name": "Utils.StateTransition.containsPath_drop", "content": "lemma containsPath_drop (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.drop n)"}, {"name": "Utils.StateTransition.zip_take_sublist", "content": "lemma zip_take_sublist (l1 l2 : List S) (n m : ℕ) :\n    ((l1.take n).zip (l2.take m)).Sublist (l1.zip l2)"}, {"name": "Utils.StateTransition.tail_take", "content": "lemma tail_take {α : Type*} (l : List α) (n : ℕ) :\n    (l.take n).tail = (l.tail).take (n - 1)"}, {"name": "Utils.StateTransition.containsPath_take", "content": "lemma containsPath_take (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.take n)"}, {"name": "Utils.StateTransition.drop_take_length_ge_two", "content": "lemma drop_take_length_ge_two {α : Type*} (path : List α) (n m : Fin path.length)\n    (h_n_lt_m : n < m) :\n    ((path.drop n.val).take (m.val - n.val + 1)).length ≥ 2"}, {"name": "Utils.StateTransition.getLast_drop_take", "content": "lemma getLast_drop_take {α : Type*} (path : List α) (n k : ℕ)\n    (h_n_lt : n < path.length)\n    (h_bound : n + k ≤ path.length)\n    (h_k_pos : k > 0) :\n    ((path.drop n).take k).getLast? = path[n + k - 1]?"}, {"name": "Utils.StateTransition.drop_take_cycle_same_endpoints", "content": "lemma drop_take_cycle_same_endpoints (path : List S) (x : S) (n m : Fin path.length)\n    (h_n_lt_m : n < m)\n    (h_x_at_n : path[n] = x)\n    (h_x_at_m : path[m] = x) :\n    ((path.drop n.val).take (m.val - n.val + 1)).head? =\n    ((path.drop n.val).take (m.val - n.val + 1)).getLast?"}, {"name": "Utils.StateTransition.containsPath_drop_take", "content": "lemma containsPath_drop_take (R : Run S) (path : List S) (n m : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath ((path.drop n).take m)"}], "local_ctx": "import Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Fintype.Basic\n\nimport Mathlib.Data.Fintype.Prod\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Piecewise\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nnamespace Utils.StateTransition\n\nvariable {S : Type*} [DecidableEq S] [Fintype S]\n\ndef Transition (S : Type*) := S × S\n\ndef Run (S : Type*) := Transition S → ℕ\n\ndef countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t\n\ndef Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t\n\ndef Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle\n\ndef Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle", "target_theorem": "lemma acyclic_containsPath_nodup (R : Run S) (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_contains : R.containsPath path) :\n    path.Nodup :=", "ground_truth_proof": ":= by\n  -- Proof by contradiction: if path has duplicates, extract a cycle\n  by_contra h_dup\n  -- If path is not Nodup, there exists an element that appears twice\n  rw [← List.exists_duplicate_iff_not_nodup] at h_dup\n  obtain ⟨x, h_duplicate⟩ := h_dup\n  -- x appears at least twice in path, at distinct positions\n  rw [List.duplicate_iff_exists_distinct_get] at h_duplicate\n  obtain ⟨n, m, h_n_lt_m, h_x_at_n, h_x_at_m⟩ := h_duplicate\n  -- Extract the subpath from index n to index m (inclusive)\n  -- This forms a cycle: path[n..m] starts and ends with x\n  -- Use take and drop: drop n first, then take (m - n + 1) elements\n  let cycle := (path.drop n.val).take (m.val - n.val + 1)\n  -- Prove this is a cycle\n  have h_n_lt_len : n.val < path.length := n.isLt\n  have h_m_lt_len : m.val < path.length := m.isLt\n  have h_cycle_len := drop_take_length_ge_two path n m h_n_lt_m\n  have h_cycle_starts_ends_with_x : cycle.head? = cycle.getLast? :=\n    drop_take_cycle_same_endpoints path x n m h_n_lt_m h_x_at_n.symm h_x_at_m.symm\n  have h_cycle_contained : R.containsPath cycle :=\n    containsPath_drop_take R path n.val (m.val - n.val + 1) h_contains\n  -- This contradicts acyclicity\n  unfold Run.isAcyclic Run.hasCycle at h_acyclic\n  push_neg at h_acyclic\n  apply h_acyclic cycle h_cycle_len h_cycle_starts_ends_with_x h_cycle_contained", "nesting_depth": 4, "transitive_dep_count": 44, "subset_aristotle": true, "category": "Applied verif."}
{"id": 145, "thm_name": "Gadgets.And.And8.completeness", "thm_stmt": "theorem completeness : Completeness (F p) elaborated Assumptions", "lean_root": "clean", "rel_path": "Clean/Gadgets/And/And8.lean", "imports": ["import Clean.Gadgets.Xor.ByteXorTable", "import Clean.Utils.Field", "import Clean.Circuit.Basic", "import Clean.Utils.Primes"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Fin.xor", "module": "Init.Data.Fin.Basic"}, {"name": "HXor", "module": "Init.Prelude"}, {"name": "HXor.hXor", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "XorOp", "module": "Init.Prelude"}, {"name": "XorOp.xor", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "UInt16", "module": "Init.Prelude"}, {"name": "UInt16.toNat", "module": "Init.Data.UInt.BasicAux"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"field_to_nat\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|field_to_nat) =>\n    `(tactic|(\n      intros\n      repeat rw [ZMod.val_add] \n      repeat rw [ZMod.val_mul] \n      repeat rw [val_eq_256]\n      try simp only [Nat.add_mod_mod, Nat.mod_add_mod, Nat.mul_mod_mod, Nat.mod_mul_mod]\n      rw [Nat.mod_eq_of_lt _]\n      repeat linarith [‹Fact (_ > 512)›.elim]))\n\nexample [Fact (p > 512)] (x y : F p) (hx : x.val < 256) (hy : y.val < 2) :\n    (x + y * 256).val = x.val + y.val * 256 := by admit /- proof elided -/"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "ByteXorTable", "content": "def ByteXorTable : Table (F p) fieldTriple := .fromStatic {\n  name := \"ByteXor\"\n  length := 256*256\n\n  row i :=\n    let (x, y) := splitTwoBytes i\n    (fromByte x, fromByte y, fromByte (x ^^^ y))\n\n  index := fun (x, y, _) => x.val * 256 + y.val\n\n  Spec := fun (x, y, z) =>\n    x.val < 256 ∧ y.val < 256 ∧ z.val = x.val ^^^ y.val\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "splitTwoBytes", "content": "def splitTwoBytes (i : Fin (256 * 256)) : Fin 256 × Fin 256 :=\n  let x := i.val / 256\n  let y := i.val % 256\n  have x_lt : x < 256 := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "fieldTriple", "content": "@[reducible]\ndef fieldTriple : TypeMap := fun F => F × F × F"}, {"name": "concatTwoBytes", "content": "def concatTwoBytes (x y : Fin 256) : Fin (256 * 256) :=\n  let i := x.val * 256 + y.val\n  have i_lt : i < 256 * 256 := by admit /- proof elided -/"}, {"name": "fieldVar", "content": "@[reducible] def fieldVar (F : Type) := field (Expression F)"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}], "lib_lemmas": [{"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.and_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.xor_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "UInt16.toNat_ofNat_of_lt", "module": "Init.Data.UInt.Lemmas"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "ZMod.val_mul_of_lt", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_sub", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_add_neg", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "val_lt_p", "content": "theorem val_lt_p {p : ℕ} (x : ℕ) : (x < p) → (x : F p).val = x"}, {"name": "natToField_eq", "content": "theorem natToField_eq {n : ℕ} {lt : n < p} (x : F p) (hx : x = natToField n lt) : x.val = n"}, {"name": "natToField_eq_natCast", "content": "theorem natToField_eq_natCast {n : ℕ} (lt : n < p) : ↑n = FieldUtils.natToField n lt"}], "used_local_defs": [{"name": "Gadgets.And.And8.Inputs", "content": "structure Inputs (F : Type) where\n  x: F\n  y: F"}, {"name": "Gadgets.And.And8.Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.val < 256 ∧ y.val < 256"}, {"name": "Gadgets.And.And8.main", "content": "def main (input : Var Inputs (F p)) : Circuit (F p) (fieldVar (F p)) := do\n  let ⟨x, y⟩ := input\n  let and ← witness fun eval => (eval x).val &&& (eval y).val\n  \n  let xor := x + y - 2*and\n  lookup ByteXorTable (x, y, xor)\n  return and"}, {"name": "Gadgets.And.And8.elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs field where\n  main\n  localLength _ := 1\n  output _ i := var ⟨i⟩"}], "used_local_lemmas": [{"name": "Gadgets.And.And8.and_times_two_add_xor", "content": "theorem and_times_two_add_xor {x y : ℕ} (hx : x < 256) (hy : y < 256) : 2 * (x &&& y) + (x ^^^ y) = x + y"}, {"name": "Gadgets.And.And8.two_and_le_add", "content": "theorem two_and_le_add {x y : ℕ} (hx : x < 256) (hy : y < 256) : 2 * (x &&& y) ≤ x + y"}, {"name": "Gadgets.And.And8.val_two", "content": "lemma val_two : (2 : F p).val = 2"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Gadgets.Xor.ByteXorTable\n\nimport Clean.Utils.Primes\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nnamespace Gadgets.And.And8\n\nopen Xor (ByteXorTable)\n\nopen FieldUtils\n\nstructure Inputs (F : Type) where\n  x: F\n  y: F\n\ndef Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.val < 256 ∧ y.val < 256\n\ndef main (input : Var Inputs (F p)) : Circuit (F p) (fieldVar (F p)) := do\n  let ⟨x, y⟩ := input\n  let and ← witness fun eval => (eval x).val &&& (eval y).val\n  \n  let xor := x + y - 2*and\n  lookup ByteXorTable (x, y, xor)\n  return and\n\ninstance elaborated : ElaboratedCircuit (F p) Inputs field where\n  main\n  localLength _ := 1\n  output _ i := var ⟨i⟩", "target_theorem": "theorem completeness : Completeness (F p) elaborated Assumptions :=", "ground_truth_proof": ":= by\n  intro i env ⟨ x_var, y_var ⟩ h_env ⟨ x, y ⟩ h_input h_assumptions\n  simp_all only [circuit_norm, main, Assumptions, ByteXorTable, Inputs.mk.injEq]\n  obtain ⟨ hx_byte, hy_byte ⟩ := h_assumptions\n  set w : F p := ZMod.val x &&& ZMod.val y\n  have hw : w = ZMod.val x &&& ZMod.val y := rfl\n  let z := x + y + -(2*w)\n\n  -- now it's pretty much the soundness proof in reverse\n  have and_byte : x.val &&& y.val < 256 := Nat.and_lt_two_pow (n:=8) x.val hy_byte\n  have p_large := p_large_enough.elim\n  have and_lt : x.val &&& y.val < p := by linarith\n  rw [natToField_eq_natCast and_lt] at hw\n  have h_and : w.val = x.val &&& y.val := natToField_eq w hw\n\n  have two_and_val : (2*w).val = 2*(x.val &&& y.val) := by\n    rw [ZMod.val_mul_of_lt, val_two, h_and]\n    rw [val_two]\n    linarith\n\n  have x_y_val : (x + y).val = x.val + y.val := by field_to_nat\n  have two_and_lt : (2*w).val ≤ (x + y).val := by\n    rw [two_and_val, x_y_val]\n    exact two_and_le_add hx_byte hy_byte\n\n  rw [←sub_eq_add_neg, ZMod.val_sub two_and_lt, x_y_val, two_and_val,\n    ←and_times_two_add_xor hx_byte hy_byte, add_comm, Nat.add_sub_cancel]", "nesting_depth": 8, "transitive_dep_count": 110, "subset_aristotle": true, "category": "Applied verif."}
{"id": 146, "thm_name": "Utils.StateTransition.acyclic_run_has_path_from_source_to_sink", "thm_stmt": "lemma acyclic_run_has_path_from_source_to_sink (R : Run S) (s d : S)\n    (h_acyclic : R.isAcyclic)\n    (h_source : R.netFlow s = 1)\n    (h_others : ∀ x, x ≠ s → x ≠ d → R.netFlow x = 0) :\n    ∃ (path : List S), path.head? = some s ∧ path.getLast? = some d ∧\n      path ≠ [] ∧ R.containsPath path ∧ path.Nodup", "lean_root": "clean", "rel_path": "Clean/Utils/SourceSinkPath.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import Mathlib.Data.Fintype.Prod", "import Mathlib.Data.List.Basic", "import Mathlib.Algebra.BigOperators.Group.Finset.Basic", "import Mathlib.Data.Finset.Basic", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise", "import Mathlib.Data.Fintype.Basic"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Sublist", "module": "Init.Data.List.Basic"}, {"name": "List.count", "module": "Init.Data.List.Basic"}, {"name": "List.drop", "module": "Init.Data.List.Basic"}, {"name": "List.tail", "module": "Init.Data.List.Basic"}, {"name": "List.take", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.cons₂", "module": "Init.Data.List.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.reduceLeDiff", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "List.getLast?", "module": "Init.Data.List.Basic"}, {"name": "List.head?", "module": "Init.Data.List.Basic"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.drop_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.tail_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zip_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.count_le", "module": "Init.Data.List.Count"}, {"name": "List.tail_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.take_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.length_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.length_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "Nat.min_eq_left", "module": "Init.Data.Nat.MinMax"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "List.getElem?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getElem?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getLast?_eq_getElem?", "module": "Init.Data.List.Lemmas"}, {"name": "List.head?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.head?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "List.duplicate_iff_exists_distinct_get", "module": "Mathlib.Data.List.NodupEquivFin"}, {"name": "List.exists_duplicate_iff_not_nodup", "module": "Mathlib.Data.List.Duplicate"}, {"name": "Finset.sum_le_univ_sum_of_nonneg", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "List.getLast_singleton", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast?_cons_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.count_pos_iff", "module": "Init.Data.List.Count"}, {"name": "Finset.sum_nonneg", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "List.countP_cons_of_pos", "module": "Init.Data.List.Count"}, {"name": "List.countP_nil", "module": "Init.Data.List.Count"}, {"name": "List.count_cons", "module": "Init.Data.List.Count"}, {"name": "List.count_nil", "module": "Init.Data.List.Count"}, {"name": "List.zipWith_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zipWith_nil_right", "module": "Init.Data.List.Basic"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.exists_cons_of_ne_nil", "module": "Init.Data.List.Lemmas"}, {"name": "List.countP_singleton", "module": "Init.Data.List.Count"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "List.zip_nil_right", "module": "Init.Data.List.Basic"}, {"name": "Nat.add_right_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "List.mem_of_mem_tail", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "List.getLast?_eq_getLast", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast_mem", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.nodup_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.mem_iff_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.ssubset_univ_iff", "module": "Mathlib.Data.Finset.BooleanAlgebra"}, {"name": "Finset.card_insert_of_notMem", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.card_lt_card", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.card_univ", "module": "Mathlib.Data.Fintype.Card"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Utils.StateTransition.Transition", "content": "def Transition (S : Type*) := S × S"}, {"name": "Utils.StateTransition.Run", "content": "def Run (S : Type*) := Transition S → ℕ"}, {"name": "Utils.StateTransition.Run.netFlow", "content": "noncomputable def Run.netFlow {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) (x : S) : ℤ :=\n  (∑ y : S, (R (x, y) : ℤ)) - (∑ y : S, (R (y, x) : ℤ))"}, {"name": "Utils.StateTransition.countTransitionInPath", "content": "def countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t"}, {"name": "Utils.StateTransition.Run.containsPath", "content": "def Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t"}, {"name": "Utils.StateTransition.Run.hasCycle", "content": "def Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle"}, {"name": "Utils.StateTransition.Run.isAcyclic", "content": "def Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle"}, {"name": "Utils.StateTransition.Run.reachable", "content": "def Run.reachable [DecidableEq S] (R : Run S) (start finish : S) : Prop :=\n  ∃ (path : List S), path.head? = some start ∧ path.getLast? = some finish ∧\n    path ≠ [] ∧ R.containsPath path"}, {"name": "Utils.StateTransition.Run.isLeaf", "content": "def Run.isLeaf (R : Run S) (root leaf : S) : Prop :=\n  R.reachable root leaf ∧ ∀ y, R (leaf, y) = 0"}], "used_local_lemmas": [{"name": "Utils.StateTransition.finset_ssubset_univ_of_not_mem", "content": "lemma finset_ssubset_univ_of_not_mem {α : Type*} [Fintype α] (s : Finset α) (x : α)\n    (h : x ∉ s) :\n    s ⊂ Finset.univ"}, {"name": "Utils.StateTransition.containsPath_has_positive_transition", "content": "lemma containsPath_has_positive_transition (R : Run S) (path : List S)\n    (h_contains : R.containsPath path) (t : Transition S)\n    (h_in : t ∈ path.zip path.tail) :\n    R t > 0"}, {"name": "Utils.StateTransition.zip_drop_sublist", "content": "lemma zip_drop_sublist (l : List S) (n : ℕ) :\n    ((l.drop n).zip (l.drop (n + 1))).Sublist (l.zip l.tail)"}, {"name": "Utils.StateTransition.containsPath_drop", "content": "lemma containsPath_drop (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.drop n)"}, {"name": "Utils.StateTransition.zip_take_sublist", "content": "lemma zip_take_sublist (l1 l2 : List S) (n m : ℕ) :\n    ((l1.take n).zip (l2.take m)).Sublist (l1.zip l2)"}, {"name": "Utils.StateTransition.tail_take", "content": "lemma tail_take {α : Type*} (l : List α) (n : ℕ) :\n    (l.take n).tail = (l.tail).take (n - 1)"}, {"name": "Utils.StateTransition.containsPath_take", "content": "lemma containsPath_take (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.take n)"}, {"name": "Utils.StateTransition.drop_take_length_ge_two", "content": "lemma drop_take_length_ge_two {α : Type*} (path : List α) (n m : Fin path.length)\n    (h_n_lt_m : n < m) :\n    ((path.drop n.val).take (m.val - n.val + 1)).length ≥ 2"}, {"name": "Utils.StateTransition.getLast_drop_take", "content": "lemma getLast_drop_take {α : Type*} (path : List α) (n k : ℕ)\n    (h_n_lt : n < path.length)\n    (h_bound : n + k ≤ path.length)\n    (h_k_pos : k > 0) :\n    ((path.drop n).take k).getLast? = path[n + k - 1]?"}, {"name": "Utils.StateTransition.drop_take_cycle_same_endpoints", "content": "lemma drop_take_cycle_same_endpoints (path : List S) (x : S) (n m : Fin path.length)\n    (h_n_lt_m : n < m)\n    (h_x_at_n : path[n] = x)\n    (h_x_at_m : path[m] = x) :\n    ((path.drop n.val).take (m.val - n.val + 1)).head? =\n    ((path.drop n.val).take (m.val - n.val + 1)).getLast?"}, {"name": "Utils.StateTransition.containsPath_drop_take", "content": "lemma containsPath_drop_take (R : Run S) (path : List S) (n m : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath ((path.drop n).take m)"}, {"name": "Utils.StateTransition.acyclic_containsPath_nodup", "content": "lemma acyclic_containsPath_nodup (R : Run S) (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_contains : R.containsPath path) :\n    path.Nodup"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton", "content": "lemma countTransitionInPath_append_singleton (path : List S) (x y : S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_not_in : (x, y) ∉ path.zip path.tail) :\n    countTransitionInPath (x, y) (path ++ [y]) = 1"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton_other", "content": "lemma countTransitionInPath_append_singleton_other (path : List S) (x y : S) (t : Transition S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_ne : t ≠ (x, y)) :\n    countTransitionInPath t (path ++ [y]) = countTransitionInPath t path"}, {"name": "Utils.StateTransition.acyclic_no_self_loop", "content": "lemma acyclic_no_self_loop (R : Run S) (s : S) (h_acyclic : R.isAcyclic) (h_edge : R (s, s) > 0) : False"}, {"name": "Utils.StateTransition.getLast_mem", "content": "lemma getLast_mem {α : Type*} (l : List α) (x : α) (h_last : l.getLast? = some x) :\n    x ∈ l"}, {"name": "Utils.StateTransition.last_not_in_zip_tail", "content": "lemma last_not_in_zip_tail {α : Type*} [DecidableEq α] (l : List α) (x : α)\n    (h_nodup : l.Nodup)\n    (h_last : l.getLast? = some x) :\n    ∀ y : α, (x, y) ∉ l.zip l.tail"}, {"name": "Utils.StateTransition.drop_of_lt_length_nonempty", "content": "lemma drop_of_lt_length_nonempty {α : Type*} (path : List α) (i : ℕ)\n    (h_i_lt : i < path.length) :\n    path.drop i ≠ []"}, {"name": "Utils.StateTransition.cycle_from_suffix_contains", "content": "lemma cycle_from_suffix_contains (R : Run S) (suffix : List S) (current y : S)\n    (h_suffix_nodup : suffix.Nodup)\n    (h_contains_suffix : R.containsPath suffix)\n    (h_suffix_nonempty : suffix ≠ [])\n    (h_suffix_last : suffix.getLast? = some current)\n    (h_edge : R (current, y) > 0) :\n    ∀ t : Transition S, countTransitionInPath t (suffix ++ [y]) ≤ R t"}, {"name": "Utils.StateTransition.path_with_back_edge_creates_cycle", "content": "lemma path_with_back_edge_creates_cycle (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_y_in_path : y ∈ path)\n    (h_edge : R (current, y) > 0) :\n    R.hasCycle"}, {"name": "Utils.StateTransition.acyclic_edge_not_in_path", "content": "lemma acyclic_edge_not_in_path (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_edge : R (current, y) > 0)\n    (h_y_in_path : y ∈ path) :\n    False"}, {"name": "Utils.StateTransition.not_mem_implies_transition_not_in_zip_tail", "content": "lemma not_mem_implies_transition_not_in_zip_tail {α : Type*} (path : List α) (x y : α)\n    (h_y_not_in : y ∉ path) :\n    (x, y) ∉ path.zip path.tail"}, {"name": "Utils.StateTransition.containsPath_append_singleton", "content": "lemma containsPath_append_singleton (R : Run S) (path : List S) (x y : S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_contains : R.containsPath path)\n    (h_y_not_in_path : y ∉ path)\n    (h_edge : R (x, y) > 0) :\n    R.containsPath (path ++ [y])"}, {"name": "Utils.StateTransition.acyclic_has_leaf_aux", "content": "lemma acyclic_has_leaf_aux (R : Run S) (root current : S)\n    (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_start : path.head? = some root)\n    (h_end : path.getLast? = some current)\n    (h_nonempty : path ≠ [])\n    (h_contains : R.containsPath path)\n    (h_has_out : ∃ y, y ∉ path ∧ R (current, y) > 0) :\n    ∃ leaf, R.isLeaf root leaf"}, {"name": "Utils.StateTransition.acyclic_has_leaf", "content": "lemma acyclic_has_leaf (R : Run S) (root : S)\n    (h_acyclic : R.isAcyclic)\n    (h_has_out : ∃ y, R (root, y) > 0) :\n    ∃ leaf, R.isLeaf root leaf"}, {"name": "Utils.StateTransition.single_le_sum_of_nonneg", "content": "lemma single_le_sum_of_nonneg {α : Type*} [Fintype α] (f : α → ℤ) (a : α)\n    (h_nonneg : ∀ x, f x ≥ 0) :\n    f a ≤ ∑ x : α, f x"}, {"name": "Utils.StateTransition.sum_pos_of_pos_element", "content": "lemma sum_pos_of_pos_element {α : Type*} [Fintype α] (f : α → ℤ) (a : α)\n    (h_pos : f a > 0)\n    (h_nonneg : ∀ x, f x ≥ 0) :\n    ∑ x : α, f x > 0"}, {"name": "Utils.StateTransition.sum_nat_cast_pos", "content": "lemma sum_nat_cast_pos {α : Type*} [Fintype α] (f : α → ℕ) (a : α)\n    (h_pos : f a > 0) :\n    ∑ x : α, (f x : ℤ) > 0"}, {"name": "Utils.StateTransition.leaf_has_negative_netFlow", "content": "lemma leaf_has_negative_netFlow (R : Run S) (root leaf : S)\n    (h_leaf : R.isLeaf root leaf)\n    (h_in : ∃ y, R (y, leaf) > 0) :\n    R.netFlow leaf < 0"}, {"name": "Utils.StateTransition.sum_zero_of_all_zero", "content": "lemma sum_zero_of_all_zero {α : Type*} [Fintype α] (f : α → ℕ) (h : ∀ x, f x = 0) :\n    ∑ x : α, (f x : ℤ) = 0"}, {"name": "Utils.StateTransition.sum_nat_cast_nonneg", "content": "lemma sum_nat_cast_nonneg {α : Type*} [Fintype α] (f : α → ℕ) :\n    0 ≤ ∑ x : α, (f x : ℤ)"}, {"name": "Utils.StateTransition.sum_nat_cast_zero_of_not_pos", "content": "lemma sum_nat_cast_zero_of_not_pos {α : Type*} [Fintype α] (f : α → ℕ)\n    (h : ∀ x, ¬(f x > 0)) :\n    ∑ x : α, (f x : ℤ) = 0"}, {"name": "Utils.StateTransition.positive_netFlow_has_outgoing_edge", "content": "lemma positive_netFlow_has_outgoing_edge (R : Run S) (s : S)\n    (h_pos : R.netFlow s > 0) :\n    ∃ y, R (s, y) > 0"}, {"name": "Utils.StateTransition.last_has_incoming_transition", "content": "lemma last_has_incoming_transition {α : Type*} (l : List α) (x : α)\n    (h_len : l.length ≥ 2)\n    (h_last : l.getLast? = some x) :\n    ∃ y, (y, x) ∈ l.zip l.tail"}, {"name": "Utils.StateTransition.path_distinct_head_last_length_ge_two", "content": "lemma path_distinct_head_last_length_ge_two {α : Type*} (path : List α) (x y : α)\n    (h_nonempty : path ≠ [])\n    (h_head : path.head? = some x)\n    (h_last : path.getLast? = some y)\n    (h_ne : x ≠ y) :\n    path.length ≥ 2"}, {"name": "Utils.StateTransition.reachable_leaf_has_incoming_edge", "content": "lemma reachable_leaf_has_incoming_edge (R : Run S) (root leaf : S)\n    (h_leaf : R.isLeaf root leaf)\n    (h_ne : root ≠ leaf) :\n    ∃ y, R (y, leaf) > 0"}, {"name": "Utils.StateTransition.unique_negative_netFlow", "content": "lemma unique_negative_netFlow (R : Run S) (s d x : S)\n    (h_source : R.netFlow s = 1)\n    (h_others : ∀ y, y ≠ s → y ≠ d → R.netFlow y = 0)\n    (h_x_neg : R.netFlow x < 0) :\n    x = d"}, {"name": "Utils.StateTransition.leaf_has_incoming_and_negative_netFlow", "content": "lemma leaf_has_incoming_and_negative_netFlow (R : Run S) (root leaf : S)\n    (h_leaf : R.isLeaf root leaf)\n    (h_root_out : ∃ y, R (root, y) > 0) :\n    R.netFlow leaf < 0"}], "local_ctx": "import Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Fintype.Basic\n\nimport Mathlib.Data.Fintype.Prod\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Piecewise\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nnamespace Utils.StateTransition\n\nvariable {S : Type*} [DecidableEq S] [Fintype S]\n\ndef Transition (S : Type*) := S × S\n\ndef Run (S : Type*) := Transition S → ℕ\n\nnoncomputable def Run.netFlow {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) (x : S) : ℤ :=\n  (∑ y : S, (R (x, y) : ℤ)) - (∑ y : S, (R (y, x) : ℤ))\n\ndef countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t\n\ndef Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t\n\ndef Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle\n\ndef Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle\n\ndef Run.reachable [DecidableEq S] (R : Run S) (start finish : S) : Prop :=\n  ∃ (path : List S), path.head? = some start ∧ path.getLast? = some finish ∧\n    path ≠ [] ∧ R.containsPath path\n\ndef Run.isLeaf (R : Run S) (root leaf : S) : Prop :=\n  R.reachable root leaf ∧ ∀ y, R (leaf, y) = 0", "target_theorem": "lemma acyclic_run_has_path_from_source_to_sink (R : Run S) (s d : S)\n    (h_acyclic : R.isAcyclic)\n    (h_source : R.netFlow s = 1)\n    (h_others : ∀ x, x ≠ s → x ≠ d → R.netFlow x = 0) :\n    ∃ (path : List S), path.head? = some s ∧ path.getLast? = some d ∧\n      path ≠ [] ∧ R.containsPath path ∧ path.Nodup :=", "ground_truth_proof": ":= by\n  -- s has positive net flow, so it has an outgoing edge\n  have h_s_out : ∃ y, R (s, y) > 0 := by\n    apply positive_netFlow_has_outgoing_edge\n    rw [h_source]\n    omega\n\n  -- Find a leaf reachable from s\n  obtain ⟨leaf, h_leaf⟩ := acyclic_has_leaf R s h_acyclic h_s_out\n\n  -- The leaf has negative net flow\n  have h_leaf_neg := leaf_has_incoming_and_negative_netFlow R s leaf h_leaf h_s_out\n\n  -- Identify leaf = d (only state with negative net flow)\n  have h_leaf_eq_d := unique_negative_netFlow R s d leaf h_source h_others h_leaf_neg\n\n  -- Extract the path from s to d\n  rw [h_leaf_eq_d] at h_leaf\n  obtain ⟨h_reach, _⟩ := h_leaf\n  obtain ⟨path, h_head, h_last, h_nonempty, h_contains⟩ := h_reach\n  use path\n  refine ⟨h_head, h_last, h_nonempty, h_contains, ?_⟩\n  exact acyclic_containsPath_nodup R path h_acyclic h_contains", "nesting_depth": 8, "transitive_dep_count": 122, "subset_aristotle": true, "category": "Applied verif."}
{"id": 147, "thm_name": "Circuit.FoldlM.operations_eq", "thm_stmt": "theorem operations_eq :\n  (Vector.foldlM circuit init xs).operations n =\n    (List.ofFn fun i => (circuit (foldlAcc n xs circuit init i) xs[i.val]).operations (n + i * constant.localLength)).flatten", "lean_root": "clean", "rel_path": "Clean/Circuit/Loops.lean", "imports": ["import Clean.Utils.Misc", "import Clean.Circuit.Subcircuit", "import Clean.Circuit.Theorems"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "Vector.mk", "module": "Init.Data.Vector.Basic"}, {"name": "List.ofFn", "module": "Init.Data.List.OfFn"}, {"name": "Vector.foldlM", "module": "Init.Data.Vector.Basic"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "induct", "content": "def induct {motive : {n : ℕ} → Vector α n → Sort u}\n    (nil : motive #v[])\n    (cons: ∀ {n : ℕ} (a : α) (as : Vector α n), motive as → motive (cons a as))\n    {n : ℕ} (v : Vector α n) : motive v :=\n  match v with\n  | ⟨ .mk [], h ⟩ => by admit /- proof elided -/\n  | ⟨ .mk (a :: as), h ⟩ => by admit /- proof elided -/"}, {"name": "cons", "content": "def cons (a : α) (v : Vector α n) : Vector α (n + 1) :=\n  ⟨ .mk (a :: v.toList), by admit /- proof elided -/\n  ⟩"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}], "lib_lemmas": [{"name": "Fin.foldl_zero", "module": "Init.Data.Fin.Fold"}, {"name": "List.foldlM_toArray", "module": "Init.Data.List.ToArray"}, {"name": "Vector.foldlM_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "List.foldlM_cons", "module": "Init.Data.List.Control"}, {"name": "Vector.toList_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "Fin.foldl_succ", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.val_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.getElem_cons_succ", "module": "Init.GetElem"}, {"name": "List.getElem_cons_zero", "module": "Init.GetElem"}, {"name": "List.getElem_toArray", "module": "Init.Data.Array.Basic"}, {"name": "Vector.getElem_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "List.flatten_cons", "module": "Init.Data.List.Basic"}, {"name": "List.ofFn_succ", "module": "Init.Data.List.OfFn"}], "repo_lemmas": [{"name": "bind_operations_eq", "content": "theorem bind_operations_eq (f : Circuit F α) (g : α → Circuit F β) (n : ℕ) :\n  (f >>= g).operations n = f.operations n ++ (g (f.output n)).operations (n + f.localLength n)"}], "used_local_defs": [{"name": "Circuit.FoldlM.prod", "content": "@[reducible]\ndef prod (circuit : β → α → Circuit F β) : β × α → Circuit F β := fun t => circuit t.1 t.2"}, {"name": "Circuit.FoldlM.foldlAcc", "content": "def foldlAcc (n : ℕ) (xs : Vector α m) (circuit : β → α → Circuit F β) (init : β) (j : Fin m) : β :=\n  Fin.foldl j (fun acc i => (circuit acc xs[i.val]).output (n + i*(circuit acc xs[i.val]).localLength)) init"}], "used_local_lemmas": [{"name": "Vector.foldlM_toList", "content": "lemma Vector.foldlM_toList (xs : Vector α n) {m : Type → Type} [Monad m] (body : β → α → m β) (init : β) :\n    xs.foldlM body init = xs.toList.foldlM body init"}, {"name": "Circuit.FoldlM.foldlM_cons", "content": "lemma foldlM_cons (x : α) :\n  (Vector.cons x xs).foldlM circuit init = (do\n    let init' ← circuit init x\n    xs.foldlM circuit init')"}, {"name": "Circuit.FoldlM.foldlAcc_zero", "content": "lemma foldlAcc_zero [NeZero m] : foldlAcc n xs circuit init 0 = init"}, {"name": "Circuit.FoldlM.foldlAcc_cons_succ", "content": "lemma foldlAcc_cons_succ (i : Fin m) (x : α) [constant : ConstantLength (prod circuit)] :\n  foldlAcc n (Vector.cons x xs) circuit init i.succ =\n    foldlAcc (n + (circuit init x).localLength n) xs circuit ((circuit init x).output n) i"}], "local_ctx": "import Clean.Circuit.Subcircuit\n\nimport Clean.Utils.Misc\n\nvariable {n m : ℕ} {F : Type} [Field F] {α β : Type}\n\nnamespace Circuit\n\nvariable {prop : Condition F}\n\nnamespace ForM\n\nvariable {circuit : α → Circuit F Unit} (xs : Vector α m) (constant : ConstantLength circuit) (n : ℕ)\n\nend ForM\n\nnamespace MapM\n\nvariable {circuit : α → Circuit F β} {xs : Vector α m} [constant: ConstantLength circuit]\n  {prop : Condition F}\n\nend MapM\n\nnamespace FoldlM\n\n@[reducible]\ndef prod (circuit : β → α → Circuit F β) : β × α → Circuit F β := fun t => circuit t.1 t.2\n\nvariable {env : Environment F} {prop : Condition F} {xs : Vector α m}\n  {circuit : β → α → Circuit F β} {init : β} {constant : ConstantLength (prod circuit)}\n\ndef foldlAcc (n : ℕ) (xs : Vector α m) (circuit : β → α → Circuit F β) (init : β) (j : Fin m) : β :=\n  Fin.foldl j (fun acc i => (circuit acc xs[i.val]).output (n + i*(circuit acc xs[i.val]).localLength)) init", "target_theorem": "theorem operations_eq :\n  (Vector.foldlM circuit init xs).operations n =\n    (List.ofFn fun i => (circuit (foldlAcc n xs circuit init i) xs[i.val]).operations (n + i * constant.localLength)).flatten :=", "ground_truth_proof": ":= by\n  induction xs using Vector.induct generalizing n init with\n  | nil => rfl\n  | cons x xs ih =>\n    rw [foldlM_cons, bind_operations_eq, ih, List.ofFn_succ, List.flatten_cons]\n    simp only [foldlAcc_cons_succ, foldlAcc_zero]\n    simp +arith [Vector.cons, add_mul, constant.localLength_eq (init, x)]", "nesting_depth": 6, "transitive_dep_count": 72, "subset_aristotle": true, "category": "Applied verif."}
{"id": 148, "thm_name": "FlatOperation.localWitnesses_toFlat", "thm_stmt": "lemma localWitnesses_toFlat {ops : Operations F} {env} :\n  (localWitnesses env ops.toFlat).toArray = (ops.localWitnesses env).toArray", "lean_root": "clean", "rel_path": "Clean/Circuit/Theorems.lean", "imports": ["import Clean.Circuit.Provable", "import Clean.Circuit.Basic"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Array", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (ops : Operations F) → Vector F ops.localLength\n  | [] => #v[]\n  | .witness _ c :: ops => c env ++ localWitnesses env ops\n  | .assert _ :: ops => localWitnesses env ops\n  | .lookup _ :: ops => localWitnesses env ops\n  | .subcircuit s :: ops => s.witnesses env ++ localWitnesses env ops"}, {"name": "Subcircuit.witnesses", "content": "@[reducible, circuit_norm]\ndef Subcircuit.witnesses (sc : Subcircuit F n) env :=\n  (FlatOperation.localWitnesses env sc.ops).cast sc.localLength_eq.symm"}, {"name": "localWitnesses", "content": "def localWitnesses (env : Environment F) : (op : Operation F) → Vector F op.localLength\n  | .witness _ c => c env\n  | .assert _ => #v[]\n  | .lookup _ => #v[]\n  | .subcircuit s => s.witnesses env"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}], "lib_lemmas": [{"name": "Array.append_assoc", "module": "Init.Data.Array.Bootstrap"}, {"name": "Array.empty_append", "module": "Init.Data.Array.Bootstrap"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "Vector.toArray_append", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.toArray_empty", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.toArray_cast", "module": "Init.Data.Vector.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "FlatOperation.localWitnesses_append", "content": "lemma localWitnesses_append {F} {a b: List (FlatOperation F)} {env} :\n    (localWitnesses env (a ++ b)).toArray = (localWitnesses env a).toArray ++ (localWitnesses env b).toArray"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nvariable {F : Type} [Field F] {α β : Type}\n\nnamespace Operations\n\nend Operations\n\nnamespace Circuit\n\nend Circuit\n\nnamespace FlatOperation", "target_theorem": "lemma localWitnesses_toFlat {ops : Operations F} {env} :\n  (localWitnesses env ops.toFlat).toArray = (ops.localWitnesses env).toArray :=", "ground_truth_proof": ":= by\n  induction ops using Operations.induct with\n  | empty => trivial\n  | witness _ _ _ ih | assert _ _ ih | lookup _ _ ih | subcircuit _ _ ih =>\n    simp only [Operations.toFlat, Operations.localLength, Operations.localWitnesses, Vector.toArray_append]\n    rw [←ih]\n    try rw [localWitnesses_append]\n    try simp only [localLength, localWitnesses, Vector.toArray_append, Subcircuit.witnesses, Vector.toArray_cast]", "nesting_depth": 9, "transitive_dep_count": 47, "subset_aristotle": true, "category": "Applied verif."}
{"id": 149, "thm_name": "Tables.KeccakInductive.tableStatement", "thm_stmt": "theorem tableStatement (output : KeccakState (F p)) : ∀ n > 0, ∀ trace, ∃ blocks, blocks.length = n - 1 ∧\n  (formalTable output).statement n trace →\n    output.Normalized ∧ output.value = absorbBlocks blocks", "lean_root": "clean", "rel_path": "Clean/Tables/KeccakInductive.lean", "imports": ["import Clean.Utils.Vector", "import Clean.Table.Basic", "import Clean.Gadgets.Keccak.AbsorbBlock", "import Clean.Types.U64", "import Clean.Circuit.Extensions", "import Clean.Table.Inductive", "import Clean.Specs.Keccak256"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "UInt64", "module": "Init.Prelude"}, {"name": "Vector.finRange", "module": "Init.Data.Vector.FinRange"}, {"name": "Vector.map", "module": "Init.Data.Vector.Basic"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Vector.push", "module": "Init.Data.Vector.Basic"}, {"name": "Fin.reduceEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "Fin.reduceFinMk", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "IO", "module": "Init.System.IO"}, {"name": "UInt32", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Not", "module": "Init.Prelude"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "Vector.foldlM", "module": "Init.Data.Vector.Basic"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.snd", "module": "Init.Prelude"}, {"name": "StateM", "module": "Init.Control.State"}, {"name": "List.Forall", "module": "Mathlib.Data.List.Defs"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem"}, {"name": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty", "content": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|simp_assign_row) =>\n    `(tactic|(\n    simp only [assignCurrRow, assignNextRow, size]\n    rw [List.finRange, List.ofFn]\n    repeat rw [Fin.foldr_succ]\n    rw [Fin.foldr_zero]\n    repeat rw [List.forM_cons]\n    rw [List.forM_nil, bind_pure_unit]\n    simp only [seval, toVars, toElements, Fin.cast_eq_self, Fin.val_zero, Fin.val_one, Fin.isValue,\n      List.getElem_toArray, List.getElem_cons_zero, List.getElem_cons_succ, Fin.succ_zero_eq_one]))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "KeccakBlock.Normalized", "content": "def KeccakBlock.Normalized (block : KeccakBlock (F p)) :=\n  ∀ i : Fin RATE, block[i.val].Normalized"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "RATE", "content": "@[reducible] def RATE := 17\nexample : RATE + CAPACITY = 25 := rfl"}, {"name": "CAPACITY", "content": "@[reducible] def CAPACITY := 8"}, {"name": "value", "content": "def value (x : U64 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3 +\n  x.x4.val * 256^4 + x.x5.val * 256^5 + x.x6.val * 256^6 + x.x7.val * 256^7"}, {"name": "U64", "content": "structure U64 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\n  x4 : T\n  x5 : T\n  x6 : T\n  x7 : T\nderiving DecidableEq"}, {"name": "KeccakRow.value", "content": "def KeccakRow.value (row : KeccakRow (F p)) := row.map U64.value"}, {"name": "map", "content": "def map {α β : Type} (x : U64 α) (f : α → β) : U64 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3, f x.x4, f x.x5, f x.x6, f x.x7 ⟩"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "InductiveTable", "content": "structure InductiveTable (F : Type) [Field F] (State Input : Type → Type) [ProvableType State] [ProvableType Input] where\n   \n  step : Var State F → Var Input F → Circuit F (Var State F)\n\n   \n  Spec : (initialState : State F) → (xs : List (Input F)) → (i : ℕ) → (xs.length = i) → (currentState : State F) → Prop\n\n   \n  InputAssumptions : ℕ → Input F → Prop := fun _ _ => True\n  InitialStateAssumptions : State F → Prop := fun _ => True\n\n  soundness : InductiveTable.Soundness F State Input Spec step\n\n  completeness : InductiveTable.Completeness F State Input InputAssumptions InitialStateAssumptions Spec step\n\n  subcircuitsConsistent : ∀ acc x, ((step acc x).operations ((size State) + (size Input))).SubcircuitsConsistent ((size State) + (size Input))\n    := by admit /- proof elided -/"}, {"name": "KeccakState.Normalized", "content": "def KeccakState.Normalized (state : KeccakState (F p)) :=\n  ∀ i : Fin 25, state[i.val].Normalized"}, {"name": "Normalized", "content": "def Normalized (x : U32 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256"}, {"name": "U32", "content": "structure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq"}, {"name": "KeccakRow.Normalized", "content": "def KeccakRow.Normalized (row : KeccakRow (F p)) :=\n  ∀ i : Fin 5, row[i.val].Normalized"}, {"name": "KeccakBlock.value", "content": "def KeccakBlock.value (block : KeccakBlock (F p)) := block.map U64.value"}, {"name": "Spec", "content": "@[reducible] def Spec (input : Input (F p)) (out_state : KeccakState (F p)) :=\n  out_state.Normalized ∧\n  out_state.value = absorbBlock input.state.value input.block.value"}, {"name": "Input", "content": "structure Input (F : Type) where\n  state : KeccakState F\n  block : KeccakBlock F"}, {"name": "KeccakBlock", "content": "@[reducible] def KeccakBlock := ProvableVector U64 RATE"}, {"name": "ProvableVector.instance", "content": "instance ProvableVector.instance : ProvableType (ProvableVector α n) where\n  size := n * size α\n  toElements x := x.map toElements |>.flatten\n  fromElements v := v.toChunks (psize α) |>.map fromElements\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "KeccakState", "content": "@[reducible] def KeccakState := ProvableVector U64 25"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  state : KeccakState F\n  d : KeccakRow F"}, {"name": "state", "content": "def state : Vector (U64 ℕ) 25 := #v[\n  ⟨67, 168, 144, 181, 2, 173, 144, 47⟩,\n  ⟨114, 52, 107, 105, 171, 22, 114, 75⟩,\n  ⟨196, 118, 22, 253, 100, 162, 87, 52⟩,\n  ⟨50, 65, 171, 81, 229, 6, 172, 155⟩,\n  ⟨178, 167, 68, 225, 82, 73, 216, 194⟩,\n  ⟨193, 5, 52, 193, 148, 168, 64, 147⟩,\n  ⟨212, 142, 107, 244, 55, 237, 100, 203⟩,\n  ⟨101, 34, 195, 62, 133, 216, 64, 34⟩,\n  ⟨240, 214, 204, 27, 17, 231, 66, 179⟩,\n  ⟨136, 37, 228, 137, 64, 208, 27, 90⟩,\n  ⟨177, 229, 130, 4, 191, 7, 25, 117⟩,\n  ⟨124, 168, 245, 7, 222, 138, 168, 16⟩,\n  ⟨115, 130, 213, 74, 217, 123, 172, 109⟩,\n  ⟨128, 149, 137, 6, 45, 133, 77, 101⟩,\n  ⟨104, 90, 153, 237, 72, 44, 164, 84⟩,\n  ⟨129, 177, 235, 28, 82, 30, 150, 201⟩,\n  ⟨52, 55, 32, 241, 142, 211, 246, 68⟩,\n  ⟨149, 124, 124, 204, 34, 220, 229, 69⟩,\n  ⟨215, 168, 47, 96, 70, 5, 220, 2⟩,\n  ⟨53, 224, 38, 18, 110, 66, 70, 9⟩,\n  ⟨213, 122, 200, 196, 186, 122, 207, 42⟩,\n  ⟨141, 103, 32, 88, 244, 160, 37, 76⟩,\n  ⟨99, 242, 138, 24, 4, 30, 100, 196⟩,\n  ⟨141, 253, 136, 54, 8, 21, 204, 152⟩,\n  ⟨93, 161, 29, 12, 44, 252, 49, 57⟩\n]"}, {"name": "absorbBlock", "content": "def absorbBlock (state : Vector ℕ 25) (block : Vector ℕ RATE) : Vector ℕ 25 :=\n  \n  let state' := Vector.mapFinRange 25 fun i => state[i] ^^^ (if _ : i.val < RATE then block[i] else 0)\n  \n  keccakPermutation state'"}, {"name": "keccakPermutation", "content": "def keccakPermutation (state : Vector ℕ 25): Vector ℕ 25 :=\n  roundConstants.foldl keccakRound state"}, {"name": "keccakRound", "content": "def keccakRound (state : Vector ℕ 25) (rc : UInt64) : Vector ℕ 25 :=\n  let theta_state := theta state\n  let rho_pi_state := rhoPi theta_state\n  let chi_state := chi rho_pi_state\n  iota chi_state rc"}, {"name": "theta", "content": "def theta (state : Vector ℕ 25) : Vector ℕ 25 :=\n  let c := thetaC state\n  let d := thetaD c\n  thetaXor state d"}, {"name": "thetaXor", "content": "def thetaXor (state : Vector ℕ 25) (d : Vector ℕ 5) : Vector ℕ 25 :=\n  #v[\n    state[0] ^^^ d[0],\n    state[1] ^^^ d[0],\n    state[2] ^^^ d[0],\n    state[3] ^^^ d[0],\n    state[4] ^^^ d[0],\n    state[5] ^^^ d[1],\n    state[6] ^^^ d[1],\n    state[7] ^^^ d[1],\n    state[8] ^^^ d[1],\n    state[9] ^^^ d[1],\n    state[10] ^^^ d[2],\n    state[11] ^^^ d[2],\n    state[12] ^^^ d[2],\n    state[13] ^^^ d[2],\n    state[14] ^^^ d[2],\n    state[15] ^^^ d[3],\n    state[16] ^^^ d[3],\n    state[17] ^^^ d[3],\n    state[18] ^^^ d[3],\n    state[19] ^^^ d[3],\n    state[20] ^^^ d[4],\n    state[21] ^^^ d[4],\n    state[22] ^^^ d[4],\n    state[23] ^^^ d[4],\n    state[24] ^^^ d[4]\n  ]"}, {"name": "thetaC", "content": "def thetaC (state : Vector ℕ 25) : Vector ℕ 5 :=\n  #v[\n    state[0] ^^^ state[1] ^^^ state[2] ^^^ state[3] ^^^ state[4],\n    state[5] ^^^ state[6] ^^^ state[7] ^^^ state[8] ^^^ state[9],\n    state[10] ^^^ state[11] ^^^ state[12] ^^^ state[13] ^^^ state[14],\n    state[15] ^^^ state[16] ^^^ state[17] ^^^ state[18] ^^^ state[19],\n    state[20] ^^^ state[21] ^^^ state[22] ^^^ state[23] ^^^ state[24]\n  ]"}, {"name": "thetaD", "content": "def thetaD (c : Vector ℕ 5) : Vector ℕ 5 :=\n  #v[\n    c[4] ^^^ (rotLeft64 c[1] 1),\n    c[0] ^^^ (rotLeft64 c[2] 1),\n    c[1] ^^^ (rotLeft64 c[3] 1),\n    c[2] ^^^ (rotLeft64 c[4] 1),\n    c[3] ^^^ (rotLeft64 c[0] 1)\n  ]"}, {"name": "rotLeft64", "content": "def rotLeft64 (value : ℕ) (left : Fin 64) : ℕ:=\n  let right := (64 - left) % 64\n  rotRight64 value right"}, {"name": "rotRight64", "content": "def rotRight64 (x : ℕ) (offset : ℕ) : ℕ :=\n  let offset := offset % 64\n  let low := x % (2^offset)\n  let high := x / (2^offset)\n  low * (2^(64-offset)) + high"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "iota", "content": "def iota (state : Vector ℕ 25) (rc : UInt64) : Vector ℕ 25 :=\n  state.set 0 ((state[0]) ^^^ rc.toFin)"}, {"name": "rhoPi", "content": "def rhoPi (state : Vector ℕ 25) : Vector ℕ 25 :=\n  #v[\n    rotLeft64 state[0] 0,\n    rotLeft64 state[15] 28,\n    rotLeft64 state[5] 1,\n    rotLeft64 state[20] 27,\n    rotLeft64 state[10] 62,\n    rotLeft64 state[6] 44,\n    rotLeft64 state[21] 20,\n    rotLeft64 state[11] 6,\n    rotLeft64 state[1] 36,\n    rotLeft64 state[16] 55,\n    rotLeft64 state[12] 43,\n    rotLeft64 state[2] 3,\n    rotLeft64 state[17] 25,\n    rotLeft64 state[7] 10,\n    rotLeft64 state[22] 39,\n    rotLeft64 state[18] 21,\n    rotLeft64 state[8] 45,\n    rotLeft64 state[23] 8,\n    rotLeft64 state[13] 15,\n    rotLeft64 state[3] 41,\n    rotLeft64 state[24] 14,\n    rotLeft64 state[14] 61,\n    rotLeft64 state[4] 18,\n    rotLeft64 state[19] 56,\n    rotLeft64 state[9] 2\n  ]"}, {"name": "chi", "content": "def chi (b : Vector ℕ 25) : Vector ℕ 25 :=\n  #v[\n    b[0] ^^^ ((not64 b[5]) &&& b[10]),\n    b[1] ^^^ ((not64 b[6]) &&& b[11]),\n    b[2] ^^^ ((not64 b[7]) &&& b[12]),\n    b[3] ^^^ ((not64 b[8]) &&& b[13]),\n    b[4] ^^^ ((not64 b[9]) &&& b[14]),\n    b[5] ^^^ ((not64 b[10]) &&& b[15]),\n    b[6] ^^^ ((not64 b[11]) &&& b[16]),\n    b[7] ^^^ ((not64 b[12]) &&& b[17]),\n    b[8] ^^^ ((not64 b[13]) &&& b[18]),\n    b[9] ^^^ ((not64 b[14]) &&& b[19]),\n    b[10] ^^^ ((not64 b[15]) &&& b[20]),\n    b[11] ^^^ ((not64 b[16]) &&& b[21]),\n    b[12] ^^^ ((not64 b[17]) &&& b[22]),\n    b[13] ^^^ ((not64 b[18]) &&& b[23]),\n    b[14] ^^^ ((not64 b[19]) &&& b[24]),\n    b[15] ^^^ ((not64 b[20]) &&& b[0]),\n    b[16] ^^^ ((not64 b[21]) &&& b[1]),\n    b[17] ^^^ ((not64 b[22]) &&& b[2]),\n    b[18] ^^^ ((not64 b[23]) &&& b[3]),\n    b[19] ^^^ ((not64 b[24]) &&& b[4]),\n    b[20] ^^^ ((not64 b[0]) &&& b[5]),\n    b[21] ^^^ ((not64 b[1]) &&& b[6]),\n    b[22] ^^^ ((not64 b[2]) &&& b[7]),\n    b[23] ^^^ ((not64 b[3]) &&& b[8]),\n    b[24] ^^^ ((not64 b[4]) &&& b[9])\n  ]"}, {"name": "not64", "content": "def not64 (a : ℕ) : ℕ := a ^^^ 0xffffffffffffffff"}, {"name": "roundConstants", "content": "def roundConstants : Vector UInt64 24 := #v[\n  0x0000000000000001, 0x0000000000008082,\n  0x800000000000808a, 0x8000000080008000,\n  0x000000000000808b, 0x0000000080000001,\n  0x8000000080008081, 0x8000000000008009,\n  0x000000000000008a, 0x0000000000000088,\n  0x0000000080008009, 0x000000008000000a,\n  0x000000008000808b, 0x800000000000008b,\n  0x8000000000008089, 0x8000000000008003,\n  0x8000000000008002, 0x8000000000000080,\n  0x000000000000800a, 0x800000008000000a,\n  0x8000000080008081, 0x8000000000008080,\n  0x0000000080000001, 0x8000000080008008\n]"}, {"name": "mapFinRange", "content": "def mapFinRange (n : ℕ) (create : Fin n → α) : Vector α n := finRange n |>.map create"}, {"name": "KeccakState.value", "content": "def KeccakState.value (state : KeccakState (F p)) := state.map U64.value"}, {"name": "value", "content": "def value (x : U32 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3"}, {"name": "Normalized", "content": "def Normalized (x : U64 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256 ∧\n  x.x4.val < 256 ∧ x.x5.val < 256 ∧ x.x6.val < 256 ∧ x.x7.val < 256"}, {"name": "Assumptions", "content": "@[reducible] def Assumptions (input : Input (F p)) :=\n  input.state.Normalized ∧ input.block.Normalized"}, {"name": "InductiveTable.Completeness", "content": "def InductiveTable.Completeness (F : Type) [Field F] (State Input : Type → Type) [ProvableType State] [ProvableType Input]\n    (InputAssumptions : ℕ → Input F → Prop) (InitialStateAssumptions : State F → Prop)\n    (Spec : (initialState : State F) → (xs : List (Input F)) → (i : ℕ) → (xs.length = i) → (currentState : State F) → Prop)\n    (step : Var State F → Var Input F → Circuit F (Var State F)) :=\n  ∀ (initialState : State F) (row_index : ℕ) (env : Environment F),\n  \n  ∀ (acc_var : Var State F) (x_var : Var Input F)\n    (acc : State F) (x : Input F) (xs : List (Input F)) (xs_len : xs.length = row_index),\n    (eval env acc_var = acc) ∧ (eval env x_var = x) →\n  \n  env.UsesLocalWitnessesCompleteness ((size State) + (size Input)) (step acc_var x_var |>.operations ((size State) + (size Input))) →\n  \n  InitialStateAssumptions initialState ∧\n  Spec initialState xs row_index xs_len acc ∧ InputAssumptions row_index x →\n  \n  Circuit.ConstraintsHold.Completeness env (step acc_var x_var |>.operations ((size State) + (size Input)))"}, {"name": "ConstraintsHold", "content": "@[circuit_norm]\ndef ConstraintsHold (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold eval ops\n  | .lookup { table, entry, .. } :: ops =>\n    table.Contains (entry.map eval) ∧ ConstraintsHold eval ops\n  | .subcircuit s :: ops =>\n    ConstraintsHoldFlat eval s.ops ∧ ConstraintsHold eval ops"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "map", "content": "def map {α β : Type} (x : U32 α) (f : α → β) : U32 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3 ⟩"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) Input KeccakState :=\n  { elaborated with Assumptions, Spec, soundness, completeness }"}, {"name": "elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Input KeccakState where\n  main\n  localLength _ := 31048\n  output _ i0 := Permutation.stateVar (i0 + 136) 23\n\n  localLength_eq _ _ := by admit /- proof elided -/"}, {"name": "main", "content": "def main (input : Var Input (F p)) : Circuit (F p) (Var KeccakState (F p)) := do\n  let { state, block } := input\n  \n  let state_rate ← Circuit.mapFinRange RATE fun i => Xor64.circuit ⟨state[i.val], block[i.val]⟩\n\n  \n  let state_capacity := Vector.mapFinRange (25 - RATE) fun i => state[RATE + i.val]\n  let state' : Vector _ 25 := state_rate ++ state_capacity\n\n  \n  Permutation.circuit state'"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) KeccakState KeccakState := {\n  elaborated with\n  Assumptions, Spec, soundness\n  \n  \n  completeness := by admit /- proof elided -/"}, {"name": "Spec", "content": "def Spec (state : KeccakState (F p)) (out_state : KeccakState (F p)) :=\n  out_state.Normalized\n  ∧ out_state.value = keccakPermutation state.value"}, {"name": "elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) KeccakState KeccakState where\n  main\n  localLength _ := 30912\n  output _ i0 := stateVar i0 23\n\n  localLength_eq state i0 := by admit /- proof elided -/"}, {"name": "main", "content": "def main (state : Var KeccakState (F p)) : Circuit (F p) (Var KeccakState (F p)) :=\n  .foldl roundConstants state\n    fun state rc => KeccakRound.circuit rc state"}, {"name": "circuit", "content": "def circuit (rc : UInt64) : FormalCircuit (F p) KeccakState KeccakState := {\n  elaborated rc with\n  Spec := Spec rc\n  Assumptions\n  soundness := soundness rc,\n  completeness := completeness rc,\n}"}, {"name": "Spec", "content": "def Spec (rc : UInt64) (state : KeccakState (F p)) (out_state : KeccakState (F p)) :=\n  out_state.Normalized\n  ∧ out_state.value = keccakRound state.value rc"}, {"name": "Assumptions", "content": "def Assumptions (state : KeccakState (F p)) := state.Normalized"}, {"name": "elaborated", "content": "instance elaborated (rc : UInt64) : ElaboratedCircuit (F p) KeccakState KeccakState where\n  main := main rc\n  localLength _ := 1288\n  output _ i0 := (Vector.mapRange 25 fun i => varFromOffset U64 (i0 + i*16 + 888) ).set 0 (varFromOffset U64 (i0 + 1280))\n\n  localLength_eq _ _ := by admit /- proof elided -/"}, {"name": "main", "content": "def main (rc : UInt64) (state : Var KeccakState (F p)) : Circuit (F p) (Var KeccakState (F p)) := do\n  let state ← Theta.circuit state\n  let state ← RhoPi.circuit state\n  let state ← Chi.circuit state\n\n  \n  let s0 ← Xor64.circuit ⟨state[0], const (U64.fromUInt64 rc)⟩\n  return state.set 0 s0"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) Inputs U64 where\n  Assumptions\n  Spec\n  soundness\n  completeness"}, {"name": "Inputs", "content": "structure Inputs (F : Type) where\n  x: U64 F\n  y: U64 F"}, {"name": "Spec", "content": "def Spec (input : Inputs (F p)) (z : U64 (F p)) :=\n  let ⟨x, y⟩ := input\n  z.value = x.value ^^^ y.value ∧ z.Normalized"}, {"name": "Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.Normalized ∧ y.Normalized"}, {"name": "fromUInt64", "content": "def fromUInt64 (x : UInt64) : U64 (F p) :=\n  decomposeNat x.toFin"}, {"name": "decomposeNat", "content": "def decomposeNat (x : ℕ) : U64 (F p) :=\n  let x0 := x % 256\n  let x1 : ℕ := (x / 256) % 256\n  let x2 : ℕ := (x / 256^2) % 256\n  let x3 : ℕ := (x / 256^3) % 256\n  let x4 : ℕ := (x / 256^4) % 256\n  let x5 : ℕ := (x / 256^5) % 256\n  let x6 : ℕ := (x / 256^6) % 256\n  let x7 : ℕ := (x / 256^7) % 256\n  ⟨ x0, x1, x2, x3, x4, x5, x6, x7 ⟩"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) KeccakState KeccakState :=\n  { elaborated with Assumptions, Spec, soundness, completeness }"}, {"name": "Assumptions", "content": "def Assumptions := KeccakState.Normalized (p:=p)"}, {"name": "elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) KeccakState KeccakState where\n  main\n  localLength _ := 400\n  localLength_eq _ _ := by admit /- proof elided -/"}, {"name": "main", "content": "def main (state : Var KeccakState (F p)) : Circuit (F p) (Var KeccakState (F p)) :=\n  .map rhoPiConstants fun (i, s) =>\n    Rotation64.circuit (-s) state[i.val]"}, {"name": "rhoPiConstants", "content": "def rhoPiConstants := rhoPiIndices.zip rhoPiShifts"}, {"name": "rhoPiIndices", "content": "def rhoPiIndices : Vector (Fin 25) 25 := #v[\n  0, 15, 5, 20, 10, 6, 21, 11, 1, 16, 12, 2, 17, 7, 22, 18, 8, 23, 13, 3, 24, 14, 4, 19, 9\n]"}, {"name": "rhoPiShifts", "content": "def rhoPiShifts : Vector (Fin 64) 25 := #v[\n  0, 28, 1, 27, 62, 44, 20, 6, 36, 55, 43, 3, 25, 10, 39, 21, 45, 8, 15, 41, 14, 61, 18, 56, 2\n]"}, {"name": "circuit", "content": "def circuit (offset : Fin 64) : FormalCircuit (F p) U64 U64 := {\n  elaborated offset with\n  Assumptions\n  Spec := Spec offset\n  soundness := soundness offset\n  completeness := completeness offset\n}"}, {"name": "Spec", "content": "def Spec (offset : Fin 64) (x : U64 (F p)) (y : U64 (F p)) :=\n  y.value = rotRight64 x.value offset.val\n  ∧ y.Normalized"}, {"name": "elaborated", "content": "def elaborated (off : Fin 64) : ElaboratedCircuit (F p) U64 U64 where\n  main := main off\n  localLength _ := 16\n  output _ i0 := output off i0"}, {"name": "main", "content": "def main (offset : Fin 64) (x : Var U64 (F p)) : Circuit (F p) (Var U64 (F p)) := do\n  let byte_offset : Fin 8 := ⟨ offset.val / 8, by admit /- proof elided -/\n  ⟩\n  let bit_offset : Fin 8 := ⟨ offset.val % 8, by admit /- proof elided -/\n  ⟩\n\n  \n  let byte_rotated ← Rotation64Bytes.circuit byte_offset x\n  Rotation64Bits.circuit bit_offset byte_rotated"}, {"name": "circuit", "content": "def circuit (offset : Fin 8) : FormalCircuit (F p) U64 U64 := {\n  elaborated offset with\n  Assumptions\n  Spec := Spec offset\n  soundness := soundness offset\n  completeness := completeness offset\n}"}, {"name": "elaborated", "content": "def elaborated (off : Fin 8) : ElaboratedCircuit (F p) U64 U64 where\n  main := main off\n  localLength _ := 16\n  output _ i0 := output off i0\n  localLength_eq _ i0 := by admit /- proof elided -/"}, {"name": "output", "content": "def output (offset : Fin 8) (i0 : ℕ) : U64 (Expression (F p)) :=\n  U64.fromLimbs (.ofFn fun ⟨i,_⟩ =>\n    (var ⟨i0 + i*2 + 1⟩) + var ⟨i0 + (i + 1) % 8 * 2⟩ * .const ((2^(8-offset.val) : ℕ) : F p))"}, {"name": "fromLimbs", "content": "def fromLimbs {F} (v : Vector F 8) : U64 F := fromElements v"}, {"name": "main", "content": "def main (offset : Fin 8) (x : Var U64 (F p)) : Circuit (F p) (Var U64 (F p)) := do\n  let parts ← Circuit.map x.toLimbs (ByteDecomposition.circuit offset)\n  let lows := parts.map Outputs.low\n  let highs := parts.map Outputs.high\n\n  let rotated := highs.zip (lows.rotate 1) |>.map fun (high, low) =>\n    high + low * ((2^(8-offset.val) : ℕ) : F p)\n\n  return U64.fromLimbs rotated"}, {"name": "Outputs", "content": "structure Outputs (F : Type) where\n  low : F\n  high : F"}, {"name": "toLimbs", "content": "def toLimbs {F} (x : U64 F) : Vector F 8 := toElements x"}, {"name": "circuit", "content": "def circuit (offset : Fin 8) : FormalCircuit (F p) field Outputs := {\n  elaborated offset with\n  main := main offset\n  Assumptions\n  Spec := Spec offset\n  soundness := soundness offset\n  completeness := completeness offset\n}"}, {"name": "Spec", "content": "def Spec (offset : Fin 8) (x : F p) (out : Outputs (F p)) :=\n  let ⟨low, high⟩ := out\n  (low.val = x.val % (2^offset.val) ∧ high.val = x.val / (2^offset.val))\n  ∧ (low.val < 2^offset.val ∧ high.val < 2^(8-offset.val))"}, {"name": "main", "content": "def main (offset : Fin 8) (x : Expression (F p)) : Circuit (F p) (Var Outputs (F p)) := do\n  let low ← witness fun env => mod (env x) (2^offset.val) (by admit /- proof elided -/\n  )\n  let high ← witness fun env => floorDiv (env x) (2^offset.val)\n\n  lookup ByteTable ((2^(8-offset.val) : F p) * low)\n  lookup ByteTable high\n\n  x === low + high * (2^offset.val : F p)\n\n  return { low, high }"}, {"name": "floorDiv", "content": "def floorDiv (x : F p) (c : ℕ+) : F p :=\n  FieldUtils.natToField (x.val / c) (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "less_than_p", "content": "def less_than_p (x : F p) : x.val < p :="}, {"name": "ByteTable", "content": "def ByteTable : Table (F p) field := .fromStatic {\n  name := \"Bytes\"\n  length := 256\n\n  row i := fromByte i\n  index x := x.val\n\n  Spec x := x.val < 256\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "mod", "content": "def mod (x : F p) (c : ℕ+) (lt : c < p) : F p :=\n  FieldUtils.natToField (x.val % c) (by admit /- proof elided -/\n  )"}, {"name": "HasAssertEq", "content": "class HasAssertEq (β : Type) (F : outParam Type) [Field F] where\n  assert_eq : β → β → Circuit F Unit"}, {"name": "Assumptions", "content": "def Assumptions (x : F p) := x.val < 256"}, {"name": "elaborated", "content": "def elaborated (offset : Fin 8) : ElaboratedCircuit (F p) field Outputs where\n  main := main offset\n  localLength _ := 2\n  output _ i0 := varFromOffset Outputs i0"}, {"name": "varFromOffset", "content": "@[explicit_provable_type]\ndef varFromOffset (α : TypeMap) [ProvableType α] (offset : ℕ) : Var α F :=\n  let vars := Vector.mapRange (size α) fun i => var ⟨offset + i⟩\n  fromVars vars"}, {"name": "mapRange", "content": "def mapRange (n : ℕ) (create : ℕ → α) : Vector α n :=\n  match n with\n  | 0 => #v[]\n  | k + 1 => mapRange k create |>.push (create k)"}, {"name": "rotate", "content": "def rotate {α : Type} {n : ℕ} (v : Vector α n) (off : ℕ) : Vector α n :=\n  ⟨(v.toList.rotate off).toArray, by admit /- proof elided -/\n  ⟩"}, {"name": "toLimbs", "content": "def toLimbs {F} (x : U32 F) : Vector F 4 := toElements x"}, {"name": "Assumptions", "content": "def Assumptions (input : U64 (F p)) := input.Normalized"}, {"name": "Spec", "content": "def Spec (offset : Fin 8) (x : U64 (F p)) (y : U64 (F p)) :=\n  y.value = rotRight64 x.value offset.val\n  ∧ y.Normalized"}, {"name": "circuit", "content": "def circuit (off : Fin 8) : FormalCircuit (F p) U64 U64 := {\n  elaborated off with\n  main := main off\n  Assumptions\n  Spec := Spec off\n  soundness := soundness off\n  completeness := completeness off\n}"}, {"name": "main", "content": "def main (offset : Fin 8) (input : Var U64 (F p)) : Circuit (F p) (Var U64 (F p)) := do\n  let ⟨x0, x1, x2, x3 , x4, x5, x6, x7⟩ := input\n\n  if offset = 0 then\n    return ⟨ x0, x1, x2, x3, x4, x5, x6, x7 ⟩\n  else if offset = 1 then\n    return ⟨ x1, x2, x3, x4, x5, x6, x7, x0 ⟩\n  else if offset = 2 then\n    return ⟨ x2, x3, x4, x5, x6, x7, x0, x1 ⟩\n  else if offset = 3 then\n    return ⟨ x3, x4, x5, x6, x7, x0, x1, x2 ⟩\n  else if offset = 4 then\n    return ⟨ x4, x5, x6, x7, x0, x1, x2, x3 ⟩\n  else if offset = 5 then\n    return ⟨ x5, x6, x7, x0, x1, x2, x3, x4 ⟩\n  else if offset = 6 then\n    return ⟨ x6, x7, x0, x1, x2, x3, x4, x5 ⟩\n  else\n    return ⟨ x7, x0, x1, x2, x3, x4, x5, x6 ⟩"}, {"name": "elaborated", "content": "instance elaborated (off : Fin 8): ElaboratedCircuit (F p) U64 U64 where\n  main := main off\n  localLength _ := 0\n  output input i0 :=\n    let ⟨x0, x1, x2, x3, x4, x5, x6, x7⟩ := input\n    match off with\n    | 0 => ⟨ x0, x1, x2, x3, x4, x5, x6, x7 ⟩\n    | 1 => ⟨ x1, x2, x3, x4, x5, x6, x7, x0 ⟩\n    | 2 => ⟨ x2, x3, x4, x5, x6, x7, x0, x1 ⟩\n    | 3 => ⟨ x3, x4, x5, x6, x7, x0, x1, x2 ⟩\n    | 4 => ⟨ x4, x5, x6, x7, x0, x1, x2, x3 ⟩\n    | 5 => ⟨ x5, x6, x7, x0, x1, x2, x3, x4 ⟩\n    | 6 => ⟨ x6, x7, x0, x1, x2, x3, x4, x5 ⟩\n    | 7 => ⟨ x7, x0, x1, x2, x3, x4, x5, x6 ⟩\n  subcircuitsConsistent x i0 := by admit /- proof elided -/"}, {"name": "Spec", "content": "def Spec (offset : Fin 8) (x : U64 (F p)) (y : U64 (F p)) :=\n  y.value = rotRight64 x.value (offset.val * 8) ∧ y.Normalized"}, {"name": "output", "content": "def output (offset : Fin 64) (i0 : ℕ) : U64 (Expression (F p)) :=\n  Rotation64Bits.output ⟨ offset.val % 8, by admit /- proof elided -/\n  ⟩ i0"}, {"name": "Spec", "content": "def Spec (state : KeccakState (F p)) (out_state : KeccakState (F p)) :=\n  out_state.Normalized\n  ∧ out_state.value = Specs.Keccak256.rhoPi state.value"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) KeccakState KeccakState :=\n { elaborated with Assumptions, Spec, soundness, completeness }"}, {"name": "Spec", "content": "def Spec (state : KeccakState (F p)) (out_state : KeccakState (F p)) : Prop :=\n  out_state.Normalized\n  ∧ out_state.value = Specs.Keccak256.theta state.value"}, {"name": "elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) KeccakState KeccakState where\n  main\n  localLength _ := 480"}, {"name": "main", "content": "def main (state : Var KeccakState (F p)) : Circuit (F p) (Var KeccakState (F p)) := do\n  let c ← ThetaC.circuit state\n  let d ← ThetaD.circuit c\n  ThetaXor.circuit ⟨state, d⟩"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) KeccakRow KeccakRow :=\n  { elaborated with Assumptions, Spec, soundness, completeness }"}, {"name": "elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) KeccakRow KeccakRow where\n  main\n  localLength _ := 120"}, {"name": "main", "content": "def main (row : Var KeccakRow (F p)) : Circuit (F p) (Var KeccakRow (F p)) := do\n  let c0 ← Rotation64.circuit (64 - 1) row[1]\n  let c0 ← Xor64.circuit ⟨row[4], c0⟩\n\n  let c1 ← Rotation64.circuit (64 - 1) row[2]\n  let c1 ← Xor64.circuit ⟨row[0], c1⟩\n\n  let c2 ← Rotation64.circuit (64 - 1) row[3]\n  let c2 ← Xor64.circuit ⟨row[1], c2⟩\n\n  let c3 ← Rotation64.circuit (64 - 1) row[4]\n  let c3 ← Xor64.circuit ⟨row[2], c3⟩\n\n  let c4 ← Rotation64.circuit (64 - 1) row[0]\n  let c4 ← Xor64.circuit ⟨row[3], c4⟩\n\n  return #v[c0, c1, c2, c3, c4]"}, {"name": "KeccakRow", "content": "@[reducible] def KeccakRow := ProvableVector U64 5"}, {"name": "Spec", "content": "def Spec (row : KeccakRow (F p)) (out : KeccakRow (F p)) : Prop :=\n  out.Normalized\n  ∧ out.value = Specs.Keccak256.thetaD row.value"}, {"name": "Assumptions", "content": "def Assumptions (state : KeccakRow (F p)) := state.Normalized"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) Inputs KeccakState :=\n  { elaborated with Assumptions, Spec, soundness, completeness }"}, {"name": "elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs KeccakState where\n  main\n  localLength _ := 200\n\n  localLength_eq _ n := by admit /- proof elided -/"}, {"name": "main", "content": "def main : Var Inputs (F p) → Circuit (F p) (Var KeccakState (F p))\n  | { state, d } => .mapFinRange 25 fun i =>\n    Xor64.circuit ⟨state[i.val], d[i.val / 5]⟩"}, {"name": "mapFinRange", "content": "def mapFinRange (m : ℕ) [NeZero m] (body : Fin m → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  Vector.mapFinRangeM m body"}, {"name": "mapFinRangeM", "content": "def mapFinRangeM (n : ℕ) {m : Type → Type} [Monad m] (f : Fin n → m β) : m (Vector β n) := (finRange n).mapM f"}, {"name": "Spec", "content": "def Spec (inputs : Inputs (F p)) (out : KeccakState (F p)) : Prop :=\n  let ⟨state, d⟩ := inputs\n  out.Normalized\n  ∧ out.value = Specs.Keccak256.thetaXor state.value d.value"}, {"name": "Assumptions", "content": "def Assumptions (inputs : Inputs (F p)) : Prop :=\n  let ⟨state, d⟩ := inputs\n  state.Normalized ∧ d.Normalized"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) KeccakState KeccakRow :=\n { elaborated with Assumptions, Spec, soundness, completeness }"}, {"name": "elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) KeccakState KeccakRow where\n  main\n  localLength _ := 160\n  localLength_eq _ _ := by admit /- proof elided -/"}, {"name": "main", "content": "def main (state : Var KeccakState (F p)) : Circuit (F p) (Var KeccakRow (F p)) :=\n  .mapFinRange 5 fun i => do\n    let c ← Xor64.circuit ⟨state[5*i.val], state[5*i.val + 1]⟩\n    let c ← Xor64.circuit ⟨c, state[5*i.val + 2]⟩\n    let c ← Xor64.circuit ⟨c, state[5*i.val + 3]⟩\n    let c ← Xor64.circuit ⟨c, state[5*i.val + 4]⟩\n    return c"}, {"name": "Spec", "content": "def Spec (state : KeccakState (F p)) (out : KeccakRow (F p)) :=\n  out.Normalized\n  ∧ out.value = Specs.Keccak256.thetaC state.value"}, {"name": "elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) KeccakState KeccakState where\n  main\n  localLength _ := 400\n  output _ i0 := Vector.mapRange 25 fun i => varFromOffset U64 (i0 + i*16 + 8)\n\n  localLength_eq state i0 := by admit /- proof elided -/"}, {"name": "main", "content": "def main (state : Var KeccakState (F p)) : Circuit (F p) (Var KeccakState (F p)) :=\n  .mapFinRange 25 fun i => do\n    let state_not ← Not.circuit (state[i + 5])\n    let state_and ← And.And64.circuit ⟨state_not, state[i + 10]⟩\n    Xor64.circuit ⟨state[i], state_and⟩"}, {"name": "Spec", "content": "def Spec (input : Inputs (F p)) (z : U64 (F p)) :=\n  let ⟨x, y⟩ := input\n  z.value = x.value &&& y.value ∧ z.Normalized"}, {"name": "circuit", "content": "def circuit : FormalCircuit (F p) U64 U64 where\n  main x := pure (not64_bytewise x)\n  Assumptions x := x.Normalized\n  Spec x z := z.value = not64 x.value ∧ z.Normalized\n\n  localLength _ := 0\n  output x _ := not64_bytewise x\n\n  soundness := by admit /- proof elided -/"}, {"name": "not64_bytewise", "content": "def not64_bytewise (x : Var U64 (F p)) : Var U64 (F p) := U64.map x (fun x => 255 - x)"}, {"name": "main", "content": "def main (args : List String) : IO Unit := do\n  match args with\n  | [steps_str, output_path] =>\n    \n    match steps_str.toNat? with\n    | some steps => generateTrace steps output_path\n    | none => IO.println \"Error: Invalid number of steps\"\n  | _ =>\n    IO.println \"Usage: lake lean TraceGen.lean "}, {"name": "Spec", "content": "def Spec (state : KeccakState (F p)) (out_state : KeccakState (F p)) :=\n  out_state.Normalized\n  ∧ out_state.value = Specs.Keccak256.chi state.value"}, {"name": "stateVar", "content": "def stateVar (n : ℕ) (i : ℕ) : Var KeccakState (F p) :=\n  Vector.mapRange 25 (fun j => varFromOffset U64 (n + i * 1288 + j * 16 + 888))\n  |>.set 0 (varFromOffset U64 (n + i * 1288 + 1280))"}, {"name": "ExplicitCircuits.from_pure", "content": "instance ExplicitCircuits.from_pure {f : α → β} : ExplicitCircuits (fun a => pure (f a) : α → Circuit F β) where\n  output a _ := f a\n  localLength _ _ := 0\n  operations _ _ := []"}, {"name": "KeccakBlock.normalized", "content": "def KeccakBlock.normalized : FormalAssertion (F p) KeccakBlock where\n  main block := .forEach block (assertion U64.AssertNormalized.circuit)\n  Assumptions _ := True\n  Spec block := block.Normalized\n  localLength_eq _ _ := by admit /- proof elided -/"}, {"name": "circuit", "content": "def circuit : FormalAssertion (F p) U64 where\n  main\n\n  Assumptions _ := True\n  Spec inputs := inputs.Normalized\n\n  soundness := by admit /- proof elided -/"}, {"name": "main", "content": "def main (inputs : Var U64 (F p)) : Circuit (F p) Unit  := do\n  let ⟨x0, x1, x2, x3, x4, x5, x6, x7⟩ := inputs\n  lookup ByteTable x0\n  lookup ByteTable x1\n  lookup ByteTable x2\n  lookup ByteTable x3\n  lookup ByteTable x4\n  lookup ByteTable x5\n  lookup ByteTable x6\n  lookup ByteTable x7"}, {"name": "assertion", "content": "@[circuit_norm]\ndef assertion (circuit : FormalAssertion F β) (b : Var β F) : Circuit F Unit :=\n  fun offset =>\n    let subcircuit := circuit.toSubcircuit offset b\n    ((), [.subcircuit subcircuit])"}, {"name": "absorbBlocks", "content": "def absorbBlocks (blocks : List (Vector ℕ RATE)) : Vector ℕ 25 :=\n  blocks.foldl absorbBlock initialState"}, {"name": "initialState", "content": "def initialState : Vector ℕ 25 := .fill 25 0"}, {"name": "foldl", "content": "def foldl {m : ℕ} [Inhabited β] [Inhabited α] (xs : Vector α m) (init : β) (body : β → α → Circuit F β)\n  (_const_out : ConstantOutput (fun (s, a) => body s a) := by admit /- proof elided -/\n    )\n  (_constant : ConstantLength (fun (s, a) => body s a) := by admit /- proof elided -/\n  )\n   : Circuit F β :=\n  xs.foldlM body init"}, {"name": "ConstantOutput", "content": "@[circuit_norm]\ndef ConstantOutput (circuit : α → Circuit F β) [Inhabited α] :=\n  ∀ (x : α) (n : ℕ), (circuit x).output n = (circuit default).output n"}, {"name": "InductiveTable.Soundness", "content": "def InductiveTable.Soundness (F : Type) [Field F] (State Input : Type → Type) [ProvableType State] [ProvableType Input]\n    (Spec : (initialState : State F) → (xs : List (Input F)) → (i : ℕ) → (xs.length = i) → (currentState : State F) → Prop)\n    (step : Var State F → Var Input F → Circuit F (Var State F)) :=\n  ∀ (initialState : State F) (row_index : ℕ) (env : Environment F),\n  \n  ∀ (acc_var : Var State F) (x_var : Var Input F)\n    (acc : State F) (x : Input F) (xs : List (Input F)) (xs_len : xs.length = row_index),\n      (eval env acc_var = acc) ∧ (eval env x_var = x) →\n    \n    Circuit.ConstraintsHold.Soundness env (step acc_var x_var |>.operations ((size State) + (size Input))) →\n    \n    Spec initialState xs row_index xs_len acc →\n    \n    Spec initialState (xs.concat x) (row_index + 1) (xs_len ▸ List.length_concat) (eval env (step acc_var x_var |>.output ((size State) + (size Input))))"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : U64 (F p) :=\n  ⟨ x.val, 0, 0, 0, 0, 0, 0, 0 ⟩"}, {"name": "traceInputs", "content": "def traceInputs {N : ℕ} (trace : TraceOfLength F (ProvablePair State Input) N) : List (Input F) :=\n  trace.val.toList.map Prod.snd"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "toList", "content": "def toList : Trace F S → List (Row F S)\n  | <+> => []\n  | rest +> row => rest.toList.concat row"}, {"name": "Trace", "content": "inductive Trace (F : Type) (S : Type → Type) [ProvableType S] where\n   \n  | empty : Trace F S\n   \n  | cons (rest : Trace F S) (row : Row F S) : Trace F S"}, {"name": "CellOffset", "content": "structure CellOffset (W : ℕ+) (S : Type → Type) [ProvableType S]  where\n  row: Fin W\n  column: Fin (size S)"}, {"name": "empty", "content": "@[reducible, table_norm, table_assignment_norm]\ndef empty : TableContext W S F where\n  circuit := []\n  assignment := .empty W"}, {"name": "TableContext", "content": "structure TableContext (W : ℕ+) (S : Type → Type) (F : Type) [Field F] [ProvableType S] where\n  circuit : Operations F\n  assignment : CellAssignment W S\nderiving Repr"}, {"name": "CellAssignment", "content": "structure CellAssignment (W : ℕ+) (S : Type → Type) [ProvableType S] where\n  offset : ℕ \n  aux_length : ℕ \n\n   \n  vars : Vector (Cell W S) offset"}, {"name": "offset", "content": "@[reducible, table_norm, table_assignment_norm]\ndef offset (table : TableContext W S F) : ℕ := table.circuit.localLength"}, {"name": "Cell", "content": "inductive Cell (W : ℕ+) (S : Type → Type) [ProvableType S] where\n  | input : CellOffset W S → Cell W S\n  | aux : ℕ → Cell W S"}, {"name": "Row", "content": "@[reducible]\ndef Row (F : Type) (S : Type → Type) [ProvableType S] := S F"}, {"name": "ProvablePair.instance", "content": "instance ProvablePair.instance {α β: TypeMap} [ProvableType α] [ProvableType β] : ProvableType (ProvablePair α β) where\n  size := size α + size β\n  toElements := fun (a, b) => toElements a ++ toElements b\n  fromElements {F} v :=\n    let a : α F := v.take (size α) |>.cast Nat.min_add_right_self |> fromElements\n    let b : β F := v.drop (size α) |>.cast (Nat.add_sub_self_left _ _) |> fromElements\n    (a, b)\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TraceOfLength", "content": "def TraceOfLength (F : Type) (S : Type → Type) [ProvableType S] (N : ℕ) : Type :=\n  { env : Trace F S // env.len = N }"}, {"name": "toList", "content": "def toList {N : ℕ} (trace : TraceOfLength F S N) : List.Vector (Row F S) N :=\n  ⟨ trace.val.toList, by admit /- proof elided -/\n  ⟩"}, {"name": "FormalTable.statement", "content": "def FormalTable.statement (table : FormalTable F S) (N : ℕ) (trace : TraceOfLength F S N) : Prop :=\n  table.Assumption N → table.Spec trace"}, {"name": "FormalTable", "content": "structure FormalTable (F : Type) [Field F] (S : Type → Type) [ProvableType S] where\n   \n  constraints : List (TableOperation S F)\n\n   \n  Assumption : ℕ → Prop := fun _ => True\n\n   \n  Spec {N : ℕ} : TraceOfLength F S N → Prop\n\n   \n  soundness :\n    ∀ (N : ℕ) (trace : TraceOfLength F S N) (env : ℕ → ℕ → Environment F),\n    Assumption N →\n    TableConstraintsHold constraints trace env →\n    Spec trace\n\n   \n  offset_consistent :\n    constraints.Forall fun cs =>\n      match cs with\n      | .boundary _ constraint => constraint.OffsetConsistent\n      | .everyRow constraint => constraint.OffsetConsistent\n      | .everyRowExceptLast constraint => constraint.OffsetConsistent\n    := by admit /- proof elided -/"}, {"name": "tail", "content": "def tail : Trace F S → Trace F S\n  | <+>  => <+>\n  | rest +> _ => rest"}, {"name": "tail", "content": "def tail {M : ℕ} (trace : TraceOfLength F S M) : TraceOfLength F S (M - 1) :=\n  ⟨ trace.val.tail, by admit /- proof elided -/\n  ⟩"}, {"name": "toFormal", "content": "def toFormal (table : InductiveTable F State Input) (input output : State F) : FormalTable F (ProvablePair State Input) where\n  constraints := table.tableConstraints input output\n  Assumption N := N > 0 ∧ table.Spec input [] 0 rfl input\n  Spec {N} trace := table.Spec input (traceInputs trace.tail) (N-1) (traceInputs_length trace.tail) output\n\n  soundness N trace env assumption constraints :=\n    table.table_soundness input output ⟨N, assumption.left⟩ trace env assumption.right constraints\n\n  offset_consistent := by admit /- proof elided -/"}, {"name": "equalityConstraint", "content": "def equalityConstraint (Input : TypeMap) [ProvableType Input] (target : State F) : SingleRowConstraint (ProvablePair State Input) F := do\n  let (actual, _) ← getCurrRow\n  actual === (const target)"}, {"name": "getCurrRow", "content": "@[table_norm, table_assignment_norm]\ndef getCurrRow : TableConstraint W S F (Var S F) := getRow 0"}, {"name": "TableConstraint", "content": "@[reducible, table_norm, table_assignment_norm]\ndef TableConstraint (W : ℕ+) (S : Type → Type) (F : Type) [Field F] [ProvableType S] :=\n  StateM (TableContext W S F)"}, {"name": "getRow", "content": "@[table_norm, table_assignment_norm]\ndef getRow (row : Fin W) : TableConstraint W S F (Var S F) :=\n  modifyGet fun ctx =>\n    let ctx' : TableContext W S F := {\n      circuit := ctx.circuit ++ [.witness (size S) fun env => .mapRange _ fun i => env.get (ctx.offset + i)],\n      assignment := ctx.assignment.pushRow row\n    }\n    (varFromOffset S ctx.offset, ctx')"}, {"name": "Row.get", "content": "@[table_norm, table_assignment_norm]\ndef Row.get (row : Row F S) (i : Fin (size S)) : F :=\n  (toElements row)[i.val]"}, {"name": "pushRow", "content": "@[table_assignment_norm]\ndef pushRow (assignment : CellAssignment W S) (row : Fin W) : CellAssignment W S :=\n  let row_vars : Vector (Cell W S) (size S) := .mapFinRange _ fun col => .input ⟨ row, col ⟩\n  {\n    offset := assignment.offset + size S\n    aux_length := assignment.aux_length\n    vars := assignment.vars ++ row_vars\n  }"}, {"name": "get", "content": "@[table_assignment_norm]\ndef get {M : ℕ} :\n    (env : TraceOfLength F S M) → (i : Fin M) → (j : Fin (size S)) → F\n  | ⟨env, h⟩, i, j => env.getLeFromBottom ⟨\n      M - 1 - i,\n      by admit /- proof elided -/\n    ⟩ j"}, {"name": "SingleRowConstraint", "content": "@[reducible]\ndef SingleRowConstraint (S : Type → Type) (F : Type) [Field F] [ProvableType S] := TableConstraint 1 S F Unit"}, {"name": "inductiveConstraint", "content": "def inductiveConstraint (table : InductiveTable F State Input) : TableConstraint 2 (ProvablePair State Input) F Unit := do\n  let (acc, x) ← getCurrRow\n  let output ← table.step acc x\n  let (output', _) ← getNextRow\n  \n  output' === output"}, {"name": "getNextRow", "content": "@[table_norm, table_assignment_norm]\ndef getNextRow : TableConstraint W S F (Var S F) := getRow 1"}, {"name": "tableConstraints", "content": "def tableConstraints (table : InductiveTable F State Input) (input_state output_state : State F) :\n  List (TableOperation (ProvablePair State Input) F) := [\n    .everyRowExceptLast table.inductiveConstraint,\n    .boundary (.fromStart 0) (equalityConstraint Input input_state),\n    .boundary (.fromEnd 0) (equalityConstraint Input output_state),\n  ]"}, {"name": "TableOperation", "content": "inductive TableOperation (S : Type → Type) (F : Type) [Field F] [ProvableType S] where\n   \n  | boundary: RowIndex → SingleRowConstraint S F → TableOperation S F\n\n   \n  | everyRow: SingleRowConstraint S F → TableOperation S F\n\n   \n  | everyRowExceptLast: TwoRowsConstraint S F → TableOperation S F"}, {"name": "RowIndex", "content": "inductive RowIndex where\n  | fromStart : ℕ → RowIndex\n  | fromEnd : ℕ → RowIndex"}, {"name": "TwoRowsConstraint", "content": "@[reducible]\ndef TwoRowsConstraint (S : Type → Type) (F : Type) [Field F] [ProvableType S] := TableConstraint 2 S F Unit"}, {"name": "TableConstraintsHold", "content": "@[table_norm]\ndef TableConstraintsHold {N : ℕ} (constraints : List (TableOperation S F))\n  (trace : TraceOfLength F S N) (env : ℕ → ℕ → Environment F) : Prop :=\n  let constraints_and_envs := constraints.mapIdx (fun i cs => (cs, env i))\n  foldl N constraints_and_envs trace.val constraints_and_envs\n  where"}, {"name": "OffsetConsistent", "content": "@[table_assignment_norm]\ndef OffsetConsistent (table : TableConstraint W S F α) : Prop :=\n  table.finalOffset = table.finalAssignment.offset"}, {"name": "finalOffset", "content": "@[reducible, table_norm, table_assignment_norm]\ndef finalOffset (table : TableConstraint W S F α) : ℕ :=\n  table .empty |>.snd.circuit.localLength"}, {"name": "finalAssignment", "content": "@[table_assignment_norm]\ndef finalAssignment (table : TableConstraint W S F α) : CellAssignment W S :=\n  table .empty |>.snd.assignment"}, {"name": "fill", "content": "def fill (n : ℕ) (a : α) : Vector α n :=\n  match n with\n  | 0 => #v[]\n  | k + 1 => (fill k a).push a"}, {"name": "len", "content": "@[table_norm, table_assignment_norm]\ndef len : Trace F S → ℕ\n  | <+> => 0\n  | rest +> _ => rest.len + 1"}, {"name": "infix:50 \" === \" => HasAssertEq.assert_eq", "content": "infix:50 \" === \" => HasAssertEq.assert_eq"}, {"name": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty", "content": "@[inherit_doc] notation:67 \"<+>\" => Trace.empty"}, {"name": "@[inherit_doc] infixl:67 \" +> \" => Trace.cons", "content": "@[inherit_doc] infixl:67 \" +> \" => Trace.cons"}], "lib_lemmas": [{"name": "Fin.val_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "Vector.getElem_map", "module": "Init.Data.Vector.Lemmas"}, {"name": "List.length_map", "module": "Init.Data.List.Lemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "fromByte_value", "content": "lemma fromByte_value {x : Fin 256} : (fromByte x).value (p:=p) = x"}, {"name": "getElem_fill", "content": "theorem getElem_fill {n} {a : α} {i : ℕ} {hi : i < n} :\n    (fill n a)[i] = a"}, {"name": "fromByte_normalized", "content": "lemma fromByte_normalized {x : Fin 256} : (fromByte x).Normalized (p:=p)"}, {"name": "toList_length", "content": "lemma toList_length : (trace : Trace F S) → trace.toList.length = trace.len\n  | <+> => rfl"}], "used_local_defs": [{"name": "Tables.KeccakInductive.table", "content": "def table : InductiveTable (F p) KeccakState KeccakBlock where\n  step state block := do\n    KeccakBlock.normalized block\n    AbsorbBlock.circuit { state, block }\n\n  Spec _ blocks i _ state : Prop :=\n    state.Normalized\n    ∧ state.value = absorbBlocks (blocks.map KeccakBlock.value)\n\n  InputAssumptions i block := block.Normalized\n\n  soundness := by admit /- proof elided -/"}, {"name": "Tables.KeccakInductive.initialState", "content": "def initialState : KeccakState (F p) := .fill 25 (U64.fromByte 0)"}, {"name": "Tables.KeccakInductive.formalTable", "content": "def formalTable (output : KeccakState (F p)) := table.toFormal initialState output"}], "used_local_lemmas": [{"name": "Tables.KeccakInductive.initialState_value", "content": "lemma initialState_value : (initialState (p:=p)).value = .fill 25 0"}, {"name": "Tables.KeccakInductive.initialState_normalized", "content": "lemma initialState_normalized : (initialState (p:=p)).Normalized"}], "local_ctx": "import Clean.Table.Inductive\n\nimport Clean.Circuit.Extensions\n\nimport Clean.Gadgets.Keccak.AbsorbBlock\n\nimport Clean.Specs.Keccak256\n\nopen Specs.Keccak256\n\nvariable {p : ℕ} [Fact p.Prime] [Fact (p > 2 ^ 16 + 2 ^ 8)]\n\nnamespace Tables.KeccakInductive\n\nopen Gadgets.Keccak256\n\ndef table : InductiveTable (F p) KeccakState KeccakBlock where\n  step state block := do\n    KeccakBlock.normalized block\n    AbsorbBlock.circuit { state, block }\n\n  Spec _ blocks i _ state : Prop :=\n    state.Normalized\n    ∧ state.value = absorbBlocks (blocks.map KeccakBlock.value)\n\n  InputAssumptions i block := block.Normalized\n\n  soundness := by admit /- proof elided -/\n\ndef initialState : KeccakState (F p) := .fill 25 (U64.fromByte 0)\n\ndef formalTable (output : KeccakState (F p)) := table.toFormal initialState output", "target_theorem": "theorem tableStatement (output : KeccakState (F p)) : ∀ n > 0, ∀ trace, ∃ blocks, blocks.length = n - 1 ∧\n  (formalTable output).statement n trace →\n    output.Normalized ∧ output.value = absorbBlocks blocks :=", "ground_truth_proof": ":= by\n  intro n hn trace\n  use (InductiveTable.traceInputs trace.tail).map KeccakBlock.value\n  intro Spec\n  simp only [formalTable, FormalTable.statement, table, InductiveTable.toFormal] at Spec\n  simp only [List.length_map, Trace.toList_length, trace.tail.prop, InductiveTable.traceInputs, hn] at Spec\n  simp only [initialState_value, initialState_normalized, absorbBlocks, Specs.Keccak256.initialState, true_and] at Spec\n  exact Spec rfl", "nesting_depth": 24, "transitive_dep_count": 288, "subset_aristotle": true, "category": "Applied verif."}
{"id": 150, "thm_name": "Environment.usesLocalWitnessesFlat_iff_extends", "thm_stmt": "theorem usesLocalWitnessesFlat_iff_extends {env : Environment F} (n : ℕ) {ops : List (FlatOperation F)}  :\n    env.UsesLocalWitnessesFlat n ops ↔ env.ExtendsVector (localWitnesses env ops) n", "lean_root": "clean", "rel_path": "Clean/Circuit/Theorems.lean", "imports": ["import Clean.Circuit.Provable", "import Clean.Circuit.Basic"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}], "used_repo_defs": [{"name": "FlatOperation.singleLocalLength", "content": "def FlatOperation.singleLocalLength : FlatOperation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Environment.UsesLocalWitnessesFlat", "content": "def Environment.UsesLocalWitnessesFlat (env : Environment F) (n : ℕ) (ops : List (FlatOperation F)) : Prop :=\n  FlatOperation.forAll n { witness n _ compute := env.ExtendsVector (compute env) n } ops"}, {"name": "Condition.applyFlat", "content": "def Condition.applyFlat (condition : Condition F) (offset : ℕ) : FlatOperation F → Prop\n  | .witness m c => condition.witness offset m c\n  | .assert e => condition.assert offset e\n  | .lookup l => condition.lookup offset l"}, {"name": "induct", "content": "def induct {motive : List (FlatOperation F) → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n    (ops : List (FlatOperation F)) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup ops)"}], "lib_lemmas": [{"name": "Vector.getElem_append", "module": "Init.Data.Vector.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "FlatOperation.forAll_empty", "content": "theorem forAll_empty {condition : Condition F} {n : ℕ} : forAll n condition [] = True"}, {"name": "FlatOperation.forAll_cons", "content": "theorem forAll_cons {condition : Condition F} {offset : ℕ} {op : FlatOperation F} {ops : List (FlatOperation F)} :\n  forAll offset condition (op :: ops) ↔\n    condition.applyFlat offset op ∧ forAll (op.singleLocalLength + offset) condition ops"}, {"name": "Environment.env_extends_witness", "content": "lemma env_extends_witness {F} {n : ℕ} {ops : List (FlatOperation F)} {env : Environment F} {m c} :\n    env.ExtendsVector (localWitnesses env (.witness m c :: ops)) n ↔\n      (env.ExtendsVector (c env) n ∧ env.ExtendsVector (localWitnesses env ops) (m + n))"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nvariable {F : Type} [Field F] {α β : Type}\n\nnamespace Operations\n\nend Operations\n\nnamespace Circuit\n\nend Circuit\n\nnamespace FlatOperation\n\nend FlatOperation\n\nnamespace Environment\n\nopen FlatOperation (localLength localWitnesses)", "target_theorem": "theorem usesLocalWitnessesFlat_iff_extends {env : Environment F} (n : ℕ) {ops : List (FlatOperation F)}  :\n    env.UsesLocalWitnessesFlat n ops ↔ env.ExtendsVector (localWitnesses env ops) n :=", "ground_truth_proof": ":= by\n  induction ops using FlatOperation.induct generalizing n with\n  | empty => simp [UsesLocalWitnessesFlat, FlatOperation.forAll_empty, ExtendsVector, localLength]\n  | witness m _ _ ih =>\n    rw [UsesLocalWitnessesFlat, FlatOperation.forAll, env_extends_witness,←ih (m + n)]\n    trivial\n  | assert | lookup =>\n    simp_all [UsesLocalWitnessesFlat, circuit_norm,\n      FlatOperation.forAll_cons, Condition.applyFlat, FlatOperation.singleLocalLength]", "nesting_depth": 7, "transitive_dep_count": 41, "subset_aristotle": true, "category": "Applied verif."}
{"id": 151, "thm_name": "Gadgets.Or.Or8.soundness", "thm_stmt": "theorem soundness : Soundness (F p) elaborated Assumptions Spec", "lean_root": "clean", "rel_path": "Clean/Gadgets/Or/Or8.lean", "imports": ["import Clean.Gadgets.Xor.ByteXorTable", "import Clean.Utils.Field", "import Clean.Circuit.Basic"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Fin.xor", "module": "Init.Data.Fin.Basic"}, {"name": "HXor", "module": "Init.Prelude"}, {"name": "HXor.hXor", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "XorOp", "module": "Init.Prelude"}, {"name": "XorOp.xor", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "Nat.bitwise", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat.reduceMod", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Nat.reducePow", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "UInt16", "module": "Init.Prelude"}, {"name": "UInt16.toNat", "module": "Init.Data.UInt.BasicAux"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"field_to_nat\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|field_to_nat) =>\n    `(tactic|(\n      intros\n      repeat rw [ZMod.val_add] \n      repeat rw [ZMod.val_mul] \n      repeat rw [val_eq_256]\n      try simp only [Nat.add_mod_mod, Nat.mod_add_mod, Nat.mul_mod_mod, Nat.mod_mul_mod]\n      rw [Nat.mod_eq_of_lt _]\n      repeat linarith [‹Fact (_ > 512)›.elim]))\n\nexample [Fact (p > 512)] (x y : F p) (hx : x.val < 256) (hy : y.val < 2) :\n    (x + y * 256).val = x.val + y.val * 256 := by admit /- proof elided -/"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "ByteXorTable", "content": "def ByteXorTable : Table (F p) fieldTriple := .fromStatic {\n  name := \"ByteXor\"\n  length := 256*256\n\n  row i :=\n    let (x, y) := splitTwoBytes i\n    (fromByte x, fromByte y, fromByte (x ^^^ y))\n\n  index := fun (x, y, _) => x.val * 256 + y.val\n\n  Spec := fun (x, y, z) =>\n    x.val < 256 ∧ y.val < 256 ∧ z.val = x.val ^^^ y.val\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "splitTwoBytes", "content": "def splitTwoBytes (i : Fin (256 * 256)) : Fin 256 × Fin 256 :=\n  let x := i.val / 256\n  let y := i.val % 256\n  have x_lt : x < 256 := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "fieldTriple", "content": "@[reducible]\ndef fieldTriple : TypeMap := fun F => F × F × F"}, {"name": "concatTwoBytes", "content": "def concatTwoBytes (x y : Fin 256) : Fin (256 * 256) :=\n  let i := x.val * 256 + y.val\n  have i_lt : i < 256 * 256 := by admit /- proof elided -/"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "fieldVar", "content": "@[reducible] def fieldVar (F : Type) := field (Expression F)"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}], "lib_lemmas": [{"name": "BitVec.toNat_ofNat", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.or_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.xor_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "UInt16.toBitVec_ofNat", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt16.toNat_ofNat_of_lt", "module": "Init.Data.UInt.Lemmas"}, {"name": "UInt16.toNat_or", "module": "Init.Data.UInt.Bitwise"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.bitwise_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "ZMod.val_add_of_lt", "module": "Mathlib.Data.ZMod.Basic"}], "repo_lemmas": [{"name": "mul_nat_val_of_dvd", "content": "theorem mul_nat_val_of_dvd {x : F p} (c : ℕ) (c_lt : c < p) {z : ℕ} :\n    (c * x).val = c * z → (c * x).val = c * x.val"}, {"name": "mul_val_of_dvd", "content": "theorem mul_val_of_dvd {x c : F p} :\n    c.val ∣ (c * x).val → (c * x).val = c.val * x.val"}, {"name": "natToField_eq", "content": "theorem natToField_eq {n : ℕ} {lt : n < p} (x : F p) (hx : x = natToField n lt) : x.val = n"}, {"name": "ext", "content": "theorem ext {x y : F p} (h : x.val = y.val) : x = y"}, {"name": "p_ne_zero", "content": "theorem p_ne_zero : p ≠ 0"}], "used_local_defs": [{"name": "Gadgets.Or.Or8.Inputs", "content": "structure Inputs (F : Type) where\n  x: F\n  y: F"}, {"name": "Gadgets.Or.Or8.Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.val < 256 ∧ y.val < 256"}, {"name": "Gadgets.Or.Or8.Spec", "content": "def Spec (input : Inputs (F p)) (z : F p) :=\n  let ⟨x, y⟩ := input\n  z.val = x.val ||| y.val ∧ z.val < 256"}, {"name": "Gadgets.Or.Or8.main", "content": "def main (input : Var Inputs (F p)) : Circuit (F p) (fieldVar (F p)) := do\n  let ⟨x, y⟩ := input\n  let or ← witness fun eval => (eval x).val ||| (eval y).val\n  \n  let xor := 2*or - x - y\n  lookup ByteXorTable (x, y, xor)\n  return or"}, {"name": "Gadgets.Or.Or8.elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs field where\n  main\n  localLength _ := 1"}], "used_local_lemmas": [{"name": "Gadgets.Or.Or8.or_times_two_sub_xor", "content": "private theorem or_times_two_sub_xor {x y : ℕ} (hx : x < 256) (hy : y < 256) :\n    2 * (x ||| y) = x + y + (x ^^^ y)"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Gadgets.Xor.ByteXorTable\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nnamespace Gadgets.Or.Or8\n\nopen Xor (ByteXorTable)\n\nopen FieldUtils\n\nstructure Inputs (F : Type) where\n  x: F\n  y: F\n\ndef Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.val < 256 ∧ y.val < 256\n\ndef Spec (input : Inputs (F p)) (z : F p) :=\n  let ⟨x, y⟩ := input\n  z.val = x.val ||| y.val ∧ z.val < 256\n\ndef main (input : Var Inputs (F p)) : Circuit (F p) (fieldVar (F p)) := do\n  let ⟨x, y⟩ := input\n  let or ← witness fun eval => (eval x).val ||| (eval y).val\n  \n  let xor := 2*or - x - y\n  lookup ByteXorTable (x, y, xor)\n  return or\n\ninstance elaborated : ElaboratedCircuit (F p) Inputs field where\n  main\n  localLength _ := 1", "target_theorem": "theorem soundness : Soundness (F p) elaborated Assumptions Spec :=", "ground_truth_proof": ":= by\n  intro i env ⟨ x_var, y_var ⟩ ⟨ x, y ⟩ h_input h_assumptions h_constraint\n  simp_all only [circuit_norm, main, Assumptions, Spec, ByteXorTable, Inputs.mk.injEq]\n  have ⟨ hx_byte, hy_byte ⟩ := h_assumptions\n  set w := env.get i\n  -- The constraint from lookup is about xor = 2*or - x - y\n  -- which in field arithmetic is 2*w + -x + -y\n  set xor := 2*w + -x + -y\n  have h_xor : xor.val = x.val ^^^ y.val := h_constraint\n  have value_goal : w.val = x.val ||| y.val := by\n    have two_or_field : 2*w = x + y + xor := by ring\n    have x_y_val : (x + y).val = x.val + y.val := by field_to_nat\n    have x_y_xor_val : (x + y + xor).val = x.val + y.val + (x.val ^^^ y.val) := by\n      -- The key insight: from 2*(x ||| y) = x + y + (x ^^^ y), we get\n      -- x + y + (x ^^^ y) = 2*(x ||| y) ≤ 2*255 = 510 < 512 < p\n      have sum_bound : (x + y).val + xor.val < p := by\n        rw [x_y_val, h_xor]\n        have : x.val + y.val + (x.val ^^^ y.val) ≤ 2 * 255 := by\n          have h := or_times_two_sub_xor hx_byte hy_byte\n          have or_le : x.val ||| y.val ≤ 255 := by\n            have : x.val ||| y.val < 256 := Nat.or_lt_two_pow (n:=8) hx_byte hy_byte\n            omega\n          linarith\n        have := p_large_enough.elim\n        omega\n\n      rw [ZMod.val_add_of_lt sum_bound, x_y_val, h_xor]\n\n    have two_or : (2*w).val = 2*(x.val ||| y.val) := by\n      rw [two_or_field, x_y_xor_val, or_times_two_sub_xor hx_byte hy_byte]\n    have two_mul_val : (2*w).val = 2*w.val := FieldUtils.mul_nat_val_of_dvd 2\n      (by linarith [p_large_enough.elim]) two_or\n    rw [two_mul_val] at two_or\n    omega\n  constructor\n  · assumption\n  simp only [value_goal]\n  show Nat.bitwise _ _ _ < 2 ^ 8\n  exact Nat.bitwise_lt_two_pow hx_byte hy_byte", "nesting_depth": 7, "transitive_dep_count": 114, "subset_aristotle": false, "category": "Applied verif."}
{"id": 152, "thm_name": "Gadgets.Not.not_bytewise_value_spec", "thm_stmt": "theorem not_bytewise_value_spec {x : U64 (F p)} (x_lt : x.Normalized) :\n    (not64_bytewise_value x).value = not64 x.value\n    ∧ (not64_bytewise_value x).Normalized", "lean_root": "clean", "rel_path": "Clean/Gadgets/Not/Not64.lean", "imports": ["import Clean.Utils.Primes", "import Clean.Utils.Bitwise", "import Clean.Types.U64", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.Prime", "module": "Mathlib.Data.Nat.Prime.Defs"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "U64", "content": "structure U64 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\n  x4 : T\n  x5 : T\n  x6 : T\n  x7 : T\nderiving DecidableEq"}, {"name": "map", "content": "def map {α β : Type} (x : U64 α) (f : α → β) : U64 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3, f x.x4, f x.x5, f x.x6, f x.x7 ⟩"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "Normalized", "content": "def Normalized (x : U64 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256 ∧\n  x.x4.val < 256 ∧ x.x5.val < 256 ∧ x.x6.val < 256 ∧ x.x7.val < 256"}, {"name": "value", "content": "def value (x : U64 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3 +\n  x.x4.val * 256^4 + x.x5.val * 256^5 + x.x6.val * 256^6 + x.x7.val * 256^7"}, {"name": "not64", "content": "def not64 (a : ℕ) : ℕ := a ^^^ 0xffffffffffffffff"}, {"name": "KeccakRow.value", "content": "def KeccakRow.value (row : KeccakRow (F p)) := row.map U64.value"}, {"name": "KeccakRow.Normalized", "content": "def KeccakRow.Normalized (row : KeccakRow (F p)) :=\n  ∀ i : Fin 5, row[i.val].Normalized"}, {"name": "BLAKE3State.Normalized", "content": "def BLAKE3State.Normalized (state : BLAKE3State (F p)) :=\n  ∀ i : Fin 16, state[i.val].Normalized"}, {"name": "KeccakBlock.value", "content": "def KeccakBlock.value (block : KeccakBlock (F p)) := block.map U64.value"}, {"name": "BLAKE3State.value", "content": "def BLAKE3State.value (state : BLAKE3State (F p)) := state.map U32.value"}, {"name": "value", "content": "def value (x : U32 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3"}, {"name": "U32", "content": "structure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq"}, {"name": "map", "content": "def map {α β : Type} (x : U32 α) (f : α → β) : U32 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3 ⟩"}, {"name": "KeccakState.value", "content": "def KeccakState.value (state : KeccakState (F p)) := state.map U64.value"}, {"name": "KeccakBlock.Normalized", "content": "def KeccakBlock.Normalized (block : KeccakBlock (F p)) :=\n  ∀ i : Fin RATE, block[i.val].Normalized"}, {"name": "RATE", "content": "@[reducible] def RATE := 17\nexample : RATE + CAPACITY = 25 := rfl"}, {"name": "CAPACITY", "content": "@[reducible] def CAPACITY := 8"}, {"name": "KeccakState.Normalized", "content": "def KeccakState.Normalized (state : KeccakState (F p)) :=\n  ∀ i : Fin 25, state[i.val].Normalized"}], "lib_lemmas": [{"name": "Nat.cast_sub", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Nat.le_pred_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Int.ofNat_lt", "module": "Init.Data.Int.Order"}, {"name": "Nat.sub_one_sub_lt_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "ZMod.val_sub", "module": "Mathlib.Data.ZMod.Basic"}], "repo_lemmas": [{"name": "not64_eq_sub", "content": "theorem not64_eq_sub {x : ℕ} (x_lt : x < 2^64) :\n    not64 x = 2^64 - 1 - x"}, {"name": "value_lt_of_normalized", "content": "omit [Fact (Nat.Prime p)] p_large_enough in\ntheorem value_lt_of_normalized {x : U64 (F p)} (hx : x.Normalized) : x.value < 2^64"}, {"name": "val_lt_p", "content": "theorem val_lt_p {p : ℕ} (x : ℕ) : (x < p) → (x : F p).val = x"}], "used_local_defs": [{"name": "Gadgets.Not.not64_bytewise_value", "content": "def not64_bytewise_value (x : U64 (F p)) : U64 (F p) := x.map (fun x => 255 - x)"}], "used_local_lemmas": [{"name": "Gadgets.Not.not_zify", "content": "theorem not_zify (n : ℕ) {x : ℕ} (hx : x < n) : ((n - 1 - x : ℕ) : ℤ) = ↑n - 1 - ↑x"}, {"name": "Gadgets.Not.not_lt", "content": "theorem not_lt (n : ℕ) {x : ℕ} (hx : x < n) : n - 1 - (x : ℤ) < n"}], "local_ctx": "import Clean.Utils.Primes\n\nimport Clean.Circuit.Basic\n\nimport Clean.Utils.Field\n\nimport Clean.Types.U64\n\nsection\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nnamespace Gadgets.Not\n\ndef not64_bytewise_value (x : U64 (F p)) : U64 (F p) := x.map (fun x => 255 - x)", "target_theorem": "theorem not_bytewise_value_spec {x : U64 (F p)} (x_lt : x.Normalized) :\n    (not64_bytewise_value x).value = not64 x.value\n    ∧ (not64_bytewise_value x).Normalized :=", "ground_truth_proof": ":= by\n\n  rw [not64_eq_sub (U64.value_lt_of_normalized x_lt)]\n\n  have h_not_val : ∀ {x : F p}, x.val < 256 → ((255 - x).val : ℤ) = 255 - ↑x.val := by\n    intro x hx\n    have val_255 : (255 : F p).val = 255 := FieldUtils.val_lt_p 255 (by linarith [p_large_enough.elim])\n    have hx' : x.val ≤ (255 : F p).val := by linarith\n    rw [ZMod.val_sub hx', val_255]\n    exact not_zify 256 hx\n\n  rw [U64.value, U64.Normalized, not64_bytewise_value, U64.map]\n  zify\n  rw [not_zify (2^64) (U64.value_lt_of_normalized x_lt), U64.value]\n  zify\n  have ⟨ hx0, hx1, hx2, hx3, hx4, hx5, hx6, hx7 ⟩ := x_lt\n  repeat rw [h_not_val (by assumption)]\n  constructor; ring\n  exact ⟨ not_lt 256 hx0, not_lt 256 hx1, not_lt 256 hx2, not_lt 256 hx3,\n      not_lt 256 hx4, not_lt 256 hx5, not_lt 256 hx6, not_lt 256 hx7 ⟩", "nesting_depth": 7, "transitive_dep_count": 70, "subset_aristotle": true, "category": "Applied verif."}
{"id": 153, "thm_name": "Gadgets.And.And8.soundness", "thm_stmt": "theorem soundness : Soundness (F p) elaborated Assumptions Spec", "lean_root": "clean", "rel_path": "Clean/Gadgets/And/And8.lean", "imports": ["import Clean.Gadgets.Xor.ByteXorTable", "import Clean.Utils.Field", "import Clean.Circuit.Basic", "import Clean.Utils.Primes"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Fin.xor", "module": "Init.Data.Fin.Basic"}, {"name": "HXor", "module": "Init.Prelude"}, {"name": "HXor.hXor", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "XorOp", "module": "Init.Prelude"}, {"name": "XorOp.xor", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "UInt16", "module": "Init.Prelude"}, {"name": "UInt16.toNat", "module": "Init.Data.UInt.BasicAux"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"field_to_nat\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|field_to_nat) =>\n    `(tactic|(\n      intros\n      repeat rw [ZMod.val_add] \n      repeat rw [ZMod.val_mul] \n      repeat rw [val_eq_256]\n      try simp only [Nat.add_mod_mod, Nat.mod_add_mod, Nat.mul_mod_mod, Nat.mod_mul_mod]\n      rw [Nat.mod_eq_of_lt _]\n      repeat linarith [‹Fact (_ > 512)›.elim]))\n\nexample [Fact (p > 512)] (x y : F p) (hx : x.val < 256) (hy : y.val < 2) :\n    (x + y * 256).val = x.val + y.val * 256 := by admit /- proof elided -/"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "ByteXorTable", "content": "def ByteXorTable : Table (F p) fieldTriple := .fromStatic {\n  name := \"ByteXor\"\n  length := 256*256\n\n  row i :=\n    let (x, y) := splitTwoBytes i\n    (fromByte x, fromByte y, fromByte (x ^^^ y))\n\n  index := fun (x, y, _) => x.val * 256 + y.val\n\n  Spec := fun (x, y, z) =>\n    x.val < 256 ∧ y.val < 256 ∧ z.val = x.val ^^^ y.val\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "splitTwoBytes", "content": "def splitTwoBytes (i : Fin (256 * 256)) : Fin 256 × Fin 256 :=\n  let x := i.val / 256\n  let y := i.val % 256\n  have x_lt : x < 256 := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "fieldTriple", "content": "@[reducible]\ndef fieldTriple : TypeMap := fun F => F × F × F"}, {"name": "concatTwoBytes", "content": "def concatTwoBytes (x y : Fin 256) : Fin (256 * 256) :=\n  let i := x.val * 256 + y.val\n  have i_lt : i < 256 * 256 := by admit /- proof elided -/"}, {"name": "fieldVar", "content": "@[reducible] def fieldVar (F : Type) := field (Expression F)"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}], "lib_lemmas": [{"name": "Nat.and_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.xor_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "UInt16.toNat_ofNat_of_lt", "module": "Init.Data.UInt.Lemmas"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.add_sub_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mul_left_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "ZMod.val_sub", "module": "Mathlib.Data.ZMod.Basic"}], "repo_lemmas": [{"name": "mul_nat_val_of_dvd", "content": "theorem mul_nat_val_of_dvd {x : F p} (c : ℕ) (c_lt : c < p) {z : ℕ} :\n    (c * x).val = c * z → (c * x).val = c * x.val"}, {"name": "mul_val_of_dvd", "content": "theorem mul_val_of_dvd {x c : F p} :\n    c.val ∣ (c * x).val → (c * x).val = c.val * x.val"}, {"name": "natToField_eq", "content": "theorem natToField_eq {n : ℕ} {lt : n < p} (x : F p) (hx : x = natToField n lt) : x.val = n"}, {"name": "ext", "content": "theorem ext {x y : F p} (h : x.val = y.val) : x = y"}, {"name": "p_ne_zero", "content": "theorem p_ne_zero : p ≠ 0"}], "used_local_defs": [{"name": "Gadgets.And.And8.Inputs", "content": "structure Inputs (F : Type) where\n  x: F\n  y: F"}, {"name": "Gadgets.And.And8.Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.val < 256 ∧ y.val < 256"}, {"name": "Gadgets.And.And8.Spec", "content": "def Spec (input : Inputs (F p)) (z : F p) :=\n  let ⟨x, y⟩ := input\n  z.val = x.val &&& y.val"}, {"name": "Gadgets.And.And8.main", "content": "def main (input : Var Inputs (F p)) : Circuit (F p) (fieldVar (F p)) := do\n  let ⟨x, y⟩ := input\n  let and ← witness fun eval => (eval x).val &&& (eval y).val\n  \n  let xor := x + y - 2*and\n  lookup ByteXorTable (x, y, xor)\n  return and"}, {"name": "Gadgets.And.And8.elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs field where\n  main\n  localLength _ := 1\n  output _ i := var ⟨i⟩"}], "used_local_lemmas": [{"name": "Gadgets.And.And8.and_times_two_add_xor", "content": "theorem and_times_two_add_xor {x y : ℕ} (hx : x < 256) (hy : y < 256) : 2 * (x &&& y) + (x ^^^ y) = x + y"}, {"name": "Gadgets.And.And8.xor_le_add", "content": "theorem xor_le_add {x y : ℕ} (hx : x < 256) (hy : y < 256) : x ^^^ y ≤ x + y"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Gadgets.Xor.ByteXorTable\n\nimport Clean.Utils.Primes\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nnamespace Gadgets.And.And8\n\nopen Xor (ByteXorTable)\n\nopen FieldUtils\n\nstructure Inputs (F : Type) where\n  x: F\n  y: F\n\ndef Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.val < 256 ∧ y.val < 256\n\ndef Spec (input : Inputs (F p)) (z : F p) :=\n  let ⟨x, y⟩ := input\n  z.val = x.val &&& y.val\n\ndef main (input : Var Inputs (F p)) : Circuit (F p) (fieldVar (F p)) := do\n  let ⟨x, y⟩ := input\n  let and ← witness fun eval => (eval x).val &&& (eval y).val\n  \n  let xor := x + y - 2*and\n  lookup ByteXorTable (x, y, xor)\n  return and\n\ninstance elaborated : ElaboratedCircuit (F p) Inputs field where\n  main\n  localLength _ := 1\n  output _ i := var ⟨i⟩", "target_theorem": "theorem soundness : Soundness (F p) elaborated Assumptions Spec :=", "ground_truth_proof": ":= by\n  intro i env ⟨ x_var, y_var ⟩ ⟨ x, y ⟩ h_input h_assumptions h_xor\n  simp_all only [circuit_norm, main, Assumptions, Spec, ByteXorTable, Inputs.mk.injEq]\n  have ⟨ hx_byte, hy_byte ⟩ := h_assumptions\n  set w := env.get i\n  set z := x + y + -(2*w)\n  show w.val = x.val &&& y.val\n\n  -- it's easier to prove something about 2*w since it features in the constraint\n  have two_and_field : 2*w = x + y - z := by ring\n\n  have x_y_val : (x + y).val = x.val + y.val := by field_to_nat\n  have z_lt : z.val ≤ (x + y).val := by\n    rw [h_xor, x_y_val]\n    exact xor_le_add hx_byte hy_byte\n  have x_y_z_val : (x + y - z).val = x.val + y.val - z.val := by\n    rw [ZMod.val_sub z_lt, x_y_val]\n\n  have two_and : (2*w).val = 2*(x.val &&& y.val) := by\n    rw [two_and_field, x_y_z_val, h_xor, ←and_times_two_add_xor hx_byte hy_byte, Nat.add_sub_cancel]\n\n  clear two_and_field x_y_val x_y_z_val h_xor z_lt\n\n  -- crucial step: since 2 divides (2*w).val, we can actually pull in .val\n  have two_mul_val : (2*w).val = 2*w.val := FieldUtils.mul_nat_val_of_dvd 2\n    (by linarith [p_large_enough.elim]) two_and\n\n  rw [two_mul_val, Nat.mul_left_cancel_iff (by linarith)] at two_and\n  exact two_and", "nesting_depth": 8, "transitive_dep_count": 109, "subset_aristotle": true, "category": "Applied verif."}
{"id": 154, "thm_name": "MemoryAccessList.isConsistentOnline_filter_of_consistentOnline", "thm_stmt": "theorem MemoryAccessList.isConsistentOnline_filter_of_consistentOnline (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted)\n    (h_consistent : MemoryAccessList.isConsistentOnline accesses h_sorted) (addr : ℕ) :\n    MemoryAccessList.isConsistentOnline (MemoryAccessList.filterAddress accesses addr) (MemoryAccessList.filterAddress_sorted accesses h_sorted addr)", "lean_root": "clean", "rel_path": "Clean/Utils/OfflineMemory.lean", "imports": ["import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Mathlib.Data.List.Sort", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Sorted", "module": "Mathlib.Deprecated.Sort"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.Sorted.filter", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Sorted.of_cons", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.filter_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.filter_nil", "module": "Init.Data.List.Basic"}, {"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "forall_true_left", "module": "Mathlib.Logic.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MemoryAccess", "content": "def MemoryAccess := ℕ × ℕ × ℕ × ℕ"}, {"name": "MemoryAccessList", "content": "def MemoryAccessList := List MemoryAccess"}, {"name": "timestamp_ordering", "content": "abbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2"}, {"name": "MemoryAccessList.isTimestampSorted", "content": "def MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering"}, {"name": "MemoryAccessList.lastWriteValue", "content": "def MemoryAccessList.lastWriteValue (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) (addr : ℕ) : ℕ := match accesses with\n  \n  | [] => 0\n  | (_t, addr', _readValue, writeValue) :: rest =>\n    if addr' = addr then\n      \n      writeValue\n    else\n      MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr"}, {"name": "MemoryAccessList.isConsistentOnline", "content": "def MemoryAccessList.isConsistentOnline (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, addr, readValue, _writeValue) :: rest =>\n    \n    readValue = MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n    ∧ MemoryAccessList.isConsistentOnline rest (List.Sorted.of_cons h)\n\nexample : MemoryAccessList.isConsistentOnline [] (by admit /- proof elided -/\n) := by admit /- proof elided -/"}, {"name": "MemoryAccessList.filterAddress", "content": "def MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)"}], "used_local_lemmas": [{"name": "MemoryAccessList.filterAddress_sorted", "content": "theorem MemoryAccessList.filterAddress_sorted (accesses : MemoryAccessList)\n    (h : accesses.isTimestampSorted) (addr : ℕ) :\n    (MemoryAccessList.filterAddress accesses addr).isTimestampSorted"}, {"name": "MemoryAccessList.lastWriteValue_filter", "content": "theorem MemoryAccessList.lastWriteValue_filter (accesses : MemoryAccessList)\n    (h_sorted : accesses.isTimestampSorted) (addr : ℕ) (h_sorted' : ((MemoryAccessList.filterAddress accesses addr).isTimestampSorted))  :\n    MemoryAccessList.lastWriteValue accesses h_sorted addr =\n    MemoryAccessList.lastWriteValue (MemoryAccessList.filterAddress accesses addr) h_sorted' addr"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Utils.Field\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Tactics\n\nimport Mathlib.Data.List.Sort\n\ndef MemoryAccess := ℕ × ℕ × ℕ × ℕ \n\ndef MemoryAccessList := List MemoryAccess\n\nabbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2\n\ndef MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering\n\ndef MemoryAccessList.lastWriteValue (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) (addr : ℕ) : ℕ := match accesses with\n  \n  | [] => 0\n  | (_t, addr', _readValue, writeValue) :: rest =>\n    if addr' = addr then\n      \n      writeValue\n    else\n      MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n\ndef MemoryAccessList.isConsistentOnline (accesses : MemoryAccessList) (h : accesses.isTimestampSorted) : Prop := match accesses with\n  | [] => True \n  | (_timestamp, addr, readValue, _writeValue) :: rest =>\n    \n    readValue = MemoryAccessList.lastWriteValue rest (List.Sorted.of_cons h) addr\n    ∧ MemoryAccessList.isConsistentOnline rest (List.Sorted.of_cons h)\n\nexample : MemoryAccessList.isConsistentOnline [] (by admit /- proof elided -/\n) := by admit /- proof elided -/\n\ndef MemoryAccessList.filterAddress (accesses : MemoryAccessList) (addr : ℕ) : MemoryAccessList :=\n  accesses.filter (fun (_timestamp, addr', _readValue, _writeValue) => addr' = addr)", "target_theorem": "theorem MemoryAccessList.isConsistentOnline_filter_of_consistentOnline (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted)\n    (h_consistent : MemoryAccessList.isConsistentOnline accesses h_sorted) (addr : ℕ) :\n    MemoryAccessList.isConsistentOnline (MemoryAccessList.filterAddress accesses addr) (MemoryAccessList.filterAddress_sorted accesses h_sorted addr) :=", "ground_truth_proof": ":= by\n  induction accesses with\n  | nil =>\n    simp only [filterAddress, List.filter_nil, isConsistentOnline]\n  | cons head tail ih =>\n    obtain ⟨t, a, r, w⟩ := head\n    simp [filterAddress, List.filter_cons, isConsistentOnline] at ⊢ h_consistent ih\n    have h_sorted' : isTimestampSorted tail := by\n      unfold isTimestampSorted at h_sorted\n      exact List.Sorted.of_cons h_sorted\n    -- is the current address the one we are filtering for?\n    by_cases h_addr : a = addr\n    ·\n      specialize ih h_sorted' (And.right h_consistent)\n      simp [h_addr, isConsistentOnline, ih]\n      have h := MemoryAccessList.lastWriteValue_filter\n      simp [h_consistent.left]\n      rw [MemoryAccessList.lastWriteValue_filter]\n      · simp [filterAddress, h_addr]\n      · have h_sorted_tail' : (MemoryAccessList.filterAddress tail addr).isTimestampSorted := by\n          simp only [filterAddress]\n          apply List.Sorted.filter\n          exact h_sorted'\n        rw [h_addr]\n        exact h_sorted_tail'\n    · simp_all only [forall_const, forall_true_left, ↓reduceIte]", "nesting_depth": 3, "transitive_dep_count": 17, "subset_aristotle": true, "category": "Applied verif."}
{"id": 155, "thm_name": "Circomlib.MultiAND.localLength_eq", "thm_stmt": "theorem localLength_eq (n : ℕ) (input : Var (fields n) (F p)) (offset : ℕ) :\n    (main input).localLength offset = n - 1", "lean_root": "clean", "rel_path": "Clean/Circomlib/Gates.lean", "imports": ["import Clean.Circuit.Theorems", "import Clean.Utils.Field", "import Clean.Circuit", "import Mathlib.Data.Nat.Bitwise", "import Clean.Gadgets.Boolean", "import Clean.Utils.Bitwise", "import Clean.Utils.BinaryOps", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "One", "module": "Init.Prelude"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "cast", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "fields", "content": "@[reducible]\ndef fields (n : ℕ) := fun F => Vector F n"}, {"name": "IsBool", "content": "def IsBool {α : Type*} [Zero α] [One α] (x : α) : Prop := x = 0 ∨ x = 1"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "fieldPair", "content": "@[reducible]\ndef fieldPair : TypeMap := fun F => F × F"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}], "lib_lemmas": [{"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.strong_induction_on", "module": "Mathlib.Data.Nat.Init"}], "repo_lemmas": [{"name": "bind_localLength_eq", "content": "theorem bind_localLength_eq (f : Circuit F α) (g : α → Circuit F β) (n : ℕ) :\n    (f >>= g).localLength n = f.localLength n + (g (f.output n)).localLength (n + f.localLength n)"}, {"name": "append_localLength", "content": "@[circuit_norm]\ntheorem append_localLength {a b: Operations F} :\n    (a ++ b).localLength = a.localLength + b.localLength"}], "used_local_defs": [{"name": "Circomlib.AND.main", "content": "def main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out"}, {"name": "Circomlib.MultiAND.main", "content": "def main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)"}], "used_local_lemmas": [], "local_ctx": "import Clean.Circuit\n\nimport Clean.Utils.Field\n\nimport Clean.Gadgets.Boolean\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Utils.Vector\n\nimport Clean.Utils.BinaryOps\n\nimport Clean.Circuit.Theorems\n\nimport Mathlib.Data.Nat.Bitwise\n\nopen IsBool\n\nnamespace Circomlib\n\nvariable {p : ℕ} [Fact p.Prime]\n\nopen Circuit (bind_output_eq bind_localLength_eq bind_forAll)\n\nopen Operations (append_localLength)\n\nopen BinaryOps (List.foldl_and_IsBool List.and_foldl_eq_foldl)\n\nnamespace XOR\n\nend XOR\n\nnamespace AND\n\ndef main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out\n\nend AND\n\nnamespace OR\n\nend OR\n\nnamespace NOT\n\nend NOT\n\nnamespace NAND\n\nend NAND\n\nnamespace NOR\n\nend NOR\n\nnamespace MultiAND\n\ndef main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)", "target_theorem": "theorem localLength_eq (n : ℕ) (input : Var (fields n) (F p)) (offset : ℕ) :\n    (main input).localLength offset = n - 1 :=", "ground_truth_proof": ":= by\n  induction n using Nat.strong_induction_on generalizing offset with\n  | _ n IH =>\n    match n with\n    | 0 =>\n      simp only [main]\n      rfl\n    | 1 =>\n      simp only [main]\n      rfl\n    | 2 =>\n      simp only [main]\n      simp only [Nat.add_one_sub_one]\n      have h := AND.circuit.localLength_eq (input[0], input[1]) offset\n      rw [show AND.circuit.localLength _ = 1 from rfl] at h\n      exact h\n    | m + 3 =>\n      let n1 := (m + 3) / 2\n      let n2 := (m + 3) - n1\n      have h_sum : n1 + n2 = m + 3 := by\n        unfold n1 n2\n        omega\n      have h_n1_lt : n1 < m + 3 := by\n        unfold n1\n        omega\n      have h_n2_lt : n2 < m + 3 := by\n        unfold n2\n        omega\n      rw [main]\n      repeat rw [bind_localLength_eq]\n      simp only [IH _ h_n1_lt, IH _ h_n2_lt]\n      simp only [Circuit.output]\n      have h_and : ∀ (inp : Expression (F p) × Expression (F p)) (off : ℕ),\n        (AND.circuit.main inp).localLength off = 1 := by\n        intro inp off\n        have := AND.circuit.localLength_eq inp off\n        rw [show AND.circuit.localLength _ = 1 from rfl] at this\n        exact this\n\n      rw [h_and]\n      conv => rhs; rw [← h_sum]\n      omega", "nesting_depth": 9, "transitive_dep_count": 87, "subset_aristotle": true, "category": "Applied verif."}
{"id": 156, "thm_name": "FlatOperation.localLength_toFlat", "thm_stmt": "lemma localLength_toFlat {ops : Operations F} :\n    localLength ops.toFlat = ops.localLength", "lean_root": "clean", "rel_path": "Clean/Circuit/Theorems.lean", "imports": ["import Clean.Circuit.Provable", "import Clean.Circuit.Basic"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}], "lib_lemmas": [{"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "forall_eq'", "module": "Init.PropLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "FlatOperation.localLength_append", "content": "lemma localLength_append {F} {a b: List (FlatOperation F)} :\n    localLength (a ++ b) = localLength a + localLength b"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nvariable {F : Type} [Field F] {α β : Type}\n\nnamespace Operations\n\nend Operations\n\nnamespace Circuit\n\nend Circuit\n\nnamespace FlatOperation", "target_theorem": "lemma localLength_toFlat {ops : Operations F} :\n    localLength ops.toFlat = ops.localLength :=", "ground_truth_proof": ":= by\n  induction ops using Operations.induct with\n  | empty => trivial\n  | witness _ _ ops ih | assert _ ops ih | lookup _ ops ih  | subcircuit _ ops ih =>\n    dsimp only [Operations.toFlat, Operations.localLength]\n    generalize ops.toFlat = flat_ops at *\n    generalize Operations.localLength ops = n at *\n    induction flat_ops using localLength.induct generalizing n with\n    | case1 => simp_all [localLength, add_comm, Subcircuit.localLength_eq]\n    | case2 m' _ ops' ih' =>\n      dsimp only [localLength, witness] at *\n      specialize ih' (n - m') (by rw [←ih]; omega)\n      simp_all +arith only [localLength_append, localLength]\n      try omega\n    | case3 ops _ ih' | case4 ops _ ih' =>\n      simp_all only [localLength_append, forall_eq', localLength]", "nesting_depth": 9, "transitive_dep_count": 43, "subset_aristotle": true, "category": "Applied verif."}
{"id": 157, "thm_name": "Gadgets.Xor64.completeness", "thm_stmt": "theorem completeness : Completeness (F p) elaborated Assumptions", "lean_root": "clean", "rel_path": "Clean/Gadgets/Xor/Xor64.lean", "imports": ["import Clean.Gadgets.Xor.ByteXorTable", "import Clean.Circuit.Provable", "import Clean.Circuit.Expression", "import Clean.Types.U64", "import Clean.Utils.Field", "import Mathlib.Data.ZMod.Basic", "import Clean.Utils.Primes", "import Mathlib.Algebra.Field.Basic", "import Clean.Circuit.Basic", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Fin.xor", "module": "Init.Data.Fin.Basic"}, {"name": "HXor", "module": "Init.Prelude"}, {"name": "HXor.hXor", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "XorOp", "module": "Init.Prelude"}, {"name": "XorOp.xor", "module": "Init.Prelude"}, {"name": "Vector.push", "module": "Init.Data.Vector.Basic"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}], "used_repo_defs": [{"name": "syntax \"let \" ident \" <== \" term : doElem", "content": "syntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem\n\nsyntax \"infer_constant_length\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "U64", "content": "structure U64 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\n  x4 : T\n  x5 : T\n  x6 : T\n  x7 : T\nderiving DecidableEq"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "ByteXorTable", "content": "def ByteXorTable : Table (F p) fieldTriple := .fromStatic {\n  name := \"ByteXor\"\n  length := 256*256\n\n  row i :=\n    let (x, y) := splitTwoBytes i\n    (fromByte x, fromByte y, fromByte (x ^^^ y))\n\n  index := fun (x, y, _) => x.val * 256 + y.val\n\n  Spec := fun (x, y, z) =>\n    x.val < 256 ∧ y.val < 256 ∧ z.val = x.val ^^^ y.val\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "splitTwoBytes", "content": "def splitTwoBytes (i : Fin (256 * 256)) : Fin 256 × Fin 256 :=\n  let x := i.val / 256\n  let y := i.val % 256\n  have x_lt : x < 256 := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "fieldTriple", "content": "@[reducible]\ndef fieldTriple : TypeMap := fun F => F × F × F"}, {"name": "concatTwoBytes", "content": "def concatTwoBytes (x y : Fin 256) : Fin (256 * 256) :=\n  let i := x.val * 256 + y.val\n  have i_lt : i < 256 * 256 := by admit /- proof elided -/"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "varFromOffset", "content": "@[explicit_provable_type]\ndef varFromOffset (α : TypeMap) [ProvableType α] (offset : ℕ) : Var α F :=\n  let vars := Vector.mapRange (size α) fun i => var ⟨offset + i⟩\n  fromVars vars"}, {"name": "mapRange", "content": "def mapRange (n : ℕ) (create : ℕ → α) : Vector α n :=\n  match n with\n  | 0 => #v[]\n  | k + 1 => mapRange k create |>.push (create k)"}, {"name": "Normalized", "content": "def Normalized (x : U64 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256 ∧\n  x.x4.val < 256 ∧ x.x5.val < 256 ∧ x.x6.val < 256 ∧ x.x7.val < 256"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}], "lib_lemmas": [{"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.xor_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Fin.forall_iff", "module": "Init.Data.Fin.Lemmas"}], "repo_lemmas": [{"name": "val_lt_p", "content": "theorem val_lt_p {p : ℕ} (x : ℕ) : (x < p) → (x : F p).val = x"}], "used_local_defs": [{"name": "Gadgets.Xor64.Inputs", "content": "structure Inputs (F : Type) where\n  x: U64 F\n  y: U64 F"}, {"name": "Gadgets.Xor64.main", "content": "def main (input : Var Inputs (F p)) : Circuit (F p) (Var U64 (F p))  := do\n  let ⟨x, y⟩ := input\n  let z ← witness fun env =>\n    let z0 := (env x.x0).val ^^^ (env y.x0).val\n    let z1 := (env x.x1).val ^^^ (env y.x1).val\n    let z2 := (env x.x2).val ^^^ (env y.x2).val\n    let z3 := (env x.x3).val ^^^ (env y.x3).val\n    let z4 := (env x.x4).val ^^^ (env y.x4).val\n    let z5 := (env x.x5).val ^^^ (env y.x5).val\n    let z6 := (env x.x6).val ^^^ (env y.x6).val\n    let z7 := (env x.x7).val ^^^ (env y.x7).val\n    U64.mk z0 z1 z2 z3 z4 z5 z6 z7\n\n  lookup ByteXorTable (x.x0, y.x0, z.x0)\n  lookup ByteXorTable (x.x1, y.x1, z.x1)\n  lookup ByteXorTable (x.x2, y.x2, z.x2)\n  lookup ByteXorTable (x.x3, y.x3, z.x3)\n  lookup ByteXorTable (x.x4, y.x4, z.x4)\n  lookup ByteXorTable (x.x5, y.x5, z.x5)\n  lookup ByteXorTable (x.x6, y.x6, z.x6)\n  lookup ByteXorTable (x.x7, y.x7, z.x7)\n  return z"}, {"name": "Gadgets.Xor64.Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.Normalized ∧ y.Normalized"}, {"name": "Gadgets.Xor64.elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs U64 where\n  main := main\n  localLength _ := 8\n  output _ i0 := varFromOffset U64 i0"}], "used_local_lemmas": [{"name": "Gadgets.Xor64.xor_val", "content": "lemma xor_val {x y : F p} (hx : x.val < 256) (hy : y.val < 256) :\n    (x.val ^^^ y.val : F p).val = x.val ^^^ y.val"}], "local_ctx": "import Mathlib.Algebra.Field.Basic\n\nimport Mathlib.Data.ZMod.Basic\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Vector\n\nimport Clean.Circuit.Expression\n\nimport Clean.Circuit.Provable\n\nimport Clean.Circuit.Basic\n\nimport Clean.Utils.Field\n\nimport Clean.Types.U64\n\nimport Clean.Gadgets.Xor.ByteXorTable\n\nsection\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nnamespace Gadgets.Xor64\n\nopen Gadgets.Xor\n\nstructure Inputs (F : Type) where\n  x: U64 F\n  y: U64 F\n\ndef main (input : Var Inputs (F p)) : Circuit (F p) (Var U64 (F p))  := do\n  let ⟨x, y⟩ := input\n  let z ← witness fun env =>\n    let z0 := (env x.x0).val ^^^ (env y.x0).val\n    let z1 := (env x.x1).val ^^^ (env y.x1).val\n    let z2 := (env x.x2).val ^^^ (env y.x2).val\n    let z3 := (env x.x3).val ^^^ (env y.x3).val\n    let z4 := (env x.x4).val ^^^ (env y.x4).val\n    let z5 := (env x.x5).val ^^^ (env y.x5).val\n    let z6 := (env x.x6).val ^^^ (env y.x6).val\n    let z7 := (env x.x7).val ^^^ (env y.x7).val\n    U64.mk z0 z1 z2 z3 z4 z5 z6 z7\n\n  lookup ByteXorTable (x.x0, y.x0, z.x0)\n  lookup ByteXorTable (x.x1, y.x1, z.x1)\n  lookup ByteXorTable (x.x2, y.x2, z.x2)\n  lookup ByteXorTable (x.x3, y.x3, z.x3)\n  lookup ByteXorTable (x.x4, y.x4, z.x4)\n  lookup ByteXorTable (x.x5, y.x5, z.x5)\n  lookup ByteXorTable (x.x6, y.x6, z.x6)\n  lookup ByteXorTable (x.x7, y.x7, z.x7)\n  return z\n\ndef Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.Normalized ∧ y.Normalized\n\ninstance elaborated : ElaboratedCircuit (F p) Inputs U64 where\n  main := main\n  localLength _ := 8\n  output _ i0 := varFromOffset U64 i0", "target_theorem": "theorem completeness : Completeness (F p) elaborated Assumptions :=", "ground_truth_proof": ":= by\n  intro i0 env input_var h_env input h_input as\n  let ⟨⟨ x0, x1, x2, x3, x4, x5, x6, x7 ⟩, ⟨ y0, y1, y2, y3, y4, y5, y6, y7 ⟩⟩ := input\n  simp only [circuit_norm, explicit_provable_type, Inputs.mk.injEq, U64.mk.injEq] at h_input\n  simp only [Assumptions, circuit_norm, U64.Normalized] at as\n  simp only [h_input, circuit_norm, main, ByteXorTable,\n    explicit_provable_type, Fin.forall_iff] at h_env ⊢\n  have h_env0 : env.get i0 = ↑(ZMod.val x0 ^^^ ZMod.val y0) := by simpa using h_env 0\n  simp_all [xor_val]", "nesting_depth": 8, "transitive_dep_count": 99, "subset_aristotle": true, "category": "Applied verif."}
{"id": 158, "thm_name": "Circuit.subcircuit_computableWitnesses", "thm_stmt": "theorem Circuit.subcircuit_computableWitnesses (circuit : FormalCircuit F β α) (input : Var β F) (n : ℕ) :\n  Environment.OnlyAccessedBelow n (eval · input) ∧ circuit.ComputableWitnesses →\n    (subcircuit circuit input).ComputableWitnesses n", "lean_root": "clean", "rel_path": "Clean/Circuit/Subcircuit.lean", "imports": ["import Clean.Circuit.Basic", "import Clean.Circuit.Theorems"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "Circuit.ComputableWitnesses", "content": "def Circuit.ComputableWitnesses (circuit : Circuit F α) (n : ℕ) :=\n  ∀ env env', (circuit.operations n).ComputableWitnesses n env env'"}, {"name": "Environment.OnlyAccessedBelow", "content": "def Environment.OnlyAccessedBelow (n : ℕ) (f : Environment F → α) :=\n  ∀ env env', env.AgreesBelow n env' → f env = f env'"}, {"name": "Operations.ComputableWitnesses", "content": "def Operations.ComputableWitnesses (ops : Operations F) (n : ℕ) (env env' : Environment F) : Prop :=\n  ops.forAllFlat n { witness n _ compute := env.AgreesBelow n env' → compute env = compute env' }"}, {"name": "Environment.AgreesBelow", "content": "def Environment.AgreesBelow (n : ℕ) (env env' : Environment F) :=\n  ∀ i < n, env.get i = env'.get i"}, {"name": "Condition.ignoreSubcircuit", "content": "def Condition.ignoreSubcircuit (condition : Condition F) : Condition F :=\n  { condition with subcircuit _ _ _ := True }"}, {"name": "Condition.implies", "content": "def Condition.implies (c c': Condition F) : Condition F where\n  witness n m compute := c.witness n m compute → c'.witness n m compute\n  assert offset e := c.assert offset e → c'.assert offset e\n  lookup offset l := c.lookup offset l → c'.lookup offset l\n  subcircuit offset _ s := c.subcircuit offset s → c'.subcircuit offset s"}, {"name": "induct", "content": "def induct {motive : List (FlatOperation F) → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n    (ops : List (FlatOperation F)) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup ops)"}, {"name": "Condition.applyFlat", "content": "def Condition.applyFlat (condition : Condition F) (offset : ℕ) : FlatOperation F → Prop\n  | .witness m c => condition.witness offset m c\n  | .assert e => condition.assert offset e\n  | .lookup l => condition.lookup offset l"}, {"name": "FlatOperation.singleLocalLength", "content": "def FlatOperation.singleLocalLength : FlatOperation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}], "lib_lemmas": [{"name": "forall_const", "module": "Init.PropLemmas"}, {"name": "imp_self", "module": "Init.Core"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "Environment.agreesBelow_of_le", "content": "lemma Environment.agreesBelow_of_le {F} {n m : ℕ} {env env' : Environment F} :\n    env.AgreesBelow n env' → m ≤ n → env.AgreesBelow m env'"}, {"name": "forAll_implies", "content": "theorem forAll_implies {c c' : Condition F} (n : ℕ) {ops : List (FlatOperation F)} :\n    (forAll n (c.implies c').ignoreSubcircuit ops) → (forAll n c ops → forAll n c' ops)"}, {"name": "forAll_cons", "content": "theorem forAll_cons {condition : Condition F} {offset : ℕ} {op : FlatOperation F} {ops : List (FlatOperation F)} :\n  forAll offset condition (op :: ops) ↔\n    condition.applyFlat offset op ∧ forAll (op.singleLocalLength + offset) condition ops"}, {"name": "forAll_empty", "content": "theorem forAll_empty {condition : Condition F} {n : ℕ} : forAll n condition [] = True"}, {"name": "forAll_toFlat_iff", "content": "lemma forAll_toFlat_iff (n : ℕ) (condition : Condition F) (ops : Operations F) :\n    FlatOperation.forAll n condition ops.toFlat ↔ ops.forAllFlat n condition"}, {"name": "forAll_append", "content": "lemma forAll_append {condition : Condition F} {ops ops' : List (FlatOperation F)} (n : ℕ) :\n  forAll n condition (ops ++ ops') ↔\n    forAll n condition ops ∧ forAll (localLength ops + n) condition ops'"}, {"name": "localLength_cons", "content": "lemma localLength_cons {F} {op : FlatOperation F} {ops : List (FlatOperation F)} :\n    localLength (op :: ops) = op.singleLocalLength + localLength ops"}], "used_local_defs": [{"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "ElaboratedCircuit.ComputableWitnesses'", "content": "def ComputableWitnesses' (circuit : ElaboratedCircuit F β α) : Prop :=\n  ∀ (n : ℕ) (input : Var β F),\n    Environment.OnlyAccessedBelow n (eval · input) →\n      (circuit.main input).ComputableWitnesses n"}, {"name": "ElaboratedCircuit.ComputableWitnesses", "content": "def ComputableWitnesses (circuit : ElaboratedCircuit F β α) : Prop :=\n  ∀ (n : ℕ) (input : Var β F) env env',\n  circuit.main input |>.operations n |>.forAllFlat n {\n    witness n _ compute :=\n      env.AgreesBelow n env' → eval env input = eval env' input → compute env = compute env' }"}], "used_local_lemmas": [{"name": "ElaboratedCircuit.computableWitnesses_implies", "content": "lemma computableWitnesses_implies {circuit : ElaboratedCircuit F β α} :\n    circuit.ComputableWitnesses → circuit.ComputableWitnesses'"}, {"name": "ElaboratedCircuit.compose_computableWitnesses", "content": "theorem compose_computableWitnesses (circuit : ElaboratedCircuit F β α) (input : Var β F) (n : ℕ) :\n  Environment.OnlyAccessedBelow n (eval · input) ∧ circuit.ComputableWitnesses →\n    (circuit.main input).ComputableWitnesses n"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Theorems\n\nvariable {F : Type} [Field F]\n\nnamespace FlatOperation\n\nopen Circuit (ConstraintsHold.Completeness ConstraintsHold)\n\nend FlatOperation\n\nvariable {α β: TypeMap} [ProvableType α] [ProvableType β]\n\nsection\n\nopen Circuit\n\nopen FlatOperation (constraintsHold_cons constraintsHold_append)\n\ndef FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :=\n\ndef FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/\n\ndef GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :=\n\nend\n\n@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])\n\nnamespace Circuit\n\nvariable {α β: TypeMap} [ProvableType α] [ProvableType β]\n\nend Circuit\n\nnamespace ElaboratedCircuit\n\ndef ComputableWitnesses' (circuit : ElaboratedCircuit F β α) : Prop :=\n  ∀ (n : ℕ) (input : Var β F),\n    Environment.OnlyAccessedBelow n (eval · input) →\n      (circuit.main input).ComputableWitnesses n\n\ndef ComputableWitnesses (circuit : ElaboratedCircuit F β α) : Prop :=\n  ∀ (n : ℕ) (input : Var β F) env env',\n  circuit.main input |>.operations n |>.forAllFlat n {\n    witness n _ compute :=\n      env.AgreesBelow n env' → eval env input = eval env' input → compute env = compute env' }\n\nend ElaboratedCircuit", "target_theorem": "theorem Circuit.subcircuit_computableWitnesses (circuit : FormalCircuit F β α) (input : Var β F) (n : ℕ) :\n  Environment.OnlyAccessedBelow n (eval · input) ∧ circuit.ComputableWitnesses →\n    (subcircuit circuit input).ComputableWitnesses n :=", "ground_truth_proof": ":= by\n  intro h env env'\n  simp only [circuit_norm, FormalCircuit.toSubcircuit, Operations.ComputableWitnesses,\n    Operations.forAllFlat, Operations.forAll_toFlat_iff]\n  exact circuit.compose_computableWitnesses input n h env env'", "nesting_depth": 12, "transitive_dep_count": 91, "subset_aristotle": true, "category": "Applied verif."}
{"id": 159, "thm_name": "Gadgets.Xor64.soundness", "thm_stmt": "theorem soundness : Soundness (F p) elaborated Assumptions Spec", "lean_root": "clean", "rel_path": "Clean/Gadgets/Xor/Xor64.lean", "imports": ["import Clean.Gadgets.Xor.ByteXorTable", "import Clean.Circuit.Provable", "import Clean.Circuit.Expression", "import Clean.Types.U64", "import Clean.Utils.Field", "import Mathlib.Data.ZMod.Basic", "import Clean.Utils.Primes", "import Clean.Utils.Bitwise", "import Mathlib.Algebra.Field.Basic", "import Clean.Circuit.Basic", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Fin.xor", "module": "Init.Data.Fin.Basic"}, {"name": "HXor", "module": "Init.Prelude"}, {"name": "HXor.hXor", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "XorOp", "module": "Init.Prelude"}, {"name": "XorOp.xor", "module": "Init.Prelude"}, {"name": "Vector.push", "module": "Init.Data.Vector.Basic"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.Prime", "module": "Mathlib.Data.Nat.Prime.Defs"}], "used_repo_defs": [{"name": "syntax \"let \" ident \" <== \" term : doElem", "content": "syntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem\n\nsyntax \"infer_constant_length\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "U64", "content": "structure U64 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\n  x4 : T\n  x5 : T\n  x6 : T\n  x7 : T\nderiving DecidableEq"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "ByteXorTable", "content": "def ByteXorTable : Table (F p) fieldTriple := .fromStatic {\n  name := \"ByteXor\"\n  length := 256*256\n\n  row i :=\n    let (x, y) := splitTwoBytes i\n    (fromByte x, fromByte y, fromByte (x ^^^ y))\n\n  index := fun (x, y, _) => x.val * 256 + y.val\n\n  Spec := fun (x, y, z) =>\n    x.val < 256 ∧ y.val < 256 ∧ z.val = x.val ^^^ y.val\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "splitTwoBytes", "content": "def splitTwoBytes (i : Fin (256 * 256)) : Fin 256 × Fin 256 :=\n  let x := i.val / 256\n  let y := i.val % 256\n  have x_lt : x < 256 := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "fieldTriple", "content": "@[reducible]\ndef fieldTriple : TypeMap := fun F => F × F × F"}, {"name": "concatTwoBytes", "content": "def concatTwoBytes (x y : Fin 256) : Fin (256 * 256) :=\n  let i := x.val * 256 + y.val\n  have i_lt : i < 256 * 256 := by admit /- proof elided -/"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "varFromOffset", "content": "@[explicit_provable_type]\ndef varFromOffset (α : TypeMap) [ProvableType α] (offset : ℕ) : Var α F :=\n  let vars := Vector.mapRange (size α) fun i => var ⟨offset + i⟩\n  fromVars vars"}, {"name": "mapRange", "content": "def mapRange (n : ℕ) (create : ℕ → α) : Vector α n :=\n  match n with\n  | 0 => #v[]\n  | k + 1 => mapRange k create |>.push (create k)"}, {"name": "Normalized", "content": "def Normalized (x : U64 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256 ∧\n  x.x4.val < 256 ∧ x.x5.val < 256 ∧ x.x6.val < 256 ∧ x.x7.val < 256"}, {"name": "value", "content": "def value (x : U64 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3 +\n  x.x4.val * 256^4 + x.x5.val * 256^5 + x.x6.val * 256^6 + x.x7.val * 256^7"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "KeccakRow.value", "content": "def KeccakRow.value (row : KeccakRow (F p)) := row.map U64.value"}, {"name": "map", "content": "def map {α β : Type} (x : U64 α) (f : α → β) : U64 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3, f x.x4, f x.x5, f x.x6, f x.x7 ⟩"}, {"name": "KeccakRow.Normalized", "content": "def KeccakRow.Normalized (row : KeccakRow (F p)) :=\n  ∀ i : Fin 5, row[i.val].Normalized"}, {"name": "BLAKE3State.Normalized", "content": "def BLAKE3State.Normalized (state : BLAKE3State (F p)) :=\n  ∀ i : Fin 16, state[i.val].Normalized"}, {"name": "KeccakBlock.value", "content": "def KeccakBlock.value (block : KeccakBlock (F p)) := block.map U64.value"}, {"name": "BLAKE3State.value", "content": "def BLAKE3State.value (state : BLAKE3State (F p)) := state.map U32.value"}, {"name": "value", "content": "def value (x : U32 (F p)) :=\n  x.x0.val + x.x1.val * 256 + x.x2.val * 256^2 + x.x3.val * 256^3"}, {"name": "U32", "content": "structure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq"}, {"name": "map", "content": "def map {α β : Type} (x : U32 α) (f : α → β) : U32 β :=\n  ⟨ f x.x0, f x.x1, f x.x2, f x.x3 ⟩"}, {"name": "KeccakState.value", "content": "def KeccakState.value (state : KeccakState (F p)) := state.map U64.value"}, {"name": "KeccakBlock.Normalized", "content": "def KeccakBlock.Normalized (block : KeccakBlock (F p)) :=\n  ∀ i : Fin RATE, block[i.val].Normalized"}, {"name": "RATE", "content": "@[reducible] def RATE := 17\nexample : RATE + CAPACITY = 25 := rfl"}, {"name": "CAPACITY", "content": "@[reducible] def CAPACITY := 8"}, {"name": "KeccakState.Normalized", "content": "def KeccakState.Normalized (state : KeccakState (F p)) :=\n  ∀ i : Fin 25, state[i.val].Normalized"}], "lib_lemmas": [{"name": "Nat.xor_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}], "repo_lemmas": [{"name": "value_xor_horner", "content": "omit [Fact (Nat.Prime p)] p_large_enough in\ntheorem value_xor_horner {x : U64 (F p)} (hx : x.Normalized) : x.value =\n    x.x0.val ^^^ 2^8 * (x.x1.val ^^^ 2^8 * (x.x2.val ^^^ 2^8 * (x.x3.val ^^^\n      2^8 * (x.x4.val ^^^ 2^8 * (x.x5.val ^^^ 2^8 * (x.x6.val ^^^ 2^8 * x.x7.val))))))"}, {"name": "value_horner", "content": "omit [Fact (Nat.Prime p)] p_large_enough in\ntheorem value_horner (x : U64 (F p)) : x.value =\n    x.x0.val + 2^8 * (x.x1.val + 2^8 * (x.x2.val + 2^8 * (x.x3.val +\n      2^8 * (x.x4.val + 2^8 * (x.x5.val + 2^8 * (x.x6.val + 2^8 * x.x7.val))))))"}, {"name": "xor_mul_two_pow", "content": "theorem xor_mul_two_pow {x y n : ℕ} : 2 ^ n * (x ^^^ y) =  2 ^ n * x ^^^  2 ^ n * y"}], "used_local_defs": [{"name": "Gadgets.Xor64.Inputs", "content": "structure Inputs (F : Type) where\n  x: U64 F\n  y: U64 F"}, {"name": "Gadgets.Xor64.main", "content": "def main (input : Var Inputs (F p)) : Circuit (F p) (Var U64 (F p))  := do\n  let ⟨x, y⟩ := input\n  let z ← witness fun env =>\n    let z0 := (env x.x0).val ^^^ (env y.x0).val\n    let z1 := (env x.x1).val ^^^ (env y.x1).val\n    let z2 := (env x.x2).val ^^^ (env y.x2).val\n    let z3 := (env x.x3).val ^^^ (env y.x3).val\n    let z4 := (env x.x4).val ^^^ (env y.x4).val\n    let z5 := (env x.x5).val ^^^ (env y.x5).val\n    let z6 := (env x.x6).val ^^^ (env y.x6).val\n    let z7 := (env x.x7).val ^^^ (env y.x7).val\n    U64.mk z0 z1 z2 z3 z4 z5 z6 z7\n\n  lookup ByteXorTable (x.x0, y.x0, z.x0)\n  lookup ByteXorTable (x.x1, y.x1, z.x1)\n  lookup ByteXorTable (x.x2, y.x2, z.x2)\n  lookup ByteXorTable (x.x3, y.x3, z.x3)\n  lookup ByteXorTable (x.x4, y.x4, z.x4)\n  lookup ByteXorTable (x.x5, y.x5, z.x5)\n  lookup ByteXorTable (x.x6, y.x6, z.x6)\n  lookup ByteXorTable (x.x7, y.x7, z.x7)\n  return z"}, {"name": "Gadgets.Xor64.Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.Normalized ∧ y.Normalized"}, {"name": "Gadgets.Xor64.Spec", "content": "def Spec (input : Inputs (F p)) (z : U64 (F p)) :=\n  let ⟨x, y⟩ := input\n  z.value = x.value ^^^ y.value ∧ z.Normalized"}, {"name": "Gadgets.Xor64.elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs U64 where\n  main := main\n  localLength _ := 8\n  output _ i0 := varFromOffset U64 i0"}], "used_local_lemmas": [{"name": "Gadgets.Xor64.soundness_to_u64", "content": "omit [Fact (Nat.Prime p)] p_large_enough in\ntheorem soundness_to_u64 {x y z : U64 (F p)}\n  (x_norm : x.Normalized) (y_norm : y.Normalized)\n  (h_eq :\n    z.x0.val = x.x0.val ^^^ y.x0.val ∧\n    z.x1.val = x.x1.val ^^^ y.x1.val ∧\n    z.x2.val = x.x2.val ^^^ y.x2.val ∧\n    z.x3.val = x.x3.val ^^^ y.x3.val ∧\n    z.x4.val = x.x4.val ^^^ y.x4.val ∧\n    z.x5.val = x.x5.val ^^^ y.x5.val ∧\n    z.x6.val = x.x6.val ^^^ y.x6.val ∧\n    z.x7.val = x.x7.val ^^^ y.x7.val) : Spec { x, y } z"}], "local_ctx": "import Mathlib.Algebra.Field.Basic\n\nimport Mathlib.Data.ZMod.Basic\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Vector\n\nimport Clean.Circuit.Expression\n\nimport Clean.Circuit.Provable\n\nimport Clean.Circuit.Basic\n\nimport Clean.Utils.Field\n\nimport Clean.Types.U64\n\nimport Clean.Gadgets.Xor.ByteXorTable\n\nsection\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nnamespace Gadgets.Xor64\n\nopen Gadgets.Xor\n\nstructure Inputs (F : Type) where\n  x: U64 F\n  y: U64 F\n\ndef main (input : Var Inputs (F p)) : Circuit (F p) (Var U64 (F p))  := do\n  let ⟨x, y⟩ := input\n  let z ← witness fun env =>\n    let z0 := (env x.x0).val ^^^ (env y.x0).val\n    let z1 := (env x.x1).val ^^^ (env y.x1).val\n    let z2 := (env x.x2).val ^^^ (env y.x2).val\n    let z3 := (env x.x3).val ^^^ (env y.x3).val\n    let z4 := (env x.x4).val ^^^ (env y.x4).val\n    let z5 := (env x.x5).val ^^^ (env y.x5).val\n    let z6 := (env x.x6).val ^^^ (env y.x6).val\n    let z7 := (env x.x7).val ^^^ (env y.x7).val\n    U64.mk z0 z1 z2 z3 z4 z5 z6 z7\n\n  lookup ByteXorTable (x.x0, y.x0, z.x0)\n  lookup ByteXorTable (x.x1, y.x1, z.x1)\n  lookup ByteXorTable (x.x2, y.x2, z.x2)\n  lookup ByteXorTable (x.x3, y.x3, z.x3)\n  lookup ByteXorTable (x.x4, y.x4, z.x4)\n  lookup ByteXorTable (x.x5, y.x5, z.x5)\n  lookup ByteXorTable (x.x6, y.x6, z.x6)\n  lookup ByteXorTable (x.x7, y.x7, z.x7)\n  return z\n\ndef Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.Normalized ∧ y.Normalized\n\ndef Spec (input : Inputs (F p)) (z : U64 (F p)) :=\n  let ⟨x, y⟩ := input\n  z.value = x.value ^^^ y.value ∧ z.Normalized\n\ninstance elaborated : ElaboratedCircuit (F p) Inputs U64 where\n  main := main\n  localLength _ := 8\n  output _ i0 := varFromOffset U64 i0", "target_theorem": "theorem soundness : Soundness (F p) elaborated Assumptions Spec :=", "ground_truth_proof": ":= by\n  intro i0 env input_var input h_input h_as h_holds\n\n  let ⟨⟨ x0_var, x1_var, x2_var, x3_var, x4_var, x5_var, x6_var, x7_var ⟩,\n       ⟨ y0_var, y1_var, y2_var, y3_var, y4_var, y5_var, y6_var, y7_var ⟩⟩ := input_var\n  let ⟨⟨ x0, x1, x2, x3, x4, x5, x6, x7 ⟩,\n       ⟨ y0, y1, y2, y3, y4, y5, y6, y7 ⟩⟩ := input\n\n  simp only [circuit_norm, explicit_provable_type, Inputs.mk.injEq, U64.mk.injEq] at h_input\n\n  simp only [circuit_norm, Assumptions] at h_as\n  obtain ⟨ x_norm, y_norm ⟩ := h_as\n\n  simp only [h_input, circuit_norm, main, ByteXorTable,\n    varFromOffset, Vector.mapRange] at h_holds\n\n  apply soundness_to_u64 x_norm y_norm\n  simp only [circuit_norm, explicit_provable_type]\n  simp [h_holds]", "nesting_depth": 8, "transitive_dep_count": 116, "subset_aristotle": true, "category": "Applied verif."}
{"id": 160, "thm_name": "Circuit.ext", "thm_stmt": "@[ext]\ntheorem ext {f g : Circuit F α}\n  (h_output : ∀ n, f.output n = g.output n)\n  (h_operations : ∀ n, f.operations n = g.operations n) :\n    f = g :=\n  ext_iff.mpr fun n => ⟨ h_output n, h_operations n ⟩", "lean_root": "clean", "rel_path": "Clean/Circuit/Theorems.lean", "imports": ["import Clean.Circuit.Provable", "import Clean.Circuit.Basic"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}], "used_repo_defs": [{"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Circuit.ext_iff", "content": "theorem ext_iff {f g : Circuit F α} :\n  (f = g) ↔ (∀ n, (f.output n = g.output n) ∧ (f.operations n = g.operations n))"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nvariable {F : Type} [Field F] {α β : Type}\n\nnamespace Operations\n\nend Operations\n\nnamespace Circuit", "target_theorem": "@[ext]\ntheorem ext {f g : Circuit F α}\n  (h_output : ∀ n, f.output n = g.output n)\n  (h_operations : ∀ n, f.operations n = g.operations n) :\n    f = g :=", "ground_truth_proof": ":=\n  ext_iff.mpr fun n => ⟨ h_output n, h_operations n ⟩", "nesting_depth": 5, "transitive_dep_count": 35, "subset_aristotle": true, "category": "Applied verif."}
{"id": 161, "thm_name": "Circomlib.MultiAND.subcircuitsConsistent", "thm_stmt": "theorem subcircuitsConsistent (n : ℕ) (input : Var (fields n) (F p)) (offset : ℕ) :\n    Operations.SubcircuitsConsistent offset ((main input).operations offset)", "lean_root": "clean", "rel_path": "Clean/Circomlib/Gates.lean", "imports": ["import Clean.Circuit.Theorems", "import Clean.Utils.Field", "import Clean.Circuit", "import Mathlib.Data.Nat.Bitwise", "import Clean.Gadgets.Boolean", "import Clean.Utils.Bitwise", "import Clean.Utils.BinaryOps", "import Clean.Circuit.Basic", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "One", "module": "Init.Prelude"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "cast", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "fields", "content": "@[reducible]\ndef fields (n : ℕ) := fun F => Vector F n"}, {"name": "IsBool", "content": "def IsBool {α : Type*} [Zero α] [One α] (x : α) : Prop := x = 0 ∨ x = 1"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "fieldPair", "content": "@[reducible]\ndef fieldPair : TypeMap := fun F => F × F"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "ExplicitCircuit.from_map", "content": "instance ExplicitCircuit.from_map {f : α → β} {g : Circuit F α}\n    (g_explicit : ExplicitCircuit g) : ExplicitCircuit (f <$> g) where\n  output n := output g n |> f\n  localLength n := localLength g n\n  operations n := operations g n\n\n  output_eq n := by admit /- proof elided -/"}, {"name": "ExplicitCircuits.from_pure", "content": "instance ExplicitCircuits.from_pure {f : α → β} : ExplicitCircuits (fun a => pure (f a) : α → Circuit F β) where\n  output a _ := f a\n  localLength _ _ := 0\n  operations _ _ := []"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}], "lib_lemmas": [{"name": "Nat.min_def", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.strong_induction_on", "module": "Mathlib.Data.Nat.Init"}], "repo_lemmas": [{"name": "bind_forAll", "content": "@[circuit_norm]\ntheorem bind_forAll {f : Circuit F α} {g : α → Circuit F β} :\n  ((f >>= g).operations n).forAll n prop ↔\n    (f.operations n).forAll n prop ∧ (((g (f.output n)).operations (n + f.localLength n)).forAll (n + f.localLength n)) prop"}, {"name": "forAll_append", "content": "@[circuit_norm]\ntheorem forAll_append {condition : Condition F} {offset : ℕ} {as bs: Operations F} :\n  forAll offset condition (as ++ bs) ↔\n    forAll offset condition as ∧ forAll (as.localLength + offset) condition bs"}, {"name": "forAll_empty", "content": "@[circuit_norm]\ntheorem forAll_empty {condition : Condition F} {n : ℕ} : forAll n condition [] = True"}, {"name": "pure_def", "content": "@[circuit_norm]\ntheorem pure_def {α} (a : α) : (pure a : Circuit F α) = fun _ => (a, [])"}], "used_local_defs": [{"name": "Circomlib.AND.main", "content": "def main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out"}, {"name": "Circomlib.MultiAND.main", "content": "def main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)"}], "used_local_lemmas": [{"name": "Circomlib.MultiAND.Circuit.subcircuitsConsistent_bind", "content": "theorem Circuit.subcircuitsConsistent_bind {α β : Type} (f : Circuit (F p) α) (g : α → Circuit (F p) β) (offset : ℕ)\n    (hf : Operations.SubcircuitsConsistent offset (f.operations offset))\n    (hg : Operations.SubcircuitsConsistent (offset + f.localLength offset)\n          ((g (f.output offset)).operations (offset + f.localLength offset))) :\n    Operations.SubcircuitsConsistent offset ((f >>= g).operations offset)"}], "local_ctx": "import Clean.Circuit\n\nimport Clean.Utils.Field\n\nimport Clean.Gadgets.Boolean\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Utils.Vector\n\nimport Clean.Utils.BinaryOps\n\nimport Clean.Circuit.Theorems\n\nimport Mathlib.Data.Nat.Bitwise\n\nopen IsBool\n\nnamespace Circomlib\n\nvariable {p : ℕ} [Fact p.Prime]\n\nopen Circuit (bind_output_eq bind_localLength_eq bind_forAll)\n\nopen Operations (append_localLength)\n\nopen BinaryOps (List.foldl_and_IsBool List.and_foldl_eq_foldl)\n\nnamespace XOR\n\nend XOR\n\nnamespace AND\n\ndef main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out\n\nend AND\n\nnamespace OR\n\nend OR\n\nnamespace NOT\n\nend NOT\n\nnamespace NAND\n\nend NAND\n\nnamespace NOR\n\nend NOR\n\nnamespace MultiAND\n\ndef main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)", "target_theorem": "theorem subcircuitsConsistent (n : ℕ) (input : Var (fields n) (F p)) (offset : ℕ) :\n    Operations.SubcircuitsConsistent offset ((main input).operations offset) :=", "ground_truth_proof": ":= by\n  induction n using Nat.strong_induction_on generalizing offset with\n  | _ n IH =>\n    match n with\n    | 0 =>\n      simp only [main, Circuit.operations, Circuit.pure_def]\n      simp only [Operations.SubcircuitsConsistent, Operations.forAll]\n    | 1 =>\n      simp only [main, Circuit.operations, Circuit.pure_def]\n      simp only [Operations.SubcircuitsConsistent, Operations.forAll]\n    | 2 =>\n      simp only [main, Circuit.operations]\n      exact AND.circuit.subcircuitsConsistent (input[0], input[1]) offset\n    | m + 3 =>\n      rw [main]\n      let n1 := (m + 3) / 2\n      let n2 := (m + 3) - n1\n      have h_n1_lt : n1 < m + 3 := by unfold n1; omega\n      have h_n2_lt : n2 < m + 3 := by unfold n2; omega\n      simp only [Circuit.operations]\n      apply Circuit.subcircuitsConsistent_bind\n      ·\n        let input1 : Var (fields n1) (F p) := input.take n1 |>.cast (by simp only [Nat.min_def, n1]; split <;> omega)\n        apply IH n1 h_n1_lt input1\n      · apply Circuit.subcircuitsConsistent_bind\n        · let input2 : Var (fields n2) (F p) := input.drop n1 |>.cast (by omega)\n          apply IH n2 h_n2_lt input2\n        · apply AND.circuit.subcircuitsConsistent", "nesting_depth": 9, "transitive_dep_count": 92, "subset_aristotle": true, "category": "Applied verif."}
{"id": 162, "thm_name": "Operations.forAll_toFlat_iff", "thm_stmt": "lemma forAll_toFlat_iff (n : ℕ) (condition : Condition F) (ops : Operations F) :\n    FlatOperation.forAll n condition ops.toFlat ↔ ops.forAllFlat n condition", "lean_root": "clean", "rel_path": "Clean/Circuit/Theorems.lean", "imports": ["import Clean.Circuit.Provable", "import Clean.Circuit.Basic"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}, {"name": "FlatOperation.singleLocalLength", "content": "def FlatOperation.singleLocalLength : FlatOperation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0"}, {"name": "Condition.applyFlat", "content": "def Condition.applyFlat (condition : Condition F) (offset : ℕ) : FlatOperation F → Prop\n  | .witness m c => condition.witness offset m c\n  | .assert e => condition.assert offset e\n  | .lookup l => condition.lookup offset l"}], "lib_lemmas": [{"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "and_assoc", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "FlatOperation.localLength_cons", "content": "lemma localLength_cons {F} {op : FlatOperation F} {ops : List (FlatOperation F)} :\n    localLength (op :: ops) = op.singleLocalLength + localLength ops"}, {"name": "FlatOperation.forAll_cons", "content": "theorem forAll_cons {condition : Condition F} {offset : ℕ} {op : FlatOperation F} {ops : List (FlatOperation F)} :\n  forAll offset condition (op :: ops) ↔\n    condition.applyFlat offset op ∧ forAll (op.singleLocalLength + offset) condition ops"}, {"name": "FlatOperation.forAll_append", "content": "lemma forAll_append {condition : Condition F} {ops ops' : List (FlatOperation F)} (n : ℕ) :\n  forAll n condition (ops ++ ops') ↔\n    forAll n condition ops ∧ forAll (localLength ops + n) condition ops'"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nvariable {F : Type} [Field F] {α β : Type}\n\nnamespace Operations\n\nend Operations\n\nnamespace Circuit\n\nend Circuit\n\nnamespace FlatOperation\n\nend FlatOperation\n\nnamespace Environment\n\nopen FlatOperation (localLength localWitnesses)\n\nend Environment\n\nnamespace Circuit\n\nend Circuit\n\nnamespace Circuit\n\nvariable {α β : Type} {n : ℕ} {prop : Condition F} {env : Environment F}\n\nend Circuit\n\nnamespace FlatOperation\n\nend FlatOperation\n\nnamespace Operations", "target_theorem": "lemma forAll_toFlat_iff (n : ℕ) (condition : Condition F) (ops : Operations F) :\n    FlatOperation.forAll n condition ops.toFlat ↔ ops.forAllFlat n condition :=", "ground_truth_proof": ":= by\n  induction ops using Operations.induct generalizing n with\n  | empty => simp only [forAllFlat, forAll, toFlat, FlatOperation.forAll]\n  | witness | assert | lookup =>\n    simp_all [forAllFlat, forAll, toFlat, FlatOperation.forAll]\n  | subcircuit s ops ih =>\n    simp_all only [forAllFlat, forAll, toFlat]\n    rw [FlatOperation.forAll_append, s.localLength_eq]\n    simp_all", "nesting_depth": 7, "transitive_dep_count": 49, "subset_aristotle": false, "category": "Applied verif."}
{"id": 163, "thm_name": "Circuit.FoldlM.forAll_iff_const", "thm_stmt": "theorem forAll_iff_const [NeZero m] (constant : ConstantLength (prod circuit))\n    (h_const_out : ConstantOutput (prod circuit)) :\n  (xs.foldlM circuit init).forAll n prop ↔\n  (circuit init (xs[0]'(NeZero.pos m))).forAll n prop ∧\n  ∀ (i : ℕ) (hi : i + 1 < m),\n    let acc := (circuit default xs[i]).output (n + i*(circuit default default).localLength);\n    (circuit acc xs[i + 1]).forAll (n + (i + 1)*(circuit default default).localLength) prop", "lean_root": "clean", "rel_path": "Clean/Circuit/Loops.lean", "imports": ["import Clean.Utils.Misc", "import Clean.Circuit.Subcircuit", "import Clean.Circuit.Theorems"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Vector.foldlM", "module": "Init.Data.Vector.Basic"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "List.ofFn", "module": "Init.Data.List.OfFn"}, {"name": "Vector.mk", "module": "Init.Data.Vector.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}], "used_repo_defs": [{"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantOutput", "content": "@[circuit_norm]\ndef ConstantOutput (circuit : α → Circuit F β) [Inhabited α] :=\n  ∀ (x : α) (n : ℕ), (circuit x).output n = (circuit default).output n\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "induct", "content": "def induct {motive : {n : ℕ} → Vector α n → Sort u}\n    (nil : motive #v[])\n    (cons: ∀ {n : ℕ} (a : α) (as : Vector α n), motive as → motive (cons a as))\n    {n : ℕ} (v : Vector α n) : motive v :=\n  match v with\n  | ⟨ .mk [], h ⟩ => by admit /- proof elided -/\n  | ⟨ .mk (a :: as), h ⟩ => by admit /- proof elided -/"}, {"name": "cons", "content": "def cons (a : α) (v : Vector α n) : Vector α (n + 1) :=\n  ⟨ .mk (a :: v.toList), by admit /- proof elided -/\n  ⟩"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}, {"name": "forAll", "content": "@[reducible, circuit_norm]\ndef forAll (circuit : Circuit F α) (n : ℕ) (prop : Condition F) :=\n  (circuit.operations n).forAll n prop"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}], "lib_lemmas": [{"name": "List.foldlM_toArray", "module": "Init.Data.List.ToArray"}, {"name": "Vector.foldlM_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "List.foldlM_cons", "module": "Init.Data.List.Control"}, {"name": "Vector.toList_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "List.flatten_cons", "module": "Init.Data.List.Basic"}, {"name": "List.ofFn_succ", "module": "Init.Data.List.OfFn"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Fin.foldl_zero", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.foldl_succ", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.val_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.getElem_cons_succ", "module": "Init.GetElem"}, {"name": "List.getElem_cons_zero", "module": "Init.GetElem"}, {"name": "List.getElem_toArray", "module": "Init.Data.Array.Basic"}, {"name": "Vector.getElem_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "iff_iff_eq", "module": "Init.Core"}, {"name": "Fin.val_last", "module": "Init.Data.Fin.Lemmas"}, {"name": "NeZero.pos", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "Fin.mk_zero'", "module": "Mathlib.Data.Fin.Basic"}], "repo_lemmas": [{"name": "bind_localLength_eq", "content": "theorem bind_localLength_eq (f : Circuit F α) (g : α → Circuit F β) (n : ℕ) :\n    (f >>= g).localLength n = f.localLength n + (g (f.output n)).localLength (n + f.localLength n)"}, {"name": "append_localLength", "content": "@[circuit_norm]\ntheorem append_localLength {a b: Operations F} :\n    (a ++ b).localLength = a.localLength + b.localLength"}, {"name": "forAll_def", "content": "lemma forAll_def {circuit : Circuit F α} {n : ℕ} :\n  circuit.forAll n prop ↔ (circuit.operations n).forAll n prop"}, {"name": "forAll_append", "content": "@[circuit_norm]\ntheorem forAll_append {condition : Condition F} {offset : ℕ} {as bs: Operations F} :\n  forAll offset condition (as ++ bs) ↔\n    forAll offset condition as ∧ forAll (as.localLength + offset) condition bs"}, {"name": "forAll_empty", "content": "@[circuit_norm]\ntheorem forAll_empty {condition : Condition F} {n : ℕ} : forAll n condition [] = True"}, {"name": "bind_operations_eq", "content": "theorem bind_operations_eq (f : Circuit F α) (g : α → Circuit F β) (n : ℕ) :\n  (f >>= g).operations n = f.operations n ++ (g (f.output n)).operations (n + f.localLength n)"}, {"name": "Fin.foldl_const", "content": "theorem Fin.foldl_const (n : ℕ) (f : Fin n → α) (init : α) :\n  Fin.foldl n (fun _ i => f i) init = match n with"}, {"name": "Fin.foldl_const_succ", "content": "theorem Fin.foldl_const_succ (n : ℕ) (f : Fin (n + 1) → α) (init : α) :\n    Fin.foldl (n + 1) (fun _ i => f i) init = f (.last n)"}], "used_local_defs": [{"name": "Circuit.FoldlM.prod", "content": "@[reducible]\ndef prod (circuit : β → α → Circuit F β) : β × α → Circuit F β := fun t => circuit t.1 t.2"}, {"name": "Circuit.FoldlM.foldlAcc", "content": "def foldlAcc (n : ℕ) (xs : Vector α m) (circuit : β → α → Circuit F β) (init : β) (j : Fin m) : β :=\n  Fin.foldl j (fun acc i => (circuit acc xs[i.val]).output (n + i*(circuit acc xs[i.val]).localLength)) init"}], "used_local_lemmas": [{"name": "Vector.foldlM_toList", "content": "lemma Vector.foldlM_toList (xs : Vector α n) {m : Type → Type} [Monad m] (body : β → α → m β) (init : β) :\n    xs.foldlM body init = xs.toList.foldlM body init"}, {"name": "Circuit.ConstantLength.length_eq_default", "content": "lemma ConstantLength.length_eq_default {circuit : α → Circuit F β} (_ : ConstantLength circuit) [Inhabited α] (a : α) (n : ℕ) :\n   (circuit a).localLength n = (circuit default).localLength 0"}, {"name": "Circuit.forAll_flatten_abstract", "content": "lemma forAll_flatten_abstract (circuit : Fin m → Circuit F β) (constant : ConstantLength circuit) :\n  Operations.forAll n prop (List.ofFn fun i => (circuit i).operations (n + i * constant.localLength)).flatten\n    ↔ ∀ (i : Fin m), (circuit i).forAll (n + i * constant.localLength) prop"}, {"name": "Circuit.FoldlM.foldlM_cons", "content": "lemma foldlM_cons (x : α) :\n  (Vector.cons x xs).foldlM circuit init = (do\n    let init' ← circuit init x\n    xs.foldlM circuit init')"}, {"name": "Circuit.FoldlM.localLength_eq", "content": "theorem localLength_eq :\n    (xs.foldlM circuit init).localLength n = m * constant.localLength"}, {"name": "Circuit.FoldlM.foldlAcc_zero", "content": "lemma foldlAcc_zero [NeZero m] : foldlAcc n xs circuit init 0 = init"}, {"name": "Circuit.FoldlM.foldlAcc_cons_succ", "content": "lemma foldlAcc_cons_succ (i : Fin m) (x : α) [constant : ConstantLength (prod circuit)] :\n  foldlAcc n (Vector.cons x xs) circuit init i.succ =\n    foldlAcc (n + (circuit init x).localLength n) xs circuit ((circuit init x).output n) i"}, {"name": "Circuit.FoldlM.operations_eq", "content": "theorem operations_eq :\n  (Vector.foldlM circuit init xs).operations n =\n    (List.ofFn fun i => (circuit (foldlAcc n xs circuit init i) xs[i.val]).operations (n + i * constant.localLength)).flatten"}, {"name": "Circuit.FoldlM.forAll_flatten_foldl", "content": "lemma forAll_flatten_foldl :\n  Operations.forAll n prop (List.ofFn fun (i : Fin m) => (circuit (foldlAcc n xs circuit init i) xs[i.val]).operations (n + i * constant.localLength)).flatten\n    ↔ ∀ (i : Fin m), (circuit (foldlAcc n xs circuit init i) xs[i.val]).forAll (n + i * constant.localLength) prop"}, {"name": "Circuit.FoldlM.forAll_iff", "content": "theorem forAll_iff {constant : ConstantLength (prod circuit)} :\n  (xs.foldlM circuit init).forAll n prop ↔\n    ∀ i : Fin m, (circuit (foldlAcc n xs circuit init i) xs[i.val]).forAll (n + i * (circuit init xs[i.val]).localLength) prop"}, {"name": "Circuit.FoldlM.foldlAcc_const_succ", "content": "theorem foldlAcc_const_succ (constant : ConstantLength (prod circuit))\n  (h_const_out : ConstantOutput fun (t : β × α) => circuit t.1 t.2)\n  (i : ℕ) (hi : i + 1 < m) :\n  foldlAcc n xs circuit init ⟨ i + 1, hi ⟩ =\n    (circuit default xs[i]).output (n + i * (circuit default default).localLength)"}], "local_ctx": "import Clean.Circuit.Subcircuit\n\nimport Clean.Utils.Misc\n\nvariable {n m : ℕ} {F : Type} [Field F] {α β : Type}\n\nnamespace Circuit\n\nvariable {prop : Condition F}\n\nnamespace ForM\n\nvariable {circuit : α → Circuit F Unit} (xs : Vector α m) (constant : ConstantLength circuit) (n : ℕ)\n\nend ForM\n\nnamespace MapM\n\nvariable {circuit : α → Circuit F β} {xs : Vector α m} [constant: ConstantLength circuit]\n  {prop : Condition F}\n\nend MapM\n\nnamespace FoldlM\n\n@[reducible]\ndef prod (circuit : β → α → Circuit F β) : β × α → Circuit F β := fun t => circuit t.1 t.2\n\nvariable {env : Environment F} {prop : Condition F} {xs : Vector α m}\n  {circuit : β → α → Circuit F β} {init : β} {constant : ConstantLength (prod circuit)}\n\ndef foldlAcc (n : ℕ) (xs : Vector α m) (circuit : β → α → Circuit F β) (init : β) (j : Fin m) : β :=\n  Fin.foldl j (fun acc i => (circuit acc xs[i.val]).output (n + i*(circuit acc xs[i.val]).localLength)) init\n\nvariable {prop : Condition F}\n\nsection\n\nvariable {env : Environment F} {prop : Condition F} {m : ℕ}\n  {Acc : ℕ → Type}\n  {circuit : β → Fin m → Circuit F β} {init : β} {constant : ConstantLength (prod circuit)}\n\nend\n\nvariable [Inhabited β] [Inhabited α] {h_const_out : ConstantOutput fun (t : β × α) => circuit t.1 t.2}", "target_theorem": "theorem forAll_iff_const [NeZero m] (constant : ConstantLength (prod circuit))\n    (h_const_out : ConstantOutput (prod circuit)) :\n  (xs.foldlM circuit init).forAll n prop ↔\n  (circuit init (xs[0]'(NeZero.pos m))).forAll n prop ∧\n  ∀ (i : ℕ) (hi : i + 1 < m),\n    let acc :=", "ground_truth_proof": ":= (circuit default xs[i]).output (n + i*(circuit default default).localLength);\n    (circuit acc xs[i + 1]).forAll (n + (i + 1)*(circuit default default).localLength) prop := by\n  rw [forAll_iff (constant:=constant)]\n  set k := (circuit default default).localLength\n  simp only\n  constructor\n  · intro h\n    constructor\n    · specialize h 0\n      simp only [Fin.val_zero] at h\n      rw [foldlAcc_zero, zero_mul, add_zero] at h\n      exact h\n    · intro i hi\n      specialize h ⟨ i + 1, hi ⟩\n      rw [foldlAcc_const_succ constant h_const_out] at h\n      convert h using 3\n      rw [ConstantLength.length_eq_default constant (init, _)]\n      rfl\n  intro h i\n  rcases i with ⟨ _ | i, hi ⟩\n  · simp only [Fin.mk_zero']\n    rw [foldlAcc_zero, zero_mul, add_zero]\n    exact h.left\n  · rw [foldlAcc_const_succ constant h_const_out]\n    convert (h.right i hi) using 3\n    rw [ConstantLength.length_eq_default constant (init, _)]\n    rfl", "nesting_depth": 8, "transitive_dep_count": 97, "subset_aristotle": true, "category": "Applied verif."}
{"id": 164, "thm_name": "Gadgets.Rotation32.Theorems.rotation32_bits_soundness", "thm_stmt": "theorem rotation32_bits_soundness {o : ℕ} (ho : o < 8) {x : U32 ℕ} :\n    (rotRight32_u32 x o).valueNat = rotRight32 x.valueNat o", "lean_root": "clean", "rel_path": "Clean/Gadgets/Rotation32/Theorems.lean", "imports": ["import Clean.Types.U32", "import Clean.Gadgets.ByteDecomposition.ByteDecomposition", "import Clean.Utils.Bitwise", "import Clean.Utils.Rotation", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.reducePow", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}], "used_repo_defs": [{"name": "U32", "content": "structure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq"}, {"name": "rotRight32", "content": "def rotRight32 (x : ℕ) (offset : ℕ) : ℕ :=\n  let offset := offset % 32\n  let low := x % (2^offset)\n  let high := x / (2^offset)\n  low * (2^(32-offset)) + high"}, {"name": "valueNat", "content": "def valueNat (x : U32 ℕ) :=\n  x.x0 + x.x1 * 256 + x.x2 * 256^2 + x.x3 * 256^3"}, {"name": "valueNat", "content": "def valueNat (x : U64 ℕ) :=\n  x.x0 + x.x1 * 256 + x.x2 * 256^2 + x.x3 * 256^3 +\n  x.x4 * 256^4 + x.x5 * 256^5 + x.x6 * 256^6 + x.x7 * 256^7"}, {"name": "U64", "content": "structure U64 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\n  x4 : T\n  x5 : T\n  x6 : T\n  x7 : T\nderiving DecidableEq"}], "lib_lemmas": [{"name": "Nat.add_div_of_dvd_left", "module": "Mathlib.Data.Nat.ModEq"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.lt_one_iff", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.pow_one", "module": "Init.Data.Nat.Basic"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pos_of_gt", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pow_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "tsub_self", "module": "Mathlib.Algebra.Order.Sub.Basic"}, {"name": "Nat.add_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.pow_mul", "module": "Init.Data.Nat.Lemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mul_right_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "mul_div_256_off", "content": "lemma mul_div_256_off {offset : ℕ} (ho : offset < 8) {x : ℕ} (i : ℕ) (h : i > 0):\n    (x * 256^i) / 2^offset = x * 256^(i-1) * 2^(8-offset)"}, {"name": "div_256_two_power", "content": "lemma div_256_two_power {offset : ℕ} (ho : offset < 8) {i : ℕ} (h : i > 0):\n    256^i / 2^offset = 256^(i-1) * 2^(8-offset)"}, {"name": "divides_256_two_power", "content": "lemma divides_256_two_power {offset : ℕ} (ho : offset < 8) {x i : ℕ} (h : i > 0):\n    (2^offset) ∣ x * (256 ^ i)"}, {"name": "mul_mod256_off", "content": "lemma mul_mod256_off {offset : ℕ} (ho : offset < 8) (x i : ℕ) (h : i > 0):\n    (x * 256^i) % 2^offset = 0"}, {"name": "shifted_decomposition_eq", "content": "lemma shifted_decomposition_eq {offset : ℕ} (ho : offset < 8) {x1 x2 : ℕ} :\n    (x1 / 2 ^ offset + x2 % 2 ^ offset * 2 ^ (8-offset)) * 256 =\n    (2^offset * (x1 / 2^offset) + (x2 % 2^offset) * 256) * 2^(8-offset)"}, {"name": "soundness_simp", "content": "lemma soundness_simp {offset : ℕ} {x y : ℕ} :\n    x % 2 ^ offset * 2 ^ (8-offset) * y + 2 ^ offset * (x / 2 ^ offset) * 2 ^ (8-offset) * y =\n    x * y * 2^ (8-offset)"}, {"name": "soundness_simp'", "content": "lemma soundness_simp' {offset : ℕ} {x : ℕ} :\n    x % 2 ^ offset * 2 ^ (8-offset) + 2 ^ offset * (x / 2 ^ offset) * 2 ^ (8-offset) =\n    x * 2^ (8-offset)"}, {"name": "shifted_decomposition_eq''", "content": "lemma shifted_decomposition_eq'' {offset : ℕ} (ho : offset < 8) {x1 x2 i : ℕ} (hi : i > 0) :\n    (x1 / 2 ^ offset + x2 % 2 ^ offset * 2 ^ (8-offset)) * 256^i =\n    (2^offset * (x1 / 2^offset) * 2^(8-offset) * 256^(i-1) +\n    (x2 % 2^offset) * 2^(8-offset) * 256^i)"}, {"name": "Nat.pow_minus_one_mul", "content": "lemma Nat.pow_minus_one_mul {x y : ℕ} (hy : y > 0) : x ^ y = x * x ^ (y - 1)"}, {"name": "shifted_decomposition_eq'", "content": "lemma shifted_decomposition_eq' {offset : ℕ} (ho : offset < 8) {x1 x2 i : ℕ} (hi : i > 0) :\n    (x1 / 2 ^ offset + x2 % 2 ^ offset * 2 ^ (8-offset)) * 256^i =\n    (2^offset * (x1 / 2^offset) + (x2 % 2^offset) * 256) * 2^(8-offset) * 256^(i-1)"}], "used_local_defs": [{"name": "Gadgets.Rotation32.Theorems.rotRight32_u32", "content": "def rotRight32_u32 : U32 ℕ → ℕ → U32 ℕ\n  | ⟨ x0, x1, x2, x3 ⟩, o => ⟨\n    (x0 / 2^o) + (x1 % 2^o) * 2^(8-o),\n    (x1 / 2^o) + (x2 % 2^o) * 2^(8-o),\n    (x2 / 2^o) + (x3 % 2^o) * 2^(8-o),\n    (x3 / 2^o) + (x0 % 2^o) * 2^(8-o),\n  ⟩"}], "used_local_lemmas": [{"name": "Gadgets.Rotation32.Theorems.h_mod32", "content": "lemma h_mod32 {o : ℕ} (ho : o < 8) {x0 x1 x2 x3 : ℕ} :\n    (x0 + x1 * 256 + x2 * 256^2 + x3 * 256^3) % 2^o = x0 % 2^o"}, {"name": "Gadgets.Rotation32.Theorems.h_div32", "content": "lemma h_div32 {o : ℕ} (ho : o < 8) {x0 x1 x2 x3: ℕ} :\n    (x0 + x1 * 256 + x2 * 256^2 + x3 * 256^3) / 2^o\n    = x0 / 2^o + x1 * 2^(8-o) + x2 * 256 * 2^(8-o) + x3 * 256^2 * 2^(8-o)"}, {"name": "Gadgets.Rotation32.Theorems.h_x0_const32", "content": "lemma h_x0_const32 {o : ℕ} (ho : o < 8) :\n    2^(8-o) * 256^3 = 2^(32-o)"}], "local_ctx": "import Clean.Utils.Field\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Utils.Rotation\n\nimport Clean.Types.U32\n\nimport Clean.Gadgets.ByteDecomposition.ByteDecomposition\n\nvariable {p : ℕ} [Fact p.Prime]\n\nvariable [p_large_enough: Fact (p > 2^16 + 2^8)]\n\nnamespace Gadgets.Rotation32.Theorems\n\nopen Gadgets.ByteDecomposition.Theorems (byteDecomposition_lift)\n\nopen Utils.Rotation\n\ndef rotRight32_u32 : U32 ℕ → ℕ → U32 ℕ\n  | ⟨ x0, x1, x2, x3 ⟩, o => ⟨\n    (x0 / 2^o) + (x1 % 2^o) * 2^(8-o),\n    (x1 / 2^o) + (x2 % 2^o) * 2^(8-o),\n    (x2 / 2^o) + (x3 % 2^o) * 2^(8-o),\n    (x3 / 2^o) + (x0 % 2^o) * 2^(8-o),\n  ⟩", "target_theorem": "theorem rotation32_bits_soundness {o : ℕ} (ho : o < 8) {x : U32 ℕ} :\n    (rotRight32_u32 x o).valueNat = rotRight32 x.valueNat o :=", "ground_truth_proof": ":= by\n  -- simplify the goal\n  simp only [rotRight32, rotRight32_u32, U32.valueNat]\n\n  have offset_mod_32 : o % 32 = o := Nat.mod_eq_of_lt (by linarith)\n  simp only [offset_mod_32]\n  rw [h_mod32 ho, h_div32 ho]\n\n  -- proof technique: we care about only what happens to x0, all \"internal\" terms remain\n  -- the same, and are just divided by 2^o\n  rw [shifted_decomposition_eq ho]\n  repeat rw [shifted_decomposition_eq'' ho (by simp only [Nat.ofNat_pos])]\n  simp only [Nat.add_one_sub_one, pow_one, add_mul, add_assoc]\n  rw [←add_assoc _ _ (_ * 256 ^ 3), soundness_simp]\n  nth_rw 4 [←add_assoc]\n  rw [Nat.mul_right_comm _ 256, soundness_simp]\n  nth_rw 2 [←add_assoc]\n  rw [Nat.mul_right_comm _ 256, soundness_simp']\n  rw [mul_assoc (x.x0 % 2 ^ o), h_x0_const32 ho]\n  ac_rfl", "nesting_depth": 3, "transitive_dep_count": 43, "subset_aristotle": true, "category": "Applied verif."}
{"id": 165, "thm_name": "Circuit.FoldlM.operations_eq_const", "thm_stmt": "theorem operations_eq_const [NeZero m] (constant : ConstantLength (prod circuit))\n    (h_const_out : ConstantOutput (prod circuit)) :\n  (Vector.foldlM circuit init xs).operations n =\n  (circuit init (xs[0]'(NeZero.pos m))).operations n ++\n  (List.ofFn fun (⟨i, _⟩ : Fin (m - 1)) =>\n    let k := (circuit default default).localLength\n    let acc := (circuit default xs[i]).output (n + i*k)\n    (circuit acc xs[i + 1]).operations (n + (i + 1)*k)).flatten", "lean_root": "clean", "rel_path": "Clean/Circuit/Loops.lean", "imports": ["import Clean.Utils.Misc", "import Clean.Circuit.Subcircuit", "import Clean.Circuit.Theorems"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "List.ofFn", "module": "Init.Data.List.OfFn"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Vector.foldlM", "module": "Init.Data.Vector.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "Vector.mk", "module": "Init.Data.Vector.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantOutput", "content": "@[circuit_norm]\ndef ConstantOutput (circuit : α → Circuit F β) [Inhabited α] :=\n  ∀ (x : α) (n : ℕ), (circuit x).output n = (circuit default).output n"}, {"name": "induct", "content": "def induct {motive : {n : ℕ} → Vector α n → Sort u}\n    (nil : motive #v[])\n    (cons: ∀ {n : ℕ} (a : α) (as : Vector α n), motive as → motive (cons a as))\n    {n : ℕ} (v : Vector α n) : motive v :=\n  match v with\n  | ⟨ .mk [], h ⟩ => by admit /- proof elided -/\n  | ⟨ .mk (a :: as), h ⟩ => by admit /- proof elided -/"}, {"name": "cons", "content": "def cons (a : α) (v : Vector α n) : Vector α (n + 1) :=\n  ⟨ .mk (a :: v.toList), by admit /- proof elided -/\n  ⟩"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}], "lib_lemmas": [{"name": "Fin.foldl_zero", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_last", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.foldlM_toArray", "module": "Init.Data.List.ToArray"}, {"name": "Vector.foldlM_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "List.foldlM_cons", "module": "Init.Data.List.Control"}, {"name": "Vector.toList_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "Fin.foldl_succ", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.val_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.getElem_cons_succ", "module": "Init.GetElem"}, {"name": "List.getElem_cons_zero", "module": "Init.GetElem"}, {"name": "List.getElem_toArray", "module": "Init.Data.Array.Basic"}, {"name": "Vector.getElem_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "List.flatten_cons", "module": "Init.Data.List.Basic"}, {"name": "List.ofFn_succ", "module": "Init.Data.List.OfFn"}, {"name": "NeZero.pos", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "Fin.cast_eq_self", "module": "Mathlib.Data.Fin.Basic"}, {"name": "Fin.succ_mk", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.append_cancel_left_eq", "module": "Init.Data.List.BasicAux"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "add_tsub_cancel_right", "module": "Mathlib.Algebra.Order.Sub.Defs"}], "repo_lemmas": [{"name": "Fin.foldl_const", "content": "theorem Fin.foldl_const (n : ℕ) (f : Fin n → α) (init : α) :\n  Fin.foldl n (fun _ i => f i) init = match n with"}, {"name": "Fin.foldl_const_succ", "content": "theorem Fin.foldl_const_succ (n : ℕ) (f : Fin (n + 1) → α) (init : α) :\n    Fin.foldl (n + 1) (fun _ i => f i) init = f (.last n)"}, {"name": "bind_operations_eq", "content": "theorem bind_operations_eq (f : Circuit F α) (g : α → Circuit F β) (n : ℕ) :\n  (f >>= g).operations n = f.operations n ++ (g (f.output n)).operations (n + f.localLength n)"}], "used_local_defs": [{"name": "Circuit.FoldlM.prod", "content": "@[reducible]\ndef prod (circuit : β → α → Circuit F β) : β × α → Circuit F β := fun t => circuit t.1 t.2"}, {"name": "Circuit.FoldlM.foldlAcc", "content": "def foldlAcc (n : ℕ) (xs : Vector α m) (circuit : β → α → Circuit F β) (init : β) (j : Fin m) : β :=\n  Fin.foldl j (fun acc i => (circuit acc xs[i.val]).output (n + i*(circuit acc xs[i.val]).localLength)) init"}], "used_local_lemmas": [{"name": "Vector.foldlM_toList", "content": "lemma Vector.foldlM_toList (xs : Vector α n) {m : Type → Type} [Monad m] (body : β → α → m β) (init : β) :\n    xs.foldlM body init = xs.toList.foldlM body init"}, {"name": "Circuit.ConstantLength.length_eq_default", "content": "lemma ConstantLength.length_eq_default {circuit : α → Circuit F β} (_ : ConstantLength circuit) [Inhabited α] (a : α) (n : ℕ) :\n   (circuit a).localLength n = (circuit default).localLength 0"}, {"name": "Circuit.FoldlM.foldlM_cons", "content": "lemma foldlM_cons (x : α) :\n  (Vector.cons x xs).foldlM circuit init = (do\n    let init' ← circuit init x\n    xs.foldlM circuit init')"}, {"name": "Circuit.FoldlM.foldlAcc_zero", "content": "lemma foldlAcc_zero [NeZero m] : foldlAcc n xs circuit init 0 = init"}, {"name": "Circuit.FoldlM.foldlAcc_cons_succ", "content": "lemma foldlAcc_cons_succ (i : Fin m) (x : α) [constant : ConstantLength (prod circuit)] :\n  foldlAcc n (Vector.cons x xs) circuit init i.succ =\n    foldlAcc (n + (circuit init x).localLength n) xs circuit ((circuit init x).output n) i"}, {"name": "Circuit.FoldlM.operations_eq", "content": "theorem operations_eq :\n  (Vector.foldlM circuit init xs).operations n =\n    (List.ofFn fun i => (circuit (foldlAcc n xs circuit init i) xs[i.val]).operations (n + i * constant.localLength)).flatten"}, {"name": "Circuit.FoldlM.foldlAcc_const_succ", "content": "theorem foldlAcc_const_succ (constant : ConstantLength (prod circuit))\n  (h_const_out : ConstantOutput fun (t : β × α) => circuit t.1 t.2)\n  (i : ℕ) (hi : i + 1 < m) :\n  foldlAcc n xs circuit init ⟨ i + 1, hi ⟩ =\n    (circuit default xs[i]).output (n + i * (circuit default default).localLength)"}], "local_ctx": "import Clean.Circuit.Subcircuit\n\nimport Clean.Utils.Misc\n\nvariable {n m : ℕ} {F : Type} [Field F] {α β : Type}\n\nnamespace Circuit\n\nvariable {prop : Condition F}\n\nnamespace ForM\n\nvariable {circuit : α → Circuit F Unit} (xs : Vector α m) (constant : ConstantLength circuit) (n : ℕ)\n\nend ForM\n\nnamespace MapM\n\nvariable {circuit : α → Circuit F β} {xs : Vector α m} [constant: ConstantLength circuit]\n  {prop : Condition F}\n\nend MapM\n\nnamespace FoldlM\n\n@[reducible]\ndef prod (circuit : β → α → Circuit F β) : β × α → Circuit F β := fun t => circuit t.1 t.2\n\nvariable {env : Environment F} {prop : Condition F} {xs : Vector α m}\n  {circuit : β → α → Circuit F β} {init : β} {constant : ConstantLength (prod circuit)}\n\ndef foldlAcc (n : ℕ) (xs : Vector α m) (circuit : β → α → Circuit F β) (init : β) (j : Fin m) : β :=\n  Fin.foldl j (fun acc i => (circuit acc xs[i.val]).output (n + i*(circuit acc xs[i.val]).localLength)) init\n\nvariable {prop : Condition F}\n\nsection\n\nvariable {env : Environment F} {prop : Condition F} {m : ℕ}\n  {Acc : ℕ → Type}\n  {circuit : β → Fin m → Circuit F β} {init : β} {constant : ConstantLength (prod circuit)}\n\nend\n\nvariable [Inhabited β] [Inhabited α] {h_const_out : ConstantOutput fun (t : β × α) => circuit t.1 t.2}", "target_theorem": "theorem operations_eq_const [NeZero m] (constant : ConstantLength (prod circuit))\n    (h_const_out : ConstantOutput (prod circuit)) :\n  (Vector.foldlM circuit init xs).operations n =\n  (circuit init (xs[0]'(NeZero.pos m))).operations n ++\n  (List.ofFn fun (⟨i, _⟩ : Fin (m - 1)) =>\n    let k := (circuit default default).localLength\n    let acc := (circuit default xs[i]).output (n + i*k)\n    (circuit acc xs[i + 1]).operations (n + (i + 1)*k)).flatten :=", "ground_truth_proof": ":= by\n  rw [operations_eq]\n  simp only\n  set k := (circuit default default).localLength\n  set k' := constant.localLength\n  have : k' = k := by simp only [k]; rw [constant.localLength_eq (_, _)]\n  rw [this]\n  rcases m with rfl | m\n  · nomatch ‹NeZero 0›\n  rw [List.ofFn_succ, List.flatten_cons]\n  simp only [foldlAcc_zero, Fin.val_zero, zero_mul, add_zero, Fin.val_succ, add_mul, one_mul,\n    add_tsub_cancel_right, Nat.add_one_sub_one, Fin.cast_eq_self, List.append_cancel_left_eq]\n  congr\n  funext i\n  rcases i with ⟨ i, hi ⟩\n  simp only [Fin.succ_mk]\n  rw [foldlAcc_const_succ constant h_const_out]", "nesting_depth": 6, "transitive_dep_count": 88, "subset_aristotle": true, "category": "Applied verif."}
{"id": 166, "thm_name": "Gadgets.Rotation64.Theorems.rotation64_bits_soundness", "thm_stmt": "theorem rotation64_bits_soundness {o : ℕ} (ho : o < 8) {x : U64 ℕ} :\n    (rotRight64_u64 x o).valueNat = rotRight64 x.valueNat o", "lean_root": "clean", "rel_path": "Clean/Gadgets/Rotation64/Theorems.lean", "imports": ["import Clean.Utils.Bitwise", "import Clean.Types.U64", "import Clean.Utils.Rotation", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.reducePow", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}], "used_repo_defs": [{"name": "U64", "content": "structure U64 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\n  x4 : T\n  x5 : T\n  x6 : T\n  x7 : T\nderiving DecidableEq"}, {"name": "valueNat", "content": "def valueNat (x : U64 ℕ) :=\n  x.x0 + x.x1 * 256 + x.x2 * 256^2 + x.x3 * 256^3 +\n  x.x4 * 256^4 + x.x5 * 256^5 + x.x6 * 256^6 + x.x7 * 256^7"}, {"name": "rotRight64", "content": "def rotRight64 (x : ℕ) (offset : ℕ) : ℕ :=\n  let offset := offset % 64\n  let low := x % (2^offset)\n  let high := x / (2^offset)\n  low * (2^(64-offset)) + high"}], "lib_lemmas": [{"name": "Nat.pow_add", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.pow_mul", "module": "Init.Data.Nat.Lemmas"}, {"name": "pow_right_inj₀", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "Nat.add_div_of_dvd_left", "module": "Mathlib.Data.Nat.ModEq"}, {"name": "Nat.add_one_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.lt_one_iff", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.ofNat_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.pow_one", "module": "Init.Data.Nat.Basic"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pos_of_gt", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "pow_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "tsub_self", "module": "Mathlib.Algebra.Order.Sub.Basic"}, {"name": "Nat.add_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mul_right_comm", "module": "Init.Data.Nat.Lemmas"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "mul_div_256_off", "content": "lemma mul_div_256_off {offset : ℕ} (ho : offset < 8) {x : ℕ} (i : ℕ) (h : i > 0):\n    (x * 256^i) / 2^offset = x * 256^(i-1) * 2^(8-offset)"}, {"name": "div_256_two_power", "content": "lemma div_256_two_power {offset : ℕ} (ho : offset < 8) {i : ℕ} (h : i > 0):\n    256^i / 2^offset = 256^(i-1) * 2^(8-offset)"}, {"name": "divides_256_two_power", "content": "lemma divides_256_two_power {offset : ℕ} (ho : offset < 8) {x i : ℕ} (h : i > 0):\n    (2^offset) ∣ x * (256 ^ i)"}, {"name": "mul_mod256_off", "content": "lemma mul_mod256_off {offset : ℕ} (ho : offset < 8) (x i : ℕ) (h : i > 0):\n    (x * 256^i) % 2^offset = 0"}, {"name": "shifted_decomposition_eq", "content": "lemma shifted_decomposition_eq {offset : ℕ} (ho : offset < 8) {x1 x2 : ℕ} :\n    (x1 / 2 ^ offset + x2 % 2 ^ offset * 2 ^ (8-offset)) * 256 =\n    (2^offset * (x1 / 2^offset) + (x2 % 2^offset) * 256) * 2^(8-offset)"}, {"name": "soundness_simp", "content": "lemma soundness_simp {offset : ℕ} {x y : ℕ} :\n    x % 2 ^ offset * 2 ^ (8-offset) * y + 2 ^ offset * (x / 2 ^ offset) * 2 ^ (8-offset) * y =\n    x * y * 2^ (8-offset)"}, {"name": "soundness_simp'", "content": "lemma soundness_simp' {offset : ℕ} {x : ℕ} :\n    x % 2 ^ offset * 2 ^ (8-offset) + 2 ^ offset * (x / 2 ^ offset) * 2 ^ (8-offset) =\n    x * 2^ (8-offset)"}, {"name": "shifted_decomposition_eq''", "content": "lemma shifted_decomposition_eq'' {offset : ℕ} (ho : offset < 8) {x1 x2 i : ℕ} (hi : i > 0) :\n    (x1 / 2 ^ offset + x2 % 2 ^ offset * 2 ^ (8-offset)) * 256^i =\n    (2^offset * (x1 / 2^offset) * 2^(8-offset) * 256^(i-1) +\n    (x2 % 2^offset) * 2^(8-offset) * 256^i)"}, {"name": "Nat.pow_minus_one_mul", "content": "lemma Nat.pow_minus_one_mul {x y : ℕ} (hy : y > 0) : x ^ y = x * x ^ (y - 1)"}, {"name": "shifted_decomposition_eq'", "content": "lemma shifted_decomposition_eq' {offset : ℕ} (ho : offset < 8) {x1 x2 i : ℕ} (hi : i > 0) :\n    (x1 / 2 ^ offset + x2 % 2 ^ offset * 2 ^ (8-offset)) * 256^i =\n    (2^offset * (x1 / 2^offset) + (x2 % 2^offset) * 256) * 2^(8-offset) * 256^(i-1)"}], "used_local_defs": [{"name": "Gadgets.Rotation64.Theorems.rotRight64_u64", "content": "def rotRight64_u64 : U64 ℕ → ℕ → U64 ℕ\n  | ⟨ x0, x1, x2, x3, x4, x5, x6, x7 ⟩, o => ⟨\n    (x0 / 2^o) + (x1 % 2^o) * 2^(8-o),\n    (x1 / 2^o) + (x2 % 2^o) * 2^(8-o),\n    (x2 / 2^o) + (x3 % 2^o) * 2^(8-o),\n    (x3 / 2^o) + (x4 % 2^o) * 2^(8-o),\n    (x4 / 2^o) + (x5 % 2^o) * 2^(8-o),\n    (x5 / 2^o) + (x6 % 2^o) * 2^(8-o),\n    (x6 / 2^o) + (x7 % 2^o) * 2^(8-o),\n    (x7 / 2^o) + (x0 % 2^o) * 2^(8-o),\n  ⟩"}], "used_local_lemmas": [{"name": "Gadgets.Rotation64.Theorems.h_mod", "content": "lemma h_mod {o : ℕ} (ho : o < 8) {x0 x1 x2 x3 x4 x5 x6 x7 : ℕ} :\n    (x0 + x1 * 256 + x2 * 256 ^ 2 + x3 * 256 ^ 3 + x4 * 256 ^ 4 + x5 * 256 ^ 5 + x6 * 256 ^ 6 + x7 * 256 ^ 7) %\n      2^o = x0 % 2^o"}, {"name": "Gadgets.Rotation64.Theorems.h_div", "content": "lemma h_div {o : ℕ} (ho : o < 8) {x0 x1 x2 x3 x4 x5 x6 x7 : ℕ} :\n    (x0 + x1 * 256 + x2 * 256^2 + x3 * 256^3 + x4 * 256^4 + x5 * 256^5 + x6 * 256^6 + x7 * 256^7) / 2 ^ o\n    = x0 / 2^o + x1 * 2^(8-o) + x2 * 256 * 2^(8-o) + x3 * 256^2 * 2^(8-o) + x4 * 256^3 * 2^(8-o) +\n    x5 * 256^4 * 2^(8-o) + x6 * 256^5 * 2^(8-o) + x7 * 256^6 * 2^(8-o)"}, {"name": "Gadgets.Rotation64.Theorems.h_x0_const", "content": "lemma h_x0_const {o : ℕ} (ho : o < 8) :\n    2^(8-o) * 256^7 = 2^(64-o)"}], "local_ctx": "import Clean.Utils.Field\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Utils.Rotation\n\nimport Clean.Types.U64\n\nvariable {p : ℕ} [Fact p.Prime]\n\nvariable [p_large_enough: Fact (p > 2^16 + 2^8)]\n\nnamespace Gadgets.Rotation64.Theorems\n\nopen Utils.Rotation\n\ndef rotRight64_u64 : U64 ℕ → ℕ → U64 ℕ\n  | ⟨ x0, x1, x2, x3, x4, x5, x6, x7 ⟩, o => ⟨\n    (x0 / 2^o) + (x1 % 2^o) * 2^(8-o),\n    (x1 / 2^o) + (x2 % 2^o) * 2^(8-o),\n    (x2 / 2^o) + (x3 % 2^o) * 2^(8-o),\n    (x3 / 2^o) + (x4 % 2^o) * 2^(8-o),\n    (x4 / 2^o) + (x5 % 2^o) * 2^(8-o),\n    (x5 / 2^o) + (x6 % 2^o) * 2^(8-o),\n    (x6 / 2^o) + (x7 % 2^o) * 2^(8-o),\n    (x7 / 2^o) + (x0 % 2^o) * 2^(8-o),\n  ⟩", "target_theorem": "theorem rotation64_bits_soundness {o : ℕ} (ho : o < 8) {x : U64 ℕ} :\n    (rotRight64_u64 x o).valueNat = rotRight64 x.valueNat o :=", "ground_truth_proof": ":= by\n  -- simplify the goal\n  simp only [rotRight64, rotRight64_u64, U64.valueNat]\n\n  have offset_mod_64 : o % 64 = o := Nat.mod_eq_of_lt (by linarith)\n  simp only [offset_mod_64]\n  rw [h_mod ho, h_div ho]\n\n  -- proof technique: we care about only what happens to x0, all \"internal\" terms remain\n  -- the same, and are just divided by 2^o\n  rw [shifted_decomposition_eq ho]\n  repeat rw [shifted_decomposition_eq'' ho (by simp only [Nat.ofNat_pos])]\n  simp only [Nat.add_one_sub_one, pow_one, add_mul, add_assoc]\n\n  -- we do a bit of expression juggling here\n  rw [←add_assoc _ _ (_ * 256 ^ 7), soundness_simp]\n  nth_rw 12 [←add_assoc]\n  rw [soundness_simp]\n  nth_rw 10 [←add_assoc]\n  rw [soundness_simp]\n  nth_rw 8 [←add_assoc]\n  rw [soundness_simp]\n  nth_rw 6 [←add_assoc]\n  rw [soundness_simp]\n  nth_rw 4 [←add_assoc]\n  nth_rw 2 [Nat.mul_right_comm]\n  rw [soundness_simp]\n  nth_rw 2 [←add_assoc]\n  rw [soundness_simp']\n  nth_rw 7 [mul_assoc]\n\n  -- at the end the terms are equal\n  rw [h_x0_const ho]\n  ac_rfl", "nesting_depth": 3, "transitive_dep_count": 41, "subset_aristotle": false, "category": "Applied verif."}
{"id": 167, "thm_name": "Utils.StateTransition.exists_path_from_source_to_sink", "thm_stmt": "theorem exists_path_from_source_to_sink\n    (R : Run S) (s d : S)\n    (h_source : R.netFlow s = 1)\n    (h_others : ∀ x, x ≠ s → x ≠ d → R.netFlow x = 0) :\n    ∃ (path : List S), path.head? = some s ∧ path.getLast? = some d ∧\n      path ≠ [] ∧ R.containsPath path ∧ path.Nodup", "lean_root": "clean", "rel_path": "Clean/Utils/SourceSinkPath.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import Mathlib.Data.Fintype.Prod", "import Mathlib.Data.List.Basic", "import Mathlib.Algebra.BigOperators.Group.Finset.Basic", "import Mathlib.Data.Finset.Basic", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise", "import Mathlib.Data.Fintype.Basic"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "List.Sublist", "module": "Init.Data.List.Basic"}, {"name": "List.count", "module": "Init.Data.List.Basic"}, {"name": "List.drop", "module": "Init.Data.List.Basic"}, {"name": "List.tail", "module": "Init.Data.List.Basic"}, {"name": "List.take", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.cons₂", "module": "Init.Data.List.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.reduceLeDiff", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "List.getLast?", "module": "Init.Data.List.Basic"}, {"name": "List.head?", "module": "Init.Data.List.Basic"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.erase", "module": "Mathlib.Data.Finset.Erase"}, {"name": "Nat.cast", "module": "Init.Data.Cast"}, {"name": "List.countP", "module": "Init.Data.List.Basic"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.drop_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.tail_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zip_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.count_le", "module": "Init.Data.List.Count"}, {"name": "List.tail_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.take_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.length_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.length_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "Nat.min_eq_left", "module": "Init.Data.Nat.MinMax"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "List.getElem?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getElem?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getLast?_eq_getElem?", "module": "Init.Data.List.Lemmas"}, {"name": "List.head?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.head?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "List.duplicate_iff_exists_distinct_get", "module": "Mathlib.Data.List.NodupEquivFin"}, {"name": "List.exists_duplicate_iff_not_nodup", "module": "Mathlib.Data.List.Duplicate"}, {"name": "Finset.sum_le_univ_sum_of_nonneg", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "List.getLast_singleton", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast?_cons_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.count_pos_iff", "module": "Init.Data.List.Count"}, {"name": "Finset.sum_nonneg", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "List.countP_cons_of_pos", "module": "Init.Data.List.Count"}, {"name": "List.countP_nil", "module": "Init.Data.List.Count"}, {"name": "List.count_cons", "module": "Init.Data.List.Count"}, {"name": "List.count_nil", "module": "Init.Data.List.Count"}, {"name": "List.zipWith_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zipWith_nil_right", "module": "Init.Data.List.Basic"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.exists_cons_of_ne_nil", "module": "Init.Data.List.Lemmas"}, {"name": "List.countP_singleton", "module": "Init.Data.List.Count"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "List.zip_nil_right", "module": "Init.Data.List.Basic"}, {"name": "Nat.add_right_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "List.mem_of_mem_tail", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "List.getLast?_eq_getLast", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast_mem", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.nodup_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.mem_iff_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.ssubset_univ_iff", "module": "Mathlib.Data.Finset.BooleanAlgebra"}, {"name": "Finset.card_insert_of_notMem", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.card_lt_card", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.card_univ", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Finset.sum_erase_add", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_le_sum", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "Finset.mem_toList", "module": "Mathlib.Data.Finset.Dedup"}, {"name": "Nat.sub_le", "module": "Init.Prelude"}, {"name": "Nat.sub_lt", "module": "Init.Prelude"}, {"name": "Finset.sum_add_distrib", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "List.countP_cons", "module": "Init.Data.List.Count"}, {"name": "Nat.cast_sum", "module": "Mathlib.Algebra.BigOperators.Ring.Finset"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "decide_eq_true_eq", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "List.getLast_cons", "module": "Init.Data.List.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Utils.StateTransition.Transition", "content": "def Transition (S : Type*) := S × S"}, {"name": "Utils.StateTransition.Run", "content": "def Run (S : Type*) := Transition S → ℕ"}, {"name": "Utils.StateTransition.Run.netFlow", "content": "noncomputable def Run.netFlow {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) (x : S) : ℤ :=\n  (∑ y : S, (R (x, y) : ℤ)) - (∑ y : S, (R (y, x) : ℤ))"}, {"name": "Utils.StateTransition.Run.size", "content": "noncomputable def Run.size {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) : ℕ :=\n  ∑ t : Transition S, R t"}, {"name": "Utils.StateTransition.countTransitionInPath", "content": "def countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t"}, {"name": "Utils.StateTransition.Run.containsPath", "content": "def Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t"}, {"name": "Utils.StateTransition.Run.hasCycle", "content": "def Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle"}, {"name": "Utils.StateTransition.Run.isAcyclic", "content": "def Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle"}, {"name": "Utils.StateTransition.Run.removeCycle", "content": "def Run.removeCycle (R : Run S) (cycle : List S) : Run S :=\n  fun t => R t - countTransitionInPath t cycle"}, {"name": "Utils.StateTransition.Run.reachable", "content": "def Run.reachable [DecidableEq S] (R : Run S) (start finish : S) : Prop :=\n  ∃ (path : List S), path.head? = some start ∧ path.getLast? = some finish ∧\n    path ≠ [] ∧ R.containsPath path"}, {"name": "Utils.StateTransition.Run.isLeaf", "content": "def Run.isLeaf (R : Run S) (root leaf : S) : Prop :=\n  R.reachable root leaf ∧ ∀ y, R (leaf, y) = 0"}, {"name": "Utils.StateTransition.countAsFirst", "content": "def countAsFirst [DecidableEq S] (xs : List S) (x : S) : ℕ :=\n  (xs.zip xs.tail).countP (fun p => p.1 = x)"}, {"name": "Utils.StateTransition.countAsSecond", "content": "def countAsSecond [DecidableEq S] (xs : List S) (x : S) : ℕ :=\n  (xs.zip xs.tail).countP (fun p => p.2 = x)"}], "used_local_lemmas": [{"name": "Utils.StateTransition.finset_ssubset_univ_of_not_mem", "content": "lemma finset_ssubset_univ_of_not_mem {α : Type*} [Fintype α] (s : Finset α) (x : α)\n    (h : x ∉ s) :\n    s ⊂ Finset.univ"}, {"name": "Utils.StateTransition.sum_decrease", "content": "lemma sum_decrease {α : Type*} [Fintype α] [DecidableEq α] (f g : α → ℕ) (a : α)\n    (h_a_decrease : g a < f a)\n    (h_others_le : ∀ x, g x ≤ f x) :\n    ∑ x : α, g x < ∑ x : α, f x"}, {"name": "Utils.StateTransition.path_has_transition", "content": "lemma path_has_transition {S : Type*} [DecidableEq S] (path : List S)\n    (h_len : path.length ≥ 2) :\n    ∃ (t : Transition S), t ∈ path.zip path.tail"}, {"name": "Utils.StateTransition.containsPath_has_positive_transition", "content": "lemma containsPath_has_positive_transition (R : Run S) (path : List S)\n    (h_contains : R.containsPath path) (t : Transition S)\n    (h_in : t ∈ path.zip path.tail) :\n    R t > 0"}, {"name": "Utils.StateTransition.zip_drop_sublist", "content": "lemma zip_drop_sublist (l : List S) (n : ℕ) :\n    ((l.drop n).zip (l.drop (n + 1))).Sublist (l.zip l.tail)"}, {"name": "Utils.StateTransition.containsPath_drop", "content": "lemma containsPath_drop (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.drop n)"}, {"name": "Utils.StateTransition.zip_take_sublist", "content": "lemma zip_take_sublist (l1 l2 : List S) (n m : ℕ) :\n    ((l1.take n).zip (l2.take m)).Sublist (l1.zip l2)"}, {"name": "Utils.StateTransition.tail_take", "content": "lemma tail_take {α : Type*} (l : List α) (n : ℕ) :\n    (l.take n).tail = (l.tail).take (n - 1)"}, {"name": "Utils.StateTransition.containsPath_take", "content": "lemma containsPath_take (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.take n)"}, {"name": "Utils.StateTransition.drop_take_length_ge_two", "content": "lemma drop_take_length_ge_two {α : Type*} (path : List α) (n m : Fin path.length)\n    (h_n_lt_m : n < m) :\n    ((path.drop n.val).take (m.val - n.val + 1)).length ≥ 2"}, {"name": "Utils.StateTransition.getLast_drop_take", "content": "lemma getLast_drop_take {α : Type*} (path : List α) (n k : ℕ)\n    (h_n_lt : n < path.length)\n    (h_bound : n + k ≤ path.length)\n    (h_k_pos : k > 0) :\n    ((path.drop n).take k).getLast? = path[n + k - 1]?"}, {"name": "Utils.StateTransition.drop_take_cycle_same_endpoints", "content": "lemma drop_take_cycle_same_endpoints (path : List S) (x : S) (n m : Fin path.length)\n    (h_n_lt_m : n < m)\n    (h_x_at_n : path[n] = x)\n    (h_x_at_m : path[m] = x) :\n    ((path.drop n.val).take (m.val - n.val + 1)).head? =\n    ((path.drop n.val).take (m.val - n.val + 1)).getLast?"}, {"name": "Utils.StateTransition.containsPath_drop_take", "content": "lemma containsPath_drop_take (R : Run S) (path : List S) (n m : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath ((path.drop n).take m)"}, {"name": "Utils.StateTransition.acyclic_containsPath_nodup", "content": "lemma acyclic_containsPath_nodup (R : Run S) (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_contains : R.containsPath path) :\n    path.Nodup"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton", "content": "lemma countTransitionInPath_append_singleton (path : List S) (x y : S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_not_in : (x, y) ∉ path.zip path.tail) :\n    countTransitionInPath (x, y) (path ++ [y]) = 1"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton_other", "content": "lemma countTransitionInPath_append_singleton_other (path : List S) (x y : S) (t : Transition S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_ne : t ≠ (x, y)) :\n    countTransitionInPath t (path ++ [y]) = countTransitionInPath t path"}, {"name": "Utils.StateTransition.countAsFirst_cons", "content": "lemma countAsFirst_cons (hd : S) (tl : List S) (x : S) :\n    countAsFirst (hd :: tl) x = (if hd = x ∧ tl ≠ [] then 1 else 0) + countAsFirst tl x"}, {"name": "Utils.StateTransition.countAsSecond_cons", "content": "lemma countAsSecond_cons (hd : S) (tl : List S) (x : S) :\n    countAsSecond (hd :: tl) x = (if tl.head? = some x then 1 else 0) + countAsSecond tl x"}, {"name": "Utils.StateTransition.countAsFirst_add_last_eq_countAsSecond_add_head", "content": "lemma countAsFirst_add_last_eq_countAsSecond_add_head (xs : List S) (x : S) :\n    countAsFirst xs x + (if xs.getLast? = some x then 1 else 0) =\n    countAsSecond xs x + (if xs.head? = some x then 1 else 0)"}, {"name": "Utils.StateTransition.cycle_balanced_at_node", "content": "lemma cycle_balanced_at_node (cycle : List S) (x : S)\n    (h_cycle : cycle.head? = cycle.getLast?) :\n    (cycle.zip cycle.tail).countP (fun p => p.1 = x) =\n    (cycle.zip cycle.tail).countP (fun p => p.2 = x)"}, {"name": "Utils.StateTransition.sum_count_pairs_fst", "content": "lemma sum_count_pairs_fst (xs : List (S × S)) (a : S) :\n    ∑ b : S, List.count (a, b) xs = List.countP (fun p => p.1 = a) xs"}, {"name": "Utils.StateTransition.sum_count_pairs_snd", "content": "lemma sum_count_pairs_snd (xs : List (S × S)) (b : S) :\n    ∑ a : S, List.count (a, b) xs = List.countP (fun p => p.2 = b) xs"}, {"name": "Utils.StateTransition.sum_countTransitionInPath_fst", "content": "lemma sum_countTransitionInPath_fst (cycle : List S) (x : S) :\n    ∑ y : S, (countTransitionInPath (x, y) cycle : ℤ) = (countAsFirst cycle x : ℤ)"}, {"name": "Utils.StateTransition.sum_countTransitionInPath_snd", "content": "lemma sum_countTransitionInPath_snd (cycle : List S) (x : S) :\n    ∑ y : S, (countTransitionInPath (y, x) cycle : ℤ) = (countAsSecond cycle x : ℤ)"}, {"name": "Utils.StateTransition.netFlow_sub", "content": "lemma netFlow_sub (R R' : Run S) (x : S)\n    (h_valid : ∀ t, R' t ≤ R t) :\n    Run.netFlow (fun t => R t - R' t) x = R.netFlow x - R'.netFlow x"}, {"name": "Utils.StateTransition.cycle_netFlow_zero", "content": "lemma cycle_netFlow_zero (cycle : List S) (x : S)\n    (h_cycle : cycle.head? = cycle.getLast?) :\n    Run.netFlow (fun t => countTransitionInPath t cycle) x = 0"}, {"name": "Utils.StateTransition.netFlow_removeCycle_eq", "content": "lemma netFlow_removeCycle_eq (R : Run S) (cycle : List S) (x : S)\n    (h_contains : R.containsPath cycle)\n    (h_cycle : cycle.head? = cycle.getLast?) :\n    (R.removeCycle cycle).netFlow x = R.netFlow x"}, {"name": "Utils.StateTransition.size_removeCycle_lt", "content": "lemma size_removeCycle_lt (R : Run S) (cycle : List S)\n    (h_len : cycle.length ≥ 2)\n    (h_contains : R.containsPath cycle)\n    (_h_cycle : cycle.head? = cycle.getLast?) :\n    (R.removeCycle cycle).size < R.size"}, {"name": "Utils.StateTransition.removeCycle_le", "content": "lemma removeCycle_le (R : Run S) (cycle : List S) (t : Transition S) :\n    (R.removeCycle cycle) t ≤ R t"}, {"name": "Utils.StateTransition.exists_smaller_run_with_same_netFlow", "content": "lemma exists_smaller_run_with_same_netFlow (R : Run S) (h_cycle : R.hasCycle) :\n    ∃ (R' : Run S), (∀ x, R'.netFlow x = R.netFlow x) ∧ R'.size < R.size ∧ (∀ t, R' t ≤ R t)"}, {"name": "Utils.StateTransition.acyclic_no_self_loop", "content": "lemma acyclic_no_self_loop (R : Run S) (s : S) (h_acyclic : R.isAcyclic) (h_edge : R (s, s) > 0) : False"}, {"name": "Utils.StateTransition.getLast_mem", "content": "lemma getLast_mem {α : Type*} (l : List α) (x : α) (h_last : l.getLast? = some x) :\n    x ∈ l"}, {"name": "Utils.StateTransition.last_not_in_zip_tail", "content": "lemma last_not_in_zip_tail {α : Type*} [DecidableEq α] (l : List α) (x : α)\n    (h_nodup : l.Nodup)\n    (h_last : l.getLast? = some x) :\n    ∀ y : α, (x, y) ∉ l.zip l.tail"}, {"name": "Utils.StateTransition.drop_of_lt_length_nonempty", "content": "lemma drop_of_lt_length_nonempty {α : Type*} (path : List α) (i : ℕ)\n    (h_i_lt : i < path.length) :\n    path.drop i ≠ []"}, {"name": "Utils.StateTransition.cycle_from_suffix_contains", "content": "lemma cycle_from_suffix_contains (R : Run S) (suffix : List S) (current y : S)\n    (h_suffix_nodup : suffix.Nodup)\n    (h_contains_suffix : R.containsPath suffix)\n    (h_suffix_nonempty : suffix ≠ [])\n    (h_suffix_last : suffix.getLast? = some current)\n    (h_edge : R (current, y) > 0) :\n    ∀ t : Transition S, countTransitionInPath t (suffix ++ [y]) ≤ R t"}, {"name": "Utils.StateTransition.path_with_back_edge_creates_cycle", "content": "lemma path_with_back_edge_creates_cycle (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_y_in_path : y ∈ path)\n    (h_edge : R (current, y) > 0) :\n    R.hasCycle"}, {"name": "Utils.StateTransition.acyclic_edge_not_in_path", "content": "lemma acyclic_edge_not_in_path (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_edge : R (current, y) > 0)\n    (h_y_in_path : y ∈ path) :\n    False"}, {"name": "Utils.StateTransition.not_mem_implies_transition_not_in_zip_tail", "content": "lemma not_mem_implies_transition_not_in_zip_tail {α : Type*} (path : List α) (x y : α)\n    (h_y_not_in : y ∉ path) :\n    (x, y) ∉ path.zip path.tail"}, {"name": "Utils.StateTransition.containsPath_append_singleton", "content": "lemma containsPath_append_singleton (R : Run S) (path : List S) (x y : S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_contains : R.containsPath path)\n    (h_y_not_in_path : y ∉ path)\n    (h_edge : R (x, y) > 0) :\n    R.containsPath (path ++ [y])"}, {"name": "Utils.StateTransition.acyclic_has_leaf_aux", "content": "lemma acyclic_has_leaf_aux (R : Run S) (root current : S)\n    (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_start : path.head? = some root)\n    (h_end : path.getLast? = some current)\n    (h_nonempty : path ≠ [])\n    (h_contains : R.containsPath path)\n    (h_has_out : ∃ y, y ∉ path ∧ R (current, y) > 0) :\n    ∃ leaf, R.isLeaf root leaf"}, {"name": "Utils.StateTransition.acyclic_has_leaf", "content": "lemma acyclic_has_leaf (R : Run S) (root : S)\n    (h_acyclic : R.isAcyclic)\n    (h_has_out : ∃ y, R (root, y) > 0) :\n    ∃ leaf, R.isLeaf root leaf"}, {"name": "Utils.StateTransition.single_le_sum_of_nonneg", "content": "lemma single_le_sum_of_nonneg {α : Type*} [Fintype α] (f : α → ℤ) (a : α)\n    (h_nonneg : ∀ x, f x ≥ 0) :\n    f a ≤ ∑ x : α, f x"}, {"name": "Utils.StateTransition.sum_pos_of_pos_element", "content": "lemma sum_pos_of_pos_element {α : Type*} [Fintype α] (f : α → ℤ) (a : α)\n    (h_pos : f a > 0)\n    (h_nonneg : ∀ x, f x ≥ 0) :\n    ∑ x : α, f x > 0"}, {"name": "Utils.StateTransition.sum_nat_cast_pos", "content": "lemma sum_nat_cast_pos {α : Type*} [Fintype α] (f : α → ℕ) (a : α)\n    (h_pos : f a > 0) :\n    ∑ x : α, (f x : ℤ) > 0"}, {"name": "Utils.StateTransition.leaf_has_negative_netFlow", "content": "lemma leaf_has_negative_netFlow (R : Run S) (root leaf : S)\n    (h_leaf : R.isLeaf root leaf)\n    (h_in : ∃ y, R (y, leaf) > 0) :\n    R.netFlow leaf < 0"}, {"name": "Utils.StateTransition.sum_zero_of_all_zero", "content": "lemma sum_zero_of_all_zero {α : Type*} [Fintype α] (f : α → ℕ) (h : ∀ x, f x = 0) :\n    ∑ x : α, (f x : ℤ) = 0"}, {"name": "Utils.StateTransition.sum_nat_cast_nonneg", "content": "lemma sum_nat_cast_nonneg {α : Type*} [Fintype α] (f : α → ℕ) :\n    0 ≤ ∑ x : α, (f x : ℤ)"}, {"name": "Utils.StateTransition.sum_nat_cast_zero_of_not_pos", "content": "lemma sum_nat_cast_zero_of_not_pos {α : Type*} [Fintype α] (f : α → ℕ)\n    (h : ∀ x, ¬(f x > 0)) :\n    ∑ x : α, (f x : ℤ) = 0"}, {"name": "Utils.StateTransition.positive_netFlow_has_outgoing_edge", "content": "lemma positive_netFlow_has_outgoing_edge (R : Run S) (s : S)\n    (h_pos : R.netFlow s > 0) :\n    ∃ y, R (s, y) > 0"}, {"name": "Utils.StateTransition.last_has_incoming_transition", "content": "lemma last_has_incoming_transition {α : Type*} (l : List α) (x : α)\n    (h_len : l.length ≥ 2)\n    (h_last : l.getLast? = some x) :\n    ∃ y, (y, x) ∈ l.zip l.tail"}, {"name": "Utils.StateTransition.path_distinct_head_last_length_ge_two", "content": "lemma path_distinct_head_last_length_ge_two {α : Type*} (path : List α) (x y : α)\n    (h_nonempty : path ≠ [])\n    (h_head : path.head? = some x)\n    (h_last : path.getLast? = some y)\n    (h_ne : x ≠ y) :\n    path.length ≥ 2"}, {"name": "Utils.StateTransition.reachable_leaf_has_incoming_edge", "content": "lemma reachable_leaf_has_incoming_edge (R : Run S) (root leaf : S)\n    (h_leaf : R.isLeaf root leaf)\n    (h_ne : root ≠ leaf) :\n    ∃ y, R (y, leaf) > 0"}, {"name": "Utils.StateTransition.unique_negative_netFlow", "content": "lemma unique_negative_netFlow (R : Run S) (s d x : S)\n    (h_source : R.netFlow s = 1)\n    (h_others : ∀ y, y ≠ s → y ≠ d → R.netFlow y = 0)\n    (h_x_neg : R.netFlow x < 0) :\n    x = d"}, {"name": "Utils.StateTransition.leaf_has_incoming_and_negative_netFlow", "content": "lemma leaf_has_incoming_and_negative_netFlow (R : Run S) (root leaf : S)\n    (h_leaf : R.isLeaf root leaf)\n    (h_root_out : ∃ y, R (root, y) > 0) :\n    R.netFlow leaf < 0"}, {"name": "Utils.StateTransition.acyclic_run_has_path_from_source_to_sink", "content": "lemma acyclic_run_has_path_from_source_to_sink (R : Run S) (s d : S)\n    (h_acyclic : R.isAcyclic)\n    (h_source : R.netFlow s = 1)\n    (h_others : ∀ x, x ≠ s → x ≠ d → R.netFlow x = 0) :\n    ∃ (path : List S), path.head? = some s ∧ path.getLast? = some d ∧\n      path ≠ [] ∧ R.containsPath path ∧ path.Nodup"}], "local_ctx": "import Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Fintype.Basic\n\nimport Mathlib.Data.Fintype.Prod\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Piecewise\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nnamespace Utils.StateTransition\n\nvariable {S : Type*} [DecidableEq S] [Fintype S]\n\ndef Transition (S : Type*) := S × S\n\ndef Run (S : Type*) := Transition S → ℕ\n\nnoncomputable def Run.netFlow {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) (x : S) : ℤ :=\n  (∑ y : S, (R (x, y) : ℤ)) - (∑ y : S, (R (y, x) : ℤ))\n\nnoncomputable def Run.size {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) : ℕ :=\n  ∑ t : Transition S, R t\n\ndef countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t\n\ndef Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t\n\ndef Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle\n\ndef Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle\n\ndef Run.removeCycle (R : Run S) (cycle : List S) : Run S :=\n  fun t => R t - countTransitionInPath t cycle\n\ndef Run.reachable [DecidableEq S] (R : Run S) (start finish : S) : Prop :=\n  ∃ (path : List S), path.head? = some start ∧ path.getLast? = some finish ∧\n    path ≠ [] ∧ R.containsPath path\n\ndef Run.isLeaf (R : Run S) (root leaf : S) : Prop :=\n  R.reachable root leaf ∧ ∀ y, R (leaf, y) = 0\n\ndef countAsFirst [DecidableEq S] (xs : List S) (x : S) : ℕ :=\n  (xs.zip xs.tail).countP (fun p => p.1 = x)\n\ndef countAsSecond [DecidableEq S] (xs : List S) (x : S) : ℕ :=\n  (xs.zip xs.tail).countP (fun p => p.2 = x)", "target_theorem": "theorem exists_path_from_source_to_sink\n    (R : Run S) (s d : S)\n    (h_source : R.netFlow s = 1)\n    (h_others : ∀ x, x ≠ s → x ≠ d → R.netFlow x = 0) :\n    ∃ (path : List S), path.head? = some s ∧ path.getLast? = some d ∧\n      path ≠ [] ∧ R.containsPath path ∧ path.Nodup :=", "ground_truth_proof": ":= by\n  -- Reduce to acyclic case by removing cycles\n  by_cases h_cyclic : R.hasCycle\n  case pos =>\n    -- R has a cycle, remove it and recurse\n    obtain ⟨R', h_netFlow_preserved, h_size_lt, h_R'_le_R⟩ := exists_smaller_run_with_same_netFlow R h_cyclic\n    -- R' has the same net flows\n    have h_source' : R'.netFlow s = 1 := by aesop\n    have h_others' : ∀ x, x ≠ s → x ≠ d → R'.netFlow x = 0 := by aesop\n    -- Recursive call with smaller run\n    obtain ⟨path, h_head, h_last, h_nonempty, h_contains', h_nodup⟩ :=\n      exists_path_from_source_to_sink R' s d h_source' h_others'\n    use path\n    refine ⟨h_head, h_last, h_nonempty, ?_, h_nodup⟩\n    -- Show R.containsPath path from R'.containsPath path\n    -- R' t ≤ R t for all t (from exists_smaller_run_with_same_netFlow)\n    intro t\n    calc countTransitionInPath t path\n      ≤ R' t := h_contains' t\n    _ ≤ R t := h_R'_le_R t\n  case neg =>\n    -- R is acyclic, use the acyclic case lemma\n    exact acyclic_run_has_path_from_source_to_sink R s d h_cyclic h_source h_others\ntermination_by R.size\ndecreasing_by\n  simp_wf\n  exact h_size_lt", "nesting_depth": 9, "transitive_dep_count": 162, "subset_aristotle": false, "category": "Applied verif."}
{"id": 168, "thm_name": "FlatOperation.onlyAccessedBelow_all", "thm_stmt": "theorem onlyAccessedBelow_all {ops : List (FlatOperation F)} (n : ℕ) :\n  forAll n { witness n _ := Environment.OnlyAccessedBelow n } ops →\n    Environment.OnlyAccessedBelow (n + localLength ops) (localWitnesses · ops)", "lean_root": "clean", "rel_path": "Clean/Circuit/Theorems.lean", "imports": ["import Clean.Circuit.Provable", "import Clean.Circuit.Basic"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Environment.OnlyAccessedBelow", "content": "def Environment.OnlyAccessedBelow (n : ℕ) (f : Environment F → α) :=\n  ∀ env env', env.AgreesBelow n env' → f env = f env'"}, {"name": "Condition.applyFlat", "content": "def Condition.applyFlat (condition : Condition F) (offset : ℕ) : FlatOperation F → Prop\n  | .witness m c => condition.witness offset m c\n  | .assert e => condition.assert offset e\n  | .lookup l => condition.lookup offset l"}, {"name": "Environment.AgreesBelow", "content": "def Environment.AgreesBelow (n : ℕ) (env env' : Environment F) :=\n  ∀ i < n, env.get i = env'.get i"}, {"name": "FlatOperation.singleLocalLength", "content": "def FlatOperation.singleLocalLength : FlatOperation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "FlatOperation.localLength_cons", "content": "lemma localLength_cons {F} {op : FlatOperation F} {ops : List (FlatOperation F)} :\n    localLength (op :: ops) = op.singleLocalLength + localLength ops"}, {"name": "FlatOperation.forAll_cons", "content": "theorem forAll_cons {condition : Condition F} {offset : ℕ} {op : FlatOperation F} {ops : List (FlatOperation F)} :\n  forAll offset condition (op :: ops) ↔\n    condition.applyFlat offset op ∧ forAll (op.singleLocalLength + offset) condition ops"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nvariable {F : Type} [Field F] {α β : Type}\n\nnamespace Operations\n\nend Operations\n\nnamespace Circuit\n\nend Circuit\n\nnamespace FlatOperation\n\nend FlatOperation\n\nnamespace Environment\n\nopen FlatOperation (localLength localWitnesses)\n\nend Environment\n\nnamespace Circuit\n\nend Circuit\n\nnamespace Circuit\n\nvariable {α β : Type} {n : ℕ} {prop : Condition F} {env : Environment F}\n\nend Circuit\n\nnamespace FlatOperation\n\nend FlatOperation\n\nnamespace Operations\n\nend Operations\n\nnamespace FlatOperation\n\nend FlatOperation\n\nnamespace FlatOperation", "target_theorem": "theorem onlyAccessedBelow_all {ops : List (FlatOperation F)} (n : ℕ) :\n  forAll n { witness n _ := Environment.OnlyAccessedBelow n } ops →\n    Environment.OnlyAccessedBelow (n + localLength ops) (localWitnesses · ops) :=", "ground_truth_proof": ":= by\n  intro h_comp env env' h_env\n  simp only\n  induction ops generalizing n with\n  | nil => simp [localWitnesses]\n  | cons op ops ih =>\n    simp_all only [forAll_cons, localLength_cons]\n    have h_ih := h_comp.right\n    replace h_comp := h_comp.left\n    replace h_ih := ih (op.singleLocalLength + n) h_ih\n    ring_nf at *\n    specialize h_ih h_env\n    clear ih\n    cases op with\n    | assert | lookup =>\n      simp_all only [Condition.applyFlat, localWitnesses]\n    | witness m c =>\n      simp_all only [Condition.applyFlat, localWitnesses,\n        Environment.OnlyAccessedBelow, Environment.AgreesBelow]\n      congr 1\n      apply h_comp env env'\n      intro i hi\n      exact h_env i (by linarith)", "nesting_depth": 4, "transitive_dep_count": 37, "subset_aristotle": true, "category": "Applied verif."}
{"id": 169, "thm_name": "Gadgets.Addition8.Theorems.soundness", "thm_stmt": "theorem soundness (x y out carry_in carry_out : F p):\n    x.val < 256 -> y.val < 256 ->\n    out.val < 256 ->\n    IsBool carry_in ->\n    IsBool carry_out ->\n    (x + y + carry_in + -out + -(carry_out * 256) = 0) ->\n    (out.val = (x.val + y.val + carry_in.val) % 256\n    ∧ carry_out.val = (x.val + y.val + carry_in.val) / 256)", "lean_root": "clean", "rel_path": "Clean/Gadgets/Addition8/Theorems.lean", "imports": ["import Clean.Gadgets.Boolean", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "One", "module": "Init.Prelude"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"field_to_nat\" : tactic", "content": "syntax \"field_to_nat\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|field_to_nat) =>\n    `(tactic|(\n      intros\n      repeat rw [ZMod.val_add] \n      repeat rw [ZMod.val_mul] \n      repeat rw [val_eq_256]\n      try simp only [Nat.add_mod_mod, Nat.mod_add_mod, Nat.mul_mod_mod, Nat.mod_mul_mod]\n      rw [Nat.mod_eq_of_lt _]\n      repeat linarith [‹Fact (_ > 512)›.elim]))\n\nexample [Fact (p > 512)] (x y : F p) (hx : x.val < 256) (hy : y.val < 2) :\n    (x + y * 256).val = x.val + y.val * 256 := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "IsBool", "content": "def IsBool {α : Type*} [Zero α] [One α] (x : α) : Prop := x = 0 ∨ x = 1"}], "lib_lemmas": [{"name": "Nat.div_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.div_eq_of_lt_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_eq_sub_mod", "module": "Init.Data.Nat.Div.Basic"}, {"name": "sub_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "eq_add_of_sub_eq", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "ZMod.val_one", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_zero", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "neg_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_add_neg", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "byte_sum_and_bit_do_not_wrap", "content": "theorem byte_sum_and_bit_do_not_wrap (x y b : F p) [Fact (p > 512)]:\n    x.val < 256 -> y.val < 256 -> b.val < 2 -> (b + x + y).val = b.val + x.val + y.val"}, {"name": "byte_sum_and_bit_do_not_wrap'", "content": "theorem byte_sum_and_bit_do_not_wrap' (x y b : F p) [Fact (p > 512)]:\n    x.val < 256 -> y.val < 256 -> b.val < 2 -> (x + y + b).val = x.val + y.val + b.val"}, {"name": "byte_sum_and_bit_lt_512", "content": "theorem byte_sum_and_bit_lt_512 (x y b : F p) [Fact (p > 512)]:\n    x.val < 256 -> y.val < 256 -> b.val < 2 -> (x + y + b).val < 512"}, {"name": "byte_plus_256_do_not_wrap", "content": "theorem byte_plus_256_do_not_wrap (x : F p) [Fact (p > 512)]:\n    x.val < 256 -> (x + 256).val = x.val + 256"}, {"name": "byte_sum_do_not_wrap", "content": "theorem byte_sum_do_not_wrap (x y : F p) [Fact (p > 512)]:\n    x.val < 256 -> y.val < 256 -> (x + y).val = x.val + y.val"}, {"name": "byte_sum_le_bound", "content": "theorem byte_sum_le_bound (x y : F p) [Fact (p > 512)]:\n    x.val < 256 -> y.val < 256 -> (x + y).val < 511"}, {"name": "val_lt_two", "content": "theorem val_lt_two {p : ℕ} [Fact p.Prime] {x : F p} (h : IsBool x) : x.val < 2"}, {"name": "val_eq_256", "content": "lemma val_eq_256 [p_large_enough: Fact (p > 512)] : (256 : F p).val = 256"}, {"name": "ext", "content": "theorem ext {x y : F p} (h : x.val = y.val) : x = y"}, {"name": "ext_iff", "content": "theorem ext_iff {x y : F p} : x = y ↔ x.val = y.val"}, {"name": "val_lt_p", "content": "theorem val_lt_p {p : ℕ} (x : ℕ) : (x < p) → (x : F p).val = x"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Gadgets.Addition8.Theorems.soundness_zero_carry", "content": "theorem soundness_zero_carry (x y out carry_in : F p):\n    x.val < 256 -> y.val < 256 -> out.val < 256  -> carry_in.val < 2 ->\n    (carry_in + x + y - out = 0 -> (out.val = (carry_in.val + x.val + y.val) % 256\n    ∧ (carry_in.val + x.val + y.val) / 256 = 0))"}, {"name": "Gadgets.Addition8.Theorems.soundness_one_carry", "content": "theorem soundness_one_carry (x y out carry_in : F p):\n    x.val < 256 -> y.val < 256 -> out.val < 256 -> carry_in.val < 2 ->\n    carry_in + x + y - out - 256 = 0 -> (out.val = (carry_in.val + x.val + y.val) % 256\n    ∧ (carry_in.val + x.val + y.val) / 256 = 1)"}], "local_ctx": "import Clean.Utils.Field\n\nimport Clean.Gadgets.Boolean\n\nnamespace Gadgets.Addition8.Theorems\n\nvariable {p : ℕ} [Fact p.Prime]\n\nvariable [p_large_enough: Fact (p > 512)]", "target_theorem": "theorem soundness (x y out carry_in carry_out : F p):\n    x.val < 256 -> y.val < 256 ->\n    out.val < 256 ->\n    IsBool carry_in ->\n    IsBool carry_out ->\n    (x + y + carry_in + -out + -(carry_out * 256) = 0) ->\n    (out.val = (x.val + y.val + carry_in.val) % 256\n    ∧ carry_out.val = (x.val + y.val + carry_in.val) / 256) :=", "ground_truth_proof": ":= by\n  intros hx hy hout carry_in_bool carry_out_bool h\n  have carry_in_bound := IsBool.val_lt_two carry_in_bool\n\n  rcases carry_out_bool with zero_carry | one_carry\n  -- case with zero carry\n  · rw [zero_carry] at h\n    simp only [zero_mul, neg_zero, add_zero] at h\n    rw [←sub_eq_add_neg] at h\n    have h_spec : carry_in + x + y - out = 0 := by\n      rw [add_comm (x + y), ←add_assoc] at h\n      assumption\n\n    have thm := soundness_zero_carry x y out carry_in hx hy hout carry_in_bound h_spec\n    apply_fun ZMod.val at zero_carry\n\n    -- now it is just a matter of shuffling terms around\n    have shuffle_terms : carry_in.val + x.val + y.val = x.val + y.val + carry_in.val := by\n      zify; ring\n    rw [ZMod.val_zero, ← thm.right] at zero_carry\n    rw [shuffle_terms] at thm\n    rw [shuffle_terms] at zero_carry\n    constructor\n    · exact thm.left\n    · exact zero_carry\n\n  -- case with one carry\n  · rw [one_carry] at h\n    simp only [one_mul] at h\n    -- rw [←sub_eq_add_neg, ←sub_eq_add_neg (carry_in + x + y)] at h\n    have h_spec : carry_in + x + y - out - 256 = 0 := by\n      rw [add_comm (x + y), ←add_assoc] at h\n      ring_nf at h; ring_nf\n      assumption\n\n    -- instantiate the sub-theorem\n    have thm := soundness_one_carry x y out carry_in hx hy hout carry_in_bound h_spec\n    apply_fun ZMod.val at one_carry\n\n    have shuffle_terms : carry_in.val + x.val + y.val = x.val + y.val + carry_in.val := by\n      zify; ring\n    rw [ZMod.val_one, ← thm.right] at one_carry\n    rw [shuffle_terms] at thm\n    rw [shuffle_terms] at one_carry\n    constructor\n    · exact thm.left\n    · exact one_carry", "nesting_depth": 3, "transitive_dep_count": 29, "subset_aristotle": true, "category": "Applied verif."}
{"id": 170, "thm_name": "MemoryAccessList.eq_of_perm_of_sorted", "thm_stmt": "lemma MemoryAccessList.eq_of_perm_of_sorted {l1 l2 l3 : MemoryAccessList} (h_l1_sorted: l1.isTimestampSorted)\n    (h_l2_sorted : l2.isAddressTimestampSorted) (h_l3_sorted : l3.isAddressTimestampSorted)\n    (h_perm1 : l1.Perm l2) (h_perm2 : l1.Perm l3) : l2 = l3", "lean_root": "clean", "rel_path": "Clean/Utils/OfflineMemory.lean", "imports": ["import Clean.Utils.Tactics", "import Clean.Circuit.Provable", "import Clean.Gadgets.Equality", "import Clean.Utils.Primes", "import Mathlib.Data.List.Sort", "import Clean.Circuit.Basic", "import Clean.Utils.Field"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Perm", "module": "Init.Data.List.Basic"}, {"name": "List.Sorted", "module": "Mathlib.Deprecated.Sort"}, {"name": "List.Pairwise", "module": "Init.Data.List.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.Pairwise.perm", "module": "Init.Data.List.Perm"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "List.Pairwise.imp", "module": "Init.Data.List.Pairwise"}, {"name": "List.Perm.nodup_iff", "module": "Init.Data.List.Perm"}, {"name": "List.Sorted.insertionSort_eq", "module": "Mathlib.Data.List.Sort"}, {"name": "List.Sorted.nodup", "module": "Mathlib.Data.List.Nodup"}, {"name": "List.eq_of_perm_of_sorted", "module": "Mathlib.Data.List.Sort"}, {"name": "List.perm_comm", "module": "Init.Data.List.Perm"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MemoryAccess", "content": "def MemoryAccess := ℕ × ℕ × ℕ × ℕ"}, {"name": "MemoryAccessList", "content": "def MemoryAccessList := List MemoryAccess"}, {"name": "timestamp_ordering", "content": "abbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2"}, {"name": "MemoryAccessList.isTimestampSorted", "content": "def MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering"}, {"name": "MemoryAccessList.timestamps_neq", "content": "def MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y"}, {"name": "MemoryAccessList.Notimestampdup", "content": "def MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses"}, {"name": "address_timestamp_ordering", "content": "abbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2"}, {"name": "address_strict_timestamp_ordering", "content": "abbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2"}, {"name": "MemoryAccessList.isAddressTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering"}, {"name": "MemoryAccessList.isAddressStrictTimestampSorted", "content": "@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering"}], "used_local_lemmas": [{"name": "MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup", "content": "theorem MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup\n    (accesses : MemoryAccessList) (h_sorted : accesses.isAddressTimestampSorted)\n    (h_no_timestamp_dup : accesses.Notimestampdup) :\n    accesses.isAddressStrictTimestampSorted"}, {"name": "MemoryAccessList.noTimestampDup_perm", "content": "theorem MemoryAccessList.noTimestampDup_perm (l1 l2 : MemoryAccessList)\n    (h_l1_nodup : l1.Notimestampdup) (h_perm : l1.Perm l2) :\n    l2.Notimestampdup"}, {"name": "MemoryAccessList.noTimestampDup_of_TimestampSorted", "content": "theorem MemoryAccessList.noTimestampDup_of_TimestampSorted\n    (accesses : MemoryAccessList) (h_sorted : accesses.isTimestampSorted) :\n    accesses.Notimestampdup"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Provable\n\nimport Clean.Gadgets.Equality\n\nimport Clean.Utils.Field\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Tactics\n\nimport Mathlib.Data.List.Sort\n\ndef MemoryAccess := ℕ × ℕ × ℕ × ℕ \n\ndef MemoryAccessList := List MemoryAccess\n\nabbrev timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, _a2, _r2, _w2), (t1, _a1, _r1, _w1) => t1 < t2\n\ndef MemoryAccessList.isTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted timestamp_ordering\n\ndef MemoryAccessList.timestamps_neq (x y: MemoryAccess) : Prop :=\n  match x, y with\n  | (t_x, _a_x, _r_x, _w_x), (t_y, _a_y, _r_y, _w_y) => t_x ≠ t_y\n\ndef MemoryAccessList.Notimestampdup (accesses : MemoryAccessList) : Prop :=\n  List.Pairwise timestamps_neq accesses\n\nabbrev address_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 ≤ t2 else a1 < a2\n\nabbrev address_strict_timestamp_ordering (x y : MemoryAccess) := match x, y with\n| (t2, a2, _, _), (t1, a1, _, _) => if a1 = a2 then t1 < t2 else a1 < a2\n\n@[reducible]\ndef MemoryAccessList.isAddressTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_timestamp_ordering\n\n@[reducible]\ndef MemoryAccessList.isAddressStrictTimestampSorted (accesses : MemoryAccessList) : Prop :=\n  accesses.Sorted address_strict_timestamp_ordering", "target_theorem": "lemma MemoryAccessList.eq_of_perm_of_sorted {l1 l2 l3 : MemoryAccessList} (h_l1_sorted: l1.isTimestampSorted)\n    (h_l2_sorted : l2.isAddressTimestampSorted) (h_l3_sorted : l3.isAddressTimestampSorted)\n    (h_perm1 : l1.Perm l2) (h_perm2 : l1.Perm l3) : l2 = l3 :=", "ground_truth_proof": ":= by\n  simp [isAddressTimestampSorted] at *\n  rw [List.perm_comm] at h_perm1\n  have l1_nodup := List.Sorted.nodup h_l1_sorted\n\n  have thm1 := List.Sorted.insertionSort_eq h_l2_sorted\n  have h_l2_nodup := (List.Perm.nodup_iff h_perm1).mpr l1_nodup\n  have h_l3_nodup := (List.Perm.nodup_iff h_perm2).mp l1_nodup\n\n  have l2_perm_l3 := List.Perm.trans h_perm1 h_perm2\n\n  have l1_notimestampdup := MemoryAccessList.noTimestampDup_of_TimestampSorted l1 h_l1_sorted\n  have l2_notimestampdup := MemoryAccessList.noTimestampDup_perm l1 l2 l1_notimestampdup (List.Perm.symm h_perm1)\n  have l3_notimestampdup := MemoryAccessList.noTimestampDup_perm l1 l3 l1_notimestampdup h_perm2\n\n  have l2_strict_sorted := MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup l2 h_l2_sorted l2_notimestampdup\n  have l3_strict_sorted := MemoryAccessList.addressStrictTimestampSorted_of_AddressTimestampSorted_noTimestampDup l3 h_l3_sorted l3_notimestampdup\n  exact List.eq_of_perm_of_sorted l2_perm_l3 l2_strict_sorted l3_strict_sorted", "nesting_depth": 3, "transitive_dep_count": 28, "subset_aristotle": false, "category": "Applied verif."}
{"id": 171, "thm_name": "Circomlib.MultiAND.main_output_binary_from_completeness", "thm_stmt": "lemma main_output_binary_from_completeness (n : ℕ) (offset : ℕ) (env : Environment (F p))\n    (input_var : Var (fields n) (F p)) (input : fields n (F p))\n    (h_eval : input = eval env input_var)\n    (h_assumptions : Assumptions n input)\n    (h_local_witnesses : env.UsesLocalWitnessesCompleteness offset ((main input_var).operations offset))\n    (h_completeness : Circuit.ConstraintsHold.Completeness env ((main input_var).operations offset)) :\n    let output := env ((main input_var).output offset)\n    IsBool output", "lean_root": "clean", "rel_path": "Clean/Circomlib/Gates.lean", "imports": ["import Clean.Circuit.Theorems", "import Clean.Circuit.Provable", "import Clean.Utils.Field", "import Clean.Circuit", "import Mathlib.Data.Nat.Bitwise", "import Clean.Gadgets.Boolean", "import Clean.Utils.Bitwise", "import Clean.Utils.BinaryOps", "import Clean.Circuit.Basic", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "One", "module": "Init.Prelude"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "Vector.forM", "module": "Init.Data.Vector.Basic"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Vector.cast", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.foldl", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.map", "module": "Init.Data.Vector.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "List.foldl", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "fields", "content": "@[reducible]\ndef fields (n : ℕ) := fun F => Vector F n"}, {"name": "IsBool", "content": "def IsBool {α : Type*} [Zero α] [One α] (x : α) : Prop := x = 0 ∨ x = 1"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "fieldPair", "content": "@[reducible]\ndef fieldPair : TypeMap := fun F => F × F"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "ConstraintsHold", "content": "@[circuit_norm]\ndef ConstraintsHold (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold eval ops\n  | .lookup { table, entry, .. } :: ops =>\n    table.Contains (entry.map eval) ∧ ConstraintsHold eval ops\n  | .subcircuit s :: ops =>\n    ConstraintsHoldFlat eval s.ops ∧ ConstraintsHold eval ops"}, {"name": "Environment.UsesLocalWitnessesCompleteness", "content": "@[circuit_norm]\ndef Environment.UsesLocalWitnessesCompleteness (env : Environment F) (offset : ℕ) : List (Operation F) → Prop\n  | [] => True\n  | .witness m c :: ops => env.ExtendsVector (c env) offset ∧ env.UsesLocalWitnessesCompleteness (offset + m) ops\n  | .assert _ :: ops => env.UsesLocalWitnessesCompleteness offset ops\n  | .lookup _ :: ops => env.UsesLocalWitnessesCompleteness offset ops\n  | .subcircuit s :: ops => s.UsesLocalWitnesses env ∧ env.UsesLocalWitnessesCompleteness (offset + s.localLength) ops"}, {"name": "main", "content": "def main {α : TypeMap} [ProvableType α] (input : Var α F × Var α F) : Circuit F Unit := do\n  let (x, y) := input\n  let diffs := (toVars x).zip (toVars y) |>.map (fun (xi, yi) => xi - yi)\n  .forEach diffs assertZero"}, {"name": "assertZero", "content": "@[circuit_norm]\ndef assertZero (e : Expression F) : Circuit F Unit := fun _ =>\n  ((), [.assert e])"}, {"name": "forEach", "content": "def forEach {m : ℕ} (xs : Vector α m) [Inhabited α] (body : α → Circuit F Unit)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F Unit :=\n  xs.forM body"}, {"name": "ExplicitCircuit.from_map", "content": "instance ExplicitCircuit.from_map {f : α → β} {g : Circuit F α}\n    (g_explicit : ExplicitCircuit g) : ExplicitCircuit (f <$> g) where\n  output n := output g n |> f\n  localLength n := localLength g n\n  operations n := operations g n\n\n  output_eq n := by admit /- proof elided -/"}, {"name": "ExplicitCircuits.from_pure", "content": "instance ExplicitCircuits.from_pure {f : α → β} : ExplicitCircuits (fun a => pure (f a) : α → Circuit F β) where\n  output a _ := f a\n  localLength _ _ := 0\n  operations _ _ := []"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}, {"name": "inductConsistent", "content": "def inductConsistent {motive : (ops : Operations F) → (n : ℕ) → ops.SubcircuitsConsistent n → Sort*}\n  (empty : ∀ n, motive [] n trivial)\n  (witness : ∀ n m c ops {h}, motive ops (m + n) h →\n    motive (.witness m c :: ops) n (by admit /- proof elided -/\n    ))\n  (assert : ∀ n e ops {h}, motive ops n h →\n    motive (.assert e :: ops) n (by admit /- proof elided -/\n    ))\n  (lookup : ∀ n l ops {h}, motive ops n h →\n    motive (.lookup l :: ops) n (by admit /- proof elided -/\n    ))\n  (subcircuit : ∀ n (s : Subcircuit F n) ops {h}, motive ops (s.localLength + n) h →\n    motive (.subcircuit s :: ops) n (by admit /- proof elided -/\n    ))\n    (ops : Operations F) (n : ℕ) (h : ops.SubcircuitsConsistent n) : motive ops n h :=\n  motive' ops n h\nwhere motive' : (ops : Operations F) → (n : ℕ) → (h : ops.SubcircuitsConsistent n) → motive ops n h\n  | [], n, _ => empty n\n  | .witness m c :: ops, n, h | .assert e :: ops, n, h | .lookup e :: ops, n, h => by admit /- proof elided -/\n    | exact witness _ _ _ _ (motive' ops _ h.right)\n    | exact assert _ _ _ (motive' ops _ h.right)\n    | exact lookup _ _ _ (motive' ops _ h.right)\n  | .subcircuit s :: ops, n', h => by admit /- proof elided -/"}, {"name": "Condition.apply", "content": "@[circuit_norm]\ndef Condition.apply (condition : Condition F) (offset : ℕ) : Operation F → Prop\n  | .witness m c => condition.witness offset m c\n  | .assert e => condition.assert offset e\n  | .lookup l => condition.lookup offset l\n  | .subcircuit s => condition.subcircuit offset s"}, {"name": "FlatOperation.singleLocalLength", "content": "def FlatOperation.singleLocalLength : FlatOperation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0"}, {"name": "Environment.UsesLocalWitnessesFlat", "content": "def Environment.UsesLocalWitnessesFlat (env : Environment F) (n : ℕ) (ops : List (FlatOperation F)) : Prop :=\n  FlatOperation.forAll n { witness n _ compute := env.ExtendsVector (compute env) n } ops"}, {"name": "Condition.applyFlat", "content": "def Condition.applyFlat (condition : Condition F) (offset : ℕ) : FlatOperation F → Prop\n  | .witness m c => condition.witness offset m c\n  | .assert e => condition.assert offset e\n  | .lookup l => condition.lookup offset l"}, {"name": "induct", "content": "def induct {motive : List (FlatOperation F) → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n    (ops : List (FlatOperation F)) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup ops)"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "forAll", "content": "@[reducible, circuit_norm]\ndef forAll (circuit : Circuit F α) (n : ℕ) (prop : Condition F) :=\n  (circuit.operations n).forAll n prop"}], "lib_lemmas": [{"name": "Nat.min_def", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.strong_induction_on", "module": "Mathlib.Data.Nat.Init"}, {"name": "List.getElem_toArray", "module": "Init.Data.Array.Basic"}, {"name": "Nat.add_zero", "module": "Init.Core"}, {"name": "Vector.getElem_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Array.foldl_toList", "module": "Init.Data.Array.Bootstrap"}, {"name": "List.foldl_cons", "module": "Init.Data.List.Basic"}, {"name": "List.foldl_nil", "module": "Init.Data.List.Basic"}, {"name": "Vector.foldl_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_map", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.toList_toArray", "module": "Init.Data.Vector.Lemmas"}, {"name": "ZMod.val_one", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Vector.foldl_append", "module": "Init.Data.Vector.Lemmas"}, {"name": "Nat.add_sub_self_left", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.min_add_right_self", "module": "Init.Data.Nat.Lemmas"}, {"name": "Vector.cast_cast", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_cast", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_drop", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.getElem_take", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.map_append", "module": "Init.Data.Vector.Lemmas"}], "repo_lemmas": [{"name": "bind_forAll", "content": "@[circuit_norm]\ntheorem bind_forAll {f : Circuit F α} {g : α → Circuit F β} :\n  ((f >>= g).operations n).forAll n prop ↔\n    (f.operations n).forAll n prop ∧ (((g (f.output n)).operations (n + f.localLength n)).forAll (n + f.localLength n)) prop"}, {"name": "forAll_append", "content": "@[circuit_norm]\ntheorem forAll_append {condition : Condition F} {offset : ℕ} {as bs: Operations F} :\n  forAll offset condition (as ++ bs) ↔\n    forAll offset condition as ∧ forAll (as.localLength + offset) condition bs"}, {"name": "forAll_empty", "content": "@[circuit_norm]\ntheorem forAll_empty {condition : Condition F} {n : ℕ} : forAll n condition [] = True"}, {"name": "pure_def", "content": "@[circuit_norm]\ntheorem pure_def {α} (a : α) : (pure a : Circuit F α) = fun _ => (a, [])"}, {"name": "can_replace_usesLocalWitnessesCompleteness", "content": "theorem can_replace_usesLocalWitnessesCompleteness {env : Environment F} {ops : Operations F} {n : ℕ} (h : ops.SubcircuitsConsistent n) :\n  env.UsesLocalWitnesses n ops → env.UsesLocalWitnessesCompleteness n ops"}, {"name": "usesLocalWitnessesFlat_iff_extends", "content": "theorem usesLocalWitnessesFlat_iff_extends {env : Environment F} (n : ℕ) {ops : List (FlatOperation F)}  :\n    env.UsesLocalWitnessesFlat n ops ↔ env.ExtendsVector (localWitnesses env ops) n"}, {"name": "forAll_cons", "content": "theorem forAll_cons {condition : Condition F} {offset : ℕ} {op : FlatOperation F} {ops : List (FlatOperation F)} :\n  forAll offset condition (op :: ops) ↔\n    condition.applyFlat offset op ∧ forAll (op.singleLocalLength + offset) condition ops"}, {"name": "forAll_empty", "content": "theorem forAll_empty {condition : Condition F} {n : ℕ} : forAll n condition [] = True"}, {"name": "env_extends_witness", "content": "lemma env_extends_witness {F} {n : ℕ} {ops : List (FlatOperation F)} {env : Environment F} {m c} :\n    env.ExtendsVector (localWitnesses env (.witness m c :: ops)) n ↔\n      (env.ExtendsVector (c env) n ∧ env.ExtendsVector (localWitnesses env ops) (m + n))"}, {"name": "forAll_cons", "content": "@[circuit_norm]\ntheorem forAll_cons {condition : Condition F} {offset : ℕ} {op : Operation F} {ops : Operations F} :\n  forAll offset condition (op :: ops) ↔\n    condition.apply offset op ∧ forAll (op.localLength + offset) condition ops"}, {"name": "toList_length_one", "content": "theorem toList_length_one {α : Type} (v : Vector α 1) :\n    v.toList = [v[0]]"}, {"name": "one_land_of_IsBool", "content": "theorem one_land_of_IsBool (a : ℕ) (h : IsBool a) : 1 &&& a = a"}, {"name": "land_one_of_IsBool", "content": "theorem land_one_of_IsBool (a : ℕ) (h : IsBool a) : a &&& 1 = a"}, {"name": "val_of_IsBool", "content": "theorem val_of_IsBool {p : ℕ} [Fact p.Prime] {x : F p} (h : IsBool x) : IsBool x.val"}, {"name": "one", "content": "@[circuit_norm]\ntheorem one {α : Type*} [Zero α] [One α] : IsBool (1 : α)"}, {"name": "zero", "content": "@[circuit_norm]\ntheorem zero {α : Type*} [Zero α] [One α] : IsBool (0 : α)"}, {"name": "eval_fieldPair", "content": "@[circuit_norm ↓]\ntheorem eval_fieldPair {F : Type} [Field F] (env : Environment F) (t : Var fieldPair F) :\n  ProvableType.eval env t = (match t with | (x, y) => (Expression.eval env x, Expression.eval env y))"}, {"name": "toList_length_two", "content": "theorem toList_length_two {α : Type} (v : Vector α 2) :\n    v.toList = [v[0], v[1]]"}, {"name": "List.foldl_and_IsBool", "content": "theorem List.foldl_and_IsBool (l : List ℕ) :\n    IsBool (List.foldl (· &&& ·) 1 l : ℕ)"}, {"name": "List.and_foldl_eq_foldl", "content": "theorem List.and_foldl_eq_foldl (a : ℕ) (orig : ℕ) (l : List ℕ) :\n    a &&& List.foldl (· &&& ·) orig l = List.foldl (· &&& ·) (a &&& orig) l"}, {"name": "eval_fields", "content": "@[circuit_norm ↓]\ntheorem eval_fields {F : Type} [Field F] (env : Environment F) (x : Var (fields n) F) :\n  ProvableType.eval env x = x.map (Expression.eval env)"}, {"name": "append_take_drop", "content": "theorem append_take_drop {v : Vector α (n + m)} :\n    (v.take n |>.cast Nat.min_add_right_self) ++ (v.drop n |>.cast (Nat.add_sub_self_left n m)) = v"}, {"name": "ext", "content": "@[ext]\ntheorem ext {f g : Circuit F α}\n  (h_output : ∀ n, f.output n = g.output n)\n  (h_operations : ∀ n, f.operations n = g.operations n) :\n    f = g"}, {"name": "ext_iff", "content": "theorem ext_iff {f g : Circuit F α} :\n  (f = g) ↔ (∀ n, (f.output n = g.output n) ∧ (f.operations n = g.operations n))"}, {"name": "ConstraintsHold.bind_soundness", "content": "@[circuit_norm] theorem ConstraintsHold.bind_soundness {f : Circuit F α} {g : α → Circuit F β} (n : ℕ) :\n  ConstraintsHold.Soundness env ((f >>= g).operations n)\n  ↔ ConstraintsHold.Soundness env (f.operations n) ∧\n    ConstraintsHold.Soundness env ((g (f.output n)).operations (n + f.localLength n))"}, {"name": "ConstraintsHold.soundness_iff_forAll", "content": "theorem ConstraintsHold.soundness_iff_forAll (n : ℕ) (env : Environment F) (ops : Operations F) :\n  ConstraintsHold.Soundness env ops ↔ ops.forAll n {\n    assert _ e := env e = 0,\n    lookup _ l := l.table.Soundness (l.entry.map env),\n    subcircuit _ _ s := s.Soundness env\n  }"}, {"name": "can_replace_soundness", "content": "theorem can_replace_soundness {ops : Operations F} {env} :\n  ConstraintsHold env ops → ConstraintsHold.Soundness env ops"}, {"name": "can_replace_completeness", "content": "theorem can_replace_completeness {env} {ops : Operations F} {n : ℕ} (h : ops.SubcircuitsConsistent n) : env.UsesLocalWitnesses n ops →\n    ConstraintsHold.Completeness env ops → ConstraintsHold env ops"}], "used_local_defs": [{"name": "Circomlib.AND.main", "content": "def main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out"}, {"name": "Circomlib.MultiAND.main", "content": "def main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)"}, {"name": "Circomlib.MultiAND.Assumptions", "content": "def Assumptions (n : ℕ) (input : fields n (F p)) : Prop :=\n  ∀ (i : ℕ) (h : i < n), IsBool input[i]"}, {"name": "Circomlib.MultiAND.Spec", "content": "def Spec (n : ℕ) (input : fields n (F p)) (output : F p) : Prop :=\n  output.val = (input.map (·.val)).foldl (· &&& ·) 1 ∧ IsBool output"}], "used_local_lemmas": [{"name": "Circomlib.MultiAND.Circuit.subcircuitsConsistent_bind", "content": "theorem Circuit.subcircuitsConsistent_bind {α β : Type} (f : Circuit (F p) α) (g : α → Circuit (F p) β) (offset : ℕ)\n    (hf : Operations.SubcircuitsConsistent offset (f.operations offset))\n    (hg : Operations.SubcircuitsConsistent (offset + f.localLength offset)\n          ((g (f.output offset)).operations (offset + f.localLength offset))) :\n    Operations.SubcircuitsConsistent offset ((f >>= g).operations offset)"}, {"name": "Circomlib.MultiAND.subcircuitsConsistent", "content": "theorem subcircuitsConsistent (n : ℕ) (input : Var (fields n) (F p)) (offset : ℕ) :\n    Operations.SubcircuitsConsistent offset ((main input).operations offset)"}, {"name": "Circomlib.MultiAND.main_usesLocalWitnesses_iff_completeness", "content": "lemma main_usesLocalWitnesses_iff_completeness (n : ℕ) (input : Var (fields n) (F p)) (offset1 offset2 : ℕ) (env : Environment (F p)) :\n    offset1 = offset2 ->\n    (env.UsesLocalWitnesses offset1 ((main input).operations offset2) ↔\n     env.UsesLocalWitnessesCompleteness offset1 ((main input).operations offset2))"}, {"name": "Circomlib.MultiAND.Vector.foldl_empty'", "content": "lemma Vector.foldl_empty' {α β : Type} (init : β) (f : β → α → β) (v : Vector α 0) :\n    Vector.foldl f init v = init"}, {"name": "Circomlib.MultiAND.Vector.foldl_and_split", "content": "lemma Vector.foldl_and_split {n1 n2 n3 : ℕ} (v : Vector ℕ n3)\n    (v1 : Vector ℕ n1) (v2 : Vector ℕ n2) (h_sum : n1 + n2 = n3)\n    (h_split : v = h_sum ▸ (v1 ++ v2)) :\n    Vector.foldl (· &&& ·) 1 v =\n    Vector.foldl (· &&& ·) 1 v1 &&& Vector.foldl (· &&& ·) 1 v2"}, {"name": "Circomlib.MultiAND.soundness_zero", "content": "lemma soundness_zero {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 0) (F p))\n    (input : fields 0 (F p)) (_h_env : input = eval env input_var)\n    (_h_assumptions : Assumptions 0 input)\n    (_h_hold : Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset)) :\n    Spec 0 input (env ((main input_var).output offset))"}, {"name": "Circomlib.MultiAND.soundness_one", "content": "lemma soundness_one {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 1) (F p))\n    (input : fields 1 (F p)) (h_env : input = eval env input_var)\n    (h_assumptions : Assumptions 1 input)\n    (_h_hold : Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset)) :\n    Spec 1 input (env ((main input_var).output offset))"}, {"name": "Circomlib.MultiAND.soundness_two", "content": "lemma soundness_two {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 2) (F p))\n    (input : fields 2 (F p)) (h_env : input = eval env input_var)\n    (h_assumptions : Assumptions 2 input)\n    (h_hold : Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset)) :\n    Spec 2 input (env ((main input_var).output offset))"}, {"name": "Circomlib.MultiAND.soundness", "content": "theorem soundness {p : ℕ} [Fact p.Prime] (n : ℕ) :\n    ∀ (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields n) (F p))\n      (input : fields n (F p)),\n    input = eval env input_var →\n    Assumptions n input →\n    Circuit.ConstraintsHold.Soundness env ((main input_var).operations offset) →\n    Spec n input (env ((main input_var).output offset))"}, {"name": "Circomlib.MultiAND.main_output_binary", "content": "lemma main_output_binary (n : ℕ) (offset : ℕ) (env : Environment (F p))\n    (input_var : Var (fields n) (F p)) (input : fields n (F p))\n    (h_eval : input = eval env input_var)\n    (h_assumptions : Assumptions n input)\n    (h_constraints : Circuit.ConstraintsHold env ((main input_var).operations offset)) :\n    let output := env ((main input_var).output offset)\n    IsBool output"}], "local_ctx": "import Clean.Circuit\n\nimport Clean.Utils.Field\n\nimport Clean.Gadgets.Boolean\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Utils.Vector\n\nimport Clean.Utils.BinaryOps\n\nimport Clean.Circuit.Theorems\n\nimport Mathlib.Data.Nat.Bitwise\n\nopen IsBool\n\nnamespace Circomlib\n\nvariable {p : ℕ} [Fact p.Prime]\n\nopen Circuit (bind_output_eq bind_localLength_eq bind_forAll)\n\nopen Operations (append_localLength)\n\nopen BinaryOps (List.foldl_and_IsBool List.and_foldl_eq_foldl)\n\nnamespace XOR\n\nend XOR\n\nnamespace AND\n\ndef main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out\n\nend AND\n\nnamespace OR\n\nend OR\n\nnamespace NOT\n\nend NOT\n\nnamespace NAND\n\nend NAND\n\nnamespace NOR\n\nend NOR\n\nnamespace MultiAND\n\ndef main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)\n\ndef Assumptions (n : ℕ) (input : fields n (F p)) : Prop :=\n  ∀ (i : ℕ) (h : i < n), IsBool input[i]\n\ndef Spec (n : ℕ) (input : fields n (F p)) (output : F p) : Prop :=\n  output.val = (input.map (·.val)).foldl (· &&& ·) 1 ∧ IsBool output", "target_theorem": "lemma main_output_binary_from_completeness (n : ℕ) (offset : ℕ) (env : Environment (F p))\n    (input_var : Var (fields n) (F p)) (input : fields n (F p))\n    (h_eval : input = eval env input_var)\n    (h_assumptions : Assumptions n input)\n    (h_local_witnesses : env.UsesLocalWitnessesCompleteness offset ((main input_var).operations offset))\n    (h_completeness : Circuit.ConstraintsHold.Completeness env ((main input_var).operations offset)) :\n    let output :=", "ground_truth_proof": ":= env ((main input_var).output offset)\n    IsBool output := by\n  apply main_output_binary\n  · assumption\n  · assumption\n  apply Circuit.can_replace_completeness (n := offset)\n  · apply subcircuitsConsistent\n  · rw [main_usesLocalWitnesses_iff_completeness]\n    · exact h_local_witnesses\n    · rfl\n  · exact h_completeness", "nesting_depth": 9, "transitive_dep_count": 165, "subset_aristotle": true, "category": "Applied verif."}
{"id": 172, "thm_name": "Circuit.MapM.operations_eq", "thm_stmt": "theorem operations_eq : (xs.mapM circuit).operations n =\n    (List.ofFn fun (i : Fin m) => (circuit xs[i.val]).operations (n + i * constant.localLength)).flatten", "lean_root": "clean", "rel_path": "Clean/Circuit/Loops.lean", "imports": ["import Clean.Utils.Misc", "import Clean.Utils.Vector", "import Clean.Circuit.Subcircuit", "import Clean.Circuit.Theorems"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector.mk", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "List.ofFn", "module": "Init.Data.List.OfFn"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "id", "module": "Init.Prelude"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "Vector.toList", "module": "Init.Data.Vector.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "Array.mapM", "module": "Init.Data.Array.Basic"}, {"name": "Array.toList", "module": "Init.Prelude"}, {"name": "Functor", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "induct", "content": "def induct {motive : {n : ℕ} → Vector α n → Sort u}\n    (nil : motive #v[])\n    (cons: ∀ {n : ℕ} (a : α) (as : Vector α n), motive as → motive (cons a as))\n    {n : ℕ} (v : Vector α n) : motive v :=\n  match v with\n  | ⟨ .mk [], h ⟩ => by admit /- proof elided -/\n  | ⟨ .mk (a :: as), h ⟩ => by admit /- proof elided -/"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "cons", "content": "def cons (a : α) (v : Vector α n) : Vector α (n + 1) :=\n  ⟨ .mk (a :: v.toList), by admit /- proof elided -/\n  ⟩"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}], "lib_lemmas": [{"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Array.toList_mapM", "module": "Init.Data.Array.Lemmas"}, {"name": "Functor.map_map", "module": "Init.Control.Lawful.Basic"}, {"name": "Vector.toArray_mapM", "module": "Init.Data.Vector.Lemmas"}, {"name": "Vector.toList_inj", "module": "Init.Data.Vector.Lemmas"}, {"name": "List.mapM_cons", "module": "Init.Data.List.Monadic"}, {"name": "map_bind", "module": "Init.Control.Lawful.Basic"}, {"name": "List.append_nil", "module": "Init.Data.List.Basic"}], "repo_lemmas": [{"name": "map_operations_eq", "content": "theorem map_operations_eq (f : Circuit F α) (g : α → β) (n : ℕ) :\n  (g <$> f).operations n = f.operations n"}, {"name": "ext_iff", "content": "theorem ext_iff {f g : Circuit F α} :\n  (f = g) ↔ (∀ n, (f.output n = g.output n) ∧ (f.operations n = g.operations n))"}, {"name": "map_output_eq", "content": "theorem map_output_eq (f : Circuit F α) (g : α → β) (n : ℕ) :\n  (g <$> f).output n = g (f.output n)"}, {"name": "toList_cons", "content": "theorem toList_cons {a : α} {v : Vector α n} : (cons a v).toList = a :: v.toList"}, {"name": "bind_operations_eq", "content": "theorem bind_operations_eq (f : Circuit F α) (g : α → Circuit F β) (n : ℕ) :\n  (f >>= g).operations n = f.operations n ++ (g (f.output n)).operations (n + f.localLength n)"}, {"name": "pure_operations_eq", "content": "theorem pure_operations_eq (a : α) (n : ℕ) :\n  (pure a : Circuit F α).operations n = []"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Vector.toList_mapM", "content": "lemma Vector.toList_mapM (xs : Vector α n) {m : Type → Type} [monad: Monad m] [LawfulMonad m] (body : α → m β) :\n    Vector.toList <$> (xs.mapM body) = xs.toList.mapM body"}, {"name": "Circuit.ofFn_flatten_cons", "content": "private lemma ofFn_flatten_cons {circuit : α → Circuit F β} (constant : ConstantLength circuit) (x : α) (xs : Vector α m) (n : ℕ) :\n  (List.ofFn fun i => (circuit (Vector.cons x xs)[i.val]).operations (n + i * constant.localLength)).flatten\n    = (circuit x).operations n ++ (List.ofFn fun i => (circuit xs[i.val]).operations (n + constant.localLength + i * constant.localLength)).flatten"}, {"name": "Circuit.MapM.ext_map_toList", "content": "lemma ext_map_toList (f g : Circuit F (Vector α n)) :\n    (fun v => v.toList) <$> f = (fun v => v.toList) <$> g → f = g"}, {"name": "Circuit.MapM.mapM_cons", "content": "lemma mapM_cons (xs : Vector α n) (body : α → Circuit F β) (x : α) :\n  (Vector.cons x xs).mapM body = do\n    let y ← body x\n    let ys ← xs.mapM body\n    return Vector.cons y ys"}], "local_ctx": "import Clean.Circuit.Subcircuit\n\nimport Clean.Utils.Misc\n\nvariable {n m : ℕ} {F : Type} [Field F] {α β : Type}\n\nnamespace Circuit\n\nvariable {prop : Condition F}\n\nnamespace ForM\n\nvariable {circuit : α → Circuit F Unit} (xs : Vector α m) (constant : ConstantLength circuit) (n : ℕ)\n\nend ForM\n\nnamespace MapM\n\nvariable {circuit : α → Circuit F β} {xs : Vector α m} [constant: ConstantLength circuit]\n  {prop : Condition F}", "target_theorem": "theorem operations_eq : (xs.mapM circuit).operations n =\n    (List.ofFn fun (i : Fin m) => (circuit xs[i.val]).operations (n + i * constant.localLength)).flatten :=", "ground_truth_proof": ":= by\n  induction xs using Vector.induct generalizing n\n  case nil => simp [pure_operations_eq]\n  case cons x xs ih =>\n    rw [mapM_cons, bind_operations_eq, bind_operations_eq, pure_operations_eq, ih,\n      constant.localLength_eq, List.append_nil, ofFn_flatten_cons]", "nesting_depth": 6, "transitive_dep_count": 70, "subset_aristotle": false, "category": "Applied verif."}
{"id": 173, "thm_name": "Utils.StateTransition.exists_smaller_run_with_same_netFlow", "thm_stmt": "lemma exists_smaller_run_with_same_netFlow (R : Run S) (h_cycle : R.hasCycle) :\n    ∃ (R' : Run S), (∀ x, R'.netFlow x = R.netFlow x) ∧ R'.size < R.size ∧ (∀ t, R' t ≤ R t)", "lean_root": "clean", "rel_path": "Clean/Utils/SourceSinkPath.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import Mathlib.Data.Fintype.Prod", "import Mathlib.Data.List.Basic", "import Mathlib.Algebra.BigOperators.Group.Finset.Basic", "import Mathlib.Data.Finset.Basic", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise", "import Mathlib.Data.Fintype.Basic"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "List.tail", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}, {"name": "Finset.erase", "module": "Mathlib.Data.Finset.Erase"}, {"name": "Nat.cast", "module": "Init.Data.Cast"}, {"name": "List.count", "module": "Init.Data.List.Basic"}, {"name": "List.countP", "module": "Init.Data.List.Basic"}, {"name": "List.getLast?", "module": "Init.Data.List.Basic"}, {"name": "List.head?", "module": "Init.Data.List.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_erase_add", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_le_sum", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "List.count_pos_iff", "module": "Init.Data.List.Count"}, {"name": "Finset.mem_toList", "module": "Mathlib.Data.Finset.Dedup"}, {"name": "Nat.sub_le", "module": "Init.Prelude"}, {"name": "Nat.sub_lt", "module": "Init.Prelude"}, {"name": "Finset.sum_add_distrib", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "List.countP_cons", "module": "Init.Data.List.Count"}, {"name": "List.count_cons", "module": "Init.Data.List.Count"}, {"name": "Nat.cast_sum", "module": "Mathlib.Algebra.BigOperators.Ring.Finset"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "List.zipWith_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "decide_eq_true_eq", "module": "Init.SimpLemmas"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "List.getLast_cons", "module": "Init.Data.List.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Utils.StateTransition.Transition", "content": "def Transition (S : Type*) := S × S"}, {"name": "Utils.StateTransition.Run", "content": "def Run (S : Type*) := Transition S → ℕ"}, {"name": "Utils.StateTransition.Run.netFlow", "content": "noncomputable def Run.netFlow {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) (x : S) : ℤ :=\n  (∑ y : S, (R (x, y) : ℤ)) - (∑ y : S, (R (y, x) : ℤ))"}, {"name": "Utils.StateTransition.Run.size", "content": "noncomputable def Run.size {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) : ℕ :=\n  ∑ t : Transition S, R t"}, {"name": "Utils.StateTransition.countTransitionInPath", "content": "def countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t"}, {"name": "Utils.StateTransition.Run.containsPath", "content": "def Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t"}, {"name": "Utils.StateTransition.Run.hasCycle", "content": "def Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle"}, {"name": "Utils.StateTransition.Run.removeCycle", "content": "def Run.removeCycle (R : Run S) (cycle : List S) : Run S :=\n  fun t => R t - countTransitionInPath t cycle"}, {"name": "Utils.StateTransition.countAsFirst", "content": "def countAsFirst [DecidableEq S] (xs : List S) (x : S) : ℕ :=\n  (xs.zip xs.tail).countP (fun p => p.1 = x)"}, {"name": "Utils.StateTransition.countAsSecond", "content": "def countAsSecond [DecidableEq S] (xs : List S) (x : S) : ℕ :=\n  (xs.zip xs.tail).countP (fun p => p.2 = x)"}], "used_local_lemmas": [{"name": "Utils.StateTransition.sum_decrease", "content": "lemma sum_decrease {α : Type*} [Fintype α] [DecidableEq α] (f g : α → ℕ) (a : α)\n    (h_a_decrease : g a < f a)\n    (h_others_le : ∀ x, g x ≤ f x) :\n    ∑ x : α, g x < ∑ x : α, f x"}, {"name": "Utils.StateTransition.path_has_transition", "content": "lemma path_has_transition {S : Type*} [DecidableEq S] (path : List S)\n    (h_len : path.length ≥ 2) :\n    ∃ (t : Transition S), t ∈ path.zip path.tail"}, {"name": "Utils.StateTransition.containsPath_has_positive_transition", "content": "lemma containsPath_has_positive_transition (R : Run S) (path : List S)\n    (h_contains : R.containsPath path) (t : Transition S)\n    (h_in : t ∈ path.zip path.tail) :\n    R t > 0"}, {"name": "Utils.StateTransition.countAsFirst_cons", "content": "lemma countAsFirst_cons (hd : S) (tl : List S) (x : S) :\n    countAsFirst (hd :: tl) x = (if hd = x ∧ tl ≠ [] then 1 else 0) + countAsFirst tl x"}, {"name": "Utils.StateTransition.countAsSecond_cons", "content": "lemma countAsSecond_cons (hd : S) (tl : List S) (x : S) :\n    countAsSecond (hd :: tl) x = (if tl.head? = some x then 1 else 0) + countAsSecond tl x"}, {"name": "Utils.StateTransition.countAsFirst_add_last_eq_countAsSecond_add_head", "content": "lemma countAsFirst_add_last_eq_countAsSecond_add_head (xs : List S) (x : S) :\n    countAsFirst xs x + (if xs.getLast? = some x then 1 else 0) =\n    countAsSecond xs x + (if xs.head? = some x then 1 else 0)"}, {"name": "Utils.StateTransition.cycle_balanced_at_node", "content": "lemma cycle_balanced_at_node (cycle : List S) (x : S)\n    (h_cycle : cycle.head? = cycle.getLast?) :\n    (cycle.zip cycle.tail).countP (fun p => p.1 = x) =\n    (cycle.zip cycle.tail).countP (fun p => p.2 = x)"}, {"name": "Utils.StateTransition.sum_count_pairs_fst", "content": "lemma sum_count_pairs_fst (xs : List (S × S)) (a : S) :\n    ∑ b : S, List.count (a, b) xs = List.countP (fun p => p.1 = a) xs"}, {"name": "Utils.StateTransition.sum_count_pairs_snd", "content": "lemma sum_count_pairs_snd (xs : List (S × S)) (b : S) :\n    ∑ a : S, List.count (a, b) xs = List.countP (fun p => p.2 = b) xs"}, {"name": "Utils.StateTransition.sum_countTransitionInPath_fst", "content": "lemma sum_countTransitionInPath_fst (cycle : List S) (x : S) :\n    ∑ y : S, (countTransitionInPath (x, y) cycle : ℤ) = (countAsFirst cycle x : ℤ)"}, {"name": "Utils.StateTransition.sum_countTransitionInPath_snd", "content": "lemma sum_countTransitionInPath_snd (cycle : List S) (x : S) :\n    ∑ y : S, (countTransitionInPath (y, x) cycle : ℤ) = (countAsSecond cycle x : ℤ)"}, {"name": "Utils.StateTransition.netFlow_sub", "content": "lemma netFlow_sub (R R' : Run S) (x : S)\n    (h_valid : ∀ t, R' t ≤ R t) :\n    Run.netFlow (fun t => R t - R' t) x = R.netFlow x - R'.netFlow x"}, {"name": "Utils.StateTransition.cycle_netFlow_zero", "content": "lemma cycle_netFlow_zero (cycle : List S) (x : S)\n    (h_cycle : cycle.head? = cycle.getLast?) :\n    Run.netFlow (fun t => countTransitionInPath t cycle) x = 0"}, {"name": "Utils.StateTransition.netFlow_removeCycle_eq", "content": "lemma netFlow_removeCycle_eq (R : Run S) (cycle : List S) (x : S)\n    (h_contains : R.containsPath cycle)\n    (h_cycle : cycle.head? = cycle.getLast?) :\n    (R.removeCycle cycle).netFlow x = R.netFlow x"}, {"name": "Utils.StateTransition.size_removeCycle_lt", "content": "lemma size_removeCycle_lt (R : Run S) (cycle : List S)\n    (h_len : cycle.length ≥ 2)\n    (h_contains : R.containsPath cycle)\n    (_h_cycle : cycle.head? = cycle.getLast?) :\n    (R.removeCycle cycle).size < R.size"}, {"name": "Utils.StateTransition.removeCycle_le", "content": "lemma removeCycle_le (R : Run S) (cycle : List S) (t : Transition S) :\n    (R.removeCycle cycle) t ≤ R t"}], "local_ctx": "import Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Fintype.Basic\n\nimport Mathlib.Data.Fintype.Prod\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Piecewise\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nnamespace Utils.StateTransition\n\nvariable {S : Type*} [DecidableEq S] [Fintype S]\n\ndef Transition (S : Type*) := S × S\n\ndef Run (S : Type*) := Transition S → ℕ\n\nnoncomputable def Run.netFlow {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) (x : S) : ℤ :=\n  (∑ y : S, (R (x, y) : ℤ)) - (∑ y : S, (R (y, x) : ℤ))\n\nnoncomputable def Run.size {S : Type*} [Fintype S] [DecidableEq S] (R : Run S) : ℕ :=\n  ∑ t : Transition S, R t\n\ndef countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t\n\ndef Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t\n\ndef Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle\n\ndef Run.removeCycle (R : Run S) (cycle : List S) : Run S :=\n  fun t => R t - countTransitionInPath t cycle\n\ndef countAsFirst [DecidableEq S] (xs : List S) (x : S) : ℕ :=\n  (xs.zip xs.tail).countP (fun p => p.1 = x)\n\ndef countAsSecond [DecidableEq S] (xs : List S) (x : S) : ℕ :=\n  (xs.zip xs.tail).countP (fun p => p.2 = x)", "target_theorem": "lemma exists_smaller_run_with_same_netFlow (R : Run S) (h_cycle : R.hasCycle) :\n    ∃ (R' : Run S), (∀ x, R'.netFlow x = R.netFlow x) ∧ R'.size < R.size ∧ (∀ t, R' t ≤ R t) :=", "ground_truth_proof": ":= by\n  -- Extract the cycle from the hypothesis\n  obtain ⟨cycle, h_len, h_cycle_prop, h_contains⟩ := h_cycle\n  -- Use R.removeCycle as the witness\n  use R.removeCycle cycle\n  constructor\n  · -- Net flow is preserved\n    intro x\n    apply netFlow_removeCycle_eq <;> aesop\n  constructor\n  · -- Size decreases\n    apply size_removeCycle_lt <;> aesop\n  · -- Each transition capacity is ≤\n    intro t\n    exact removeCycle_le R cycle t", "nesting_depth": 6, "transitive_dep_count": 62, "subset_aristotle": false, "category": "Applied verif."}
{"id": 174, "thm_name": "Utils.StateTransition.acyclic_has_leaf", "thm_stmt": "lemma acyclic_has_leaf (R : Run S) (root : S)\n    (h_acyclic : R.isAcyclic)\n    (h_has_out : ∃ y, R (root, y) > 0) :\n    ∃ leaf, R.isLeaf root leaf", "lean_root": "clean", "rel_path": "Clean/Utils/SourceSinkPath.lean", "imports": ["import Mathlib.Algebra.Order.BigOperators.Group.Finset", "import Mathlib.Data.Fintype.Prod", "import Mathlib.Data.List.Basic", "import Mathlib.Algebra.BigOperators.Group.Finset.Basic", "import Mathlib.Data.Finset.Basic", "import Mathlib.Algebra.BigOperators.Ring.Finset", "import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise", "import Mathlib.Data.Fintype.Basic"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List.count", "module": "Init.Data.List.Basic"}, {"name": "List.tail", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "List.Sublist", "module": "Init.Data.List.Basic"}, {"name": "List.drop", "module": "Init.Data.List.Basic"}, {"name": "List.take", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.cons₂", "module": "Init.Data.List.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Nat.reduceLeDiff", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Nat"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.countP_cons_of_pos", "module": "Init.Data.List.Count"}, {"name": "List.countP_nil", "module": "Init.Data.List.Count"}, {"name": "List.count_cons", "module": "Init.Data.List.Count"}, {"name": "List.count_nil", "module": "Init.Data.List.Count"}, {"name": "List.zipWith_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zipWith_nil_right", "module": "Init.Data.List.Basic"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.exists_cons_of_ne_nil", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast?_cons_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.tail_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zip_cons_cons", "module": "Init.Data.List.Basic"}, {"name": "List.countP_singleton", "module": "Init.Data.List.Count"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "List.zip_nil_right", "module": "Init.Data.List.Basic"}, {"name": "Nat.add_right_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "List.mem_of_mem_tail", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "List.drop_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.Sublist.count_le", "module": "Init.Data.List.Count"}, {"name": "List.tail_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.take_succ_cons", "module": "Init.Data.List.Basic"}, {"name": "List.length_drop", "module": "Init.Data.List.TakeDrop"}, {"name": "List.length_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "Nat.min_eq_left", "module": "Init.Data.Nat.MinMax"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "List.getElem?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getElem?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.getLast?_eq_getElem?", "module": "Init.Data.List.Lemmas"}, {"name": "List.head?_drop", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.head?_take", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "List.duplicate_iff_exists_distinct_get", "module": "Mathlib.Data.List.NodupEquivFin"}, {"name": "List.exists_duplicate_iff_not_nodup", "module": "Mathlib.Data.List.Duplicate"}, {"name": "List.getLast?_eq_getLast", "module": "Init.Data.List.Lemmas"}, {"name": "List.getLast_mem", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.nodup_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.mem_iff_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.ssubset_univ_iff", "module": "Mathlib.Data.Finset.BooleanAlgebra"}, {"name": "Finset.card_insert_of_notMem", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.card_lt_card", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.card_univ", "module": "Mathlib.Data.Fintype.Card"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Utils.StateTransition.Transition", "content": "def Transition (S : Type*) := S × S"}, {"name": "Utils.StateTransition.Run", "content": "def Run (S : Type*) := Transition S → ℕ"}, {"name": "Utils.StateTransition.countTransitionInPath", "content": "def countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t"}, {"name": "Utils.StateTransition.Run.containsPath", "content": "def Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t"}, {"name": "Utils.StateTransition.Run.hasCycle", "content": "def Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle"}, {"name": "Utils.StateTransition.Run.isAcyclic", "content": "def Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle"}, {"name": "Utils.StateTransition.Run.reachable", "content": "def Run.reachable [DecidableEq S] (R : Run S) (start finish : S) : Prop :=\n  ∃ (path : List S), path.head? = some start ∧ path.getLast? = some finish ∧\n    path ≠ [] ∧ R.containsPath path"}, {"name": "Utils.StateTransition.Run.isLeaf", "content": "def Run.isLeaf (R : Run S) (root leaf : S) : Prop :=\n  R.reachable root leaf ∧ ∀ y, R (leaf, y) = 0"}], "used_local_lemmas": [{"name": "Utils.StateTransition.finset_ssubset_univ_of_not_mem", "content": "lemma finset_ssubset_univ_of_not_mem {α : Type*} [Fintype α] (s : Finset α) (x : α)\n    (h : x ∉ s) :\n    s ⊂ Finset.univ"}, {"name": "Utils.StateTransition.zip_drop_sublist", "content": "lemma zip_drop_sublist (l : List S) (n : ℕ) :\n    ((l.drop n).zip (l.drop (n + 1))).Sublist (l.zip l.tail)"}, {"name": "Utils.StateTransition.containsPath_drop", "content": "lemma containsPath_drop (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.drop n)"}, {"name": "Utils.StateTransition.zip_take_sublist", "content": "lemma zip_take_sublist (l1 l2 : List S) (n m : ℕ) :\n    ((l1.take n).zip (l2.take m)).Sublist (l1.zip l2)"}, {"name": "Utils.StateTransition.tail_take", "content": "lemma tail_take {α : Type*} (l : List α) (n : ℕ) :\n    (l.take n).tail = (l.tail).take (n - 1)"}, {"name": "Utils.StateTransition.containsPath_take", "content": "lemma containsPath_take (R : Run S) (path : List S) (n : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath (path.take n)"}, {"name": "Utils.StateTransition.drop_take_length_ge_two", "content": "lemma drop_take_length_ge_two {α : Type*} (path : List α) (n m : Fin path.length)\n    (h_n_lt_m : n < m) :\n    ((path.drop n.val).take (m.val - n.val + 1)).length ≥ 2"}, {"name": "Utils.StateTransition.getLast_drop_take", "content": "lemma getLast_drop_take {α : Type*} (path : List α) (n k : ℕ)\n    (h_n_lt : n < path.length)\n    (h_bound : n + k ≤ path.length)\n    (h_k_pos : k > 0) :\n    ((path.drop n).take k).getLast? = path[n + k - 1]?"}, {"name": "Utils.StateTransition.drop_take_cycle_same_endpoints", "content": "lemma drop_take_cycle_same_endpoints (path : List S) (x : S) (n m : Fin path.length)\n    (h_n_lt_m : n < m)\n    (h_x_at_n : path[n] = x)\n    (h_x_at_m : path[m] = x) :\n    ((path.drop n.val).take (m.val - n.val + 1)).head? =\n    ((path.drop n.val).take (m.val - n.val + 1)).getLast?"}, {"name": "Utils.StateTransition.containsPath_drop_take", "content": "lemma containsPath_drop_take (R : Run S) (path : List S) (n m : ℕ)\n    (h_contains : R.containsPath path) :\n    R.containsPath ((path.drop n).take m)"}, {"name": "Utils.StateTransition.acyclic_containsPath_nodup", "content": "lemma acyclic_containsPath_nodup (R : Run S) (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_contains : R.containsPath path) :\n    path.Nodup"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton", "content": "lemma countTransitionInPath_append_singleton (path : List S) (x y : S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_not_in : (x, y) ∉ path.zip path.tail) :\n    countTransitionInPath (x, y) (path ++ [y]) = 1"}, {"name": "Utils.StateTransition.countTransitionInPath_append_singleton_other", "content": "lemma countTransitionInPath_append_singleton_other (path : List S) (x y : S) (t : Transition S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_ne : t ≠ (x, y)) :\n    countTransitionInPath t (path ++ [y]) = countTransitionInPath t path"}, {"name": "Utils.StateTransition.acyclic_no_self_loop", "content": "lemma acyclic_no_self_loop (R : Run S) (s : S) (h_acyclic : R.isAcyclic) (h_edge : R (s, s) > 0) : False"}, {"name": "Utils.StateTransition.getLast_mem", "content": "lemma getLast_mem {α : Type*} (l : List α) (x : α) (h_last : l.getLast? = some x) :\n    x ∈ l"}, {"name": "Utils.StateTransition.last_not_in_zip_tail", "content": "lemma last_not_in_zip_tail {α : Type*} [DecidableEq α] (l : List α) (x : α)\n    (h_nodup : l.Nodup)\n    (h_last : l.getLast? = some x) :\n    ∀ y : α, (x, y) ∉ l.zip l.tail"}, {"name": "Utils.StateTransition.drop_of_lt_length_nonempty", "content": "lemma drop_of_lt_length_nonempty {α : Type*} (path : List α) (i : ℕ)\n    (h_i_lt : i < path.length) :\n    path.drop i ≠ []"}, {"name": "Utils.StateTransition.cycle_from_suffix_contains", "content": "lemma cycle_from_suffix_contains (R : Run S) (suffix : List S) (current y : S)\n    (h_suffix_nodup : suffix.Nodup)\n    (h_contains_suffix : R.containsPath suffix)\n    (h_suffix_nonempty : suffix ≠ [])\n    (h_suffix_last : suffix.getLast? = some current)\n    (h_edge : R (current, y) > 0) :\n    ∀ t : Transition S, countTransitionInPath t (suffix ++ [y]) ≤ R t"}, {"name": "Utils.StateTransition.path_with_back_edge_creates_cycle", "content": "lemma path_with_back_edge_creates_cycle (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_y_in_path : y ∈ path)\n    (h_edge : R (current, y) > 0) :\n    R.hasCycle"}, {"name": "Utils.StateTransition.acyclic_edge_not_in_path", "content": "lemma acyclic_edge_not_in_path (R : Run S) (path : List S) (current y : S)\n    (h_acyclic : R.isAcyclic)\n    (h_end : path.getLast? = some current)\n    (h_contains : R.containsPath path)\n    (h_edge : R (current, y) > 0)\n    (h_y_in_path : y ∈ path) :\n    False"}, {"name": "Utils.StateTransition.not_mem_implies_transition_not_in_zip_tail", "content": "lemma not_mem_implies_transition_not_in_zip_tail {α : Type*} (path : List α) (x y : α)\n    (h_y_not_in : y ∉ path) :\n    (x, y) ∉ path.zip path.tail"}, {"name": "Utils.StateTransition.containsPath_append_singleton", "content": "lemma containsPath_append_singleton (R : Run S) (path : List S) (x y : S)\n    (h_nonempty : path ≠ [])\n    (h_last : path.getLast? = some x)\n    (h_contains : R.containsPath path)\n    (h_y_not_in_path : y ∉ path)\n    (h_edge : R (x, y) > 0) :\n    R.containsPath (path ++ [y])"}, {"name": "Utils.StateTransition.acyclic_has_leaf_aux", "content": "lemma acyclic_has_leaf_aux (R : Run S) (root current : S)\n    (path : List S)\n    (h_acyclic : R.isAcyclic)\n    (h_start : path.head? = some root)\n    (h_end : path.getLast? = some current)\n    (h_nonempty : path ≠ [])\n    (h_contains : R.containsPath path)\n    (h_has_out : ∃ y, y ∉ path ∧ R (current, y) > 0) :\n    ∃ leaf, R.isLeaf root leaf"}], "local_ctx": "import Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Fintype.Basic\n\nimport Mathlib.Data.Fintype.Prod\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Basic\n\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Group.Finset.Piecewise\n\nimport Mathlib.Algebra.BigOperators.Ring.Finset\n\nnamespace Utils.StateTransition\n\nvariable {S : Type*} [DecidableEq S] [Fintype S]\n\ndef Transition (S : Type*) := S × S\n\ndef Run (S : Type*) := Transition S → ℕ\n\ndef countTransitionInPath [DecidableEq S] (t : Transition S) (path : List S) : ℕ :=\n  (path.zip path.tail).count t\n\ndef Run.containsPath [DecidableEq S] (R : Run S) (path : List S) : Prop :=\n  ∀ t : Transition S, countTransitionInPath t path ≤ R t\n\ndef Run.hasCycle [DecidableEq S] (R : Run S) : Prop :=\n  ∃ (cycle : List S), cycle.length ≥ 2 ∧\n    cycle.head? = cycle.getLast? ∧\n    R.containsPath cycle\n\ndef Run.isAcyclic [DecidableEq S] (R : Run S) : Prop :=\n  ¬R.hasCycle\n\ndef Run.reachable [DecidableEq S] (R : Run S) (start finish : S) : Prop :=\n  ∃ (path : List S), path.head? = some start ∧ path.getLast? = some finish ∧\n    path ≠ [] ∧ R.containsPath path\n\ndef Run.isLeaf (R : Run S) (root leaf : S) : Prop :=\n  R.reachable root leaf ∧ ∀ y, R (leaf, y) = 0", "target_theorem": "lemma acyclic_has_leaf (R : Run S) (root : S)\n    (h_acyclic : R.isAcyclic)\n    (h_has_out : ∃ y, R (root, y) > 0) :\n    ∃ leaf, R.isLeaf root leaf :=", "ground_truth_proof": ":= by\n  -- Start with path = [root]\n  have h_root_start : [root].head? = some root := by simp\n  have h_root_end : [root].getLast? = some root := by simp\n  have h_root_nonempty : [root] ≠ [] := by simp\n  have h_root_contains : R.containsPath [root] := by\n    intro t\n    simp [countTransitionInPath]\n\n  -- Apply the auxiliary lemma\n  obtain ⟨y, h_pos⟩ := h_has_out\n  apply acyclic_has_leaf_aux R root root [root] h_acyclic h_root_start h_root_end h_root_nonempty h_root_contains\n  use y\n  constructor\n  · intro h_mem\n    simp at h_mem\n    rw [h_mem] at h_pos\n    exact acyclic_no_self_loop R root h_acyclic h_pos\n  · exact h_pos", "nesting_depth": 8, "transitive_dep_count": 98, "subset_aristotle": false, "category": "Applied verif."}
{"id": 175, "thm_name": "Circuit.constraintsHold_toFlat_iff", "thm_stmt": "theorem Circuit.constraintsHold_toFlat_iff : ∀ {ops : Operations F}, ∀ {env : Environment F},\n    ConstraintsHoldFlat env ops.toFlat ↔ ConstraintsHold env ops", "lean_root": "clean", "rel_path": "Clean/Circuit/Subcircuit.lean", "imports": ["import Clean.Circuit.Basic", "import Clean.Circuit.Theorems"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "ConstraintsHold", "content": "@[circuit_norm]\ndef ConstraintsHold (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold eval ops\n  | .lookup { table, entry, .. } :: ops =>\n    table.Contains (entry.map eval) ∧ ConstraintsHold eval ops\n  | .subcircuit s :: ops =>\n    ConstraintsHoldFlat eval s.ops ∧ ConstraintsHold eval ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "induct", "content": "def induct {motive : Operations F → Sort*}\n  (empty : motive [])\n  (witness : ∀ m c ops, motive ops → motive (.witness m c :: ops))\n  (assert : ∀ e ops, motive ops → motive (.assert e :: ops))\n  (lookup : ∀ l ops, motive ops → motive (.lookup l :: ops))\n  (subcircuit : ∀ {n} (s : Subcircuit F n) ops, motive ops → motive (.subcircuit s :: ops))\n    (ops : Operations F) : motive ops :=\n  match ops with\n  | [] => empty\n  | .witness m c :: ops => witness m c ops (induct empty witness assert lookup subcircuit ops)\n  | .assert e :: ops => assert e ops (induct empty witness assert lookup subcircuit ops)\n  | .lookup l :: ops => lookup l ops (induct empty witness assert lookup subcircuit ops)\n  | .subcircuit s :: ops => subcircuit s ops (induct empty witness assert lookup subcircuit ops)"}], "lib_lemmas": [{"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "FlatOperation.constraintsHold_cons", "content": "lemma constraintsHold_cons : ∀ {op : FlatOperation F}, ∀ {ops : List (FlatOperation F)}, ∀ {env : Environment F},\n    ConstraintsHoldFlat env (op :: ops) ↔ ConstraintsHoldFlat env [op] ∧ ConstraintsHoldFlat env ops"}, {"name": "FlatOperation.constraintsHold_append", "content": "lemma constraintsHold_append : ∀ {a b: List (FlatOperation F)}, ∀ {env : Environment F},\n    ConstraintsHoldFlat env (a ++ b) ↔ ConstraintsHoldFlat env a ∧ ConstraintsHoldFlat env b"}], "local_ctx": "import Clean.Circuit.Basic\n\nimport Clean.Circuit.Theorems\n\nvariable {F : Type} [Field F]\n\nnamespace FlatOperation\n\nopen Circuit (ConstraintsHold.Completeness ConstraintsHold)\n\nend FlatOperation\n\nvariable {α β: TypeMap} [ProvableType α] [ProvableType β]\n\nsection\n\nopen Circuit\n\nopen FlatOperation (constraintsHold_cons constraintsHold_append)", "target_theorem": "theorem Circuit.constraintsHold_toFlat_iff : ∀ {ops : Operations F}, ∀ {env : Environment F},\n    ConstraintsHoldFlat env ops.toFlat ↔ ConstraintsHold env ops :=", "ground_truth_proof": ":= by\n  intro ops env\n  induction ops using Operations.induct with\n  | empty => trivial\n  -- we can handle all non-empty cases at once\n  | witness | assert | lookup | subcircuit =>\n    dsimp only [Operations.toFlat]\n    try rw [constraintsHold_cons]\n    try rw [constraintsHold_append]\n    simp_all only [ConstraintsHold, ConstraintsHoldFlat, and_true, true_and]", "nesting_depth": 8, "transitive_dep_count": 75, "subset_aristotle": false, "category": "Applied verif."}
{"id": 176, "thm_name": "Circuit.FoldlM.output_eq", "thm_stmt": "theorem output_eq :\n  (xs.foldlM circuit init).output n =\n    Fin.foldl m (fun acc i => (circuit acc xs[i.val]).output (n + i * constant.localLength)) init", "lean_root": "clean", "rel_path": "Clean/Circuit/Loops.lean", "imports": ["import Clean.Utils.Misc", "import Clean.Circuit.Subcircuit", "import Clean.Circuit.Theorems"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector.mk", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.foldl", "module": "Init.Data.Fin.Fold"}, {"name": "Vector.foldlM", "module": "Init.Data.Vector.Basic"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "induct", "content": "def induct {motive : {n : ℕ} → Vector α n → Sort u}\n    (nil : motive #v[])\n    (cons: ∀ {n : ℕ} (a : α) (as : Vector α n), motive as → motive (cons a as))\n    {n : ℕ} (v : Vector α n) : motive v :=\n  match v with\n  | ⟨ .mk [], h ⟩ => by admit /- proof elided -/\n  | ⟨ .mk (a :: as), h ⟩ => by admit /- proof elided -/"}, {"name": "cons", "content": "def cons (a : α) (v : Vector α n) : Vector α (n + 1) :=\n  ⟨ .mk (a :: v.toList), by admit /- proof elided -/\n  ⟩"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}], "lib_lemmas": [{"name": "Fin.foldl_succ", "module": "Init.Data.Fin.Fold"}, {"name": "Fin.val_succ", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.val_zero", "module": "Init.Data.Fin.Lemmas"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "List.foldlM_toArray", "module": "Init.Data.List.ToArray"}, {"name": "Vector.foldlM_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "List.foldlM_cons", "module": "Init.Data.List.Control"}, {"name": "Vector.toList_mk", "module": "Init.Data.Vector.Lemmas"}], "repo_lemmas": [{"name": "bind_output_eq", "content": "theorem bind_output_eq (f : Circuit F α) (g : α → Circuit F β) (n : ℕ) :\n  (f >>= g).output n = (g (f.output n)).output (n + f.localLength n)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Vector.foldlM_toList", "content": "lemma Vector.foldlM_toList (xs : Vector α n) {m : Type → Type} [Monad m] (body : β → α → m β) (init : β) :\n    xs.foldlM body init = xs.toList.foldlM body init"}, {"name": "Circuit.FoldlM.foldlM_cons", "content": "lemma foldlM_cons (x : α) :\n  (Vector.cons x xs).foldlM circuit init = (do\n    let init' ← circuit init x\n    xs.foldlM circuit init')"}, {"name": "Circuit.FoldlM.finFoldl_cons_succ", "content": "lemma finFoldl_cons_succ (x : α) :\n  Fin.foldl (m + 1) (fun acc i => (circuit acc (Vector.cons x xs)[i.val]).output (n + i * constant.localLength)) init\n    = Fin.foldl m (fun acc i => (circuit acc xs[i.val]).output (n + constant.localLength + i * constant.localLength)) ((circuit init x).output n)"}], "local_ctx": "import Clean.Circuit.Subcircuit\n\nimport Clean.Utils.Misc\n\nvariable {n m : ℕ} {F : Type} [Field F] {α β : Type}\n\nnamespace Circuit\n\nvariable {prop : Condition F}\n\nnamespace ForM\n\nvariable {circuit : α → Circuit F Unit} (xs : Vector α m) (constant : ConstantLength circuit) (n : ℕ)\n\nend ForM\n\nnamespace MapM\n\nvariable {circuit : α → Circuit F β} {xs : Vector α m} [constant: ConstantLength circuit]\n  {prop : Condition F}\n\nend MapM\n\nnamespace FoldlM\n\nvariable {env : Environment F} {prop : Condition F} {xs : Vector α m}\n  {circuit : β → α → Circuit F β} {init : β} {constant : ConstantLength (prod circuit)}", "target_theorem": "theorem output_eq :\n  (xs.foldlM circuit init).output n =\n    Fin.foldl m (fun acc i => (circuit acc xs[i.val]).output (n + i * constant.localLength)) init :=", "ground_truth_proof": ":= by\n  induction xs using Vector.induct generalizing init n\n  case nil => rfl\n  case cons x xs ih =>\n    rw [foldlM_cons, bind_output_eq, ih, constant.localLength_eq (init, x), finFoldl_cons_succ]", "nesting_depth": 8, "transitive_dep_count": 60, "subset_aristotle": true, "category": "Applied verif."}
{"id": 177, "thm_name": "Circuit.ForM.operations_eq", "thm_stmt": "theorem operations_eq :\n  (xs.forM circuit).operations n =\n    (List.ofFn fun (i : Fin m) => (circuit xs[i.val]).operations (n + i * constant.localLength)).flatten", "lean_root": "clean", "rel_path": "Clean/Circuit/Loops.lean", "imports": ["import Clean.Utils.Misc", "import Clean.Circuit.Subcircuit", "import Clean.Circuit.Theorems"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector.mk", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "List.ofFn", "module": "Init.Data.List.OfFn"}, {"name": "Vector.forM", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "induct", "content": "def induct {motive : {n : ℕ} → Vector α n → Sort u}\n    (nil : motive #v[])\n    (cons: ∀ {n : ℕ} (a : α) (as : Vector α n), motive as → motive (cons a as))\n    {n : ℕ} (v : Vector α n) : motive v :=\n  match v with\n  | ⟨ .mk [], h ⟩ => by admit /- proof elided -/\n  | ⟨ .mk (a :: as), h ⟩ => by admit /- proof elided -/"}, {"name": "cons", "content": "def cons (a : α) (v : Vector α n) : Vector α (n + 1) :=\n  ⟨ .mk (a :: v.toList), by admit /- proof elided -/\n  ⟩"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "toList", "content": "def toList : Operations F → List (Operation F) := id"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}], "lib_lemmas": [{"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "List.forM_eq_forM", "module": "Init.Data.List.Control"}, {"name": "List.forM_toArray", "module": "Init.Data.List.ToArray"}, {"name": "Vector.forM_eq_forM", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.forM_mk", "module": "Init.Data.Vector.Lemmas"}, {"name": "List.forM_cons", "module": "Init.Data.List.Control"}, {"name": "Vector.toList_mk", "module": "Init.Data.Vector.Lemmas"}], "repo_lemmas": [{"name": "bind_operations_eq", "content": "theorem bind_operations_eq (f : Circuit F α) (g : α → Circuit F β) (n : ℕ) :\n  (f >>= g).operations n = f.operations n ++ (g (f.output n)).operations (n + f.localLength n)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Vector.forM_toList", "content": "lemma Vector.forM_toList (xs : Vector α n) {m : Type → Type} [Monad m] (body : α → m Unit) :\n    xs.forM body = forM xs.toList body"}, {"name": "Circuit.ofFn_flatten_cons", "content": "private lemma ofFn_flatten_cons {circuit : α → Circuit F β} (constant : ConstantLength circuit) (x : α) (xs : Vector α m) (n : ℕ) :\n  (List.ofFn fun i => (circuit (Vector.cons x xs)[i.val]).operations (n + i * constant.localLength)).flatten\n    = (circuit x).operations n ++ (List.ofFn fun i => (circuit xs[i.val]).operations (n + constant.localLength + i * constant.localLength)).flatten"}], "local_ctx": "import Clean.Circuit.Subcircuit\n\nimport Clean.Utils.Misc\n\nvariable {n m : ℕ} {F : Type} [Field F] {α β : Type}\n\nnamespace Circuit\n\nvariable {prop : Condition F}\n\nnamespace ForM\n\nvariable {circuit : α → Circuit F Unit} (xs : Vector α m) (constant : ConstantLength circuit) (n : ℕ)", "target_theorem": "theorem operations_eq :\n  (xs.forM circuit).operations n =\n    (List.ofFn fun (i : Fin m) => (circuit xs[i.val]).operations (n + i * constant.localLength)).flatten :=", "ground_truth_proof": ":= by\n  induction xs using Vector.induct generalizing n\n  case nil => rfl\n  case cons x xs ih =>\n    rw [ofFn_flatten_cons, Vector.forM_toList, Vector.cons, Vector.toList_mk, List.forM_cons, ←Vector.forM_toList,\n      bind_operations_eq, ih, constant.localLength_eq]", "nesting_depth": 6, "transitive_dep_count": 56, "subset_aristotle": false, "category": "Applied verif."}
{"id": 178, "thm_name": "Gadgets.Xor32.completeness", "thm_stmt": "theorem completeness : Completeness (F p) elaborated Assumptions", "lean_root": "clean", "rel_path": "Clean/Gadgets/Xor/Xor32.lean", "imports": ["import Clean.Gadgets.Xor.ByteXorTable", "import Clean.Circuit.Provable", "import Clean.Circuit.Expression", "import Clean.Utils.Field", "import Clean.Types.U32", "import Mathlib.Data.ZMod.Basic", "import Clean.Utils.Primes", "import Mathlib.Algebra.Field.Basic", "import Clean.Circuit.Basic", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "False.elim", "module": "Init.Prelude"}, {"name": "Fin.xor", "module": "Init.Data.Fin.Basic"}, {"name": "HXor", "module": "Init.Prelude"}, {"name": "HXor.hXor", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "XorOp", "module": "Init.Prelude"}, {"name": "XorOp.xor", "module": "Init.Prelude"}, {"name": "Vector.push", "module": "Init.Data.Vector.Basic"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}], "used_repo_defs": [{"name": "syntax \"let \" ident \" <== \" term : doElem", "content": "syntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem\n\nsyntax \"infer_constant_length\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "U32", "content": "structure U32 (T : Type) where\n  x0 : T\n  x1 : T\n  x2 : T\n  x3 : T\nderiving DecidableEq"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "ByteXorTable", "content": "def ByteXorTable : Table (F p) fieldTriple := .fromStatic {\n  name := \"ByteXor\"\n  length := 256*256\n\n  row i :=\n    let (x, y) := splitTwoBytes i\n    (fromByte x, fromByte y, fromByte (x ^^^ y))\n\n  index := fun (x, y, _) => x.val * 256 + y.val\n\n  Spec := fun (x, y, z) =>\n    x.val < 256 ∧ y.val < 256 ∧ z.val = x.val ^^^ y.val\n\n  contains_iff := by admit /- proof elided -/"}, {"name": "splitTwoBytes", "content": "def splitTwoBytes (i : Fin (256 * 256)) : Fin 256 × Fin 256 :=\n  let x := i.val / 256\n  let y := i.val % 256\n  have x_lt : x < 256 := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "fromByte", "content": "def fromByte (x : Fin 256) : F p :=\n  FieldUtils.natToField x.val (by admit /- proof elided -/\n  )"}, {"name": "natToField", "content": "def natToField (n : ℕ) (lt : n < p) : F p :=\n  match p with\n  | 0 => False.elim (Nat.not_lt_zero n lt)\n  | _ + 1 => ⟨ n, lt ⟩"}, {"name": "fieldTriple", "content": "@[reducible]\ndef fieldTriple : TypeMap := fun F => F × F × F"}, {"name": "concatTwoBytes", "content": "def concatTwoBytes (x y : Fin 256) : Fin (256 * 256) :=\n  let i := x.val * 256 + y.val\n  have i_lt : i < 256 * 256 := by admit /- proof elided -/"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "varFromOffset", "content": "@[explicit_provable_type]\ndef varFromOffset (α : TypeMap) [ProvableType α] (offset : ℕ) : Var α F :=\n  let vars := Vector.mapRange (size α) fun i => var ⟨offset + i⟩\n  fromVars vars"}, {"name": "mapRange", "content": "def mapRange (n : ℕ) (create : ℕ → α) : Vector α n :=\n  match n with\n  | 0 => #v[]\n  | k + 1 => mapRange k create |>.push (create k)"}, {"name": "Normalized", "content": "def Normalized (x : U32 (F p)) :=\n  x.x0.val < 256 ∧ x.x1.val < 256 ∧ x.x2.val < 256 ∧ x.x3.val < 256"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}], "lib_lemmas": [{"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.xor_lt_two_pow", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Fin.forall_iff", "module": "Init.Data.Fin.Lemmas"}], "repo_lemmas": [{"name": "val_lt_p", "content": "theorem val_lt_p {p : ℕ} (x : ℕ) : (x < p) → (x : F p).val = x"}], "used_local_defs": [{"name": "Gadgets.Xor32.Inputs", "content": "structure Inputs (F : Type) where\n  x: U32 F\n  y: U32 F"}, {"name": "Gadgets.Xor32.main", "content": "def main (input : Var Inputs (F p)) : Circuit (F p) (Var U32 (F p))  := do\n  let ⟨x, y⟩ := input\n  let z ← witness fun env =>\n    let z0 := (env x.x0).val ^^^ (env y.x0).val\n    let z1 := (env x.x1).val ^^^ (env y.x1).val\n    let z2 := (env x.x2).val ^^^ (env y.x2).val\n    let z3 := (env x.x3).val ^^^ (env y.x3).val\n    U32.mk z0 z1 z2 z3\n\n  lookup ByteXorTable (x.x0, y.x0, z.x0)\n  lookup ByteXorTable (x.x1, y.x1, z.x1)\n  lookup ByteXorTable (x.x2, y.x2, z.x2)\n  lookup ByteXorTable (x.x3, y.x3, z.x3)\n  return z"}, {"name": "Gadgets.Xor32.Assumptions", "content": "def Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.Normalized ∧ y.Normalized"}, {"name": "Gadgets.Xor32.elaborated", "content": "instance elaborated : ElaboratedCircuit (F p) Inputs U32 where\n  main := main\n  localLength _ := 4\n  output _ i0 := varFromOffset U32 i0"}], "used_local_lemmas": [{"name": "Gadgets.Xor32.xor_val", "content": "lemma xor_val {x y : F p} (hx : x.val < 256) (hy : y.val < 256) :\n  (x.val ^^^ y.val : F p).val = x.val ^^^ y.val"}], "local_ctx": "import Mathlib.Algebra.Field.Basic\n\nimport Mathlib.Data.ZMod.Basic\n\nimport Clean.Utils.Primes\n\nimport Clean.Utils.Vector\n\nimport Clean.Circuit.Expression\n\nimport Clean.Circuit.Provable\n\nimport Clean.Circuit.Basic\n\nimport Clean.Utils.Field\n\nimport Clean.Types.U32\n\nimport Clean.Gadgets.Xor.ByteXorTable\n\nsection\n\nvariable {p : ℕ} [Fact p.Prime] [p_large_enough: Fact (p > 512)]\n\nnamespace Gadgets.Xor32\n\nopen Gadgets.Xor\n\nstructure Inputs (F : Type) where\n  x: U32 F\n  y: U32 F\n\ndef main (input : Var Inputs (F p)) : Circuit (F p) (Var U32 (F p))  := do\n  let ⟨x, y⟩ := input\n  let z ← witness fun env =>\n    let z0 := (env x.x0).val ^^^ (env y.x0).val\n    let z1 := (env x.x1).val ^^^ (env y.x1).val\n    let z2 := (env x.x2).val ^^^ (env y.x2).val\n    let z3 := (env x.x3).val ^^^ (env y.x3).val\n    U32.mk z0 z1 z2 z3\n\n  lookup ByteXorTable (x.x0, y.x0, z.x0)\n  lookup ByteXorTable (x.x1, y.x1, z.x1)\n  lookup ByteXorTable (x.x2, y.x2, z.x2)\n  lookup ByteXorTable (x.x3, y.x3, z.x3)\n  return z\n\ndef Assumptions (input : Inputs (F p)) :=\n  let ⟨x, y⟩ := input\n  x.Normalized ∧ y.Normalized\n\ninstance elaborated : ElaboratedCircuit (F p) Inputs U32 where\n  main := main\n  localLength _ := 4\n  output _ i0 := varFromOffset U32 i0", "target_theorem": "theorem completeness : Completeness (F p) elaborated Assumptions :=", "ground_truth_proof": ":= by\n  intro i0 env input_var h_env input h_input as\n  let ⟨⟨ x0_var, x1_var, x2_var, x3_var ⟩,\n       ⟨ y0_var, y1_var, y2_var, y3_var ⟩⟩ := input_var\n  let ⟨⟨ x0, x1, x2, x3 ⟩,\n       ⟨ y0, y1, y2, y3 ⟩⟩ := input\n  simp only [circuit_norm, explicit_provable_type, Inputs.mk.injEq, U32.mk.injEq] at h_input\n\n  simp only [Assumptions, circuit_norm, U32.Normalized] at as\n  obtain ⟨ x_bytes, y_bytes ⟩ := as\n  obtain ⟨ x0_byte, x1_byte, x2_byte, x3_byte ⟩ := x_bytes\n  obtain ⟨ y0_byte, y1_byte, y2_byte, y3_byte ⟩ := y_bytes\n\n  simp only [h_input, circuit_norm, main, ByteXorTable,\n    explicit_provable_type, Fin.forall_iff] at h_env ⊢\n  have h_env0 : env.get i0 = ↑(ZMod.val x0 ^^^ ZMod.val y0) := by simpa using h_env 0\n  simp_all [xor_val]", "nesting_depth": 7, "transitive_dep_count": 99, "subset_aristotle": true, "category": "Applied verif."}
{"id": 179, "thm_name": "Circomlib.MultiAND.completeness_two", "thm_stmt": "lemma completeness_two {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 2) (F p))\n    (input : fields 2 (F p))\n    (h_local_witnesses : env.UsesLocalWitnessesCompleteness offset ((main input_var).operations offset))\n    (h_env : input = eval env input_var)\n    (h_assumptions : Assumptions 2 input) :\n    Circuit.ConstraintsHold.Completeness env ((main input_var).operations offset)", "lean_root": "clean", "rel_path": "Clean/Circomlib/Gates.lean", "imports": ["import Clean.Circuit.Theorems", "import Clean.Utils.Field", "import Clean.Circuit", "import Mathlib.Data.Nat.Bitwise", "import Clean.Gadgets.Boolean", "import Clean.Utils.Bitwise", "import Clean.Utils.BinaryOps", "import Clean.Utils.Vector"], "used_lib_defs": [{"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "One", "module": "Init.Prelude"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}], "used_repo_defs": [{"name": "syntax \"infer_constant_length\" : tactic", "content": "syntax \"infer_constant_length\" : tactic\n\nsyntax \"let \" ident \" <== \" term : doElem\n\nsyntax \"let \" ident \" : \" term \" <== \" term : doElem"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x <== $e) => `(doElem| let $x ← HasAssignEq.assignEq $e)\n  | `(doElem| let $x : $t <== $e) => `(doElem| let $x : $t ← HasAssignEq.assignEq $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|infer_constant_length) => `(tactic|(\n    apply ConstantLength.fromConstantLength\n    try simp only [circuit_norm]\n    try intros\n    try ac_rfl))\n\nexample :\n  let add (x : Expression F) := do\n    let y : Expression F ← witness fun _ => 1\n    let z ← witness fun eval => eval (x + y)\n    assertZero (x + y - z)\n    pure z\n  ConstantLength add := by admit /- proof elided -/"}, {"name": "F", "content": "def F p := ZMod p"}, {"name": "fields", "content": "@[reducible]\ndef fields (n : ℕ) := fun F => Vector F n"}, {"name": "IsBool", "content": "def IsBool {α : Type*} [Zero α] [One α] (x : α) : Prop := x = 0 ∨ x = 1"}, {"name": "map", "content": "def map {m : ℕ} (xs : Vector α m) (body : α → Circuit F β)\n    (_constant : ConstantLength body := by admit /- proof elided -/\n    ) : Circuit F (Vector β m) :=\n  xs.mapM body"}, {"name": "Circuit", "content": "def Circuit (F : Type) [Field F] (α : Type) := ℕ → α × List (Operation F)"}, {"name": "Operation", "content": "inductive Operation (F : Type) [Field F] where\n  | witness : (m : ℕ) → (compute : Environment F → Vector F m) → Operation F\n  | assert : Expression F → Operation F\n  | lookup : Lookup F → Operation F\n  | subcircuit : {n : ℕ} → Subcircuit F n → Operation F"}, {"name": "Condition", "content": "structure Condition (F : Type) [Field F] where\n  witness (offset : ℕ) : (m : ℕ) → (Environment F → Vector F m) → Prop := fun _ _ => True\n  assert (offset : ℕ) (_ : Expression F) : Prop := True\n  lookup (offset : ℕ) (_ : Lookup F) : Prop := True\n  subcircuit (offset : ℕ) {m : ℕ} (_ : Subcircuit F m) : Prop := True"}, {"name": "FlatOperation", "content": "inductive FlatOperation (F : Type) where\n  | witness : (m : ℕ) → (Environment F → Vector F m) → FlatOperation F\n  | assert : Expression F → FlatOperation F\n  | lookup : Lookup F → FlatOperation F"}, {"name": "Subcircuit", "content": "structure Subcircuit (F : Type) [Field F] (offset : ℕ) where\n  ops : List (FlatOperation F)\n\n  \n  \n  \n  Soundness : Environment F → Prop\n  Completeness : Environment F → Prop\n  UsesLocalWitnesses : Environment F → Prop\n\n  \n  \n  localLength : ℕ\n\n  \n  imply_soundness : ∀ env,\n    ConstraintsHoldFlat env ops → Soundness env\n\n  \n  implied_by_completeness : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    Completeness env → ConstraintsHoldFlat env ops\n\n  \n  imply_usesLocalWitnesses : ∀ env, env.ExtendsVector (localWitnesses env ops) offset →\n    UsesLocalWitnesses env\n\n  \n  localLength_eq : localLength = FlatOperation.localLength ops"}, {"name": "localWitnesses", "content": "@[circuit_norm]\ndef localWitnesses (env : Environment F) : (l : List (FlatOperation F)) → Vector F (localLength l)\n  | [] => #v[]\n  | witness _ compute :: ops => compute env ++ localWitnesses env ops\n  | assert _ :: ops | lookup _ :: ops => localWitnesses env ops"}, {"name": "lookup", "content": "@[circuit_norm]\ndef lookup {Row : TypeMap} [ProvableType Row] (table : Table F Row)  (entry : Row (Expression F)) : Circuit F Unit := fun _ =>\n  ((), [.lookup { table := table.toRaw, entry := toElements entry }])"}, {"name": "Table.toRaw", "content": "@[circuit_norm]\ndef Table.toRaw (table : Table F Row) : RawTable F where\n  name := table.name\n  arity := size Row\n  Contains row := table.Contains (fromElements row)\n  Soundness row := table.Soundness (fromElements row)\n  Completeness row := table.Completeness (fromElements row)\n  imply_soundness row := table.imply_soundness (fromElements row)\n  implied_by_completeness row := table.implied_by_completeness (fromElements row)"}, {"name": "RawTable", "content": "structure RawTable (F : Type) where\n  name : String\n  arity : ℕ\n  Contains : Vector F arity → Prop\n  Soundness : Vector F arity → Prop\n  Completeness : Vector F arity → Prop\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "StaticTable", "content": "structure StaticTable (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n  length : ℕ\n  row : Fin length → Row F\n  \n  \n  \n  index : Row F → ℕ\n  Spec : Row F → Prop\n  contains_iff : ∀ t, (∃ i, t = row i) ↔ Spec t"}, {"name": "Contains", "content": "def Contains (table : StaticTable F Row) (row : Row F) :=\n  ∃ i : Fin table.length, row = table.row i"}, {"name": "ProvableType.fromStruct", "content": "instance ProvableType.fromStruct {α : TypeMap} [ProvableStruct α] : ProvableType α where\n  size := combinedSize α\n  toElements x :=\n    toComponents x |> componentsToElements (components α) |>.cast combinedSize_eq.symm\n  fromElements v :=\n    v.cast combinedSize_eq |> componentsFromElements (components α) |> fromComponents\n  fromElements_toElements x := by admit /- proof elided -/"}, {"name": "TypeMap", "content": "@[reducible]\ndef TypeMap := Type → Type"}, {"name": "Table", "content": "structure Table (F : Type) (Row : TypeMap) [ProvableType Row] where\n  name : String\n   \n  Contains : Row F → Prop\n\n   \n  Soundness : Row F → Prop\n  Completeness : Row F → Prop\n\n  imply_soundness : ∀ row, Contains row → Soundness row\n  implied_by_completeness : ∀ row, Completeness row → Contains row"}, {"name": "Expression", "content": "inductive Expression (F : Type) where\n  | var : Variable F -> Expression F\n  | const : F -> Expression F\n  | add : Expression F -> Expression F -> Expression F\n  | mul : Expression F -> Expression F -> Expression F"}, {"name": "Variable", "content": "structure Variable (F : Type) where\n  index : ℕ"}, {"name": "const", "content": "def const (x : α F) : Var α F :=\n  let values : Vector F _ := toElements x\n  fromVars (values.map .const)"}, {"name": "Var", "content": "@[reducible] def Var (M : TypeMap) (F : Type) := M (Expression F)"}, {"name": "fromVars", "content": "@[circuit_norm]\ndef fromVars (vars : Vector (Expression F) (size M)) := fromElements vars"}, {"name": "Lookup", "content": "structure Lookup (F : Type) where\n  table : RawTable F\n  entry : Vector (Expression F) table.arity"}, {"name": "Witnessable", "content": "class Witnessable (F : Type) [Field F] (value : outParam TypeMap) (var : TypeMap) [ProvableType value] where\n  witness : ((Environment F) → value F) → Circuit F (var F)\n  var_eq : var F = value (Expression F) := by admit /- proof elided -/"}, {"name": "Environment", "content": "structure Environment (F : Type) where\n  get : ℕ → F"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : List (FlatOperation F) → ℕ\n  | [] => 0\n  | witness m _ :: ops => m + localLength ops\n  | assert _ :: ops | lookup _ :: ops => localLength ops"}, {"name": "ConstraintsHoldFlat", "content": "def ConstraintsHoldFlat (eval : Environment F) : List (FlatOperation F) → Prop\n  | [] => True\n  | op :: ops => match op with\n    | assert e => (eval e = 0) ∧ ConstraintsHoldFlat eval ops\n    | lookup { table, entry } =>\n      table.Contains (entry.map eval) ∧ ConstraintsHoldFlat eval ops\n    | _ => ConstraintsHoldFlat eval ops"}, {"name": "ConstantLength", "content": "class ConstantLength (circuit : α → Circuit F β) where\n  localLength : ℕ\n  localLength_eq : ∀ (a : α) (n : ℕ), (circuit a).localLength n = localLength"}, {"name": "ConstantLength.fromConstantLength", "content": "def ConstantLength.fromConstantLength {circuit : α → Circuit F β} [Inhabited α]\n    (h : ∀ (a : α) n, (circuit a).localLength n = (circuit default).localLength 0) : ConstantLength circuit where\n  localLength := (circuit default).localLength 0\n  localLength_eq a n := h a n"}, {"name": "HasAssignEq", "content": "class HasAssignEq (β : Type) (F : outParam Type) [Field F] where\n  assignEq : β → Circuit F β"}, {"name": "FormalCircuit", "content": "structure FormalCircuit (F : Type) [Field F] (Input Output : TypeMap) [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions (_ : Input F) : Prop := True\n  Spec : Input F → Output F → Prop\n  soundness : Soundness F elaborated Assumptions Spec\n  completeness : Completeness F elaborated Assumptions"}, {"name": "ElaboratedCircuit", "content": "class ElaboratedCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output] where\n  main : Var Input F → Circuit F (Var Output F)\n\n   \n  localLength : Var Input F → ℕ\n\n   \n  localLength_eq : ∀ input offset, (main input).localLength offset = localLength input\n    := by admit /- proof elided -/"}, {"name": "GeneralFormalCircuit", "content": "structure GeneralFormalCircuit (F : Type) (Input Output : TypeMap) [Field F] [ProvableType Input] [ProvableType Output]\n    extends elaborated : ElaboratedCircuit F Input Output where\n  Assumptions : Input F → Prop \n  Spec : Input F → Output F → Prop \n  soundness : GeneralFormalCircuit.Soundness F elaborated Spec\n  completeness : GeneralFormalCircuit.Completeness F elaborated Assumptions"}, {"name": "Soundness", "content": "@[circuit_norm]\ndef Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "operations", "content": "@[reducible, circuit_norm]\ndef operations (circuit : Circuit F α) (offset : ℕ) : Operations F :=\n  (circuit offset).2"}, {"name": "Operations", "content": "@[reducible, circuit_norm]\ndef Operations (F : Type) [Field F] := List (Operation F)"}, {"name": "output", "content": "@[reducible, circuit_norm]\ndef output (circuit : Circuit F α) (offset : ℕ) : α :=\n  (circuit offset).1"}, {"name": "ConstraintsHold.Soundness", "content": "@[circuit_norm]\ndef ConstraintsHold.Soundness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Soundness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Soundness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Soundness (entry.map eval) ∧ ConstraintsHold.Soundness eval ops\n  | .subcircuit s :: ops =>\n    s.Soundness eval ∧ ConstraintsHold.Soundness eval ops"}, {"name": "GeneralFormalCircuit.Soundness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Spec : Input F → Output F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  let output := eval env (circuit.output input_var offset)\n  Spec input output"}, {"name": "eval", "content": "@[explicit_provable_type]\ndef eval (env : Environment F) (x : Var α F) : α F :=\n  let vars := toVars x\n  let values := vars.map (Expression.eval env)\n  fromElements values"}, {"name": "toVars", "content": "@[circuit_norm]\ndef toVars (var : M (Expression F)) := toElements var"}, {"name": "eval", "content": "@[circuit_norm]\ndef eval (env : Environment F) : Expression F → F\n  | var v => env.get v.index\n  | const c => c\n  | add x y => eval env x + eval env y\n  | mul x y => eval env x * eval env y"}, {"name": "FormalAssertion.Soundness", "content": "@[circuit_norm]\ndef FormalAssertion.Soundness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env,\n  \n  ∀ input_var : Var Input F, ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Soundness env (circuit.main input_var |>.operations offset) →\n  \n  Spec input"}, {"name": "unit", "content": "@[reducible]\ndef unit (_ : Type) := Unit"}, {"name": "subcircuit", "content": "@[circuit_norm]\ndef subcircuit (circuit : FormalCircuit F β α) (b : Var β F) : Circuit F (Var α F) :=\n  fun offset =>\n    let a := circuit.output b offset\n    let subcircuit := circuit.toSubcircuit offset b\n    (a, [.subcircuit subcircuit])"}, {"name": "FormalAssertion.toSubcircuit", "content": "def FormalAssertion.toSubcircuit (circuit : FormalAssertion F β)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  {\n    ops := ops.toFlat,\n    Soundness env := circuit.Assumptions (eval env input_var) → circuit.Spec (eval env input_var),\n    Completeness env := circuit.Assumptions (eval env input_var) ∧ circuit.Spec (eval env input_var),\n    UsesLocalWitnesses _ := True,\n    localLength := circuit.localLength input_var\n\n    imply_soundness := by admit /- proof elided -/"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operation F → ℕ\n  | .witness m _ => m\n  | .assert _ => 0\n  | .lookup _ => 0\n  | .subcircuit s => s.localLength"}, {"name": "localLength", "content": "@[circuit_norm]\ndef localLength : Operations F → ℕ\n  | [] => 0\n  | .witness m _ :: ops => m + localLength ops\n  | .assert _ :: ops => localLength ops\n  | .lookup _ :: ops => localLength ops\n  | .subcircuit s :: ops => s.localLength + localLength ops"}, {"name": "FormalAssertion", "content": "structure FormalAssertion (F : Type) (Input : TypeMap) [Field F] [ProvableType Input]\n    extends elaborated : ElaboratedCircuit F Input unit where\n  Assumptions : Input F → Prop\n  Spec : Input F → Prop\n  soundness : FormalAssertion.Soundness F elaborated Assumptions Spec\n  completeness : FormalAssertion.Completeness F elaborated Assumptions Spec\n\n  \n  localLength _ := 0\n  \n  output _ _ := ()"}, {"name": "SubcircuitsConsistent", "content": "@[circuit_norm]\ndef SubcircuitsConsistent (offset : ℕ) (ops : Operations F) := ops.forAll offset {\n  subcircuit offset {n} _ := n = offset\n}\n\n @[circuit_norm]\ndef forAll (offset : ℕ) (condition : Condition F) : Operations F → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops\n  | .subcircuit s :: ops => condition.subcircuit offset s ∧ forAll (s.localLength + offset) condition ops"}, {"name": "FlatOperation.forAll", "content": "def FlatOperation.forAll (offset : ℕ) (condition : Condition F) : List (FlatOperation F) → Prop\n  | [] => True\n  | .witness m c :: ops => condition.witness offset m c ∧ forAll (m + offset) condition ops\n  | .assert e :: ops => condition.assert offset e ∧ forAll offset condition ops\n  | .lookup l :: ops => condition.lookup offset l ∧ forAll offset condition ops"}, {"name": "localLength", "content": "@[reducible, circuit_norm]\ndef localLength (circuit : Circuit F α) (offset := 0) : ℕ :=\n  Operations.localLength (circuit offset).2"}, {"name": "FormalAssertion.Completeness", "content": "@[circuit_norm]\ndef FormalAssertion.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input unit)\n    (Assumptions : Input F → Prop) (Spec : Input F → Prop) :=\n  \n  ∀ offset, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input → Spec input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "ConstraintsHold.Completeness", "content": "@[circuit_norm]\ndef ConstraintsHold.Completeness (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold.Completeness eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold.Completeness eval ops\n  | .lookup { table, entry } :: ops =>\n    table.Completeness (entry.map eval) ∧ ConstraintsHold.Completeness eval ops\n  | .subcircuit s :: ops =>\n    s.Completeness eval ∧ ConstraintsHold.Completeness eval ops"}, {"name": "Completeness", "content": "@[circuit_norm]\ndef Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output)\n    (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "GeneralFormalCircuit.Completeness", "content": "@[circuit_norm]\ndef GeneralFormalCircuit.Completeness (F : Type) [Field F] (circuit : ElaboratedCircuit F Input Output) (Assumptions : Input F → Prop) :=\n  \n  ∀ offset : ℕ, ∀ env, ∀ input_var : Var Input F,\n  env.UsesLocalWitnessesCompleteness offset (circuit.main input_var |>.operations offset) →\n  \n  ∀ input : Input F, eval env input_var = input →\n  Assumptions input →\n  \n  ConstraintsHold.Completeness env (circuit.main input_var |>.operations offset)"}, {"name": "toFlat", "content": "def toFlat : Operations F → List (FlatOperation F)\n  | [] => []\n  | .witness m c :: ops => .witness m c :: toFlat ops\n  | .assert e :: ops => .assert e :: toFlat ops\n  | .lookup l :: ops => .lookup l :: toFlat ops\n  | .subcircuit s :: ops => s.ops ++ toFlat ops"}, {"name": "Environment.UsesLocalWitnesses", "content": "def Environment.UsesLocalWitnesses (env : Environment F) (offset : ℕ) (ops : Operations F) : Prop :=\n  ops.forAllFlat offset { witness n _ compute := env.ExtendsVector (compute env) n }"}, {"name": "Environment.ExtendsVector", "content": "@[circuit_norm]\ndef Environment.ExtendsVector (env : Environment F) (wit : Vector F n) (offset : ℕ) : Prop :=\n  ∀ i : Fin n, env.get (offset + i.val) = wit[i.val]"}, {"name": "Operations.forAllFlat", "content": "def Operations.forAllFlat (n : ℕ) (condition : Condition F) (ops : Operations F) : Prop :=\n  forAll n { condition with subcircuit n _ s := FlatOperation.forAll n condition s.ops } ops"}, {"name": "GeneralFormalCircuit.toSubcircuit", "content": "def GeneralFormalCircuit.toSubcircuit (circuit : GeneralFormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n      let input := eval env input_var\n      let output := eval env (circuit.output input_var n)\n      ConstraintsHoldFlat env ops.toFlat → circuit.Spec input output :="}, {"name": "FormalCircuit.toSubcircuit", "content": "def FormalCircuit.toSubcircuit (circuit : FormalCircuit F β α)\n    (n : ℕ) (input_var : Var β F) : Subcircuit F n :=\n  let ops := circuit.main input_var |>.operations n\n  have h_consistent : ops.SubcircuitsConsistent n := circuit.subcircuitsConsistent input_var n\n\n  have imply_soundness : ∀ env : Environment F,\n    let input := eval env input_var\n    let output := eval env (circuit.output input_var n)\n    ConstraintsHoldFlat env ops.toFlat → circuit.Assumptions input → circuit.Spec input output :="}, {"name": "fieldPair", "content": "@[reducible]\ndef fieldPair : TypeMap := fun F => F × F"}, {"name": "field", "content": "@[reducible] def field : TypeMap := id"}, {"name": "ConstraintsHold", "content": "@[circuit_norm]\ndef ConstraintsHold (eval : Environment F) : List (Operation F) → Prop\n  | [] => True\n  | .witness _ _ :: ops => ConstraintsHold eval ops\n  | .assert e :: ops => eval e = 0 ∧ ConstraintsHold eval ops\n  | .lookup { table, entry, .. } :: ops =>\n    table.Contains (entry.map eval) ∧ ConstraintsHold eval ops\n  | .subcircuit s :: ops =>\n    ConstraintsHoldFlat eval s.ops ∧ ConstraintsHold eval ops"}, {"name": "Environment.UsesLocalWitnessesCompleteness", "content": "@[circuit_norm]\ndef Environment.UsesLocalWitnessesCompleteness (env : Environment F) (offset : ℕ) : List (Operation F) → Prop\n  | [] => True\n  | .witness m c :: ops => env.ExtendsVector (c env) offset ∧ env.UsesLocalWitnessesCompleteness (offset + m) ops\n  | .assert _ :: ops => env.UsesLocalWitnessesCompleteness offset ops\n  | .lookup _ :: ops => env.UsesLocalWitnessesCompleteness offset ops\n  | .subcircuit s :: ops => s.UsesLocalWitnesses env ∧ env.UsesLocalWitnessesCompleteness (offset + s.localLength) ops"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Circomlib.AND.main", "content": "def main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out"}, {"name": "Circomlib.MultiAND.main", "content": "def main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)"}, {"name": "Circomlib.MultiAND.Assumptions", "content": "def Assumptions (n : ℕ) (input : fields n (F p)) : Prop :=\n  ∀ (i : ℕ) (h : i < n), IsBool input[i]"}, {"name": "Circomlib.MultiAND.Spec", "content": "def Spec (n : ℕ) (input : fields n (F p)) (output : F p) : Prop :=\n  output.val = (input.map (·.val)).foldl (· &&& ·) 1 ∧ IsBool output"}], "used_local_lemmas": [], "local_ctx": "import Clean.Circuit\n\nimport Clean.Utils.Field\n\nimport Clean.Gadgets.Boolean\n\nimport Clean.Utils.Bitwise\n\nimport Clean.Utils.Vector\n\nimport Clean.Utils.BinaryOps\n\nimport Clean.Circuit.Theorems\n\nimport Mathlib.Data.Nat.Bitwise\n\nopen IsBool\n\nnamespace Circomlib\n\nvariable {p : ℕ} [Fact p.Prime]\n\nopen Circuit (bind_output_eq bind_localLength_eq bind_forAll)\n\nopen Operations (append_localLength)\n\nopen BinaryOps (List.foldl_and_IsBool List.and_foldl_eq_foldl)\n\nnamespace XOR\n\nend XOR\n\nnamespace AND\n\ndef main (input : Expression (F p) × Expression (F p)) := do\n  let a := input.1\n  let b := input.2\n  let out <== a*b\n  return out\n\nend AND\n\nnamespace OR\n\nend OR\n\nnamespace NOT\n\nend NOT\n\nnamespace NAND\n\nend NAND\n\nnamespace NOR\n\nend NOR\n\nnamespace MultiAND\n\ndef main : {n : ℕ} → Vector (Expression (F p)) n → Circuit (F p) (Expression (F p))\n  | 0, _ =>\n    return (1 : F p)\n  | 1, input =>\n    return input[0]\n  | 2, input =>\n    AND.circuit.main (input[0], input[1])\n  | n + 3, input => do\n    let n1 := (n + 3) / 2\n    let n2 := (n + 3) - n1\n\n    let input1 : Vector (Expression (F p)) n1 := input.take n1 |>.cast (by admit /- proof elided -/\n    )\n    let input2 : Vector (Expression (F p)) n2 := input.drop n1 |>.cast (by admit /- proof elided -/\n    )\n\n    let out1 ← main input1\n    let out2 ← main input2\n\n    AND.circuit.main (out1, out2)\n\ndef Assumptions (n : ℕ) (input : fields n (F p)) : Prop :=\n  ∀ (i : ℕ) (h : i < n), IsBool input[i]\n\ndef Spec (n : ℕ) (input : fields n (F p)) (output : F p) : Prop :=\n  output.val = (input.map (·.val)).foldl (· &&& ·) 1 ∧ IsBool output", "target_theorem": "lemma completeness_two {p : ℕ} [Fact p.Prime]\n    (offset : ℕ) (env : Environment (F p)) (input_var : Var (fields 2) (F p))\n    (input : fields 2 (F p))\n    (h_local_witnesses : env.UsesLocalWitnessesCompleteness offset ((main input_var).operations offset))\n    (h_env : input = eval env input_var)\n    (h_assumptions : Assumptions 2 input) :\n    Circuit.ConstraintsHold.Completeness env ((main input_var).operations offset) :=", "ground_truth_proof": ":= by\n  simp only [main, circuit_norm] at h_local_witnesses ⊢\n\n  have h_binary0 : IsBool input[0] := h_assumptions 0 (by norm_num)\n  have h_binary1 : IsBool input[1] := h_assumptions 1 (by norm_num)\n\n  apply AND.circuit.completeness\n  · exact h_local_witnesses\n  · subst h_env\n    rfl\n  · simp only [Assumptions] at h_assumptions\n    constructor\n    · have h_eval0 : env input_var[0] = input[0] :=\n        by simp[h_env, circuit_norm]\n      change IsBool (env input_var[0])\n      rw [h_eval0]\n      exact h_binary0\n    · have h_eval1 : env input_var[1] = input[1] :=\n        by simp[h_env, circuit_norm]\n      change IsBool (env input_var[1])\n      rw [h_eval1]\n      exact h_binary1", "nesting_depth": 9, "transitive_dep_count": 85, "subset_aristotle": false, "category": "Applied verif."}
{"id": 180, "thm_name": "xfor_inv_lemma", "thm_stmt": "lemma xfor_inv_lemma (I : Int -> hProp) (a b : Int)\n  (F : val -> formula)\n  (Q : val -> hProp) :\n  structural_pred F ->\n  a <= b ->\n    (∃ H',\n      H ==> I a ∗ H' ∧\n      (∀ i, a <= i ∧ i < b -> I i ==> F i fun _ => I (i + 1)) ∧\n      (fun _ => I b ∗ H') ===> Q) ->\n    H ==> wpgen_for a b F Q", "lean_root": "splean", "rel_path": "SPLean/Theories/WP1.lean", "imports": ["import SPLean.Theories.XChange", "import Mathlib.Data.List.Indexes", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Lean", "import SPLean.Theories.WPUtil"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "L", "module": "Archive.Hairer"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "OfNat.ofNat", "module": "Init.Prelude"}, {"name": "instOfNat", "module": "Init.Data.Int.Basic"}, {"name": "instOfNatNat", "module": "Init.Prelude"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "scase_if", "module": "Ssreflect.Elim"}, {"name": "shave", "module": "Ssreflect.Have"}, {"name": "srw", "module": "Ssreflect.Rewrite"}], "used_repo_defs": [{"name": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hexists xs b\n\nsyntax \"fun\" ident+ \" => \" lang : lang"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty\n\nsyntax \" := \" : bop\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang"}, {"name": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hforall xs b"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang"}, {"name": "macro \"xsimp\" : tactic =>", "content": "macro \"xsimp\" : tactic =>\n  `(tactic| (\n    xsimp_start\n    repeat xsimp_step\n    try rev_pure\n    try hide_mvars\n    try hsimp\n    rotate_left\n\n  ))\n\nsyntax \" <= \" : bop"}, {"name": "macro \"xval\" : tactic => do", "content": "macro \"xval\" : tactic => do\n  `(tactic| (xstruct_if_needed; apply xval_lemma))"}, {"name": "macro \"xval\" : tactic => `(tactic| (xwp; xval))", "content": "macro \"xval\" : tactic => `(tactic| (xwp; xval))"}, {"name": "macro \"xseq\" : tactic => do", "content": "macro \"xseq\" : tactic => do\n  `(tactic| (xstruct_if_needed; apply xseq_lemma))"}, {"name": "macro \"xchange\" l:term : tactic =>", "content": "macro \"xchange\" l:term : tactic =>\n  `(tactic| (xchange_core $l; xsimp))"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules", "content": "macro_rules\n  | `({ $P }[$t:lang]{$v, $Q}) => `(triple [lang| $t] $P (fun $v => $Q))\n  | `({ $P }[$t:lang]{$Q})     => `(triple [lang| $t] $P (fun _ => $Q))\n  | `(WP[$t:lang]{$v, $Q})     => `(wp [lang| $t] (fun $v => $Q))\n  | `(WP[$t:lang]{$Q})         => `(wp [lang| $t] (fun _ => $Q))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic| xstep $(t)? ) => `(tactic| (xwp; xapp $(t)?))"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "HWand", "content": "class HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hWand : α → β → γ"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "hpure", "content": "def hpure (P : Prop) : hProp :=\n  hexists (fun (_ : P) => emp)"}, {"name": "hexists", "content": "def hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}, {"name": "infixr:55 \" -∗ \" => HWand.hWand", "content": "infixr:55 \" -∗ \" => HWand.hWand"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "Int.le_induction_down", "module": "Mathlib.Data.Int.Init"}], "repo_lemmas": [{"name": "himpl_trans", "content": "lemma himpl_trans H2 H1 H3 :\n  (H1 ==> H2) → (H2 ==> H3) → (H1 ==> H3)"}], "used_local_defs": [{"name": "Theories.formula", "content": "abbrev formula := (val → hProp) → hProp"}, {"name": "Theories.mkstruct", "content": "def mkstruct (F : formula) :=\n  fun (Q : val -> hProp) ↦ ∃ʰ Q', F Q' ∗ (Q' -∗ Q)"}, {"name": "Theories.structural", "content": "def structural (F : formula) :=\n  forall Q, mkstruct F Q ==> F Q"}, {"name": "Theories.structural_pred", "content": "def structural_pred (S : α -> formula) :=\n  ∀ x, structural $ S x"}, {"name": "Theories.wpgen_val", "content": "def wpgen_val (v : val) : formula :=\n  fun Q ↦ Q v"}, {"name": "Theories.wpgen_seq", "content": "def wpgen_seq (F1 F2 : formula) : formula :=\n  fun Q ↦ F1 (fun _ ↦ F2 Q)"}, {"name": "Theories.wpgen_for", "content": "def wpgen_for (v₁ v₂ : trm) (F1 : val -> formula) : formula :=\n  mkstruct fun Q =>\n    ∃ʰ n₁ n₂ : Int, ⌜v₁ = n₁⌝ ∗ ⌜v₂ = n₂⌝ ∗\n      h∀ (S : Int -> formula),\n        (let F i :=\n          if i < n₂ then\n            wpgen_seq (F1 (val_int i)) (S (i + 1))\n          else wpgen_val val_unit\n        ⌜structural_pred S /\\ ∀ i, F i ===> S i⌝ -∗ S n₁ Q )"}], "used_local_lemmas": [{"name": "Theories.mkstruct_erase", "content": "lemma mkstruct_erase Q F :\n  F Q ==> mkstruct F Q"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Indexes\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WPUtil\n\nopen trm val prim\n\nnamespace Theories\n\nabbrev formula := (val → hProp) → hProp\n\ndef mkstruct (F : formula) :=\n  fun (Q : val -> hProp) ↦ ∃ʰ Q', F Q' ∗ (Q' -∗ Q)\n\ndef structural (F : formula) :=\n  forall Q, mkstruct F Q ==> F Q\n\ndef structural_pred (S : α -> formula) :=\n  ∀ x, structural $ S x\n\ndef wpgen_val (v : val) : formula :=\n  fun Q ↦ Q v\n\ndef wpgen_seq (F1 F2 : formula) : formula :=\n  fun Q ↦ F1 (fun _ ↦ F2 Q)\n\ndef wpgen_for (v₁ v₂ : trm) (F1 : val -> formula) : formula :=\n  mkstruct fun Q =>\n    ∃ʰ n₁ n₂ : Int, ⌜v₁ = n₁⌝ ∗ ⌜v₂ = n₂⌝ ∗\n      h∀ (S : Int -> formula),\n        (let F i :=\n          if i < n₂ then\n            wpgen_seq (F1 (val_int i)) (S (i + 1))\n          else wpgen_val val_unit\n        ⌜structural_pred S /\\ ∀ i, F i ===> S i⌝ -∗ S n₁ Q )\n\nsection tactics\n\nopen Lean Elab Tactic\n\nsection xapp\n\nend xapp\n\nend tactics\n\nopen AList\n\nsection funs_fixs_eval_like\n\nvariable (xs : List var) (vs : List val) (t : trm) (v0 : trm)\n  (heqt : t = trm_apps v0 ts)\n  (hconv : trms_to_vals ts = vs)\n  (hform : var_funs xs vs.length)   -- NOTE: can be relaxed to `vs.length ≤ xs.length`\n\nvariable (f : var) (hf : f ∉ xs)\n\nend funs_fixs_eval_like\n\nend Theories\n\nopen Theories", "target_theorem": "lemma xfor_inv_lemma (I : Int -> hProp) (a b : Int)\n  (F : val -> formula)\n  (Q : val -> hProp) :\n  structural_pred F ->\n  a <= b ->\n    (∃ H',\n      H ==> I a ∗ H' ∧\n      (∀ i, a <= i ∧ i < b -> I i ==> F i fun _ => I (i + 1)) ∧\n      (fun _ => I b ∗ H') ===> Q) ->\n    H ==> wpgen_for a b F Q :=", "ground_truth_proof": ":= by\n  move=> sF L ![H' Ma Mb Mc]\n  unfold wpgen_for\n  apply himpl_trans; rotate_left; apply mkstruct_erase\n  unfold_let\n  xsimp[a,b]=> //== ls\n  srw OfNat.ofNat instOfNat instOfNatNat /== => hs\n  -- shave-> ls hs: i + (OfNat.ofNat 1) = i + 1; sdone\n  shave P: ∀ i, a <= i ∧ i <= b -> I i ==> S i fun _ => I b\n  { move=> i [/[swap] iLb]\n    apply (Int.le_induction_down _ _ _ iLb)\n    { move=> ?\n      xchange (hs b)=> /==\n      sby xval }\n    move=> i ? ih ?\n    xchange hs\n    scase_if=> // ?; rotate_left; omega\n    xseq\n    xchange Mb;\n    srw OfNat.ofNat instOfNat instOfNatNat /==\n    omega\n    apply himpl_trans; rotate_left; apply sF\n    unfold mkstruct; xsimp; apply ih; omega }\n  xchange Ma; xchange P; omega\n  apply himpl_trans; rotate_left; apply ls\n  unfold mkstruct; xsimp; apply Mc", "nesting_depth": 8, "transitive_dep_count": 40, "subset_aristotle": false, "category": "Framework"}
{"id": 181, "thm_name": "Matrix.PosSemidef.PosDef_iff_det_ne_zero", "thm_stmt": "lemma PosSemidef.PosDef_iff_det_ne_zero [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosSemidef) :\n    M.PosDef ↔ M.det ≠ 0", "lean_root": "CvxLean", "rel_path": "CvxLean/Lib/Math/Subadditivity.lean", "imports": ["import Mathlib.LinearAlgebra.Matrix.DotProduct", "import CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef", "import CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum", "import CvxLean.Lib.Math.LinearAlgebra.Eigenspace", "import Mathlib.LinearAlgebra.Matrix.LDL", "import Mathlib.LinearAlgebra.Matrix.PosDef", "import Mathlib.LinearAlgebra.Matrix.Spectrum", "import Mathlib.LinearAlgebra.Eigenspace.Basic"], "used_lib_defs": [{"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Matrix.IsHermitian", "module": "Mathlib.LinearAlgebra.Matrix.Hermitian"}, {"name": "A", "module": "examples.CircleOptimisation"}, {"name": "Matrix.diagonal", "module": "Mathlib.Data.Matrix.Diagonal"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Invertible", "module": "Mathlib.Algebra.Group.Invertible.Defs"}, {"name": "Matrix.invertibleOfIsUnitDet", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Float", "module": "Init.Data.Float"}, {"name": "Fin.elim0", "module": "Init.Data.Fin.Basic"}, {"name": "Float.ofNat", "module": "Init.Data.OfScientific"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "EuclideanSpace", "module": "Mathlib.Analysis.InnerProductSpace.PiL2"}, {"name": "RCLike", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "x", "module": "doc.literate.literate_lean_test"}, {"name": "Matrix.PosSemidef", "module": "Mathlib.LinearAlgebra.Matrix.PosDef"}, {"name": "Real", "module": "Mathlib.Data.Real.Basic"}, {"name": "Matrix.PosSemidef.sqrt", "module": "Mathlib.Analysis.Matrix.Order"}, {"name": "RCLike.ofReal", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "Matrix.toLin'", "module": "Mathlib.LinearAlgebra.Matrix.ToLin"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Module.End", "module": "Mathlib.Algebra.Module.LinearMap.End"}, {"name": "Module.End.HasEigenvector", "module": "Mathlib.LinearAlgebra.Eigenspace.Basic"}, {"name": "OrthonormalBasis", "module": "Mathlib.Analysis.InnerProductSpace.PiL2"}, {"name": "Pi.basisFun", "module": "Mathlib.LinearAlgebra.StdBasis"}], "used_repo_defs": [{"name": "PosDef.Invertible", "content": "noncomputable instance PosDef.Invertible [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) :\n    Invertible M :=\n  invertibleOfIsUnitDet M (isUnit_iff_ne_zero.2 hM.det_ne_zero)"}, {"name": "det", "content": "def det {n : ℕ} (A : Matrix (Fin n) (Fin n) Float) : Float :=\n  if h : 0 < n then\n    if n == 1 then A ⟨0, h⟩ ⟨0, h⟩ else\n      (List.finRange n).foldl (fun s i =>\n        s + (-1) ^ (Float.ofNat i.val) * A i ⟨0, h⟩ * det (minor A i ⟨0, h⟩)) 0\n  else 0"}, {"name": "minor", "content": "def minor (A : Matrix (Fin n) (Fin n) Float) (a b : Fin n) :\n    Matrix (Fin n.pred) (Fin n.pred) Float :=\n  match n with\n  | 0 => fun _ => Fin.elim0\n  | _ + 1 => minorAux A a b"}, {"name": "minorAux", "content": "private def minorAux (A : Matrix (Fin n.succ) (Fin n.succ) Float) (a b : Fin n.succ) :\n    Matrix (Fin n) (Fin n) Float :=\n  fun i j =>\n    let i' : Fin n.succ := if i.val < a.val then i else i.succ;\n    let j' : Fin n.succ := if j.val < b.val then j else j.succ;\n    A i' j'"}], "lib_lemmas": [{"name": "Real.sqrt_nonneg", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "Matrix.mul_assoc", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.one_mul", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.diagonal_mul_diagonal", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.mul_one", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Real.sqrt_mul", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "Real.sqrt_mul_self", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "Matrix.det_mul", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "Matrix.dotProduct_mulVec", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.exists_mulVec_eq_zero_iff", "module": "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv"}, {"name": "Matrix.mulVec_mulVec", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.mul_zero", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.transpose_transpose", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Matrix.vecMul_transpose", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "RCLike.re_to_real", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "inner_self_eq_zero", "module": "Mathlib.Analysis.InnerProductSpace.Basic"}, {"name": "lt_of_le_of_ne'", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "PosSemidef_diagonal", "content": "lemma PosSemidef_diagonal [DecidableEq n] {f : n → ℝ} (hf : ∀ i, 0 ≤ f i) :\n    (diagonal f).PosSemidef"}, {"name": "PosSemidef.conjTranspose_mul_mul", "content": "lemma PosSemidef.conjTranspose_mul_mul (M N : Matrix n n 𝕜) (hM : M.PosSemidef) :\n    (Nᴴ * M * N).PosSemidef"}, {"name": "spectral_theorem", "content": "theorem spectral_theorem (xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)) (as : n → ℝ)\n    (hxs : ∀ j, Module.End.HasEigenvector (Matrix.toLin' A) (as j) (xs j)) :\n    xs.toBasis.toMatrix (Pi.basisFun 𝕜 n) * A =\n    diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix (Pi.basisFun 𝕜 n)"}, {"name": "PosDef.det_ne_zero", "content": "lemma PosDef.det_ne_zero [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : M.det ≠ 0"}], "used_local_defs": [{"name": "Matrix.IsHermitian.sqrt", "content": "noncomputable def IsHermitian.sqrt {A : Matrix n n ℝ} (hA : A.IsHermitian) : Matrix n n ℝ :=\n  hA.eigenvectorMatrix * Matrix.diagonal (fun i => (hA.eigenvalues i).sqrt) * hA.eigenvectorMatrixᵀ"}], "used_local_lemmas": [{"name": "Matrix.IsHermitian.eigenvectorMatrix_inv_mul", "content": "lemma eigenvectorMatrix_inv_mul : hA.eigenvectorMatrixInv * hA.eigenvectorMatrix = 1"}, {"name": "Matrix.IsHermitian.spectral_theorem''", "content": "theorem spectral_theorem'' :\n    hA.eigenvectorMatrix * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * hA.eigenvectorMatrixᴴ =\n    A"}, {"name": "Matrix.conjTranspose_eq_transpose", "content": "lemma conjTranspose_eq_transpose {m n : Type _} {A : Matrix m n ℝ} : Aᴴ = Aᵀ"}, {"name": "Matrix.PosSemidef.sqrt_mul_sqrt", "content": "@[simp]\nlemma PosSemidef.sqrt_mul_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :\n    hA.1.sqrt * hA.1.sqrt = A"}, {"name": "Matrix.PosSemidef.PosSemidef_sqrt", "content": "lemma PosSemidef.PosSemidef_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :\n    hA.1.sqrt.PosSemidef"}], "local_ctx": "import Mathlib.LinearAlgebra.Matrix.PosDef\n\nimport Mathlib.LinearAlgebra.Matrix.Spectrum\n\nimport Mathlib.LinearAlgebra.Eigenspace.Basic\n\nimport Mathlib.LinearAlgebra.Matrix.LDL\n\nimport Mathlib.LinearAlgebra.Matrix.DotProduct\n\nimport CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef\n\nimport CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum\n\nimport CvxLean.Lib.Math.LinearAlgebra.Eigenspace\n\nnamespace Finset\n\nopen BigOperators\n\nend Finset\n\nnamespace Matrix\n\nvariable {n : Type _} [Fintype n] [DecidableEq n] [LinearOrder n] [LocallyFiniteOrderBot n]\n\nopen BigOperators Matrix\n\nnamespace IsHermitian\n\nvariable {𝕜 : Type _} [DecidableEq 𝕜] [RCLike 𝕜] {A : Matrix n n 𝕜} (hA : A.IsHermitian)\n\nend IsHermitian\n\nnoncomputable def IsHermitian.sqrt {A : Matrix n n ℝ} (hA : A.IsHermitian) : Matrix n n ℝ :=\n  hA.eigenvectorMatrix * Matrix.diagonal (fun i => (hA.eigenvalues i).sqrt) * hA.eigenvectorMatrixᵀ", "target_theorem": "lemma PosSemidef.PosDef_iff_det_ne_zero [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosSemidef) :\n    M.PosDef ↔ M.det ≠ 0 :=", "ground_truth_proof": ":= by\n  refine' ⟨PosDef.det_ne_zero, _⟩; intro hdet; refine' ⟨hM.1, _⟩\n  intros x hx\n  apply lt_of_le_of_ne' (hM.2 x)\n  rw [← hM.sqrt_mul_sqrt, ← mulVec_mulVec, dotProduct_mulVec, ← transpose_transpose hM.1.sqrt,\n    vecMul_transpose, transpose_transpose, ← conjTranspose_eq_transpose,\n    hM.PosSemidef_sqrt.1.eq]\n  simp only [RCLike.re_to_real, star, id]\n  change @inner ℝ (EuclideanSpace ℝ _) _ (hM.1.sqrt.mulVec x) (hM.1.sqrt.mulVec x) ≠ 0\n  intro hinner\n  have sqrtMdet0 : hM.1.sqrt.det = 0 := by\n    refine' exists_mulVec_eq_zero_iff.1 ⟨x, hx, _⟩\n    rw [inner_self_eq_zero.1 hinner]\n  rw [← hM.sqrt_mul_sqrt, det_mul, sqrtMdet0, mul_zero] at hdet\n  apply hdet rfl", "nesting_depth": 4, "transitive_dep_count": 58, "subset_aristotle": false, "category": "Applied verif."}
{"id": 182, "thm_name": "Theories.wp_ref", "thm_stmt": "lemma wp_ref x v t Q :\n  (h∀ p, (p ~~> v) -∗ wp (subst x p t) (Q ∗ ∃ʰ v', (p ~~> v'))) ==>\n  wp (trm_ref x v t) Q", "lean_root": "splean", "rel_path": "SPLean/Theories/WP1.lean", "imports": ["import SPLean.Theories.XChange", "import Mathlib.Data.List.Indexes", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Lean", "import SPLean.Theories.WPUtil"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}], "used_repo_defs": [{"name": "syntax \"if \" lang \"then \" lang \"end \" : lang", "content": "syntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"fun\" ident+ \" => \" lang : lang"}, {"name": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hforall xs b"}, {"name": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hexists xs b"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "hsingle", "content": "def hsingle (p : loc) (v : val) : hProp :=\n  fun h => (h = Finmap.singleton p v)"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "HWand", "content": "class HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hWand : α → β → γ"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "hforall", "content": "def hforall {A} (J : A → hProp) : hProp :=\n  fun h => forall x, J x h"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:60 \" ~~> \" => hsingle", "content": "infixr:60 \" ~~> \" => hsingle"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}, {"name": "infixr:55 \" -∗ \" => HWand.hWand", "content": "infixr:55 \" -∗ \" => HWand.hWand"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "Finmap.union_comm_of_disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.Disjoint.symm", "module": "Mathlib.Data.Finmap"}], "repo_lemmas": [{"name": "eval_conseq", "content": "lemma eval_conseq s t Q1 Q2 :\n  eval s t Q1 →\n  Q1 ===> Q2 →\n  eval s t Q2"}, {"name": "hwand_inv", "content": "lemma hwand_inv h1 h2 H1 H2 :\n  (H1 -∗ H2) h2 →\n  H1 h1 →\n  Finmap.Disjoint h1 h2 →\n  H2 (h1 ∪ h2)"}, {"name": "hsingl_inv", "content": "lemma hsingl_inv p v h :\n  (p ~~> v) h →\n  h = Finmap.singleton p v"}, {"name": "hforall_inv", "content": "lemma hforall_inv A (J : A → hProp) h :\n  (hforall J) h → forall x, J x h"}, {"name": "union_singleton_eq_insert", "content": "lemma union_singleton_eq_insert (h : state) :\n  Finmap.singleton p v ∪ h = h.insert p v"}, {"name": "insert_delete_id", "content": "lemma insert_delete_id (h : state) (p : loc) :\n  p ∉ h →\n  h = (h.insert p v).erase p"}], "used_local_defs": [{"name": "Theories.wp", "content": "def wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q"}], "used_local_lemmas": [], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Indexes\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WPUtil\n\nopen trm val prim\n\nnamespace Theories\n\ndef wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q", "target_theorem": "lemma wp_ref x v t Q :\n  (h∀ p, (p ~~> v) -∗ wp (subst x p t) (Q ∗ ∃ʰ v', (p ~~> v'))) ==>\n  wp (trm_ref x v t) Q :=", "ground_truth_proof": ":=\nby\n  move=> > /hforall_inv hwp\n  apply (eval.eval_ref _ _ _ _ _ (fun v' s' ↦ v' = v ∧ s' = h))=> //== > ?\n  apply (eval_conseq _ _ (Q ∗ ∃ʰ v', p ~~> v' ))\n  { move: (hwp p)=> {hwp} /(hwand_inv (Finmap.singleton p v))\n    srw union_singleton_eq_insert=> wp\n    apply wp=> //\n    sby unfold Finmap.Disjoint=> > /== -> }\n  move=> > s ![>] ? [v'] /= /hsingl_inv -> hdis ->\n  srw Finmap.union_comm_of_disjoint=> //\n  srw union_singleton_eq_insert -insert_delete_id=> //\n  move: hdis=> /Finmap.Disjoint.symm\n  sby unfold Finmap.Disjoint", "nesting_depth": 5, "transitive_dep_count": 49, "subset_aristotle": false, "category": "Framework"}
{"id": 183, "thm_name": "exact_imp_eval", "thm_stmt": "lemma exact_imp_eval :\n  evalExact s t Q → eval s t Q", "lean_root": "splean", "rel_path": "SPLean/Theories/Lang.lean", "imports": ["import Mathlib.Data.Finmap", "import SPLean.Common.Heap", "import Mathlib.Data.Real.Basic", "import SPLean.Common.Util"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "Int.natAbs", "module": "Init.Data.Int.Basic"}, {"name": "Finmap.lookup", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "seq", "module": "Talk.DemoLeanSSR"}, {"name": "Finmap.insert", "module": "Mathlib.Data.Finmap"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}], "used_repo_defs": [{"name": "syntax \"fun\" ident+ \" => \" lang : lang", "content": "syntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" <= \" : bop\n\nsyntax \" >= \" : bop\n\nsyntax \"not\" : uop\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"ref\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"fix\" ident ident+ \" => \" lang : lang\n\nsyntax \"for\" ident \" in \" \"[\" lang \" : \" lang \"]\" \" {\"  (ppLine lang) ( \" }\") : lang\n\nsyntax \"while\" lang  \" {\"  (ppLine lang) ( \" }\") : lang\n\nsyntax \"alloc\" lang \" as \" ident \" in\" ppDedent(ppLine lang) : lang"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "null", "content": "def null : loc := 0"}, {"name": "val", "content": "  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "trm", "content": "  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "trm_is_val", "content": "abbrev trm_is_val : trm → Prop\n  | trm_val _ => true\n  | _         => false"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "is_true", "content": "noncomputable def is_true (P : Prop) : Bool :=\n  if P then true else false"}, {"name": "read_state", "content": "def read_state (p : loc) (h : state) :=\n  match Finmap.lookup p h with\n  | some v => v\n  | none   => default"}, {"name": "evalunop", "content": "inductive evalunop : prim → val → (val → Prop) → Prop where\n  | evalunop_neg : forall b1,\n      evalunop val_neg (val_bool b1) (fun v => v = val_bool (¬ b1))\n  | evalunop_opp : forall n1,\n      evalunop val_opp (val_int n1) (fun v => v = val_int (- n1))\n  | evalunop_oppr : forall r1,\n      evalunop val_opp (val_real r1) (fun v => v = val_real (- r1))"}, {"name": "evalbinop", "content": "inductive evalbinop : val → val → val → (val->Prop) → Prop where\n  | evalbinop_eq : forall v1 v2,\n      evalbinop val_eq v1 v2 (fun v => v = val_bool (is_true (v1 = v2)))\n  | evalbinop_neq : forall v1 v2,\n      evalbinop val_neq v1 v2 (fun v => v = val_bool (is_true (v1 ≠ v2)))\n  | evalbinop_add : forall n1 n2,\n      evalbinop val_add (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 + n2))\n  | evalbinop_addr : forall r₁ r₂,\n      evalbinop val_add (val_real r₁) (val_real r₂)\n        (fun v => v = val_real (r₁ + r₂))\n  | evalbinop_sub : forall n1 n2,\n      evalbinop val_sub (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 - n2))\n  | evalbinop_subr : forall r1 r2,\n      evalbinop val_sub (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 - r2))\n  | evalbinop_mul : forall n1 n2,\n      evalbinop val_mul (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 * n2))\n  | evalbinop_mulr : forall r1 r2,\n      evalbinop val_mul (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 * r2))\n  | evalbinop_div : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_div (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 / n2))\n  | evalbinop_divr : forall r1 r2,\n      ¬(r2 = 0) →\n      evalbinop val_div (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 / r2))\n  | evalbinop_mod : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_mod (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 % n2))\n  | evalbinop_le : forall n1 n2,\n      evalbinop val_le (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 <= n2))\n  | evalbinop_ler : forall r1 r2,\n      evalbinop val_le (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 <= r2))\n  | evalbinop_lt : forall n1 n2,\n      evalbinop val_lt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 < n2))\n  | evalbinop_ltr : forall r1 r2,\n      evalbinop val_lt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 < r2))\n  | evalbinop_ge : forall n1 n2,\n      evalbinop val_ge (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 >= n2))\n  | evalbinop_ger : forall r1 r2,\n      evalbinop val_ge (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 >= r2))\n  | evalbinop_gt : forall n1 n2,\n      evalbinop val_gt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 > n2))\n  | evalbinop_gtr : forall r1 r2,\n      evalbinop val_gt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 > r2))\n\n  \n  \n  \n  \n  | evalbinop_ptr_add : forall (p1 : loc) (p2 : Int) n,\n      p2 = p1 + n ->\n      evalbinop val_ptr_add (val_loc p1) (val_int n)\n        (fun v => v = val_loc (Int.natAbs p2))"}, {"name": "purepost", "content": "def purepost (s : state) (P : val → Prop) : val → state → Prop :=\n  fun v s' => P v ∧ s' = s"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "conseq", "content": "def conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=\n  match vs with\n  | [] => ∅\n  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))"}, {"name": "make_list", "content": "def make_list {A} (n : Nat) (v : A) : List A :=\n  match n with\n  | 0      => []\n  | n' + 1 => v :: make_list n' v"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}], "used_local_lemmas": [], "local_ctx": "import Mathlib.Data.Finmap\n\nimport Mathlib.Data.Real.Basic\n\nimport SPLean.Common.Util\n\nimport SPLean.Common.Heap\n\nopen Classical\n\ninductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\ndef null : loc := 0\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm\n\nend\n\nabbrev state := Finmap (λ _ : loc ↦ val)\n\nsection\n\nopen trm\n\nabbrev trm_is_val : trm → Prop\n  | trm_val _ => true\n  | _         => false\n\ndef subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n\nnoncomputable def is_true (P : Prop) : Bool :=\n  if P then true else false\n\nopen val\n\nopen prim\n\ndef read_state (p : loc) (h : state) :=\n  match Finmap.lookup p h with\n  | some v => v\n  | none   => default\n\ninductive evalunop : prim → val → (val → Prop) → Prop where\n  | evalunop_neg : forall b1,\n      evalunop val_neg (val_bool b1) (fun v => v = val_bool (¬ b1))\n  | evalunop_opp : forall n1,\n      evalunop val_opp (val_int n1) (fun v => v = val_int (- n1))\n  | evalunop_oppr : forall r1,\n      evalunop val_opp (val_real r1) (fun v => v = val_real (- r1))\n\ninductive evalbinop : val → val → val → (val->Prop) → Prop where\n  | evalbinop_eq : forall v1 v2,\n      evalbinop val_eq v1 v2 (fun v => v = val_bool (is_true (v1 = v2)))\n  | evalbinop_neq : forall v1 v2,\n      evalbinop val_neq v1 v2 (fun v => v = val_bool (is_true (v1 ≠ v2)))\n  | evalbinop_add : forall n1 n2,\n      evalbinop val_add (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 + n2))\n  | evalbinop_addr : forall r₁ r₂,\n      evalbinop val_add (val_real r₁) (val_real r₂)\n        (fun v => v = val_real (r₁ + r₂))\n  | evalbinop_sub : forall n1 n2,\n      evalbinop val_sub (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 - n2))\n  | evalbinop_subr : forall r1 r2,\n      evalbinop val_sub (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 - r2))\n  | evalbinop_mul : forall n1 n2,\n      evalbinop val_mul (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 * n2))\n  | evalbinop_mulr : forall r1 r2,\n      evalbinop val_mul (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 * r2))\n  | evalbinop_div : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_div (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 / n2))\n  | evalbinop_divr : forall r1 r2,\n      ¬(r2 = 0) →\n      evalbinop val_div (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 / r2))\n  | evalbinop_mod : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_mod (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 % n2))\n  | evalbinop_le : forall n1 n2,\n      evalbinop val_le (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 <= n2))\n  | evalbinop_ler : forall r1 r2,\n      evalbinop val_le (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 <= r2))\n  | evalbinop_lt : forall n1 n2,\n      evalbinop val_lt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 < n2))\n  | evalbinop_ltr : forall r1 r2,\n      evalbinop val_lt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 < r2))\n  | evalbinop_ge : forall n1 n2,\n      evalbinop val_ge (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 >= n2))\n  | evalbinop_ger : forall r1 r2,\n      evalbinop val_ge (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 >= r2))\n  | evalbinop_gt : forall n1 n2,\n      evalbinop val_gt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 > n2))\n  | evalbinop_gtr : forall r1 r2,\n      evalbinop val_gt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 > r2))\n\n  \n  \n  \n  \n  | evalbinop_ptr_add : forall (p1 : loc) (p2 : Int) n,\n      p2 = p1 + n ->\n      evalbinop val_ptr_add (val_loc p1) (val_int n)\n        (fun v => v = val_loc (Int.natAbs p2))\n\nsection eval\n\ndef purepost (s : state) (P : val → Prop) : val → state → Prop :=\n  fun v s' => P v ∧ s' = s\n\ndef purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s\n\nvariable (Q : val → state → Prop)\n\ndef conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=\n  match vs with\n  | [] => ∅\n  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))\n\ndef make_list {A} (n : Nat) (v : A) : List A :=\n  match n with\n  | 0      => []\n  | n' + 1 => v :: make_list n' v\n\ninductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q\n\ninductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q", "target_theorem": "lemma exact_imp_eval :\n  evalExact s t Q → eval s t Q :=", "ground_truth_proof": ":= by\n  elim=> >\n  { sby constructor }\n  { sby constructor }\n  { sby constructor }\n  { move=> * ; sby constructor }\n  { move=> * ; sby apply eval.eval_app_arg2 }\n  { move=> * ; sby apply eval.eval_app_fun }\n  { move=> * ; sby apply eval.eval_app_fix }\n  { move=> ??? h ; apply (eval.eval_seq Q1)=>// ; exact h }\n  { move=> * ; sby constructor }\n  { move=> * ; sby constructor }\n  { move=> * ; apply eval.eval_unop=> //\n    sby unfold purepostin purepost }\n  { move=> * ; apply eval.eval_binop=> //\n    sby unfold purepostin purepost }\n  { move=> * ; sby apply eval.eval_ref }\n  { move=> * ; sby apply eval.eval_get }\n  { move=> * ; sby apply eval.eval_set }\n  { move=> * ; sby apply eval.eval_alloc_arg }\n  { move=> * ; sby apply eval.eval_alloc }\n  { move=> ?ih ; sby constructor }\n  move=> * ; sby constructor", "nesting_depth": 6, "transitive_dep_count": 36, "subset_aristotle": false, "category": "Framework"}
{"id": 184, "thm_name": "Matrix.LogDetAtom.cond_elim", "thm_stmt": "lemma LogDetAtom.cond_elim {t : n → ℝ} {Y Z D : Matrix n n ℝ} (ht : ∀ i, (t i).exp ≤ Y.diag i)\n    (hD : D = Matrix.diagonal (Y.diag)) (hZ : Z = Y.toUpperTri)\n    (hPSD : (fromBlocks D Z Zᵀ A).PosSemidef) : A.PosDef", "lean_root": "CvxLean", "rel_path": "CvxLean/Lib/Math/LogDet.lean", "imports": ["import CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri", "import CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef", "import Mathlib.LinearAlgebra.Matrix.LDL", "import CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL", "import CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular", "import CvxLean.Lib.Math.SchurComplement", "import CvxLean.Lib.Math.Subadditivity"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Float", "module": "Init.Data.Float"}, {"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Invertible", "module": "Mathlib.Algebra.Group.Invertible.Defs"}, {"name": "Matrix.invertibleOfIsUnitDet", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Fin.elim0", "module": "Init.Data.Fin.Basic"}, {"name": "Float.ofNat", "module": "Init.Data.OfScientific"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "LinearOrder", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Matrix.diagonal", "module": "Mathlib.Data.Matrix.Diagonal"}, {"name": "Matrix.fromBlocks", "module": "Mathlib.Data.Matrix.Block"}, {"name": "Matrix.PosSemidef", "module": "Mathlib.LinearAlgebra.Matrix.PosDef"}, {"name": "Matrix.det", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Matrix.IsHermitian", "module": "Mathlib.LinearAlgebra.Matrix.Hermitian"}, {"name": "A", "module": "examples.CircleOptimisation"}, {"name": "Matrix.PosSemidef.sqrt", "module": "Mathlib.Analysis.Matrix.Order"}, {"name": "RCLike", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "RCLike.ofReal", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Matrix.vecMul", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "IsUnit", "module": "Mathlib.Algebra.Group.Units.Defs"}, {"name": "Matrix.mulVec", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.toLin'", "module": "Mathlib.LinearAlgebra.Matrix.ToLin"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Module.End", "module": "Mathlib.Algebra.Module.LinearMap.End"}, {"name": "Module.End.HasEigenvector", "module": "Mathlib.LinearAlgebra.Eigenspace.Basic"}, {"name": "LT", "module": "Init.Prelude"}, {"name": "Matrix.BlockTriangular", "module": "Mathlib.LinearAlgebra.Matrix.Block"}, {"name": "id", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "toUpperTri", "content": "def toUpperTri (A : Matrix (Fin n) (Fin n) Float) : Matrix (Fin n) (Fin n) Float :=\n  fun i j => if i ≤ j then A i j else 0"}, {"name": "PosDef.Invertible", "content": "noncomputable instance PosDef.Invertible [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) :\n    Invertible M :=\n  invertibleOfIsUnitDet M (isUnit_iff_ne_zero.2 hM.det_ne_zero)"}, {"name": "diag", "content": "def diag {n} {α} (M : Matrix n n α) : n → α :=\n  fun i => M i i"}, {"name": "det", "content": "def det {n : ℕ} (A : Matrix (Fin n) (Fin n) Float) : Float :=\n  if h : 0 < n then\n    if n == 1 then A ⟨0, h⟩ ⟨0, h⟩ else\n      (List.finRange n).foldl (fun s i =>\n        s + (-1) ^ (Float.ofNat i.val) * A i ⟨0, h⟩ * det (minor A i ⟨0, h⟩)) 0\n  else 0"}, {"name": "minor", "content": "def minor (A : Matrix (Fin n) (Fin n) Float) (a b : Fin n) :\n    Matrix (Fin n.pred) (Fin n.pred) Float :=\n  match n with\n  | 0 => fun _ => Fin.elim0\n  | _ + 1 => minorAux A a b"}, {"name": "minorAux", "content": "private def minorAux (A : Matrix (Fin n.succ) (Fin n.succ) Float) (a b : Fin n.succ) :\n    Matrix (Fin n) (Fin n) Float :=\n  fun i j =>\n    let i' : Fin n.succ := if i.val < a.val then i else i.succ;\n    let j' : Fin n.succ := if j.val < b.val then j else j.succ;\n    A i' j'"}, {"name": "toUpperTri", "content": "def toUpperTri {m α : Type _} [LinearOrder m] [Zero α] (A : Matrix m m α) :\n    Matrix m m α :=\n  fun i j => if i ≤ j then A i j else 0"}, {"name": "IsHermitian.sqrt", "content": "noncomputable def IsHermitian.sqrt {A : Matrix n n ℝ} (hA : A.IsHermitian) : Matrix n n ℝ :=\n  hA.eigenvectorMatrix * Matrix.diagonal (fun i => (hA.eigenvalues i).sqrt) * hA.eigenvectorMatrixᵀ"}, {"name": "SymmMatrixVariable", "content": "structure SymmMatrixVariable where\n  index  : Nat\n  name   : String\n  I      : Nat\n  J      : Nat\n  primal : Option Float\n  dual   : Option Float"}, {"name": "", "content": "instance : ToString SymmMatrixVariable where\n  toString smv :=\n    s!\"{smv.index} | \" ++\n    s!\"{smv.name} | \" ++\n    s!\"{smv.I} | \" ++\n    s!\"{smv.J} | \" ++\n    s!\"{smv.primal} | \" ++\n    s!\"{smv.dual}\""}, {"name": "Result", "content": "structure Result where\n  summary        : Summary\n  constraints    : List Constraint\n  vars           : List Variable\n  symmMatrixVars : List SymmMatrixVariable"}, {"name": "", "content": "instance : ToString Result where\n  toString res :=\n    (toString res.summary) ++ \"\\n\" ++\n    \"CONSTRAINTS \\n\" ++\n    \"INDEX | NAME | AT | ACTIVITY | LOWER LIMIT | UPPER LIMIT | DUAL LOWER | DUAL UPPER \\n\" ++\n    (res.constraints.foldl (fun acc s => acc ++ (toString s) ++ \"\\n\") \"\") ++ \"\\n\" ++\n    \"VARIABLES \\n\" ++\n    \"INDEX | NAME | AT | ACTIVITY | LOWER LIMIT | UPPER LIMIT | DUAL LOWER | DUAL UPPER | \" ++\n    \"CONIC DUAL \\n\" ++\n    (res.vars.foldl (fun acc s => acc ++ (toString s) ++ \"\\n\") \"\") ++ \"\\n\" ++\n    \"SYMMETRIC MATRIX VARIABLES \\n\" ++\n    \"INDEX | NAME | I | J | PRIMAL | DUAL \\n\" ++\n    (res.symmMatrixVars.foldl (fun acc s => acc ++ (toString s) ++ \"\\n\") \"\") ++ \"\\n\""}, {"name": "symmMatrixVarsTitle", "content": "private def symmMatrixVarsTitle : Parsec Unit :=\n  skipString \"SYMMETRIC MATRIX VARIABLES\" <* endOfLine"}, {"name": "symmMatrixVarsHeaders", "content": "private def symmMatrixVarsHeaders : Parsec Unit :=\n  skipString \"INDEX\"  <* ws' <*\n  skipString \"NAME\"   <* ws' <*\n  skipString \"I\"      <* ws' <*\n  skipString \"J\"      <* ws' <*\n  skipString \"PRIMAL\" <* ws' <*\n  skipString \"DUAL\"   <* ws' <* endOfLine"}, {"name": "symmMatrixVar", "content": "private def symmMatrixVar : Parsec SymmMatrixVariable :=\n  SymmMatrixVariable.mk <$>\n  constraintElem nat         <*>\n  constraintElem string      <*>\n  constraintElem nat         <*>\n  constraintElem nat         <*>\n  constraintElem optionFloat <*>\n  constraintElem optionFloat <* ws' <* endOfLine"}, {"name": "symmMatrixVars", "content": "private def symmMatrixVars : Parsec (List SymmMatrixVariable) :=\n  (symmMatrixVarsTitle  *>\n  symmMatrixVarsHeaders *>\n  Array.data <$> many symmMatrixVar) <|> pure []"}, {"name": "result", "content": "def result : Parsec Result :=\n  Result.mk <$>\n  summary        <* ws <*>\n  constraints    <* ws <*>\n  vars           <* ws <*>\n  symmMatrixVars <* ws"}, {"name": "inferDimension", "content": "unsafe def inferDimension (ty : Expr) : MetaM (List (Nat × Nat)) :=\n  match ty.consumeMData with\n  | .const ``Real _ =>\n      return [(1, 1)]\n  | .forallE _ (.app (.const ``Fin _) nExpr) e _ =>\n    match e with\n    | .const ``Real _  => do\n      let n : Nat ← evalExpr Nat (mkConst ``Nat) nExpr\n      return [(n, 1)]\n    | .forallE _ (.app (.const ``Fin _) mExpr) (.const ``Real _) _ => do\n      let n : Nat ← evalExpr Nat (mkConst ``Nat) nExpr\n      let m : Nat ← evalExpr Nat (mkConst ``Nat) mExpr\n      return [(n, m)]\n    | _ => throwSolveError \"could not infer dimension of {ty}.\"\n  | .app (.app (.app M FinN) FinM) R => do\n    match (M, FinN, FinM, R) with\n    | (.const ``Matrix _, .app (.const ``Fin _) nExpr, .app (.const ``Fin _) mExpr,\n        .const ``Real _) =>\n      let n : Nat ← evalExpr Nat (mkConst ``Nat) nExpr\n      let m : Nat ← evalExpr Nat (mkConst ``Nat) mExpr\n      return [(n, m)]\n    | _ => throwSolveError \"could not infer dimension of {ty}.\"\n  | .app (.app (.const ``Prod _) tyl) tyr => do\n    let l ← inferDimension tyl\n    let r ← inferDimension tyr\n    return (l ++ r)\n  | _ => throwSolveError \"could not infer dimension of {ty}.\""}, {"name": "ofProblemData", "content": "unsafe def ofProblemData (minExpr : MinimizationExpr) (data : ProblemData)\n    (sections : ScalarAffineSections) : MetaM CBF.Problem := do\n  let totalDim ← getTotalDim minExpr\n\n  let mut cbf := CBF.Problem.empty\n  cbf := cbf.addScalarVariable (CBF.Cone.mk CBF.ConeType.F totalDim)\n\n  if h : data.objective.isSome then\n    let sa := data.objective.get h\n    let AEnc := CBF.EncodedMatrixList.fromArray #[sa.A]\n    let aEnc := CBF.EncodedVector.fromArray sa.a\n    let bEnc := CBF.EncodedValue.mk (some sa.b)\n    cbf := cbf.setObjectivePSDVariablesCoord AEnc\n    cbf := cbf.setObjectiveScalarVariablesCoord aEnc\n    cbf := cbf.setObjectiveShiftCoord bEnc\n\n  for (sa, sct) in data.scalarAffineConstraints do\n    let coneType := translateCone sct\n    let cone := CBF.Cone.mk coneType 1\n    let AEnc := CBF.EncodedMatrixList.fromArray #[sa.A]\n    let aEnc := CBF.EncodedVector.fromArray sa.a\n    let bEnc := sa.b\n    cbf := cbf.addScalarValuedAffineConstraint cone AEnc aEnc bEnc\n\n  for ma in data.matrixAffineConstraints do\n    let HEnc := CBF.EncodedMatrixList.fromArray ma.H\n    let DEnc := CBF.EncodedMatrix.fromArray ma.D\n    cbf := cbf.addMatrixValuedAffineConstraint ma.n HEnc DEnc\n\n  \n  let n := cbf.scalarConstraints.n\n  let cones := cbf.scalarConstraints.cones\n  let groupedCones ← groupCones sections cones\n  cbf := cbf.setScalarConstraints (CBF.ConeProduct.mk n groupedCones.length groupedCones)\n\n  return cbf"}, {"name": "SimpleMatrixVarSol", "content": "structure SimpleMatrixVarSol where\n  name  : String\n  I     : Nat\n  J     : Nat\n  value : Option Float"}, {"name": "", "content": "instance : ToString SimpleMatrixVarSol where\n  toString v :=\n    match v.value with\n    | some x => v.name ++ \"[\" ++ toString v.I ++ \", \" ++ toString v.J ++ \"]* = \" ++ toString x\n    | none   => \"none\""}, {"name": "SolutionData", "content": "structure SolutionData where\n  status         : String\n  varsSols       : List SimpleVarSol\n  matrixVarsSols : List SimpleMatrixVarSol"}, {"name": "", "content": "instance : ToString SolutionData where\n  toString s :=\n    \"Status: \" ++ s.status ++ \"\\n\" ++\n    \"Variables: (\" ++ \", \".intercalate (s.varsSols.map toString) ++ \")\\n\" ++\n    \"Matrix variables: (\" ++ \", \".intercalate (s.matrixVarsSols.map toString) ++ \")\""}, {"name": "p", "content": "def p :=\n  optimization (x y : ℝ)\n    minimize -2 * x\n    subject to\n      c₁ : 0 ≤ x\n      c₂ : 1 < y\n      c₃ : log (y - 1) ≤ 2 * sqrt x + 1\n      c₄ : 3 * x + 5 * y ≤ 10"}, {"name": "--", "content": "-- Transform (proving `p ≡ q`).\n\nequivalence* eqv/q : p :="}, {"name": "const", "content": "def const (k : α) : Matrix m n α :=\n  fun _ _ => k"}, {"name": "[Preorder", "content": "instance [Preorder α] : Preorder (Matrix m n α) where\n  le := fun A B => ∀ i j, A i j ≤ B i j\n  le_refl := fun _ _ _ => le_refl _\n  le_trans := fun _ _ _ hAB hBC i j => le_trans (hAB i j) (hBC i j)\n  lt_iff_le_not_le := fun _ _ => refl _"}, {"name": "abs", "content": "def abs (A : Matrix m n ℝ) : Matrix m n ℝ :=\n  fun i j => |A i j|"}, {"name": "sum", "content": "def sum [Fintype m] [AddCommMonoid α] (X : Matrix m m α) : α :=\n  ∑ i, (∑ j, X i j)"}, {"name": "toArray", "content": "def toArray (A : Matrix (Fin n) (Fin n) Float) : Array (Array Float) :=\n  (Array.range n).map <| fun i =>\n    if h : i < n then Vec.Computable.toArray (A ⟨i, h⟩) else Array.mk <| List.replicate n 0"}, {"name": "dotProduct", "content": "def dotProduct (v w : Fin n → Float) : Float :=\n  (Array.zipWith (Vec.Computable.toArray v) (Vec.Computable.toArray w) Float.mul).foldl Float.add 0"}, {"name": "mulVec", "content": "def mulVec (M : Matrix (Fin n) (Fin m) Float) (v : (Fin m) → Float) : Fin n → Float :=\n  fun i => (fun j => M i j) ⬝ᵥᶜ v"}, {"name": "vecMul", "content": "def vecMul (x : Fin n → Float) (M : Matrix (Fin n) (Fin m) Float) : Fin m → Float :=\n  fun j => x ⬝ᵥᶜ fun i => M i j"}, {"name": "transpose", "content": "def transpose {m n} {α} (M : Matrix m n α) : Matrix n m α :=\n  fun i j => M j i"}, {"name": "diag", "content": "def diag {n} {α} (M : Matrix n n α) : n → α :=\n  fun i => M i i"}, {"name": "mul", "content": "def mul (M : Matrix (Fin l) (Fin m) Float) (N : Matrix (Fin m) (Fin n) Float) :\n    Matrix (Fin l) (Fin n) Float :=\n  fun i k => (fun j => M i j) ⬝ᵥᶜ (fun j => N j k)"}, {"name": "sum", "content": "def sum (A : Matrix (Fin n) (Fin n) Float) : Float :=\n  Vec.Computable.sum (fun i => Vec.Computable.sum (A i))"}, {"name": "trace", "content": "def trace (A : Matrix (Fin n) (Fin n) Float) : Float :=\n  (Vec.Computable.toArray (fun i => A i i)).foldl (· + ·) 0"}, {"name": "covarianceMatrix", "content": "def covarianceMatrix {N n : ℕ} (Y : Matrix (Fin N) (Fin n) Float) (i j : Fin n) : Float :=\n  Vec.Computable.sum (fun k => (Y k i) * (Y k j)) / (OfNat.ofNat N)"}, {"name": "diagonal", "content": "def diagonal (x : Fin n → Float) : Matrix (Fin n) (Fin n) Float :=\n  fun i j => (if i = j then x i else 0)"}, {"name": "fromBlocks", "content": "def fromBlocks {l : Type} {m : Type} {n : Type} {o : Type} {α : Type} :\n    Matrix n l α → Matrix n m α → Matrix o l α → Matrix o m α → Matrix (n ⊕ o) (l ⊕ m) α :=\n  fun A B C D i j =>\n    match i with\n    | Sum.inl i =>\n      match j with\n      | Sum.inl j => A i j\n      | Sum.inr j => B i j\n    | Sum.inr i =>\n      match j with\n      | Sum.inl j => C i j\n      | Sum.inr j => D i j"}, {"name": "minorAux", "content": "private def minorAux (A : Matrix (Fin n.succ) (Fin n.succ) Float) (a b : Fin n.succ) :\n    Matrix (Fin n) (Fin n) Float :=\n  fun i j =>\n    let i' : Fin n.succ := if i.val < a.val then i else i.succ;\n    let j' : Fin n.succ := if j.val < b.val then j else j.succ;\n    A i' j'"}, {"name": "minor", "content": "def minor (A : Matrix (Fin n) (Fin n) Float) (a b : Fin n) :\n    Matrix (Fin n.pred) (Fin n.pred) Float :=\n  match n with\n  | 0 => fun _ => Fin.elim0\n  | _ + 1 => minorAux A a b"}, {"name": "det", "content": "def det {n : ℕ} (A : Matrix (Fin n) (Fin n) Float) : Float :=\n  if h : 0 < n then\n    if n == 1 then A ⟨0, h⟩ ⟨0, h⟩ else\n      (List.finRange n).foldl (fun s i =>\n        s + (-1) ^ (Float.ofNat i.val) * A i ⟨0, h⟩ * det (minor A i ⟨0, h⟩)) 0\n  else 0"}, {"name": "inv", "content": "def inv (A : Matrix (Fin n) (Fin n) Float) : Matrix (Fin n) (Fin n) Float :=\n  (1 / det A) • adjugate A"}, {"name": "adjugate", "content": "def adjugate (A : Matrix (Fin n) (Fin n) Float) : Matrix (Fin n) (Fin n) Float :=\n  transpose (cofactor A)"}, {"name": "cofactor", "content": "def cofactor (A : Matrix (Fin n) (Fin n) Float) : Matrix (Fin n) (Fin n) Float :=\n  fun i j => (-1) ^ (Float.ofNat (i.val + j.val)) * (A i j)"}, {"name": "transpose", "content": "def transpose {m n} {α} (M : Matrix m n α) : Matrix n m α :=\n  fun i j => M j i"}, {"name": "upperTriangular", "content": "def upperTriangular [LT m] (M : Matrix m m R) :=\n  M.BlockTriangular id"}, {"name": "zeroCone", "content": "def zeroCone (x : ℝ) : Prop :=\n  x = 0"}, {"name": "Vec.zeroCone", "content": "def Vec.zeroCone {n} [Fintype n] (x : n → ℝ) : Prop :=\n  ∀ i, Real.zeroCone (x i)"}, {"name": "Matrix.zeroCone", "content": "def Matrix.zeroCone {n m} [Fintype n] [Fintype m] (M : Matrix n m ℝ) : Prop :=\n  ∀ i j, Real.zeroCone (M i j)"}, {"name": "addMatrixAffineConstraint", "content": "def addMatrixAffineConstraint (data : ProblemData) (H : Array (Array (Array Float)))\n    (D : Array (Array Float)) : ProblemData :=\n  let constraint := MatrixAffine.mk D.size H D\n  { data with matrixAffineConstraints := data.matrixAffineConstraints.push constraint }"}, {"name": "upperTriangular", "content": "def upperTriangular [LT m] (M : Matrix m m R) :=\n  M.BlockTriangular id"}, {"name": "lowerTriangular", "content": "def lowerTriangular [LT m] (M : Matrix m m R) :=\n  M.BlockTriangular OrderDual.toDual"}, {"name": "Matrix.PSDCone", "content": "def Matrix.PSDCone {n} [Fintype n] (A : Matrix n n ℝ) : Prop :=\n  Matrix.PosSemidef A"}], "lib_lemmas": [{"name": "congr_fun", "module": "Batteries.Logic"}, {"name": "Matrix.fromBlocks_mulVec", "module": "Mathlib.Data.Matrix.Block"}, {"name": "Finset.prod_ne_zero_iff", "module": "Mathlib.Algebra.BigOperators.GroupWithZero.Finset"}, {"name": "Matrix.det_diagonal", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "lt_of_lt_of_le", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "ne_of_gt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Matrix.det_isEmpty", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "Matrix.det_of_upperTriangular", "module": "Mathlib.LinearAlgebra.Matrix.Block"}, {"name": "le_of_add_le_of_nonneg_left", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "not_nonempty_iff", "module": "Mathlib.Logic.IsEmpty"}], "repo_lemmas": [{"name": "Real.one_sub_div_exp_pos_of_pos", "content": "lemma Real.one_sub_div_exp_pos_of_pos {x : ℝ} : 0 < x → 0 < 1 - 1 / Real.exp x"}, {"name": "evalOneSubDivExp", "content": "@[positivity (1 - (1 / (Real.exp (_ : ℝ))))]\ndef evalOneSubDivExp : PositivityExt where eval {_ _α} zα pα e := do\n  let (.app (.app _sub _one) (.app (.app _div _one') (.app _exp (x : Q(ℝ))))) ←\n    withReducible (whnf e) | throwError \"not (1 - 1 / Real.exp x)\"\n  match ← core zα pα x with\n  | .positive pa =>\n      let pa' ← mkAppM ``Real.one_sub_div_exp_pos_of_pos #[pa]\n      pure (.positive pa')\n  | _ =>\n      pure .none"}, {"name": "PosSemidef.PosDef_iff_det_ne_zero", "content": "lemma PosSemidef.PosDef_iff_det_ne_zero [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosSemidef) :\n    M.PosDef ↔ M.det ≠ 0"}, {"name": "PosSemidef.PosSemidef_sqrt", "content": "lemma PosSemidef.PosSemidef_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :\n    hA.1.sqrt.PosSemidef"}, {"name": "PosSemidef.add", "content": "lemma PosSemidef.add {M N : Matrix n n 𝕜} (hM : M.PosSemidef) (hN : N.PosSemidef) :\n    (M + N).PosSemidef"}, {"name": "PosSemidef.sqrt_mul_sqrt", "content": "@[simp]\nlemma PosSemidef.sqrt_mul_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :\n    hA.1.sqrt * hA.1.sqrt = A"}, {"name": "PosSemidef.conjTranspose_mul_mul", "content": "lemma PosSemidef.conjTranspose_mul_mul (M N : Matrix n n 𝕜) (hM : M.PosSemidef) :\n    (Nᴴ * M * N).PosSemidef"}, {"name": "eigenvectorMatrix_inv_mul", "content": "lemma eigenvectorMatrix_inv_mul : hA.eigenvectorMatrixInv * hA.eigenvectorMatrix = 1"}, {"name": "spectral_theorem''", "content": "theorem spectral_theorem'' :\n    hA.eigenvectorMatrix * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * hA.eigenvectorMatrixᴴ =\n    A"}, {"name": "conjTranspose_eq_transpose", "content": "lemma conjTranspose_eq_transpose {m n : Type _} {A : Matrix m n ℝ} : Aᴴ = Aᵀ"}, {"name": "IsHermitian.fromBlocks₁₁", "content": "lemma IsHermitian.fromBlocks₁₁ [Fintype m] [DecidableEq m] {A : Matrix m m 𝕜} (B : Matrix m n 𝕜)\n    (D : Matrix n n 𝕜) (hA : A.IsHermitian) :\n    (Matrix.fromBlocks A B Bᴴ D).IsHermitian ↔ (D - Bᴴ * A⁻¹ * B).IsHermitian"}, {"name": "IsHermitian.fromBlocks₂₂", "content": "lemma IsHermitian.fromBlocks₂₂ [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜) (B : Matrix m n 𝕜)\n    {D : Matrix n n 𝕜} (hD : D.IsHermitian) :\n    (Matrix.fromBlocks A B Bᴴ D).IsHermitian ↔ (A - B * D⁻¹ * Bᴴ).IsHermitian"}, {"name": "PosSemidef.fromBlocks₁₁", "content": "lemma PosSemidef.fromBlocks₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}\n    (B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (hA : A.PosDef) [Invertible A] :\n    (fromBlocks A B Bᴴ D).PosSemidef ↔ (D - Bᴴ * A⁻¹ * B).PosSemidef"}, {"name": "PosSemidef.fromBlocks₂₂", "content": "lemma PosSemidef.fromBlocks₂₂ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜)\n    (B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (hD : D.PosDef) [Invertible D] :\n    (fromBlocks A B Bᴴ D).PosSemidef ↔ (A - B * D⁻¹ * Bᴴ).PosSemidef"}, {"name": "schur_complement_eq₁₁", "content": "lemma schur_complement_eq₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}\n    (B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (x : m → 𝕜) (y : n → 𝕜) [Invertible A]\n    (hA : A.IsHermitian) :\n    vecMul (star (x ⊕ᵥ y)) (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =\n      vecMul (star (x + (A⁻¹ * B).mulVec y)) A ⬝ᵥ (x + (A⁻¹ * B).mulVec y) +\n        vecMul (star y) (D - Bᴴ * A⁻¹ * B) ⬝ᵥ y"}, {"name": "schur_complement_eq₂₂", "content": "lemma schur_complement_eq₂₂ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜)\n    (B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (x : m → 𝕜) (y : n → 𝕜) [Invertible D]\n    (hD : D.IsHermitian) :\n    vecMul (star (x ⊕ᵥ y)) (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =\n      vecMul (star ((D⁻¹ * Bᴴ).mulVec x + y)) D ⬝ᵥ ((D⁻¹ * Bᴴ).mulVec x + y) +\n        vecMul (star x) (A - B * D⁻¹ * Bᴴ) ⬝ᵥ x"}, {"name": "IsHermitian.nonsingular_inv", "content": "lemma IsHermitian.nonsingular_inv [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.IsHermitian)\n    (hMdet : IsUnit M.det) : M⁻¹.IsHermitian"}, {"name": "PosDef.isUnit_det", "content": "lemma PosDef.isUnit_det [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : IsUnit M.det"}, {"name": "PosDef.nonsingular_inv", "content": "lemma PosDef.nonsingular_inv [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : M⁻¹.PosDef"}, {"name": "PosDef.det_ne_zero", "content": "lemma PosDef.det_ne_zero [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : M.det ≠ 0"}, {"name": "PosDef.PosDef_sqrt", "content": "lemma PosDef.PosDef_sqrt {A : Matrix n n ℝ} (hA : A.PosDef) : hA.1.sqrt.PosDef"}, {"name": "PosDef.conjTranspose_mul_mul", "content": "lemma PosDef.conjTranspose_mul_mul [DecidableEq n] (M N : Matrix n n 𝕜) (hM : M.PosDef)\n    (hN : N.det ≠ 0) : (Nᴴ * M * N).PosDef"}, {"name": "PosSemiDef.IsSymm", "content": "lemma PosSemiDef.IsSymm {n} {A : Matrix (Fin n) (Fin n) ℝ} (hA : PosSemidef A) : IsSymm A"}, {"name": "PosSemidef.det_nonneg", "content": "lemma PosSemidef.det_nonneg {M : Matrix n n ℝ} (hM : M.PosSemidef) [DecidableEq n] : 0 ≤ det M"}, {"name": "isUnit_det_of_PosDef_inv", "content": "lemma isUnit_det_of_PosDef_inv [DecidableEq n] {M : Matrix n n ℝ} (h : M⁻¹.PosDef) :\n    IsUnit M.det"}, {"name": "PosDef_inv_iff_PosDef", "content": "lemma PosDef_inv_iff_PosDef [DecidableEq n] (M : Matrix n n ℝ) : M⁻¹.PosDef ↔ M.PosDef\n    \nlemma conj_symm {x : n → 𝕜} {M : Matrix n n 𝕜} (hM : M.IsHermitian) :\n    star (star x ⬝ᵥ mulVec M x) = star x ⬝ᵥ mulVec M x"}, {"name": "det_add_det_le_det_add", "content": "lemma det_add_det_le_det_add [Nonempty n] (A B : Matrix n n ℝ) (hA : A.PosSemidef)\n    (hB : B.PosSemidef) : A.det + B.det ≤ (A + B).det"}, {"name": "det_add_det_le_det_add'", "content": "lemma det_add_det_le_det_add' [Nonempty n] (A B : Matrix n n ℝ) (hA : A.PosDef)\n    (hB : B.PosSemidef) : A.det + B.det ≤ (A + B).det"}, {"name": "sum_exp_eq_sum_div", "content": "lemma sum_exp_eq_sum_div {n} (x : Fin n → ℝ) (t : ℝ) :\n    Vec.sum (Vec.exp (x - Vec.const n t)) = (Vec.sum (Vec.exp x)) / (Real.exp t)"}, {"name": "sum_exp_pos", "content": "lemma sum_exp_pos {n} (hn : 0 < n) (x : Fin n → ℝ) :\n    0 < Vec.sum (Vec.exp x)"}, {"name": "eigenspace_add", "content": "lemma eigenspace_add {f g : End R M} {a b : R} :\n    eigenspace f a ⊓ eigenspace g b ≤ eigenspace (f + g) (a + b)"}, {"name": "eigenspace_one", "content": "lemma eigenspace_one : eigenspace (1 : End R M) 1 = ⊤"}, {"name": "has_eigenvector_add", "content": "lemma has_eigenvector_add {f g : End R M} {a b : R} {x : M} (hf : HasEigenvector f a x)\n    (hg : HasEigenvector g b x) : HasEigenvector (f + g) (a + b) x"}, {"name": "has_eigenvector_one", "content": "lemma has_eigenvector_one {x : M} (hx : x ≠ 0) : HasEigenvector (1 : End R M) 1 x"}, {"name": "IsHermitian.has_eigenvector_one_add", "content": "lemma IsHermitian.has_eigenvector_one_add {A : Matrix n n ℝ} (hA : A.IsHermitian) (i : n) :\n    Module.End.HasEigenvector\n      (1 + Matrix.toLin' A) (1 + (hA.eigenvalues i)) ((hA.eigenvectorBasis) i)"}, {"name": "det_add_det_le_det_add'", "content": "lemma det_add_det_le_det_add' [Nonempty n] (A B : Matrix n n ℝ) (hA : A.PosDef)\n    (hB : B.PosSemidef) : A.det + B.det ≤ (A + B).det"}, {"name": "one_add_prod_le_prod_one_add", "content": "lemma one_add_prod_le_prod_one_add {n : Type _} [Fintype n] [Nonempty n]\n    (f : n → ℝ) (hf : ∀ i, 0 ≤ f i) : 1 + (∏ i, f i) ≤ ∏ i, (1 + f i)"}, {"name": "BlockTriangular.zero", "content": "lemma BlockTriangular.zero : BlockTriangular (0 : Matrix m m R) b"}, {"name": "BlockTriangular_diagonal", "content": "lemma BlockTriangular_diagonal [DecidableEq m] (d : m → R) :\n    BlockTriangular (diagonal d) b"}, {"name": "BlockTriangular_blockDiagonal'", "content": "lemma BlockTriangular_blockDiagonal' [DecidableEq α]\n    (d : ∀ (i : α), Matrix (m' i) (m' i) R) :\n    BlockTriangular (blockDiagonal'  d) Sigma.fst"}, {"name": "BlockTriangular_blockDiagonal", "content": "lemma BlockTriangular_blockDiagonal [DecidableEq α] (d : α → Matrix m m R) :\n    BlockTriangular (blockDiagonal d) Prod.snd"}, {"name": "toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular", "content": "lemma toBlock_inverse_mul_toBlock_eq_one_of_BlockTriangular [LinearOrder α]\n    [Invertible M] (hM : BlockTriangular M b) (k : α) :\n    M⁻¹.toBlock (fun i => b i < k) (fun i => b i < k) *\n  M.toBlock (fun i => b i < k) (fun i => b i < k) = 1"}, {"name": "inv_toBlock_of_BlockTriangular", "content": "lemma inv_toBlock_of_BlockTriangular [LinearOrder α]\n    [Invertible M] (hM : BlockTriangular M b) (k : α) :\n    (M.toBlock (fun i => b i < k) (fun i => b i < k))⁻¹ =\n    M⁻¹.toBlock (fun i => b i < k) (fun i => b i < k)"}, {"name": "toSquareBlock_inv_mul_toSquareBlock_eq_one", "content": "lemma toSquareBlock_inv_mul_toSquareBlock_eq_one [LinearOrder α]\n    [Invertible M] (hM : BlockTriangular M b) (k : α) :\n    M⁻¹.toSquareBlock b k * M.toSquareBlock b k = 1"}, {"name": "toUpperTri_zero", "content": "theorem toUpperTri_zero {m : Type _} [LinearOrder m] :\n    Matrix.toUpperTri (0 : Matrix m m ℝ) = 0"}, {"name": "toUpperTri_smul", "content": "theorem toUpperTri_smul {m : Type _} [LinearOrder m]\n    (A : Matrix m m ℝ) (κ : ℝ) :\n    κ • Matrix.toUpperTri A = Matrix.toUpperTri (κ • A)"}, {"name": "toUpperTri_add", "content": "theorem toUpperTri_add {m : Type _} [LinearOrder m]\n    (A B : Matrix m m ℝ) :\n    Matrix.toUpperTri (A + B) = Matrix.toUpperTri A + Matrix.toUpperTri B"}, {"name": "upperTriangular_toUpperTri", "content": "lemma upperTriangular_toUpperTri {m : Type _} [LinearOrder m]\n    (A : Matrix m m ℝ) : A.toUpperTri.upperTriangular"}, {"name": "upperTriangular.toUpperTri_eq", "content": "lemma upperTriangular.toUpperTri_eq {A : Matrix n n ℝ}\n    (hA : upperTriangular A) : A.toUpperTri = A"}, {"name": "Real.log_nonneg_of_ge_one", "content": "lemma Real.log_nonneg_of_ge_one {x : ℝ} : 0 ≤ x - 1 → 0 ≤ log x"}, {"name": "Real.log_pos_of_gt_one", "content": "lemma Real.log_pos_of_gt_one {x : ℝ} : 0 < x - 1 → 0 < log x"}, {"name": "Real.one_sub_one_div_sq_nonneg_of_le_one", "content": "lemma Real.one_sub_one_div_sq_nonneg_of_le_one {x : ℝ} :\n    0 < x → 0 ≤ 1 - x → 0 ≤ (1 / x ^ (2 : ℝ)) - 1"}, {"name": "Real.one_sub_sq_nonneg_of_le_one", "content": "lemma Real.one_sub_sq_nonneg_of_le_one {x : ℝ} : 0 ≤ x → 0 ≤ 1 - x → 0 ≤ 1 - x ^ (2 : ℝ)"}, {"name": "Real.exp_sub_one_pos_of_pos", "content": "lemma Real.exp_sub_one_pos_of_pos {x : ℝ} : 0 < x → 0 < Real.exp x - 1"}, {"name": "Real.one_sub_div_exp_pos_of_pos", "content": "lemma Real.one_sub_div_exp_pos_of_pos {x : ℝ} : 0 < x → 0 < 1 - 1 / Real.exp x"}, {"name": "Real.scaled_sq_diff_pos_of_pos", "content": "lemma Real.scaled_sq_diff_pos_of_pos {a x : ℝ} :\n    0 < a - 1 → 0 < x → 0 < (a * x) ^ (2 : ℝ) - x ^ (2 : ℝ)\n  \nlemma det_eq_prod_eigenvalues (xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)) (as : n → ℝ)\n    (hxs : ∀ j, Module.End.HasEigenvector (Matrix.toLin' A) (as j) (xs j)) : det A = ∏ i, as i"}, {"name": "IsHermitian.hasEigenvector_eigenvectorBasis", "content": "lemma IsHermitian.hasEigenvector_eigenvectorBasis (hA : A.IsHermitian) (i : n) :\n    Module.End.HasEigenvector (Matrix.toLin' A) (hA.eigenvalues i) (hA.eigenvectorBasis i)"}, {"name": "log_eq_log", "content": "lemma log_eq_log {x y : ℝ} (hx : 0 < x) (hy : 0 < y) : x = y ↔ log x = log y"}, {"name": "exp_iff_expCone", "content": "theorem exp_iff_expCone (t x : ℝ) : exp x ≤ t ↔ expCone x 1 t"}, {"name": "PosSemidef_zero", "content": "@[simp]\nlemma PosSemidef_zero : PosSemidef (0 : Matrix n n 𝕜)"}, {"name": "LogDetAtom.feasibility_PosDef", "content": "lemma LogDetAtom.feasibility_PosDef {D Z : Matrix n n ℝ} (hD : D = LDL.diag hA)\n    (hZ : Z = LDL.diag hA * (LDL.lower hA)ᵀ) : (fromBlocks D Z Zᵀ A).PosSemidef"}, {"name": "LogDetAtom.feasibility_PosDef'", "content": "lemma LogDetAtom.feasibility_PosDef' {D Z Y : Matrix n n ℝ} (hY : Y = LDL.diag hA * (LDL.lower hA)ᵀ)\n    (hD : D = diagonal Y.diag) (hZ : Z = Y.toUpperTri) : (fromBlocks D Z Zᵀ A).PosSemidef"}, {"name": "LDL.diagEntries_pos", "content": "lemma LDL.diagEntries_pos {A : Matrix n n ℝ} (hA: A.PosDef) (i : n) :\n    0 < LDL.diagEntries hA i"}, {"name": "LogDetAtom.solution_eq_atom", "content": "lemma LogDetAtom.solution_eq_atom {A : Matrix n n ℝ} (hA: A.PosDef) :\n    ∑ i, Real.log (LDL.diagEntries hA i) = Real.log (A.det)"}, {"name": "LogDetAtom.feasibility_exp", "content": "lemma LogDetAtom.feasibility_exp {A : Matrix n n ℝ} (hA: A.PosDef) (i : n) :\n    LDL.diagEntries hA i ≤ ((LDL.diag hA) * ((LDL.lower hA)ᵀ)).diag i"}, {"name": "Real.inverse_eq_inv", "content": "lemma Real.inverse_eq_inv (a : ℝ) : Ring.inverse a = a⁻¹"}, {"name": "PosSemidef_diagonal", "content": "lemma PosSemidef_diagonal [DecidableEq n] {f : n → ℝ} (hf : ∀ i, 0 ≤ f i) :\n    (diagonal f).PosSemidef\nlemma PosDef_diagonal [DecidableEq n] {f : n → ℝ} (hf : ∀ i, 0 < f i) : (diagonal f).PosDef\n\n\nlemma upperTriangular_inv_of_upperTriangular [Fintype m] [LinearOrder m]\n    [Invertible M] (hM : upperTriangular M) : upperTriangular M⁻¹"}, {"name": "lowerTriangular_inv_of_lowerTriangular", "content": "lemma lowerTriangular_inv_of_lowerTriangular [Fintype m] [LinearOrder m]\n    [Invertible M] (hM : lowerTriangular M) : lowerTriangular M⁻¹"}, {"name": "upperTriangular.mul", "content": "lemma upperTriangular.mul [Fintype m] [LinearOrder m] (hM : upperTriangular M)\n    (hN : upperTriangular N) : upperTriangular (M * N)"}, {"name": "lowerTriangular.mul", "content": "lemma lowerTriangular.mul [Fintype m] [LinearOrder m] (hM : lowerTriangular M)\n    (hN : lowerTriangular N) : lowerTriangular (M * N)"}, {"name": "lowerTriangular.transpose", "content": "lemma lowerTriangular.transpose [Fintype m] [LinearOrder m]\n    (hM : lowerTriangular M) : upperTriangular Mᵀ"}, {"name": "upperTriangular.transpose", "content": "lemma upperTriangular.transpose [Fintype m] [LinearOrder m]\n    (hM : upperTriangular M) : lowerTriangular Mᵀ"}, {"name": "diag_inv_mul_diag_eq_one_of_upperTriangular", "content": "lemma diag_inv_mul_diag_eq_one_of_upperTriangular [Fintype m] [LinearOrder m]\n    [Invertible M] (hM : upperTriangular M) (k : m) :\n    M⁻¹ k k * M k k = 1"}, {"name": "diag_inv_mul_diag_eq_one_of_lowerTriangular", "content": "lemma diag_inv_mul_diag_eq_one_of_lowerTriangular [Fintype m] [LinearOrder m]\n    [Invertible M] (hM : lowerTriangular M) (k : m) : M⁻¹ k k * M k k = 1"}, {"name": "det_lowerInv", "content": "@[simp]\nlemma det_lowerInv : (lowerInv hS).det = 1"}, {"name": "det_lower", "content": "@[simp]\nlemma det_lower : (lower hS).det = 1"}, {"name": "lower_eq_to_matrix", "content": "lemma lower_eq_to_matrix :\n    lower hS =\n      ((@gramSchmidtBasis 𝕜 (n → 𝕜) _\n        (NormedAddCommGroup.ofMatrix hS.transpose)\n        (InnerProductSpace.ofMatrix hS.transpose)\n  n _ _ _ (Pi.basisFun 𝕜 n)).toMatrix (Pi.basisFun 𝕜 n))ᵀ"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Matrix.IsHermitian₁₁_of_IsHermitian_toBlock", "content": "lemma IsHermitian₁₁_of_IsHermitian_toBlock {A B C D : Matrix n n ℝ}\n    (h : (fromBlocks A B C D).IsHermitian) : IsHermitian A"}, {"name": "Matrix.IsHermitian₂₂_of_IsHermitian_toBlock", "content": "lemma IsHermitian₂₂_of_IsHermitian_toBlock {A B C D : Matrix n n ℝ}\n    (h : (fromBlocks A B C D).IsHermitian) : IsHermitian D"}, {"name": "Matrix.PosSemidef₁₁_of_PosSemidef_toBlock", "content": "lemma PosSemidef₁₁_of_PosSemidef_toBlock {A B C D : Matrix n n ℝ}\n    (h : (fromBlocks A B C D).PosSemidef) : PosSemidef A"}, {"name": "Matrix.PosSemidef₂₂_of_PosSemidef_toBlock", "content": "lemma PosSemidef₂₂_of_PosSemidef_toBlock {A B C D : Matrix n n ℝ}\n    (h : (fromBlocks A B C D).PosSemidef) :\n    PosSemidef D"}, {"name": "Matrix.LogDetAtom.optimality_D_posdef", "content": "lemma LogDetAtom.optimality_D_posdef {t : n → ℝ} {Y Z D : Matrix n n ℝ}\n    (ht : ∀ i, (t i).exp ≤ Y.diag i) (hD : D = Matrix.diagonal (Y.diag)) (_hZ : Z = Y.toUpperTri)\n    (hPSD : (fromBlocks D Z Zᵀ A).PosSemidef) : D.PosDef"}, {"name": "Matrix.LogDetAtom.optimality_Ddet_le_Adet", "content": "lemma LogDetAtom.optimality_Ddet_le_Adet {t : n → ℝ} {Y Z D : Matrix n n ℝ}\n    (ht : ∀ i, (t i).exp ≤ Y.diag i) (hD : D = Matrix.diagonal (Y.diag)) (hZ : Z = Y.toUpperTri)\n    (hPSD : (fromBlocks D Z Zᵀ A).PosSemidef) : D.det ≤ A.det"}], "local_ctx": "import Mathlib.LinearAlgebra.Matrix.LDL\n\nimport CvxLean.Lib.Math.SchurComplement\n\nimport CvxLean.Lib.Math.Subadditivity\n\nimport CvxLean.Lib.Math.LinearAlgebra.Matrix.Triangular\n\nimport CvxLean.Lib.Math.LinearAlgebra.Matrix.ToUpperTri\n\nimport CvxLean.Lib.Math.LinearAlgebra.Matrix.LDL\n\nnamespace Matrix\n\nopen Matrix BigOperators\n\nvariable {n : Type} [Fintype n] [LinearOrder n] [LocallyFiniteOrderBot n]\n\nvariable {𝕜 : Type} [RCLike 𝕜]\n\nvariable {A : Matrix n n ℝ} (hA : A.PosDef)\n\nopen scoped Matrix ComplexOrder", "target_theorem": "lemma LogDetAtom.cond_elim {t : n → ℝ} {Y Z D : Matrix n n ℝ} (ht : ∀ i, (t i).exp ≤ Y.diag i)\n    (hD : D = Matrix.diagonal (Y.diag)) (hZ : Z = Y.toUpperTri)\n    (hPSD : (fromBlocks D Z Zᵀ A).PosSemidef) : A.PosDef :=", "ground_truth_proof": ":= by\n  have h_D_pd : D.PosDef := LogDetAtom.optimality_D_posdef ht hD hZ hPSD\n  have h_A_psd : A.PosSemidef := PosSemidef₂₂_of_PosSemidef_toBlock hPSD\n  have h_Ddet_le_Adet : D.det ≤ A.det := LogDetAtom.optimality_Ddet_le_Adet ht hD hZ hPSD\n  have h_Adet_pos : 0 < A.det := lt_of_lt_of_le h_D_pd.det_pos h_Ddet_le_Adet\n  rw [h_A_psd.PosDef_iff_det_ne_zero]\n  apply ne_of_gt h_Adet_pos", "nesting_depth": 6, "transitive_dep_count": 87, "subset_aristotle": false, "category": "Applied verif."}
{"id": 185, "thm_name": "Tm.ren_den_eq", "thm_stmt": "theorem Tm.ren_den_eq (e : Γ ⊢ τ) : ∀ {Δ}, (r : Ren Γ Δ) → ⟦e.ren r⟧ = (⟦e⟧) ∘' (⟦r⟧)", "lean_root": "pcf-lean", "rel_path": "PCF/Denotation.lean", "imports": ["import PCF.Utility", "import «PCF».Flat", "import PCF.Domain", "import «PCF».Context"], "used_lib_defs": [{"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.fst", "module": "Init.Prelude"}, {"name": "Prod.snd", "module": "Init.Prelude"}, {"name": "Trans.trans", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}, {"name": "Con", "module": "Mathlib.GroupTheory.Congruence.Defs"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Eq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"⟦\" τ \" ty⟧\" => Ty.den τ", "content": "notation:max \"⟦\" τ \" ty⟧\" => Ty.den τ"}, {"name": "notation:max \"⟦\" Γ \" cx⟧\" => Ev Γ", "content": "notation:max \"⟦\" Γ \" cx⟧\" => Ev Γ"}, {"name": "notation:100 \"⟦\" t \"⟧\" => Tm.den t", "content": "notation:100 \"⟦\" t \"⟧\" => Tm.den t"}, {"name": "notation:100 \"⟦\" r \"⟧\" => Ren.den r", "content": "notation:100 \"⟦\" r \"⟧\" => Ren.den r"}, {"name": "notation:100 \"⟦\" σ \"⟧\" => Sb.den σ", "content": "notation:100 \"⟦\" σ \"⟧\" => Sb.den σ"}, {"name": "notation:100 \"⟦\" C \" con⟧\" => Con.den C", "content": "notation:100 \"⟦\" C \" con⟧\" => Con.den C"}, {"name": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x", "content": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x"}, {"name": "notation:max \"⨆\" => Domain.sup", "content": "notation:max \"⨆\" => Domain.sup"}, {"name": "notation:max \"⋆\" => Order.refl", "content": "notation:max \"⋆\" => Order.refl"}, {"name": "notation:max \"⊥\" => Domain.bot", "content": "notation:max \"⊥\" => Domain.bot"}, {"name": "Mono", "content": "structure Mono (α) (β) [Order α] [Order β] where\n  act : α → β\n  act' : is_monotone act"}, {"name": "[Order", "content": "instance [Order α] [Order β] : CoeFun (Mono α β) (fun _ => α → β) where\n  coe f := f.act"}, {"name": "Tm", "content": "inductive Tm : Cx → Ty → Type\n  | var : ∀ τ, Γ ∋ τ → Tm Γ τ\n  | true : Tm Γ .bool\n  | false : Tm Γ .bool\n  | zero : Tm Γ .nat\n  | succ : Tm Γ .nat → Tm Γ .nat\n  | pred : Tm Γ .nat → Tm Γ .nat\n  | zero? : Tm Γ .nat → Tm Γ .bool\n  | cond : Tm Γ .bool → Tm Γ τ → Tm Γ τ → Tm Γ τ\n  | fn : Tm (Γ ∷ τ) υ → Tm Γ (τ ⇒ υ)\n  | app : Tm Γ (τ ⇒ υ) → Tm Γ τ → Tm Γ υ\n  | fix : Tm Γ (τ ⇒ τ) → Tm Γ τ"}, {"name": "Cont.eval", "content": "def Cont.eval {α : Type i} {β : Type j} [Order α] [Order β] [Domain α] [Domain β]\n  : Cont (Cont α β × α) β := ⟨\n    Mono.eval_cont,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Cont.fst", "content": "def Cont.fst [Order α] [Order β] [Domain α] [Domain β] : Cont (α × β) α :=\n  ⟨⟨Prod.fst, And.left⟩, Domain.sup_is_mono (fun _ ↦ ⋆)⟩"}, {"name": "Cont", "content": "structure Cont (α) (β) [Order α] [Order β] [Domain α] [Domain β] where\n  fn : Mono α β\n  sub : ∀ {c : Chain α}, fn.act (⨆ c) ⊑ ⨆ (fn ∘ c)"}, {"name": "[Order", "content": "instance [Order α] [Order β] [Domain α] [Domain β] : CoeFun (Cont α β) (fun _ => α → β) where\n  coe f := f.fn.act"}, {"name": "[Order", "content": "instance [Order α] [Order β] [Domain α] [Domain β] : Order (Cont α β) where\n  R := fun x y ↦ x.fn.act ⊑ y.fn.act\n  refl := fun _ ↦ ⋆\n  trans := fun p q ↦ p ⬝ q\n  anti := fun p q ↦ Cont.ext (p ⇄! q)"}, {"name": "[Order", "content": "instance [Order α] [Order β] [Domain α] [Domain β] : Domain (Cont α β) where\n  bot := ⟨⟨fun _ ↦ ⊥, fun _ ↦ Domain.is_bot⟩, Domain.is_bot⟩\n  sup := fun c ↦ ⟨Mono.sup_cont c, by admit /- proof elided -/⟩\n  is_bot := fun _ ↦ Domain.is_bot\n  is_bound := by admit /- proof elided -/\n  is_least := by admit /- proof elided -/"}, {"name": "Domain", "content": "class Domain (α) [Order α] where\n  bot : α\n  sup : (c : Chain α) → α\n  is_bot {x} : bot ⊑ x\n  is_bound (c) (n): c.act n ⊑ sup c\n  is_least (c) {d} : ({n : _} → c.act n ⊑ d) → sup c ⊑ d"}, {"name": "Order", "content": "class Order (α) where\n  R : α → α → Prop\n  refl {x} : R x x\n  trans {x y z} : R x y → R y z → R x z\n  anti {x y} : R x y → R y x → x = y"}, {"name": "[o", "content": "instance [o : Order α] : Trans o.R o.R o.R where\n  trans := o.trans"}, {"name": "[Order", "content": "instance [Order α] [Order β] : Order (α × β) where\n  R := fun ⟨a₀, b₀⟩ ⟨a₁, b₁⟩ ↦ a₀ ⊑ a₁ ∧ b₀ ⊑ b₁\n  refl := ⟨⋆, ⋆⟩\n  trans := fun ⟨p₀, p₁⟩ ⟨q₀, q₁⟩ ↦ ⟨p₀ ⬝ q₀, p₁ ⬝ q₁⟩\n  anti := fun ⟨p₀, p₁⟩ ⟨q₀, q₁⟩ ↦ Prod.ext (p₀ ⇄! q₀) (p₁ ⇄! q₁)"}, {"name": "[Order", "content": "instance [Order α] [Order β] [Domain α] [Domain β] : Domain (α × β) where\n  bot := ⟨⊥, ⊥⟩\n  sup := fun c ↦ ⟨⨆ c.fst, ⨆ c.snd⟩\n  is_bot := ⟨Domain.is_bot, Domain.is_bot⟩\n  is_bound := fun c n ↦ ⟨Domain.is_bound c.fst n, Domain.is_bound c.snd n⟩\n  is_least := fun c _ p ↦ ⟨Domain.is_least c.fst p.left, Domain.is_least c.snd p.right⟩"}, {"name": "Chain", "content": "def Chain (α : Type i) [Order α] := Mono Nat α"}, {"name": "[Order", "content": "instance [Order α] : CoeFun (Chain α) (fun _ => Nat → α) where\n  coe f := f.act"}, {"name": "is_monotone", "content": "def is_monotone [Order α] [Order β] (f : α → β) := ∀ {x y : α}, x ⊑ y → f x ⊑ f y"}, {"name": "Cont.snd", "content": "def Cont.snd [Order α] [Order β] [Domain α] [Domain β] : Cont (α × β) β :=\n  ⟨⟨Prod.snd, And.right⟩, Domain.sup_is_mono (fun _ ↦ ⋆)⟩"}, {"name": "Chain.apply", "content": "def Chain.apply [Order α] [Order β] [Domain α] [Domain β] (c : Chain (Cont α β)) (a : α) : Chain β\n  := Mono.apply c a"}, {"name": "Mono.apply", "content": "def Mono.apply [Order α] [Order β] [Domain α] [Domain β] (c : Mono Nat (Cont α β)) (a : α) : Chain β where\n  act := fun n ↦ (c n) a\n  act' := fun a_b ↦ (c • a_b) _"}, {"name": "Mono.eval_cont", "content": "def Mono.eval_cont {α : Type i} {β : Type j} [Order α] [Order β] [Domain α] [Domain β]\n  : Mono (Cont α β × α) β :=\n  ⟨fun x ↦ x.fst x.snd, fun {x y} p ↦ (x.fst • p.right) ⬝ (p.left y.snd)⟩"}, {"name": "Mono.from_cont", "content": "def Mono.from_cont [Order α] [Order β] [Domain α] [Domain β] : Mono (Cont α β) (Mono α β) :=\n  ⟨Cont.fn, fun p a ↦ p a⟩"}, {"name": "Mono.sup", "content": "def Mono.sup [Order α] [Domain α] : Mono (Chain α) α :=\n  ⟨⨆, Domain.sup_is_mono⟩"}, {"name": "Mono.sup_cont", "content": "def Mono.sup_cont [Order α] [Order β] [Domain α] [Domain β] (c : Chain (Cont α β)) : Mono α β\n  := ⟨\n    fun a ↦ ⨆ (c.apply a),\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Cont.comp'", "content": "def Cont.comp' [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] (f : Cont β γ) (g : Cont α β)\n  : Cont α γ\n  := ⟨\n    ⟨fun x ↦ f (g x), fun x_y ↦ f • g • x_y⟩,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.eval", "content": "def Mono.eval {α : Type i} {β : Type j} [Order α] [Order β] : Mono (Mono α β × α) β :=\n  ⟨fun x ↦ x.fst x.snd, fun {x y} p ↦ (x.fst • p.right) ⬝ (p.left y.snd)⟩"}, {"name": "Chain.fst", "content": "def Chain.fst [Order α] [Order β] (c : Chain (α × β)) : Chain α :=\n  ⟨fun n ↦ (c n).fst, fun p ↦ by admit /- proof elided -/\n  ⟩"}, {"name": "Chain.snd", "content": "def Chain.snd [Order α] [Order β] (c : Chain (α × β)) : Chain β :=\n  ⟨fun n ↦ (c n).snd, fun p ↦ by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.const", "content": "def Mono.const [Order α] [Order β] (b : β) : Mono α β := ⟨fun _ ↦ b, fun _ ↦ ⋆⟩"}, {"name": "Mono.comp", "content": "def Mono.comp {α : Type i} {β : Type j} {γ : Type k} [Order α] [Order β] [Order γ]\n  : Mono (Mono β γ × Mono α β) (Mono α γ) := ⟨\n    fun h ↦ ⟨fun x ↦ h.fst (h.snd x), fun x_y ↦ h.fst • (h.snd • x_y)⟩,\n    fun {h₀ h₁} h a ↦ (h₀.fst • h.right a) ⬝ (h.left (h₁.snd a))\n  ⟩"}, {"name": "Mono.pair", "content": "def Mono.pair [Order α] [Order β] [Order γ]\n  (f : Mono γ α) (g : Mono γ β) : Mono γ (α × β) :=\n    ⟨fun c ↦ ⟨f c, g c⟩, fun p ↦ ⟨f • p, g • p⟩⟩"}, {"name": "Cont.const", "content": "def Cont.const [Order α] [Order β] [Domain α] [Domain β] (b : β) : Cont α β := ⟨Mono.const b, fun {c} ↦ by admit /- proof elided -/\n  ⟩"}, {"name": "Domain.sup_of_const", "content": "def Domain.sup_of_const [Order α] [Domain α] (a : α) : ⨆ (Mono.const a) = a :=\n  (by admit /- proof elided -/\n  ) ⇄! (Domain.is_bound (Mono.const a) 0)"}, {"name": "Cont.flat", "content": "def Cont.flat (f : α → β) : (Cont (Flat α) (Flat β)) := (Mono.flat f).promote_trivial"}, {"name": "Flat", "content": "inductive Flat (α : Type) : Type where\n  | none : Flat α\n  | some : α → Flat α"}, {"name": "Mono.flat", "content": "def Mono.flat (f : α → β) : (Mono (Flat α) (Flat β)) := ⟨\n    lift_flat f,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "lift_flat", "content": "def lift_flat (f : α → β) : Flat α → Flat β\n| .none => .none\n| .some x => .some (f x)"}, {"name": "Cont.cond", "content": "def Cont.cond [Order α] [Domain α] : Cont (Flat Bool) (Cont (α × α) α) := ⟨\n  cond',\n  by admit /- proof elided -/\n⟩"}, {"name": "cond'", "content": "def cond' [Order α] [Domain α] : Mono (Flat Bool) (Cont (α × α) α) := ⟨\n  fun b ↦ (\n    match b with\n    | .none => Cont.const ⊥\n    | .some true => Cont.fst\n    | .some false => Cont.snd\n  ),\n  by admit /- proof elided -/\n⟩"}, {"name": "Cont.pred", "content": "def Cont.pred : Cont (Flat Nat) (Flat Nat) := Mono.pred.promote_trivial"}, {"name": "Mono.pred", "content": "def Mono.pred : Mono (Flat Nat) (Flat Nat) := ⟨\n    Nat.partial_pred,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Nat.partial_pred", "content": "def Nat.partial_pred : Flat Nat → Flat Nat :=\n  fun n ↦ match n with\n  | .some (.succ n) => .some n\n  | _ => .none"}, {"name": "Cont.curry", "content": "def Cont.curry {α : Type i} {β : Type j}\n  [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont (α × β) γ) : Cont α (Cont β γ) := ⟨\n    ⟨\n      fun a ↦ ⟨\n        ⟨\n          fun b ↦ f (a, b),\n          fun b' ↦ f • ⟨⋆, b'⟩\n        ⟩,\n        by admit /- proof elided -/\n      ⟩,\n      fun a' b ↦ f • ⟨a', ⋆⟩\n    ⟩,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Cont.pair", "content": "def Cont.pair [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont γ α) (g : Cont γ β) : Cont γ (α × β) := ⟨\n    ⟨fun c ↦ ⟨f c, g c⟩, fun p ↦ ⟨f • p, g • p⟩⟩,\n    ⟨f.sub ⬝ Domain.sup_is_mono (fun _ ↦ ⋆), g.sub ⬝ Domain.sup_is_mono (fun _ ↦ ⋆)⟩\n  ⟩"}, {"name": "Cont.uncurry", "content": "def Cont.uncurry {α : Type i} {β : Type j}\n  [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont α (Cont β γ)) : Cont (α × β) γ  := ⟨\n    Mono.uncurry_cont f,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.uncurry_cont", "content": "def Mono.uncurry_cont {α : Type i} {β : Type j}\n  [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont α (Cont β γ)) : Mono (α × β) γ := ⟨\n    fun ⟨a, b⟩ ↦ (f a) b,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Cont.fix'", "content": "def Cont.fix' [Order α] [Domain α] : Cont (Cont α α) α := ⟨\n    fix_mono,\n    by\n      intro f\n      apply fix_is_least_prefixed\n      calc ⨆ f (⨆ (fix_mono ∘ f))\n        _ = ⨆ (f.apply (⨆ (fix_mono ∘ f))) := rfl\n        _ ⊑ ⨆ (Mono.sup ∘ Mono.comp ∘ Mono.pair (Mono.from_cont ∘ f) (Mono.const (fix_mono ∘ f))) :="}, {"name": "fix_is_prefixed", "content": "def fix_is_prefixed [Order α] [Domain α] (f : Cont α α) : is_prefixed f (⨆ f.iterations) :="}, {"name": "sup_succ", "content": "def sup_succ [Order α] [Domain α] {c : Chain α} : ⨆ (c ∘ Mono.succ) ⊑ ⨆ c :="}, {"name": "Mono.succ", "content": "def Mono.succ : Mono Nat Nat := ⟨Nat.succ, Nat.succ_le_succ⟩"}, {"name": "Cont.iterations", "content": "def Cont.iterations [Order α] [Domain α] (f : Cont α α) : Chain α := ⟨\n    fun n ↦ Cont.iter n f ⊥,\n    increasing_implies_monotone (fun n ↦ iter n f ⊥) (by admit /- proof elided -/\n    )\n  ⟩"}, {"name": "Cont.iter", "content": "def Cont.iter [Order α] [Domain α] : Nat → Cont α α → Cont α α\n| 0 => fun _ ↦ Cont.id\n| .succ n => fun f ↦ f ∘ iter n f"}, {"name": "Cont.id", "content": "def Cont.id [Order α] [Domain α] : Cont α α := ⟨Mono.id, ⋆⟩"}, {"name": "Mono.id", "content": "def Mono.id [Order α] : Mono α α\n  := ⟨Function.id, Function.id⟩"}, {"name": "Function.id", "content": "@[inline] def Function.id {α : Sort u} (a : α) : α := a"}, {"name": "increasing_implies_monotone", "content": "def increasing_implies_monotone [Order α] (f : Nat → α) : (∀ n, f n ⊑ f n.succ) → is_monotone f :="}, {"name": "is_prefixed", "content": "def is_prefixed [Order α] [Domain α] (f : Cont α α) (a : α) := f a ⊑ a"}, {"name": "Cont.fix", "content": "def Cont.fix [Order α] [Domain α] (f : Cont α α) := ⨆ f.iterations"}, {"name": "fix_is_least_prefixed", "content": "def fix_is_least_prefixed [Order α] [Domain α] (f : Cont α α) (a : α) (h : is_prefixed f a)\n  : f.fix ⊑ a :="}, {"name": "Cont.fix_mono", "content": "def Cont.fix_mono [Order α] [Domain α] : Mono (Cont α α) α := ⟨\n    Cont.fix,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Nat.zero?", "content": "def Nat.zero? : Nat → Bool\n| .zero => true\n| _ => false"}, {"name": "Cont.swap", "content": "def Cont.swap [Order α] [Domain α] [Order β] [Domain β] : Cont (α × β) (β × α) := ⟨\n  Mono.swap,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.swap", "content": "def Mono.swap [Order α] [Order β] : Mono (α × β) (β × α) := ⟨\n    fun p ↦ ⟨p.snd, p.fst⟩,\n    fun ⟨a', b'⟩ ↦ ⟨b', a'⟩\n  ⟩"}, {"name": "Cont.assoc_swap_assoc", "content": "def Cont.assoc_swap_assoc {α : Type i} {β : Type j}\n  [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] : Cont ((α × β) × γ) ((α × γ) × β) := ⟨\n    Mono.assoc_swap_assoc,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.assoc_swap_assoc", "content": "def Mono.assoc_swap_assoc {α : Type i} {β : Type j}\n  [Order α] [Order β] [Order γ] : Mono ((α × β) × γ) ((α × γ) × β) := ⟨\n    fun p ↦ ⟨⟨p.fst.fst, p.snd⟩, p.fst.snd⟩,\n    fun ⟨⟨a', b'⟩, c'⟩ ↦ ⟨⟨a', c'⟩, b'⟩\n  ⟩"}, {"name": "Ty", "content": "inductive Ty\n  | bool\n  | nat\n  | pow : Ty → Ty → Ty"}, {"name": "DomainType", "content": "structure DomainType : Type (i + 1) :=\n  carrier : Type i\n  order : Order carrier\n  domain : Domain carrier"}, {"name": "", "content": "instance : Coe DomainType Type where\n  coe D := D.carrier"}, {"name": "(τ", "content": "instance (τ : DomainType) : Order (τ) := τ.order"}, {"name": "(τ", "content": "instance (τ : DomainType) : Domain (τ) := τ.domain"}, {"name": "Cx", "content": "inductive Cx\n  | nil\n  | cons : Cx -> Ty -> Cx"}, {"name": "Ren", "content": "def Ren Γ Δ := ∀ τ, Γ ∋ τ → Δ ∋ τ"}, {"name": "Var", "content": "inductive Var : Cx → Ty → Type\n  | z : ∀ {Γ : Cx}, Var (Γ ∷ τ) τ\n  | s : ∀ {Γ : Cx} {υ : Ty} τ, Var Γ τ → Var (Γ ∷ υ) τ"}, {"name": "infixr:100 \" ⇒ \" => Ty.pow", "content": "infixr:100 \" ⇒ \" => Ty.pow"}, {"name": "infixl:70 \" ∷ \"  => Cx.cons", "content": "infixl:70 \" ∷ \"  => Cx.cons"}, {"name": "infix:70 \" ∋ \" => Var", "content": "infix:70 \" ∋ \" => Var"}, {"name": "infix:70 \" ⊢ \" => Tm", "content": "infix:70 \" ⊢ \" => Tm"}, {"name": "infix:100 \" ⊑ \" => Order.R", "content": "infix:100 \" ⊑ \" => Order.R"}, {"name": "notation:max \"⋆\" => Order.refl", "content": "notation:max \"⋆\" => Order.refl"}, {"name": "infix:100 \" ⇄! \" => Order.anti", "content": "infix:100 \" ⇄! \" => Order.anti"}, {"name": "infixl:100 \" • \" => Mono.act'", "content": "infixl:100 \" • \" => Mono.act'"}, {"name": "notation:max \"⊥\" => Domain.bot", "content": "notation:max \"⊥\" => Domain.bot"}, {"name": "notation:max \"⨆\" => Domain.sup", "content": "notation:max \"⨆\" => Domain.sup"}, {"name": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x", "content": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x"}, {"name": "infix:100 \" ∘ \" => Cont.comp'", "content": "infix:100 \" ∘ \" => Cont.comp'"}, {"name": "infixr:100 \" ∘' \" => Cont.comp'", "content": "infixr:100 \" ∘' \" => Cont.comp'"}, {"name": "infixl:70 \" ∷ᵣ \" => Ren.keep", "content": "infixl:70 \" ∷ᵣ \" => Ren.keep"}, {"name": "Var.tm", "content": "def Var.tm (x : Γ ∋ τ) : Γ ⊢ τ := Tm.var τ x"}, {"name": "Var.succ", "content": "def Var.succ (x : Γ ∋ τ) {υ : Ty} : (Γ ∷ υ) ∋ τ := Var.s τ x"}, {"name": "Var.ren", "content": "def Var.ren (v : Γ ∋ τ) (r : Ren Γ Δ) := r τ v"}, {"name": "Ren.keep", "content": "def Ren.keep (r : Ren Γ Δ) (τ : Ty) : Ren (Γ ∷ τ) (Δ ∷ τ) :=\n  fun υ v => match v with\n  | .z     => .z\n  | .s _ x => (x.ren r).succ"}, {"name": "Tm.ren", "content": "def Tm.ren (t : Γ ⊢ τ) (r : Ren Γ Δ) : Δ ⊢ τ :=\n  match t with\n  | .var τ x    => (x.ren r).tm\n  | .true       => .true\n  | .false      => .false\n  | .zero       => .zero\n  | .succ e     => (e.ren r).succ\n  | .pred e     => (e.ren r).pred\n  | .zero? e    => (e.ren r).zero?\n  | .cond s t f => (s.ren r).cond (t.ren r) (f.ren r)\n  | .fn e       => (e.ren (r ∷ᵣ _)).fn\n  | .app f a    => (f.ren r).app (a.ren r)\n  | .fix f      => (f.ren r).fix"}], "lib_lemmas": [{"name": "congrArg", "module": "Init.Prelude"}, {"name": "funext {α : Sort u} {β : α → Sort v} {f g : (x : α) → β x} (h : ∀ x, f x = g x) : f = g", "module": ""}], "repo_lemmas": [{"name": "Domain.sup_is_mono", "content": "theorem Domain.sup_is_mono [Order α] [Domain α] {c d : Chain α} (p : c ⊑ d) : ⨆ c ⊑ ⨆ d"}, {"name": "Mono.ext", "content": "@[ext] theorem Mono.ext [Order α] [Order β] {f g : Mono α β} (p : f.act = g.act) : f = g"}, {"name": "Cont.ext", "content": "@[ext] theorem Cont.ext [Order α] [Order β] [Domain α] [Domain β]\n  {f g : Cont α β} (p : f.fn.act = g.fn.act) : f = g"}, {"name": "congrArg2", "content": "theorem congrArg2\n  {α₀ : Sort u₀} {α₁ : Sort u₁} {β : Sort v} {a₀ a₀' : α₀} {a₁ a₁' : α₁}\n  (f : α₀ → α₁ → β) (h₀ : Eq a₀ a₀') (h₁ : Eq a₁ a₁') : Eq (f a₀ a₁) (f a₀' a₁')"}, {"name": "Cont.pair_after", "content": "theorem Cont.pair_after [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] [Order δ] [Domain δ]\n  (f : Cont γ α) (g : Cont γ β) (h : Cont δ γ) : (f ∘' h).pair (g ∘' h) = (f.pair g) ∘' h"}], "used_local_defs": [{"name": "Ty.den", "content": "noncomputable def Ty.den : Ty → DomainType\n  | .bool => ⟨Flat Bool, _, inferInstance⟩\n  | .nat => ⟨Flat Nat, _, inferInstance⟩\n  | .pow T₀ T₁ => by admit /- proof elided -/"}, {"name": "Ev", "content": "def Ev (Γ : Cx) : Type := ∀ τ, Var Γ τ → ↑⟦τ ty⟧"}, {"name": "Ev.push", "content": "def Ev.push {Γ : Cx} (ρ : ⟦Γ cx⟧) {τ : Ty} (d : ↑⟦τ ty⟧) : ⟦Γ ∷ τ cx⟧ :=\n  fun {τ} x ↦ match x with\n  | .z => d\n  | .s τ x => ρ τ x"}, {"name": "Ev.from", "content": "def Ev.from {Γ : Cx} {τ : Ty} : Cont (⟦Γ cx⟧ × ⟦τ ty⟧) (⟦Γ ∷ τ cx⟧) := ⟨\n  ⟨\n    fun ⟨ρ, d⟩ υ x ↦ ρ.push d υ x,\n    by admit /- proof elided -/\n  ⟩,\n  by admit /- proof elided -/\n⟩"}, {"name": "Tm.den", "content": "noncomputable def Tm.den : (Γ ⊢ τ) → Cont (⟦Γ cx⟧) (⟦τ ty⟧)\n  | .var τ x => ⟨⟨fun ρ ↦ ρ τ x, fun ρ₀_ρ₁ ↦ ρ₀_ρ₁ τ x⟩, ⋆⟩\n  | .true => Cont.const (.some .true)\n  | .false => Cont.const (.some .false)\n  | .zero => Cont.const (.some 0)\n  | .succ e => Cont.flat (Nat.succ) ∘ e.den\n  | .pred e => Cont.pred ∘ e.den\n  | .zero? e => Cont.flat (Nat.zero?) ∘ e.den\n  | .cond s t f  => Cont.uncurry (Cont.cond) ∘ Cont.pair s.den (Cont.pair t.den f.den)\n  | .fn e  => Cont.curry (e.den ∘ Ev.from)\n  | .app f e => Cont.eval ∘ (Cont.pair f.den e.den)\n  | .fix f => Cont.fix' ∘ f.den"}, {"name": "Ren.den", "content": "noncomputable def Ren.den (r : Ren Γ Δ) : Cont (⟦Δ cx⟧) (⟦Γ cx⟧) :=\n  ⟨⟨fun ρ _ x ↦ (⟦(x.ren r).tm⟧) ρ, fun ρ' _ x ↦ (⟦(x.ren r).tm⟧) • ρ'⟩, fun _ x ↦ (⟦(x.ren r).tm⟧).sub⟩"}, {"name": "Sb.den", "content": "noncomputable def Sb.den (σ : Sb Γ Δ) : Cont (⟦Δ cx⟧) (⟦Γ cx⟧) :=\n  ⟨⟨fun ρ _ x ↦ (⟦x.sub σ⟧) ρ, fun ρ' _ x ↦ (⟦x.sub σ⟧) • ρ'⟩, fun _ x ↦ (⟦x.sub σ⟧).sub⟩"}, {"name": "Con.den", "content": "noncomputable def Con.den : Con Δ υ Γ τ → Cont (⟦Γ cx⟧ × Cont (⟦Δ cx⟧) (⟦υ ty⟧)) ⟦τ ty⟧\n  | id         => Cont.uncurry Cont.id ∘' Cont.swap\n  | comp C₀ C₁ => Cont.uncurry (Cont.curry (C₁.den ∘' Cont.swap)\n                             ∘' Cont.curry (C₀.den ∘' Cont.swap)) ∘' Cont.swap\n  | sub C σ => Cont.uncurry ((Cont.curry C.den) ∘' (⟦σ⟧))\n  | succ C => Cont.flat (Nat.succ) ∘' C.den\n  | pred C => Cont.pred ∘' C.den\n  | zero? C => Cont.flat (Nat.zero?) ∘' C.den\n  | fn C   => Cont.curry ((Cont.uncurry (Cont.curry C.den ∘' Ev.from)) ∘' Cont.assoc_swap_assoc)\n  | cond_s C t f => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair C.den (Cont.pair ((⟦t⟧) ∘' Cont.fst) ((⟦f⟧) ∘' Cont.fst))\n  | cond_t s C f => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair ((⟦s⟧) ∘' Cont.fst) (Cont.pair C.den ((⟦f⟧) ∘' Cont.fst))\n  | cond_f s t C => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair ((⟦s⟧) ∘' Cont.fst) (Cont.pair ((⟦t⟧) ∘' Cont.fst) C.den)\n  | app_f C a => Cont.eval ∘' (Cont.pair C.den ((⟦a⟧) ∘' Cont.fst))\n  | app_a f C => Cont.eval ∘' (Cont.pair ((⟦f⟧) ∘' Cont.fst) C.den)\n  | fix C => Cont.fix' ∘' C.den"}], "used_local_lemmas": [], "local_ctx": "import «PCF».Flat\n\nimport «PCF».Context\n\nnoncomputable def Ty.den : Ty → DomainType\n  | .bool => ⟨Flat Bool, _, inferInstance⟩\n  | .nat => ⟨Flat Nat, _, inferInstance⟩\n  | .pow T₀ T₁ => by admit /- proof elided -/\n\nnotation:max \"⟦\" τ \" ty⟧\" => Ty.den τ\n\ndef Ev (Γ : Cx) : Type := ∀ τ, Var Γ τ → ↑⟦τ ty⟧\n\nnotation:max \"⟦\" Γ \" cx⟧\" => Ev Γ\n\ndef Ev.push {Γ : Cx} (ρ : ⟦Γ cx⟧) {τ : Ty} (d : ↑⟦τ ty⟧) : ⟦Γ ∷ τ cx⟧ :=\n  fun {τ} x ↦ match x with\n  | .z => d\n  | .s τ x => ρ τ x\n\ndef Ev.from {Γ : Cx} {τ : Ty} : Cont (⟦Γ cx⟧ × ⟦τ ty⟧) (⟦Γ ∷ τ cx⟧) := ⟨\n  ⟨\n    fun ⟨ρ, d⟩ υ x ↦ ρ.push d υ x,\n    by admit /- proof elided -/\n  ⟩,\n  by admit /- proof elided -/\n⟩\n\nnoncomputable def Tm.den : (Γ ⊢ τ) → Cont (⟦Γ cx⟧) (⟦τ ty⟧)\n  | .var τ x => ⟨⟨fun ρ ↦ ρ τ x, fun ρ₀_ρ₁ ↦ ρ₀_ρ₁ τ x⟩, ⋆⟩\n  | .true => Cont.const (.some .true)\n  | .false => Cont.const (.some .false)\n  | .zero => Cont.const (.some 0)\n  | .succ e => Cont.flat (Nat.succ) ∘ e.den\n  | .pred e => Cont.pred ∘ e.den\n  | .zero? e => Cont.flat (Nat.zero?) ∘ e.den\n  | .cond s t f  => Cont.uncurry (Cont.cond) ∘ Cont.pair s.den (Cont.pair t.den f.den)\n  | .fn e  => Cont.curry (e.den ∘ Ev.from)\n  | .app f e => Cont.eval ∘ (Cont.pair f.den e.den)\n  | .fix f => Cont.fix' ∘ f.den\n\nnotation:100 \"⟦\" t \"⟧\" => Tm.den t\n\nnoncomputable def Ren.den (r : Ren Γ Δ) : Cont (⟦Δ cx⟧) (⟦Γ cx⟧) :=\n  ⟨⟨fun ρ _ x ↦ (⟦(x.ren r).tm⟧) ρ, fun ρ' _ x ↦ (⟦(x.ren r).tm⟧) • ρ'⟩, fun _ x ↦ (⟦(x.ren r).tm⟧).sub⟩\n\nnotation:100 \"⟦\" r \"⟧\" => Ren.den r\n\nnoncomputable def Sb.den (σ : Sb Γ Δ) : Cont (⟦Δ cx⟧) (⟦Γ cx⟧) :=\n  ⟨⟨fun ρ _ x ↦ (⟦x.sub σ⟧) ρ, fun ρ' _ x ↦ (⟦x.sub σ⟧) • ρ'⟩, fun _ x ↦ (⟦x.sub σ⟧).sub⟩\n\nnotation:100 \"⟦\" σ \"⟧\" => Sb.den σ\n\nnoncomputable def Con.den : Con Δ υ Γ τ → Cont (⟦Γ cx⟧ × Cont (⟦Δ cx⟧) (⟦υ ty⟧)) ⟦τ ty⟧\n  | id         => Cont.uncurry Cont.id ∘' Cont.swap\n  | comp C₀ C₁ => Cont.uncurry (Cont.curry (C₁.den ∘' Cont.swap)\n                             ∘' Cont.curry (C₀.den ∘' Cont.swap)) ∘' Cont.swap\n  | sub C σ => Cont.uncurry ((Cont.curry C.den) ∘' (⟦σ⟧))\n  | succ C => Cont.flat (Nat.succ) ∘' C.den\n  | pred C => Cont.pred ∘' C.den\n  | zero? C => Cont.flat (Nat.zero?) ∘' C.den\n  | fn C   => Cont.curry ((Cont.uncurry (Cont.curry C.den ∘' Ev.from)) ∘' Cont.assoc_swap_assoc)\n  | cond_s C t f => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair C.den (Cont.pair ((⟦t⟧) ∘' Cont.fst) ((⟦f⟧) ∘' Cont.fst))\n  | cond_t s C f => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair ((⟦s⟧) ∘' Cont.fst) (Cont.pair C.den ((⟦f⟧) ∘' Cont.fst))\n  | cond_f s t C => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair ((⟦s⟧) ∘' Cont.fst) (Cont.pair ((⟦t⟧) ∘' Cont.fst) C.den)\n  | app_f C a => Cont.eval ∘' (Cont.pair C.den ((⟦a⟧) ∘' Cont.fst))\n  | app_a f C => Cont.eval ∘' (Cont.pair ((⟦f⟧) ∘' Cont.fst) C.den)\n  | fix C => Cont.fix' ∘' C.den\n\nnotation:100 \"⟦\" C \" con⟧\" => Con.den C", "target_theorem": "theorem Tm.ren_den_eq (e : Γ ⊢ τ) : ∀ {Δ}, (r : Ren Γ Δ) → ⟦e.ren r⟧ = (⟦e⟧) ∘' (⟦r⟧) :=", "ground_truth_proof": ":= by\n  induction e with\n  | fn e Φ =>\n    intro _ r\n    calc ⟦e.fn.ren r⟧\n      _ = Cont.curry ((⟦e.ren (r.keep _)⟧) ∘ Ev.from) := rfl\n      _ = Cont.curry (((⟦e⟧) ∘' ⟦r.keep _⟧) ∘ Ev.from) := by rw [Φ (r.keep _)]\n      _ = (⟦e.fn⟧) ∘' ⟦r⟧ := by {\n        apply Cont.ext ∘ funext\n        intro ρ\n        apply Cont.ext ∘ funext\n        intro d\n        have p : (⟦r.keep _⟧) (Ev.from (ρ, d)) = Ev.from ((⟦r⟧) ρ, d) := by {\n          funext τ x\n          cases x with\n          | z => rfl\n          | s x => rfl\n        }\n        calc ((((⟦e⟧) ∘' ⟦r.keep _⟧) ∘' Ev.from).curry ρ) d\n          _ = (⟦e⟧) ((⟦r.keep _⟧) (Ev.from (ρ, d))) := rfl\n          _ = (⟦e⟧) (Ev.from ((⟦r⟧) ρ, d)) := by rw [p]\n          _ = ((⟦e.fn⟧) ((⟦r⟧) ρ)) d := rfl\n      }\n  | var | true | false | zero => intros; rfl\n  | succ _ Φ | pred _ Φ | zero? _ Φ | fix _ Φ => intro _ r; exact congrArg _ (Φ r)\n  | app f a Φf Φa =>\n    intro _ r; exact congrArg2 (fun f a ↦ Cont.eval ∘' Cont.pair f a) (Φf r) (Φa r)\n  | cond s t f Φs Φt Φf =>\n    intro _ r\n    calc ⟦(s.cond t f).ren r⟧\n      _ = Cont.uncurry (Cont.cond) ∘' Cont.pair (⟦s.ren r⟧) (Cont.pair (⟦t.ren r⟧) (⟦f.ren r⟧)) := rfl\n      _ = Cont.uncurry (Cont.cond) ∘' Cont.pair ((⟦s⟧) ∘' ⟦r⟧) (Cont.pair ((⟦t⟧) ∘' ⟦r⟧) ((⟦f⟧) ∘' ⟦r⟧))\n        := by rw [Φs, Φt, Φf]\n      _ = Cont.uncurry (Cont.cond) ∘' Cont.pair (⟦s⟧) ((Cont.pair (⟦t⟧) (⟦f⟧))) ∘' ⟦r⟧\n        := by rw [Cont.pair_after (⟦t⟧) (⟦f⟧) (⟦r⟧), Cont.pair_after (⟦s⟧) _ (⟦r⟧)]\n      _ = (⟦s.cond t f⟧) ∘' ⟦r⟧ := rfl", "nesting_depth": 12, "transitive_dep_count": 89, "subset_aristotle": false, "category": "Semantics"}
{"id": 186, "thm_name": "wtRename", "thm_stmt": "theorem wtRename {ξ : ℕ → ℕ} {Γ Δ} {a A : Term}\n  (hξ : Δ ⊢ ξ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ rename ξ a ∶ rename ξ A", "lean_root": "TTBFL", "rel_path": "src/safety.lean", "imports": ["import «src».typing", "import src.syntactics", "import src.reduction", "import src.typing"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}, {"name": "Nat.sub", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "substRename", "content": "def substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )"}, {"name": "renameSubst", "content": "def renameSubst ξ σ : ∀ s, rename ξ (subst σ s) = subst (rename ξ ∘ σ) s :=\n  renameSubst' _ _ (rename ξ ∘ σ) (by admit /- proof elided -/\n  )"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "In", "content": "inductive In : Nat → Term → Ctxt → Prop where\n  | here {Γ A} : In 0 (rename succ A) (Γ ∷ A)\n  | there {Γ x A B} : In x A Γ → In (succ x) (rename succ A) (Γ ∷ B)"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "infix:40 (priority := 1001) \"≈\" => Eqv", "content": "infix:40 (priority := 1001) \"≈\" => Eqv"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}], "lib_lemmas": [{"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "renameLiftRename", "content": "theorem renameLiftRename ξ a : rename succ (rename ξ a) = rename (lift ξ) (rename succ a)"}, {"name": "liftSucc", "content": "omit lc in\ntheorem liftSucc ξ : ∀ x, (lift ξ ∘ succ) x = (succ ∘ ξ) x"}, {"name": "renameComp", "content": "theorem renameComp ξ ζ s : rename ξ (rename ζ s) = rename (ξ ∘ ζ) s"}, {"name": "renameComp'", "content": "theorem renameComp' ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) : ∀ s, (rename ξ ∘ rename ζ) s = rename ς s"}, {"name": "liftComp", "content": "omit lc in\ntheorem liftComp ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) :\n  ∀ x, (lift ξ ∘ lift ζ) x = lift ς x"}, {"name": "renameExt", "content": "theorem renameExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ s, rename ξ s = rename ζ s"}, {"name": "liftExt", "content": "omit lc in\ntheorem liftExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ x, lift ξ x = lift ζ x"}, {"name": "wtfPiInvA", "content": "theorem wtfPiInvA {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j"}, {"name": "wtfPiInvA𝒰", "content": "theorem wtfPiInvA𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j ∧ 𝒰 j ≈ 𝒰'"}, {"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}, {"name": "convRename", "content": "theorem convRename {a b} ξ : a ⇔ b → rename ξ a ⇔ rename ξ b"}, {"name": "parsRename", "content": "theorem parsRename {a b} ξ (r : a ⇒⋆ b) : rename ξ a ⇒⋆ rename ξ b"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "renameDist", "content": "theorem renameDist ξ a s : subst (rename ξ a +: var) (rename (lift ξ) s) = rename ξ (subst (a +: var) s)"}, {"name": "substExt", "content": "theorem substExt σ τ (h : ∀ x, σ x = τ x) : ∀ s, subst σ s = subst τ s"}, {"name": "upExt", "content": "theorem upExt σ τ (h : ∀ x, σ x = τ x) : ∀ x, (⇑ σ) x = (⇑ τ) x"}, {"name": "wRenameLift", "content": "theorem wRenameLift {ξ : ℕ → ℕ} {Γ Δ A}\n  (h : Δ ⊢ ξ ∶ Γ) :\n  Δ ∷ (rename ξ A) ⊢ lift ξ ∶ Γ ∷ A"}, {"name": "inHere", "content": "theorem inHere {Γ A A'} (e : A' = rename succ A) : (Γ ∷ A) ∋ 0 ∶ A'"}, {"name": "inThere", "content": "theorem inThere {Γ x A A' B} (h : Γ ∋ x ∶ A) (e : A' = rename succ A) : Γ ∷ B ∋ succ x ∶ A'"}, {"name": "convEqv", "content": "theorem convEqv {a b} : a ⇔ b → a ≈ b"}, {"name": "parsEqv", "content": "theorem parsEqv {a b} (r : a ⇒⋆ b) : a ≈ b"}, {"name": "parEqv", "content": "theorem parEqv {a b} (r : a ⇒ b) : a ≈ b"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import «src».typing\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]", "target_theorem": "theorem wtRename {ξ : ℕ → ℕ} {Γ Δ} {a A : Term}\n  (hξ : Δ ⊢ ξ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ rename ξ a ∶ rename ξ A :=", "ground_truth_proof": ":= by\n  induction h generalizing ξ Δ\n  case var => constructor; assumption; apply_rules [hξ]\n  case 𝒰 ih => exact Wt.𝒰 (ih hξ hΔ)\n  case pi ihA ihB =>\n    let ihA' := ihA hξ hΔ\n    refine Wt.pi ihA' ?_\n    rw [renameLiftRename]\n    exact ihB (wRenameLift hξ) (Wf.cons hΔ ihA')\n  case abs ihPi ihA ihb =>\n    let ihPi' := ihPi hξ hΔ\n    refine Wt.abs ihPi' (ihA hξ hΔ) ?_\n    let ⟨k, hA⟩ := wtfPiInvA ihPi'\n    exact ihb (wRenameLift hξ) (Wf.cons hΔ hA)\n  case app ihb iha => rw [← renameDist]; exact Wt.app (ihb hξ hΔ) (iha hξ hΔ)\n  case mty ih => exact Wt.mty (ih hξ hΔ)\n  case exf ihb ihA => exact Wt.exf (ihb hξ hΔ) (ihA hξ hΔ)\n  case lvl iha ihj => exact Wt.lvl (iha hξ hΔ) (ihj hξ hΔ)\n  case lof => constructor <;> assumption\n  case trans ihi ihj => exact Wt.trans (ihi hξ hΔ) (ihj hξ hΔ)\n  case conv e _ _ iha ihA =>\n    exact Wt.conv (convEqv (convRename ξ (eqvConv e))) (iha hξ hΔ) (ihA hξ hΔ)\n  case sub ihj ihA => exact Wt.sub (ihj hξ hΔ) (ihA hξ hΔ)", "nesting_depth": 5, "transitive_dep_count": 51, "subset_aristotle": false, "category": "Type systems"}
{"id": 187, "thm_name": "Matrix.PosDef.nonsingular_inv", "thm_stmt": "lemma PosDef.nonsingular_inv [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : M⁻¹.PosDef", "lean_root": "CvxLean", "rel_path": "CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean", "imports": ["import Mathlib.LinearAlgebra.Matrix.PosDef", "import Mathlib.Algebra.Star.Pi"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Invertible", "module": "Mathlib.Algebra.Group.Invertible.Defs"}, {"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Matrix.invertibleOfIsUnitDet", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.det", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "Matrix.mulVec", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "x", "module": "doc.literate.literate_lean_test"}, {"name": "v", "module": "examples.BallisticWidget"}, {"name": "IsUnit", "module": "Mathlib.Algebra.Group.Units.Defs"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Matrix.nondegenerate_iff_det_ne_zero", "module": "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv"}, {"name": "star_eq_zero", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "star_star", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "Matrix.conjTranspose_nonsing_inv", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.inv_eq_right_inv", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.mul_nonsing_inv", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.dotProduct_mulVec", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.star_dotProduct", "module": "Mathlib.LinearAlgebra.Matrix.ConjTranspose"}, {"name": "Matrix.star_mulVec", "module": "Mathlib.LinearAlgebra.Matrix.ConjTranspose"}, {"name": "Matrix.det_ne_zero_of_left_inverse", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.eq_zero_of_mulVec_eq_zero", "module": "Mathlib.LinearAlgebra.Matrix.Nondegenerate"}, {"name": "Matrix.isHermitian_inv", "module": "Mathlib.LinearAlgebra.Matrix.Hermitian"}, {"name": "Matrix.mulVec_mulVec", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.one_mulVec", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "isUnit_iff_ne_zero", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Matrix.PosDef", "content": "noncomputable instance PosDef.Invertible [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) :\n    Invertible M :=\n  invertibleOfIsUnitDet M (isUnit_iff_ne_zero.2 hM.det_ne_zero)"}], "used_local_lemmas": [{"name": "Matrix.PosDef.det_ne_zero", "content": "lemma PosDef.det_ne_zero [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : M.det ≠ 0"}, {"name": "Matrix.IsHermitian.nonsingular_inv", "content": "lemma IsHermitian.nonsingular_inv [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.IsHermitian)\n    (hMdet : IsUnit M.det) : M⁻¹.IsHermitian"}, {"name": "Matrix.conj_symm", "content": "lemma conj_symm {x : n → 𝕜} {M : Matrix n n 𝕜} (hM : M.IsHermitian) :\n    star (star x ⬝ᵥ mulVec M x) = star x ⬝ᵥ mulVec M x"}], "local_ctx": "import Mathlib.LinearAlgebra.Matrix.PosDef\n\nimport Mathlib.Algebra.Star.Pi\n\nnamespace Matrix\n\nvariable {m n : Type _} [Fintype m] [Fintype n]\n\nvariable {𝕜 : Type _}\n\nvariable [NormedField 𝕜] [PartialOrder 𝕜] [StarRing 𝕜] [StarOrderedRing 𝕜]\n\nvariable [RCLike 𝕜]\n\nnoncomputable instance PosDef.Invertible [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) :\n    Invertible M :=\n  invertibleOfIsUnitDet M (isUnit_iff_ne_zero.2 hM.det_ne_zero)", "target_theorem": "lemma PosDef.nonsingular_inv [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : M⁻¹.PosDef :=", "ground_truth_proof": ":= by\n  refine' ⟨hM.1.nonsingular_inv (isUnit_iff_ne_zero.2 hM.det_ne_zero), _⟩\n  intros x hx\n  have hMMinv := mul_nonsing_inv _ (isUnit_iff_ne_zero.2 hM.det_ne_zero)\n  have hMinvdet : M⁻¹.det ≠ 0 := det_ne_zero_of_left_inverse hMMinv\n  have hres :=\n    hM.2 (M⁻¹.mulVec x) (fun h => hx (eq_zero_of_mulVec_eq_zero hMinvdet h))\n  rw [mulVec_mulVec, hMMinv, one_mulVec, star_dotProduct] at hres\n  rw [conj_symm ((@isHermitian_inv _ _ _ _ _ _ M hM.Invertible).2 hM.1)] at hres\n  exact hres", "nesting_depth": 3, "transitive_dep_count": 29, "subset_aristotle": false, "category": "Applied verif."}
{"id": 188, "thm_name": "Matrix.PosSemidef.sqrt_mul_sqrt", "thm_stmt": "@[simp]\nlemma PosSemidef.sqrt_mul_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :\n    hA.1.sqrt * hA.1.sqrt = A :=\n  calc\n    hA.1.sqrt * hA.1.sqrt =\n      hA.1.eigenvectorMatrix * (Matrix.diagonal (fun i => (hA.1.eigenvalues i).sqrt)\n      * (hA.1.eigenvectorMatrixᵀ * hA.1.eigenvectorMatrix)\n      * Matrix.diagonal (fun i => (hA.1.eigenvalues i).sqrt)) * hA.1.eigenvectorMatrixᵀ", "lean_root": "CvxLean", "rel_path": "CvxLean/Lib/Math/Subadditivity.lean", "imports": ["import Mathlib.LinearAlgebra.Matrix.DotProduct", "import CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef", "import CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum", "import CvxLean.Lib.Math.LinearAlgebra.Eigenspace", "import Mathlib.LinearAlgebra.Matrix.LDL", "import Mathlib.LinearAlgebra.Matrix.PosDef", "import Mathlib.LinearAlgebra.Matrix.Spectrum", "import Mathlib.LinearAlgebra.Eigenspace.Basic"], "used_lib_defs": [{"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Matrix.IsHermitian", "module": "Mathlib.LinearAlgebra.Matrix.Hermitian"}, {"name": "A", "module": "examples.CircleOptimisation"}, {"name": "Matrix.diagonal", "module": "Mathlib.Data.Matrix.Diagonal"}, {"name": "Matrix.PosSemidef", "module": "Mathlib.LinearAlgebra.Matrix.PosDef"}, {"name": "Matrix.PosSemidef.sqrt", "module": "Mathlib.Analysis.Matrix.Order"}, {"name": "Real", "module": "Mathlib.Data.Real.Basic"}, {"name": "RCLike", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "RCLike.ofReal", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "EuclideanSpace", "module": "Mathlib.Analysis.InnerProductSpace.PiL2"}, {"name": "Matrix.toLin'", "module": "Mathlib.LinearAlgebra.Matrix.ToLin"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Module.End", "module": "Mathlib.Algebra.Module.LinearMap.End"}, {"name": "Module.End.HasEigenvector", "module": "Mathlib.LinearAlgebra.Eigenspace.Basic"}, {"name": "OrthonormalBasis", "module": "Mathlib.Analysis.InnerProductSpace.PiL2"}, {"name": "Pi.basisFun", "module": "Mathlib.LinearAlgebra.StdBasis"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Matrix.mul_assoc", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.one_mul", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.diagonal_mul_diagonal", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.mul_one", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Real.sqrt_mul", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "Real.sqrt_mul_self", "module": "Mathlib.Data.Real.Sqrt"}], "repo_lemmas": [{"name": "spectral_theorem", "content": "theorem spectral_theorem (xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)) (as : n → ℝ)\n    (hxs : ∀ j, Module.End.HasEigenvector (Matrix.toLin' A) (as j) (xs j)) :\n    xs.toBasis.toMatrix (Pi.basisFun 𝕜 n) * A =\n    diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix (Pi.basisFun 𝕜 n)"}], "used_local_defs": [{"name": "Matrix.IsHermitian.sqrt", "content": "noncomputable def IsHermitian.sqrt {A : Matrix n n ℝ} (hA : A.IsHermitian) : Matrix n n ℝ :=\n  hA.eigenvectorMatrix * Matrix.diagonal (fun i => (hA.eigenvalues i).sqrt) * hA.eigenvectorMatrixᵀ"}], "used_local_lemmas": [{"name": "Matrix.IsHermitian.eigenvectorMatrix_inv_mul", "content": "lemma eigenvectorMatrix_inv_mul : hA.eigenvectorMatrixInv * hA.eigenvectorMatrix = 1"}, {"name": "Matrix.IsHermitian.spectral_theorem''", "content": "theorem spectral_theorem'' :\n    hA.eigenvectorMatrix * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * hA.eigenvectorMatrixᴴ =\n    A"}, {"name": "Matrix.conjTranspose_eq_transpose", "content": "lemma conjTranspose_eq_transpose {m n : Type _} {A : Matrix m n ℝ} : Aᴴ = Aᵀ"}], "local_ctx": "import Mathlib.LinearAlgebra.Matrix.PosDef\n\nimport Mathlib.LinearAlgebra.Matrix.Spectrum\n\nimport Mathlib.LinearAlgebra.Eigenspace.Basic\n\nimport Mathlib.LinearAlgebra.Matrix.LDL\n\nimport Mathlib.LinearAlgebra.Matrix.DotProduct\n\nimport CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef\n\nimport CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum\n\nimport CvxLean.Lib.Math.LinearAlgebra.Eigenspace\n\nnamespace Finset\n\nopen BigOperators\n\nend Finset\n\nnamespace Matrix\n\nvariable {n : Type _} [Fintype n] [DecidableEq n] [LinearOrder n] [LocallyFiniteOrderBot n]\n\nopen BigOperators Matrix\n\nnamespace IsHermitian\n\nvariable {𝕜 : Type _} [DecidableEq 𝕜] [RCLike 𝕜] {A : Matrix n n 𝕜} (hA : A.IsHermitian)\n\nend IsHermitian\n\nnoncomputable def IsHermitian.sqrt {A : Matrix n n ℝ} (hA : A.IsHermitian) : Matrix n n ℝ :=\n  hA.eigenvectorMatrix * Matrix.diagonal (fun i => (hA.eigenvalues i).sqrt) * hA.eigenvectorMatrixᵀ", "target_theorem": "@[simp]\nlemma PosSemidef.sqrt_mul_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :\n    hA.1.sqrt * hA.1.sqrt = A :=", "ground_truth_proof": ":=\n  calc\n    hA.1.sqrt * hA.1.sqrt =\n      hA.1.eigenvectorMatrix * (Matrix.diagonal (fun i => (hA.1.eigenvalues i).sqrt)\n      * (hA.1.eigenvectorMatrixᵀ * hA.1.eigenvectorMatrix)\n      * Matrix.diagonal (fun i => (hA.1.eigenvalues i).sqrt)) * hA.1.eigenvectorMatrixᵀ := by\n      simp [IsHermitian.sqrt, Matrix.mul_assoc]\n    _ = A := by\n      rw [← conjTranspose_eq_transpose, hA.1.conjTranspose_eigenvectorMatrix,\n        hA.1.eigenvectorMatrix_inv_mul, Matrix.mul_one, diagonal_mul_diagonal,\n        ← hA.1.conjTranspose_eigenvectorMatrix]\n      convert hA.1.spectral_theorem''\n      rw [← Real.sqrt_mul (hA.eigenvalues_nonneg _), Real.sqrt_mul_self (hA.eigenvalues_nonneg _)]\n      simp", "nesting_depth": 3, "transitive_dep_count": 27, "subset_aristotle": false, "category": "Applied verif."}
{"id": 189, "thm_name": "log_prod_gaussianPdf", "thm_stmt": "lemma log_prod_gaussianPdf {N n : ℕ} (y : Fin N → Fin n → ℝ) (R : Matrix (Fin n) (Fin n) ℝ)\n    (hR : R.PosDef) : log (∏ i : Fin N, gaussianPdf R (y i)) =\n    ∑ i : Fin N, (- (log (sqrt ((2 * π) ^ n)) + log (sqrt R.det)) + - R⁻¹.quadForm (y i) / 2)", "lean_root": "CvxLean", "rel_path": "CvxLean/Lib/Math/CovarianceEstimation.lean", "imports": ["import CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Float", "module": "Init.Data.Float"}, {"name": "Array.range", "module": "Init.Data.Array.Basic"}, {"name": "Array.zipWith", "module": "Init.Data.Array.Basic"}, {"name": "Float.add", "module": "Init.Data.Float"}, {"name": "Float.mul", "module": "Init.Data.Float"}, {"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Fin.elim0", "module": "Init.Data.Fin.Basic"}, {"name": "Float.ofNat", "module": "Init.Data.OfScientific"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "Real.exp", "module": "Mathlib.Analysis.Complex.Exponential"}, {"name": "Real.sqrt", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Invertible", "module": "Mathlib.Algebra.Group.Invertible.Defs"}, {"name": "Matrix.invertibleOfIsUnitDet", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Real.log", "module": "Mathlib.Analysis.SpecialFunctions.Log.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.sum", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}], "used_repo_defs": [{"name": "mulVec", "content": "def mulVec (M : Matrix (Fin n) (Fin m) Float) (v : (Fin m) → Float) : Fin n → Float :=\n  fun i => (fun j => M i j) ⬝ᵥᶜ v"}, {"name": "dotProduct", "content": "def dotProduct (v w : Fin n → Float) : Float :=\n  (Array.zipWith (Vec.Computable.toArray v) (Vec.Computable.toArray w) Float.mul).foldl Float.add 0"}, {"name": "toArray", "content": "def toArray (x : Fin n → Float) : Array Float :=\n  (Array.range n).map (fun i => if h : i < n then x ⟨i, h⟩ else 0)"}, {"name": "det", "content": "def det {n : ℕ} (A : Matrix (Fin n) (Fin n) Float) : Float :=\n  if h : 0 < n then\n    if n == 1 then A ⟨0, h⟩ ⟨0, h⟩ else\n      (List.finRange n).foldl (fun s i =>\n        s + (-1) ^ (Float.ofNat i.val) * A i ⟨0, h⟩ * det (minor A i ⟨0, h⟩)) 0\n  else 0"}, {"name": "minor", "content": "def minor (A : Matrix (Fin n) (Fin n) Float) (a b : Fin n) :\n    Matrix (Fin n.pred) (Fin n.pred) Float :=\n  match n with\n  | 0 => fun _ => Fin.elim0\n  | _ + 1 => minorAux A a b"}, {"name": "minorAux", "content": "private def minorAux (A : Matrix (Fin n.succ) (Fin n.succ) Float) (a b : Fin n.succ) :\n    Matrix (Fin n) (Fin n) Float :=\n  fun i j =>\n    let i' : Fin n.succ := if i.val < a.val then i else i.succ;\n    let j' : Fin n.succ := if j.val < b.val then j else j.succ;\n    A i' j'"}, {"name": "PosDef.Invertible", "content": "noncomputable instance PosDef.Invertible [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) :\n    Invertible M :=\n  invertibleOfIsUnitDet M (isUnit_iff_ne_zero.2 hM.det_ne_zero)"}, {"name": "infixl:72 \" ⬝ᵥᶜ \" => Matrix.Computable.dotProduct", "content": "infixl:72 \" ⬝ᵥᶜ \" => Matrix.Computable.dotProduct"}], "lib_lemmas": [{"name": "Real.exp_pos", "module": "Mathlib.Analysis.Complex.Exponential"}, {"name": "Real.pi_pos", "module": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic"}, {"name": "Real.sqrt_pos", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "div_pos", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "mul_pos", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "pow_pos", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "zero_lt_one", "module": "Mathlib.Algebra.Order.ZeroLEOne"}, {"name": "Real.exp_ne_zero", "module": "Mathlib.Analysis.Complex.Exponential"}, {"name": "Real.log_div", "module": "Mathlib.Analysis.SpecialFunctions.Log.Basic"}, {"name": "Real.log_exp", "module": "Mathlib.Analysis.SpecialFunctions.Log.Basic"}, {"name": "Real.log_mul", "module": "Mathlib.Analysis.SpecialFunctions.Log.Basic"}, {"name": "Real.log_one", "module": "Mathlib.Analysis.SpecialFunctions.Log.Basic"}, {"name": "Real.log_prod", "module": "Mathlib.Analysis.SpecialFunctions.Log.Basic"}, {"name": "Real.rpow_eq_pow", "module": "Mathlib.Analysis.SpecialFunctions.Pow.Real"}, {"name": "Real.sqrt_mul", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "div_ne_zero", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "le_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "mul_nonneg", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "ne_of_gt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "pow_nonneg", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Matrix.quadForm", "content": "def Matrix.quadForm {n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) (x : Fin n → ℝ) : ℝ :=\n  x ⬝ᵥ R.mulVec x"}, {"name": "gaussianPdf", "content": "def gaussianPdf {n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) (x : Fin n → ℝ) : ℝ :=\n  (1 / sqrt (((2 * π) ^ n) * R.det)) * exp (- R⁻¹.quadForm x / 2)"}], "used_local_lemmas": [{"name": "gaussianPdf_pos", "content": "lemma gaussianPdf_pos {n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) (y : Fin n → ℝ) (h : R.PosDef) :\n    0 < gaussianPdf R y"}], "local_ctx": "import CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef\n\nnoncomputable section CovarianceEstimation\n\nopen Real\n\nopen Matrix\n\nopen BigOperators\n\ndef Matrix.quadForm {n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) (x : Fin n → ℝ) : ℝ :=\n  x ⬝ᵥ R.mulVec x\n\ndef gaussianPdf {n : ℕ} (R : Matrix (Fin n) (Fin n) ℝ) (x : Fin n → ℝ) : ℝ :=\n  (1 / sqrt (((2 * π) ^ n) * R.det)) * exp (- R⁻¹.quadForm x / 2)", "target_theorem": "lemma log_prod_gaussianPdf {N n : ℕ} (y : Fin N → Fin n → ℝ) (R : Matrix (Fin n) (Fin n) ℝ)\n    (hR : R.PosDef) : log (∏ i : Fin N, gaussianPdf R (y i)) =\n    ∑ i : Fin N, (- (log (sqrt ((2 * π) ^ n)) + log (sqrt R.det)) + - R⁻¹.quadForm (y i) / 2) :=", "ground_truth_proof": ":= by\n  have : ∀ i,\n    i ∈ Finset.univ → gaussianPdf R (y i) ≠ 0 := fun i _ => ne_of_gt (gaussianPdf_pos _ _ hR)\n  have sqrt_2_pi_n_R_det_ne_zero: sqrt ((2 * π) ^ n * R.det) ≠ 0 := by\n    refine' ne_of_gt (sqrt_pos.2 (mul_pos _ hR.det_pos))\n    exact (pow_pos (mul_pos (by positivity) pi_pos) _)\n  rw [log_prod Finset.univ (fun i => gaussianPdf R (y i)) this]\n  unfold gaussianPdf\n  apply congr_arg (Finset.sum Finset.univ)\n  ext i\n  rw [log_mul, log_div, sqrt_mul, log_mul, log_exp, log_one, zero_sub]\n  simp [rpow_eq_pow]\n  exact ne_of_gt (sqrt_pos.2 (pow_pos (mul_pos (by positivity) pi_pos) _))\n  exact ne_of_gt (sqrt_pos.2 hR.det_pos)\n  exact pow_nonneg (mul_nonneg (by positivity) (le_of_lt pi_pos)) _\n  norm_num\n  exact sqrt_2_pi_n_R_det_ne_zero\n  exact div_ne_zero (by norm_num) sqrt_2_pi_n_R_det_ne_zero\n  exact exp_ne_zero _", "nesting_depth": 5, "transitive_dep_count": 52, "subset_aristotle": false, "category": "Applied verif."}
{"id": 190, "thm_name": "wp_mono_part", "thm_stmt": "@[local simp]\nlemma wp_mono_part (x : NonDetT DivM α) (post₁ post₂ : α -> Prop) :\n  (post₁ ≤ post₂) → ([totl|wp x post₁]) ≤ ([part| wp x post₂])", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetTheory.lean", "imports": ["import Loom.MonadAlgebras.WP.Basic", "import Loom.MonadAlgebras.NonDetT.Basic", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "LE", "module": "Init.Prelude"}, {"name": "OrderBot", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "OrderTop", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "ForInStep", "module": "Init.Core"}, {"name": "ForInStep.yield", "module": "Init.Core"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.MonadEnv", "module": "Lean.Environment"}, {"name": "Lean.SimpleScopedEnvExtension", "module": "Lean.ScopedEnvExtension"}, {"name": "Lean.SimplePersistentEnvExtension", "module": "Lean.EnvExtension"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "open", "content": "notation \"[part|\" t \"]\" => open PartialCorrectness PartialCorrectness.DemonicChoice in t"}, {"name": "open", "content": "notation \"[part|\" t \"]\" => open ExceptionAsSuccess in t"}, {"name": "open", "content": "notation \"[totl|\" t \"]\" => open TotalCorrectness TotalCorrectness.DemonicChoice in t"}, {"name": "open", "content": "notation \"[totl|\" t \"]\" => open ExceptionAsFailure in t"}, {"name": "LE.pure", "content": "noncomputable def LE.pure {l : Type u} [inst: LE l] [OrderTop l] [OrderBot l] : Prop -> l := fun p =>\n  if p then ⊤ else ⊥"}, {"name": "DivM", "content": "inductive DivM (α : Type u) where\n  | res (x : α)\n  | div"}, {"name": "spec", "content": "def spec (pre : l) (post : α -> l) : Cont l α :=\n  fun p => pre ⊓ ⌜post ≤ p⌝"}, {"name": "NonDetT", "content": "inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} {α : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m α -> Cont l α\n  | .pure ret => pure ret\n  | .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | .pickCont τ p f => fun post => let p : Set τ := p; ⨅ a ∈ (p : Set τ), wp (f a) post"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f =>\n  fun post =>\n    let p : Set τ := p;\n    ⨅ a ∈ (p : Set τ), wp (f a) post\n  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l),\n      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓\n      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)"}, {"name": "NonDetT.μ", "content": "def NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}, {"name": "NonDetT.bind", "content": "def NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f"}, {"name": "WPGen.bind", "content": "def WPGen.bind {x : m α} {f : α -> m β} (wpg : WPGen x) (wpgf : ∀ a, WPGen (f a)) :\n  WPGen (x >>= f) where\n  get := fun post => wpg.get (fun a => (wpgf a).get post)\n  prop := by admit /- proof elided -/"}, {"name": "_root_.Lean.SimpleScopedEnvExtension.get", "content": "private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "Context", "content": "structure Context where\n  ref         : Syntax\n   \n  m           : Syntax\n   \n  returnType  : Syntax\n  mutableVars : VarSet := {}\n  insideFor   : Bool := false"}, {"name": "_root_.Lean.SimplePersistentEnvExtension.get", "content": "private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "WPGen", "content": "structure WPGen (x : m α) where\n  get : Cont l α\n  \n  prop : ∀ post, get post <= wp x post"}, {"name": "_root_.Lean.EnvExtension.get", "content": "private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "CCPOBotLawful", "content": "class CCPOBotLawful (m : Type u -> Type v) [∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] where\n  prop {α} : CCPOBot.compBot (m := m) (α := α) = Lean.Order.bot"}], "lib_lemmas": [{"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "TotalCorrectness.DivM.wp_eq", "content": "lemma TotalCorrectness.DivM.wp_eq (α : Type) (x : DivM α) (post : α -> Prop) :\n  wp x post =\n    match x with"}, {"name": "PartialCorrectness.DivM.wp_eq", "content": "lemma PartialCorrectness.DivM.wp_eq (α : Type) (x : DivM α) (post : α -> Prop) :\n  wp x post =\n    match x with"}, {"name": "NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}, {"name": "NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α β : Type u} (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Loom.MonadAlgebras.NonDetT.Extract\n\nnotation \"[totl|\" t \"]\" => open TotalCorrectness TotalCorrectness.DemonicChoice in t\n\nnotation \"[part|\" t \"]\" => open PartialCorrectness PartialCorrectness.DemonicChoice in t", "target_theorem": "@[local simp]\nlemma wp_mono_part (x : NonDetT DivM α) (post₁ post₂ : α -> Prop) :\n  (post₁ ≤ post₂) → ([totl|wp x post₁]) ≤ ([part| wp x post₂]) :=", "ground_truth_proof": ":= by\n    intro le\n    simp [loomLogicSimp]\n    simp [loomLogicSimp] at le\n    unhygienic induction x <;> try simp [loomLogicSimp]\n    { exact le ret }\n    { simp [[totl| DivM.wp_eq], [part| DivM.wp_eq]]\n      intro wp1\n      match x_1 with\n      | .div => trivial\n      | .res a => simp; simp at wp1; exact f_ih a post₁ post₂ le wp1 }\n    { intro _ _ wp1 i hi\n      exact f_ih i post₁ post₂ le (wp1 i hi) }\n    intro x x_1 h1 sp1\n    exists x\n    constructor\n    { intro b xb\n      have hbx := h1 b xb\n      simp [←[totl| NonDetT.wp_eq_wp]] at hbx\n      simp [←[part| NonDetT.wp_eq_wp]]\n      exact f_ih b (fun x_2 ↦\n        match x_2 with\n        | ForInStep.yield b' => x (ForInStep.yield b') ∧ x_1 b' < x_1 b\n        | ForInStep.done b' => x (ForInStep.done b'))\n        x\n        (by\n          intro x2 hx2\n          match x2 with\n          | .yield b' => simp at hx2; simp [hx2]\n          | .done b' => simp at hx2; simp [hx2] )\n        hbx }\n    simp [spec, LE.pure] at sp1\n    simp [spec, LE.pure, sp1]\n    exact le_trans sp1.right (by\n      simp [loomLogicSimp, ←[totl| NonDetT.wp_eq_wp], ←[part| NonDetT.wp_eq_wp]]\n      intro x1; exact cont_ih x1 post₁ post₂ le )", "nesting_depth": 4, "transitive_dep_count": 43, "subset_aristotle": false, "category": "Framework"}
{"id": 191, "thm_name": "Matrix.det_add_det_le_det_add", "thm_stmt": "lemma det_add_det_le_det_add [Nonempty n] (A B : Matrix n n ℝ) (hA : A.PosSemidef)\n    (hB : B.PosSemidef) : A.det + B.det ≤ (A + B).det", "lean_root": "CvxLean", "rel_path": "CvxLean/Lib/Math/Subadditivity.lean", "imports": ["import Mathlib.LinearAlgebra.Matrix.DotProduct", "import CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef", "import CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum", "import CvxLean.Lib.Math.LinearAlgebra.Eigenspace", "import Mathlib.LinearAlgebra.Matrix.LDL", "import Mathlib.LinearAlgebra.Matrix.PosDef", "import Mathlib.LinearAlgebra.Matrix.Spectrum", "import Mathlib.LinearAlgebra.Eigenspace.Basic"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Float", "module": "Init.Data.Float"}, {"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Fin.elim0", "module": "Init.Data.Fin.Basic"}, {"name": "Float.ofNat", "module": "Init.Data.OfScientific"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "Matrix.PosSemidef", "module": "Mathlib.LinearAlgebra.Matrix.PosDef"}, {"name": "Matrix.det", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "A", "module": "examples.CircleOptimisation"}, {"name": "Matrix.IsHermitian", "module": "Mathlib.LinearAlgebra.Matrix.Hermitian"}, {"name": "Matrix.diagonal", "module": "Mathlib.Data.Matrix.Diagonal"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Invertible", "module": "Mathlib.Algebra.Group.Invertible.Defs"}, {"name": "Matrix.invertibleOfIsUnitDet", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "EuclideanSpace", "module": "Mathlib.Analysis.InnerProductSpace.PiL2"}, {"name": "RCLike", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "x", "module": "doc.literate.literate_lean_test"}, {"name": "Real", "module": "Mathlib.Data.Real.Basic"}, {"name": "Matrix.PosSemidef.sqrt", "module": "Mathlib.Analysis.Matrix.Order"}, {"name": "RCLike.ofReal", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "Matrix.toLin'", "module": "Mathlib.LinearAlgebra.Matrix.ToLin"}, {"name": "Module", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "Module.End", "module": "Mathlib.Algebra.Module.LinearMap.End"}, {"name": "Module.End.HasEigenvector", "module": "Mathlib.LinearAlgebra.Eigenspace.Basic"}, {"name": "OrthonormalBasis", "module": "Mathlib.Analysis.InnerProductSpace.PiL2"}, {"name": "Pi.basisFun", "module": "Mathlib.LinearAlgebra.StdBasis"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "IsUnit", "module": "Mathlib.Algebra.Group.Units.Defs"}, {"name": "Module.End.eigenspace", "module": "Mathlib.LinearAlgebra.Eigenspace.Basic"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}], "used_repo_defs": [{"name": "det", "content": "def det {n : ℕ} (A : Matrix (Fin n) (Fin n) Float) : Float :=\n  if h : 0 < n then\n    if n == 1 then A ⟨0, h⟩ ⟨0, h⟩ else\n      (List.finRange n).foldl (fun s i =>\n        s + (-1) ^ (Float.ofNat i.val) * A i ⟨0, h⟩ * det (minor A i ⟨0, h⟩)) 0\n  else 0"}, {"name": "minor", "content": "def minor (A : Matrix (Fin n) (Fin n) Float) (a b : Fin n) :\n    Matrix (Fin n.pred) (Fin n.pred) Float :=\n  match n with\n  | 0 => fun _ => Fin.elim0\n  | _ + 1 => minorAux A a b"}, {"name": "minorAux", "content": "private def minorAux (A : Matrix (Fin n.succ) (Fin n.succ) Float) (a b : Fin n.succ) :\n    Matrix (Fin n) (Fin n) Float :=\n  fun i j =>\n    let i' : Fin n.succ := if i.val < a.val then i else i.succ;\n    let j' : Fin n.succ := if j.val < b.val then j else j.succ;\n    A i' j'"}, {"name": "PosDef.Invertible", "content": "noncomputable instance PosDef.Invertible [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) :\n    Invertible M :=\n  invertibleOfIsUnitDet M (isUnit_iff_ne_zero.2 hM.det_ne_zero)"}], "lib_lemmas": [{"name": "Real.sqrt_nonneg", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "Matrix.mul_assoc", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.one_mul", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.diagonal_mul_diagonal", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.mul_one", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Real.sqrt_mul", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "Real.sqrt_mul_self", "module": "Mathlib.Data.Real.Sqrt"}, {"name": "Matrix.det_mul", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "Matrix.dotProduct_mulVec", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.exists_mulVec_eq_zero_iff", "module": "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv"}, {"name": "Matrix.mulVec_mulVec", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.mul_zero", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.transpose_transpose", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Matrix.vecMul_transpose", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "RCLike.re_to_real", "module": "Mathlib.Analysis.RCLike.Basic"}, {"name": "inner_self_eq_zero", "module": "Mathlib.Analysis.InnerProductSpace.Basic"}, {"name": "lt_of_le_of_ne'", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Finset.powerset_univ", "module": "Mathlib.Data.Fintype.Powerset"}, {"name": "Finset.prod_add", "module": "Mathlib.Algebra.BigOperators.Ring.Finset"}, {"name": "Finset.prod_nonneg", "module": "Mathlib.Algebra.Order.BigOperators.Ring.Finset"}, {"name": "Finset.sum_le_univ_sum_of_nonneg", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "Finset.sum_pair", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.univ_nonempty", "module": "Mathlib.Data.Finset.BooleanAlgebra"}, {"name": "Matrix.add_mul", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.det_conj", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.det_one", "module": "Mathlib.LinearAlgebra.Matrix.Determinant.Basic"}, {"name": "Matrix.isUnit_iff_isUnit_det", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.mul_add", "module": "Mathlib.Data.Matrix.Mul"}, {"name": "Matrix.mul_inv_rev", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.mul_nonsing_inv", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.nonsing_inv_mul", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "Matrix.toLin'_mul", "module": "Mathlib.LinearAlgebra.Matrix.ToLin"}, {"name": "Matrix.toLin'_one", "module": "Mathlib.LinearAlgebra.Matrix.ToLin"}, {"name": "add_left_inj", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "isUnit_iff_ne_zero", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "map_add", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_add", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_eq_mul_left_iff", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "mul_le_mul_left", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "symm", "module": "Mathlib.Order.Defs.Unbundled"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "PosSemidef_diagonal", "content": "lemma PosSemidef_diagonal [DecidableEq n] {f : n → ℝ} (hf : ∀ i, 0 ≤ f i) :\n    (diagonal f).PosSemidef"}, {"name": "PosSemidef.conjTranspose_mul_mul", "content": "lemma PosSemidef.conjTranspose_mul_mul (M N : Matrix n n 𝕜) (hM : M.PosSemidef) :\n    (Nᴴ * M * N).PosSemidef"}, {"name": "spectral_theorem", "content": "theorem spectral_theorem (xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)) (as : n → ℝ)\n    (hxs : ∀ j, Module.End.HasEigenvector (Matrix.toLin' A) (as j) (xs j)) :\n    xs.toBasis.toMatrix (Pi.basisFun 𝕜 n) * A =\n    diagonal (RCLike.ofReal ∘ as) * xs.toBasis.toMatrix (Pi.basisFun 𝕜 n)"}, {"name": "PosDef.det_ne_zero", "content": "lemma PosDef.det_ne_zero [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : M.det ≠ 0"}, {"name": "has_eigenvector_one", "content": "lemma has_eigenvector_one {x : M} (hx : x ≠ 0) : HasEigenvector (1 : End R M) 1 x"}, {"name": "eigenspace_one", "content": "lemma eigenspace_one : eigenspace (1 : End R M) 1 = ⊤"}, {"name": "has_eigenvector_add", "content": "lemma has_eigenvector_add {f g : End R M} {a b : R} {x : M} (hf : HasEigenvector f a x)\n    (hg : HasEigenvector g b x) : HasEigenvector (f + g) (a + b) x"}, {"name": "eigenspace_add", "content": "lemma eigenspace_add {f g : End R M} {a b : R} :\n    eigenspace f a ⊓ eigenspace g b ≤ eigenspace (f + g) (a + b)"}, {"name": "PosSemidef.mul_mul_of_IsHermitian", "content": "lemma PosSemidef.mul_mul_of_IsHermitian {M N : Matrix n n 𝕜} (hM : M.PosSemidef)\n    (hN : N.IsHermitian) : (N * M * N).PosSemidef"}, {"name": "PosDef.conjTranspose_mul_mul", "content": "lemma PosDef.conjTranspose_mul_mul [DecidableEq n] (M N : Matrix n n 𝕜) (hM : M.PosDef)\n    (hN : N.det ≠ 0) : (Nᴴ * M * N).PosDef"}, {"name": "det_eq_prod_eigenvalues", "content": "lemma det_eq_prod_eigenvalues (xs : OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n)) (as : n → ℝ)\n    (hxs : ∀ j, Module.End.HasEigenvector (Matrix.toLin' A) (as j) (xs j)) : det A = ∏ i, as i"}, {"name": "IsHermitian.nonsingular_inv", "content": "lemma IsHermitian.nonsingular_inv [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.IsHermitian)\n    (hMdet : IsUnit M.det) : M⁻¹.IsHermitian"}], "used_local_defs": [{"name": "Matrix.IsHermitian.sqrt", "content": "noncomputable def IsHermitian.sqrt {A : Matrix n n ℝ} (hA : A.IsHermitian) : Matrix n n ℝ :=\n  hA.eigenvectorMatrix * Matrix.diagonal (fun i => (hA.eigenvalues i).sqrt) * hA.eigenvectorMatrixᵀ"}], "used_local_lemmas": [{"name": "Finset.one_add_prod_le_prod_one_add", "content": "lemma one_add_prod_le_prod_one_add {n : Type _} [Fintype n] [Nonempty n]\n    (f : n → ℝ) (hf : ∀ i, 0 ≤ f i) : 1 + (∏ i, f i) ≤ ∏ i, (1 + f i)"}, {"name": "Matrix.IsHermitian.eigenvectorMatrix_inv_mul", "content": "lemma eigenvectorMatrix_inv_mul : hA.eigenvectorMatrixInv * hA.eigenvectorMatrix = 1"}, {"name": "Matrix.IsHermitian.spectral_theorem''", "content": "theorem spectral_theorem'' :\n    hA.eigenvectorMatrix * diagonal (RCLike.ofReal ∘ hA.eigenvalues) * hA.eigenvectorMatrixᴴ =\n    A"}, {"name": "Matrix.conjTranspose_eq_transpose", "content": "lemma conjTranspose_eq_transpose {m n : Type _} {A : Matrix m n ℝ} : Aᴴ = Aᵀ"}, {"name": "Matrix.PosSemidef.sqrt_mul_sqrt", "content": "@[simp]\nlemma PosSemidef.sqrt_mul_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :\n    hA.1.sqrt * hA.1.sqrt = A"}, {"name": "Matrix.PosSemidef.PosSemidef_sqrt", "content": "lemma PosSemidef.PosSemidef_sqrt {A : Matrix n n ℝ} (hA : A.PosSemidef) :\n    hA.1.sqrt.PosSemidef"}, {"name": "Matrix.IsHermitian.has_eigenvector_one_add", "content": "lemma IsHermitian.has_eigenvector_one_add {A : Matrix n n ℝ} (hA : A.IsHermitian) (i : n) :\n    Module.End.HasEigenvector\n      (1 + Matrix.toLin' A) (1 + (hA.eigenvalues i)) ((hA.eigenvectorBasis) i)"}, {"name": "Matrix.PosSemidef.PosDef_iff_det_ne_zero", "content": "lemma PosSemidef.PosDef_iff_det_ne_zero [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosSemidef) :\n    M.PosDef ↔ M.det ≠ 0"}, {"name": "Matrix.det_add_det_le_det_add'", "content": "lemma det_add_det_le_det_add' [Nonempty n] (A B : Matrix n n ℝ) (hA : A.PosDef)\n    (hB : B.PosSemidef) : A.det + B.det ≤ (A + B).det"}], "local_ctx": "import Mathlib.LinearAlgebra.Matrix.PosDef\n\nimport Mathlib.LinearAlgebra.Matrix.Spectrum\n\nimport Mathlib.LinearAlgebra.Eigenspace.Basic\n\nimport Mathlib.LinearAlgebra.Matrix.LDL\n\nimport Mathlib.LinearAlgebra.Matrix.DotProduct\n\nimport CvxLean.Lib.Math.LinearAlgebra.Matrix.PosDef\n\nimport CvxLean.Lib.Math.LinearAlgebra.Matrix.Spectrum\n\nimport CvxLean.Lib.Math.LinearAlgebra.Eigenspace\n\nnamespace Finset\n\nopen BigOperators\n\nend Finset\n\nnamespace Matrix\n\nvariable {n : Type _} [Fintype n] [DecidableEq n] [LinearOrder n] [LocallyFiniteOrderBot n]\n\nopen BigOperators Matrix\n\nnamespace IsHermitian\n\nvariable {𝕜 : Type _} [DecidableEq 𝕜] [RCLike 𝕜] {A : Matrix n n 𝕜} (hA : A.IsHermitian)\n\nend IsHermitian\n\nnoncomputable def IsHermitian.sqrt {A : Matrix n n ℝ} (hA : A.IsHermitian) : Matrix n n ℝ :=\n  hA.eigenvectorMatrix * Matrix.diagonal (fun i => (hA.eigenvalues i).sqrt) * hA.eigenvectorMatrixᵀ", "target_theorem": "lemma det_add_det_le_det_add [Nonempty n] (A B : Matrix n n ℝ) (hA : A.PosSemidef)\n    (hB : B.PosSemidef) : A.det + B.det ≤ (A + B).det :=", "ground_truth_proof": ":= by\n  by_cases hA' : A.det = 0\n  { by_cases hB' : B.det = 0\n    { simp [hA', hB']\n      apply (hA.add hB).det_nonneg }\n    { rw [add_comm A B, add_comm A.det B.det]\n      apply det_add_det_le_det_add' _ _ (hB.PosDef_iff_det_ne_zero.2 hB') hA } }\n  { apply det_add_det_le_det_add' _ _ (hA.PosDef_iff_det_ne_zero.2 hA') hB }", "nesting_depth": 5, "transitive_dep_count": 105, "subset_aristotle": false, "category": "Applied verif."}
{"id": 192, "thm_name": "Vec.le_div_iff", "thm_stmt": "lemma le_div_iff (hc : StrongLT 0 c) : a ≤ b / c ↔ a * c ≤ b", "lean_root": "CvxLean", "rel_path": "CvxLean/Lib/Math/Data/Vec.lean", "imports": ["import CvxLean.Lib.Math.Data.Real", "import Mathlib.Analysis.NormedSpace.PiLp", "import CvxLean.Lib.Math.Data.Fin", "import Mathlib.Analysis.InnerProductSpace.PiL2"], "used_lib_defs": [{"name": "StrongLT", "module": "Mathlib.Order.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Mathlib.Analysis.NormedSpace.PiLp\n\nimport Mathlib.Analysis.InnerProductSpace.PiL2\n\nimport CvxLean.Lib.Math.Data.Real\n\nimport CvxLean.Lib.Math.Data.Fin\n\nnamespace Vec\n\nvariable {m : Type u} {n : Type v} [Fintype m] [Fintype n] {α : Type w}\n\nsection AddCommMonoid\n\nvariable [AddCommMonoid α] {m : Nat} {n : Nat} (x : Fin m → α) (y : Fin n → α)\n\nopen BigOperators\n\nopen FinsetInterval\n\nend AddCommMonoid\n\nnoncomputable section Real\n\nopen Real BigOperators\n\nvariable (x y : m → ℝ)\n\nend Real\n\nsection RealLemmas\n\nvariable {a b c : m → ℝ}", "target_theorem": "lemma le_div_iff (hc : StrongLT 0 c) : a ≤ b / c ↔ a * c ≤ b :=", "ground_truth_proof": ":= by\n  constructor\n  · intro h i; have hi := h i; simp at hi;\n    rw [_root_.le_div_iff (hc i)] at hi; exact hi\n  · intro h i; have hi := h i; simp at hi;\n    dsimp; rw [_root_.le_div_iff (hc i)]; exact hi", "nesting_depth": 1, "transitive_dep_count": 1, "subset_aristotle": false, "category": "Applied verif."}
{"id": 193, "thm_name": "Vec.div_le_iff", "thm_stmt": "lemma div_le_iff (hb : StrongLT 0 b) : a / b ≤ c ↔ a ≤ c * b", "lean_root": "CvxLean", "rel_path": "CvxLean/Lib/Math/Data/Vec.lean", "imports": ["import CvxLean.Lib.Math.Data.Real", "import Mathlib.Analysis.NormedSpace.PiLp", "import CvxLean.Lib.Math.Data.Fin", "import Mathlib.Analysis.InnerProductSpace.PiL2"], "used_lib_defs": [{"name": "StrongLT", "module": "Mathlib.Order.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Mathlib.Analysis.NormedSpace.PiLp\n\nimport Mathlib.Analysis.InnerProductSpace.PiL2\n\nimport CvxLean.Lib.Math.Data.Real\n\nimport CvxLean.Lib.Math.Data.Fin\n\nnamespace Vec\n\nvariable {m : Type u} {n : Type v} [Fintype m] [Fintype n] {α : Type w}\n\nsection AddCommMonoid\n\nvariable [AddCommMonoid α] {m : Nat} {n : Nat} (x : Fin m → α) (y : Fin n → α)\n\nopen BigOperators\n\nopen FinsetInterval\n\nend AddCommMonoid\n\nnoncomputable section Real\n\nopen Real BigOperators\n\nvariable (x y : m → ℝ)\n\nend Real\n\nsection RealLemmas\n\nvariable {a b c : m → ℝ}", "target_theorem": "lemma div_le_iff (hb : StrongLT 0 b) : a / b ≤ c ↔ a ≤ c * b :=", "ground_truth_proof": ":= by\n  constructor\n  · intro h i; have hi := h i; simp at hi;\n    rw [_root_.div_le_iff (hb i)] at hi; exact hi\n  · intro h i; have hi := h i; simp at hi;\n    dsimp; rw [_root_.div_le_iff (hb i)]; exact hi", "nesting_depth": 1, "transitive_dep_count": 1, "subset_aristotle": false, "category": "Applied verif."}
{"id": 194, "thm_name": "DemonicChoice.ExtractNonDet.extract_refines_wp", "thm_stmt": "lemma ExtractNonDet.extract_refines_wp (s : NonDetT m α) (inst : ExtractNonDet Findable s) :\n  wp s post ⊓ s.prop ⊤ <= wp s.extract post", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/NonDetT'/Extract.lean", "imports": ["import Loom.MonadAlgebras.WP.Gen", "import Loom.MonadAlgebras.WP.Liberal", "import Mathlib.Order.CompleteBooleanAlgebra", "import Mathlib.Logic.Function.Basic", "import Mathlib.Data.W.Basic", "import Loom.MonadAlgebras.NonDetT'.Basic", "import Loom.MonadAlgebras.WP.Basic", "import Mathlib.Order.Lattice", "import Mathlib.Data.FinEnum", "import Mathlib.Order.Basic", "import Loom/MonadAlgebras/NonDetT/Findable.lean"], "used_lib_defs": [{"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Encodable", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Encodable.decode", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "BooleanAlgebra", "module": "Mathlib.Order.BooleanAlgebra.Defs"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "inline", "module": "Init.Core"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.MonadEnv", "module": "Lean.Environment"}, {"name": "Lean.SimpleScopedEnvExtension", "module": "Lean.ScopedEnvExtension"}, {"name": "Lean.SimplePersistentEnvExtension", "module": "Lean.EnvExtension"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "MonadNonDet", "content": "class MonadNonDet (m : Type u → Type v) where\n  pick : (τ : Type u) → [Inhabited τ] → m τ\n  \n  pickSuchThat : (τ : Type u) → (p : τ → Prop) → [Findable p] → m τ\n  assume : (as : Prop) → [Decidable as] → m PUnit.{u+1}\n  \n  rep {α : Type u} : α → (α → m (ForInStep α)) → m α"}, {"name": "wlp", "content": "def wlp (c : m α) (post : α -> l) : l := iwp c post ⊔ wp c post"}, {"name": "iwp", "content": "abbrev iwp (c : m α) : Cont l α := Cont.inv (wp c)"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "Cont.inv", "content": "def Cont.inv {t : Type v} {α : Type u} [BooleanAlgebra t] (wp : Cont t α) : Cont t α :=\n  fun f => (wp fun x => (f x)ᶜ)ᶜ"}, {"name": "Cont", "content": "abbrev Cont (t : Type v) (α : Type u) := (α -> t) -> t"}, {"name": "NonDetT", "content": "inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α"}, {"name": "CCPOBot", "content": "class CCPOBot (m : Type u -> Type v)  where\n  compBot {α} : m α"}, {"name": "WPGen", "content": "structure WPGen (x : m α) where\n  get : Cont l α\n  \n  prop : ∀ post, get post <= wp x post"}, {"name": "CCPOBotLawful", "content": "class CCPOBotLawful (m : Type u -> Type v) [∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] where\n  prop {α} : CCPOBot.compBot (m := m) (α := α) = Lean.Order.bot"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} {α : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m α -> Cont l α\n  | .pure ret => pure ret\n  | .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | .pickCont τ p f => fun post => let p : Set τ := p; ⨅ a ∈ (p : Set τ), wp (f a) post"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "NonDetT.μ", "content": "def NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "NonDetT.bind", "content": "def NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f"}, {"name": "WPGen.bind", "content": "def WPGen.bind {x : m α} {f : α -> m β} (wpg : WPGen x) (wpgf : ∀ a, WPGen (f a)) :\n  WPGen (x >>= f) where\n  get := fun post => wpg.get (fun a => (wpgf a).get post)\n  prop := by admit /- proof elided -/"}, {"name": "_root_.Lean.SimpleScopedEnvExtension.get", "content": "private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "Context", "content": "structure Context where\n  ref         : Syntax\n   \n  m           : Syntax\n   \n  returnType  : Syntax\n  mutableVars : VarSet := {}\n  insideFor   : Bool := false"}, {"name": "_root_.Lean.SimplePersistentEnvExtension.get", "content": "private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "_root_.Lean.EnvExtension.get", "content": "private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "NonDetT.pickSuchThat", "content": "def NonDetT.pickSuchThat (τ : Type u) (p : τ → Prop) [Findable p] : NonDetT m τ :=\n  NonDetT.pickCont τ p pure"}, {"name": "{l", "content": "instance {l σ : Type u} : MonadLift (Cont l) (Cont (σ -> l)) where\n  monadLift x := fun f s => x (f · s)"}], "lib_lemmas": [{"name": "iInf_const", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "iInf_inf_eq", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "iInf_le_of_le", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "iSup_const", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "inf_assoc", "module": "Mathlib.Order.Lattice"}, {"name": "inf_comm", "module": "Mathlib.Order.Lattice"}, {"name": "inf_le_of_left_le", "module": "Mathlib.Order.Lattice"}], "repo_lemmas": [{"name": "wp_pure", "content": "lemma wp_pure (x : α) (post : α -> l) : wp (m := m) (pure x) post = post x"}, {"name": "wp_bind", "content": "lemma wp_bind {β} (x : m α) (f : α -> m β) (post : β -> l) :\n    wp (x >>= f) post = wp x (fun x => wp (f x) post)"}, {"name": "wlp_join_wp", "content": "lemma wlp_join_wp (c : m α) (post post' : α -> l) :\n  wlp c post ⊓ wp c post' = wp c (fun x => post x ⊓ post' x)"}, {"name": "wlp_himp", "content": "lemma wlp_himp (c : m α) (post post' : α -> l) :\n  wp c (fun x => post' x ⇨ post x) = wlp c post' ⇨ wp c post"}, {"name": "wp_wlp", "content": "omit [LawfulMonad m] in\nlemma wp_wlp (c : m α) (post : α -> l) :\n  wp c post <= wlp c post"}, {"name": "NonDetT.wp_vis", "content": "@[simp]\nlemma NonDetT.wp_vis {β : Type u} (x : m β) (f : β → NonDetT m α) post :\n  _root_.wp (NonDetT.vis x f) post = _root_.wp x fun a => _root_.wp (f a) post"}, {"name": "NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}, {"name": "NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α β : Type u} (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}, {"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}, {"name": "NonDetT.wp_pickCont", "content": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post"}, {"name": "NonDetT.wp_pickCont", "content": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⨆ a, ⌜p a⌝ ⊓ _root_.wp (f a) post"}, {"name": "meet_himp", "content": "lemma meet_himp (x x' y z : l) :\n  x = x' ->\n  (x ⇨ y) ⊓ (x' ⇨ z) = x ⇨ (y ⊓ z)"}], "used_local_defs": [{"name": "findNat", "content": "def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0"}, {"name": "find", "content": "def find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode"}, {"name": "WeakFindable", "content": "class WeakFindable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_some_p : find () = some x -> p x"}, {"name": "Findable", "content": "class Findable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_none : (find ()).isNone -> ∀ x, ¬ p x\n  find_some_p : find () = some x -> p x"}, {"name": "WeakFindable", "content": "instance WeakFindable.of_Findable {α : Type u} (p : α -> Prop) [Findable p] : WeakFindable p where\n  find := Findable.find p\n  find_some_p := Findable.find_some_p"}, {"name": "_inst_α", "content": "instance {p : α -> Prop} [Encodable α] [DecidablePred p] : Findable p where\n  find := fun _ => find p\n  find_none := find_none p\n  find_some_p := find_some_p p _"}, {"name": "_inst_α", "content": "@[instance high]\ninstance {p : α -> Prop} [FinEnum α] [DecidablePred p] : Findable p where\n  find := fun _ => FinEnum.toList α |>.find? p\n  find_none := by admit /- proof elided -/"}, {"name": "ExtractNonDet", "content": "inductive ExtractNonDet (findable : {τ : Type u} -> (τ -> Prop) -> Type u) {m} : {α : Type u} -> NonDetT m α -> Type _ where\n  | pure {α} : ∀ (x : α), ExtractNonDet findable (NonDetT.pure x)\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) :\n      (∀ y, ExtractNonDet findable (f y)) → ExtractNonDet findable (.vis x f)\n  | pickSuchThat {α} (τ : Type u) (p : τ -> Prop) (f : τ → NonDetT m α)\n    {_ : findable p}\n     : (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont τ p f)\n  | assume {α} (p : PUnit -> Prop) (f : PUnit → NonDetT m α) {_ : Decidable (p .unit)} :\n    (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont PUnit p f)"}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pure' : ExtractNonDet findable (Pure.pure (f := NonDetT m) x) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.liftM (x : m α) :\n  ExtractNonDet findable (liftM (n := NonDetT m) x) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.assume' {p : Prop} [Decidable p] : ExtractNonDet findable (MonadNonDet.assume (m :=  NonDetT m) p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pickSuchThat' {τ : Type u} (p : τ -> Prop) [Findable p] :\n  ExtractNonDet Findable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pickSuchThat_weak {τ : Type u} (p : τ -> Prop) [WeakFindable p] :\n  ExtractNonDet WeakFindable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.if {p : Prop} {dec : Decidable p} {x y : NonDetT m α}\n  (_ : ExtractNonDet findable x) (_ : ExtractNonDet findable y) :\n  ExtractNonDet findable (if p then x else y) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.ForIn_list {xs : List α} {init : β} {f : α → β → NonDetT m (ForInStep β)}\n  (_ : ∀ a b, ExtractNonDet findable (f a b)) :\n  ExtractNonDet findable (forIn xs init f) :="}, {"name": "NonDetT.extractGen", "content": "@[simp, inline]\ndef NonDetT.extractGen {findable : {τ : Type u} -> (τ -> Prop) -> Type u} {α : Type u}\n  (findOf : ∀ {τ : Type u} (p : τ -> Prop), findable p -> Unit -> Option τ)\n  : (s : NonDetT m α) -> (ex : ExtractNonDet findable s := by admit /- proof elided -/\n  ) -> m α\n  | .pure x, _ => Pure.pure x\n  | .vis x f, .vis _ _ _ => liftM x >>= (fun x => extractGen findOf (f x))\n  | .pickCont _ p f, .pickSuchThat _ _ _ _ =>\n    match findOf p ‹_› () with\n    | none => CCPOBot.compBot\n    | some x =>  extractGen findOf (f x)\n  | .pickCont _ p f, .assume _ _ _ =>\n    if p .unit then\n      extractGen findOf (f .unit)\n    else CCPOBot.compBot"}, {"name": "NonDetT.extract", "content": "@[inline]\ndef NonDetT.extract {α : Type u} (s : NonDetT m α) (ex : ExtractNonDet Findable s := by admit /- proof elided -/\n) : m α :=\n  NonDetT.extractGen Findable.find s"}, {"name": "NonDetT.prop", "content": "abbrev NonDetT.prop {α : Type u} : (s : NonDetT m α) -> Cont l α\n  | .pure x => Pure.pure x\n  | .vis x f => fun post => wlp x fun y => NonDetT.prop (f y) post\n  | .pickCont _ p f => fun post =>\n    (⨅ t, ⌜p t⌝ ⇨ NonDetT.prop (f t) post) ⊓ (⨆ t, ⌜p t⌝)"}, {"name": "Extractable", "content": "structure Extractable (x : NonDetT m α) where\n  cond : Cont l α\n  prop : ∀ post, cond post <= x.prop post"}, {"name": "ExtractNonDet.prop", "content": "def ExtractNonDet.prop {α : Type u} (s : NonDetT m α) :  ExtractNonDet WeakFindable s -> l\n  | .pure x => ⊤\n  | .vis x f ex => wlp x fun y => (ex y).prop\n  | .pickSuchThat _ p f ex => ⨅ t ∈ WeakFindable.find p (), (ex t).prop\n  | .assume p f ex =>\n      if p .unit then\n        (ex .unit).prop\n      else ⊤"}], "used_local_lemmas": [], "local_ctx": "import Mathlib.Logic.Function.Basic\n\nimport Mathlib.Order.CompleteBooleanAlgebra\n\nimport Mathlib.Order.Lattice\n\nimport Mathlib.Order.Basic\n\nimport Mathlib.Data.W.Basic\n\nimport Mathlib.Data.FinEnum\n\nimport Loom.MonadAlgebras.WP.Gen\n\nimport Loom.MonadAlgebras.WP.Liberal\n\nimport Loom.MonadAlgebras.NonDetT'.Basic\n\nopen Lean.Order\n\ndef findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0\n\ndef find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode\n\nclass WeakFindable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_some_p : find () = some x -> p x\n\nclass Findable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_none : (find ()).isNone -> ∀ x, ¬ p x\n  find_some_p : find () = some x -> p x\n\ninstance WeakFindable.of_Findable {α : Type u} (p : α -> Prop) [Findable p] : WeakFindable p where\n  find := Findable.find p\n  find_some_p := Findable.find_some_p\n\ninstance {p : α -> Prop} [Encodable α] [DecidablePred p] : Findable p where\n  find := fun _ => find p\n  find_none := find_none p\n  find_some_p := find_some_p p _\n\n@[instance high]\ninstance {p : α -> Prop} [FinEnum α] [DecidablePred p] : Findable p where\n  find := fun _ => FinEnum.toList α |>.find? p\n  find_none := by admit /- proof elided -/\n\ninductive ExtractNonDet (findable : {τ : Type u} -> (τ -> Prop) -> Type u) {m} : {α : Type u} -> NonDetT m α -> Type _ where\n  | pure {α} : ∀ (x : α), ExtractNonDet findable (NonDetT.pure x)\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) :\n      (∀ y, ExtractNonDet findable (f y)) → ExtractNonDet findable (.vis x f)\n  | pickSuchThat {α} (τ : Type u) (p : τ -> Prop) (f : τ → NonDetT m α)\n    {_ : findable p}\n     : (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont τ p f)\n  | assume {α} (p : PUnit -> Prop) (f : PUnit → NonDetT m α) {_ : Decidable (p .unit)} :\n    (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont PUnit p f)\n\ninstance ExtractNonDet.pure' : ExtractNonDet findable (Pure.pure (f := NonDetT m) x) :=\n\ninstance ExtractNonDet.liftM (x : m α) :\n  ExtractNonDet findable (liftM (n := NonDetT m) x) :=\n\ninstance ExtractNonDet.assume' {p : Prop} [Decidable p] : ExtractNonDet findable (MonadNonDet.assume (m :=  NonDetT m) p) :=\n\ninstance ExtractNonDet.pickSuchThat' {τ : Type u} (p : τ -> Prop) [Findable p] :\n  ExtractNonDet Findable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=\n\ninstance ExtractNonDet.pickSuchThat_weak {τ : Type u} (p : τ -> Prop) [WeakFindable p] :\n  ExtractNonDet WeakFindable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=\n\ninstance ExtractNonDet.if {p : Prop} {dec : Decidable p} {x y : NonDetT m α}\n  (_ : ExtractNonDet findable x) (_ : ExtractNonDet findable y) :\n  ExtractNonDet findable (if p then x else y) :=\n\ninstance ExtractNonDet.ForIn_list {xs : List α} {init : β} {f : α → β → NonDetT m (ForInStep β)}\n  (_ : ∀ a b, ExtractNonDet findable (f a b)) :\n  ExtractNonDet findable (forIn xs init f) :=\n\nvariable [Monad m] [CCPOBot m] [CompleteBooleanAlgebra l] [MAlgOrdered m l] [MAlgDet m l] [LawfulMonad m]\n\n@[simp, inline]\ndef NonDetT.extractGen {findable : {τ : Type u} -> (τ -> Prop) -> Type u} {α : Type u}\n  (findOf : ∀ {τ : Type u} (p : τ -> Prop), findable p -> Unit -> Option τ)\n  : (s : NonDetT m α) -> (ex : ExtractNonDet findable s := by admit /- proof elided -/\n  ) -> m α\n  | .pure x, _ => Pure.pure x\n  | .vis x f, .vis _ _ _ => liftM x >>= (fun x => extractGen findOf (f x))\n  | .pickCont _ p f, .pickSuchThat _ _ _ _ =>\n    match findOf p ‹_› () with\n    | none => CCPOBot.compBot\n    | some x =>  extractGen findOf (f x)\n  | .pickCont _ p f, .assume _ _ _ =>\n    if p .unit then\n      extractGen findOf (f .unit)\n    else CCPOBot.compBot\n\n@[inline]\ndef NonDetT.extract {α : Type u} (s : NonDetT m α) (ex : ExtractNonDet Findable s := by admit /- proof elided -/\n) : m α :=\n  NonDetT.extractGen Findable.find s\n\nabbrev NonDetT.prop {α : Type u} : (s : NonDetT m α) -> Cont l α\n  | .pure x => Pure.pure x\n  | .vis x f => fun post => wlp x fun y => NonDetT.prop (f y) post\n  | .pickCont _ p f => fun post =>\n    (⨅ t, ⌜p t⌝ ⇨ NonDetT.prop (f t) post) ⊓ (⨆ t, ⌜p t⌝)\n\nstructure Extractable (x : NonDetT m α) where\n  cond : Cont l α\n  prop : ∀ post, cond post <= x.prop post\n\ndef ExtractNonDet.prop {α : Type u} (s : NonDetT m α) :  ExtractNonDet WeakFindable s -> l\n  | .pure x => ⊤\n  | .vis x f ex => wlp x fun y => (ex y).prop\n  | .pickSuchThat _ p f ex => ⨅ t ∈ WeakFindable.find p (), (ex t).prop\n  | .assume p f ex =>\n      if p .unit then\n        (ex .unit).prop\n      else ⊤\n\nnamespace DemonicChoice", "target_theorem": "lemma ExtractNonDet.extract_refines_wp (s : NonDetT m α) (inst : ExtractNonDet Findable s) :\n  wp s post ⊓ s.prop ⊤ <= wp s.extract post :=", "ground_truth_proof": ":= by\n  unhygienic induction inst\n  { simp [wp_pure, NonDetT.extract] }\n  { simp [NonDetT.wp_vis, NonDetT.prop]; rw [inf_comm, wlp_join_wp]\n    simp [NonDetT.extract, wp_bind]\n    apply wp_cons; aesop (add norm inf_comm) }\n  { simp [NonDetT.wp_pickCont, NonDetT.prop, NonDetT.extract]; split\n    { have := Findable.find_none (p := p) (by simpa);\n      have : (∀ x, p x = False) := by simpa\n      simp [this] }\n    rw [<-inf_assoc]; refine inf_le_of_left_le ?_\n    rw [← @iInf_inf_eq]; simp [meet_himp _ _ _ _ rfl]\n    rename_i y _\n    refine iInf_le_of_le y ?_\n    have := Findable.find_some_p (p := p) (by assumption)\n    simp [this]; apply a_ih }\n  simp [NonDetT.wp_pickCont, NonDetT.prop, NonDetT.extract]\n  have : ∀ a : PUnit.{u_1 + 1}, a = .unit := by simp\n  simp [this, iInf_const]; split_ifs <;> simp [*, iSup_const]\n  apply a_ih", "nesting_depth": 5, "transitive_dep_count": 77, "subset_aristotle": true, "category": "Framework"}
{"id": 195, "thm_name": "Matrix.isUnit_det_of_PosDef_inv", "thm_stmt": "lemma isUnit_det_of_PosDef_inv [DecidableEq n] {M : Matrix n n ℝ} (h : M⁻¹.PosDef) :\n    IsUnit M.det", "lean_root": "CvxLean", "rel_path": "CvxLean/Lib/Math/LinearAlgebra/Matrix/PosDef.lean", "imports": ["import Mathlib.LinearAlgebra.Matrix.PosDef", "import Mathlib.Algebra.Star.Pi"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Invertible", "module": "Mathlib.Algebra.Group.Invertible.Defs"}, {"name": "Matrix", "module": "Mathlib.LinearAlgebra.Matrix.Defs"}, {"name": "Matrix.invertibleOfIsUnitDet", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}, {"name": "IsUnit", "module": "Mathlib.Algebra.Group.Units.Defs"}, {"name": "v", "module": "examples.BallisticWidget"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Matrix.nondegenerate_iff_det_ne_zero", "module": "Mathlib.LinearAlgebra.Matrix.ToLinearEquiv"}, {"name": "star_eq_zero", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "star_star", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "isUnit_iff_ne_zero", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "IsUnit.ne_zero", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "Matrix.det_nonsing_inv", "module": "Mathlib.LinearAlgebra.Matrix.NonsingularInverse"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Matrix.PosDef", "content": "noncomputable instance PosDef.Invertible [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) :\n    Invertible M :=\n  invertibleOfIsUnitDet M (isUnit_iff_ne_zero.2 hM.det_ne_zero)"}], "used_local_lemmas": [{"name": "Matrix.PosDef.det_ne_zero", "content": "lemma PosDef.det_ne_zero [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : M.det ≠ 0"}, {"name": "Matrix.PosDef.isUnit_det", "content": "lemma PosDef.isUnit_det [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) : IsUnit M.det"}], "local_ctx": "import Mathlib.LinearAlgebra.Matrix.PosDef\n\nimport Mathlib.Algebra.Star.Pi\n\nnamespace Matrix\n\nvariable {m n : Type _} [Fintype m] [Fintype n]\n\nvariable {𝕜 : Type _}\n\nvariable [NormedField 𝕜] [PartialOrder 𝕜] [StarRing 𝕜] [StarOrderedRing 𝕜]\n\nvariable [RCLike 𝕜]\n\nnoncomputable instance PosDef.Invertible [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) :\n    Invertible M :=\n  invertibleOfIsUnitDet M (isUnit_iff_ne_zero.2 hM.det_ne_zero)", "target_theorem": "lemma isUnit_det_of_PosDef_inv [DecidableEq n] {M : Matrix n n ℝ} (h : M⁻¹.PosDef) :\n    IsUnit M.det :=", "ground_truth_proof": ":= by\n  apply isUnit_iff_ne_zero.2\n  have := h.isUnit_det\n  rw [det_nonsing_inv, isUnit_ring_inverse] at this\n  apply IsUnit.ne_zero this", "nesting_depth": 3, "transitive_dep_count": 15, "subset_aristotle": false, "category": "Applied verif."}
{"id": 196, "thm_name": "Lipmaa.soundness", "thm_stmt": "lemma soundness\n    {F : Type} [Field F]\n    {n_stmt n_wit n_var : ℕ}\n    {u_stmt : Fin n_stmt → (Polynomial F) }\n    {u_wit : Fin n_wit → (Polynomial F) }\n    {v_stmt : Fin n_stmt → (Polynomial F) }\n    {v_wit : Fin n_wit → (Polynomial F) }\n    {w_stmt : Fin n_stmt → (Polynomial F) }\n    {w_wit : Fin n_wit → (Polynomial F) }\n    {r : Fin n_wit → F} :\n    (AGMProofSystemInstantiation.soundness\n      F\n      (Lipmaa\n        (F := F) (n_stmt := n_stmt) (n_wit := n_wit) (n_var := n_var)\n        (u_stmt := u_stmt) (u_wit := u_wit) (v_stmt := v_stmt)\n        (v_wit := v_wit) (w_stmt := w_stmt) (w_wit := w_wit) (r := r))\n      (Fin n_wit -> F)\n      (fun (stmt : Fin n_stmt → F) (wit : Fin n_wit -> F) =>\n        let t : Polynomial F :=\n          ∏ i in (Finset.univ : Finset (Fin n_wit)), (Polynomial.X - Polynomial.C (r i));\n        (((List.sum (List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)))\n            + (List.sum (List.map (fun i => Polynomial.C (wit i) * u_wit i) (List.finRange n_wit))))\n            *\n          ((List.sum (List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)))\n            + (List.sum (List.map (fun i => Polynomial.C (wit i) * v_wit i) (List.finRange n_wit))))\n            -\n          ((List.sum (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)))\n            + (List.sum (List.map (fun i => Polynomial.C (wit i) * w_wit i) (List.finRange n_wit)))))\n            %ₘ t = 0\n      )\n      (fun prover i => prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i))\n    )", "lean_root": "formal-snarks-project", "rel_path": "FormalSnarksProject/SNARKs/Lipmaa/Soundness.lean", "imports": ["import FormalSnarksProject.ToMathlib.OptionEquivRight", "import FormalSnarksProject.SNARKs.Lipmaa.Defs"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "MvPolynomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Finsupp.toFun", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "MvPolynomial.C", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.X", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "OracleSpec", "module": "VCVio.OracleComp.OracleSpec"}, {"name": "OracleComp", "module": "VCVio.OracleComp.OracleComp"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "inline", "module": "Init.Core"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "List.sum", "module": "Init.Data.List.Basic"}, {"name": "AlgEquiv", "module": "Mathlib.Algebra.Algebra.Equiv"}, {"name": "EquivLike", "module": "Mathlib.Data.FunLike.Equiv"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.single", "module": "Mathlib.Data.Finsupp.Single"}, {"name": "Function.comp", "module": "Init.Prelude"}, {"name": "MvPolynomial.coeff", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.optionEquivRight", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "Sum", "module": "Init.Core"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.comp", "module": "Mathlib.Algebra.Ring.Hom.Defs"}], "used_repo_defs": [{"name": "syntax \"integral_domain_tactic\" : tactic", "content": "syntax \"integral_domain_tactic\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n| `(tactic| simplify_mvpoly_option_eqn) =>\n  `(tactic|\n    simp only [monomial_zero', List.singleton_append, List.cons_append, List.append_assoc,\n      List.map_cons, Sum.elim_inl, Sum.elim_inr, List.map_append, List.map_map, List.sum_cons,\n      List.sum_append, List.map_nil, List.sum_nil, add_zero, Sum.elim_lam_const_lam_const, map_one,\n      one_mul, map_zero, zero_mul, map_neg, neg_mul, neg_add_rev, zero_add, mul_zero,\n      \n      Function.comp, List.sum_map_zero] at eqn;\n    simp only [mul_add, add_mul, List.sum_map_add] at eqn;\n    \n    simp only [\n      \n      mul_assoc,\n      \n      mul_left_comm (C _) (X (some _)) _, mul_left_comm (List.sum _) (X (some _)) _,\n      mul_comm (C _) (X (some _)), mul_comm (List.sum _) (X (some _)),\n      \n      neg_mul, mul_neg,\n      \n      List.sum_map_mul_right, List.sum_map_mul_left] at eqn;\n\n    \n    trace \"Converting to MvPolynomial over Polynomials\";\n    replace eqn := congr_arg (MvPolynomial.optionEquivRight F Vars) eqn;\n    simp only [AlgEquiv.map_add, AlgEquiv.map_zero, AlgEquiv.map_mul, AlgEquiv.map_one,\n      AlgEquiv.map_neg, AlgEquiv.list_map_sum, AlgEquiv.map_pow] at eqn;\n    simp only [MvPolynomial.optionEquivRight_C, MvPolynomial.optionEquivRight_X_none, MvPolynomial.optionEquivRight_X_some, optionEquivRight_to_MvPolynomial_Option] at eqn;\n\n    \n    simp only [←MvPolynomial.C_mul, ←MvPolynomial.C_pow, ←MvPolynomial.C_add,\n      MvPolynomial.sum_map_C] at eqn;\n\n    simp only [MvPolynomial.X, C_apply, MvPolynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, mul_add, add_mul] at eqn\n  )"}, {"name": "macro_rules", "content": "macro_rules\n| `(tactic| polynomial_ext) =>\n  `(tactic| sorry) "}, {"name": "macro_rules", "content": "macro_rules\n| `(tactic| integral_domain_tactic) =>\n  `(tactic|\n      trace \"Call to integral_domain_tactic\";\n      \n      \n      \n      simp_all (config := {decide := false, failIfUnchanged := false}) only [\n            \n            false_or, or_false, true_or, or_true, not_true, not_false_iff,\n            \n            add_zero, zero_add, mul_zero, zero_mul, mul_one, one_mul, neg_zero,\n            \n            neg_eq_zero, add_eq_zero_iff_eq_neg,\n            eq_self_iff_true, eq_zero_of_zero_eq, one_ne_zero, mul_ne_zero_iff, zero_sub_eq_iff,\n            \n            mul_eq_zero];\n      first\n        \n        | done\n        \n        | cases_or _ ∨ _\n          all_goals integral_domain_tactic\n        \n        | skip\n   )"}, {"name": "Lipmaa", "content": "noncomputable def Lipmaa\n     \n    {F : Type} [Field F]\n     \n    {n_stmt n_wit n_var : ℕ}\n     \n    {u_stmt : Fin n_stmt → (Polynomial F) }\n    {u_wit : Fin n_wit → (Polynomial F) }\n    {v_stmt : Fin n_stmt → (Polynomial F) }\n    {v_wit : Fin n_wit → (Polynomial F) }\n    {w_stmt : Fin n_stmt → (Polynomial F) }\n    {w_wit : Fin n_wit → (Polynomial F) }\n     \n    {r : Fin n_wit → F} :\n    AGMProofSystemInstantiation F :=\n  let t : Polynomial F :=\n    ∏ i in (Finset.univ : Finset (Fin n_wit)), (Polynomial.X - Polynomial.C (r i));\n  {\n    Stmt := Fin n_stmt -> F\n    Sample := Option Vars\n    SRSElements_G1 := @SRS_Elements_G1_Idx n_stmt n_wit n_var\n    ListSRSElements_G1 :=\n      [SRS_Elements_G1_Idx.α]\n      ++ [SRS_Elements_G1_Idx.β]\n      ++ [SRS_Elements_G1_Idx.δ]\n      ++ ((List.finRange n_var).map fun i => SRS_Elements_G1_Idx.x_pow i)\n      ++ ((List.finRange (n_var - 1)).map fun i => SRS_Elements_G1_Idx.x_pow_times_t i)\n      ++ ((List.finRange n_stmt).map fun i => SRS_Elements_G1_Idx.y i)\n      ++ ((List.finRange n_wit).map fun i => SRS_Elements_G1_Idx.q i)\n    SRSElements_G2 := @SRS_Elements_G2_Idx n_stmt n_wit n_var\n    ListSRSElements_G2 :=\n      [SRS_Elements_G2_Idx.β]\n      ++ [SRS_Elements_G2_Idx.γ]\n      ++ [SRS_Elements_G2_Idx.δ]\n      ++ ((List.finRange n_var).map fun i => SRS_Elements_G2_Idx.x_pow i)\n    SRSElementValue_G1 := fun SRS_idx => match SRS_idx with\n      | SRS_Elements_G1_Idx.α => (X Vars_y ^ 5) * (X Vars_y ^ 1) * (X Vars_y ^ 75)\n      | SRS_Elements_G1_Idx.β => (X Vars_y ^ 5) * (X Vars_y ^ 1) * (X Vars_y ^ 25)\n      | SRS_Elements_G1_Idx.δ => (X Vars_y ^ 5) * (X Vars_y ^ 1) * (X Vars_y ^ 1)\n      | SRS_Elements_G1_Idx.x_pow i => (X Vars_y ^ 5) * (X Vars_y ^ 1) * X Vars_x ^ (i : ℕ)\n      | SRS_Elements_G1_Idx.x_pow_times_t i => (X Vars_y ^ 5)\n                                                  * X Vars_x ^ (i : ℕ)\n                                                  * to_MvPolynomial_Option Vars t\n      | SRS_Elements_G1_Idx.y i => (((X Vars_y ^ 25) * (X Vars_y ^ 1)) * ( (to_MvPolynomial_Option Vars (u_stmt i))))\n                                      +\n                                      ((X Vars_y ^ 75) * (X Vars_y ^ 1)) * (to_MvPolynomial_Option Vars (v_stmt i))\n                                      +\n                                      (X Vars_y ^ 1) * (to_MvPolynomial_Option Vars (w_stmt i))\n      | SRS_Elements_G1_Idx.q i => ((X Vars_y ^ 25) * (X Vars_y ^ 5)) * ( to_MvPolynomial_Option Vars (u_wit i))\n                                      +\n                                      ((X Vars_y ^ 75) * (X Vars_y ^ 5)) * (to_MvPolynomial_Option Vars (v_wit i))\n                                      +\n                                      (X Vars_y ^ 5) * to_MvPolynomial_Option Vars (w_wit i)\n      \n    SRSElementValue_G2 := fun SRS_idx => match SRS_idx with\n      | SRS_Elements_G2_Idx.β => (X Vars_y ^ 5) * (X Vars_y ^ 1) * (X Vars_y ^ 25)\n      | SRS_Elements_G2_Idx.γ => (X Vars_y ^ 5) * (X Vars_y ^ 1) * (X Vars_y ^ 5)\n      | SRS_Elements_G2_Idx.δ => (X Vars_y ^ 5) * (X Vars_y ^ 1) * (X Vars_y ^ 1)\n      | SRS_Elements_G2_Idx.x_pow i => (X Vars_y ^ 5) * (X Vars_y ^ 1) * X Vars_x ^ (i : ℕ)\n    Proof_G1 := Proof_G1_Idx\n    ListProof_G1 := [Proof_G1_Idx.A, Proof_G1_Idx.C]\n    Proof_G2 := Proof_G2_Idx\n    ListProof_G2 := [Proof_G2_Idx.B]\n    EqualityChecks := Unit\n    Pairings := fun _ => PairingsIdx\n    ListPairings := fun _ => [PairingsIdx.ab, PairingsIdx.αβ, PairingsIdx.stmtγ, PairingsIdx.cδ]\n    verificationPairingSRS_G1 := fun stmt _ i SRS_idx => match i with\n      | PairingsIdx.ab => match SRS_idx with\n        | SRS_Elements_G1_Idx.α => 0\n        | SRS_Elements_G1_Idx.β => 0\n        | SRS_Elements_G1_Idx.δ => 0\n        | SRS_Elements_G1_Idx.x_pow _ => 0\n        | SRS_Elements_G1_Idx.x_pow_times_t _ => 0\n        | SRS_Elements_G1_Idx.y _ => 0\n        | SRS_Elements_G1_Idx.q _ => 0\n      | PairingsIdx.αβ => match SRS_idx with\n        | SRS_Elements_G1_Idx.α => 1\n        | SRS_Elements_G1_Idx.β => 0\n        | SRS_Elements_G1_Idx.δ => 0\n        | SRS_Elements_G1_Idx.x_pow _ => 0\n        | SRS_Elements_G1_Idx.x_pow_times_t _ => 0\n        | SRS_Elements_G1_Idx.y _ => 0\n        | SRS_Elements_G1_Idx.q _ => 0\n      | PairingsIdx.stmtγ => match SRS_idx with\n        | SRS_Elements_G1_Idx.α => 0\n        | SRS_Elements_G1_Idx.β => 0\n        | SRS_Elements_G1_Idx.δ => 0\n        | SRS_Elements_G1_Idx.x_pow _ => 0\n        | SRS_Elements_G1_Idx.x_pow_times_t _ => 0\n        | SRS_Elements_G1_Idx.y i => stmt i\n        | SRS_Elements_G1_Idx.q _ => 0\n      | PairingsIdx.cδ => match SRS_idx with\n        | SRS_Elements_G1_Idx.α => 0\n        | SRS_Elements_G1_Idx.β => 0\n        | SRS_Elements_G1_Idx.δ => 0\n        | SRS_Elements_G1_Idx.x_pow _ => 0\n        | SRS_Elements_G1_Idx.x_pow_times_t _ => 0\n        | SRS_Elements_G1_Idx.y _ => 0\n        | SRS_Elements_G1_Idx.q i => 0\n    verificationPairingSRS_G2 := fun stmt _ i SRS_idx => match i with\n      | PairingsIdx.ab => match SRS_idx with\n        | SRS_Elements_G2_Idx.β => 0\n        | SRS_Elements_G2_Idx.γ => 0\n        | SRS_Elements_G2_Idx.δ => 0\n        | SRS_Elements_G2_Idx.x_pow _ => 0\n      | PairingsIdx.αβ => match SRS_idx with\n        | SRS_Elements_G2_Idx.β => 1\n        | SRS_Elements_G2_Idx.γ => 0\n        | SRS_Elements_G2_Idx.δ => 0\n        | SRS_Elements_G2_Idx.x_pow _ => 0\n      | PairingsIdx.stmtγ => match SRS_idx with\n        | SRS_Elements_G2_Idx.β => 0\n        | SRS_Elements_G2_Idx.γ => 1\n        | SRS_Elements_G2_Idx.δ => 0\n        | SRS_Elements_G2_Idx.x_pow _ => 0\n      | PairingsIdx.cδ => match SRS_idx with\n        | SRS_Elements_G2_Idx.β => 0\n        | SRS_Elements_G2_Idx.γ => 0\n        | SRS_Elements_G2_Idx.δ => 1\n        | SRS_Elements_G2_Idx.x_pow _ => 0\n    verificationPairingProof_G1 := fun stmt _ i pf => match i with\n      | PairingsIdx.ab => match pf with\n        | Proof_G1_Idx.A => 1\n        | Proof_G1_Idx.C => 0\n      | PairingsIdx.αβ => match pf with\n        | Proof_G1_Idx.A => 0\n        | Proof_G1_Idx.C => 0\n      | PairingsIdx.stmtγ => match pf with\n        | Proof_G1_Idx.A => 0\n        | Proof_G1_Idx.C => 0\n      | PairingsIdx.cδ => match pf with\n        | Proof_G1_Idx.A => 0\n        | Proof_G1_Idx.C => 1\n    verificationPairingProof_G2 := fun stmt _ i pf => match i with\n      | PairingsIdx.ab => match pf with\n        | Proof_G2_Idx.B => -1\n      | PairingsIdx.αβ => match pf with\n        | Proof_G2_Idx.B => 0\n      | PairingsIdx.stmtγ => match pf with\n        | Proof_G2_Idx.B => 0\n      | PairingsIdx.cδ => match pf with\n        | Proof_G2_Idx.B => 0\n  }"}, {"name": "PairingsIdx", "content": "inductive PairingsIdx : Type where\n  | ab : PairingsIdx\n  | αβ : PairingsIdx\n  | stmtγ : PairingsIdx\n  | cδ : PairingsIdx"}, {"name": "SRS_Elements_G2_Idx", "content": "inductive SRS_Elements_G2_Idx {n_stmt n_wit n_var : ℕ} : Type where\n  | β : SRS_Elements_G2_Idx\n  | γ : SRS_Elements_G2_Idx\n  | δ : SRS_Elements_G2_Idx\n  | x_pow : Fin n_var → SRS_Elements_G2_Idx"}, {"name": "SRS_Elements_G1_Idx", "content": "inductive SRS_Elements_G1_Idx {n_stmt n_wit n_var : ℕ} : Type where\n  | α : SRS_Elements_G1_Idx\n  | β : SRS_Elements_G1_Idx\n  | δ : SRS_Elements_G1_Idx\n  | x_pow : Fin n_var → SRS_Elements_G1_Idx\n  | x_pow_times_t : Fin (n_var - 1) → SRS_Elements_G1_Idx\n  | y : Fin n_stmt → SRS_Elements_G1_Idx\n  | q : Fin n_wit → SRS_Elements_G1_Idx"}, {"name": "Proof_G2_Idx", "content": "inductive Proof_G2_Idx : Type where\n  | B : Proof_G2_Idx"}, {"name": "Proof_G1_Idx", "content": "inductive Proof_G1_Idx : Type where\n  | A : Proof_G1_Idx\n  | C : Proof_G1_Idx"}, {"name": "Vars", "content": "inductive Vars : Type where\n  | y : Vars\nderiving Repr, BEq\n\nlocal notation \"Vars_y\" => some Vars.y\nlocal notation \"Vars_x\" => none"}, {"name": "AGMProofSystemInstantiation", "content": "structure AGMProofSystemInstantiation (F : Type) [Field F] where\n  Stmt Sample SRSElements Proof EqualityChecks : Type\n\n  ListSRSElements : List SRSElements\n  SRSElementValue : SRSElements → MvPolynomial Sample F\n  ListProof : List Proof\n  Pairings : EqualityChecks → Type\n  ListPairings : (k : EqualityChecks) → List (Pairings k)\n  verificationPairingSRS_G1 : Stmt -> (k : EqualityChecks) → Pairings k → SRSElements → F\n  verificationPairingSRS_G2 : Stmt -> (k : EqualityChecks) → Pairings k → SRSElements → F\n  verificationPairingProof_G1 : Stmt -> (k : EqualityChecks) → Pairings k → Proof → F\n  verificationPairingProof_G2 : Stmt -> (k : EqualityChecks) → Pairings k → Proof → F"}, {"name": "to_MvPolynomial_Option", "content": "noncomputable def to_MvPolynomial_Option {F : Type} [Field F] (V : Type) :\n    Polynomial F →+* MvPolynomial (Option V) F\n  where\n    toFun p := Polynomial.eval₂ (MvPolynomial.C) (MvPolynomial.X none) p\n    map_one' := by admit /- proof elided -/"}, {"name": "soundness", "content": "def soundness (F : Type) [Field F]\n    (𝓟 : AGMProofSystemInstantiation F)\n    (Wit : Type) (relation : 𝓟.Stmt -> Wit -> Prop)\n    (extractor : 𝓟.Prover -> Wit) : Prop :=\n   ∀ stmt : 𝓟.Stmt,\n    ∀ prover : 𝓟.Prover,\n      𝓟.verify prover stmt -> relation stmt (extractor prover)"}, {"name": "verify", "content": "def verify {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (stmt : 𝓟.Stmt) : Prop :=\n  (\n    ∀ check_idx : 𝓟.EqualityChecks, 𝓟.check_poly prover stmt check_idx = 0\n  )\n  ∧\n  ∀ pfs ∈ 𝓟.Identified_Proof_Elems,\n    𝓟.proof_element_as_poly prover pfs.fst = 𝓟.proof_element_as_poly prover pfs.snd"}, {"name": "check_poly", "content": "noncomputable def check_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (stmt : 𝓟.Stmt) (check_idx : 𝓟.EqualityChecks) :\n    MvPolynomial 𝓟.Sample F :=\n  (\n  (𝓟.ListPairings check_idx).map fun pairing =>\n    𝓟.pairing_poly prover stmt check_idx pairing\n  ).sum"}, {"name": "pairing_poly", "content": "noncomputable def pairing_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (stmt : 𝓟.Stmt) (check_idx : 𝓟.EqualityChecks) (pairing : 𝓟.Pairings check_idx) :\n    MvPolynomial 𝓟.Sample F :=\n  (\n    ( \n      \n      (\n        (𝓟.ListProof.map fun pf_elem => \n          C (𝓟.verificationPairingProof stmt check_idx pairing pf_elem) \n            *\n            \n            𝓟.proof_element_as_poly prover pf_elem).sum\n      )\n      +\n      ( \n        (𝓟.ListSRSElements.map fun SRS_elem =>\n          C (𝓟.verificationPairingSRS stmt check_idx pairing SRS_elem) * (𝓟.SRSElementValue SRS_elem)).sum\n      )\n    )\n    *\n    ( \n      \n      (\n        (𝓟.ListProof.map fun pf_elem => \n          C (𝓟.verificationPairingProof stmt check_idx pairing pf_elem) \n            *\n            \n            𝓟.proof_element_as_poly prover pf_elem).sum\n      )\n      +\n      ( \n        (𝓟.ListSRSElements.map fun SRS_elem =>\n          C (𝓟.verificationPairingSRS stmt check_idx pairing SRS_elem) * (𝓟.SRSElementValue SRS_elem)).sum\n      )\n    )\n  )"}, {"name": "Prover", "content": "def Prover (F : Type) [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) : Type :=\n  (𝓟.Proof -> 𝓟.SRSElements -> F) × (𝓟.Proof -> 𝓟.SRSElements -> F)"}, {"name": "proof_element_as_poly", "content": "noncomputable def proof_element_as_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (pf_elem : 𝓟.Proof) :\n    MvPolynomial (𝓟.Sample) F :=\n  (𝓟.ListSRSElements.map fun SRS_elem =>\n          MvPolynomial.C (prover.fst pf_elem SRS_elem) * (𝓟.SRSElementValue SRS_elem)).sum"}, {"name": "proof_element_G1_as_poly", "content": "noncomputable def proof_element_G1_as_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (pf_elem : 𝓟.Proof_G1) :\n    MvPolynomial (𝓟.Sample) F :=\n  (𝓟.ListSRSElements_G1.map fun SRS_elem =>\n          MvPolynomial.C (prover.fst pf_elem SRS_elem) * (𝓟.SRSElementValue_G1 SRS_elem)).sum"}, {"name": "proof_element_G2_as_poly", "content": "noncomputable def proof_element_G2_as_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (pf_elem : 𝓟.Proof_G2) :\n    MvPolynomial (𝓟.Sample) F :=\n  (𝓟.ListSRSElements_G2.map fun SRS_elem =>\n          MvPolynomial.C (prover.snd pf_elem SRS_elem) * (𝓟.SRSElementValue_G2 SRS_elem)).sum"}], "lib_lemmas": [{"name": "dvd_mul_right", "module": "Mathlib.Algebra.Divisibility.Basic"}, {"name": "EquivLike.apply_eq_iff_eq", "module": "Mathlib.Data.FunLike.Equiv"}, {"name": "Finsupp.add_apply", "module": "Mathlib.Algebra.Group.Finsupp"}, {"name": "Finsupp.single_apply", "module": "Mathlib.Data.Finsupp.Single"}, {"name": "Finsupp.smul_single'", "module": "Mathlib.Data.Finsupp.SMul"}, {"name": "List.append_assoc", "module": "Init.Data.List.Basic"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.map_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_cons", "module": "Init.Data.List.Basic"}, {"name": "List.map_map", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_nil", "module": "Init.Data.List.Basic"}, {"name": "List.singleton_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.sum_append", "module": "Batteries.Data.List.Lemmas"}, {"name": "List.sum_cons", "module": "Init.Data.List.Basic"}, {"name": "List.sum_map_add", "module": "Mathlib.Algebra.BigOperators.Group.List.Basic"}, {"name": "List.sum_map_mul_left", "module": "Mathlib.Algebra.BigOperators.Ring.List"}, {"name": "List.sum_map_mul_right", "module": "Mathlib.Algebra.BigOperators.Ring.List"}, {"name": "List.sum_map_zero", "module": "Mathlib.Algebra.BigOperators.Group.List.Defs"}, {"name": "List.sum_nil", "module": "Init.Data.List.Basic"}, {"name": "MvPolynomial.C_add", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.C_apply", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.C_mul", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.C_pow", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_add", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_monomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_neg", "module": "Mathlib.Algebra.MvPolynomial.CommRing"}, {"name": "MvPolynomial.coeff_zero", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.monomial_mul", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.monomial_pow", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.monomial_zero'", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.optionEquivRight_C", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "MvPolynomial.optionEquivRight_X_none", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "MvPolynomial.optionEquivRight_X_some", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "Polynomial.monic_X_sub_C", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Polynomial.monic_prod_of_monic", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Sum.elim_inl", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.elim_inr", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.elim_lam_const_lam_const", "module": "Init.Data.Sum.Lemmas"}, {"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "map_neg", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_one", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_zero", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_add", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_left_comm", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_neg", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "neg_add_rev", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "neg_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "neg_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_pow", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_eq_iff_eq_add'", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "Vars.finsupp_eq_ext", "content": "lemma Vars.finsupp_eq_ext (f g : Vars →₀ ℕ) : f = g ↔\n    f Vars.y = g Vars.y"}, {"name": "MvPolynomial.sum_map_C", "content": "lemma MvPolynomial.sum_map_C {σ A R : Type} [CommSemiring R] (l : List A) (f : A → R) :\n    (l.map (fun (x : A) => C (σ := σ) (f x))).sum = C ((l.map f).sum)"}, {"name": "AlgEquiv.list_map_sum", "content": "theorem AlgEquiv.list_map_sum {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂}\n    [CommSemiring R] [Semiring A₁] [Semiring A₂] [Algebra R A₁] [Algebra R A₂]\n    (e : A₁ ≃ₐ[R] A₂) {ι : Type u_1} (f : ι → A₁) (l : List ι) :\n    e (l.map (fun (x : ι) => f x)).sum = (l.map fun (x : ι) => e (f x)).sum"}, {"name": "optionEquivRight_to_MvPolynomial_Option", "content": "lemma optionEquivRight_to_MvPolynomial_Option {F V : Type} [Field F] (p : Polynomial F) :\n    (MvPolynomial.optionEquivRight F V) (to_MvPolynomial_Option V p) = C p"}, {"name": "optionEquivRight_comp_to_MvPolynomial_Option", "content": "lemma optionEquivRight_comp_to_MvPolynomial_Option {F V : Type} [Field F] :\n    RingHom.comp (MvPolynomial.optionEquivRight F V).toRingEquiv.toRingHom (to_MvPolynomial_Option (F := F) V) = C"}, {"name": "to_MvPolynomial_Option_C", "content": "lemma to_MvPolynomial_Option_C {F V : Type} [Field F] (r : F) :\n    to_MvPolynomial_Option V (Polynomial.C r) = MvPolynomial.C r"}, {"name": "Polynomial.hom_congr_vars", "content": "theorem Polynomial.hom_congr_vars {R : Type u} {S : Type v}\n    [CommSemiring R] [CommSemiring S]\n    {f₁ : Polynomial R →+* S} {f₂ : Polynomial R →+* S}\n    (hC : RingHom.comp f₁ Polynomial.C = RingHom.comp f₂ Polynomial.C)\n    (hv : f₁ (Polynomial.X) = f₂ (Polynomial.X)) :\n    f₁ = f₂"}, {"name": "to_MvPolynomial_Option_X", "content": "lemma to_MvPolynomial_Option_X {F V : Type} [Field F] :\n    to_MvPolynomial_Option V (Polynomial.X) = MvPolynomial.X (R := F) none"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Lipmaa.Polynomial.mul_self_modByMonic", "content": "lemma Polynomial.mul_self_modByMonic {F : Type} [Field F] (t p : Polynomial F) (mt : t.Monic) : (t * p) %ₘ t = 0"}], "local_ctx": "import FormalSnarksProject.SNARKs.Lipmaa.Defs\n\nopen scoped BigOperators Classical\n\nsection Lipmaa\n\nopen MvPolynomial Option AGMProofSystemInstantiation\n\nnamespace Lipmaa\n\nsection soundness", "target_theorem": "lemma soundness\n    {F : Type} [Field F]\n    {n_stmt n_wit n_var : ℕ}\n    {u_stmt : Fin n_stmt → (Polynomial F) }\n    {u_wit : Fin n_wit → (Polynomial F) }\n    {v_stmt : Fin n_stmt → (Polynomial F) }\n    {v_wit : Fin n_wit → (Polynomial F) }\n    {w_stmt : Fin n_stmt → (Polynomial F) }\n    {w_wit : Fin n_wit → (Polynomial F) }\n    {r : Fin n_wit → F} :\n    (AGMProofSystemInstantiation.soundness\n      F\n      (Lipmaa\n        (F := F) (n_stmt := n_stmt) (n_wit := n_wit) (n_var := n_var)\n        (u_stmt := u_stmt) (u_wit := u_wit) (v_stmt := v_stmt)\n        (v_wit := v_wit) (w_stmt := w_stmt) (w_wit := w_wit) (r := r))\n      (Fin n_wit -> F)\n      (fun (stmt : Fin n_stmt → F) (wit : Fin n_wit -> F) =>\n        let t : Polynomial F :=\n          ∏ i in (Finset.univ : Finset (Fin n_wit)), (Polynomial.X - Polynomial.C (r i));\n        (((List.sum (List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)))\n            + (List.sum (List.map (fun i => Polynomial.C (wit i) * u_wit i) (List.finRange n_wit))))\n            *\n          ((List.sum (List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)))\n            + (List.sum (List.map (fun i => Polynomial.C (wit i) * v_wit i) (List.finRange n_wit))))\n            -\n          ((List.sum (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)))\n            + (List.sum (List.map (fun i => Polynomial.C (wit i) * w_wit i) (List.finRange n_wit)))))\n            %ₘ t = 0\n      )\n      (fun prover i => prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i))\n    ) :=", "ground_truth_proof": ":= by\n\n  unfold soundness verify check_poly pairing_poly proof_element_G1_as_poly proof_element_G2_as_poly\n\n  -- TODO namespcace AGMProofSystemInstantiation eliminate\n  intros stmt prover eqns'\n  rcases eqns' with ⟨eqns, null⟩\n  intro t\n  have eqn := eqns ()\n  clear eqns null\n\n  -- let C_m := fun i => prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)\n  -- let C_h := fun x => prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)\n\n  suffices\n      ((List.sum (List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)))\n      + (List.sum (List.map (fun i => Polynomial.C (prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * u_wit i) (List.finRange n_wit))))\n      *\n      ((List.sum (List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)))\n      + (List.sum (List.map (fun i => Polynomial.C (prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * v_wit i) (List.finRange n_wit))))\n      =\n      ((List.sum (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)))\n      + (List.sum (List.map (fun i => Polynomial.C (prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * w_wit i) (List.finRange n_wit))))\n      +\n      List.sum (List.map (fun x : Fin (n_var - 1) => Polynomial.C (prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * (Polynomial.X ^ (x : ℕ) * t)) (List.finRange (n_var - 1))) by\n\n    rw [<-sub_eq_iff_eq_add'] at this\n    have h := congr_arg (fun x => x %ₘ t) this\n    simp only at h\n    simp\n    rw [h]\n    clear this h\n\n    simp only [mul_comm _ (t), <-mul_assoc]\n    simp only [mul_assoc, List.sum_map_mul_right, List.sum_map_mul_left]\n\n    apply Polynomial.mul_self_modByMonic\n    apply Polynomial.monic_prod_of_monic\n    intro i hi\n    exact Polynomial.monic_X_sub_C (r i)\n    done\n\n\n\n  -- Step 1: Obtain the coefficient equations of the MvPolynomials\n  simp_rw [Lipmaa] at eqn\n  -- All I want is a tactic that will apply the following simplifications to eqn in sequence.\n  -- TODO can I write a tactic taking a nested list of simp lemmas?\n  -- Can I combine all of these?\n  simp only [monomial_zero', List.singleton_append, List.cons_append, List.append_assoc,\n    List.map_cons, Sum.elim_inl, Sum.elim_inr, List.map_append, List.map_map, List.sum_cons,\n    List.sum_append, List.map_nil, List.sum_nil, add_zero, Sum.elim_lam_const_lam_const, map_one,\n    one_mul, map_zero, zero_mul, map_neg, neg_mul, neg_add_rev, zero_add, mul_zero,\n    -- Note: everything above is @simp tagged\n    Function.comp, List.sum_map_zero] at eqn\n\n  simp only [mul_add, add_mul, List.sum_map_add] at eqn\n\n  -- Move all the X (some _) terms to the left, and out of sums\n  simp only [\n    -- Associativity to obtain a right-leaning tree\n    mul_assoc,\n    -- Commutativity lemmas to move X (some _) to the left\n    mul_left_comm (C _) (X (some _)) _, mul_left_comm (List.sum _) (X (some _)) _,\n    mul_comm (C _) (X (some _)), mul_comm (List.sum _) (X (some _)),\n    -- Commutativity lemmas to move X (some _) ^ _ to the left\n    mul_left_comm (C _) (X (some _) ^ _) _, mul_left_comm (List.sum _) (X (some _) ^ _) _,\n    mul_comm (C _) (X (some _) ^ _), mul_comm (List.sum _) (X (some _) ^ _),\n    -- Move negations to the bottom\n    neg_mul, mul_neg,\n    -- Move constant multiplications (which the X (some _) terms should be) out of sums\n    List.sum_map_mul_right, List.sum_map_mul_left] at eqn\n\n  -- Apply MvPolynomial.optionEquivRight *here*, so that we can treat polynomials in Vars_X as constants\n  trace \"Converting to MvPolynomial over Polynomials\"\n  -- replace eqn := congr_arg (MvPolynomial.optionEquivRight F Vars) eqn\n  simp only [←(EquivLike.apply_eq_iff_eq (optionEquivRight _ _))] at eqn\n  simp only [AlgEquiv.map_add, AlgEquiv.map_zero, AlgEquiv.map_mul, AlgEquiv.map_one,\n    AlgEquiv.map_neg, AlgEquiv.list_map_sum, AlgEquiv.map_pow] at eqn\n  simp only [optionEquivRight_C, optionEquivRight_X_none, optionEquivRight_X_some, optionEquivRight_to_MvPolynomial_Option] at eqn\n\n  -- Move Cs back out so we can recognize the monomials\n  simp only [←C_mul, ←C_pow, ←C_add,\n    sum_map_C] at eqn\n\n  simp only [X, C_apply, monomial_mul, monomial_pow, one_mul, mul_one, add_zero, zero_add, mul_add, add_mul] at eqn\n\n  trace \"Applying individual coefficients\"\n\n  have h0012 := congr_arg (coeff (Finsupp.single Vars.y (75 * 0 + 25 * 0 + 5 * 1 + 1 * 2))) eqn\n  have h0021 := congr_arg (coeff (Finsupp.single Vars.y (75 * 0 + 25 * 0 + 5 * 2 + 1 * 1))) eqn\n  have h0022 := congr_arg (coeff (Finsupp.single Vars.y (75 * 0 + 25 * 0 + 5 * 2 + 1 * 2))) eqn\n  have h0112 := congr_arg (coeff (Finsupp.single Vars.y (75 * 0 + 25 * 1 + 5 * 1 + 1 * 2))) eqn\n  have h0121 := congr_arg (coeff (Finsupp.single Vars.y (75 * 0 + 25 * 1 + 5 * 2 + 1 * 1))) eqn\n  have h0122 := congr_arg (coeff (Finsupp.single Vars.y (75 * 0 + 25 * 1 + 5 * 2 + 1 * 2))) eqn\n  have h0212 := congr_arg (coeff (Finsupp.single Vars.y (75 * 0 + 25 * 2 + 5 * 1 + 1 * 2))) eqn\n  have h0221 := congr_arg (coeff (Finsupp.single Vars.y (75 * 0 + 25 * 2 + 5 * 2 + 1 * 1))) eqn\n  have h0222 := congr_arg (coeff (Finsupp.single Vars.y (75 * 0 + 25 * 2 + 5 * 2 + 1 * 2))) eqn\n  have h1022 := congr_arg (coeff (Finsupp.single Vars.y (75 * 1 + 25 * 0 + 5 * 2 + 1 * 2))) eqn\n  have h1112 := congr_arg (coeff (Finsupp.single Vars.y (75 * 1 + 25 * 1 + 5 * 1 + 1 * 2))) eqn\n  have h1121 := congr_arg (coeff (Finsupp.single Vars.y (75 * 1 + 25 * 1 + 5 * 2 + 1 * 1))) eqn\n  have h1122 := congr_arg (coeff (Finsupp.single Vars.y (75 * 1 + 25 * 1 + 5 * 2 + 1 * 2))) eqn\n\n  -- have h1023 := congr_arg (coeff (Finsupp.single Vars.y (75 * 1 + 25 * 0 + 5 * 2 + 1 * 3))) eqn\n\n\n  clear eqn\n\n  trace \"Distribute coefficient-taking over terms\"\n  simp only [coeff_monomial, coeff_add, coeff_neg, coeff_zero] at h0012 h0021 h0022 h0112 h0121 h0122 h0212 h0221 h0222 h1022 h1112 h1121 h1122\n\n  -- done\n\n  trace \"Simplifying coefficient expressions\"\n  simp only [Vars.finsupp_eq_ext, Finsupp.smul_single', Finsupp.single_apply, Finsupp.add_apply] at h0012 h0021 h0022 h0112 h0121 h0122 h0212 h0221 h0222 h1022 h1112 h1121 h1122\n\n  trace \"Determine which coefficients are nonzero\"\n  simp (config := {decide := true}) only [ite_false, ite_true] at h0012 h0021 h0022 h0112 h0121 h0122 h0212 h0221 h0222 h1022 h1112 h1121 h1122\n  trace \"Remove zeros\"\n  simp only [neg_zero, add_zero, zero_add, one_pow, mul_one, one_mul] at h0012 h0021 h0022 h0112 h0121 h0122 h0212 h0221 h0222 h1022 h1112 h1121 h1122\n\n\n  -- Step 2: Recursively simplify and case-analyze the equations\n  -- dsimp only\n\n\n  -- Set statements so that the equations are easier to read\n  -- /-\n  set sum_u_stmt := (List.sum (List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)))\n  set sum_v_stmt := (List.sum (List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)))\n  set sum_w_stmt := (List.sum (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)))\n\n  set A_1 := Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)\n  set A_2 := Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β)\n  set A_3 := Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ)\n  set sum_A_x := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) * Polynomial.X ^ (x : ℕ))\n          (List.finRange n_var))\n\n  set sum_A_u_stmt := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *\n              u_stmt x)\n          (List.finRange n_stmt))\n  set sum_A_v_stmt := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *\n              v_stmt x)\n          (List.finRange n_stmt))\n  set sum_A_w_stmt := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *\n              w_stmt x)\n          (List.finRange n_stmt))\n  set sum_A_u_wit := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *\n              u_wit x)\n          (List.finRange n_wit))\n  set sum_A_v_wit := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *\n              v_wit x)\n          (List.finRange n_wit))\n  set sum_A_w_wit := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *\n              w_wit x)\n          (List.finRange n_wit))\n  set sum_A_x_t := (List.sum\n                  (List.map\n                    (fun x =>\n                      Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *\n                        (Polynomial.X ^ (x : ℕ) * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))\n                    (List.finRange (n_var - 1))))\n  set B_1 := Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.β))\n  set B_2 := Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.γ))\n  set B_3 := Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.δ))\n\n  set sum_B_x := List.sum\n                    (List.map\n                      (fun x =>\n                        Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ (x : ℕ))\n                      (List.finRange n_var))\n\n  set C_1 := Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α)\n  set C_2 := Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β)\n  set C_3 := Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ)\n  set sum_C_u_wit := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *\n              u_wit x)\n          (List.finRange n_wit))\n  set sum_C_v_wit := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *\n              v_wit x)\n          (List.finRange n_wit))\n  set sum_C_w_wit := List.sum\n        (List.map\n          (fun x =>\n            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *\n              w_wit x)\n          (List.finRange n_wit))\n  set sum_C_x_t := List.sum\n        (List.map\n          (fun x : Fin (n_var - 1) =>\n            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * (Polynomial.X ^ (x : ℕ) * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))\n          (List.finRange (n_var - 1)))\n\n  -- clear_value sum_A_x sum_A_x_t sum_B_x sum_C_x_t\n  clear_value sum_u_stmt sum_v_stmt sum_w_stmt A_1 A_2 A_3 sum_A_x sum_A_u_stmt sum_A_v_stmt sum_A_w_stmt sum_A_u_wit sum_A_v_wit sum_A_w_wit sum_A_x_t B_1 B_2 B_3 sum_B_x C_1 C_2 C_3 sum_C_u_wit sum_C_v_wit sum_C_w_wit sum_C_x_t\n  -- -/\n  -- done\n\n  integral_domain_tactic\n\n  save\n\n  skip\n  polyrith\n\n  polyrith", "nesting_depth": 6, "transitive_dep_count": 124, "subset_aristotle": false, "category": "Applied verif."}
{"id": 197, "thm_name": "ToySnark.soundness", "thm_stmt": "lemma soundness\n    {F : Type} [Field F] :\n    (AGMProofSystemInstantiation.soundness\n      F\n      (ToySnark\n        (F := F))\n      (WitEntries -> F)\n      (fun (stmt : StmtEntries → F) (wit : WitEntries -> F) =>\n        wit WitEntries.A * stmt StmtEntries.y = stmt StmtEntries.z -- - wit WitEntries.I\n        ∨\n        wit WitEntries.B * stmt StmtEntries.x = stmt StmtEntries.z -- - wit WitEntries.I\n      )\n      (fun prover i => prover.fst Proof_G1_Idx.Pf (if i = WitEntries.A then .α else .β))\n\n    )", "lean_root": "formal-snarks-project", "rel_path": "FormalSnarksProject/SNARKs/ToySnark.lean", "imports": ["import Mathlib.Algebra.Polynomial.Div", "import FormalSnarksProject.ToMathlib.OptionEquivRight", "import Mathlib.Algebra.MvPolynomial.Equiv", "import FormalSnarksProject.Models.AGMProofSystemInstantiation", "import FormalSnarksProject.SoundnessTactic.SoundnessProver"], "used_lib_defs": [{"name": "Empty", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "MvPolynomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.X", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "MvPolynomial.C", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "OracleSpec", "module": "VCVio.OracleComp.OracleSpec"}, {"name": "OracleComp", "module": "VCVio.OracleComp.OracleComp"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "inline", "module": "Init.Core"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "AlgEquiv", "module": "Mathlib.Algebra.Algebra.Equiv"}, {"name": "Finsupp", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Finsupp.single", "module": "Mathlib.Data.Finsupp.Single"}, {"name": "Function.comp", "module": "Init.Prelude"}, {"name": "List.replace", "module": "Init.Data.List.Basic"}, {"name": "List.sum", "module": "Init.Data.List.Basic"}, {"name": "MvPolynomial.coeff", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.optionEquivRight", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Sum", "module": "Init.Core"}, {"name": "DFunLike", "module": "Mathlib.Data.FunLike.Basic"}, {"name": "CommSemiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Algebra", "module": "Mathlib.Algebra.Algebra.Defs"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Finsupp.toFun", "module": "Mathlib.Data.Finsupp.Defs"}, {"name": "Polynomial.eval₂", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "RingHom", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "RingHom.comp", "module": "Mathlib.Algebra.Ring.Hom.Defs"}, {"name": "Polynomial.C", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}], "used_repo_defs": [{"name": "syntax \"integral_domain_tactic\" : tactic", "content": "syntax \"integral_domain_tactic\" : tactic"}, {"name": "macro_rules", "content": "macro_rules\n| `(tactic| simplify_mvpoly_option_eqn) =>\n  `(tactic|\n    simp only [monomial_zero', List.singleton_append, List.cons_append, List.append_assoc,\n      List.map_cons, Sum.elim_inl, Sum.elim_inr, List.map_append, List.map_map, List.sum_cons,\n      List.sum_append, List.map_nil, List.sum_nil, add_zero, Sum.elim_lam_const_lam_const, map_one,\n      one_mul, map_zero, zero_mul, map_neg, neg_mul, neg_add_rev, zero_add, mul_zero,\n      \n      Function.comp, List.sum_map_zero] at eqn;\n    simp only [mul_add, add_mul, List.sum_map_add] at eqn;\n    \n    simp only [\n      \n      mul_assoc,\n      \n      mul_left_comm (C _) (X (some _)) _, mul_left_comm (List.sum _) (X (some _)) _,\n      mul_comm (C _) (X (some _)), mul_comm (List.sum _) (X (some _)),\n      \n      neg_mul, mul_neg,\n      \n      List.sum_map_mul_right, List.sum_map_mul_left] at eqn;\n\n    \n    trace \"Converting to MvPolynomial over Polynomials\";\n    replace eqn := congr_arg (MvPolynomial.optionEquivRight F Vars) eqn;\n    simp only [AlgEquiv.map_add, AlgEquiv.map_zero, AlgEquiv.map_mul, AlgEquiv.map_one,\n      AlgEquiv.map_neg, AlgEquiv.list_map_sum, AlgEquiv.map_pow] at eqn;\n    simp only [MvPolynomial.optionEquivRight_C, MvPolynomial.optionEquivRight_X_none, MvPolynomial.optionEquivRight_X_some, optionEquivRight_to_MvPolynomial_Option] at eqn;\n\n    \n    simp only [←MvPolynomial.C_mul, ←MvPolynomial.C_pow, ←MvPolynomial.C_add,\n      MvPolynomial.sum_map_C] at eqn;\n\n    simp only [MvPolynomial.X, C_apply, MvPolynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, mul_add, add_mul] at eqn\n  )"}, {"name": "macro_rules", "content": "macro_rules\n| `(tactic| polynomial_ext) =>\n  `(tactic| sorry) "}, {"name": "macro_rules", "content": "macro_rules\n| `(tactic| integral_domain_tactic) =>\n  `(tactic|\n      trace \"Call to integral_domain_tactic\";\n      \n      \n      \n      simp_all (config := {decide := false, failIfUnchanged := false}) only [\n            \n            false_or, or_false, true_or, or_true, not_true, not_false_iff,\n            \n            add_zero, zero_add, mul_zero, zero_mul, mul_one, one_mul, neg_zero,\n            \n            neg_eq_zero, add_eq_zero_iff_eq_neg,\n            eq_self_iff_true, eq_zero_of_zero_eq, one_ne_zero, mul_ne_zero_iff, zero_sub_eq_iff,\n            \n            mul_eq_zero];\n      first\n        \n        | done\n        \n        | cases_or _ ∨ _\n          all_goals integral_domain_tactic\n        \n        | skip\n   )"}, {"name": "Vars", "content": "inductive Vars : Type where\n  | α : Vars\n  | β_v : Vars\n  | β_w : Vars\n  | β_y : Vars\n  | γ : Vars\nderiving Repr, BEq\n\nlocal notation \"poly_α\" => X (some Vars.α)\nlocal notation \"poly_β_v\" => X (some Vars.β_v)\nlocal notation \"poly_β_w\" => X (some Vars.β_w)\nlocal notation \"poly_β_y\" => X (some Vars.β_y)\nlocal notation \"poly_γ\" => X (some Vars.γ)\nlocal notation \"poly_s\" => X (none)"}, {"name": "Vars", "content": "inductive Vars : Type where\n  | r_v : Vars\n  | r_w : Vars\n  | α_v : Vars\n  | α_w : Vars\n  | α_y : Vars\n  | β : Vars\n  | γ : Vars\nderiving Repr, BEq\n\nlocal notation \"poly_r_v\" => X (some Vars.r_v)\nlocal notation \"poly_r_w\" => X (some Vars.r_w)\nlocal notation \"poly_α_v\" => X (some Vars.α_v)\nlocal notation \"poly_α_w\" => X (some Vars.α_w)\nlocal notation \"poly_α_y\" => X (some Vars.α_y)\nlocal notation \"poly_β\" => X (some Vars.β)\nlocal notation \"poly_γ\" => X (some Vars.γ)\nlocal notation \"poly_s\" => X (none)"}, {"name": "SRS_Elements_G1_Idx", "content": "inductive SRS_Elements_G1_Idx {n_stmt n_wit n_var : ℕ} : Type where\n  | α : SRS_Elements_G1_Idx\n  | β : SRS_Elements_G1_Idx\n  | δ : SRS_Elements_G1_Idx\n  | x_pow : Fin n_var → SRS_Elements_G1_Idx\n  | x_pow_times_t : Fin (n_var - 1) → SRS_Elements_G1_Idx\n  | y : Fin n_stmt → SRS_Elements_G1_Idx\n  | q : Fin n_wit → SRS_Elements_G1_Idx"}, {"name": "SRS_Elements_G2_Idx", "content": "inductive SRS_Elements_G2_Idx {n_stmt n_wit n_var : ℕ} : Type where\n  | β : SRS_Elements_G2_Idx\n  | γ : SRS_Elements_G2_Idx\n  | δ : SRS_Elements_G2_Idx\n  | x_pow : Fin n_var → SRS_Elements_G2_Idx"}, {"name": "AGMProofSystemInstantiation", "content": "structure AGMProofSystemInstantiation (F : Type) [Field F] where\n  Stmt Sample SRSElements Proof EqualityChecks : Type\n\n  ListSRSElements : List SRSElements\n  SRSElementValue : SRSElements → MvPolynomial Sample F\n  ListProof : List Proof\n  Pairings : EqualityChecks → Type\n  ListPairings : (k : EqualityChecks) → List (Pairings k)\n  verificationPairingSRS_G1 : Stmt -> (k : EqualityChecks) → Pairings k → SRSElements → F\n  verificationPairingSRS_G2 : Stmt -> (k : EqualityChecks) → Pairings k → SRSElements → F\n  verificationPairingProof_G1 : Stmt -> (k : EqualityChecks) → Pairings k → Proof → F\n  verificationPairingProof_G2 : Stmt -> (k : EqualityChecks) → Pairings k → Proof → F"}, {"name": "proof_element_G1_as_poly", "content": "noncomputable def proof_element_G1_as_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (pf_elem : 𝓟.Proof_G1) :\n    MvPolynomial (𝓟.Sample) F :=\n  (𝓟.ListSRSElements_G1.map fun SRS_elem =>\n          MvPolynomial.C (prover.fst pf_elem SRS_elem) * (𝓟.SRSElementValue_G1 SRS_elem)).sum"}, {"name": "Prover", "content": "def Prover (F : Type) [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) : Type :=\n  (𝓟.Proof -> 𝓟.SRSElements -> F) × (𝓟.Proof -> 𝓟.SRSElements -> F)"}, {"name": "proof_element_G2_as_poly", "content": "noncomputable def proof_element_G2_as_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (pf_elem : 𝓟.Proof_G2) :\n    MvPolynomial (𝓟.Sample) F :=\n  (𝓟.ListSRSElements_G2.map fun SRS_elem =>\n          MvPolynomial.C (prover.snd pf_elem SRS_elem) * (𝓟.SRSElementValue_G2 SRS_elem)).sum"}, {"name": "soundness", "content": "def soundness (F : Type) [Field F]\n    (𝓟 : AGMProofSystemInstantiation F)\n    (Wit : Type) (relation : 𝓟.Stmt -> Wit -> Prop)\n    (extractor : 𝓟.Prover -> Wit) : Prop :=\n   ∀ stmt : 𝓟.Stmt,\n    ∀ prover : 𝓟.Prover,\n      𝓟.verify prover stmt -> relation stmt (extractor prover)"}, {"name": "verify", "content": "def verify {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (stmt : 𝓟.Stmt) : Prop :=\n  (\n    ∀ check_idx : 𝓟.EqualityChecks, 𝓟.check_poly prover stmt check_idx = 0\n  )\n  ∧\n  ∀ pfs ∈ 𝓟.Identified_Proof_Elems,\n    𝓟.proof_element_as_poly prover pfs.fst = 𝓟.proof_element_as_poly prover pfs.snd"}, {"name": "check_poly", "content": "noncomputable def check_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (stmt : 𝓟.Stmt) (check_idx : 𝓟.EqualityChecks) :\n    MvPolynomial 𝓟.Sample F :=\n  (\n  (𝓟.ListPairings check_idx).map fun pairing =>\n    𝓟.pairing_poly prover stmt check_idx pairing\n  ).sum"}, {"name": "pairing_poly", "content": "noncomputable def pairing_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (stmt : 𝓟.Stmt) (check_idx : 𝓟.EqualityChecks) (pairing : 𝓟.Pairings check_idx) :\n    MvPolynomial 𝓟.Sample F :=\n  (\n    ( \n      \n      (\n        (𝓟.ListProof.map fun pf_elem => \n          C (𝓟.verificationPairingProof stmt check_idx pairing pf_elem) \n            *\n            \n            𝓟.proof_element_as_poly prover pf_elem).sum\n      )\n      +\n      ( \n        (𝓟.ListSRSElements.map fun SRS_elem =>\n          C (𝓟.verificationPairingSRS stmt check_idx pairing SRS_elem) * (𝓟.SRSElementValue SRS_elem)).sum\n      )\n    )\n    *\n    ( \n      \n      (\n        (𝓟.ListProof.map fun pf_elem => \n          C (𝓟.verificationPairingProof stmt check_idx pairing pf_elem) \n            *\n            \n            𝓟.proof_element_as_poly prover pf_elem).sum\n      )\n      +\n      ( \n        (𝓟.ListSRSElements.map fun SRS_elem =>\n          C (𝓟.verificationPairingSRS stmt check_idx pairing SRS_elem) * (𝓟.SRSElementValue SRS_elem)).sum\n      )\n    )\n  )"}, {"name": "proof_element_as_poly", "content": "noncomputable def proof_element_as_poly {F : Type} [Field F]\n    (𝓟 : AGMProofSystemInstantiation F) (prover : 𝓟.Prover) (pf_elem : 𝓟.Proof) :\n    MvPolynomial (𝓟.Sample) F :=\n  (𝓟.ListSRSElements.map fun SRS_elem =>\n          MvPolynomial.C (prover.fst pf_elem SRS_elem) * (𝓟.SRSElementValue SRS_elem)).sum"}, {"name": "to_MvPolynomial_Option", "content": "noncomputable def to_MvPolynomial_Option {F : Type} [Field F] (V : Type) :\n    Polynomial F →+* MvPolynomial (Option V) F\n  where\n    toFun p := Polynomial.eval₂ (MvPolynomial.C) (MvPolynomial.X none) p\n    map_one' := by admit /- proof elided -/"}], "lib_lemmas": [{"name": "DFunLike.ext_iff", "module": "Mathlib.Data.FunLike.Basic"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "Finsupp.add_apply", "module": "Mathlib.Algebra.Group.Finsupp"}, {"name": "Finsupp.single_apply", "module": "Mathlib.Data.Finsupp.Single"}, {"name": "List.append_assoc", "module": "Init.Data.List.Basic"}, {"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.map_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_cons", "module": "Init.Data.List.Basic"}, {"name": "List.map_map", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_nil", "module": "Init.Data.List.Basic"}, {"name": "List.singleton_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.sum_append", "module": "Batteries.Data.List.Lemmas"}, {"name": "List.sum_cons", "module": "Init.Data.List.Basic"}, {"name": "List.sum_map_add", "module": "Mathlib.Algebra.BigOperators.Group.List.Basic"}, {"name": "List.sum_map_mul_left", "module": "Mathlib.Algebra.BigOperators.Ring.List"}, {"name": "List.sum_map_mul_right", "module": "Mathlib.Algebra.BigOperators.Ring.List"}, {"name": "List.sum_map_zero", "module": "Mathlib.Algebra.BigOperators.Group.List.Defs"}, {"name": "List.sum_nil", "module": "Init.Data.List.Basic"}, {"name": "MvPolynomial.C_apply", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_add", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_monomial", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.coeff_neg", "module": "Mathlib.Algebra.MvPolynomial.CommRing"}, {"name": "MvPolynomial.coeff_zero", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.monomial_mul", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.monomial_zero'", "module": "Mathlib.Algebra.MvPolynomial.Basic"}, {"name": "MvPolynomial.optionEquivRight_C", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "MvPolynomial.optionEquivRight_X_none", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "MvPolynomial.optionEquivRight_X_some", "module": "Mathlib.Algebra.MvPolynomial.Equiv"}, {"name": "Polynomial.C_add", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_inj", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.C_mul", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Sum.elim_inl", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.elim_inr", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.elim_lam_const_lam_const", "module": "Init.Data.Sum.Lemmas"}, {"name": "add_eq_zero", "module": "Mathlib.Algebra.Group.Units.Basic"}, {"name": "add_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "add_neg_eq_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_self", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "map_neg", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_one", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "map_zero", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "mul_add", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "mul_eq_zero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "mul_left_comm", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_neg", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "neg_add_rev", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "neg_mul", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "neg_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "one_ne_zero", "module": "Mathlib.Algebra.NeZero"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_ne_one", "module": "Mathlib.Algebra.NeZero"}], "repo_lemmas": [{"name": "MvPolynomial.sum_map_C", "content": "lemma MvPolynomial.sum_map_C {σ A R : Type} [CommSemiring R] (l : List A) (f : A → R) :\n    (l.map (fun (x : A) => C (σ := σ) (f x))).sum = C ((l.map f).sum)"}, {"name": "AlgEquiv.list_map_sum", "content": "theorem AlgEquiv.list_map_sum {R : Type uR} {A₁ : Type uA₁} {A₂ : Type uA₂}\n    [CommSemiring R] [Semiring A₁] [Semiring A₂] [Algebra R A₁] [Algebra R A₂]\n    (e : A₁ ≃ₐ[R] A₂) {ι : Type u_1} (f : ι → A₁) (l : List ι) :\n    e (l.map (fun (x : ι) => f x)).sum = (l.map fun (x : ι) => e (f x)).sum"}, {"name": "optionEquivRight_to_MvPolynomial_Option", "content": "lemma optionEquivRight_to_MvPolynomial_Option {F V : Type} [Field F] (p : Polynomial F) :\n    (MvPolynomial.optionEquivRight F V) (to_MvPolynomial_Option V p) = C p"}, {"name": "optionEquivRight_comp_to_MvPolynomial_Option", "content": "lemma optionEquivRight_comp_to_MvPolynomial_Option {F V : Type} [Field F] :\n    RingHom.comp (MvPolynomial.optionEquivRight F V).toRingEquiv.toRingHom (to_MvPolynomial_Option (F := F) V) = C"}, {"name": "to_MvPolynomial_Option_C", "content": "lemma to_MvPolynomial_Option_C {F V : Type} [Field F] (r : F) :\n    to_MvPolynomial_Option V (Polynomial.C r) = MvPolynomial.C r"}, {"name": "Polynomial.hom_congr_vars", "content": "theorem Polynomial.hom_congr_vars {R : Type u} {S : Type v}\n    [CommSemiring R] [CommSemiring S]\n    {f₁ : Polynomial R →+* S} {f₂ : Polynomial R →+* S}\n    (hC : RingHom.comp f₁ Polynomial.C = RingHom.comp f₂ Polynomial.C)\n    (hv : f₁ (Polynomial.X) = f₂ (Polynomial.X)) :\n    f₁ = f₂"}, {"name": "to_MvPolynomial_Option_X", "content": "lemma to_MvPolynomial_Option_X {F V : Type} [Field F] :\n    to_MvPolynomial_Option V (Polynomial.X) = MvPolynomial.X (R := F) none"}], "used_local_defs": [{"name": "ToySnark.Vars", "content": "inductive Vars : Type where\n  | α : Vars\n  | β : Vars\nderiving Repr, BEq"}, {"name": "ToySnark.StmtEntries", "content": "inductive StmtEntries : Type where\n  | x : StmtEntries\n  | y : StmtEntries\n  | z : StmtEntries\nderiving Repr, BEq"}, {"name": "ToySnark.WitEntries", "content": "inductive WitEntries : Type where\n  | A : WitEntries\n  | B : WitEntries\nderiving Repr, BEq\n\nlocal notation \"Vars_α\" => some Vars.α\nlocal notation \"Vars_β\" => some Vars.β\nlocal notation \"Vars_x\" => none"}, {"name": "ToySnark.Proof_G1_Idx", "content": "inductive Proof_G1_Idx : Type where\n  | Pf : Proof_G1_Idx"}, {"name": "ToySnark.Proof_G2_Idx", "content": "def Proof_G2_Idx : Type := Empty"}, {"name": "ToySnark.PairingsIdx", "content": "inductive PairingsIdx : Type where\n  | lhs : PairingsIdx\n  | rhs : PairingsIdx"}, {"name": "ToySnark.SRS_Elements_G2_Idx", "content": "inductive SRS_Elements_G2_Idx : Type where\n  | α : SRS_Elements_G2_Idx\n  | β : SRS_Elements_G2_Idx"}, {"name": "ToySnark.ToySnark", "content": "noncomputable def ToySnark\n     \n    {F : Type} [Field F] :\n    AGMProofSystemInstantiation F :=\n  {\n    Stmt := StmtEntries -> F\n    Sample := Option Vars\n    SRSElements_G1 := SRS_Elements_G1_Idx\n    ListSRSElements_G1 :=\n      [.α, .β]\n    SRSElements_G2 := SRS_Elements_G2_Idx\n    ListSRSElements_G2 :=\n      [.α, .β]\n    SRSElementValue_G1 := fun SRS_idx => match SRS_idx with\n      | SRS_Elements_G1_Idx.α => MvPolynomial.X Vars_α\n      | SRS_Elements_G1_Idx.β => MvPolynomial.X Vars_β\n    SRSElementValue_G2 := fun SRS_idx => match SRS_idx with\n      | SRS_Elements_G2_Idx.α => MvPolynomial.X Vars_α\n      | SRS_Elements_G2_Idx.β => MvPolynomial.X Vars_β\n    Proof_G1 := Proof_G1_Idx\n    ListProof_G1 := [Proof_G1_Idx.Pf]\n    Proof_G2 := Proof_G2_Idx\n    ListProof_G2 := []\n    EqualityChecks := Unit\n    Pairings := fun _ => PairingsIdx\n    ListPairings := fun _ => [PairingsIdx.lhs, PairingsIdx.rhs]\n    verificationPairingSRS_G1 := fun stmt _ i SRS_idx => match i with\n      | PairingsIdx.lhs => 0\n      | PairingsIdx.rhs => match SRS_idx with\n        | SRS_Elements_G1_Idx.α => stmt StmtEntries.z\n        | SRS_Elements_G1_Idx.β => 0\n    verificationPairingSRS_G2 := fun stmt _ i SRS_idx => match i with\n      | PairingsIdx.lhs => match SRS_idx with\n        | SRS_Elements_G2_Idx.α => stmt StmtEntries.x\n        | SRS_Elements_G2_Idx.β => stmt StmtEntries.y\n      | PairingsIdx.rhs => match SRS_idx with\n        | SRS_Elements_G2_Idx.α => 0\n        | SRS_Elements_G2_Idx.β => -1\n    verificationPairingProof_G1 := fun stmt _ i pf => match i with\n      | PairingsIdx.lhs => match pf with\n        | Proof_G1_Idx.Pf => 1\n      | PairingsIdx.rhs => match pf with\n        | Proof_G1_Idx.Pf => 0\n    verificationPairingProof_G2 := fun stmt _ i pf => 0\n  }"}], "used_local_lemmas": [{"name": "ToySnark.Vars.finsupp_eq_ext", "content": "lemma Vars.finsupp_eq_ext (f g : Vars →₀ ℕ) : f = g ↔\n    f Vars.α = g Vars.α\n    ∧ f Vars.β = g Vars.β"}], "local_ctx": "import FormalSnarksProject.Models.AGMProofSystemInstantiation\n\nimport Mathlib.Algebra.Polynomial.Div\n\nimport FormalSnarksProject.ToMathlib.OptionEquivRight\n\nimport Mathlib.Algebra.MvPolynomial.Equiv\n\nimport FormalSnarksProject.SoundnessTactic.SoundnessProver\n\nopen scoped BigOperators Classical\n\nsection ToySnark\n\nopen MvPolynomial Option List\n\nnamespace ToySnark\n\ninductive Vars : Type where\n  | α : Vars\n  | β : Vars\nderiving Repr, BEq\n\ninductive StmtEntries : Type where\n  | x : StmtEntries\n  | y : StmtEntries\n  | z : StmtEntries\nderiving Repr, BEq\n\ninductive WitEntries : Type where\n  | A : WitEntries\n  | B : WitEntries\nderiving Repr, BEq\n\nlocal notation \"Vars_α\" => some Vars.α\nlocal notation \"Vars_β\" => some Vars.β\nlocal notation \"Vars_x\" => none\n\ninductive Proof_G1_Idx : Type where\n  | Pf : Proof_G1_Idx\n\ndef Proof_G2_Idx : Type := Empty\n\ninductive PairingsIdx : Type where\n  | lhs : PairingsIdx\n  | rhs : PairingsIdx\n\ninductive SRS_Elements_G2_Idx : Type where\n  | α : SRS_Elements_G2_Idx\n  | β : SRS_Elements_G2_Idx\n\nnoncomputable def ToySnark\n     \n    {F : Type} [Field F] :\n    AGMProofSystemInstantiation F :=\n  {\n    Stmt := StmtEntries -> F\n    Sample := Option Vars\n    SRSElements_G1 := SRS_Elements_G1_Idx\n    ListSRSElements_G1 :=\n      [.α, .β]\n    SRSElements_G2 := SRS_Elements_G2_Idx\n    ListSRSElements_G2 :=\n      [.α, .β]\n    SRSElementValue_G1 := fun SRS_idx => match SRS_idx with\n      | SRS_Elements_G1_Idx.α => MvPolynomial.X Vars_α\n      | SRS_Elements_G1_Idx.β => MvPolynomial.X Vars_β\n    SRSElementValue_G2 := fun SRS_idx => match SRS_idx with\n      | SRS_Elements_G2_Idx.α => MvPolynomial.X Vars_α\n      | SRS_Elements_G2_Idx.β => MvPolynomial.X Vars_β\n    Proof_G1 := Proof_G1_Idx\n    ListProof_G1 := [Proof_G1_Idx.Pf]\n    Proof_G2 := Proof_G2_Idx\n    ListProof_G2 := []\n    EqualityChecks := Unit\n    Pairings := fun _ => PairingsIdx\n    ListPairings := fun _ => [PairingsIdx.lhs, PairingsIdx.rhs]\n    verificationPairingSRS_G1 := fun stmt _ i SRS_idx => match i with\n      | PairingsIdx.lhs => 0\n      | PairingsIdx.rhs => match SRS_idx with\n        | SRS_Elements_G1_Idx.α => stmt StmtEntries.z\n        | SRS_Elements_G1_Idx.β => 0\n    verificationPairingSRS_G2 := fun stmt _ i SRS_idx => match i with\n      | PairingsIdx.lhs => match SRS_idx with\n        | SRS_Elements_G2_Idx.α => stmt StmtEntries.x\n        | SRS_Elements_G2_Idx.β => stmt StmtEntries.y\n      | PairingsIdx.rhs => match SRS_idx with\n        | SRS_Elements_G2_Idx.α => 0\n        | SRS_Elements_G2_Idx.β => -1\n    verificationPairingProof_G1 := fun stmt _ i pf => match i with\n      | PairingsIdx.lhs => match pf with\n        | Proof_G1_Idx.Pf => 1\n      | PairingsIdx.rhs => match pf with\n        | Proof_G1_Idx.Pf => 0\n    verificationPairingProof_G2 := fun stmt _ i pf => 0\n  }\n\nsection soundness", "target_theorem": "lemma soundness\n    {F : Type} [Field F] :\n    (AGMProofSystemInstantiation.soundness\n      F\n      (ToySnark\n        (F := F))\n      (WitEntries -> F)\n      (fun (stmt : StmtEntries → F) (wit : WitEntries -> F) =>\n        wit WitEntries.A * stmt StmtEntries.y = stmt StmtEntries.z -- - wit WitEntries.I\n        ∨\n        wit WitEntries.B * stmt StmtEntries.x = stmt StmtEntries.z -- - wit WitEntries.I\n      )\n      (fun prover i => prover.fst Proof_G1_Idx.Pf (if i = WitEntries.A then .α else .β))\n\n    ) :=", "ground_truth_proof": ":= by\n  unfold AGMProofSystemInstantiation.soundness AGMProofSystemInstantiation.verify AGMProofSystemInstantiation.proof_element_G1_as_poly AGMProofSystemInstantiation.proof_element_G2_as_poly\n  intros stmt prover eqns'\n  rcases eqns' with ⟨eqns, null⟩\n  have eqn := eqns ()\n  clear eqns null\n\n  -- Step 1: Obtain the coefficient equations of the mv_polynomials\n  simp_rw [ToySnark] at eqn\n  simp only [monomial_zero', List.singleton_append, List.cons_append, List.append_assoc,\n    List.map_cons, Sum.elim_inl, Sum.elim_inr, List.map_append, List.map_map, List.sum_cons,\n    List.sum_append, List.map_nil, List.sum_nil, add_zero, Sum.elim_lam_const_lam_const, map_one,\n    one_mul, map_zero, zero_mul, map_neg, neg_mul, neg_add_rev, zero_add, mul_zero,\n    -- Note: everything above is @simp tagged\n    Function.comp, List.sum_map_zero] at eqn\n\n  simp only [mul_add, add_mul, List.sum_map_add] at eqn\n\n  -- Move all the X (some _) terms to the left, and out of sums\n  simp only [\n    -- Associativity to obtain a right-leaning tree\n    mul_assoc,\n    -- Commutativity lemmas to move X (some _) to the left\n    mul_left_comm (C _) (X (some _)) _, mul_left_comm (List.sum _) (X (some _)) _,\n    mul_comm (C _) (X (some _)), mul_comm (List.sum _) (X (some _)),\n    -- Move negations to the bottom\n    neg_mul, mul_neg,\n    -- Move constant multiplications (which the X (some _) terms should be) out of sums\n    List.sum_map_mul_right, List.sum_map_mul_left] at eqn\n\n\n  -- I apply MvPolynomial.optionEquivRight *here*,\n  -- so that we can treat polynomials in Vars_X as constants\n  trace \"Converting to MvPolynomial over Polynomials\"\n  replace eqn := congr_arg (MvPolynomial.optionEquivRight F Vars) eqn\n  simp only [AlgEquiv.map_add, AlgEquiv.map_zero, AlgEquiv.map_mul, AlgEquiv.map_one,\n    AlgEquiv.map_neg, AlgEquiv.list_map_sum, AlgEquiv.map_pow] at eqn\n  simp only [MvPolynomial.optionEquivRight_C, MvPolynomial.optionEquivRight_X_none, MvPolynomial.optionEquivRight_X_some, optionEquivRight_to_MvPolynomial_Option] at eqn\n  -- Move Cs back out so we can recognize the monomials\n  -- simp only [←MvPolynomial.C_mul, ←MvPolynomial.C_pow, ←MvPolynomial.C_add,\n  --   MvPolynomial.sum_map_C] at eqn\n\n  simp only [MvPolynomial.X, C_apply, MvPolynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, mul_add, add_mul] at eqn\n\n  trace \"Applying individual coefficients\"\n\n\n  have h20 := congr_arg (coeff (Finsupp.single Vars.α 2 + Finsupp.single Vars.β 0)) eqn\n  have h11 := congr_arg (coeff (Finsupp.single Vars.α 1 + Finsupp.single Vars.β 1)) eqn\n  have h02 := congr_arg (coeff (Finsupp.single Vars.α 0 + Finsupp.single Vars.β 2)) eqn\n\n  clear eqn\n\n  trace \"Distribute coefficient-taking over terms\"\n  simp only [coeff_monomial, coeff_add, coeff_neg, coeff_zero] at h20 h11 h02\n\n  trace \"Simplifying coefficient expressions\"\n  simp only [Vars.finsupp_eq_ext, Finsupp.single_apply, Finsupp.add_apply] at h20 h11 h02\n\n  simp [ite_true, ite_self, add_zero, ite_false, and_self, zero_add,\n    one_ne_zero, and_false, false_and, add_eq_zero, mul_eq_zero,\n    add_right_eq_self, zero_ne_one, and_true, true_and, neg_zero] at h20 h11 h02 ⊢\n\n\n  -- Completely remove references to Polynomial\n  simp only [add_neg_eq_zero, Polynomial.C_inj, ←Polynomial.C_add, ←Polynomial.C_mul] at h20 h11 h02\n\n  integral_domain_tactic", "nesting_depth": 5, "transitive_dep_count": 129, "subset_aristotle": false, "category": "Applied verif."}
{"id": 198, "thm_name": "Intmax.theorem1", "thm_stmt": "theorem theorem1 : ¬adversaryWon (attackGame requests) := λ contra ↦ by\n  /-\n    PAPER: Suppose an adversary and a challenger have interacted in Attack game 1.\n    We will show that either the resulting contract balance is positive (the adver-\n    sary lost the game), or the adversary has been able to either break the bind-\n    ing property of the authenticated dictionary scheme or found a collision of the\n    hash function H.\n  -/\n\n  /-\n    The attack game plays out the same regardless of validity of requests.\n  -/\n  rw [attackGame_eq_attackGameBlocks!_normalise, attackGameBlocks_eq_attackGameR] at contra\n  set requests! := normalise requests with eqRequests\n  /-\n    PAPER: Let B∗ = (Bi)i∈[n] be the contract state after the attack game\n  -/\n  set Bstar := attackGameR requests! [] with eqBstar\n  /-\n    As such, we can consider a state with only valid requests.\n  -/\n  have hValid : ∀ request ∈ (normalise requests), request.isValid", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Theorem1.lean", "imports": ["import FVIntmax.Lemma5", "import FVIntmax.Wheels.AuthenticatedDictionary", "import FVIntmax.Request", "import FVIntmax.AttackGame", "import FVIntmax.Wheels", "import FVIntmax.Propositions", "import Mathlib", "import FVIntmax.Balance", "import FVIntmax.Lemma4", "import FVIntmax.Wheels.Dictionary", "import FVIntmax.Wheels.SignatureAggregation", "import FVIntmax.Lemma3"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "IsGLB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "And", "module": "Init.Prelude"}, {"name": "IsGreatest", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "InfSet", "module": "Mathlib.Order.SetNotation"}, {"name": "iInf", "module": "Mathlib.Order.SetNotation"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "Classical.arbitrary", "module": "Mathlib.Logic.Nonempty"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.reduceAdd", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int"}, {"name": "Int.reduceNeg", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Int"}, {"name": "IsLUB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "List.Ico", "module": "Mathlib.Data.List.Intervals"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "List.lookup", "module": "Init.Data.List.Basic"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "List.take", "module": "Init.Data.List.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.snd", "module": "Init.Prelude"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "absurd", "module": "Init.Prelude"}, {"name": "Lean.Parser.Tactic.replace", "module": "Init.Tactics"}, {"name": "List.foldl", "module": "Init.Prelude"}, {"name": "IsLeast", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "LE", "module": "Init.Prelude"}, {"name": "LE.le", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Preorder.toLE", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "List.concat", "module": "Init.Prelude"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Monotone", "module": "Mathlib.Order.Monotone.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}, {"name": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂", "content": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂"}, {"name": "notation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂", "content": "notation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "Merge", "content": "def Merge (D₁ D₂ : Dict α ω) : Dict α ω := D\n  where D := λ x ↦ First (D₁ x) (D₂ x)"}, {"name": "First", "content": "def First (x₁ x₂ : Option α) : Option α :=\n  match x₁, x₂ with\n  | .some x, .none => .some x\n  | .some x, .some _ => .some x\n  | .none, .some y => .some y\n  | .none, .none => .none"}, {"name": "getBalanceProof", "content": "def getBalanceProof (requests : List (Request K₁ K₂ C Sigma Pi V))\n                    (h₀ : ∀ request ∈ requests, request.isValid)\n                    (i : Fin (attackGameR requests []).length)\n                    (h₁ : (attackGameR requests [])[i].isWithdrawalBlock) :\n                    BalanceProof K₁ K₂ C Pi V :=\n  let request := requests[i]'(by admit /- proof elided -/\n  )\n  have : request.getWithdrawal.isSome := by admit /- proof elided -/"}, {"name": "attackGameR", "content": "def attackGameR : Scontract K₁ K₂ V C Sigma :=\n  σ ++ attackGameRGo requests σ"}, {"name": "attackGameRGo", "content": "def attackGameRGo (requests : List (Request K₁ K₂ C Sigma Pi V))\n                  (σ : Scontract K₁ K₂ V C Sigma) : Scontract K₁ K₂ V C Sigma :=\n  match requests with\n  | [] => []\n  | hd :: tl => hd.toBlock! σ :: attackGameRGo tl (σ.appendBlock! hd)"}, {"name": "appendBlock!", "content": "def appendBlock! (σ : Scontract K₁ K₂ V C Sigma)\n                 (request : Request K₁ K₂ C Sigma Pi V) : Scontract K₁ K₂ V C Sigma :=\n  σ ++ [request.toBlock! σ]"}, {"name": "toBlock!", "content": "def toBlock! (σ : Scontract K₁ K₂ V C Sigma)\n             (request : Request K₁ K₂ C Sigma Pi V) : Block K₁ K₂ C Sigma V :=\n  match request with\n  | .deposit r v            => .deposit r v\n  | .transfer a e c s sigma => .transfer a e c s sigma\n  | .withdrawal π           => .withdrawal (π.toBalanceF σ)"}, {"name": "Request", "content": "inductive Request where\n  | deposit (recipient : K₂) (amount : V₊)\n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n  | withdrawal (π : BalanceProof K₁ K₂ C Pi V)"}, {"name": "BalanceProof.toBalanceF", "content": "def BalanceProof.toBalanceF (π : BalanceProof K₁ K₂ C Pi V)\n                            (σ : Scontract K₁ K₂ V C Sigma) : K₁ → V₊ :=\n  λ k : K₁ ↦ ⟨Bal π σ k, by admit /- proof elided -/\n  ⟩"}, {"name": "Bal", "content": "def Bal (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) : S K₁ K₂ V :=\n  fStar (TransactionsInBlocks π bs) (.initial K₁ K₂ V)"}, {"name": "TransactionsInBlocks", "content": "def TransactionsInBlocks\n  (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) : List (Τ K₁ K₂ V) :=\n  (bs.map (TransactionsInBlock π)).flatten"}, {"name": "TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}, {"name": "TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "fStar", "content": "def fStar (Ts : List (Τ K₁ K₂ V)) (s₀ : S K₁ K₂ V) : S K₁ K₂ V :=\n  Ts.foldl f s₀"}, {"name": "f", "content": "def f (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨\n    λ k ↦\n      have : InfSet V := infV b T k\n      ⨅ x : boundedBelow b T, fc x.1 k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "infV", "content": "def infV (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) :\n  InfSet V where\n    sInf := λ s ↦ if s = V' b T k\n                  then (exists_inf b T).1 k\n                  else 0"}, {"name": "exists_inf", "content": "def exists_inf (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : { s : S K₁ K₂ V // ∀ k, IsGLB (V' b T k) (s k) } :=\n  ⟨\n    f' b T,\n    λ k ↦\n      have f'_codomain : (f' b T) k ∈ V' b T k := by admit /- proof elided -/\n  ⟩"}, {"name": "fc", "content": "def fc (τcXb : Τc K₁ K₂ V × S K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨λ k : Kbar K₁ K₂ ↦\n    match τcXb with\n    | ⟨⟨⟨⟨s, r, v⟩, _⟩, hτ⟩, b⟩ =>\n      let v' := v' (v.get hτ) b s\n      b k + (e r - e s) k • v',\n  by admit /- proof elided -/\n  ⟩"}, {"name": "e", "content": "def e (i : Kbar K₁ K₂) : Kbar K₁ K₂ → ℤ := λ j ↦ if i = j then 1 else 0"}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "Τc", "content": "abbrev Τc (K₁ K₂ V : Type) [PreWithZero V] : Type := { τ : Τ K₁ K₂ V // τ.isComplete }"}, {"name": "boundedBelow", "content": "abbrev boundedBelow (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) :=\n  { a : Τc K₁ K₂ V × S K₁ K₂ V | (T, b) ≤ (↑a.1, a.2) }"}, {"name": "f'", "content": "def f' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V := \n  ⟨\n    λ k ↦\n      match h : T with\n      | ⟨(_, _, .some _), hT⟩ => fc (⟨T, by admit /- proof elided -/\n      ⟩, b) k\n      | ⟨(s, _, .none), _⟩ => if k = s then 0 else b k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "V'", "content": "def V' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) : Set V :=\n  { v : V | v ∈ (fc · k) '' boundedBelow b T }"}, {"name": "isComplete", "content": "def isComplete (τ : Τ K₁ K₂ V) :=\n  match τ with | ⟨(_, _, v), _⟩ => v.isSome"}, {"name": "initial", "content": "def initial : Scontract K₁ K₂ V C Sigma := []"}, {"name": "Scontract", "content": "abbrev Scontract (K₁ K₂ V : Type) [PreWithZero V] (C Sigma : Type) :=\n  List (Block K₁ K₂ C Sigma V)"}, {"name": "Request.isValid", "content": "def Request.isValid (request : Request K₁ K₂ C Sigma Pi V) : Bool :=\n  match request with\n   \n  | deposit .. =>\n      true\n   \n  | transfer aggregator extradata commitment senders sigma =>\n      let isValidSA := SA.Verify senders (commitment, aggregator, extradata) sigma\n      let isValidAggregator := aggregator = AgreedUponAggregator\n      isValidSA ∧ isValidAggregator\n   \n  | withdrawal π =>\n      π.Verify (M := (C × K₁ × ExtraDataT))"}, {"name": "AgreedUponAggregator", "content": "def AgreedUponAggregator {K₁ : Type} [Nonempty K₁] : K₁ := Classical.arbitrary _"}, {"name": "SignatureAggregation", "content": "class SignatureAggregation (M Kₚ Kₛ Sigma : Type) where\n  KeyGen    : SimpleRandom.Seed → Kₚ × Kₛ\n  Sign      : Kₛ → M → Sigma\n  Aggregate : List (Kₚ × Sigma) → Sigma\n  Verify    : List Kₚ → M → Sigma → Bool\n\n   \n  Correctness : ∀ (l : List (Kₚ × Kₛ)) (_ : ∀ pair ∈ l, pair ←ᵣ KeyGen) (m : M),\n                  let (kₚs, kₛs) := l.unzip\n                  Verify kₚs m (Aggregate (kₚs.zip (kₛs.map (Sign · m)))) = true\n\n   \n  Unforgeability :\n    ComputationallyInfeasible <|\n      ∃ (k : List (Kₚ × Kₛ)) (m : M) (σ : Sigma),\n        let kₚs := k.map Prod.fst\n        Verify kₚs m σ = true ∧\n        \n        \n        ∃ userIdx : Fin k.length,\n          let (honestₚ, honestₛ) := k[userIdx]\n          honestₚ ∈ kₚs ∧ ∃ key : Kₛ, key ≠ honestₛ ∧ Sign key m = σ"}, {"name": "ADScheme", "content": "class ADScheme (K : Type)\n               (M : Type)\n               (C Pi : Type) where\n  Commit : Dict K M → CommitT C K Pi\n  Verify : Pi → K → M → C → Bool \n\n   \n  correct_keys_eq : ∀ {dict : Dict K M}, (Commit dict).keys = dict.keys \n\n   \n  correct_consistent :\n    ∀ {dict : Dict K M} {key : K} (h : key ∈ dict.keys), \n      Verify (Commit dict)[key] key dict[key] (Commit dict).commitment = true \n\n   \n  binding : ComputationallyInfeasible <|\n              ∃ (c : C) (k : K) (m₁ m₂ : M) (π₁ π₂ : Pi),\n                Verify π₁ k m₁ c = true ∧\n                Verify π₂ k m₂ c = true ∧\n                m₁ ≠ m₂"}, {"name": "Verify", "content": "def Verify (π : BalanceProof K₁ K₂ C Pi V)\n           [AD : ADScheme K₂ M C Pi] : Bool :=\n  ∀ x, (h : x ∈ Dict.keys π) → let ((π', salt), t) := (π x).get (by admit /- proof elided -/\n  )\n                               AD.Verify π' x.2 (H _ (t, salt)) x.1\n\nnoncomputable opaque H {α : Type} (ω : Type) [Nonempty ω] (x : α) : ω := Classical.arbitrary _\n\nopaque ComputationallyInfeasible (p : Prop) : Prop := p"}, {"name": "attackGame", "content": "def attackGame : Scontract K₁ K₂ V C Sigma :=\n  attackGameBlocks' requests []"}, {"name": "attackGameBlocks'", "content": "def attackGameBlocks' : Scontract K₁ K₂ V C Sigma :=\n  requests.foldl Scontract.appendBlock σ"}, {"name": "appendBlock", "content": "def appendBlock (σ : Scontract K₁ K₂ V C Sigma)\n                (request : Request K₁ K₂ C Sigma Pi V) : Scontract K₁ K₂ V C Sigma :=\n  (request.toBlock σ).elim σ (σ ++ [·])"}, {"name": "toBlock", "content": "def toBlock (σ : Scontract K₁ K₂ V C Sigma)\n            (request : Request K₁ K₂ C Sigma Pi V) : Option (Block K₁ K₂ C Sigma V) :=\n  if ¬request.isValid\n  then .none\n  else .some <|\n  match request with\n   \n  | .deposit r v            => .deposit r v\n   \n  | .transfer a e c s sigma => .transfer a e c s sigma\n   \n  | .withdrawal π           => .withdrawal (π.toBalanceF σ)"}, {"name": "isValid", "content": "def isValid (s : S' K₁ K₂ V) := ∀ k : Kbar K₁ K₂, k matches .Source ∨ 0 ≤ s k"}, {"name": "isValid", "content": "def isValid (τ : Τ' K₁ K₂ V) :=\n  match τ with\n  | (s, r, v) => s ≠ r ∧ (s matches .Source → v.isSome)\n\n@[aesop norm (rule_sets := [Intmax.aesop_valid])]"}, {"name": "Request.getWithdrawal", "content": "def Request.getWithdrawal (request : Request K₁ K₂ C Sigma Pi V) : Option (BalanceProof K₁ K₂ C Pi V) :=\n  match request with\n  | withdrawal π => .some π\n  | transfer .. | deposit .. => .none"}, {"name": "adversaryWon", "content": "def adversaryWon (blocks : Scontract K₁ K₂ V C Sigma) : Prop :=\n  ¬0 ≤ computeBalance blocks"}, {"name": "computeBalance", "content": "def computeBalance (blocks : Scontract K₁ K₂ V C Sigma) : V :=\n  computeBalance' blocks 0"}, {"name": "computeBalance'", "content": "def computeBalance' (blocks : Scontract K₁ K₂ V C Sigma) (acc : V) : V :=\n  blocks.foldl Block.updateBalance acc"}, {"name": "Block.updateBalance", "content": "def Block.updateBalance (bal : V) (block : Block K₁ K₂ C Sigma V) : V :=\n  match block with\n   \n  | .deposit _ v  => bal + v\n   \n  | .transfer ..  => bal\n   \n  | .withdrawal B => bal - ∑ k : K₁, (B k).1\n\naxiom computationallyInfeasible_axiom : ∀ {p : Prop}, ComputationallyInfeasible p → ¬p"}, {"name": "isπ", "content": "def isπ (requests : List (Request K₁ K₂ C Sigma Pi V)) :=\n  ∀ (h₀ : ∀ request ∈ requests, request.isValid)\n    (i : Fin (attackGameR requests []).length)\n    (h : (attackGameR requests [])[i].isWithdrawalBlock),\n    (attackGameR requests [])[i].getWithdrawal h =\n    let π := getBalanceProof requests h₀ i h\n    let σ := (attackGameR requests []).take i.1\n    π.toBalanceF σ"}, {"name": "Block.isValid", "content": "def Block.isValid (block : Block K₁ K₂ C Sigma V) (π : BalanceProof K₁ K₂ C Pi V) : Bool :=\n  match block with\n   \n  | .deposit .. => true\n   \n  | .transfer aggregator extradata commitment senders sigma =>\n      let isValidSA := SA.Verify senders (commitment, aggregator, extradata) sigma\n      let isValidAggregator := aggregator = AgreedUponAggregator\n      isValidSA ∧ isValidAggregator\n   \n  | .withdrawal _ => π.Verify (M := (C × K₁ × ExtraDataT))"}, {"name": "BalanceProof.initial", "content": "def BalanceProof.initial : BalanceProof K₁ K₂ C Pi V := λ _ ↦ .none"}, {"name": "Setoid'", "content": "class Setoid' (X : Type) extends Preorder X where\n  isEquiv : IsEquivRel (X := X)"}, {"name": "IsEquivRel", "content": "def IsEquivRel {X : Type} [Preorder X] := ∀ a b : X, a ≤ b ↔ a ≅ b"}, {"name": "iso", "content": "def iso {X : Type} [Preorder X] (a b : X) := a ≤ b ∧ b ≤ a"}, {"name": "trivialPreorder", "content": "def trivialPreorder {α : Type} : Preorder α :=\n  {\n    lt := λ _ _ ↦ False\n    le := λ _ _ ↦ True\n    le_refl := by admit /- proof elided -/"}, {"name": "le", "content": "def le (v₁ v₂ : Vector α n) :=\n  ∀ x ∈ (v₁.1.zip v₂.1), x.1 ≤ x.2"}, {"name": "aggregateWithdrawals'", "content": "def aggregateWithdrawals' (σ : Scontract K₁ K₂ V C Sigma) : V :=\n  ∑ (i : Fin σ.length × K₁),\n    if h : σ[i.1].isWithdrawalBlock\n    then (σ[i.1].getWithdrawal h) i.2\n    else 0"}, {"name": "computeBalanceSum", "content": "def computeBalanceSum (σ : Scontract K₁ K₂ V C Sigma) :=\n  let v_deposited : V := aggregateDeposits σ\n  let v_withdrawn : V := aggregateWithdrawals σ\n  v_deposited - v_withdrawn"}, {"name": "aggregateWithdrawals", "content": "def aggregateWithdrawals (σ : Scontract K₁ K₂ V C Sigma) : V :=\n  ∑ i : Fin σ.length,\n    if h : σ[i].isWithdrawalBlock\n    then ∑ (k : K₁), (σ[i.1].getWithdrawal h) k\n    else 0"}, {"name": "aggregateDeposits", "content": "def aggregateDeposits (σ : Scontract K₁ K₂ V C Sigma) : V :=\n  ∑ i : Fin σ.length,\n    if h : σ[i].isDepositBlock\n    then (σ[i.1].getDeposit h).2.1\n    else 0"}, {"name": "normalise", "content": "def normalise (requests : List (Request K₁ K₂ C Sigma Pi V)) : List (Request K₁ K₂ C Sigma Pi V) :=\n  requests.filter Request.isValid"}, {"name": "getDeposit", "content": "def getDeposit (b : Block K₁ K₂ C Sigma V) (_h : b.isDepositBlock) : K₂ × V₊ :=\n  match b with | deposit r v => (r, v)"}, {"name": "value", "content": "def value (τ : Τ K₁ K₂ V) : Option V₊ := τ.1.2.2"}, {"name": "BalFixed", "content": "private def BalFixed (bs : List (Block K₁ K₂ C Sigma V)) : BalanceProof K₁ K₂ C Pi V → S K₁ K₂ V :=\n  λ π ↦ fStarFixed (s₀ := S.initial K₁ K₂ V) (TransactionsInBlocksFixed π bs)"}, {"name": "fStarFixed", "content": "private def fStarFixed {n : ℕ}\n  (Ts : Vector (Τ K₁ K₂ V) n) (s₀ : S K₁ K₂ V) : S K₁ K₂ V :=\n  fStar Ts.1 s₀"}, {"name": "TransactionsInBlocksFixed", "content": "private def TransactionsInBlocksFixed (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) :\n  Vector (Τ K₁ K₂ V) (length_of_TransactionsInBlocks (Pi := Pi) bs).1 :=\n  ⟨TransactionsInBlocks π bs, by admit /- proof elided -/\n  ⟩"}, {"name": "length_of_TransactionsInBlocks", "content": "private abbrev length_of_TransactionsInBlocks (bs : List (Block K₁ K₂ C Sigma V)) :\n  { n : ℕ // n = (TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length } :=\n  ⟨(TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length, rfl⟩"}, {"name": "initial", "content": "def initial (K₁ K₂ V : Type) [PreWithZero V] : S K₁ K₂ V :=\n  ⟨S'.initial K₁ K₂ V, S'.isValid_initial⟩"}, {"name": "Bal'", "content": "private def Bal' (bs : List (Block K₁ K₂ C Sigma V)) : BalanceProof K₁ K₂ C Pi V → S K₁ K₂ V :=\n  fStar (s₀ := S.initial K₁ K₂ V) ∘ TransactionsInBlocks (bs := bs)"}, {"name": "BalFixed'", "content": "private def BalFixed' (bs : List (Block K₁ K₂ C Sigma V)) : BalanceProof K₁ K₂ C Pi V → S K₁ K₂ V :=\n  fStarFixed (n := (length_of_TransactionsInBlocks (Pi := Pi) bs).1)\n             (s₀ := S.initial K₁ K₂ V) ∘ TransactionsInBlocksFixed (Pi := Pi) (bs := bs)"}, {"name": "v'", "content": "def v' (v : V₊) (b : S K₁ K₂ V) (s : Kbar K₁ K₂) : V₊ :=\n  match h : s with\n  | .Source => v\n  | .key _  => ⟨v ⊓ b s, by admit /- proof elided -/\n  ⟩"}, {"name": "reindex", "content": "@[simp]\nprivate def reindex : (a : ℕ) → a ∈ Finset.range (k + 1) \\ {0} → ℕ :=\n  λ a _ ↦ a.pred"}, {"name": "getWithdrawal", "content": "def getWithdrawal (b : Block K₁ K₂ C Sigma V) (_h : b.isWithdrawalBlock) : K₁ → V₊ :=\n  match b with | .withdrawal B => B"}, {"name": "attackGameBlocks!", "content": "def attackGameBlocks! (requests : List (Request K₁ K₂ C Sigma Pi V)) : Scontract K₁ K₂ V C Sigma :=\n  attackGameBlocks'! requests []"}, {"name": "attackGameBlocks'!", "content": "def attackGameBlocks'! : Scontract K₁ K₂ V C Sigma :=\n  requests.foldl Scontract.appendBlock! σ"}, {"name": "attackGameBlocks'!r", "content": "def attackGameBlocks'!r (requests : List (Request K₁ K₂ C Sigma Pi V))\n                        (σ : Scontract K₁ K₂ V C Sigma) : Scontract K₁ K₂ V C Sigma :=\n  match requests with\n  | [] => σ\n  | hd :: tl => attackGameBlocks'!r tl (σ.appendBlock! hd)"}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "List.foldl_assoc", "module": "Init.Data.List.Lemmas"}, {"name": "List.foldl_append", "module": "Init.Data.List.Lemmas"}, {"name": "Set.mem_insert_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_singleton_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "Set.mem_union", "module": "Mathlib.Data.Set.Basic"}, {"name": "and_imp", "module": "Init.SimpLemmas"}, {"name": "forall_eq", "module": "Init.PropLemmas"}, {"name": "forall_eq_or_imp", "module": "Init.PropLemmas"}, {"name": "List.concat_cons", "module": "Init.Data.List.Lemmas"}, {"name": "Classical.not_forall", "module": "Init.Classical"}, {"name": "Classical.not_imp", "module": "Init.Classical"}, {"name": "Fin.getElem_fin", "module": "Init.GetElem"}, {"name": "Fin.is_le'", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.lt_def", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.lt_of_le_of_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.lt_of_lt_of_le", "module": "Init.Data.Fin.Lemmas"}, {"name": "Finset.exists_minimal", "module": "Mathlib.Order.Preorder.Finite"}, {"name": "Finset.filter_nonempty_iff", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Finset.mem_filter", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "List.getElem_mem", "module": "Init.GetElem"}, {"name": "List.getElem_of_mem", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_take_iff_getElem", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.take_add", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.take_concat_get", "module": "Init.Data.List.TakeDrop"}, {"name": "Nat.exists_eq_add_of_le", "module": "Init.Data.Nat.Lemmas"}, {"name": "Option.ne_none_iff_exists", "module": "Init.Data.Option.Lemmas"}, {"name": "Set.toFinset_setOf", "module": "Mathlib.Data.Fintype.Sets"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "eq_or_lt_of_le", "module": "Mathlib.Order.Basic"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "inf_of_le_left", "module": "Mathlib.Order.Lattice"}, {"name": "lt_add_iff_pos_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "lt_irrefl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_and", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "zero_lt_one", "module": "Mathlib.Algebra.Order.ZeroLEOne"}, {"name": "Option.isSome_iff_ne_none", "module": "Init.Data.Option.Lemmas"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "List.lookup_graph", "module": "Mathlib.Data.List.Basic"}, {"name": "mem_upperBounds", "module": "Mathlib.Order.Bounds.Basic"}, {"name": "iInf_subtype", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "Decidable.not_not", "module": "Init.PropLemmas"}, {"name": "List.eq_nil_of_length_eq_zero", "module": "Init.Data.List.Lemmas"}, {"name": "List.length_filter_le", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_eq_nil_iff", "module": "Init.Data.List.Lemmas"}, {"name": "Nat.le_of_lt_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pred_eq_sub_one", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.sub_one_lt_of_le", "module": "Init.Data.Nat.Lemmas"}, {"name": "eq_mpr_eq_cast", "module": "Init.PropLemmas"}, {"name": "id_eq", "module": "Init.Prelude"}, {"name": "not_exists", "module": "Init.PropLemmas"}], "repo_lemmas": [{"name": "proposition6", "content": "lemma proposition6 [Setoid' Y] {D₁ D₂ : Dict X Y} :\n  (∃ join, IsLUB {D₁, D₂} join) ↔ ∀ x, D₁ x ≠ .none ∧ D₂ x ≠ .none → D₁ x ≅ D₂ x"}, {"name": "proposition4", "content": "lemma proposition4 [Setoid' X] {x y : Option X} :\n  (∃ join : Option X, IsLUB {x, y, .none} join) ↔ (x ≠ .none ∧ y ≠ .none → x ≅ y)"}, {"name": "proposition2", "content": "lemma proposition2 [Setoid' X] {x y : X} :\n  (∃ join : X, IsLUB {x, y} join) ↔ x ≅ y"}, {"name": "iso_rfl", "content": "@[simp, refl]\nlemma iso_rfl : a ≅ a"}, {"name": "iso_trans", "content": "@[trans]\nlemma iso_trans : (a ≅ b) → (b ≅ c) → a ≅ c"}, {"name": "proposition2'", "content": "lemma proposition2' [Setoid' X] {join x y : X} (h : IsLUB {x, y} join) :\n  (x ≅ join) ∧ y ≅ join"}, {"name": "proposition3'", "content": "lemma proposition3' : \n  (∃ join : X, IsLUB {x, y} join) ↔ (∃ join : Option X, IsLUB {.some x, .some y, .none} join)"}, {"name": "proposition5", "content": "lemma proposition5 [Preorder Y] {f g : X → Y} {join : X → Y} :\n  IsLUB {f, g} join ↔ ∀ x : X, IsLUB {f x, g x} (join x)"}, {"name": "proposition6_aux", "content": "lemma proposition6_aux [Setoid' Y] {D₁ D₂ : Dict X Y}\n  (h : ∀ k, D₁ k ≠ .none ∧ D₂ k ≠ .none → D₁ k ≅ D₂ k) : IsLUB {D₁, D₂} (Dict.Merge D₁ D₂)"}, {"name": "proposition6'", "content": "lemma proposition6' [Setoid' Y] {D₁ D₂ join : Dict X Y} (h : IsLUB {D₁, D₂} join) :\n  join ≅ Dict.Merge D₁ D₂"}, {"name": "iso_symm", "content": "@[symm]\nlemma iso_symm : (a ≅ b) ↔ b ≅ a"}, {"name": "proposition4'", "content": "lemma proposition4' [Setoid' X] {join x y : Option X} (h : IsLUB {x, y, .none} join) :\n  join ≅ Dict.First x y"}, {"name": "proposition5'", "content": "lemma proposition5' [Preorder Y] {f g h join' : X → Y}\n  (h₀ : IsLUB {f, g} join')\n  (h₁ : ∀ x, h x ≅ join' x) :\n  join' ≅ h"}, {"name": "discretePreorder_eq_equality_Pi_Prod_ExtraDataT", "content": "@[simp]\nlemma discretePreorder_eq_equality_Pi_Prod_ExtraDataT {a b : (Pi × ExtraDataT)} : a ≤ b"}, {"name": "BalanceProof.snd_discrete", "content": "@[simp]\nlemma BalanceProof.snd_discrete {x y : TransactionBatch K₁ K₂ V} :\n  @LE.le (TransactionBatch K₁ K₂ V) Preorder.toLE x y ↔ x = y"}, {"name": "Merge_assoc", "content": "lemma Merge_assoc {D₃ : Dict α ω} :\n  Merge (Merge D₁ D₂) D₃ = Merge D₁ (Merge D₂ D₃)"}, {"name": "none_iso_some", "content": "@[simp]\nlemma none_iso_some : (.none ≅ some x) ↔ False"}, {"name": "lemma5", "content": "lemma lemma5 (π : BalanceProof K₁ K₂ C Pi V) :\n  Bal π σ .Source =\n  (∑ (i : Fin σ.length) (k : K₁),\n     if h : σ[i].isWithdrawalBlock\n     then let w := σ[i].getWithdrawal h\n          w k ⊓ Bal π (σ.take i.1) k\n     else 0)\n  -\n  aggregateDeposits σ"}, {"name": "lemma5_aux", "content": "private lemma lemma5_aux {len : ℕ} {σ : Scontract K₁ K₂ V C Sigma}\n  (hlen : len = σ.length) :\n  (Bal π σ) Kbar.Source =\n  (∑ x ∈ (Finset.univ : Finset (Fin σ.length)) ×ˢ Finset.univ,\n      if h : σ[x.1].isWithdrawalBlock then\n        (σ[x.1].getWithdrawal h x.2).1 ⊓ ((Bal π (List.take (x.1.1) σ)) x.2)\n      else 0) -\n  ∑ i : Fin (List.length σ), if h : σ[i].isDepositBlock then (σ[↑i].getDeposit h).2 else 0"}, {"name": "lemma4", "content": "lemma lemma4 (h : π₁ ≤ π₂) : Bal π₁ bs ≤ Bal π₂ bs"}, {"name": "Bal'_eq_Bal", "content": "private lemma Bal'_eq_Bal : Bal' bs π = Bal π bs"}, {"name": "BalFixed_eq_BalFixed'", "content": "private lemma BalFixed_eq_BalFixed' : BalFixed bs π = BalFixed' bs π"}, {"name": "monotone_TransactionsInBlocksFixed", "content": "lemma monotone_TransactionsInBlocksFixed :\n  Monotone λ (π : BalanceProof K₁ K₂ C Pi V) ↦ TransactionsInBlocksFixed π Bstar"}, {"name": "TransactionsInBlocksFixed_le_of_TransactionsInBlocks", "content": "lemma TransactionsInBlocksFixed_le_of_TransactionsInBlocks\n  (h : ∀ i : Fin (length_of_TransactionsInBlocks bs).1,\n    (TransactionsInBlocks π bs)[i]'(by blast with π i) ≤\n    (TransactionsInBlocks π' bs)[i]'(by blast with π' i)) :\n  TransactionsInBlocksFixed π bs ≤ TransactionsInBlocksFixed π' bs"}, {"name": "v_transactionsInBlocks", "content": "lemma v_transactionsInBlocks {s r v v'} {eq₁ eq₂} {i}\n  (h : π ≤ π')\n  (h₀ : i < (TransactionsInBlocks π Bstar).length)\n  (h₁ : (TransactionsInBlocks π Bstar)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' Bstar)[i]'(by blast with π) = ⟨(s, r, v'), eq₂⟩) :\n  v ≤ v'"}, {"name": "delta_TransactionsInBlock_transfer", "content": "private lemma delta_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }}\n  (h : π ≤ π') : \n  ∀ i : ℕ, (hlen : i < (TransactionsInBlock_transfer π b).length) →\n    (TransactionsInBlock_transfer π b)[i]'hlen =\n    (TransactionsInBlock_transfer π' b)[i]'(by rwa [length_TransactionsInBlock_transfer _ π]) ∨\n    ((TransactionsInBlock_transfer π b)[i]'hlen).1.2.2.isNone"}, {"name": "senderReceiver_transactionsInBlocks", "content": "lemma senderReceiver_transactionsInBlocks {s r v} {s' r' v'} {eq₁ eq₂} {i}\n  (h₀ : i < (TransactionsInBlocks π bs).length)\n  (h₁ : (TransactionsInBlocks π bs)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' bs)[i]'(by blast with π) = ⟨(s', r', v'), eq₂⟩) :\n  s = s' ∧ r = r'"}, {"name": "monotone_fStarFixed", "content": "lemma monotone_fStarFixed :\n  Monotone λ (Ts : Vector (Τ K₁ K₂ V) n) ↦ Intmax.fStarFixed Ts (S.initial K₁ K₂ V)"}, {"name": "monotone_fStarFixed_aux", "content": "private theorem monotone_fStarFixed_aux (h : v₁ ≤ v₂) (h₂ : b₁ ≤ b₂) :\n                                        v₁.1.foldl f b₁ ≤ v₂.1.foldl f b₂"}, {"name": "monotone_f", "content": "lemma monotone_f (h₁ : b₁ ≤ b₂) (h₂ : T₁ ≤ T₂) : f b₁ T₁ k ≤ f b₂ T₂ k"}, {"name": "lemma3", "content": "lemma lemma3 : Bal π bs .Source ≤ 0"}, {"name": "sum_fStar_le_zero_aux", "content": "private lemma sum_fStar_le_zero_aux (h : ∑ (k : Kbar K₁ K₂), b k ≤ 0) :\n  ∑ (k : Kbar K₁ K₂), fStar Tstar b k ≤ 0"}, {"name": "sum_f_le_sum", "content": "lemma sum_f_le_sum : ∑ (k : Kbar K₁ K₂), f b T k ≤ ∑ (k : Kbar K₁ K₂), b k"}, {"name": "sum_fc_eq_sum", "content": "@[simp]\nlemma sum_fc_eq_sum : ∑ (k : Kbar K₁ K₂), fc (Tc, b) k = ∑ (k : Kbar K₁ K₂), b k"}, {"name": "aggregateWithdrawals_eq_aggregateWithdrawals'", "content": "lemma aggregateWithdrawals_eq_aggregateWithdrawals' {σ : Scontract K₁ K₂ V C Sigma} :\n  aggregateWithdrawals σ = aggregateWithdrawals' σ"}, {"name": "keys_Merge_right", "content": "lemma keys_Merge_right {x : α}\n                       (h₁ : x ∉ D₁) (h₂ : x ∈ D₂) : Merge D₁ D₂ x = D₂ x"}, {"name": "mem_iff_isSome", "content": "lemma mem_iff_isSome {m : Dict α ω} {x : α} : x ∈ m ↔ (m x).isSome"}, {"name": "keys_Merge_left", "content": "lemma keys_Merge_left {x : α} (h : x ∈ D₁) : Merge D₁ D₂ x = D₁ x"}, {"name": "computeBalance_eq_sum", "content": "lemma computeBalance_eq_sum : computeBalance σ = computeBalanceSum σ"}, {"name": "computeBalance_eq_sum_aux", "content": "private lemma computeBalance_eq_sum_aux : computeBalance' σ v = v + computeBalanceSum σ"}, {"name": "computeBalance'_eq_zero", "content": "lemma computeBalance'_eq_zero : computeBalance' σ v = v + computeBalance' σ 0"}, {"name": "Block.updateBalance_eq_zero", "content": "lemma Block.updateBalance_eq_zero :\n  block.updateBalance v = v + block.updateBalance 0"}, {"name": "computeBalance'_cons", "content": "@[simp]\nlemma computeBalance'_cons : computeBalance' (hd :: σ) v =\n                             computeBalance' σ (Block.updateBalance v hd)"}, {"name": "reindex_mem", "content": "private lemma reindex_mem :\n  ∀ (a : ℕ) (ha : a ∈ Finset.range (k + 1) \\ {0}), reindex a ha ∈ Finset.range k"}, {"name": "reindex_inj", "content": "private lemma reindex_inj :\n  ∀ (a₁ : ℕ) (ha₁ : a₁ ∈ Finset.range (k + 1) \\ {0})\n    (a₂ : ℕ) (ha₂ : a₂ ∈ Finset.range (k + 1) \\ {0}),\n  reindex a₁ ha₁ = reindex a₂ ha₂ → a₁ = a₂"}, {"name": "mem_dict_iff_mem_keys", "content": "lemma mem_dict_iff_mem_keys {dict : Dict α ω} : k ∈ dict ↔ k ∈ dict.keys"}, {"name": "Function.injective_of_CryptInjective", "content": "theorem Function.injective_of_CryptInjective\n  {α ω : Type} [Nonempty ω] {f : α → ω} [inj : Injective f] : Function.Injective f"}, {"name": "getElem_Ico_of_lt", "content": "lemma getElem_Ico_of_lt {m n : ℕ} (h : n < m) : (List.Ico 0 m)[n]'(by simp [h]) = n"}, {"name": "attackGame_eq_attackGameBlocks!_normalise", "content": "lemma attackGame_eq_attackGameBlocks!_normalise :\n  attackGame requests = attackGameBlocks! (normalise requests)"}, {"name": "attackGameBlocks'_eq_attackGameBlocks'!_normalise", "content": "lemma attackGameBlocks'_eq_attackGameBlocks'!_normalise :\n  attackGameBlocks' requests σ = attackGameBlocks'! (normalise requests) σ"}, {"name": "appendBlock_eq_appendBlock!_of_isValid", "content": "lemma appendBlock_eq_appendBlock!_of_isValid (h : request.isValid) :\n  appendBlock σ request = appendBlock! σ request"}, {"name": "appendBlock_eq_id_of_not_isValid", "content": "lemma appendBlock_eq_id_of_not_isValid (h : ¬request.isValid) :\n  appendBlock σ request = σ"}, {"name": "attackGameBlocks_eq_attackGameR", "content": "lemma attackGameBlocks_eq_attackGameR :\n  attackGameBlocks! requests = attackGameR requests []"}, {"name": "attackGameBlocks'r_eq_attackGameBlocks'", "content": "lemma attackGameBlocks'r_eq_attackGameBlocks' :\n  attackGameBlocks'! requests σ = attackGameBlocks'!r requests σ"}, {"name": "attackGameR_eq_attackGameBlocks'", "content": "lemma attackGameR_eq_attackGameBlocks' :\n  attackGameR requests σ = attackGameBlocks'!r requests σ"}, {"name": "appendBlock!_def", "content": "lemma appendBlock!_def : σ.appendBlock! request = σ ++ [request.toBlock! σ]"}], "used_local_defs": [{"name": "Intmax.mergeR''", "content": "def mergeR'' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  match πs with\n  | [] => acc\n  | π :: πs => Dict.Merge acc (mergeR'' πs π)"}, {"name": "Intmax.mergeR'''", "content": "def mergeR''' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  πs.foldl Dict.Merge acc"}, {"name": "Intmax.BalanceProof.compat", "content": "def BalanceProof.compat (π₁ π₂ : BalanceProof K₁ K₂ C Pi V) : Prop :=\n  ∀ k, π₁ k ≠ none ∧ π₂ k ≠ none → π₁ k ≅ π₂ k"}], "used_local_lemmas": [{"name": "Intmax.mergeR''_eq_foldl", "content": "lemma mergeR''_eq_foldl :\n  mergeR'' πs acc = mergeR''' πs acc"}, {"name": "Intmax.mergeR''_cons", "content": "@[simp]\nlemma mergeR''_cons :\n  mergeR'' (π :: πs) acc = Dict.Merge acc (mergeR'' πs π)"}, {"name": "Intmax.mergeR''_append", "content": "@[simp]\nlemma mergeR''_append :\n  mergeR'' (πs ++ πs') acc = mergeR'' πs' (mergeR'' πs acc)"}, {"name": "Intmax.mergeR''_concat", "content": "@[simp]\nlemma mergeR''_concat {π} {πs : List (BalanceProof K₁ K₂ C Pi V)} {acc} :\n  mergeR'' (πs.concat π) acc = ((mergeR'' πs acc) <+> π)"}, {"name": "Intmax.existsLUB_iff_compat", "content": "lemma existsLUB_iff_compat :\n  (∃ join, IsLUB {π₁, π₂} join) ↔ π₁ <≅> π₂"}, {"name": "Intmax.compat_comm", "content": "lemma compat_comm : (π₁ <≅> π₂) ↔ π₂ <≅> π₁"}, {"name": "Intmax.merge_le", "content": "lemma merge_le (h₁ : π₁ ≤ π₃) (h₂ : π₂ ≤ π₃) : π₁ <+> π₂ ≤ π₃"}, {"name": "Intmax.isLUB_union_Merge_of_isLUB_isLUB_compat", "content": "lemma isLUB_union_Merge_of_isLUB_isLUB_compat {A B : Set (BalanceProof K₁ K₂ C Pi V)}\n  (h₁ : IsLUB A j₁) (h₂ : IsLUB B j₂) (h₃ : j₁ <≅> j₂) : IsLUB (A ∪ B) (j₁ <+> j₂)"}, {"name": "Intmax.mergeR''_eq_none'", "content": "@[simp]\nlemma mergeR''_eq_none' :\n  (mergeR'' πs acc) K = none ↔ acc K = none ∧ ∀ π ∈ πs, π K = none"}, {"name": "Intmax.merge_K", "content": "lemma merge_K : (π <+> acc) K = Dict.First (π K) (acc K)"}, {"name": "Intmax.mergeR''_eq_some", "content": "@[simp]\nlemma mergeR''_eq_some {x} :\n  acc K = some x → (mergeR'' πs acc) K = acc K"}, {"name": "Intmax.iso_K_merge_left_of_ne_none", "content": "lemma iso_K_merge_left_of_ne_none (h : π K ≠ none) : π K ≅ (π <+> acc) K"}, {"name": "Intmax.merge_left_none_eq_right", "content": "lemma merge_left_none_eq_right (h : π K = none) : (π <+> acc) K = acc K"}, {"name": "Intmax.verify_merge_of_valid", "content": "lemma verify_merge_of_valid {π₁ π₂ : BalanceProof K₁ K₂ C Pi V}\n                            (h₁ : π₁.Verify (M := (C × K₁ × ExtraDataT)))\n                            (h₂ : π₂.Verify (M := (C × K₁ × ExtraDataT))) :\n                            BalanceProof.Verify (M := (C × K₁ × ExtraDataT)) (Dict.Merge π₁ π₂)"}, {"name": "Intmax.valid_mergeR''_aux", "content": "private lemma valid_mergeR''_aux {π : BalanceProof K₁ K₂ C Pi V}\n                                 {πs : List (BalanceProof K₁ K₂ C Pi V)}\n                                 {n : ℕ}\n  (hn : n < πs.length.succ)\n  (h₀ : π.Verify (M := (C × K₁ × ExtraDataT)))\n  (h : ∀ π ∈ πs, π.Verify (M := (C × K₁ × ExtraDataT))) :\n  (mergeR'' (πs.take n) π).Verify (M := (C × K₁ × ExtraDataT))"}, {"name": "Intmax.valid_mergeR''", "content": "lemma valid_mergeR'' {πs : List (BalanceProof K₁ K₂ C Pi V)} {n : ℕ}\n  (hn : n < πs.length.succ)\n  (h : ∀ π ∈ πs, π.Verify (M := (C × K₁ × ExtraDataT))) :\n  (mergeR'' (πs.take n) .initial).Verify (M := (C × K₁ × ExtraDataT))"}, {"name": "Intmax.Merge_split", "content": "lemma Merge_split (h₀ : 0 < i) (h₁ : i ≤ πs.length) :\n  mergeR'' (πs.take i) acc =\n  mergeR'' (πs.take (i - 1)) acc <+> πs[i - 1]"}, {"name": "Intmax.merge_lem_aux", "content": "private lemma merge_lem_aux :\n  mergeR'' (π :: πs) acc = acc <+> π <+> (mergeR'' πs BalanceProof.initial)"}, {"name": "Intmax.merge_lem", "content": "lemma merge_lem :\n  mergeR'' (π :: πs) BalanceProof.initial = π <+> (mergeR'' πs BalanceProof.initial)"}, {"name": "Intmax.compat_lem", "content": "lemma compat_lem {π π' π'': BalanceProof K₁ K₂ C Pi V} :\n  π <≅> π' → π <≅> π'' → π <≅> (π' <+> π'')"}, {"name": "Intmax.compat_merge_of_compat", "content": "lemma compat_merge_of_compat :\n  (∀ π', π' ∈ πs → π <≅> π') → π <≅> (mergeR'' πs .initial)"}, {"name": "Intmax.prop6_general_aux", "content": "private lemma prop6_general_aux :\n  (∀ π π' : BalanceProof K₁ K₂ C Pi V, π ∈ πs ∧ π' ∈ πs → π <≅> π') →\n     IsLUB {π | π ∈ πs} (mergeR'' πs .initial)"}, {"name": "Intmax.prop6_general", "content": "lemma prop6_general (h : ∀ i : Fin πs.length,\n                     IsLUB {mergeR'' (πs.take i.1) .initial, πs[i]} (mergeR'' (πs.take (i.1 + 1)) .initial))\n  : IsLUB {π | π ∈ πs} (mergeR'' πs .initial)"}, {"name": "Intmax.batch?_neq_of_mem", "content": "lemma batch?_neq_of_mem {π₁k π₂k : Option ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V)}\n  (h₀ : π₁k ≠ .none ∧ π₂k ≠ .none)\n  (h : ¬(π₁k ≅ π₂k)) : (π₁k.get (Option.isSome_iff_ne_none.2 h₀.1)).2 ≠\n                       (π₂k.get (Option.isSome_iff_ne_none.2 h₀.2)).2"}, {"name": "Intmax.not_adversaryWon_attackGame_of_exists_LUB", "content": "omit Hinj in\nprivate theorem not_adversaryWon_attackGame_of_exists_LUB\n  (eqrequests : requests! = normalise requests)\n  (hValid     : ∀ request ∈ requests!, request.isValid)\n  (eqI        : I = (List.finRange (attackGameR requests! []).length).filter ((attackGameR requests! [])[·].isWithdrawalBlock))\n  (hI         : ∀ {i : Fin (attackGameR requests! []).length}, i ∈ I ↔ (attackGameR requests! [])[i].isWithdrawalBlock)\n  (hgetπ      : getπ =\n                λ i : {i : Fin (attackGameR requests! []).length // i ∈ I} ↦\n                  (i, getBalanceProof requests! hValid ⟨i.1.1, i.1.2⟩ (hI.1 i.2)))\n  (eqπs       : πs = I.attach.map getπ)\n  (hlen       : πs.length ≤ n)\n  (hπs        : ∀ (i : {i // i ∈ I}), (List.lookup i πs).isSome)\n  (eqπproofs  : πproofs = πs.map Prod.snd)\n  (πproofslen : πproofs.length = πs.length)\n  (isπ        : ∀ (i : Fin (List.length (attackGameR requests! [])))\n                  (h : (attackGameR requests! [])[i.1].isWithdrawalBlock),\n                  (attackGameR requests! [])[i.1].getWithdrawal h =\n                  (getBalanceProof requests! hValid i h).toBalanceF ((attackGameR requests! []).take i.1))\n  (contra     : ¬0 ≤ computeBalanceSum (attackGameR requests! []))\n  (existsLUB  : ∃ join, IsLUB {π | π ∈ πproofs} join) : False"}], "local_ctx": "import FVIntmax.AttackGame\n\nimport FVIntmax.Lemma3\n\nimport FVIntmax.Lemma4\n\nimport FVIntmax.Lemma5\n\nimport FVIntmax.Propositions\n\nimport FVIntmax.Request\n\nimport FVIntmax.Wheels\n\nimport FVIntmax.Wheels.AuthenticatedDictionary\n\nimport FVIntmax.Wheels.SignatureAggregation\n\nimport Mathlib\n\nnamespace Intmax\n\nopen Classical\n\nnoncomputable section Intmax\n\nnoncomputable section theorem1\n\nsection AttackGame\n\nvariable {Sigma Pi M : Type}\n         {C : Type} [Nonempty C]\n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {K₁ : Type} [Nonempty K₁] [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         {Kₚ : Type} [Nonempty Kₚ]\n         {Kₛ : Type} [Nonempty Kₛ]\n\ndef mergeR'' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  match πs with\n  | [] => acc\n  | π :: πs => Dict.Merge acc (mergeR'' πs π)\n\nsection MergeLemmas\n\nvariable {acc π : BalanceProof K₁ K₂ C Pi V} {πs πs : List (BalanceProof K₁ K₂ C Pi V)}\n\ndef mergeR''' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  πs.foldl Dict.Merge acc\n\nend MergeLemmas\n\ndef BalanceProof.compat (π₁ π₂ : BalanceProof K₁ K₂ C Pi V) : Prop :=\n  ∀ k, π₁ k ≠ none ∧ π₂ k ≠ none → π₁ k ≅ π₂ k\n\nnotation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂\n\nnotation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂\n\nsection Compat\n\nvariable {π₁ π₂ π₃ : BalanceProof K₁ K₂ C Pi V}\n\nend Compat\n\nsection MergeLemmas\n\nvariable {π acc : BalanceProof K₁ K₂ C Pi V}\n         {πs : List (BalanceProof K₁ K₂ C Pi V)}\n\nend MergeLemmas\n\nvariable [AD : ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n\nsection MergeLemmas\n\nvariable {π acc : BalanceProof K₁ K₂ C Pi V}\n         {πs : List (BalanceProof K₁ K₂ C Pi V)}\n\nend MergeLemmas\n\nvariable [SA : SignatureAggregation (C × K₁ × ExtraDataT) K₂ KₛT Sigma]\n         [Hinj : CryptoAssumptions.Injective (H (α := TransactionBatch K₁ K₂ V × ExtraDataT) (ω := (C × K₁ × ExtraDataT)))]\n         {requests : List (Request K₁ K₂ C Sigma Pi V)}\n         (isπ : isπ (normalise requests))\n\nopen Lean.Elab.Tactic Lean.Parser.Tactic in", "target_theorem": "theorem theorem1 : ¬adversaryWon (attackGame requests) :=", "ground_truth_proof": ":= λ contra ↦ by\n  /-\n    PAPER: Suppose an adversary and a challenger have interacted in Attack game 1.\n    We will show that either the resulting contract balance is positive (the adver-\n    sary lost the game), or the adversary has been able to either break the bind-\n    ing property of the authenticated dictionary scheme or found a collision of the\n    hash function H.\n  -/\n\n  /-\n    The attack game plays out the same regardless of validity of requests.\n  -/\n  rw [attackGame_eq_attackGameBlocks!_normalise, attackGameBlocks_eq_attackGameR] at contra\n  set requests! := normalise requests with eqRequests\n  /-\n    PAPER: Let B∗ = (Bi)i∈[n] be the contract state after the attack game\n  -/\n  set Bstar := attackGameR requests! [] with eqBstar\n  /-\n    As such, we can consider a state with only valid requests.\n  -/\n  have hValid : ∀ request ∈ (normalise requests), request.isValid := by unfold normalise; aesop\n  let n := Bstar.length\n  have hValidπ {i : Fin n} {h₀} {h₁} {π} (h : (requests![i.1]'h₀).getWithdrawal.get h₁ = π) :\n    π.Verify (M := (C × K₁ × ExtraDataT)) := by\n    rcases i with ⟨i, hi⟩\n    unfold Request.isValid at hValid\n    set request := requests![i] with eqRequest\n    specialize hValid request (by simp [eqRequest, requests!])\n    have hValid_withdrawal {h₀} (h : (requests![i]'h₀).getWithdrawal.isSome) :\n      requests![i]'h₀ matches .withdrawal .. := by\n      simp [Request.getWithdrawal] at h\n      aesop\n    aesop\n  have hn : n = requests!.length := by simp [n, eqBstar]\n  /-\n    PAPER: let I ⊆ [n] be the indices of the withdrawal blocks in B∗\n  -/\n  let I : List (Fin n) := (List.finRange n).filter (Bstar[·].isWithdrawalBlock)\n  have hI : ∀ i, i ∈ I ↔ Bstar[i].isWithdrawalBlock := by aesop\n  let getπ : {i : Fin n // i ∈ I} → ({i : Fin n // i ∈ I} × BalanceProof K₁ K₂ C Pi V) :=\n    λ i ↦\n      have lenEq : Bstar.length = n := by simp [n, eqBstar]\n      have hi₁ : i.1.1 < Bstar.length := by rw [lenEq]; exact i.1.2\n      (i, getBalanceProof requests! hValid ⟨i.1.1, hi₁⟩ ((hI i.1).1 i.2))\n  let πs : List ({i : Fin n // i ∈ I} × BalanceProof K₁ K₂ C Pi V) := I.attach.map getπ\n  have lenπs : πs.length ≤ n := by\n    simp [πs, I, n]\n    simp_rw [show Bstar.length = (List.finRange (List.length Bstar)).length by aesop]\n    exact List.length_filter_le _ _\n  have hπs : ∀ i : {i : Fin n // i ∈ I}, (πs.lookup i).isSome := λ i ↦\n    have : i ∈ I.attach := by rcases i with ⟨i, hi⟩; aesop\n    by simp [πs, getπ, List.lookup_graph _ this]\n  /-\n    PAPER: (πi)i∈I be the balance proofs used in the withdrawal request\n  -/\n  let πproofs := πs.map Prod.snd\n  have lenπs': πproofs.length = πs.length := by simp [πproofs]\n  have validπs {π : BalanceProof K₁ K₂ C Pi V} (h : π ∈ πproofs) :\n               π.Verify (M := (C × K₁ × ExtraDataT)) := by\n    simp [πs, πproofs] at h; rcases h with ⟨π, _, hπ⟩\n    exact hValidπ hπ\n  unfold Intmax.isπ at isπ; specialize isπ hValid; dsimp at isπ\n  /-\n    PAPER: The resulting contract balance can be computed by adding all deposited amounts and subtracting all withdrawn amounts:\n    NB we prove \n  -/\n  dsimp [adversaryWon] at contra; simp [computeBalance_eq_sum] at contra\n  /-\n    PAPER: We now have two possibilities, either the balance proofs (πi)i∈I have a join in Π or they don’t.\n  -/\n  by_cases eq : ∃ join : BalanceProof K₁ K₂ C Pi V, IsLUB {π | π ∈ πproofs} join\n  · -- PAPER: Suppose they have a join π ∈ Π. Then we have cf. `not_adversaryWon_attackGame_of_exists_LUB`.\n    exact not_adversaryWon_attackGame_of_exists_LUB\n            eqRequests\n            hValid\n            (I := I) (by simp only [Fin.getElem_fin, I])\n            (by simp only [hI]; intros; trivial)\n            (getπ := getπ) (by simp only [getπ])\n            (πs := πs) (by simp only [πs])\n            lenπs\n            hπs\n            (πproofs := πproofs) (by simp only [πproofs])\n            lenπs'\n            isπ\n            contra\n            eq\n  · /-\n      PAPER: Now, suppose the balance proofs (πi)i∈I do not have a join in Π.\n    -/\n    /-\n      PAPER: Let ik be the k′th index in I (so that I = {i1, i2, . . . , im}, where m = |I|).\n             Then, let (π′ k)k∈{1,...,m} be the balance proofs defined recursively as π′1 = πi1 and π′\n             k = Merge(π′k−1, πik ).\n\n      NB we define `πs'` to be an ordered sequence of balance proofs with the `.initial` balance proof\n      at the head position. Note further we use a slightly different merging scheme; one can\n      compare `mergeR` and `mergeR''`, where the former is the definition from the paper, while\n      the latter is the one we use.\n    -/\n    let πs' := List.Ico 0 (πs.length + 1) |>.map λ i ↦ mergeR'' (πproofs.take i) .initial\n    have lenπs'': πs'.length = πs.length + 1 := by simp [πs', lenπs']\n    /-\n      PAPER: Clearly, these merged balance proofs are valid, since each of the original bal-\n             ance proofs are valid (otherwise they wouldn’t be accepted by the rollup con-\n             tract), and since the merge of two valid balance proofs is again valid.\n\n      NB the relevant proof is `valid_mergeR''`.\n    -/\n    have hπs' : ∀ π ∈ πs', π.Verify (M := (C × K₁ × ExtraDataT)) := by\n      simp [πs', πproofs]; intros n hn\n      exact valid_mergeR'' (by simp; omega) (λ _ hx ↦ validπs hx)\n    have recπs' : ∀ {i : ℕ} (hi : i < πs'.length), πs'[i] = mergeR'' (πproofs.take i) .initial :=\n      by simp [πs', πproofs]; intros i hi; rw [List.getElem_Ico_of_lt hi]\n    let m := πs'.length.pred\n    have hm : m = πproofs.length := by simp [m, lenπs', lenπs'']\n    set π'ₘ := πs'[m]'(by simp [m]; omega) with eqπ'ₘ\n    simp only [not_exists] at eq\n    /-\n      With no withdrawal requests, one can derive an immediate contradiction by showing that\n      the `.initial` balance proof is the least upper bound, which had been assumed not to exist.\n    -/\n    by_cases isempty : πs.length = 0\n    · have : πproofs = [] := List.map_eq_nil_iff.2 (List.eq_nil_of_length_eq_zero isempty)\n      simp_rw [this] at eq\n      specialize eq .initial\n      simp at eq\n    · /-\n        PAPER: Now, we argue that there must be an index k ∈ {1, . . . , m} such that π′k is not the join\n        of π′k−1 and πik in Π, since if not, the final merged balance proof π′m would\n        be a join of (πi)i∈I (by Proposition 6), which we have assumed not to exist.\n      -/\n      have idx : ∃ i : {n : ℕ // 0 < n ∧ n < πs'.length},\n                 ¬IsLUB {\n                    πs'[i.1-1],\n                    πproofs[i.1-1]'(Nat.sub_one_lt_of_le i.2.1 (Nat.le_of_lt_add_one (by rw [lenπs', lenπs''.symm]; exact i.2.2)))\n                  } (πs'[i.1]'i.2.2) := by\n        by_contra c; simp at c\n        specialize eq π'ₘ; simp only [eqπ'ₘ] at eq\n        simp_rw [recπs' (i := m), show List.take m πproofs = πproofs by simp [hm]] at eq\n        have : IsLUB {π | π ∈ πproofs} (mergeR'' πproofs .initial) := by\n          apply prop6_general\n          rintro ⟨i, hi⟩\n          simp only\n          simp_rw [recπs'] at c; specialize c (i + 1) (by omega); simp at c\n          exact c\n        exact absurd this eq\n      rcases idx with ⟨⟨i, hi₁, hi₂⟩, lubi⟩; simp only at lubi\n      /-\n        PAPER: It then follows from Proposition 6 that there is a key (C, s) ∈ AD.C × K2 such that π′\n        k−1(C, s)̸ ≃ πik (C, s).\n      -/\n      let π₁! := πs'[i-1]'(by omega)\n      let π₂! := πproofs[i-1]'(by simp [hm.symm, m]; omega)\n      have eq₁ : ∃ key : {k : C × K₂ // π₁! k ≠ .none ∧ π₂! k ≠ .none}, ¬(((π₁! key.1) ≅ (π₂! key.1))) := by\n        simp only [π₁!, π₂!, id_eq, Int.Nat.cast_ofNat_Int, Int.reduceNeg, Nat.pred_eq_sub_one, Int.reduceAdd,\n          eq_mpr_eq_cast, Subtype.exists]\n        by_contra c; simp only [Int.reduceNeg, not_exists, Decidable.not_not] at c\n        apply proposition6_aux at c\n        simp_rw [recπs' (i := i)] at lubi\n        rw [Merge_split hi₁ (by omega)] at lubi\n        nth_rw 2 [recπs' (i := i - 1)] at c\n        contradiction\n      rcases eq₁ with ⟨⟨key, mem₁, mem₂⟩, hkey⟩\n      set π₁ := (π₁! key).get (Option.isSome_iff_ne_none.2 mem₁) with eqπ₁\n      set π₂ := (π₂! key).get (Option.isSome_iff_ne_none.2 mem₂) with eqπ₂\n      rcases key with ⟨c, s⟩\n      rcases π₁ with ⟨⟨π, salt⟩, t⟩\n      /-\n        PAPER: Also, since both balance proofs are valid, as remarked earlier, we have\n        AD.Verify(π, s, H(t, salt), C) and AD.Verify(π′, s, H(t′, salt′), C).\n      -/\n      have π₁valid : AD.Verify π s (H _ (t, salt)) c := by\n        have : π₁!.Verify (M := (C × K₁ × ExtraDataT)) := hπs' _ (by simp [π₁!])\n        simp [BalanceProof.Verify] at this; simp_rw [←Dict.mem_dict_iff_mem_keys] at this\n        specialize this c s (Option.isSome_iff_ne_none.2 mem₁)\n        convert this <;> rw [←eqπ₁]\n      rcases π₂ with ⟨⟨π', salt'⟩, t'⟩\n      have π₂valid : AD.Verify π' s (H _ (t', salt')) c := by\n        have : π₂!.Verify (M := (C × K₁ × ExtraDataT)) := validπs (by simp [π₂!])\n        simp [BalanceProof.Verify] at this; simp_rw [←Dict.mem_dict_iff_mem_keys] at this\n        specialize this c s (Option.isSome_iff_ne_none.2 mem₂)\n        convert this <;> rw [←eqπ₂]\n      have tneq : t ≠ t' := by apply batch?_neq_of_mem (by simp; exact ⟨mem₁, mem₂⟩) at hkey; simp [←eqπ₁, ←eqπ₂] at hkey; exact hkey\n      by_cases hashEq : H (ω := (C × K₁ × ExtraDataT)) (t, salt) = H _ (t', salt')\n      · /-\n          PAPER: It follows that that either H(t, salt) = H(t′, salt′)\n          meaning that we have found a hash collision\n\n          NB this is shown by contradiction as the hash function had been assumed injective.\n        -/\n        have : Function.Injective (H (ω := (C × K₁ × ExtraDataT))) :=\n          Intmax.CryptoAssumptions.Function.injective_of_CryptInjective (inj := Hinj) -- AXIOMATISED\n        have : (t, salt) = (t', salt') := this hashEq\n        injection this\n        contradiction\n      · /-\n          PAPER: or H(t, salt)̸ = H(t′, salt′), which means we have broken the binding property\n          of the authenticated dictionary scheme\n\n          NB this is shown by contradiction as breaking the binding property of the authenticated dictionary\n          had been assumed computationally infeasible and the `computationallyInfeasible_axiom`\n          yields the absolute impossibility of this ocurring.\n        -/\n        have binding := AD.binding\n        apply computationallyInfeasible_axiom at binding -- AXIOMATISED\n        simp at binding\n        specialize binding _ _ _ _ _ _ π₁valid _ _ _ _ π₂valid\n        rcases H (C × K₁ × ExtraDataT) (t, salt) with ⟨c, k₁, extra⟩\n        set hash₁ := H (C × K₁ × ExtraDataT) (t, salt) with eqhash₁\n        set hash₂ := H (C × K₁ × ExtraDataT) (t', salt') with eqhash₂\n        rcases hash₁ with ⟨c₁, k₁, ed₁⟩; rcases hash₂ with ⟨c₂, k₂, ed₂⟩\n        dsimp at binding hashEq\n        simp [binding] at hashEq", "nesting_depth": 14, "transitive_dep_count": 290, "subset_aristotle": false, "category": "Applied verif."}
{"id": 199, "thm_name": "Intmax.prop6_general", "thm_stmt": "lemma prop6_general (h : ∀ i : Fin πs.length,\n                     IsLUB {mergeR'' (πs.take i.1) .initial, πs[i]} (mergeR'' (πs.take (i.1 + 1)) .initial))\n  : IsLUB {π | π ∈ πs} (mergeR'' πs .initial)", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Theorem1.lean", "imports": ["import FVIntmax.Lemma5", "import FVIntmax.Wheels.AuthenticatedDictionary", "import FVIntmax.Request", "import FVIntmax.AttackGame", "import FVIntmax.Wheels", "import FVIntmax.Propositions", "import Mathlib", "import FVIntmax.Balance", "import FVIntmax.Lemma4", "import FVIntmax.Wheels.Dictionary", "import FVIntmax.Wheels.SignatureAggregation", "import FVIntmax.Lemma3"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "IsLUB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "List.take", "module": "Init.Data.List.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Lean.Parser.Tactic.replace", "module": "Init.Tactics"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "absurd", "module": "Init.Prelude"}, {"name": "List.foldl", "module": "Init.Prelude"}, {"name": "IsLeast", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "LE", "module": "Init.Prelude"}, {"name": "LE.le", "module": "Init.Prelude"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Preorder.toLE", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "List.concat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}, {"name": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂", "content": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂"}, {"name": "notation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂", "content": "notation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "Merge", "content": "def Merge (D₁ D₂ : Dict α ω) : Dict α ω := D\n  where D := λ x ↦ First (D₁ x) (D₂ x)"}, {"name": "First", "content": "def First (x₁ x₂ : Option α) : Option α :=\n  match x₁, x₂ with\n  | .some x, .none => .some x\n  | .some x, .some _ => .some x\n  | .none, .some y => .some y\n  | .none, .none => .none"}, {"name": "BalanceProof.initial", "content": "def BalanceProof.initial : BalanceProof K₁ K₂ C Pi V := λ _ ↦ .none"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "Setoid'", "content": "class Setoid' (X : Type) extends Preorder X where\n  isEquiv : IsEquivRel (X := X)"}, {"name": "IsEquivRel", "content": "def IsEquivRel {X : Type} [Preorder X] := ∀ a b : X, a ≤ b ↔ a ≅ b"}, {"name": "iso", "content": "def iso {X : Type} [Preorder X] (a b : X) := a ≤ b ∧ b ≤ a"}, {"name": "trivialPreorder", "content": "def trivialPreorder {α : Type} : Preorder α :=\n  {\n    lt := λ _ _ ↦ False\n    le := λ _ _ ↦ True\n    le_refl := by admit /- proof elided -/"}, {"name": "le", "content": "def le (v₁ v₂ : Vector α n) :=\n  ∀ x ∈ (v₁.1.zip v₂.1), x.1 ≤ x.2"}, {"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "List.foldl_assoc", "module": "Init.Data.List.Lemmas"}, {"name": "List.foldl_append", "module": "Init.Data.List.Lemmas"}, {"name": "Set.mem_insert_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_singleton_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "Set.mem_union", "module": "Mathlib.Data.Set.Basic"}, {"name": "and_imp", "module": "Init.SimpLemmas"}, {"name": "forall_eq", "module": "Init.PropLemmas"}, {"name": "forall_eq_or_imp", "module": "Init.PropLemmas"}, {"name": "List.concat_cons", "module": "Init.Data.List.Lemmas"}, {"name": "Classical.not_forall", "module": "Init.Classical"}, {"name": "Classical.not_imp", "module": "Init.Classical"}, {"name": "Fin.getElem_fin", "module": "Init.GetElem"}, {"name": "Fin.is_le'", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.lt_def", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.lt_of_le_of_lt", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.lt_of_lt_of_le", "module": "Init.Data.Fin.Lemmas"}, {"name": "Finset.exists_minimal", "module": "Mathlib.Order.Preorder.Finite"}, {"name": "Finset.filter_nonempty_iff", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Finset.mem_filter", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "List.getElem_mem", "module": "Init.GetElem"}, {"name": "List.getElem_of_mem", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_take_iff_getElem", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.take_add", "module": "Init.Data.List.Nat.TakeDrop"}, {"name": "List.take_concat_get", "module": "Init.Data.List.TakeDrop"}, {"name": "Nat.exists_eq_add_of_le", "module": "Init.Data.Nat.Lemmas"}, {"name": "Option.ne_none_iff_exists", "module": "Init.Data.Option.Lemmas"}, {"name": "Set.toFinset_setOf", "module": "Mathlib.Data.Fintype.Sets"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "eq_or_lt_of_le", "module": "Mathlib.Order.Basic"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "inf_of_le_left", "module": "Mathlib.Order.Lattice"}, {"name": "lt_add_iff_pos_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "lt_irrefl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_and", "module": "Init.SimpLemmas"}, {"name": "not_false_eq_true", "module": "Init.SimpLemmas"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "zero_lt_one", "module": "Mathlib.Algebra.Order.ZeroLEOne"}], "repo_lemmas": [{"name": "proposition6", "content": "lemma proposition6 [Setoid' Y] {D₁ D₂ : Dict X Y} :\n  (∃ join, IsLUB {D₁, D₂} join) ↔ ∀ x, D₁ x ≠ .none ∧ D₂ x ≠ .none → D₁ x ≅ D₂ x"}, {"name": "proposition4", "content": "lemma proposition4 [Setoid' X] {x y : Option X} :\n  (∃ join : Option X, IsLUB {x, y, .none} join) ↔ (x ≠ .none ∧ y ≠ .none → x ≅ y)"}, {"name": "proposition2", "content": "lemma proposition2 [Setoid' X] {x y : X} :\n  (∃ join : X, IsLUB {x, y} join) ↔ x ≅ y"}, {"name": "iso_rfl", "content": "@[simp, refl]\nlemma iso_rfl : a ≅ a"}, {"name": "iso_trans", "content": "@[trans]\nlemma iso_trans : (a ≅ b) → (b ≅ c) → a ≅ c"}, {"name": "proposition2'", "content": "lemma proposition2' [Setoid' X] {join x y : X} (h : IsLUB {x, y} join) :\n  (x ≅ join) ∧ y ≅ join"}, {"name": "proposition3'", "content": "lemma proposition3' : \n  (∃ join : X, IsLUB {x, y} join) ↔ (∃ join : Option X, IsLUB {.some x, .some y, .none} join)"}, {"name": "proposition5", "content": "lemma proposition5 [Preorder Y] {f g : X → Y} {join : X → Y} :\n  IsLUB {f, g} join ↔ ∀ x : X, IsLUB {f x, g x} (join x)"}, {"name": "proposition6_aux", "content": "lemma proposition6_aux [Setoid' Y] {D₁ D₂ : Dict X Y}\n  (h : ∀ k, D₁ k ≠ .none ∧ D₂ k ≠ .none → D₁ k ≅ D₂ k) : IsLUB {D₁, D₂} (Dict.Merge D₁ D₂)"}, {"name": "proposition6'", "content": "lemma proposition6' [Setoid' Y] {D₁ D₂ join : Dict X Y} (h : IsLUB {D₁, D₂} join) :\n  join ≅ Dict.Merge D₁ D₂"}, {"name": "iso_symm", "content": "@[symm]\nlemma iso_symm : (a ≅ b) ↔ b ≅ a"}, {"name": "proposition4'", "content": "lemma proposition4' [Setoid' X] {join x y : Option X} (h : IsLUB {x, y, .none} join) :\n  join ≅ Dict.First x y"}, {"name": "proposition5'", "content": "lemma proposition5' [Preorder Y] {f g h join' : X → Y}\n  (h₀ : IsLUB {f, g} join')\n  (h₁ : ∀ x, h x ≅ join' x) :\n  join' ≅ h"}, {"name": "discretePreorder_eq_equality_Pi_Prod_ExtraDataT", "content": "@[simp]\nlemma discretePreorder_eq_equality_Pi_Prod_ExtraDataT {a b : (Pi × ExtraDataT)} : a ≤ b"}, {"name": "BalanceProof.snd_discrete", "content": "@[simp]\nlemma BalanceProof.snd_discrete {x y : TransactionBatch K₁ K₂ V} :\n  @LE.le (TransactionBatch K₁ K₂ V) Preorder.toLE x y ↔ x = y"}, {"name": "Merge_assoc", "content": "lemma Merge_assoc {D₃ : Dict α ω} :\n  Merge (Merge D₁ D₂) D₃ = Merge D₁ (Merge D₂ D₃)"}, {"name": "none_iso_some", "content": "@[simp]\nlemma none_iso_some : (.none ≅ some x) ↔ False"}], "used_local_defs": [{"name": "Intmax.mergeR''", "content": "def mergeR'' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  match πs with\n  | [] => acc\n  | π :: πs => Dict.Merge acc (mergeR'' πs π)"}, {"name": "Intmax.mergeR'''", "content": "def mergeR''' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  πs.foldl Dict.Merge acc"}, {"name": "Intmax.BalanceProof.compat", "content": "def BalanceProof.compat (π₁ π₂ : BalanceProof K₁ K₂ C Pi V) : Prop :=\n  ∀ k, π₁ k ≠ none ∧ π₂ k ≠ none → π₁ k ≅ π₂ k"}], "used_local_lemmas": [{"name": "Intmax.mergeR''_eq_foldl", "content": "lemma mergeR''_eq_foldl :\n  mergeR'' πs acc = mergeR''' πs acc"}, {"name": "Intmax.mergeR''_cons", "content": "@[simp]\nlemma mergeR''_cons :\n  mergeR'' (π :: πs) acc = Dict.Merge acc (mergeR'' πs π)"}, {"name": "Intmax.mergeR''_append", "content": "@[simp]\nlemma mergeR''_append :\n  mergeR'' (πs ++ πs') acc = mergeR'' πs' (mergeR'' πs acc)"}, {"name": "Intmax.mergeR''_concat", "content": "@[simp]\nlemma mergeR''_concat {π} {πs : List (BalanceProof K₁ K₂ C Pi V)} {acc} :\n  mergeR'' (πs.concat π) acc = ((mergeR'' πs acc) <+> π)"}, {"name": "Intmax.existsLUB_iff_compat", "content": "lemma existsLUB_iff_compat :\n  (∃ join, IsLUB {π₁, π₂} join) ↔ π₁ <≅> π₂"}, {"name": "Intmax.compat_comm", "content": "lemma compat_comm : (π₁ <≅> π₂) ↔ π₂ <≅> π₁"}, {"name": "Intmax.merge_le", "content": "lemma merge_le (h₁ : π₁ ≤ π₃) (h₂ : π₂ ≤ π₃) : π₁ <+> π₂ ≤ π₃"}, {"name": "Intmax.isLUB_union_Merge_of_isLUB_isLUB_compat", "content": "lemma isLUB_union_Merge_of_isLUB_isLUB_compat {A B : Set (BalanceProof K₁ K₂ C Pi V)}\n  (h₁ : IsLUB A j₁) (h₂ : IsLUB B j₂) (h₃ : j₁ <≅> j₂) : IsLUB (A ∪ B) (j₁ <+> j₂)"}, {"name": "Intmax.mergeR''_eq_none'", "content": "@[simp]\nlemma mergeR''_eq_none' :\n  (mergeR'' πs acc) K = none ↔ acc K = none ∧ ∀ π ∈ πs, π K = none"}, {"name": "Intmax.merge_K", "content": "lemma merge_K : (π <+> acc) K = Dict.First (π K) (acc K)"}, {"name": "Intmax.mergeR''_eq_some", "content": "@[simp]\nlemma mergeR''_eq_some {x} :\n  acc K = some x → (mergeR'' πs acc) K = acc K"}, {"name": "Intmax.iso_K_merge_left_of_ne_none", "content": "lemma iso_K_merge_left_of_ne_none (h : π K ≠ none) : π K ≅ (π <+> acc) K"}, {"name": "Intmax.merge_left_none_eq_right", "content": "lemma merge_left_none_eq_right (h : π K = none) : (π <+> acc) K = acc K"}, {"name": "Intmax.merge_lem_aux", "content": "private lemma merge_lem_aux :\n  mergeR'' (π :: πs) acc = acc <+> π <+> (mergeR'' πs BalanceProof.initial)"}, {"name": "Intmax.merge_lem", "content": "lemma merge_lem :\n  mergeR'' (π :: πs) BalanceProof.initial = π <+> (mergeR'' πs BalanceProof.initial)"}, {"name": "Intmax.compat_lem", "content": "lemma compat_lem {π π' π'': BalanceProof K₁ K₂ C Pi V} :\n  π <≅> π' → π <≅> π'' → π <≅> (π' <+> π'')"}, {"name": "Intmax.compat_merge_of_compat", "content": "lemma compat_merge_of_compat :\n  (∀ π', π' ∈ πs → π <≅> π') → π <≅> (mergeR'' πs .initial)"}, {"name": "Intmax.prop6_general_aux", "content": "private lemma prop6_general_aux :\n  (∀ π π' : BalanceProof K₁ K₂ C Pi V, π ∈ πs ∧ π' ∈ πs → π <≅> π') →\n     IsLUB {π | π ∈ πs} (mergeR'' πs .initial)"}], "local_ctx": "import FVIntmax.AttackGame\n\nimport FVIntmax.Lemma3\n\nimport FVIntmax.Lemma4\n\nimport FVIntmax.Lemma5\n\nimport FVIntmax.Propositions\n\nimport FVIntmax.Request\n\nimport FVIntmax.Wheels\n\nimport FVIntmax.Wheels.AuthenticatedDictionary\n\nimport FVIntmax.Wheels.SignatureAggregation\n\nimport Mathlib\n\nnamespace Intmax\n\nopen Classical\n\nnoncomputable section Intmax\n\nnoncomputable section theorem1\n\nsection AttackGame\n\nvariable {Sigma Pi M : Type}\n         {C : Type} [Nonempty C]\n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {K₁ : Type} [Nonempty K₁] [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         {Kₚ : Type} [Nonempty Kₚ]\n         {Kₛ : Type} [Nonempty Kₛ]\n\ndef mergeR'' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  match πs with\n  | [] => acc\n  | π :: πs => Dict.Merge acc (mergeR'' πs π)\n\nsection MergeLemmas\n\nvariable {acc π : BalanceProof K₁ K₂ C Pi V} {πs πs : List (BalanceProof K₁ K₂ C Pi V)}\n\ndef mergeR''' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  πs.foldl Dict.Merge acc\n\nend MergeLemmas\n\ndef BalanceProof.compat (π₁ π₂ : BalanceProof K₁ K₂ C Pi V) : Prop :=\n  ∀ k, π₁ k ≠ none ∧ π₂ k ≠ none → π₁ k ≅ π₂ k\n\nnotation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂\n\nnotation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂\n\nsection Compat\n\nvariable {π₁ π₂ π₃ : BalanceProof K₁ K₂ C Pi V}\n\nend Compat\n\nsection MergeLemmas\n\nvariable {π acc : BalanceProof K₁ K₂ C Pi V}\n         {πs : List (BalanceProof K₁ K₂ C Pi V)}\n\nend MergeLemmas\n\nvariable [AD : ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n\nsection MergeLemmas\n\nvariable {π acc : BalanceProof K₁ K₂ C Pi V}\n         {πs : List (BalanceProof K₁ K₂ C Pi V)}", "target_theorem": "lemma prop6_general (h : ∀ i : Fin πs.length,\n                     IsLUB {mergeR'' (πs.take i.1) .initial, πs[i]} (mergeR'' (πs.take (i.1 + 1)) .initial))\n  : IsLUB {π | π ∈ πs} (mergeR'' πs .initial) :=", "ground_truth_proof": ":= by\n  replace h : ∀ (i : ℕ) (h : i < πs.length),\n                IsLUB {mergeR'' (List.take i πs) .initial, πs[↑i]}\n                      (mergeR'' (List.take (i + 1) πs) .initial) := λ i hi ↦ h ⟨i, hi⟩\n  apply prop6_general_aux\n  by_contra contra\n  simp at contra\n  let min₁ : Finset (Fin πs.length) := {n | ∃ i, n < i ∧ ¬(πs[n] <≅> πs[i])}\n  have wa_ne : min₁.Nonempty := by\n    rcases contra with ⟨π, ⟨π_in_πs, ⟨π', ⟨π'_in_πs, h⟩⟩⟩⟩\n    obtain ⟨π_ind, π_ind_lim, π_in_πs⟩ := List.getElem_of_mem π_in_πs\n    obtain ⟨π'_ind, π'_ind_lim, π'_in_πs⟩ := List.getElem_of_mem π'_in_πs\n    by_cases h' : π_ind < π'_ind\n    · simp_all only [List.getElem_mem, Set.toFinset_setOf, gt_iff_lt, Fin.getElem_fin, min₁]\n      rw [Finset.filter_nonempty_iff]; simp only [Finset.mem_univ, true_and]\n      use ⟨π_ind, by omega⟩; use ⟨π'_ind, by omega⟩\n      aesop\n    · rewrite [not_lt] at h'\n      rcases eq_or_lt_of_le h' with h' | h'\n      · exfalso; aesop\n      · simp_all only [List.getElem_mem, Set.toFinset_setOf, gt_iff_lt, Fin.getElem_fin, min₁]\n        rw [Finset.filter_nonempty_iff]; simp only [Finset.mem_univ, true_and]\n        use ⟨π'_ind, by omega⟩; use ⟨π_ind, by omega⟩\n        subst π_in_πs; subst π'_in_πs\n        simp [h']; rw [compat_comm]; assumption\n  have wa_min : ∃ idx, idx ∈ min₁ ∧ ∀ k ∈ min₁, idx ≤ k := by\n    rcases Finset.exists_minimal min₁ wa_ne with ⟨idx, h₁, h₂⟩; use idx; aesop\n  rcases wa_min with ⟨idx, idx_min_wa⟩\n  let π := πs[idx]\n  let min₂ : Finset (Fin πs.length) := {idx' | idx < idx' ∧ ¬(πs[idx'] <≅> πs[idx])}\n  have min₂_ne : min₂.Nonempty := by\n    rw [Finset.filter_nonempty_iff]; simp_rw [compat_comm]; aesop\n  have min₂_min : ∃ idx', idx' ∈ min₂ ∧ ∀ k ∈ min₂, idx' ≤ k := by\n    rcases Finset.exists_minimal min₂ min₂_ne with ⟨idx, h₁, h₂⟩; use idx; aesop\n  rcases min₂_min with ⟨idx', idx'_min_min₂⟩\n  have idx_lt_idx' : idx < idx' := by simp [min₂] at idx'_min_min₂; tauto\n  have eq₁' : ∀ i, idx ≤ i ∧ i < idx' → πs[i] <≅> πs[idx] := by\n    by_contra h\n    simp only [Fin.getElem_fin, and_imp, not_forall, Classical.not_imp] at h\n    rcases h with ⟨i, h, h', h''⟩\n    have : i ∈ min₂ := by\n      dsimp [min₂]\n      simp only [Finset.mem_filter, Finset.mem_univ, h'', not_false_eq_true, and_true, true_and]\n      rcases eq_or_lt_of_le h <;> aesop\n    exact absurd (Fin.lt_of_le_of_lt (idx'_min_min₂.2 i this) h') (lt_irrefl _)\n  have eq₁'' : ∀ i, i < idx → πs[i] <≅> πs[idx] := by\n    intros i h\n    by_contra contra\n    have : i ∈ min₁ := by\n      dsimp [min₁]\n      simp only [gt_iff_lt, Set.toFinset_setOf, Finset.mem_filter, Finset.mem_univ, true_and]\n      use idx\n      exact ⟨h, contra⟩\n    exact absurd (Fin.lt_of_lt_of_le h (idx_min_wa.2 i this)) (lt_irrefl _)\n  have eq₁ : ∀ i, i < idx' → πs[i] <≅> πs[idx] := by\n    intros i h; by_cases h' : idx ≤ i <;> aesop\n  have eq₂ : ¬(πs[idx] <≅> πs[idx']) := by\n    rw [compat_comm]; aesop\n  have eq₃ : ∃ k : C × K₂, ¬(πs[idx] k ≅ πs[idx'] k) ∧ πs[idx] k ≠ .none ∧ πs[idx'] k ≠ .none := by\n    unfold BalanceProof.compat at eq₂\n    simp only [Fin.getElem_fin, ne_eq, and_imp, Prod.forall, not_forall, Classical.not_imp] at eq₂\n    rcases eq₂ with ⟨k₁, k₂, h₁, h₂, h⟩\n    use ⟨k₁, k₂⟩\n    exact ⟨h, h₁, h₂⟩\n  rcases eq₃ with ⟨k, eq₃⟩\n  have eq₄ : ∀ i, i < idx' → πs[idx] k ≅ πs[i] k ∨ πs[i] k = .none := by\n    intros i h\n    unfold BalanceProof.compat at eq₁ eq₃\n    simp only [Fin.getElem_fin] at eq₃\n    specialize eq₁ i h k\n    rw [iso_symm]\n    by_cases h' : πs[i] k = .none <;> tauto\n  specialize h idx'.1 (by simp)\n  have eq₅ : mergeR'' (πs.take (idx.1 + 1)) .initial k ≅ πs[idx] k := by\n    have eq₆' : ∀ i, i < idx.1 + 1 ∧ i < πs.length →\n      mergeR'' (List.take i πs) .initial k ≅ (πs[idx]) k ∨ mergeR'' (List.take i πs) .initial k = .none := by\n      intros i h\n      induction i with\n      | zero => right; rfl\n      | succ i ih =>\n        have : i < πs.length := by linarith\n        rw [←List.take_concat_get _ _ this, mergeR''_concat]\n        rcases ih ⟨by linarith, by linarith⟩ with ih | ih\n        · left\n          refine iso_trans ?_ ih\n          rw [iso_symm]\n          apply iso_K_merge_left_of_ne_none\n          by_contra h\n          rw [h] at ih\n          obtain ⟨_, h⟩ := Option.ne_none_iff_exists.1 eq₃.2.1\n          rwa [←h, none_iso_some] at ih\n        · rw [merge_left_none_eq_right ih, iso_symm,\n              show πs[i] = πs[((⟨i, by linarith⟩) : Fin πs.length)] from rfl]\n          apply eq₄\n          simp only [Fin.lt_def] at idx_lt_idx' ⊢\n          exact lt_trans (by omega) idx_lt_idx'\n    rw [←List.take_concat_get (h := idx.2), mergeR''_concat]\n    rcases eq₆' idx.1 ⟨by simp only [lt_add_iff_pos_right, zero_lt_one], idx.2⟩ with h | h\n    · refine iso_trans ?_ h\n      rw [iso_symm]\n      apply iso_K_merge_left_of_ne_none\n      by_contra h'\n      obtain ⟨_, h''⟩ := Option.ne_none_iff_exists.1 eq₃.2.1\n      rwa [h', ←h'', none_iso_some] at h\n    · rw [merge_left_none_eq_right h]\n      rfl\n  have eq₆ : mergeR'' (πs.take (idx.1 + 1)) .initial k ≅ mergeR'' (πs.take idx'.1) .initial k := by\n    have : idx.1 + 1 ≤ idx'.1 := by\n      rw [Fin.lt_def] at idx_lt_idx'\n      linarith\n    rcases Nat.exists_eq_add_of_le this with ⟨i, h⟩\n    rw [h, List.take_add (m := idx.1 + 1), mergeR''_append]\n    have : ∃ r, mergeR'' (List.take (idx.1 + 1) πs) BalanceProof.initial k = .some r := by\n      by_contra h\n      simp at h\n      have : mergeR'' (List.take (idx.1 + 1) πs) BalanceProof.initial k = .none := by\n        by_contra h'\n        have h' := Option.ne_none_iff_exists.1 h'\n        rcases h' with ⟨x, h'⟩\n        specialize h x.1.1 x.1.2 x.2\n        apply h\n        rw [←h']\n      rw [this] at eq₅\n      rcases Option.ne_none_iff_exists.1 eq₃.2.1 with ⟨x, h''⟩\n      rw [←h'', none_iso_some] at eq₅\n      exact eq₅\n    rcases this with ⟨_, h⟩\n    rw [mergeR''_eq_some h]\n  have eq₇ : (πs[idx]) k ≅ mergeR'' (List.take idx'.1 πs) BalanceProof.initial k :=\n    iso_trans (eq₅.symm) eq₆\n  have eq₈ : ¬(mergeR'' (πs.take idx'.1) .initial <≅> πs[idx']) := by\n    unfold BalanceProof.compat\n    simp only [ne_eq, mergeR''_eq_none', not_and, not_forall, Classical.not_imp, Fin.getElem_fin, and_imp]\n    use k; simp at eq₃\n    simp [eq₃, BalanceProof.initial]\n    refine' ⟨_, λ contra' ↦ absurd (iso_trans eq₇ contra') eq₃.1⟩\n    use π; simp [π]\n    refine' ⟨_, eq₃.2.1⟩\n    rw [List.mem_take_iff_getElem]\n    simp only [Fin.is_le', inf_of_le_left]\n    use idx.1\n    simp [min₂] at idx'_min_min₂\n    use idx'_min_min₂.1.1\n  unfold BalanceProof.compat at eq₈\n  rw [←proposition6] at eq₈\n  exact absurd (by tauto) eq₈", "nesting_depth": 9, "transitive_dep_count": 120, "subset_aristotle": false, "category": "Applied verif."}
{"id": 200, "thm_name": "Intmax.senderReceiver_transactionsInBlocks", "thm_stmt": "lemma senderReceiver_transactionsInBlocks {s r v} {s' r' v'} {eq₁ eq₂} {i}\n  (h₀ : i < (TransactionsInBlocks π bs).length)\n  (h₁ : (TransactionsInBlocks π bs)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' bs)[i]'(by blast with π) = ⟨(s', r', v'), eq₂⟩) :\n  s = s' ∧ r = r'", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Lemma4.lean", "imports": ["import FVIntmax.Balance"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "TransactionsInBlocks", "content": "def TransactionsInBlocks\n  (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) : List (Τ K₁ K₂ V) :=\n  (bs.map (TransactionsInBlock π)).flatten"}, {"name": "TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}, {"name": "TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "v'", "content": "def v' (v : V₊) (b : S K₁ K₂ V) (s : Kbar K₁ K₂) : V₊ :=\n  match h : s with\n  | .Source => v\n  | .key _  => ⟨v ⊓ b s, by admit /- proof elided -/\n  ⟩"}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "List.ext_get_iff", "module": "Mathlib.Data.List.Basic"}], "repo_lemmas": [{"name": "receiver_transactionsInBlocks", "content": "lemma receiver_transactionsInBlocks {bs : List (Block K₁ K₂ C Sigma V)}\n                                    {π₁ π₂ : BalanceProof K₁ K₂ C Pi V} :\n  (TransactionsInBlocks π₁ bs).map (λ s ↦ s.1.2.1) =\n  (TransactionsInBlocks π₂ bs).map (λ s ↦ s.1.2.1)"}, {"name": "receiver_transactionsInBlock", "content": "lemma receiver_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.2.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.2.1)"}, {"name": "length_transactionsInBlock", "content": "lemma length_transactionsInBlock :\n  (TransactionsInBlock π₁ b).length = (TransactionsInBlock π₂ b).length"}, {"name": "length_TransactionsInBlock_transfer", "content": "lemma length_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }} :\n  ∀ (π₁ π₂ : BalanceProof K₁ K₂ C Pi V),\n    (TransactionsInBlock_transfer π₁ b).length =\n    (TransactionsInBlock_transfer π₂ b).length"}, {"name": "length_transactionsInBlocks", "content": "lemma length_transactionsInBlocks {bs : List (Block K₁ K₂ C Sigma V)}\n                                  {π₁ π₂ : BalanceProof K₁ K₂ C Pi V} :\n  (TransactionsInBlocks π₁ bs).length = (TransactionsInBlocks π₂ bs).length"}, {"name": "sender_transactionsInBlocks", "content": "lemma sender_transactionsInBlocks {bs : List (Block K₁ K₂ C Sigma V)}\n                                  {π₁ π₂ : BalanceProof K₁ K₂ C Pi V} :\n  (TransactionsInBlocks π₁ bs).map (λ s ↦ s.1.1) =\n  (TransactionsInBlocks π₂ bs).map (λ s ↦ s.1.1)"}, {"name": "sender_transactionsInBlock", "content": "lemma sender_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.1)"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import FVIntmax.Balance\n\nnamespace Intmax\n\nopen Mathlib\n\nnoncomputable section Lemma4\n\nsection HicSuntDracones\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type} [AddCommGroup V] [Lattice V]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nsection HelperFunctionsToAppeaseLean\n\nopen Mathlib\n\nopen Lean.Elab.Tactic in\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nend\n\nend HelperFunctionsToAppeaseLean", "target_theorem": "lemma senderReceiver_transactionsInBlocks {s r v} {s' r' v'} {eq₁ eq₂} {i}\n  (h₀ : i < (TransactionsInBlocks π bs).length)\n  (h₁ : (TransactionsInBlocks π bs)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' bs)[i]'(by blast with π) = ⟨(s', r', v'), eq₂⟩) :\n  s = s' ∧ r = r' :=", "ground_truth_proof": ":= by\n  have eq₁ := sender_transactionsInBlocks (Sigma := Sigma) (bs := bs) (π₁ := π) (π₂ := π')\n  have eq₂ := receiver_transactionsInBlocks (Sigma := Sigma) (bs := bs) (π₁ := π) (π₂ := π')\n  simp [List.ext_get_iff, length_transactionsInBlocks (π₁ := π') (π₂ := π)] at eq₁ eq₂\n  specialize eq₁ i h₀ h₀; specialize eq₂ i h₀ h₀\n  aesop", "nesting_depth": 7, "transitive_dep_count": 49, "subset_aristotle": false, "category": "Applied verif."}
{"id": 201, "thm_name": "Intmax.isLUB_union_Merge_of_isLUB_isLUB_compat", "thm_stmt": "lemma isLUB_union_Merge_of_isLUB_isLUB_compat {A B : Set (BalanceProof K₁ K₂ C Pi V)}\n  (h₁ : IsLUB A j₁) (h₂ : IsLUB B j₂) (h₃ : j₁ <≅> j₂) : IsLUB (A ∪ B) (j₁ <+> j₂)", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Theorem1.lean", "imports": ["import FVIntmax.Lemma5", "import FVIntmax.Wheels.AuthenticatedDictionary", "import FVIntmax.Request", "import FVIntmax.AttackGame", "import FVIntmax.Wheels", "import FVIntmax.Propositions", "import Mathlib", "import FVIntmax.Balance", "import FVIntmax.Lemma4", "import FVIntmax.Wheels.SignatureAggregation", "import FVIntmax.Lemma3"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "IsLUB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "IsLeast", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "LE", "module": "Init.Prelude"}, {"name": "LE.le", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Preorder.toLE", "module": "Mathlib.Order.Defs.PartialOrder"}], "used_repo_defs": [{"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}, {"name": "notation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂", "content": "notation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂"}, {"name": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂", "content": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "Merge", "content": "def Merge (D₁ D₂ : Dict α ω) : Dict α ω := D\n  where D := λ x ↦ First (D₁ x) (D₂ x)"}, {"name": "First", "content": "def First (x₁ x₂ : Option α) : Option α :=\n  match x₁, x₂ with\n  | .some x, .none => .some x\n  | .some x, .some _ => .some x\n  | .none, .some y => .some y\n  | .none, .none => .none"}, {"name": "Setoid'", "content": "class Setoid' (X : Type) extends Preorder X where\n  isEquiv : IsEquivRel (X := X)"}, {"name": "IsEquivRel", "content": "def IsEquivRel {X : Type} [Preorder X] := ∀ a b : X, a ≤ b ↔ a ≅ b"}, {"name": "iso", "content": "def iso {X : Type} [Preorder X] (a b : X) := a ≤ b ∧ b ≤ a"}, {"name": "trivialPreorder", "content": "def trivialPreorder {α : Type} : Preorder α :=\n  {\n    lt := λ _ _ ↦ False\n    le := λ _ _ ↦ True\n    le_refl := by admit /- proof elided -/"}, {"name": "le", "content": "def le (v₁ v₂ : Vector α n) :=\n  ∀ x ∈ (v₁.1.zip v₂.1), x.1 ≤ x.2"}, {"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "Set.mem_insert_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_singleton_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "Set.mem_union", "module": "Mathlib.Data.Set.Basic"}, {"name": "and_imp", "module": "Init.SimpLemmas"}, {"name": "forall_eq", "module": "Init.PropLemmas"}, {"name": "forall_eq_or_imp", "module": "Init.PropLemmas"}], "repo_lemmas": [{"name": "proposition6", "content": "lemma proposition6 [Setoid' Y] {D₁ D₂ : Dict X Y} :\n  (∃ join, IsLUB {D₁, D₂} join) ↔ ∀ x, D₁ x ≠ .none ∧ D₂ x ≠ .none → D₁ x ≅ D₂ x"}, {"name": "proposition4", "content": "lemma proposition4 [Setoid' X] {x y : Option X} :\n  (∃ join : Option X, IsLUB {x, y, .none} join) ↔ (x ≠ .none ∧ y ≠ .none → x ≅ y)"}, {"name": "proposition2", "content": "lemma proposition2 [Setoid' X] {x y : X} :\n  (∃ join : X, IsLUB {x, y} join) ↔ x ≅ y"}, {"name": "iso_rfl", "content": "@[simp, refl]\nlemma iso_rfl : a ≅ a"}, {"name": "iso_trans", "content": "@[trans]\nlemma iso_trans : (a ≅ b) → (b ≅ c) → a ≅ c"}, {"name": "proposition2'", "content": "lemma proposition2' [Setoid' X] {join x y : X} (h : IsLUB {x, y} join) :\n  (x ≅ join) ∧ y ≅ join"}, {"name": "proposition3'", "content": "lemma proposition3' : \n  (∃ join : X, IsLUB {x, y} join) ↔ (∃ join : Option X, IsLUB {.some x, .some y, .none} join)"}, {"name": "proposition5", "content": "lemma proposition5 [Preorder Y] {f g : X → Y} {join : X → Y} :\n  IsLUB {f, g} join ↔ ∀ x : X, IsLUB {f x, g x} (join x)"}, {"name": "proposition6_aux", "content": "lemma proposition6_aux [Setoid' Y] {D₁ D₂ : Dict X Y}\n  (h : ∀ k, D₁ k ≠ .none ∧ D₂ k ≠ .none → D₁ k ≅ D₂ k) : IsLUB {D₁, D₂} (Dict.Merge D₁ D₂)"}, {"name": "proposition6'", "content": "lemma proposition6' [Setoid' Y] {D₁ D₂ join : Dict X Y} (h : IsLUB {D₁, D₂} join) :\n  join ≅ Dict.Merge D₁ D₂"}, {"name": "iso_symm", "content": "@[symm]\nlemma iso_symm : (a ≅ b) ↔ b ≅ a"}, {"name": "proposition4'", "content": "lemma proposition4' [Setoid' X] {join x y : Option X} (h : IsLUB {x, y, .none} join) :\n  join ≅ Dict.First x y"}, {"name": "proposition5'", "content": "lemma proposition5' [Preorder Y] {f g h join' : X → Y}\n  (h₀ : IsLUB {f, g} join')\n  (h₁ : ∀ x, h x ≅ join' x) :\n  join' ≅ h"}, {"name": "discretePreorder_eq_equality_Pi_Prod_ExtraDataT", "content": "@[simp]\nlemma discretePreorder_eq_equality_Pi_Prod_ExtraDataT {a b : (Pi × ExtraDataT)} : a ≤ b"}, {"name": "BalanceProof.snd_discrete", "content": "@[simp]\nlemma BalanceProof.snd_discrete {x y : TransactionBatch K₁ K₂ V} :\n  @LE.le (TransactionBatch K₁ K₂ V) Preorder.toLE x y ↔ x = y"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Intmax.existsLUB_iff_compat", "content": "lemma existsLUB_iff_compat :\n  (∃ join, IsLUB {π₁, π₂} join) ↔ π₁ <≅> π₂"}, {"name": "Intmax.merge_le", "content": "lemma merge_le (h₁ : π₁ ≤ π₃) (h₂ : π₂ ≤ π₃) : π₁ <+> π₂ ≤ π₃"}], "local_ctx": "import FVIntmax.AttackGame\n\nimport FVIntmax.Lemma3\n\nimport FVIntmax.Lemma4\n\nimport FVIntmax.Lemma5\n\nimport FVIntmax.Propositions\n\nimport FVIntmax.Request\n\nimport FVIntmax.Wheels\n\nimport FVIntmax.Wheels.AuthenticatedDictionary\n\nimport FVIntmax.Wheels.SignatureAggregation\n\nimport Mathlib\n\nnamespace Intmax\n\nopen Classical\n\nnoncomputable section Intmax\n\nnoncomputable section theorem1\n\nsection AttackGame\n\nvariable {Sigma Pi M : Type}\n         {C : Type} [Nonempty C]\n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {K₁ : Type} [Nonempty K₁] [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         {Kₚ : Type} [Nonempty Kₚ]\n         {Kₛ : Type} [Nonempty Kₛ]\n\nsection MergeLemmas\n\nvariable {acc π : BalanceProof K₁ K₂ C Pi V} {πs πs : List (BalanceProof K₁ K₂ C Pi V)}\n\nend MergeLemmas\n\nnotation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂\n\nnotation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂\n\nsection Compat\n\nvariable {π₁ π₂ π₃ : BalanceProof K₁ K₂ C Pi V}\n\nend Compat", "target_theorem": "lemma isLUB_union_Merge_of_isLUB_isLUB_compat {A B : Set (BalanceProof K₁ K₂ C Pi V)}\n  (h₁ : IsLUB A j₁) (h₂ : IsLUB B j₂) (h₃ : j₁ <≅> j₂) : IsLUB (A ∪ B) (j₁ <+> j₂) :=", "ground_truth_proof": ":= by\n  have h₃'' := h₃\n  obtain ⟨j, h₃⟩ := existsLUB_iff_compat.2 h₃\n  split_ands\n  · simp only [IsLUB, IsLeast, upperBounds, Set.mem_insert_iff, Set.mem_singleton_iff,\n    forall_eq_or_imp, forall_eq, Set.mem_setOf_eq, lowerBounds, and_imp, Set.mem_union] at h₁ h₂ h₃ ⊢\n    rcases h₁ with ⟨h₁, h₁'⟩\n    rcases h₂ with ⟨h₂, h₂'⟩\n    rintro D₁ (hD₁ | hD₂)\n    · simp [-Prod.forall, (·≤·)]\n      intros x\n      unfold Dict.Merge Dict.Merge.D Dict.First\n      specialize h₁ hD₁\n      simp [-Prod.forall, (·≤·)] at h₁\n      specialize h₁ x\n      set d₁ := D₁ x with eqX\n      set d₂ := j₁ x with eqY\n      set d₃ := j₂ x with eqZ\n      rcases d₁ with _ | d₁ <;> rcases d₂ with _ | d₂ <;> rcases d₃ with _ | d₃ <;> simp\n      · simp at h₁\n      · simp at h₁\n      · simp at h₁\n        exact h₁\n      · simp at h₁\n        exact h₁\n    · simp only [LE.le, discretePreorder_eq_equality_Pi_Prod_ExtraDataT, BalanceProof.snd_discrete,]\n      intros x\n      unfold Dict.Merge Dict.Merge.D Dict.First\n      have eq₂ : D₁ ≤ j₂ := h₂ hD₂\n      simp [-Prod.forall, (·≤·)] at eq₂\n      specialize eq₂ x\n      set d₁ := D₁ x with eqX\n      set d₂ := j₁ x with eqY\n      set d₃ := j₂ x with eqZ\n      rcases d₁ with _ | d₁ <;> rcases d₂ with _ | d₂ <;> rcases d₃ with _ | d₃ <;> simp\n      · simp at eq₂\n      · simp at eq₂\n        exact eq₂\n      · simp at eq₂\n      · simp at eq₂\n        specialize h₃'' x\n        rw [←eqY, ←eqZ] at h₃''\n        simp at h₃''\n        exact iso_trans eq₂ h₃''.symm\n  · exact λ _ hπ ↦ merge_le (h₁.right λ _ hd ↦ hπ (by tauto))\n                            (h₂.right λ _ hd ↦ hπ (by tauto))", "nesting_depth": 7, "transitive_dep_count": 52, "subset_aristotle": false, "category": "Applied verif."}
{"id": 202, "thm_name": "Intmax.Vec.le_trans", "thm_stmt": "lemma le_trans (h₁ : v₁ ≤ v₂) (h₂ : v₂ ≤ v₃) : v₁ ≤ v₃", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Wheels.lean", "imports": ["import Mathlib.Logic.Embedding.Basic", "import FVIntmax.Wheels.Wheels", "import Mathlib.Tactic", "import Mathlib.Algebra.Order.Ring.Unbundled.Nonneg", "import Mathlib.Data.Finmap", "import Mathlib.Data.Set.Image", "import Mathlib.Data.Finite.Defs", "import Mathlib.Data.List.Intervals"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Intmax.CryptoAssumptions.Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "Intmax.Vec.le", "content": "def le (v₁ v₂ : Vector α n) :=\n  ∀ x ∈ (v₁.1.zip v₂.1), x.1 ≤ x.2"}], "used_local_lemmas": [], "local_ctx": "import Mathlib.Algebra.Order.Ring.Unbundled.Nonneg\n\nimport Mathlib.Data.Finite.Defs\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Intervals\n\nimport Mathlib.Data.Set.Image\n\nimport Mathlib.Logic.Embedding.Basic\n\nimport Mathlib.Tactic\n\nimport FVIntmax.Wheels.Wheels\n\nnamespace Intmax\n\nnamespace CryptoAssumptions\n\nsection Hashing\n\nclass Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)\n\nend Hashing\n\nend CryptoAssumptions\n\nsection UniquelyIndexed\n\nprefix:max \"!\" => UniqueTokenT\n\nnamespace UniquelyIndexed\n\nend UniquelyIndexed\n\nend UniquelyIndexed\n\nsection NonNeg\n\npostfix:max \"₊\" => NonNeg\n\nsection Nonneg\n\nvariable {α : Type} [PreWithZero α] {v : α₊}\n\nend Nonneg\n\nend NonNeg\n\nsection Order\n\nsection VectorPreorder\n\nopen Mathlib\n\nvariable {α : Type} [Preorder α] {n : ℕ}\n         {v₁ v₂ v₃ : Vector α n}\n\nnamespace Vec\n\ndef le (v₁ v₂ : Vector α n) :=\n  ∀ x ∈ (v₁.1.zip v₂.1), x.1 ≤ x.2", "target_theorem": "lemma le_trans (h₁ : v₁ ≤ v₂) (h₂ : v₂ ≤ v₃) : v₁ ≤ v₃ :=", "ground_truth_proof": ":= by\n  dsimp [(·≤·), le] at *\n  rcases v₁ with ⟨l₁, hl₁⟩\n  rcases v₂ with ⟨l₂, hl₂⟩\n  rcases v₃ with ⟨l₃, hl₃⟩\n  simp at *\n  induction' l₂ with hd₂ tl₂ ih generalizing l₁ l₃ n\n  · rcases l₃; exact h₁; simp_all; omega\n  · rcases l₃ with _ | ⟨hd₃, tl₃⟩ <;> [simp; skip]\n    rcases l₁ with _ | ⟨hd₁, tl₁⟩ <;> simp at *\n    intros a b h\n    rcases h with ⟨h₃, h₄⟩ | h\n    · transitivity hd₂ <;> tauto\n    · specialize @ih tl₁.length tl₁ rfl (by aesop) tl₃ (by aesop)\n      aesop", "nesting_depth": 2, "transitive_dep_count": 3, "subset_aristotle": false, "category": "Applied verif."}
{"id": 203, "thm_name": "Intmax.monotone_TransactionsInBlocksFixed", "thm_stmt": "lemma monotone_TransactionsInBlocksFixed :\n  Monotone λ (π : BalanceProof K₁ K₂ C Pi V) ↦ TransactionsInBlocksFixed π Bstar", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Lemma4.lean", "imports": ["import FVIntmax.Wheels.Dictionary", "import FVIntmax.Wheels", "import FVIntmax.Balance"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Classical.arbitrary", "module": "Mathlib.Logic.Nonempty"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Monotone", "module": "Mathlib.Order.Monotone.Defs"}, {"name": "Finset.sort", "module": "Mathlib.Data.Finset.Sort"}, {"name": "LE", "module": "Init.Prelude"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Subtype", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "TransactionsInBlocks", "content": "def TransactionsInBlocks\n  (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) : List (Τ K₁ K₂ V) :=\n  (bs.map (TransactionsInBlock π)).flatten"}, {"name": "TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}, {"name": "TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "le", "content": "def le (v₁ v₂ : Vector α n) :=\n  ∀ x ∈ (v₁.1.zip v₂.1), x.1 ≤ x.2"}, {"name": "v'", "content": "def v' (v : V₊) (b : S K₁ K₂ V) (s : Kbar K₁ K₂) : V₊ :=\n  match h : s with\n  | .Source => v\n  | .key _  => ⟨v ⊓ b s, by admit /- proof elided -/\n  ⟩"}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "not_and_or", "module": "Mathlib.Logic.Basic"}, {"name": "congr_fun", "module": "Batteries.Logic"}, {"name": "List.ext_get_iff", "module": "Mathlib.Data.List.Basic"}], "repo_lemmas": [{"name": "le_of_ext_le", "content": "lemma le_of_ext_le {α : Type} [Preorder α] {v₁ v₂ : Vector α n}\n  (h : ∀ i : Fin n, v₁.1[i] ≤ v₂.1[i]) : v₁ ≤ v₂"}, {"name": "mem_dict_iff_mem_keys", "content": "lemma mem_dict_iff_mem_keys {dict : Dict α ω} : k ∈ dict ↔ k ∈ dict.keys"}, {"name": "eq_of_BalanceProof_le", "content": "lemma eq_of_BalanceProof_le (h : π ≤ π') (h₁ : k ∈ π) (h₂ : k ∈ π') :\n  ((π k).get h₁).2 = ((π' k).get h₂).2"}, {"name": "mem_of_BalanceProof_le", "content": "lemma mem_of_BalanceProof_le (h : π ≤ π') (h₁ : k ∈ π) : k ∈ π'"}, {"name": "notin_of_BalanceProof_le_notin", "content": "lemma notin_of_BalanceProof_le_notin (h : π ≤ π') (h₁ : k ∉ π') : k ∉ π"}, {"name": "length_TransactionsInBlock_transfer", "content": "lemma length_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }} :\n  ∀ (π₁ π₂ : BalanceProof K₁ K₂ C Pi V),\n    (TransactionsInBlock_transfer π₁ b).length =\n    (TransactionsInBlock_transfer π₂ b).length"}, {"name": "length_transactionsInBlock", "content": "lemma length_transactionsInBlock :\n  (TransactionsInBlock π₁ b).length = (TransactionsInBlock π₂ b).length"}, {"name": "map_join_unnecessarily_specific", "content": "lemma map_join_unnecessarily_specific\n  {α β γ δ Pi : Type}\n  [LE δ]\n  [LE Pi]\n  {l : List α}\n  {P : (β × γ × δ) → Prop}\n  {π π' : Pi}\n  {f : Pi → α → List (Subtype P)}\n  {i : ℕ}\n  (h₀ : (List.length ∘ f π) = (List.length ∘ f π'))\n  (h₁ : ∀ (a : α)\n          (i : ℕ) (h : i < (f π a).length),\n          (f π a)[i].1.2.2 ≤ ((f π' a)[i]'(by apply congr_fun at h₀; aesop)).1.2.2)\n  (h) :\n  ((List.map (f π) l).flatten[i]'h).1.2.2 ≤\n  ((List.map (f π') l).flatten[i]'(by aesop)).1.2.2"}, {"name": "map_eq_project_triple", "content": "lemma map_eq_project_triple {β γ δ : Type}\n                            {s : β} {r : γ} {v : δ}\n                            {i : ℕ}\n                            {P : (β × γ × δ) → Prop}\n                            {l : List (Subtype P)}\n                            {h₀}\n                            {h : i < l.length} : \n  l[i]'h = ⟨(s, r, v), h₀⟩ → (l[i]'h).1.2.2 = v"}, {"name": "receiver_transactionsInBlocks", "content": "lemma receiver_transactionsInBlocks {bs : List (Block K₁ K₂ C Sigma V)}\n                                    {π₁ π₂ : BalanceProof K₁ K₂ C Pi V} :\n  (TransactionsInBlocks π₁ bs).map (λ s ↦ s.1.2.1) =\n  (TransactionsInBlocks π₂ bs).map (λ s ↦ s.1.2.1)"}, {"name": "receiver_transactionsInBlock", "content": "lemma receiver_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.2.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.2.1)"}, {"name": "length_transactionsInBlocks", "content": "lemma length_transactionsInBlocks {bs : List (Block K₁ K₂ C Sigma V)}\n                                  {π₁ π₂ : BalanceProof K₁ K₂ C Pi V} :\n  (TransactionsInBlocks π₁ bs).length = (TransactionsInBlocks π₂ bs).length"}, {"name": "sender_transactionsInBlocks", "content": "lemma sender_transactionsInBlocks {bs : List (Block K₁ K₂ C Sigma V)}\n                                  {π₁ π₂ : BalanceProof K₁ K₂ C Pi V} :\n  (TransactionsInBlocks π₁ bs).map (λ s ↦ s.1.1) =\n  (TransactionsInBlocks π₂ bs).map (λ s ↦ s.1.1)"}, {"name": "sender_transactionsInBlock", "content": "lemma sender_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.1)"}], "used_local_defs": [{"name": "Intmax.length_of_TransactionsInBlocks", "content": "private abbrev length_of_TransactionsInBlocks (bs : List (Block K₁ K₂ C Sigma V)) :\n  { n : ℕ // n = (TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length } :=\n  ⟨(TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length, rfl⟩"}, {"name": "Intmax.TransactionsInBlocksFixed", "content": "private def TransactionsInBlocksFixed (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) :\n  Vector (Τ K₁ K₂ V) (length_of_TransactionsInBlocks (Pi := Pi) bs).1 :=\n  ⟨TransactionsInBlocks π bs, by admit /- proof elided -/\n  ⟩"}], "used_local_lemmas": [{"name": "Intmax.TransactionsInBlocksFixed_le_of_TransactionsInBlocks", "content": "lemma TransactionsInBlocksFixed_le_of_TransactionsInBlocks\n  (h : ∀ i : Fin (length_of_TransactionsInBlocks bs).1,\n    (TransactionsInBlocks π bs)[i]'(by blast with π i) ≤\n    (TransactionsInBlocks π' bs)[i]'(by blast with π' i)) :\n  TransactionsInBlocksFixed π bs ≤ TransactionsInBlocksFixed π' bs"}, {"name": "Intmax.senderReceiver_transactionsInBlocks", "content": "lemma senderReceiver_transactionsInBlocks {s r v} {s' r' v'} {eq₁ eq₂} {i}\n  (h₀ : i < (TransactionsInBlocks π bs).length)\n  (h₁ : (TransactionsInBlocks π bs)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' bs)[i]'(by blast with π) = ⟨(s', r', v'), eq₂⟩) :\n  s = s' ∧ r = r'"}, {"name": "Intmax.delta_TransactionsInBlock_transfer", "content": "private lemma delta_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }}\n  (h : π ≤ π') : \n  ∀ i : ℕ, (hlen : i < (TransactionsInBlock_transfer π b).length) →\n    (TransactionsInBlock_transfer π b)[i]'hlen =\n    (TransactionsInBlock_transfer π' b)[i]'(by rwa [length_TransactionsInBlock_transfer _ π]) ∨\n    ((TransactionsInBlock_transfer π b)[i]'hlen).1.2.2.isNone"}, {"name": "Intmax.v_transactionsInBlocks", "content": "lemma v_transactionsInBlocks {s r v v'} {eq₁ eq₂} {i}\n  (h : π ≤ π')\n  (h₀ : i < (TransactionsInBlocks π Bstar).length)\n  (h₁ : (TransactionsInBlocks π Bstar)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' Bstar)[i]'(by blast with π) = ⟨(s, r, v'), eq₂⟩) :\n  v ≤ v'"}], "local_ctx": "import FVIntmax.Balance\n\nnamespace Intmax\n\nopen Mathlib\n\nnoncomputable section Lemma4\n\nsection HicSuntDracones\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type} [AddCommGroup V] [Lattice V]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nsection HelperFunctionsToAppeaseLean\n\nopen Mathlib\n\nprivate abbrev length_of_TransactionsInBlocks (bs : List (Block K₁ K₂ C Sigma V)) :\n  { n : ℕ // n = (TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length } :=\n  ⟨(TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length, rfl⟩\n\nopen Lean.Elab.Tactic in\n\nprivate def TransactionsInBlocksFixed (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) :\n  Vector (Τ K₁ K₂ V) (length_of_TransactionsInBlocks (Pi := Pi) bs).1 :=\n  ⟨TransactionsInBlocks π bs, by admit /- proof elided -/\n  ⟩\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nend\n\nend HelperFunctionsToAppeaseLean\n\nend\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs Bstar : List (Block K₁ K₂ C Sigma V)}", "target_theorem": "lemma monotone_TransactionsInBlocksFixed :\n  Monotone λ (π : BalanceProof K₁ K₂ C Pi V) ↦ TransactionsInBlocksFixed π Bstar :=", "ground_truth_proof": ":= by\n  intros π π' h\n  dsimp\n  apply TransactionsInBlocksFixed_le_of_TransactionsInBlocks; rintro ⟨i, hi⟩; simp\n  generalize eq : (TransactionsInBlocks π Bstar)[i]'(by blast with π) = Tstar\n  generalize eq' : (TransactionsInBlocks π' Bstar)[i]'(by clear eq; blast with π') = Tstar'\n  rcases Tstar with ⟨⟨s, r, v⟩, h₁⟩\n  rcases Tstar' with ⟨⟨s', r', v'⟩, h₂⟩\n  obtain ⟨⟨⟩, ⟨⟩⟩ := senderReceiver_transactionsInBlocks _ eq eq'; simp\n  exact v_transactionsInBlocks h _ eq eq'", "nesting_depth": 9, "transitive_dep_count": 73, "subset_aristotle": false, "category": "Applied verif."}
{"id": 204, "thm_name": "Intmax.compat_merge_of_compat", "thm_stmt": "lemma compat_merge_of_compat :\n  (∀ π', π' ∈ πs → π <≅> π') → π <≅> (mergeR'' πs .initial)", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Theorem1.lean", "imports": ["import FVIntmax.Lemma5", "import FVIntmax.Wheels.AuthenticatedDictionary", "import FVIntmax.Request", "import FVIntmax.AttackGame", "import FVIntmax.Wheels", "import FVIntmax.Propositions", "import Mathlib", "import FVIntmax.Lemma4", "import FVIntmax.Wheels.Dictionary", "import FVIntmax.Wheels.SignatureAggregation", "import FVIntmax.Lemma3"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}, {"name": "notation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂", "content": "notation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂"}, {"name": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂", "content": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "Merge", "content": "def Merge (D₁ D₂ : Dict α ω) : Dict α ω := D\n  where D := λ x ↦ First (D₁ x) (D₂ x)"}, {"name": "First", "content": "def First (x₁ x₂ : Option α) : Option α :=\n  match x₁, x₂ with\n  | .some x, .none => .some x\n  | .some x, .some _ => .some x\n  | .none, .some y => .some y\n  | .none, .none => .none"}, {"name": "BalanceProof.initial", "content": "def BalanceProof.initial : BalanceProof K₁ K₂ C Pi V := λ _ ↦ .none"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Merge_assoc", "content": "lemma Merge_assoc {D₃ : Dict α ω} :\n  Merge (Merge D₁ D₂) D₃ = Merge D₁ (Merge D₂ D₃)"}], "used_local_defs": [{"name": "Intmax.mergeR''", "content": "def mergeR'' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  match πs with\n  | [] => acc\n  | π :: πs => Dict.Merge acc (mergeR'' πs π)"}, {"name": "Intmax.BalanceProof.compat", "content": "def BalanceProof.compat (π₁ π₂ : BalanceProof K₁ K₂ C Pi V) : Prop :=\n  ∀ k, π₁ k ≠ none ∧ π₂ k ≠ none → π₁ k ≅ π₂ k"}], "used_local_lemmas": [{"name": "Intmax.merge_lem_aux", "content": "private lemma merge_lem_aux :\n  mergeR'' (π :: πs) acc = acc <+> π <+> (mergeR'' πs BalanceProof.initial)"}, {"name": "Intmax.merge_lem", "content": "lemma merge_lem :\n  mergeR'' (π :: πs) BalanceProof.initial = π <+> (mergeR'' πs BalanceProof.initial)"}, {"name": "Intmax.compat_lem", "content": "lemma compat_lem {π π' π'': BalanceProof K₁ K₂ C Pi V} :\n  π <≅> π' → π <≅> π'' → π <≅> (π' <+> π'')"}], "local_ctx": "import FVIntmax.AttackGame\n\nimport FVIntmax.Lemma3\n\nimport FVIntmax.Lemma4\n\nimport FVIntmax.Lemma5\n\nimport FVIntmax.Propositions\n\nimport FVIntmax.Request\n\nimport FVIntmax.Wheels\n\nimport FVIntmax.Wheels.AuthenticatedDictionary\n\nimport FVIntmax.Wheels.SignatureAggregation\n\nimport Mathlib\n\nnamespace Intmax\n\nopen Classical\n\nnoncomputable section Intmax\n\nnoncomputable section theorem1\n\nsection AttackGame\n\nvariable {Sigma Pi M : Type}\n         {C : Type} [Nonempty C]\n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {K₁ : Type} [Nonempty K₁] [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         {Kₚ : Type} [Nonempty Kₚ]\n         {Kₛ : Type} [Nonempty Kₛ]\n\ndef mergeR'' (πs : List (BalanceProof K₁ K₂ C Pi V)) (acc : BalanceProof K₁ K₂ C Pi V) : BalanceProof K₁ K₂ C Pi V :=\n  match πs with\n  | [] => acc\n  | π :: πs => Dict.Merge acc (mergeR'' πs π)\n\nsection MergeLemmas\n\nvariable {acc π : BalanceProof K₁ K₂ C Pi V} {πs πs : List (BalanceProof K₁ K₂ C Pi V)}\n\nend MergeLemmas\n\ndef BalanceProof.compat (π₁ π₂ : BalanceProof K₁ K₂ C Pi V) : Prop :=\n  ∀ k, π₁ k ≠ none ∧ π₂ k ≠ none → π₁ k ≅ π₂ k\n\nnotation:51 π₁:52 \" <≅> \" π₂:52 => BalanceProof.compat π₁ π₂\n\nnotation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂\n\nsection Compat\n\nvariable {π₁ π₂ π₃ : BalanceProof K₁ K₂ C Pi V}\n\nend Compat\n\nsection MergeLemmas\n\nvariable {π acc : BalanceProof K₁ K₂ C Pi V}\n         {πs : List (BalanceProof K₁ K₂ C Pi V)}\n\nend MergeLemmas\n\nvariable [AD : ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n\nsection MergeLemmas\n\nvariable {π acc : BalanceProof K₁ K₂ C Pi V}\n         {πs : List (BalanceProof K₁ K₂ C Pi V)}", "target_theorem": "lemma compat_merge_of_compat :\n  (∀ π', π' ∈ πs → π <≅> π') → π <≅> (mergeR'' πs .initial) :=", "ground_truth_proof": ":= by\n  induction πs generalizing π with\n  | nil =>\n    intros π\n    unfold BalanceProof.compat BalanceProof.initial\n    simp\n  | cons π πs ih =>\n    intros π_1 h\n    rw [merge_lem]\n    apply compat_lem <;> aesop", "nesting_depth": 5, "transitive_dep_count": 22, "subset_aristotle": false, "category": "Applied verif."}
{"id": 205, "thm_name": "Intmax.aggregateDeposits_cons", "thm_stmt": "@[simp]\nlemma aggregateDeposits_cons {hd} {tl : List (Block K₁ K₂ C Sigma V)} :\n  aggregateDeposits (hd :: tl) =\n  (if h : hd.isDepositBlock\n   then (hd.getDeposit h).2.1\n   else 0) +\n  aggregateDeposits tl", "lean_root": "FVIntmax", "rel_path": "FVIntmax/AttackGame.lean", "imports": ["import FVIntmax.Wheels.AuthenticatedDictionary", "import FVIntmax.Request", "import FVIntmax.Block", "import FVIntmax.Wheels.SignatureAggregation", "import FVIntmax.BalanceProof", "import FVIntmax.Wheels"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}], "used_repo_defs": [{"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "Scontract", "content": "abbrev Scontract (K₁ K₂ V : Type) [PreWithZero V] (C Sigma : Type) :=\n  List (Block K₁ K₂ C Sigma V)"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "getDeposit", "content": "def getDeposit (b : Block K₁ K₂ C Sigma V) (_h : b.isDepositBlock) : K₂ × V₊ :=\n  match b with | deposit r v => (r, v)"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "Finset.mem_range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Finset.sum_bij", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.sum_dite_of_true", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise"}, {"name": "Finset.sum_eq_sum_diff_singleton_add", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise"}, {"name": "Finset.sum_fin_eq_sum_range", "module": "Mathlib.Data.Fintype.BigOperators"}, {"name": "Finset.univ_eq_attach", "module": "Mathlib.Data.Fintype.Sets"}, {"name": "List.getElem_cons_zero", "module": "Init.GetElem"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "dif_pos", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Intmax.aggregateDeposits", "content": "def aggregateDeposits (σ : Scontract K₁ K₂ V C Sigma) : V :=\n  ∑ i : Fin σ.length,\n    if h : σ[i].isDepositBlock\n    then (σ[i.1].getDeposit h).2.1\n    else 0"}, {"name": "Intmax.reindex", "content": "@[simp]\nprivate def reindex : (a : ℕ) → a ∈ Finset.range (k + 1) \\ {0} → ℕ :=\n  λ a _ ↦ a.pred"}], "used_local_lemmas": [{"name": "Intmax.reindex_mem", "content": "private lemma reindex_mem :\n  ∀ (a : ℕ) (ha : a ∈ Finset.range (k + 1) \\ {0}), reindex a ha ∈ Finset.range k"}, {"name": "Intmax.reindex_inj", "content": "private lemma reindex_inj :\n  ∀ (a₁ : ℕ) (ha₁ : a₁ ∈ Finset.range (k + 1) \\ {0})\n    (a₂ : ℕ) (ha₂ : a₂ ∈ Finset.range (k + 1) \\ {0}),\n  reindex a₁ ha₁ = reindex a₂ ha₂ → a₁ = a₂"}], "local_ctx": "import FVIntmax.Wheels.AuthenticatedDictionary\n\nimport FVIntmax.Wheels.SignatureAggregation\n\nimport FVIntmax.BalanceProof\n\nimport FVIntmax.Block\n\nimport FVIntmax.Request\n\nimport FVIntmax.Wheels\n\nnamespace Intmax\n\nnoncomputable section Intmax\n\nsection RollupContract\n\nopen Classical\n\nsection\n\nvariable {C : Type} [Nonempty C]\n\n         {V : Type} [Lattice V] [AddCommGroup V]\n         \n         {K₁ : Type} [Finite K₁] [Nonempty K₁]\n         {K₂ : Type} [Finite K₂]\n         {Sigma : Type}\n         [ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n         [SA : SignatureAggregation (C × K₁ × ExtraDataT) K₂ KₛT Sigma]\n\nend\n\nsection\n\nvariable {C : Type}\n         {V : Type} [Lattice V] [AddCommGroup V]\n         {K₁ : Type} [Finite K₁]\n         {K₂ : Type}\n         {Sigma : Type}\n         {block : Block K₁ K₂ C Sigma V}\n         {v : V}\n\nsection Lemmas\n\nend Lemmas\n\nnamespace Scontract\n\nsection\n\nvariable [LinearOrder K₁] [Nonempty K₁]\n         [LinearOrder K₂] [Finite K₂]\n         [Nonempty C]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         [ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n         [SignatureAggregation (C × K₁ × ExtraDataT) K₂ KₛT Sigma]\n\nsection appendBlock\n\nvariable {σ : Scontract K₁ K₂ V C Sigma} {request : Request K₁ K₂ C Sigma Pi V}\n\nend appendBlock\n\nend\n\nsection\n\nvariable [LinearOrder K₁]\n         [Finite K₂] [LinearOrder K₂]\n         {σ : Scontract K₁ K₂ V C Sigma}\n         {request : Request K₁ K₂ C Sigma Pi V}\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\nend\n\nend Scontract\n\nend\n\nend RollupContract\n\nsection AttackGameDefs\n\nvariable {K₁ : Type} [Finite K₁] [LinearOrder K₁] [Nonempty K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n\n         {V : Type} [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\n         {Sigma : Type}\n         {C : Type} [Nonempty C]\n\n         [ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n         [SignatureAggregation (C × K₁ × ExtraDataT) K₂ KₛT Sigma]\n\n         (requests : List (Request K₁ K₂ C Sigma Pi V))\n         (σ : Scontract K₁ K₂ V C Sigma)\n\ndef aggregateDeposits (σ : Scontract K₁ K₂ V C Sigma) : V :=\n  ∑ i : Fin σ.length,\n    if h : σ[i].isDepositBlock\n    then (σ[i.1].getDeposit h).2.1\n    else 0\n\nend AttackGameDefs\n\nsection AttackGameLemmas\n\nvariable {K₁ K₂ Sigma C : Type}\n         {V : Type} [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n\nsection computeBalanceSum\n\nvariable {k : ℕ}\n\n@[simp]\nprivate def reindex : (a : ℕ) → a ∈ Finset.range (k + 1) \\ {0} → ℕ :=\n  λ a _ ↦ a.pred\n\nend computeBalanceSum", "target_theorem": "@[simp]\nlemma aggregateDeposits_cons {hd} {tl : List (Block K₁ K₂ C Sigma V)} :\n  aggregateDeposits (hd :: tl) =\n  (if h : hd.isDepositBlock\n   then (hd.getDeposit h).2.1\n   else 0) +\n  aggregateDeposits tl :=", "ground_truth_proof": ":= by\n  simp [aggregateDeposits]\n  rw [\n    Finset.sum_fin_eq_sum_range,\n    Finset.sum_eq_sum_diff_singleton_add (i := 0),\n    dif_pos (show 0 < tl.length + 1 by omega)\n  ]\n  simp_rw [List.getElem_cons_zero (h := _)]; case h => exact Finset.mem_range.2 (by omega)\n  let F : ℕ → V := λ i ↦\n    if h : i < tl.length\n    then if h_1 : tl[i].isDepositBlock = true\n         then (tl[i].getDeposit h_1).2\n         else 0\n    else 0\n  rw [Finset.sum_bij (t := Finset.range tl.length)\n                     (g := F)\n                     (i := reindex)\n                     (hi := reindex_mem)\n                     (i_inj := reindex_inj)\n                     (i_surj := λ b hb ↦ by use b.succ; simp; exact Finset.mem_range.1 hb)\n                     (h := λ n hn ↦ by simp at hn; simp [hn]\n                                       rcases n with _ | n <;> [omega; simp [F, reindex]]\n                                       nth_rw 2 [dif_pos (by rcases hn with ⟨hn, _⟩; omega)])]\n  unfold F\n  rw [Finset.sum_dite_of_true (λ _ ↦ (Finset.mem_range.1 ·)), Finset.sum_fin_eq_sum_range, Finset.univ_eq_attach]\n  nth_rw 2 [Finset.sum_dite_of_true]; case h => exact λ _ ↦ (Finset.mem_range.1 ·)\n  rw [add_comm]; simp", "nesting_depth": 5, "transitive_dep_count": 28, "subset_aristotle": false, "category": "Applied verif."}
{"id": 206, "thm_name": "Intmax.f_transfer_source''", "thm_stmt": "lemma f_transfer_source''\n  (h : b.isTransferBlock) (h₁ : T ∈ TransactionsInBlock π b) : \n  (f σ T) .Source = σ .Source := f_transfer_source ⟨⟨b, h⟩, h₁⟩", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Balance.lean", "imports": ["import FVIntmax.BalanceProof", "import FVIntmax.Wheels", "import FVIntmax.Propositions", "import Mathlib", "import FVIntmax.Key", "import FVIntmax.Block", "import Mathlib.Algebra.Group.Int", "import FVIntmax.Wheels.Dictionary", "import FVIntmax.State", "import FVIntmax.Transaction"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "IsGLB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "And", "module": "Init.Prelude"}, {"name": "IsGreatest", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "InfSet", "module": "Mathlib.Order.SetNotation"}, {"name": "iInf", "module": "Mathlib.Order.SetNotation"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.mk", "module": "Init.Prelude"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "Τc", "content": "abbrev Τc (K₁ K₂ V : Type) [PreWithZero V] : Type := { τ : Τ K₁ K₂ V // τ.isComplete }"}, {"name": "isComplete", "content": "def isComplete (τ : Τ K₁ K₂ V) :=\n  match τ with | ⟨(_, _, v), _⟩ => v.isSome"}, {"name": "value", "content": "def value (τ : Τ K₁ K₂ V) : Option V₊ := τ.1.2.2"}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "Set.image_eq_range", "module": "Mathlib.Data.Set.Image"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "Finset.mem_filter", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Finset.mem_sort", "module": "Mathlib.Data.Finset.Sort"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "List.mem_attach", "module": "Init.Data.List.Attach"}, {"name": "List.mem_map", "module": "Init.Data.List.Lemmas"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "exists_and_left", "module": "Init.PropLemmas"}, {"name": "exists_const", "module": "Init.PropLemmas"}, {"name": "exists_eq", "module": "Init.PropLemmas"}, {"name": "exists_eq_right", "module": "Init.PropLemmas"}, {"name": "exists_eq_right_right", "module": "Init.PropLemmas"}, {"name": "exists_false", "module": "Init.PropLemmas"}, {"name": "exists_prop", "module": "Init.PropLemmas"}, {"name": "exists_true_left", "module": "Init.PropLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "ite_not", "module": "Init.PropLemmas"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Intmax.TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}, {"name": "Intmax.e", "content": "def e (i : Kbar K₁ K₂) : Kbar K₁ K₂ → ℤ := λ j ↦ if i = j then 1 else 0"}, {"name": "Intmax.fc", "content": "def fc (τcXb : Τc K₁ K₂ V × S K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨λ k : Kbar K₁ K₂ ↦\n    match τcXb with\n    | ⟨⟨⟨⟨s, r, v⟩, _⟩, hτ⟩, b⟩ =>\n      let v' := v' (v.get hτ) b s\n      b k + (e r - e s) k • v',\n  by admit /- proof elided -/\n  ⟩"}, {"name": "Intmax.boundedBelow", "content": "abbrev boundedBelow (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) :=\n  { a : Τc K₁ K₂ V × S K₁ K₂ V | (T, b) ≤ (↑a.1, a.2) }"}, {"name": "Intmax.V'", "content": "def V' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) : Set V :=\n  { v : V | v ∈ (fc · k) '' boundedBelow b T }"}, {"name": "Intmax.f'", "content": "def f' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V := \n  ⟨\n    λ k ↦\n      match h : T with\n      | ⟨(_, _, .some _), hT⟩ => fc (⟨T, by admit /- proof elided -/\n      ⟩, b) k\n      | ⟨(s, _, .none), _⟩ => if k = s then 0 else b k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Intmax.exists_inf", "content": "def exists_inf (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : { s : S K₁ K₂ V // ∀ k, IsGLB (V' b T k) (s k) } :=\n  ⟨\n    f' b T,\n    λ k ↦\n      have f'_codomain : (f' b T) k ∈ V' b T k := by admit /- proof elided -/\n  ⟩"}, {"name": "Intmax.infV", "content": "def infV (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) :\n  InfSet V where\n    sInf := λ s ↦ if s = V' b T k\n                  then (exists_inf b T).1 k\n                  else 0"}, {"name": "Intmax.f", "content": "def f (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨\n    λ k ↦\n      have : InfSet V := infV b T k\n      ⨅ x : boundedBelow b T, fc x.1 k,\n    by admit /- proof elided -/\n  ⟩"}], "used_local_lemmas": [{"name": "Intmax.V'_eq_range", "content": "private lemma V'_eq_range {b : S K₁ K₂ V} {T : Τ K₁ K₂ V} {k : Kbar K₁ K₂} :\n  V' b T k =\n  Set.range λ (x : { x : (Τc K₁ K₂ V × S K₁ K₂ V) // (T, b) ≤ (↑x.1, x.2) }) ↦ fc ↑x k"}, {"name": "Intmax.f_eq_f'", "content": "lemma f_eq_f' : f = f' (K₁ := K₁) (K₂ := K₂) (V := V)"}, {"name": "Intmax.f_transfer_source", "content": "lemma f_transfer_source\n  (h : ∃ block : {b : Block K₁ K₂ C Sigma V // b.isTransferBlock},\n         T ∈ TransactionsInBlock π block.1) :\n  (f σ T) .Source = σ .Source"}], "local_ctx": "import Mathlib\n\nimport Mathlib.Algebra.Group.Int\n\nimport FVIntmax.BalanceProof\n\nimport FVIntmax.Block\n\nimport FVIntmax.Key\n\nimport FVIntmax.Propositions\n\nimport FVIntmax.State\n\nimport FVIntmax.Transaction\n\nimport FVIntmax.Wheels\n\nimport FVIntmax.Wheels.Dictionary\n\nnamespace Intmax\n\nnoncomputable section\n\nopen Classical\n\nsection Balance\n\nvariable {Pi K₁ K₂ V C Sigma : Type}\n\nsection Extraction\n\nsection Deposit\n\nvariable [PreWithZero V]\n\ndef TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/\n\nend Deposit\n\nsection Transfer\n\nvariable [Finite K₁] [Finite K₂]\n         [LinearOrder K₁] [LinearOrder K₂] [PreWithZero V]\n\ndef TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/\n\nend Transfer\n\nsection Withdrawal\n\nvariable [LinearOrder K₁] [Finite K₁] [PreWithZero V]\n\ndef TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/\n\nend Withdrawal\n\nvariable [Finite K₁] [LinearOrder K₁]\n         [Finite K₂] [LinearOrder K₂]\n         [PreWithZero V]\n         {b : Block K₁ K₂ C Sigma V}\n         {bs : List (Block K₁ K₂ C Sigma V)}\n         {π₁ π₂ : BalanceProof K₁ K₂ C Pi V}\n\ndef TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b\n\nend Extraction\n\nsection e\n\ndef e (i : Kbar K₁ K₂) : Kbar K₁ K₂ → ℤ := λ j ↦ if i = j then 1 else 0\n\nvariable {i j : Kbar K₁ K₂}\n\nend e\n\nsection WithStructuredTypes\n\nsection v'\n\nvariable [Zero V] [Lattice V] -- NB `PreWithZero V` is implied as `CompleteLattice V` gives `Preorder V`.\n\nvariable {v : V₊} {b : S K₁ K₂ V} {s : Kbar K₁ K₂}\n\nend v'\n\nsection Fc\n\nvariable [Lattice V]\n         [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\ndef fc (τcXb : Τc K₁ K₂ V × S K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨λ k : Kbar K₁ K₂ ↦\n    match τcXb with\n    | ⟨⟨⟨⟨s, r, v⟩, _⟩, hτ⟩, b⟩ =>\n      let v' := v' (v.get hτ) b s\n      b k + (e r - e s) k • v',\n  by admit /- proof elided -/\n  ⟩\n\nvariable {τc : Τc K₁ K₂ V} {b : S K₁ K₂ V}\n\nend Fc\n\nsection Order\n\nvariable [Lattice V] [AddCommGroup V]\n\nend Order\n\nsection BoundedBelow\n\nvariable [Lattice V] [AddCommGroup V]\n\nabbrev boundedBelow (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) :=\n  { a : Τc K₁ K₂ V × S K₁ K₂ V | (T, b) ≤ (↑a.1, a.2) }\n\nend BoundedBelow\n\nsection LGroup\n\nvariable [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\ndef V' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) : Set V :=\n  { v : V | v ∈ (fc · k) '' boundedBelow b T }\n\nsection f\n\ndef f' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V := \n  ⟨\n    λ k ↦\n      match h : T with\n      | ⟨(_, _, .some _), hT⟩ => fc (⟨T, by admit /- proof elided -/\n      ⟩, b) k\n      | ⟨(s, _, .none), _⟩ => if k = s then 0 else b k,\n    by admit /- proof elided -/\n  ⟩\n\ndef exists_inf (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : { s : S K₁ K₂ V // ∀ k, IsGLB (V' b T k) (s k) } :=\n  ⟨\n    f' b T,\n    λ k ↦\n      have f'_codomain : (f' b T) k ∈ V' b T k := by admit /- proof elided -/\n  ⟩\n\ndef infV (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) :\n  InfSet V where\n    sInf := λ s ↦ if s = V' b T k\n                  then (exists_inf b T).1 k\n                  else 0\n\ndef f (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨\n    λ k ↦\n      have : InfSet V := infV b T k\n      ⨅ x : boundedBelow b T, fc x.1 k,\n    by admit /- proof elided -/\n  ⟩\n\nend f\n\nsection fStar\n\nvariable {s : S K₁ K₂ V}\n\nend fStar\n\nvariable [Finite K₁] [LinearOrder K₁]\n         [Finite K₂] [LinearOrder K₂]\n\nsection LocalProperties\n\nvariable {σ : S K₁ K₂ V}\n         {π : BalanceProof K₁ K₂ C Pi V}\n         {T : Τ K₁ K₂ V}\n         {b : Block K₁ K₂ C Sigma V}\n         {Sigma : Type}", "target_theorem": "lemma f_transfer_source''\n  (h : b.isTransferBlock) (h₁ : T ∈ TransactionsInBlock π b) : \n  (f σ T) .Source = σ .Source :=", "ground_truth_proof": ":= f_transfer_source ⟨⟨b, h⟩, h₁⟩", "nesting_depth": 6, "transitive_dep_count": 90, "subset_aristotle": false, "category": "Applied verif."}
{"id": 207, "thm_name": "Intmax.sum_f_le_sum", "thm_stmt": "lemma sum_f_le_sum : ∑ (k : Kbar K₁ K₂), f b T k ≤ ∑ (k : Kbar K₁ K₂), b k", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Lemma3.lean", "imports": ["import FVIntmax.Balance"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "IsGLB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "And", "module": "Init.Prelude"}, {"name": "IsGreatest", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "InfSet", "module": "Mathlib.Order.SetNotation"}, {"name": "iInf", "module": "Mathlib.Order.SetNotation"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}], "used_repo_defs": [{"name": "isComplete", "content": "def isComplete (τ : Τ K₁ K₂ V) :=\n  match τ with | ⟨(_, _, v), _⟩ => v.isSome"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "Τc", "content": "abbrev Τc (K₁ K₂ V : Type) [PreWithZero V] : Type := { τ : Τ K₁ K₂ V // τ.isComplete }"}, {"name": "V'", "content": "def V' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) : Set V :=\n  { v : V | v ∈ (fc · k) '' boundedBelow b T }"}, {"name": "boundedBelow", "content": "abbrev boundedBelow (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) :=\n  { a : Τc K₁ K₂ V × S K₁ K₂ V | (T, b) ≤ (↑a.1, a.2) }"}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "fc", "content": "def fc (τcXb : Τc K₁ K₂ V × S K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨λ k : Kbar K₁ K₂ ↦\n    match τcXb with\n    | ⟨⟨⟨⟨s, r, v⟩, _⟩, hτ⟩, b⟩ =>\n      let v' := v' (v.get hτ) b s\n      b k + (e r - e s) k • v',\n  by admit /- proof elided -/\n  ⟩"}, {"name": "e", "content": "def e (i : Kbar K₁ K₂) : Kbar K₁ K₂ → ℤ := λ j ↦ if i = j then 1 else 0"}, {"name": "f", "content": "def f (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨\n    λ k ↦\n      have : InfSet V := infV b T k\n      ⨅ x : boundedBelow b T, fc x.1 k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "infV", "content": "def infV (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) :\n  InfSet V where\n    sInf := λ s ↦ if s = V' b T k\n                  then (exists_inf b T).1 k\n                  else 0"}, {"name": "exists_inf", "content": "def exists_inf (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : { s : S K₁ K₂ V // ∀ k, IsGLB (V' b T k) (s k) } :=\n  ⟨\n    f' b T,\n    λ k ↦\n      have f'_codomain : (f' b T) k ∈ V' b T k := by admit /- proof elided -/\n  ⟩"}, {"name": "f'", "content": "def f' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V := \n  ⟨\n    λ k ↦\n      match h : T with\n      | ⟨(_, _, .some _), hT⟩ => fc (⟨T, by admit /- proof elided -/\n      ⟩, b) k\n      | ⟨(s, _, .none), _⟩ => if k = s then 0 else b k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "Finset.sum_add_distrib", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_smul", "module": "Mathlib.Algebra.Module.BigOperators"}, {"name": "Finset.sum_le_sum", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}], "repo_lemmas": [{"name": "f_IsGLB_of_V'", "content": "lemma f_IsGLB_of_V' {b : S K₁ K₂ V} {T : Τ K₁ K₂ V} {k : Kbar K₁ K₂} :\n  IsGLB (V' b T k) (f b T k)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Intmax.sum_fc_eq_sum", "content": "@[simp]\nlemma sum_fc_eq_sum : ∑ (k : Kbar K₁ K₂), fc (Tc, b) k = ∑ (k : Kbar K₁ K₂), b k"}], "local_ctx": "import FVIntmax.Balance\n\nnamespace Intmax\n\nsection Lemma3\n\nvariable\n  {Pi C Sigma : Type}\n  {K₁ : Type} [Finite K₁]\n  {K₂ : Type} [Finite K₂]\n  {V : Type} [Lattice V] [AddCommGroup V]\n             [CovariantClass V V (· + ·) (· ≤ ·)]\n             [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\nvariable {Tc : Τc K₁ K₂ V} {T : Τ K₁ K₂ V} {b : S K₁ K₂ V} {Tstar : List (Τ K₁ K₂ V)}", "target_theorem": "lemma sum_f_le_sum : ∑ (k : Kbar K₁ K₂), f b T k ≤ ∑ (k : Kbar K₁ K₂), b k :=", "ground_truth_proof": ":= by\n  by_cases eq : T.isComplete\n  · conv_rhs => rw [←sum_fc_eq_sum (Tc := ⟨T, eq⟩)]\n    refine' Finset.sum_le_sum (λ k _ ↦ _)\n    have fcInV' : fc (⟨T, eq⟩, b) k ∈ V' b T k := by\n      dsimp [V']\n      rw [Set.mem_image]\n      use (⟨T, eq⟩, b)\n      simp\n    exact f_IsGLB_of_V'.1 fcInV'\n  · rcases T with ⟨⟨s, r, v⟩, h⟩\n    let Tc : Τc K₁ K₂ V := ⟨⟨(s, r, some 0), by valid⟩, by simp⟩\n    conv_rhs => rw [←sum_fc_eq_sum (Tc := Tc)]\n    refine' (Finset.sum_le_sum (λ k _ ↦ _))\n    have fcInV' : fc (Tc, b) k ∈ V' b ⟨(s, r, v), h⟩ k := by\n      dsimp [V']\n      rw [Set.mem_image]\n      use (⟨⟨(s, r, some 0), by valid⟩, by valid⟩, b)\n      have : v = none := by aesop\n      simp [this, (·≤·)]\n    exact f_IsGLB_of_V'.1 fcInV'", "nesting_depth": 5, "transitive_dep_count": 38, "subset_aristotle": false, "category": "Applied verif."}
{"id": 208, "thm_name": "Intmax.v_transactionsInBlocks", "thm_stmt": "lemma v_transactionsInBlocks {s r v v'} {eq₁ eq₂} {i}\n  (h : π ≤ π')\n  (h₀ : i < (TransactionsInBlocks π Bstar).length)\n  (h₁ : (TransactionsInBlocks π Bstar)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' Bstar)[i]'(by blast with π) = ⟨(s, r, v'), eq₂⟩) :\n  v ≤ v'", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Lemma4.lean", "imports": ["import FVIntmax.Wheels.Dictionary", "import FVIntmax.Wheels", "import FVIntmax.Balance"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.sort", "module": "Mathlib.Data.Finset.Sort"}, {"name": "LE", "module": "Init.Prelude"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Subtype", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "v'", "content": "def v' (v : V₊) (b : S K₁ K₂ V) (s : Kbar K₁ K₂) : V₊ :=\n  match h : s with\n  | .Source => v\n  | .key _  => ⟨v ⊓ b s, by admit /- proof elided -/\n  ⟩"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "TransactionsInBlocks", "content": "def TransactionsInBlocks\n  (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) : List (Τ K₁ K₂ V) :=\n  (bs.map (TransactionsInBlock π)).flatten"}, {"name": "TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}, {"name": "TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "not_and_or", "module": "Mathlib.Logic.Basic"}, {"name": "congr_fun", "module": "Batteries.Logic"}], "repo_lemmas": [{"name": "mem_dict_iff_mem_keys", "content": "lemma mem_dict_iff_mem_keys {dict : Dict α ω} : k ∈ dict ↔ k ∈ dict.keys"}, {"name": "eq_of_BalanceProof_le", "content": "lemma eq_of_BalanceProof_le (h : π ≤ π') (h₁ : k ∈ π) (h₂ : k ∈ π') :\n  ((π k).get h₁).2 = ((π' k).get h₂).2"}, {"name": "mem_of_BalanceProof_le", "content": "lemma mem_of_BalanceProof_le (h : π ≤ π') (h₁ : k ∈ π) : k ∈ π'"}, {"name": "notin_of_BalanceProof_le_notin", "content": "lemma notin_of_BalanceProof_le_notin (h : π ≤ π') (h₁ : k ∉ π') : k ∉ π"}, {"name": "length_TransactionsInBlock_transfer", "content": "lemma length_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }} :\n  ∀ (π₁ π₂ : BalanceProof K₁ K₂ C Pi V),\n    (TransactionsInBlock_transfer π₁ b).length =\n    (TransactionsInBlock_transfer π₂ b).length"}, {"name": "length_transactionsInBlock", "content": "lemma length_transactionsInBlock :\n  (TransactionsInBlock π₁ b).length = (TransactionsInBlock π₂ b).length"}, {"name": "map_join_unnecessarily_specific", "content": "lemma map_join_unnecessarily_specific\n  {α β γ δ Pi : Type}\n  [LE δ]\n  [LE Pi]\n  {l : List α}\n  {P : (β × γ × δ) → Prop}\n  {π π' : Pi}\n  {f : Pi → α → List (Subtype P)}\n  {i : ℕ}\n  (h₀ : (List.length ∘ f π) = (List.length ∘ f π'))\n  (h₁ : ∀ (a : α)\n          (i : ℕ) (h : i < (f π a).length),\n          (f π a)[i].1.2.2 ≤ ((f π' a)[i]'(by apply congr_fun at h₀; aesop)).1.2.2)\n  (h) :\n  ((List.map (f π) l).flatten[i]'h).1.2.2 ≤\n  ((List.map (f π') l).flatten[i]'(by aesop)).1.2.2"}, {"name": "map_eq_project_triple", "content": "lemma map_eq_project_triple {β γ δ : Type}\n                            {s : β} {r : γ} {v : δ}\n                            {i : ℕ}\n                            {P : (β × γ × δ) → Prop}\n                            {l : List (Subtype P)}\n                            {h₀}\n                            {h : i < l.length} : \n  l[i]'h = ⟨(s, r, v), h₀⟩ → (l[i]'h).1.2.2 = v"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Intmax.delta_TransactionsInBlock_transfer", "content": "private lemma delta_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }}\n  (h : π ≤ π') : \n  ∀ i : ℕ, (hlen : i < (TransactionsInBlock_transfer π b).length) →\n    (TransactionsInBlock_transfer π b)[i]'hlen =\n    (TransactionsInBlock_transfer π' b)[i]'(by rwa [length_TransactionsInBlock_transfer _ π]) ∨\n    ((TransactionsInBlock_transfer π b)[i]'hlen).1.2.2.isNone"}], "local_ctx": "import FVIntmax.Balance\n\nnamespace Intmax\n\nopen Mathlib\n\nnoncomputable section Lemma4\n\nsection HicSuntDracones\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type} [AddCommGroup V] [Lattice V]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nsection HelperFunctionsToAppeaseLean\n\nopen Mathlib\n\nopen Lean.Elab.Tactic in\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nend\n\nend HelperFunctionsToAppeaseLean\n\nend\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs Bstar : List (Block K₁ K₂ C Sigma V)}", "target_theorem": "lemma v_transactionsInBlocks {s r v v'} {eq₁ eq₂} {i}\n  (h : π ≤ π')\n  (h₀ : i < (TransactionsInBlocks π Bstar).length)\n  (h₁ : (TransactionsInBlocks π Bstar)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' Bstar)[i]'(by blast with π) = ⟨(s, r, v'), eq₂⟩) :\n  v ≤ v' :=", "ground_truth_proof": ":= by\n  unfold TransactionsInBlocks at h₁ h₂\n  apply List.map_eq_project_triple at h₁\n  apply List.map_eq_project_triple at h₂\n  rw [←h₁, ←h₂]\n  apply List.map_join_unnecessarily_specific (by ext x; simp; rw [length_transactionsInBlock])\n  intros b i h₃\n  unfold TransactionsInBlock\n  match h' : b with\n  | Block.transfer .. =>\n    rcases delta_TransactionsInBlock_transfer (b := ⟨b, by aesop⟩) h i (by aesop) <;>\n           [aesop; (simp [(·≤·)]; aesop)]\n  | Block.deposit .. | Block.withdrawal .. => simp", "nesting_depth": 7, "transitive_dep_count": 57, "subset_aristotle": false, "category": "Applied verif."}
{"id": 209, "thm_name": "Intmax.lemma3", "thm_stmt": "lemma lemma3 : Bal π bs .Source ≤ 0", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Lemma3.lean", "imports": ["import FVIntmax.Balance"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "IsGLB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "And", "module": "Init.Prelude"}, {"name": "IsGreatest", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "InfSet", "module": "Mathlib.Order.SetNotation"}, {"name": "iInf", "module": "Mathlib.Order.SetNotation"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "fStar", "content": "def fStar (Ts : List (Τ K₁ K₂ V)) (s₀ : S K₁ K₂ V) : S K₁ K₂ V :=\n  Ts.foldl f s₀"}, {"name": "f", "content": "def f (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨\n    λ k ↦\n      have : InfSet V := infV b T k\n      ⨅ x : boundedBelow b T, fc x.1 k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "infV", "content": "def infV (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) :\n  InfSet V where\n    sInf := λ s ↦ if s = V' b T k\n                  then (exists_inf b T).1 k\n                  else 0"}, {"name": "exists_inf", "content": "def exists_inf (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : { s : S K₁ K₂ V // ∀ k, IsGLB (V' b T k) (s k) } :=\n  ⟨\n    f' b T,\n    λ k ↦\n      have f'_codomain : (f' b T) k ∈ V' b T k := by admit /- proof elided -/\n  ⟩"}, {"name": "fc", "content": "def fc (τcXb : Τc K₁ K₂ V × S K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨λ k : Kbar K₁ K₂ ↦\n    match τcXb with\n    | ⟨⟨⟨⟨s, r, v⟩, _⟩, hτ⟩, b⟩ =>\n      let v' := v' (v.get hτ) b s\n      b k + (e r - e s) k • v',\n  by admit /- proof elided -/\n  ⟩"}, {"name": "e", "content": "def e (i : Kbar K₁ K₂) : Kbar K₁ K₂ → ℤ := λ j ↦ if i = j then 1 else 0"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "Τc", "content": "abbrev Τc (K₁ K₂ V : Type) [PreWithZero V] : Type := { τ : Τ K₁ K₂ V // τ.isComplete }"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "boundedBelow", "content": "abbrev boundedBelow (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) :=\n  { a : Τc K₁ K₂ V × S K₁ K₂ V | (T, b) ≤ (↑a.1, a.2) }"}, {"name": "f'", "content": "def f' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V := \n  ⟨\n    λ k ↦\n      match h : T with\n      | ⟨(_, _, .some _), hT⟩ => fc (⟨T, by admit /- proof elided -/\n      ⟩, b) k\n      | ⟨(s, _, .none), _⟩ => if k = s then 0 else b k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "V'", "content": "def V' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) : Set V :=\n  { v : V | v ∈ (fc · k) '' boundedBelow b T }"}, {"name": "isComplete", "content": "def isComplete (τ : Τ K₁ K₂ V) :=\n  match τ with | ⟨(_, _, v), _⟩ => v.isSome"}, {"name": "TransactionsInBlocks", "content": "def TransactionsInBlocks\n  (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) : List (Τ K₁ K₂ V) :=\n  (bs.map (TransactionsInBlock π)).flatten"}, {"name": "TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}, {"name": "TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "Bal", "content": "def Bal (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) : S K₁ K₂ V :=\n  fStar (TransactionsInBlocks π bs) (.initial K₁ K₂ V)"}, {"name": "initial", "content": "def initial : Scontract K₁ K₂ V C Sigma := []"}, {"name": "Scontract", "content": "abbrev Scontract (K₁ K₂ V : Type) [PreWithZero V] (C Sigma : Type) :=\n  List (Block K₁ K₂ C Sigma V)"}, {"name": "initial", "content": "def initial (K₁ K₂ V : Type) [PreWithZero V] : S K₁ K₂ V :=\n  ⟨S'.initial K₁ K₂ V, S'.isValid_initial⟩"}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "Finset.sum_add_distrib", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_smul", "module": "Mathlib.Algebra.Module.BigOperators"}, {"name": "Finset.sum_le_sum", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Finset.sum_nonneg", "module": "Mathlib.Algebra.Order.BigOperators.Group.Finset"}, {"name": "Finset.sum_singleton", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "sub_nonpos", "module": "Mathlib.Algebra.Order.Group.Unbundled.Basic"}], "repo_lemmas": [{"name": "f_IsGLB_of_V'", "content": "lemma f_IsGLB_of_V' {b : S K₁ K₂ V} {T : Τ K₁ K₂ V} {k : Kbar K₁ K₂} :\n  IsGLB (V' b T k) (f b T k)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Intmax.sum_fc_eq_sum", "content": "@[simp]\nlemma sum_fc_eq_sum : ∑ (k : Kbar K₁ K₂), fc (Tc, b) k = ∑ (k : Kbar K₁ K₂), b k"}, {"name": "Intmax.sum_f_le_sum", "content": "lemma sum_f_le_sum : ∑ (k : Kbar K₁ K₂), f b T k ≤ ∑ (k : Kbar K₁ K₂), b k"}, {"name": "Intmax.sum_fStar_le_zero_aux", "content": "private lemma sum_fStar_le_zero_aux (h : ∑ (k : Kbar K₁ K₂), b k ≤ 0) :\n  ∑ (k : Kbar K₁ K₂), fStar Tstar b k ≤ 0"}], "local_ctx": "import FVIntmax.Balance\n\nnamespace Intmax\n\nsection Lemma3\n\nvariable\n  {Pi C Sigma : Type}\n  {K₁ : Type} [Finite K₁]\n  {K₂ : Type} [Finite K₂]\n  {V : Type} [Lattice V] [AddCommGroup V]\n             [CovariantClass V V (· + ·) (· ≤ ·)]\n             [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\nvariable {Tc : Τc K₁ K₂ V} {T : Τ K₁ K₂ V} {b : S K₁ K₂ V} {Tstar : List (Τ K₁ K₂ V)}\n\nvariable [LinearOrder K₁] [LinearOrder K₂]\n         {π : BalanceProof K₁ K₂ C Pi V}\n         {bs : List (Block K₁ K₂ C Sigma V)}", "target_theorem": "lemma lemma3 : Bal π bs .Source ≤ 0 :=", "ground_truth_proof": ":= by\n  dsimp [Bal]\n  generalize eq : TransactionsInBlocks π _ = blocks\n  generalize eq₁ : S.initial K₁ K₂ V = s₀\n  generalize eq₂ : fStar blocks s₀ = f\n  have : ∑ x ∈ {Kbar.Source}, f x = \n         ∑ x : Kbar K₁ K₂, f x - ∑ x ∈ Finset.univ \\ {Kbar.Source}, f x := by simp\n  rw [←Finset.sum_singleton (a := Kbar.Source) (f := f), this, sub_nonpos]\n  have := sum_fStar_le_zero_aux (Tstar := blocks) (b := s₀)\n  have eq₃ : ∑ x : Kbar K₁ K₂, f x ≤ 0 := by aesop\n  have eq₄ : 0 ≤ ∑ x ∈ Finset.univ \\ {Kbar.Source}, f x := Finset.sum_nonneg λ i ↦ by rcases i <;> aesop\n  exact le_trans eq₃ eq₄", "nesting_depth": 9, "transitive_dep_count": 75, "subset_aristotle": false, "category": "Applied verif."}
{"id": 210, "thm_name": "Intmax.proposition4", "thm_stmt": "lemma proposition4 [Setoid' X] {x y : Option X} :\n  (∃ join : Option X, IsLUB {x, y, .none} join) ↔ (x ≠ .none ∧ y ≠ .none → x ≅ y)", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Propositions.lean", "imports": ["import Mathlib.Order.Bounds.Basic", "import FVIntmax.Wheels.Dictionary", "import Aesop", "import Mathlib.Order.Bounds.Defs", "import Mathlib.Order.Defs"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "IsLUB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "IsLeast", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ici", "module": "Mathlib.Order.Interval.Set.Defs"}], "used_repo_defs": [{"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}], "lib_lemmas": [{"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "mem_upperBounds", "module": "Mathlib.Order.Bounds.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Intmax.iso", "content": "def iso {X : Type} [Preorder X] (a b : X) := a ≤ b ∧ b ≤ a"}, {"name": "Intmax.IsEquivRel", "content": "def IsEquivRel {X : Type} [Preorder X] := ∀ a b : X, a ≤ b ↔ a ≅ b"}, {"name": "Intmax.Setoid'", "content": "class Setoid' (X : Type) extends Preorder X where\n  isEquiv : IsEquivRel (X := X)"}], "used_local_lemmas": [{"name": "Intmax.iso_rfl", "content": "@[simp, refl]\nlemma iso_rfl : a ≅ a"}, {"name": "Intmax.iso_trans", "content": "@[trans]\nlemma iso_trans : (a ≅ b) → (b ≅ c) → a ≅ c"}, {"name": "Intmax.proposition2", "content": "lemma proposition2 [Setoid' X] {x y : X} :\n  (∃ join : X, IsLUB {x, y} join) ↔ x ≅ y"}, {"name": "Intmax.proposition2'", "content": "lemma proposition2' [Setoid' X] {join x y : X} (h : IsLUB {x, y} join) :\n  (x ≅ join) ∧ y ≅ join"}, {"name": "Intmax.proposition3'", "content": "lemma proposition3' : \n  (∃ join : X, IsLUB {x, y} join) ↔ (∃ join : Option X, IsLUB {.some x, .some y, .none} join)"}], "local_ctx": "import Mathlib.Order.Bounds.Basic\n\nimport Mathlib.Order.Bounds.Defs\n\nimport Mathlib.Order.Defs\n\nimport Aesop\n\nimport FVIntmax.Wheels.Dictionary\n\nnamespace Intmax\n\ndef iso {X : Type} [Preorder X] (a b : X) := a ≤ b ∧ b ≤ a\n\nnotation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b\n\nsection iso\n\nvariable {X : Type} [Preorder X]\n         {a b c : X}\n\nend iso\n\ndef IsEquivRel {X : Type} [Preorder X] := ∀ a b : X, a ≤ b ↔ a ≅ b\n\nclass Setoid' (X : Type) extends Preorder X where\n  isEquiv : IsEquivRel (X := X)\n\nsection Automation\n\nvariable {α : Type} [Preorder α]\n         {x y : α}\n\nend Automation\n\nsection Propositions\n\nvariable {X Y : Type}\n\nsection proposition3\n\nvariable [Preorder X]\n         {x? y? join? join'? : Option X}\n         {x y join join' : X}\n\nend proposition3\n\nsection Automation\n\nvariable {X : Type} [Setoid' X]\n         {x y : X}\n         {x? y? : Option X}\n\nend Automation", "target_theorem": "lemma proposition4 [Setoid' X] {x y : Option X} :\n  (∃ join : Option X, IsLUB {x, y, .none} join) ↔ (x ≠ .none ∧ y ≠ .none → x ≅ y) :=", "ground_truth_proof": ":= by\n  refine' Iff.intro (λ h ⟨h₁, h₂⟩ ↦ _) (λ h ↦ _)\n  · rcases x with _ | x <;> rcases y with _ | y <;> [rfl; simp at h₁; simp at h₂; skip]\n    simp\n    rw [←proposition3'] at h\n    rcases h with ⟨join, h⟩\n    obtain ⟨h₁, h₂⟩ := proposition2' h\n    exact iso_trans h₁ h₂.symm\n  · rcases x with _ | x <;> rcases y with _ | y\n    · use .none; simp\n    · use .some y; simp\n    · use .some x; simp\n    · simp at h\n      simp_rw [←proposition3']\n      exact proposition2.2 h", "nesting_depth": 3, "transitive_dep_count": 19, "subset_aristotle": false, "category": "Applied verif."}
{"id": 211, "thm_name": "Intmax.sender_transactionsInBlock", "thm_stmt": "lemma sender_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.1)", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Balance.lean", "imports": ["import FVIntmax.BalanceProof", "import FVIntmax.Wheels", "import FVIntmax.Propositions", "import Mathlib", "import FVIntmax.Key", "import FVIntmax.Block", "import Mathlib.Algebra.Group.Int", "import FVIntmax.Wheels.Dictionary", "import FVIntmax.State", "import FVIntmax.Transaction"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Sum", "module": "Init.Core"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "Finset.length_sort", "module": "Mathlib.Data.Finset.Sort"}, {"name": "List.length_attach", "module": "Init.Data.List.Attach"}, {"name": "List.length_map", "module": "Init.Data.List.Lemmas"}, {"name": "exists_and_left", "module": "Init.PropLemmas"}, {"name": "exists_prop", "module": "Init.PropLemmas"}, {"name": "ite_not", "module": "Init.PropLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "List.ext_get", "module": "Init.Data.List.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Intmax.TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}], "used_local_lemmas": [{"name": "Intmax.length_TransactionsInBlock_transfer", "content": "lemma length_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }} :\n  ∀ (π₁ π₂ : BalanceProof K₁ K₂ C Pi V),\n    (TransactionsInBlock_transfer π₁ b).length =\n    (TransactionsInBlock_transfer π₂ b).length"}, {"name": "Intmax.length_transactionsInBlock", "content": "lemma length_transactionsInBlock :\n  (TransactionsInBlock π₁ b).length = (TransactionsInBlock π₂ b).length"}], "local_ctx": "import Mathlib\n\nimport Mathlib.Algebra.Group.Int\n\nimport FVIntmax.BalanceProof\n\nimport FVIntmax.Block\n\nimport FVIntmax.Key\n\nimport FVIntmax.Propositions\n\nimport FVIntmax.State\n\nimport FVIntmax.Transaction\n\nimport FVIntmax.Wheels\n\nimport FVIntmax.Wheels.Dictionary\n\nnamespace Intmax\n\nnoncomputable section\n\nopen Classical\n\nsection Balance\n\nvariable {Pi K₁ K₂ V C Sigma : Type}\n\nsection Extraction\n\nsection Deposit\n\nvariable [PreWithZero V]\n\ndef TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/\n\nend Deposit\n\nsection Transfer\n\nvariable [Finite K₁] [Finite K₂]\n         [LinearOrder K₁] [LinearOrder K₂] [PreWithZero V]\n\ndef TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/\n\nend Transfer\n\nsection Withdrawal\n\nvariable [LinearOrder K₁] [Finite K₁] [PreWithZero V]\n\ndef TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/\n\nend Withdrawal\n\nvariable [Finite K₁] [LinearOrder K₁]\n         [Finite K₂] [LinearOrder K₂]\n         [PreWithZero V]\n         {b : Block K₁ K₂ C Sigma V}\n         {bs : List (Block K₁ K₂ C Sigma V)}\n         {π₁ π₂ : BalanceProof K₁ K₂ C Pi V}\n\ndef TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b", "target_theorem": "lemma sender_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.1) :=", "ground_truth_proof": ":= by\n  apply List.ext_get (by simp; rw [length_transactionsInBlock])\n  intros n h₁ h₂\n  simp; unfold TransactionsInBlock\n  match b with\n  | Block.deposit ..    => simp [TransactionsInBlock_deposit]\n  | Block.transfer ..   => simp [TransactionsInBlock_transfer]\n  | Block.withdrawal .. => simp [TransactionsInBlock_withdrawal]", "nesting_depth": 5, "transitive_dep_count": 48, "subset_aristotle": false, "category": "Applied verif."}
{"id": 212, "thm_name": "Intmax.receiver_transactionsInBlock", "thm_stmt": "lemma receiver_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.2.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.2.1)", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Balance.lean", "imports": ["import FVIntmax.BalanceProof", "import FVIntmax.Wheels", "import FVIntmax.Propositions", "import Mathlib", "import FVIntmax.Key", "import FVIntmax.Block", "import Mathlib.Algebra.Group.Int", "import FVIntmax.Wheels.Dictionary", "import FVIntmax.State", "import FVIntmax.Transaction"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Sum", "module": "Init.Core"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "Finset.length_sort", "module": "Mathlib.Data.Finset.Sort"}, {"name": "List.length_attach", "module": "Init.Data.List.Attach"}, {"name": "List.length_map", "module": "Init.Data.List.Lemmas"}, {"name": "exists_and_left", "module": "Init.PropLemmas"}, {"name": "exists_prop", "module": "Init.PropLemmas"}, {"name": "ite_not", "module": "Init.PropLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "List.ext_get", "module": "Init.Data.List.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Intmax.TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}], "used_local_lemmas": [{"name": "Intmax.length_TransactionsInBlock_transfer", "content": "lemma length_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }} :\n  ∀ (π₁ π₂ : BalanceProof K₁ K₂ C Pi V),\n    (TransactionsInBlock_transfer π₁ b).length =\n    (TransactionsInBlock_transfer π₂ b).length"}, {"name": "Intmax.length_transactionsInBlock", "content": "lemma length_transactionsInBlock :\n  (TransactionsInBlock π₁ b).length = (TransactionsInBlock π₂ b).length"}], "local_ctx": "import Mathlib\n\nimport Mathlib.Algebra.Group.Int\n\nimport FVIntmax.BalanceProof\n\nimport FVIntmax.Block\n\nimport FVIntmax.Key\n\nimport FVIntmax.Propositions\n\nimport FVIntmax.State\n\nimport FVIntmax.Transaction\n\nimport FVIntmax.Wheels\n\nimport FVIntmax.Wheels.Dictionary\n\nnamespace Intmax\n\nnoncomputable section\n\nopen Classical\n\nsection Balance\n\nvariable {Pi K₁ K₂ V C Sigma : Type}\n\nsection Extraction\n\nsection Deposit\n\nvariable [PreWithZero V]\n\ndef TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/\n\nend Deposit\n\nsection Transfer\n\nvariable [Finite K₁] [Finite K₂]\n         [LinearOrder K₁] [LinearOrder K₂] [PreWithZero V]\n\ndef TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/\n\nend Transfer\n\nsection Withdrawal\n\nvariable [LinearOrder K₁] [Finite K₁] [PreWithZero V]\n\ndef TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/\n\nend Withdrawal\n\nvariable [Finite K₁] [LinearOrder K₁]\n         [Finite K₂] [LinearOrder K₂]\n         [PreWithZero V]\n         {b : Block K₁ K₂ C Sigma V}\n         {bs : List (Block K₁ K₂ C Sigma V)}\n         {π₁ π₂ : BalanceProof K₁ K₂ C Pi V}\n\ndef TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b", "target_theorem": "lemma receiver_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.2.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.2.1) :=", "ground_truth_proof": ":= by\n  apply List.ext_get (by simp; rw [length_transactionsInBlock])\n  intros n h₁ h₂\n  simp; unfold TransactionsInBlock\n  match b with\n  | Block.deposit ..    => simp [TransactionsInBlock_deposit]\n  | Block.transfer ..   => simp [TransactionsInBlock_transfer]\n  | Block.withdrawal .. => simp [TransactionsInBlock_withdrawal]", "nesting_depth": 5, "transitive_dep_count": 48, "subset_aristotle": false, "category": "Applied verif."}
{"id": 213, "thm_name": "Intmax.f_withdrawal_block_source", "thm_stmt": "lemma f_withdrawal_block_source (h : b.isWithdrawalBlock) :\n  ((TransactionsInBlock π b).foldl f σ) .Source = σ .Source + ∑ k : K₁, (b.getWithdrawal h k).1 ⊓ σ k", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Balance.lean", "imports": ["import FVIntmax.BalanceProof", "import FVIntmax.Wheels", "import FVIntmax.Propositions", "import Mathlib", "import FVIntmax.Key", "import FVIntmax.Block", "import Mathlib.Algebra.Group.Int", "import FVIntmax.Wheels.Dictionary", "import FVIntmax.State", "import FVIntmax.Transaction"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "IsGLB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "And", "module": "Init.Prelude"}, {"name": "IsGreatest", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "InfSet", "module": "Mathlib.Order.SetNotation"}, {"name": "iInf", "module": "Mathlib.Order.SetNotation"}, {"name": "LinearOrder", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "List.foldl", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Set.range", "module": "Mathlib.Data.Set.Operations"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "Τc", "content": "abbrev Τc (K₁ K₂ V : Type) [PreWithZero V] : Type := { τ : Τ K₁ K₂ V // τ.isComplete }"}, {"name": "isComplete", "content": "def isComplete (τ : Τ K₁ K₂ V) :=\n  match τ with | ⟨(_, _, v), _⟩ => v.isSome"}, {"name": "getWithdrawal", "content": "def getWithdrawal (b : Block K₁ K₂ C Sigma V) (_h : b.isWithdrawalBlock) : K₁ → V₊ :=\n  match b with | .withdrawal B => B"}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "Finset.filter_eq'", "module": "Mathlib.Data.Finset.Basic"}, {"name": "Finset.filter_or", "module": "Mathlib.Data.Finset.Basic"}, {"name": "Finset.mem_univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_const_zero", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Defs"}, {"name": "Finset.sum_ite", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Piecewise"}, {"name": "Finset.sum_singleton", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "Finset.sum_union", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "List.foldl_cons", "module": "Init.Data.List.Basic"}, {"name": "List.map_cons", "module": "Init.Data.List.Basic"}, {"name": "List.mem_cons", "module": "Init.Data.List.Lemmas"}, {"name": "Option.get_some", "module": "Init.Data.Option.Basic"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_left_inj", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_right_inj", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ite_smul", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "neg_smul", "module": "Mathlib.Algebra.Module.Defs"}, {"name": "not_or", "module": "Init.PropLemmas"}, {"name": "one_smul", "module": "Mathlib.Algebra.Group.Action.Defs"}, {"name": "sub_zero", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_smul", "module": "Mathlib.Algebra.GroupWithZero.Action.Defs"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Set.image_eq_range", "module": "Mathlib.Data.Set.Image"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "Finset.mem_sort", "module": "Mathlib.Data.Finset.Sort"}, {"name": "List.bind_eq_flatMap", "module": "Mathlib.Data.List.Basic"}, {"name": "List.flatMap_singleton'", "module": "Init.Data.List.Lemmas"}, {"name": "List.flatMap_subtype", "module": "Init.Data.List.Attach"}, {"name": "List.pure_def", "module": "Mathlib.Data.List.Monad"}, {"name": "List.unattach_attach", "module": "Init.Data.List.Attach"}, {"name": "Set.setOf_true", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.toFinset_univ", "module": "Mathlib.Data.Fintype.Sets"}, {"name": "exists_eq", "module": "Init.PropLemmas"}, {"name": "forall_const", "module": "Init.PropLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Intmax.TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "Intmax.TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}, {"name": "Intmax.e", "content": "def e (i : Kbar K₁ K₂) : Kbar K₁ K₂ → ℤ := λ j ↦ if i = j then 1 else 0"}, {"name": "Intmax.v'", "content": "def v' (v : V₊) (b : S K₁ K₂ V) (s : Kbar K₁ K₂) : V₊ :=\n  match h : s with\n  | .Source => v\n  | .key _  => ⟨v ⊓ b s, by admit /- proof elided -/\n  ⟩"}, {"name": "Intmax.fc", "content": "def fc (τcXb : Τc K₁ K₂ V × S K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨λ k : Kbar K₁ K₂ ↦\n    match τcXb with\n    | ⟨⟨⟨⟨s, r, v⟩, _⟩, hτ⟩, b⟩ =>\n      let v' := v' (v.get hτ) b s\n      b k + (e r - e s) k • v',\n  by admit /- proof elided -/\n  ⟩"}, {"name": "Intmax.boundedBelow", "content": "abbrev boundedBelow (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) :=\n  { a : Τc K₁ K₂ V × S K₁ K₂ V | (T, b) ≤ (↑a.1, a.2) }"}, {"name": "Intmax.V'", "content": "def V' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) : Set V :=\n  { v : V | v ∈ (fc · k) '' boundedBelow b T }"}, {"name": "Intmax.f'", "content": "def f' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V := \n  ⟨\n    λ k ↦\n      match h : T with\n      | ⟨(_, _, .some _), hT⟩ => fc (⟨T, by admit /- proof elided -/\n      ⟩, b) k\n      | ⟨(s, _, .none), _⟩ => if k = s then 0 else b k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Intmax.exists_inf", "content": "def exists_inf (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : { s : S K₁ K₂ V // ∀ k, IsGLB (V' b T k) (s k) } :=\n  ⟨\n    f' b T,\n    λ k ↦\n      have f'_codomain : (f' b T) k ∈ V' b T k := by admit /- proof elided -/\n  ⟩"}, {"name": "Intmax.infV", "content": "def infV (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) :\n  InfSet V where\n    sInf := λ s ↦ if s = V' b T k\n                  then (exists_inf b T).1 k\n                  else 0"}, {"name": "Intmax.f", "content": "def f (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨\n    λ k ↦\n      have : InfSet V := infV b T k\n      ⨅ x : boundedBelow b T, fc x.1 k,\n    by admit /- proof elided -/\n  ⟩"}], "used_local_lemmas": [{"name": "Intmax.e_def", "content": "@[simp]\nlemma e_def : e i = λ j ↦ if i = j then 1 else 0"}, {"name": "Intmax.v'_key_eq_meet", "content": "@[simp]\nlemma v'_key_eq_meet {k : Key K₁ K₂} : v' v b (Kbar.key k) = ⟨v ⊓ b k, by simp⟩"}, {"name": "Intmax.V'_eq_range", "content": "private lemma V'_eq_range {b : S K₁ K₂ V} {T : Τ K₁ K₂ V} {k : Kbar K₁ K₂} :\n  V' b T k =\n  Set.range λ (x : { x : (Τc K₁ K₂ V × S K₁ K₂ V) // (T, b) ≤ (↑x.1, x.2) }) ↦ fc ↑x k"}, {"name": "Intmax.f_eq_f'", "content": "lemma f_eq_f' : f = f' (K₁ := K₁) (K₂ := K₂) (V := V)"}, {"name": "Intmax.f_withdrawal_block_source_aux", "content": "omit [Finite K₂] [LinearOrder K₂] in\nprivate lemma f_withdrawal_block_source_aux {l : List K₁}\n  (h₀ : l.Nodup) (h : b.isWithdrawalBlock) :\n  (List.foldl f' σ\n    (List.map (λ s : K₁ ↦ ⟨(s, Kbar.Source, some (b.getWithdrawal h s)), by unfold Τ'.isValid; aesop⟩) l)).1\n                            .Source = σ .Source + ∑ x : K₁, if x ∈ l then (↑(b.getWithdrawal h x) ⊓ σ x) else 0"}], "local_ctx": "import Mathlib\n\nimport Mathlib.Algebra.Group.Int\n\nimport FVIntmax.BalanceProof\n\nimport FVIntmax.Block\n\nimport FVIntmax.Key\n\nimport FVIntmax.Propositions\n\nimport FVIntmax.State\n\nimport FVIntmax.Transaction\n\nimport FVIntmax.Wheels\n\nimport FVIntmax.Wheels.Dictionary\n\nnamespace Intmax\n\nnoncomputable section\n\nopen Classical\n\nsection Balance\n\nvariable {Pi K₁ K₂ V C Sigma : Type}\n\nsection Extraction\n\nsection Deposit\n\nvariable [PreWithZero V]\n\ndef TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/\n\nend Deposit\n\nsection Transfer\n\nvariable [Finite K₁] [Finite K₂]\n         [LinearOrder K₁] [LinearOrder K₂] [PreWithZero V]\n\ndef TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/\n\nend Transfer\n\nsection Withdrawal\n\nvariable [LinearOrder K₁] [Finite K₁] [PreWithZero V]\n\ndef TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/\n\nend Withdrawal\n\nvariable [Finite K₁] [LinearOrder K₁]\n         [Finite K₂] [LinearOrder K₂]\n         [PreWithZero V]\n         {b : Block K₁ K₂ C Sigma V}\n         {bs : List (Block K₁ K₂ C Sigma V)}\n         {π₁ π₂ : BalanceProof K₁ K₂ C Pi V}\n\ndef TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b\n\nend Extraction\n\nsection e\n\ndef e (i : Kbar K₁ K₂) : Kbar K₁ K₂ → ℤ := λ j ↦ if i = j then 1 else 0\n\nvariable {i j : Kbar K₁ K₂}\n\nend e\n\nsection WithStructuredTypes\n\nsection v'\n\nvariable [Zero V] [Lattice V] -- NB `PreWithZero V` is implied as `CompleteLattice V` gives `Preorder V`.\n\ndef v' (v : V₊) (b : S K₁ K₂ V) (s : Kbar K₁ K₂) : V₊ :=\n  match h : s with\n  | .Source => v\n  | .key _  => ⟨v ⊓ b s, by admit /- proof elided -/\n  ⟩\n\nvariable {v : V₊} {b : S K₁ K₂ V} {s : Kbar K₁ K₂}\n\nend v'\n\nsection Fc\n\nvariable [Lattice V]\n         [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\ndef fc (τcXb : Τc K₁ K₂ V × S K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨λ k : Kbar K₁ K₂ ↦\n    match τcXb with\n    | ⟨⟨⟨⟨s, r, v⟩, _⟩, hτ⟩, b⟩ =>\n      let v' := v' (v.get hτ) b s\n      b k + (e r - e s) k • v',\n  by admit /- proof elided -/\n  ⟩\n\nvariable {τc : Τc K₁ K₂ V} {b : S K₁ K₂ V}\n\nend Fc\n\nsection Order\n\nvariable [Lattice V] [AddCommGroup V]\n\nend Order\n\nsection BoundedBelow\n\nvariable [Lattice V] [AddCommGroup V]\n\nabbrev boundedBelow (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) :=\n  { a : Τc K₁ K₂ V × S K₁ K₂ V | (T, b) ≤ (↑a.1, a.2) }\n\nend BoundedBelow\n\nsection LGroup\n\nvariable [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\ndef V' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) : Set V :=\n  { v : V | v ∈ (fc · k) '' boundedBelow b T }\n\nsection f\n\ndef f' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V := \n  ⟨\n    λ k ↦\n      match h : T with\n      | ⟨(_, _, .some _), hT⟩ => fc (⟨T, by admit /- proof elided -/\n      ⟩, b) k\n      | ⟨(s, _, .none), _⟩ => if k = s then 0 else b k,\n    by admit /- proof elided -/\n  ⟩\n\ndef exists_inf (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : { s : S K₁ K₂ V // ∀ k, IsGLB (V' b T k) (s k) } :=\n  ⟨\n    f' b T,\n    λ k ↦\n      have f'_codomain : (f' b T) k ∈ V' b T k := by admit /- proof elided -/\n  ⟩\n\ndef infV (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) :\n  InfSet V where\n    sInf := λ s ↦ if s = V' b T k\n                  then (exists_inf b T).1 k\n                  else 0\n\ndef f (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨\n    λ k ↦\n      have : InfSet V := infV b T k\n      ⨅ x : boundedBelow b T, fc x.1 k,\n    by admit /- proof elided -/\n  ⟩\n\nend f\n\nsection fStar\n\nvariable {s : S K₁ K₂ V}\n\nend fStar\n\nvariable [Finite K₁] [LinearOrder K₁]\n         [Finite K₂] [LinearOrder K₂]\n\nsection LocalProperties\n\nvariable {σ : S K₁ K₂ V}\n         {π : BalanceProof K₁ K₂ C Pi V}\n         {T : Τ K₁ K₂ V}\n         {b : Block K₁ K₂ C Sigma V}\n         {Sigma : Type}", "target_theorem": "lemma f_withdrawal_block_source (h : b.isWithdrawalBlock) :\n  ((TransactionsInBlock π b).foldl f σ) .Source = σ .Source + ∑ k : K₁, (b.getWithdrawal h k).1 ⊓ σ k :=", "ground_truth_proof": ":= by\n  simp only [TransactionsInBlock]\n  split <;> [simp at h; simp at h; skip]\n  next B =>\n  simp only [f_eq_f', TransactionsInBlock_withdrawal, List.pure_def, List.bind_eq_flatMap,\n    exists_eq, Set.setOf_true, Set.toFinset_univ, Finset.mem_sort, Finset.mem_univ, forall_const,\n    List.flatMap_subtype, List.unattach_attach, List.flatMap_singleton', Block.getWithdrawal]\n  have : (Block.withdrawal B).getWithdrawal h = B := by ext k; simp [Block.getWithdrawal]\n  simp_rw [←this]\n  rw [f_withdrawal_block_source_aux (by simp)]\n  simp [Finset.mem_sort]", "nesting_depth": 6, "transitive_dep_count": 105, "subset_aristotle": false, "category": "Applied verif."}
{"id": 214, "thm_name": "Intmax.monotone_f", "thm_stmt": "lemma monotone_f (h₁ : b₁ ≤ b₂) (h₂ : T₁ ≤ T₂) : f b₁ T₁ k ≤ f b₂ T₂ k", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Lemma4.lean", "imports": ["import FVIntmax.Balance"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "IsGLB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "And", "module": "Init.Prelude"}, {"name": "IsGreatest", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "InfSet", "module": "Mathlib.Order.SetNotation"}, {"name": "iInf", "module": "Mathlib.Order.SetNotation"}], "used_repo_defs": [{"name": "f", "content": "def f (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨\n    λ k ↦\n      have : InfSet V := infV b T k\n      ⨅ x : boundedBelow b T, fc x.1 k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "infV", "content": "def infV (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) :\n  InfSet V where\n    sInf := λ s ↦ if s = V' b T k\n                  then (exists_inf b T).1 k\n                  else 0"}, {"name": "exists_inf", "content": "def exists_inf (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : { s : S K₁ K₂ V // ∀ k, IsGLB (V' b T k) (s k) } :=\n  ⟨\n    f' b T,\n    λ k ↦\n      have f'_codomain : (f' b T) k ∈ V' b T k := by admit /- proof elided -/\n  ⟩"}, {"name": "fc", "content": "def fc (τcXb : Τc K₁ K₂ V × S K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨λ k : Kbar K₁ K₂ ↦\n    match τcXb with\n    | ⟨⟨⟨⟨s, r, v⟩, _⟩, hτ⟩, b⟩ =>\n      let v' := v' (v.get hτ) b s\n      b k + (e r - e s) k • v',\n  by admit /- proof elided -/\n  ⟩"}, {"name": "e", "content": "def e (i : Kbar K₁ K₂) : Kbar K₁ K₂ → ℤ := λ j ↦ if i = j then 1 else 0"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "Τc", "content": "abbrev Τc (K₁ K₂ V : Type) [PreWithZero V] : Type := { τ : Τ K₁ K₂ V // τ.isComplete }"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "boundedBelow", "content": "abbrev boundedBelow (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) :=\n  { a : Τc K₁ K₂ V × S K₁ K₂ V | (T, b) ≤ (↑a.1, a.2) }"}, {"name": "f'", "content": "def f' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V := \n  ⟨\n    λ k ↦\n      match h : T with\n      | ⟨(_, _, .some _), hT⟩ => fc (⟨T, by admit /- proof elided -/\n      ⟩, b) k\n      | ⟨(s, _, .none), _⟩ => if k = s then 0 else b k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "V'", "content": "def V' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) : Set V :=\n  { v : V | v ∈ (fc · k) '' boundedBelow b T }"}, {"name": "isComplete", "content": "def isComplete (τ : Τ K₁ K₂ V) :=\n  match τ with | ⟨(_, _, v), _⟩ => v.isSome"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "le_isGLB_iff", "module": "Mathlib.Order.Bounds.Basic"}, {"name": "mem_lowerBounds", "module": "Mathlib.Order.Bounds.Basic"}], "repo_lemmas": [{"name": "V'_sset_V'_of_le", "content": "lemma V'_sset_V'_of_le {b₁ b₂ : S K₁ K₂ V} {T₁ T₂ : Τ K₁ K₂ V} {k : Kbar K₁ K₂}\n  (h : b₁ ≤ b₂) (h₁ : T₁ ≤ T₂) : \n  V' b₂ T₂ k ⊆ V' b₁ T₁ k"}, {"name": "boundedBelow_sset_boundedBelow_of_le", "content": "lemma boundedBelow_sset_boundedBelow_of_le {b₁ b₂ : S K₁ K₂ V} {T₁ T₂ : Τ K₁ K₂ V}\n  (h : b₁ ≤ b₂) (h₁ : T₁ ≤ T₂) : boundedBelow b₂ T₂ ⊆ boundedBelow b₁ T₁"}, {"name": "f_IsGLB_of_V'", "content": "lemma f_IsGLB_of_V' {b : S K₁ K₂ V} {T : Τ K₁ K₂ V} {k : Kbar K₁ K₂} :\n  IsGLB (V' b T k) (f b T k)"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import FVIntmax.Balance\n\nnamespace Intmax\n\nopen Mathlib\n\nnoncomputable section Lemma4\n\nsection HicSuntDracones\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type} [AddCommGroup V] [Lattice V]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nsection HelperFunctionsToAppeaseLean\n\nopen Mathlib\n\nopen Lean.Elab.Tactic in\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nend\n\nend HelperFunctionsToAppeaseLean\n\nend\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs Bstar : List (Block K₁ K₂ C Sigma V)}\n\nend\n\nend HicSuntDracones\n\nsection\n\nvariable {n : ℕ}\n         {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁]\n         {K₂ : Type} [Finite K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs Bstar : List (Block K₁ K₂ C Sigma V)}\n\nsection Monotone\n\nvariable {b₁ b₂ : S K₁ K₂ V}\n         {T₁ T₂ : Τ K₁ K₂ V}\n         {k : Kbar K₁ K₂}\n         {v₁ v₂ : Vector (Τ K₁ K₂ V) n}", "target_theorem": "lemma monotone_f (h₁ : b₁ ≤ b₂) (h₂ : T₁ ≤ T₂) : f b₁ T₁ k ≤ f b₂ T₂ k :=", "ground_truth_proof": ":= by\n  obtain ⟨inf₁, -⟩ := f_IsGLB_of_V' (b := b₁) (T := T₁) (k := k)\n  have inf₂ := f_IsGLB_of_V' (b := b₂) (T := T₂) (k := k)\n  have := V'_sset_V'_of_le (k := k) h₁ h₂\n  rw [le_isGLB_iff inf₂, mem_lowerBounds]\n  aesop", "nesting_depth": 8, "transitive_dep_count": 36, "subset_aristotle": false, "category": "Applied verif."}
{"id": 215, "thm_name": "Intmax.proposition6", "thm_stmt": "lemma proposition6 [Setoid' Y] {D₁ D₂ : Dict X Y} :\n  (∃ join, IsLUB {D₁, D₂} join) ↔ ∀ x, D₁ x ≠ .none ∧ D₂ x ≠ .none → D₁ x ≅ D₂ x", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Propositions.lean", "imports": ["import Mathlib.Order.Bounds.Basic", "import FVIntmax.Wheels.Dictionary", "import Aesop", "import Mathlib.Order.Bounds.Defs", "import Mathlib.Order.Defs"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "IsLUB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "IsLeast", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Iff", "module": "Init.Core"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Ici", "module": "Mathlib.Order.Interval.Set.Defs"}, {"name": "Function.eval", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}], "used_repo_defs": [{"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}, {"name": "Merge", "content": "def Merge (D₁ D₂ : Dict α ω) : Dict α ω := D\n  where D := λ x ↦ First (D₁ x) (D₂ x)"}, {"name": "First", "content": "def First (x₁ x₂ : Option α) : Option α :=\n  match x₁, x₂ with\n  | .some x, .none => .some x\n  | .some x, .some _ => .some x\n  | .none, .some y => .some y\n  | .none, .none => .none"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂", "content": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂"}], "lib_lemmas": [{"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "mem_upperBounds", "module": "Mathlib.Order.Bounds.Basic"}, {"name": "isLUB_pi", "module": "Mathlib.Order.Bounds.Image"}], "repo_lemmas": [{"name": "mem_iff_isSome", "content": "lemma mem_iff_isSome {m : Dict α ω} {x : α} : x ∈ m ↔ (m x).isSome"}], "used_local_defs": [{"name": "Intmax.iso", "content": "def iso {X : Type} [Preorder X] (a b : X) := a ≤ b ∧ b ≤ a"}, {"name": "Intmax.IsEquivRel", "content": "def IsEquivRel {X : Type} [Preorder X] := ∀ a b : X, a ≤ b ↔ a ≅ b"}, {"name": "Intmax.Setoid'", "content": "class Setoid' (X : Type) extends Preorder X where\n  isEquiv : IsEquivRel (X := X)"}], "used_local_lemmas": [{"name": "Intmax.iso_rfl", "content": "@[simp, refl]\nlemma iso_rfl : a ≅ a"}, {"name": "Intmax.iso_trans", "content": "@[trans]\nlemma iso_trans : (a ≅ b) → (b ≅ c) → a ≅ c"}, {"name": "Intmax.proposition2", "content": "lemma proposition2 [Setoid' X] {x y : X} :\n  (∃ join : X, IsLUB {x, y} join) ↔ x ≅ y"}, {"name": "Intmax.proposition2'", "content": "lemma proposition2' [Setoid' X] {join x y : X} (h : IsLUB {x, y} join) :\n  (x ≅ join) ∧ y ≅ join"}, {"name": "Intmax.proposition3'", "content": "lemma proposition3' : \n  (∃ join : X, IsLUB {x, y} join) ↔ (∃ join : Option X, IsLUB {.some x, .some y, .none} join)"}, {"name": "Intmax.proposition4", "content": "lemma proposition4 [Setoid' X] {x y : Option X} :\n  (∃ join : Option X, IsLUB {x, y, .none} join) ↔ (x ≠ .none ∧ y ≠ .none → x ≅ y)"}, {"name": "Intmax.proposition5", "content": "lemma proposition5 [Preorder Y] {f g : X → Y} {join : X → Y} :\n  IsLUB {f, g} join ↔ ∀ x : X, IsLUB {f x, g x} (join x)"}, {"name": "Intmax.proposition6_aux", "content": "lemma proposition6_aux [Setoid' Y] {D₁ D₂ : Dict X Y}\n  (h : ∀ k, D₁ k ≠ .none ∧ D₂ k ≠ .none → D₁ k ≅ D₂ k) : IsLUB {D₁, D₂} (Dict.Merge D₁ D₂)"}], "local_ctx": "import Mathlib.Order.Bounds.Basic\n\nimport Mathlib.Order.Bounds.Defs\n\nimport Mathlib.Order.Defs\n\nimport Aesop\n\nimport FVIntmax.Wheels.Dictionary\n\nnamespace Intmax\n\ndef iso {X : Type} [Preorder X] (a b : X) := a ≤ b ∧ b ≤ a\n\nnotation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b\n\nsection iso\n\nvariable {X : Type} [Preorder X]\n         {a b c : X}\n\nend iso\n\ndef IsEquivRel {X : Type} [Preorder X] := ∀ a b : X, a ≤ b ↔ a ≅ b\n\nclass Setoid' (X : Type) extends Preorder X where\n  isEquiv : IsEquivRel (X := X)\n\nsection Automation\n\nvariable {α : Type} [Preorder α]\n         {x y : α}\n\nend Automation\n\nsection Propositions\n\nvariable {X Y : Type}\n\nsection proposition3\n\nvariable [Preorder X]\n         {x? y? join? join'? : Option X}\n         {x y join join' : X}\n\nend proposition3\n\nsection Automation\n\nvariable {X : Type} [Setoid' X]\n         {x y : X}\n         {x? y? : Option X}\n\nend Automation", "target_theorem": "lemma proposition6 [Setoid' Y] {D₁ D₂ : Dict X Y} :\n  (∃ join, IsLUB {D₁, D₂} join) ↔ ∀ x, D₁ x ≠ .none ∧ D₂ x ≠ .none → D₁ x ≅ D₂ x :=", "ground_truth_proof": ":= by\n  refine' ⟨λ h ↦ _, λ h ↦ _⟩\n  simp_rw [proposition5] at h\n  simp_rw [←proposition4]\n  aesop\n  use D₁.Merge D₂\n  exact proposition6_aux h", "nesting_depth": 4, "transitive_dep_count": 30, "subset_aristotle": false, "category": "Applied verif."}
{"id": 216, "thm_name": "Intmax.proposition6'", "thm_stmt": "lemma proposition6' [Setoid' Y] {D₁ D₂ join : Dict X Y} (h : IsLUB {D₁, D₂} join) :\n  join ≅ Dict.Merge D₁ D₂", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Propositions.lean", "imports": ["import Mathlib.Order.Bounds.Basic", "import FVIntmax.Wheels.Dictionary", "import Aesop", "import Mathlib.Order.Bounds.Defs", "import Mathlib.Order.Defs"], "used_lib_defs": [{"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "IsLUB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Function.eval", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.image", "module": "Mathlib.Data.Set.Defs"}, {"name": "Iff", "module": "Init.Core"}, {"name": "IsLeast", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "Set.Ici", "module": "Mathlib.Order.Interval.Set.Defs"}], "used_repo_defs": [{"name": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b", "content": "notation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b"}, {"name": "Merge", "content": "def Merge (D₁ D₂ : Dict α ω) : Dict α ω := D\n  where D := λ x ↦ First (D₁ x) (D₂ x)"}, {"name": "First", "content": "def First (x₁ x₂ : Option α) : Option α :=\n  match x₁, x₂ with\n  | .some x, .none => .some x\n  | .some x, .some _ => .some x\n  | .none, .some y => .some y\n  | .none, .none => .none"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂", "content": "notation:65 π₁:65 \" <+> \" π₂:66 => Dict.Merge π₁ π₂"}], "lib_lemmas": [{"name": "isLUB_pi", "module": "Mathlib.Order.Bounds.Image"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "mem_upperBounds", "module": "Mathlib.Order.Bounds.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Intmax.iso", "content": "def iso {X : Type} [Preorder X] (a b : X) := a ≤ b ∧ b ≤ a"}, {"name": "Intmax.IsEquivRel", "content": "def IsEquivRel {X : Type} [Preorder X] := ∀ a b : X, a ≤ b ↔ a ≅ b"}, {"name": "Intmax.Setoid'", "content": "class Setoid' (X : Type) extends Preorder X where\n  isEquiv : IsEquivRel (X := X)"}], "used_local_lemmas": [{"name": "Intmax.iso_rfl", "content": "@[simp, refl]\nlemma iso_rfl : a ≅ a"}, {"name": "Intmax.iso_symm", "content": "@[symm]\nlemma iso_symm : (a ≅ b) ↔ b ≅ a"}, {"name": "Intmax.iso_trans", "content": "@[trans]\nlemma iso_trans : (a ≅ b) → (b ≅ c) → a ≅ c"}, {"name": "Intmax.proposition2", "content": "lemma proposition2 [Setoid' X] {x y : X} :\n  (∃ join : X, IsLUB {x, y} join) ↔ x ≅ y"}, {"name": "Intmax.proposition2'", "content": "lemma proposition2' [Setoid' X] {join x y : X} (h : IsLUB {x, y} join) :\n  (x ≅ join) ∧ y ≅ join"}, {"name": "Intmax.proposition3'", "content": "lemma proposition3' : \n  (∃ join : X, IsLUB {x, y} join) ↔ (∃ join : Option X, IsLUB {.some x, .some y, .none} join)"}, {"name": "Intmax.proposition4", "content": "lemma proposition4 [Setoid' X] {x y : Option X} :\n  (∃ join : Option X, IsLUB {x, y, .none} join) ↔ (x ≠ .none ∧ y ≠ .none → x ≅ y)"}, {"name": "Intmax.proposition4'", "content": "lemma proposition4' [Setoid' X] {join x y : Option X} (h : IsLUB {x, y, .none} join) :\n  join ≅ Dict.First x y"}, {"name": "Intmax.proposition5", "content": "lemma proposition5 [Preorder Y] {f g : X → Y} {join : X → Y} :\n  IsLUB {f, g} join ↔ ∀ x : X, IsLUB {f x, g x} (join x)"}, {"name": "Intmax.proposition5'", "content": "lemma proposition5' [Preorder Y] {f g h join' : X → Y}\n  (h₀ : IsLUB {f, g} join')\n  (h₁ : ∀ x, h x ≅ join' x) :\n  join' ≅ h"}], "local_ctx": "import Mathlib.Order.Bounds.Basic\n\nimport Mathlib.Order.Bounds.Defs\n\nimport Mathlib.Order.Defs\n\nimport Aesop\n\nimport FVIntmax.Wheels.Dictionary\n\nnamespace Intmax\n\ndef iso {X : Type} [Preorder X] (a b : X) := a ≤ b ∧ b ≤ a\n\nnotation:51 (priority := high) a:52 \" ≅ \" b:52 => iso a b\n\nsection iso\n\nvariable {X : Type} [Preorder X]\n         {a b c : X}\n\nend iso\n\ndef IsEquivRel {X : Type} [Preorder X] := ∀ a b : X, a ≤ b ↔ a ≅ b\n\nclass Setoid' (X : Type) extends Preorder X where\n  isEquiv : IsEquivRel (X := X)\n\nsection Automation\n\nvariable {α : Type} [Preorder α]\n         {x y : α}\n\nend Automation\n\nsection Propositions\n\nvariable {X Y : Type}\n\nsection proposition3\n\nvariable [Preorder X]\n         {x? y? join? join'? : Option X}\n         {x y join join' : X}\n\nend proposition3\n\nsection Automation\n\nvariable {X : Type} [Setoid' X]\n         {x y : X}\n         {x? y? : Option X}\n\nend Automation", "target_theorem": "lemma proposition6' [Setoid' Y] {D₁ D₂ join : Dict X Y} (h : IsLUB {D₁, D₂} join) :\n  join ≅ Dict.Merge D₁ D₂ :=", "ground_truth_proof": ":= by\n  unfold Dict.Merge Dict.Merge.D\n  apply proposition5' (h₀ := h)\n  intros x\n  rw [iso_symm]\n  apply proposition4'\n  revert x\n  rw [proposition5] at h\n  simpa", "nesting_depth": 5, "transitive_dep_count": 30, "subset_aristotle": false, "category": "Applied verif."}
{"id": 217, "thm_name": "Intmax.computeBalance'_eq_zero", "thm_stmt": "lemma computeBalance'_eq_zero : computeBalance' σ v = v + computeBalance' σ 0", "lean_root": "FVIntmax", "rel_path": "FVIntmax/AttackGame.lean", "imports": ["import FVIntmax.Wheels.AuthenticatedDictionary", "import FVIntmax.Request", "import FVIntmax.Block", "import FVIntmax.Wheels.SignatureAggregation", "import FVIntmax.BalanceProof", "import FVIntmax.Wheels"], "used_lib_defs": [{"name": "Sigma", "module": "Init.Core"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "Scontract", "content": "abbrev Scontract (K₁ K₂ V : Type) [PreWithZero V] (C Sigma : Type) :=\n  List (Block K₁ K₂ C Sigma V)"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "sub_eq_add_neg", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Intmax.Block.updateBalance", "content": "def Block.updateBalance (bal : V) (block : Block K₁ K₂ C Sigma V) : V :=\n  match block with\n   \n  | .deposit _ v  => bal + v\n   \n  | .transfer ..  => bal\n   \n  | .withdrawal B => bal - ∑ k : K₁, (B k).1"}, {"name": "Intmax.computeBalance'", "content": "def computeBalance' (blocks : Scontract K₁ K₂ V C Sigma) (acc : V) : V :=\n  blocks.foldl Block.updateBalance acc"}], "used_local_lemmas": [{"name": "Intmax.Block.updateBalance_eq_zero", "content": "lemma Block.updateBalance_eq_zero :\n  block.updateBalance v = v + block.updateBalance 0"}, {"name": "Intmax.computeBalance'_cons", "content": "@[simp]\nlemma computeBalance'_cons : computeBalance' (hd :: σ) v =\n                             computeBalance' σ (Block.updateBalance v hd)"}], "local_ctx": "import FVIntmax.Wheels.AuthenticatedDictionary\n\nimport FVIntmax.Wheels.SignatureAggregation\n\nimport FVIntmax.BalanceProof\n\nimport FVIntmax.Block\n\nimport FVIntmax.Request\n\nimport FVIntmax.Wheels\n\nnamespace Intmax\n\nnoncomputable section Intmax\n\nsection RollupContract\n\nopen Classical\n\nsection\n\nvariable {C : Type} [Nonempty C]\n\n         {V : Type} [Lattice V] [AddCommGroup V]\n         \n         {K₁ : Type} [Finite K₁] [Nonempty K₁]\n         {K₂ : Type} [Finite K₂]\n         {Sigma : Type}\n         [ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n         [SA : SignatureAggregation (C × K₁ × ExtraDataT) K₂ KₛT Sigma]\n\ndef Block.updateBalance (bal : V) (block : Block K₁ K₂ C Sigma V) : V :=\n  match block with\n   \n  | .deposit _ v  => bal + v\n   \n  | .transfer ..  => bal\n   \n  | .withdrawal B => bal - ∑ k : K₁, (B k).1\n\nend\n\nsection\n\nvariable {C : Type}\n         {V : Type} [Lattice V] [AddCommGroup V]\n         {K₁ : Type} [Finite K₁]\n         {K₂ : Type}\n         {Sigma : Type}\n         {block : Block K₁ K₂ C Sigma V}\n         {v : V}\n\nsection Lemmas\n\nend Lemmas\n\nnamespace Scontract\n\nsection\n\nvariable [LinearOrder K₁] [Nonempty K₁]\n         [LinearOrder K₂] [Finite K₂]\n         [Nonempty C]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         [ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n         [SignatureAggregation (C × K₁ × ExtraDataT) K₂ KₛT Sigma]\n\nsection appendBlock\n\nvariable {σ : Scontract K₁ K₂ V C Sigma} {request : Request K₁ K₂ C Sigma Pi V}\n\nend appendBlock\n\nend\n\nsection\n\nvariable [LinearOrder K₁]\n         [Finite K₂] [LinearOrder K₂]\n         {σ : Scontract K₁ K₂ V C Sigma}\n         {request : Request K₁ K₂ C Sigma Pi V}\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\nend\n\nend Scontract\n\nend\n\nend RollupContract\n\nsection AttackGameDefs\n\nvariable {K₁ : Type} [Finite K₁] [LinearOrder K₁] [Nonempty K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n\n         {V : Type} [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n\n         {Sigma : Type}\n         {C : Type} [Nonempty C]\n\n         [ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n         [SignatureAggregation (C × K₁ × ExtraDataT) K₂ KₛT Sigma]\n\n         (requests : List (Request K₁ K₂ C Sigma Pi V))\n         (σ : Scontract K₁ K₂ V C Sigma)\n\ndef computeBalance' (blocks : Scontract K₁ K₂ V C Sigma) (acc : V) : V :=\n  blocks.foldl Block.updateBalance acc\n\nend AttackGameDefs\n\nsection AttackGameLemmas\n\nvariable {K₁ K₂ Sigma C : Type}\n         {V : Type} [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]\n\nsection computeBalanceSum\n\nvariable {k : ℕ}\n\nend computeBalanceSum\n\nvariable [Finite K₁]\n\nend AttackGameLemmas\n\nsection ComputeLemmas\n\nvariable {K₁ : Type} [Finite K₁] {K₂ Sigma C : Type}\n         {V : Type} [Lattice V] [AddCommGroup V]\n         {σ : Scontract K₁ K₂ V C Sigma}\n         [ADScheme K₂ (C × K₁ × ExtraDataT) C Pi]", "target_theorem": "lemma computeBalance'_eq_zero : computeBalance' σ v = v + computeBalance' σ 0 :=", "ground_truth_proof": ":= by\n  induction' σ with hd tl ih generalizing v\n  · simp\n  · rw [computeBalance'_cons, computeBalance'_cons, ih, Block.updateBalance_eq_zero]\n    nth_rw 2 [ih]\n    exact add_assoc v _ _", "nesting_depth": 3, "transitive_dep_count": 15, "subset_aristotle": false, "category": "Applied verif."}
{"id": 218, "thm_name": "evalFuel_sound", "thm_stmt": "theorem evalFuel_sound  : evalFuel n c = some v → c ⤳⋆ v", "lean_root": "IntroEffects", "rel_path": "IntroEffects/Eval.lean", "imports": ["import IntroEffects.SmallStep"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "scoped syntax num : embedded", "content": "scoped syntax num : embedded"}, {"name": "instantiateComp", "content": "def instantiateComp (what : Value) (comp : Computation) : Computation :=\n  instantiateComputationLvl what 0 comp"}, {"name": "instantiateComputationLvl", "content": "def instantiateComputationLvl (what : Value) (level : Nat) : Computation → Computation\n| .ret v => .ret <| instantiateValueLvl what level v\n| .opCall op v c => .opCall op (instantiateValueLvl what level v) (instantiateComputationLvl what (level+1) c)\n| .bind c₁ c₂ => .bind (instantiateComputationLvl what level c₁) (instantiateComputationLvl what (level+1) c₂)\n| .ite v c₁ c₂ => .ite (instantiateValueLvl what level v) (instantiateComputationLvl what level c₁) (instantiateComputationLvl what level c₂)\n| .app v₁ v₂ => .app (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .handle v c => .handle (instantiateValueLvl what level v) (instantiateComputationLvl what level c)\n| .join v₁ v₂ => .join (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .fst v => .fst (instantiateValueLvl what level v)\n| .snd v => .snd (instantiateValueLvl what level v)\n| .add v₁ v₂ => .add (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .sub v₁ v₂ => .sub (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .max v₁ v₂ => .max (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .lt v₁ v₂ => .lt (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .mul v₁ v₂ => .mul (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .eq v₁ v₂ => .eq (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\ntermination_by c => sizeOf c\ndecreasing_by\n  all_goals simp\n  all_goals grind"}, {"name": "instantiateValueLvl", "content": "def instantiateValueLvl (what : Value) (level : Nat) : Value → Value\n| var@(.var (.bvar bvarLevel)) => if bvarLevel = level then what else var\n| .lam c => .lam <| instantiateComputationLvl what (level + 1) c\n| .hdl h => .hdl <| instantiateHandlerLvl what level h\n| .pair v₁ v₂ => .pair (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .recfun c => .recfun <| instantiateComputationLvl what (level + 2) c\n| .string s => .string s\n| .bool b => .bool b\n| .unit => .unit\n| .var v => .var v\n| .num n => .num n\ntermination_by v => sizeOf v\ndecreasing_by\n  all_goals simp\n  all_goals grind"}, {"name": "instantiateHandlerLvl", "content": "def instantiateHandlerLvl (what : Value) (level : Nat) : Handler → Handler\n| ⟨ret, ops'⟩ =>\n    ⟨instantiateComputationLvl what (level+1) ret, instantiateOps what level ops'⟩\ntermination_by h => sizeOf h\ndecreasing_by\n  all_goals simp\n  all_goals grind"}, {"name": "instantiateOps", "content": "def instantiateOps (what : Value) (level : Nat) : List OpClause → List OpClause\n| [] => []\n| ⟨op, body⟩ :: ls =>\n  ⟨op, instantiateComputationLvl what (level+2) body⟩ :: instantiateOps what level ls\ntermination_by l => sizeOf l"}, {"name": "OpClause", "content": "structure OpClause where\n  op : Name\n   \n  body : Computation\nderiving Repr, BEq"}, {"name": "Computation", "content": "inductive Computation where\n| ret : Value → Computation\n \n| opCall : Name → Value → Computation → Computation\n \n| bind : Computation → Computation → Computation\n| ite (val : Value) (trueBranch falseBranch : Computation)\n| app : Value → Value → Computation\n \n| handle : Value → Computation → Computation\n| join : Value → Value → Computation\n| fst : Value → Computation\n| snd : Value → Computation\n| add : Value → Value → Computation\n| sub : Value → Value → Computation\n| mul : Value → Value → Computation\n| max : Value → Value → Computation\n| lt : Value → Value → Computation\n| eq : Value → Value → Computation\nderiving BEq"}, {"name": "Value", "content": "inductive Value where\n| var : Var → Value\n| bool : Bool → Value\n| string : String → Value\n| num : Int → Value\n| unit : Value\n| pair : Value → Value → Value\n \n| lam : Computation → Value\n \n| recfun : Computation → Value\n| hdl : Handler → Value\nderiving BEq"}, {"name": "Handler", "content": "structure Handler where\n   \n  ret : Computation\n  ops : List OpClause\nderiving Repr, BEq"}, {"name": "Var", "content": "inductive Var where\n| fvar : Name → Var\n| bvar : Nat → Var\nderiving Repr, DecidableEq"}, {"name": "Name", "content": "abbrev Name := String"}, {"name": "Step", "content": "@[grind cases]\ninductive Step : Computation → Computation → Prop\n \n| beta v body : Step (.app (.lam body) v) (instantiateComp v body)\n \n| recBeta v body : Step (.app (.recfun body) v) (instantiate2 (.recfun body) v body)\n \n| iteTrue c₁ c₂ : Step (.ite (.bool true) c₁ c₂) c₁\n \n| iteFalse c₁ c₂ : Step (.ite (.bool false) c₁ c₂) c₂\n \n| bindStep c₁ c₁' c₂ (h : Step c₁ c₁') : Step (.bind c₁ c₂) (.bind c₁' c₂)\n \n| bindReturn v c : Step (.bind (.ret v) c) (instantiateComp v c)\n \n| bindOp op v body c : Step (.bind (.opCall op v body) c) (.opCall op v (.bind body c))\n \n| handleInner h c₁ c₂ (h₁ : Step c₁ c₂) : Step (.handle (.hdl h) c₁) (.handle (.hdl h) c₂)\n \n| handleReturn h v :\n  Step (.handle (.hdl h) (.ret v)) (instantiateComp v h.ret)\n \n| handleOpHit h op v body c (hop : h.lookup op = some ⟨op, c⟩) :\n  Step (.handle (.hdl h) (.opCall op v body))\n    (instantiate2 v (.lam (.handle (.hdl h) body)) c)\n \n| handleOpMiss h op v body (hop : h.lookup op = none) :\n  Step (.handle (.hdl h) (.opCall op v body)) (.opCall op v (.handle (.hdl h) body))\n \n| join s₁ s₂ : Step (.join (.string s₁) (.string s₂)) (.ret (.string (strAppend s₁ s₂)))\n \n| fstStep v₁ v₂ : Step (.fst (.pair v₁ v₂)) (.ret v₁)\n \n| sndStep v₁ v₂ : Step (.snd (.pair v₁ v₂)) (.ret v₂)\n| add v₁ v₂ : Step (.add (.num v₁) (.num v₂)) (.ret (.num (v₁ + v₂)))\n| sub v₁ v₂ : Step (.sub (.num v₁) (.num v₂)) (.ret (.num (v₁ - v₂)))\n| max v₁ v₂ : Step (.max (.num v₁) (.num v₂)) (.ret (.num (Max.max v₁ v₂)))\n| lt v₁ v₂ : Step (.lt (.num v₁) (.num v₂)) (.ret (.bool (v₁ < v₂)))\n| mul v₁ v₂ : Step (.mul (.num v₁) (.num v₂)) (.ret (.num (v₁ * v₂)))\n| eq v₁ v₂ : Step (.eq v₁ v₂) (.ret (.bool (v₁ == v₂)))"}, {"name": "instantiate2", "content": "def instantiate2 (arg cont : Value) (body : Computation) : Computation :=\n  instantiateComputationLvl arg 1 (instantiateComputationLvl cont 0 body)"}, {"name": "strAppend", "content": "def strAppend : String → String → String\n| \"\", s => s\n| s, \"\" => s\n| s₁, s₂ => s₁ ++ \" \" ++ s₂"}, {"name": "StepStar", "content": "@[grind cases]\ninductive StepStar : Computation → Computation → Prop\n| refl (c) : StepStar c c\n| trans : Step c₁ c₂ → StepStar c₂ c₃ → StepStar c₁ c₃"}, {"name": "Handler.lookup", "content": "def Handler.lookup (hdl : Handler) (name : Name) :=\n  hdl.ops.find? (·.op == name)"}, {"name": "infix:50 \" ⤳ \" => Step", "content": "infix:50 \" ⤳ \" => Step"}, {"name": "infix:50 \" ⤳⋆ \" => StepStar", "content": "infix:50 \" ⤳⋆ \" => StepStar"}], "lib_lemmas": [{"name": "List.find?_some", "module": "Init.Data.List.Find"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "evalSingleStep", "content": "def evalSingleStep : Computation → Option Computation\n \n| .app (.lam body) v => some <| instantiateComp v body\n \n| .app (.recfun body) v => some <| instantiate2 (.recfun body) v body\n| .ite (.bool true) c₁ _ => some c₁\n| .ite (.bool false) _ c₂ => some c₂\n \n| .bind (.ret v) c => some <| instantiateComp v c\n \n| .bind (.opCall op v body) c => some <| .opCall op v (.bind body c)\n| .bind c₁ c₂ => (evalSingleStep c₁).map (fun c₁' => .bind c₁' c₂)\n   \n| .handle (.hdl h) (.ret v) => some <| instantiateComp v h.ret\n| .handle (.hdl h) (.opCall op v body) =>\n  match h.lookup op with\n   \n  | some ⟨_, c⟩ =>\n    let cont := .lam (.handle (.hdl h) body)\n    some <| instantiate2 v cont c\n   \n  | none => some <| .opCall op v (.handle (.hdl h) body)\n| .handle (.hdl h) c =>\n  (evalSingleStep c).map (fun c' => .handle (.hdl h) c')\n| .join (.string s₁) (.string s₂) => some <| .ret (.string (strAppend s₁ s₂))\n| .fst (.pair v₁ _) => some <| .ret v₁\n| .snd (.pair _ v₂) => some <| .ret v₂\n| .add (.num v₁) (.num v₂) => some <| .ret (.num (v₁ + v₂))\n| .sub (.num v₁) (.num v₂) => some <| .ret (.num (v₁ - v₂))\n| .max (.num v₁) (.num v₂) => some <| .ret (.num (Max.max v₁ v₂))\n| .lt (.num v₁) (.num v₂) => some <| .ret (.bool (v₁ < v₂))\n| .mul (.num v₁) (.num v₂) => some <| .ret (.num (v₁ * v₂))\n| .eq v₁ v₂ => some <| .ret (.bool (v₁ == v₂))\n| _ => none"}, {"name": "evalFuel", "content": "def evalFuel : Nat → Computation → Option Computation\n| 0, _ => none\n| _+1, .ret v => some <| .ret v\n| _+1, .opCall n v c => some <| .opCall n v c\n| n+1, c =>\n  match evalSingleStep c with\n  | some c' => evalFuel n c'\n  | none => none"}], "used_local_lemmas": [{"name": "evalSingleStep_sound", "content": "theorem evalSingleStep_sound {c c' : Computation} :\n    evalSingleStep c = some c' → c ⤳ c'"}], "local_ctx": "import IntroEffects.SmallStep\n\ndef evalSingleStep : Computation → Option Computation\n \n| .app (.lam body) v => some <| instantiateComp v body\n \n| .app (.recfun body) v => some <| instantiate2 (.recfun body) v body\n| .ite (.bool true) c₁ _ => some c₁\n| .ite (.bool false) _ c₂ => some c₂\n \n| .bind (.ret v) c => some <| instantiateComp v c\n \n| .bind (.opCall op v body) c => some <| .opCall op v (.bind body c)\n| .bind c₁ c₂ => (evalSingleStep c₁).map (fun c₁' => .bind c₁' c₂)\n   \n| .handle (.hdl h) (.ret v) => some <| instantiateComp v h.ret\n| .handle (.hdl h) (.opCall op v body) =>\n  match h.lookup op with\n   \n  | some ⟨_, c⟩ =>\n    let cont := .lam (.handle (.hdl h) body)\n    some <| instantiate2 v cont c\n   \n  | none => some <| .opCall op v (.handle (.hdl h) body)\n| .handle (.hdl h) c =>\n  (evalSingleStep c).map (fun c' => .handle (.hdl h) c')\n| .join (.string s₁) (.string s₂) => some <| .ret (.string (strAppend s₁ s₂))\n| .fst (.pair v₁ _) => some <| .ret v₁\n| .snd (.pair _ v₂) => some <| .ret v₂\n| .add (.num v₁) (.num v₂) => some <| .ret (.num (v₁ + v₂))\n| .sub (.num v₁) (.num v₂) => some <| .ret (.num (v₁ - v₂))\n| .max (.num v₁) (.num v₂) => some <| .ret (.num (Max.max v₁ v₂))\n| .lt (.num v₁) (.num v₂) => some <| .ret (.bool (v₁ < v₂))\n| .mul (.num v₁) (.num v₂) => some <| .ret (.num (v₁ * v₂))\n| .eq v₁ v₂ => some <| .ret (.bool (v₁ == v₂))\n| _ => none\n\ndef evalFuel : Nat → Computation → Option Computation\n| 0, _ => none\n| _+1, .ret v => some <| .ret v\n| _+1, .opCall n v c => some <| .opCall n v c\n| n+1, c =>\n  match evalSingleStep c with\n  | some c' => evalFuel n c'\n  | none => none", "target_theorem": "theorem evalFuel_sound  : evalFuel n c = some v → c ⤳⋆ v :=", "ground_truth_proof": ":= by\n  induction h : n generalizing c with\n  | zero => simp [evalFuel]\n  | succ n ih =>\n    cases hstep : evalSingleStep c with\n    | none =>\n      cases c <;> try grind [evalFuel]\n      all_goals\n      ( simp [evalFuel]\n        intro h; rw [←h]; constructor)\n    | some c' =>\n      cases c with\n      | ret | opCall =>\n        simp [evalFuel]\n        intro h; rw [←h]; constructor\n      | _ =>\n        simp [evalFuel, hstep]\n        intro h\n        exact StepStar.trans (evalSingleStep_sound hstep) (evalFuel_sound h)", "nesting_depth": 8, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Semantics"}
{"id": 219, "thm_name": "evalFuelRet_complete_aux", "thm_stmt": "theorem evalFuelRet_complete_aux (h : c ⤳⋆ r) :\n    ∀v, r= .ret v → ∃n, evalFuel n c = some (.ret v)", "lean_root": "IntroEffects", "rel_path": "IntroEffects/Eval.lean", "imports": ["import IntroEffects.SmallStep"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "scoped syntax num : embedded", "content": "scoped syntax num : embedded"}, {"name": "instantiateComp", "content": "def instantiateComp (what : Value) (comp : Computation) : Computation :=\n  instantiateComputationLvl what 0 comp"}, {"name": "instantiateComputationLvl", "content": "def instantiateComputationLvl (what : Value) (level : Nat) : Computation → Computation\n| .ret v => .ret <| instantiateValueLvl what level v\n| .opCall op v c => .opCall op (instantiateValueLvl what level v) (instantiateComputationLvl what (level+1) c)\n| .bind c₁ c₂ => .bind (instantiateComputationLvl what level c₁) (instantiateComputationLvl what (level+1) c₂)\n| .ite v c₁ c₂ => .ite (instantiateValueLvl what level v) (instantiateComputationLvl what level c₁) (instantiateComputationLvl what level c₂)\n| .app v₁ v₂ => .app (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .handle v c => .handle (instantiateValueLvl what level v) (instantiateComputationLvl what level c)\n| .join v₁ v₂ => .join (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .fst v => .fst (instantiateValueLvl what level v)\n| .snd v => .snd (instantiateValueLvl what level v)\n| .add v₁ v₂ => .add (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .sub v₁ v₂ => .sub (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .max v₁ v₂ => .max (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .lt v₁ v₂ => .lt (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .mul v₁ v₂ => .mul (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .eq v₁ v₂ => .eq (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\ntermination_by c => sizeOf c\ndecreasing_by\n  all_goals simp\n  all_goals grind"}, {"name": "instantiateValueLvl", "content": "def instantiateValueLvl (what : Value) (level : Nat) : Value → Value\n| var@(.var (.bvar bvarLevel)) => if bvarLevel = level then what else var\n| .lam c => .lam <| instantiateComputationLvl what (level + 1) c\n| .hdl h => .hdl <| instantiateHandlerLvl what level h\n| .pair v₁ v₂ => .pair (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .recfun c => .recfun <| instantiateComputationLvl what (level + 2) c\n| .string s => .string s\n| .bool b => .bool b\n| .unit => .unit\n| .var v => .var v\n| .num n => .num n\ntermination_by v => sizeOf v\ndecreasing_by\n  all_goals simp\n  all_goals grind"}, {"name": "instantiateHandlerLvl", "content": "def instantiateHandlerLvl (what : Value) (level : Nat) : Handler → Handler\n| ⟨ret, ops'⟩ =>\n    ⟨instantiateComputationLvl what (level+1) ret, instantiateOps what level ops'⟩\ntermination_by h => sizeOf h\ndecreasing_by\n  all_goals simp\n  all_goals grind"}, {"name": "instantiateOps", "content": "def instantiateOps (what : Value) (level : Nat) : List OpClause → List OpClause\n| [] => []\n| ⟨op, body⟩ :: ls =>\n  ⟨op, instantiateComputationLvl what (level+2) body⟩ :: instantiateOps what level ls\ntermination_by l => sizeOf l"}, {"name": "OpClause", "content": "structure OpClause where\n  op : Name\n   \n  body : Computation\nderiving Repr, BEq"}, {"name": "Computation", "content": "inductive Computation where\n| ret : Value → Computation\n \n| opCall : Name → Value → Computation → Computation\n \n| bind : Computation → Computation → Computation\n| ite (val : Value) (trueBranch falseBranch : Computation)\n| app : Value → Value → Computation\n \n| handle : Value → Computation → Computation\n| join : Value → Value → Computation\n| fst : Value → Computation\n| snd : Value → Computation\n| add : Value → Value → Computation\n| sub : Value → Value → Computation\n| mul : Value → Value → Computation\n| max : Value → Value → Computation\n| lt : Value → Value → Computation\n| eq : Value → Value → Computation\nderiving BEq"}, {"name": "Value", "content": "inductive Value where\n| var : Var → Value\n| bool : Bool → Value\n| string : String → Value\n| num : Int → Value\n| unit : Value\n| pair : Value → Value → Value\n \n| lam : Computation → Value\n \n| recfun : Computation → Value\n| hdl : Handler → Value\nderiving BEq"}, {"name": "Handler", "content": "structure Handler where\n   \n  ret : Computation\n  ops : List OpClause\nderiving Repr, BEq"}, {"name": "Var", "content": "inductive Var where\n| fvar : Name → Var\n| bvar : Nat → Var\nderiving Repr, DecidableEq"}, {"name": "Name", "content": "abbrev Name := String"}, {"name": "Step", "content": "@[grind cases]\ninductive Step : Computation → Computation → Prop\n \n| beta v body : Step (.app (.lam body) v) (instantiateComp v body)\n \n| recBeta v body : Step (.app (.recfun body) v) (instantiate2 (.recfun body) v body)\n \n| iteTrue c₁ c₂ : Step (.ite (.bool true) c₁ c₂) c₁\n \n| iteFalse c₁ c₂ : Step (.ite (.bool false) c₁ c₂) c₂\n \n| bindStep c₁ c₁' c₂ (h : Step c₁ c₁') : Step (.bind c₁ c₂) (.bind c₁' c₂)\n \n| bindReturn v c : Step (.bind (.ret v) c) (instantiateComp v c)\n \n| bindOp op v body c : Step (.bind (.opCall op v body) c) (.opCall op v (.bind body c))\n \n| handleInner h c₁ c₂ (h₁ : Step c₁ c₂) : Step (.handle (.hdl h) c₁) (.handle (.hdl h) c₂)\n \n| handleReturn h v :\n  Step (.handle (.hdl h) (.ret v)) (instantiateComp v h.ret)\n \n| handleOpHit h op v body c (hop : h.lookup op = some ⟨op, c⟩) :\n  Step (.handle (.hdl h) (.opCall op v body))\n    (instantiate2 v (.lam (.handle (.hdl h) body)) c)\n \n| handleOpMiss h op v body (hop : h.lookup op = none) :\n  Step (.handle (.hdl h) (.opCall op v body)) (.opCall op v (.handle (.hdl h) body))\n \n| join s₁ s₂ : Step (.join (.string s₁) (.string s₂)) (.ret (.string (strAppend s₁ s₂)))\n \n| fstStep v₁ v₂ : Step (.fst (.pair v₁ v₂)) (.ret v₁)\n \n| sndStep v₁ v₂ : Step (.snd (.pair v₁ v₂)) (.ret v₂)\n| add v₁ v₂ : Step (.add (.num v₁) (.num v₂)) (.ret (.num (v₁ + v₂)))\n| sub v₁ v₂ : Step (.sub (.num v₁) (.num v₂)) (.ret (.num (v₁ - v₂)))\n| max v₁ v₂ : Step (.max (.num v₁) (.num v₂)) (.ret (.num (Max.max v₁ v₂)))\n| lt v₁ v₂ : Step (.lt (.num v₁) (.num v₂)) (.ret (.bool (v₁ < v₂)))\n| mul v₁ v₂ : Step (.mul (.num v₁) (.num v₂)) (.ret (.num (v₁ * v₂)))\n| eq v₁ v₂ : Step (.eq v₁ v₂) (.ret (.bool (v₁ == v₂)))"}, {"name": "instantiate2", "content": "def instantiate2 (arg cont : Value) (body : Computation) : Computation :=\n  instantiateComputationLvl arg 1 (instantiateComputationLvl cont 0 body)"}, {"name": "strAppend", "content": "def strAppend : String → String → String\n| \"\", s => s\n| s, \"\" => s\n| s₁, s₂ => s₁ ++ \" \" ++ s₂"}, {"name": "StepStar", "content": "@[grind cases]\ninductive StepStar : Computation → Computation → Prop\n| refl (c) : StepStar c c\n| trans : Step c₁ c₂ → StepStar c₂ c₃ → StepStar c₁ c₃"}, {"name": "infix:50 \" ⤳ \" => Step", "content": "infix:50 \" ⤳ \" => Step"}, {"name": "infix:50 \" ⤳⋆ \" => StepStar", "content": "infix:50 \" ⤳⋆ \" => StepStar"}], "lib_lemmas": [{"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "evalSingleStep", "content": "def evalSingleStep : Computation → Option Computation\n \n| .app (.lam body) v => some <| instantiateComp v body\n \n| .app (.recfun body) v => some <| instantiate2 (.recfun body) v body\n| .ite (.bool true) c₁ _ => some c₁\n| .ite (.bool false) _ c₂ => some c₂\n \n| .bind (.ret v) c => some <| instantiateComp v c\n \n| .bind (.opCall op v body) c => some <| .opCall op v (.bind body c)\n| .bind c₁ c₂ => (evalSingleStep c₁).map (fun c₁' => .bind c₁' c₂)\n   \n| .handle (.hdl h) (.ret v) => some <| instantiateComp v h.ret\n| .handle (.hdl h) (.opCall op v body) =>\n  match h.lookup op with\n   \n  | some ⟨_, c⟩ =>\n    let cont := .lam (.handle (.hdl h) body)\n    some <| instantiate2 v cont c\n   \n  | none => some <| .opCall op v (.handle (.hdl h) body)\n| .handle (.hdl h) c =>\n  (evalSingleStep c).map (fun c' => .handle (.hdl h) c')\n| .join (.string s₁) (.string s₂) => some <| .ret (.string (strAppend s₁ s₂))\n| .fst (.pair v₁ _) => some <| .ret v₁\n| .snd (.pair _ v₂) => some <| .ret v₂\n| .add (.num v₁) (.num v₂) => some <| .ret (.num (v₁ + v₂))\n| .sub (.num v₁) (.num v₂) => some <| .ret (.num (v₁ - v₂))\n| .max (.num v₁) (.num v₂) => some <| .ret (.num (Max.max v₁ v₂))\n| .lt (.num v₁) (.num v₂) => some <| .ret (.bool (v₁ < v₂))\n| .mul (.num v₁) (.num v₂) => some <| .ret (.num (v₁ * v₂))\n| .eq v₁ v₂ => some <| .ret (.bool (v₁ == v₂))\n| _ => none"}, {"name": "evalFuel", "content": "def evalFuel : Nat → Computation → Option Computation\n| 0, _ => none\n| _+1, .ret v => some <| .ret v\n| _+1, .opCall n v c => some <| .opCall n v c\n| n+1, c =>\n  match evalSingleStep c with\n  | some c' => evalFuel n c'\n  | none => none"}], "used_local_lemmas": [{"name": "evalSingleStep_complete", "content": "theorem evalSingleStep_complete {c c' : Computation} :\n    c ⤳ c' → evalSingleStep c = (some c')"}, {"name": "evalFuel_step", "content": "theorem evalFuel_step (h : c ⤳ c') : evalFuel (n + 1) c = evalFuel n c'"}], "local_ctx": "import IntroEffects.SmallStep\n\ndef evalSingleStep : Computation → Option Computation\n \n| .app (.lam body) v => some <| instantiateComp v body\n \n| .app (.recfun body) v => some <| instantiate2 (.recfun body) v body\n| .ite (.bool true) c₁ _ => some c₁\n| .ite (.bool false) _ c₂ => some c₂\n \n| .bind (.ret v) c => some <| instantiateComp v c\n \n| .bind (.opCall op v body) c => some <| .opCall op v (.bind body c)\n| .bind c₁ c₂ => (evalSingleStep c₁).map (fun c₁' => .bind c₁' c₂)\n   \n| .handle (.hdl h) (.ret v) => some <| instantiateComp v h.ret\n| .handle (.hdl h) (.opCall op v body) =>\n  match h.lookup op with\n   \n  | some ⟨_, c⟩ =>\n    let cont := .lam (.handle (.hdl h) body)\n    some <| instantiate2 v cont c\n   \n  | none => some <| .opCall op v (.handle (.hdl h) body)\n| .handle (.hdl h) c =>\n  (evalSingleStep c).map (fun c' => .handle (.hdl h) c')\n| .join (.string s₁) (.string s₂) => some <| .ret (.string (strAppend s₁ s₂))\n| .fst (.pair v₁ _) => some <| .ret v₁\n| .snd (.pair _ v₂) => some <| .ret v₂\n| .add (.num v₁) (.num v₂) => some <| .ret (.num (v₁ + v₂))\n| .sub (.num v₁) (.num v₂) => some <| .ret (.num (v₁ - v₂))\n| .max (.num v₁) (.num v₂) => some <| .ret (.num (Max.max v₁ v₂))\n| .lt (.num v₁) (.num v₂) => some <| .ret (.bool (v₁ < v₂))\n| .mul (.num v₁) (.num v₂) => some <| .ret (.num (v₁ * v₂))\n| .eq v₁ v₂ => some <| .ret (.bool (v₁ == v₂))\n| _ => none\n\ndef evalFuel : Nat → Computation → Option Computation\n| 0, _ => none\n| _+1, .ret v => some <| .ret v\n| _+1, .opCall n v c => some <| .opCall n v c\n| n+1, c =>\n  match evalSingleStep c with\n  | some c' => evalFuel n c'\n  | none => none", "target_theorem": "theorem evalFuelRet_complete_aux (h : c ⤳⋆ r) :\n    ∀v, r= .ret v → ∃n, evalFuel n c = some (.ret v) :=", "ground_truth_proof": ":= by\n  induction h with\n  | refl c' =>\n    intro v hv\n    exact ⟨1, by grind [evalFuel]⟩\n  | @trans c1 c2 c3 hStep hTail ih =>\n    intro v hv\n    obtain ⟨n, ihFuel⟩ := ih v hv\n    refine ⟨n+1, ?_⟩\n    simp [evalFuel_step hStep, ihFuel]", "nesting_depth": 8, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Semantics"}
{"id": 220, "thm_name": "evalFuelOpCall_complete_aux", "thm_stmt": "theorem evalFuelOpCall_complete_aux (h : c ⤳⋆ r) :\n    ∀v, r = .opCall name v comp → ∃n, evalFuel n c = some (.opCall name v comp)", "lean_root": "IntroEffects", "rel_path": "IntroEffects/Eval.lean", "imports": ["import IntroEffects.SmallStep"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "scoped syntax num : embedded", "content": "scoped syntax num : embedded"}, {"name": "instantiateComp", "content": "def instantiateComp (what : Value) (comp : Computation) : Computation :=\n  instantiateComputationLvl what 0 comp"}, {"name": "instantiateComputationLvl", "content": "def instantiateComputationLvl (what : Value) (level : Nat) : Computation → Computation\n| .ret v => .ret <| instantiateValueLvl what level v\n| .opCall op v c => .opCall op (instantiateValueLvl what level v) (instantiateComputationLvl what (level+1) c)\n| .bind c₁ c₂ => .bind (instantiateComputationLvl what level c₁) (instantiateComputationLvl what (level+1) c₂)\n| .ite v c₁ c₂ => .ite (instantiateValueLvl what level v) (instantiateComputationLvl what level c₁) (instantiateComputationLvl what level c₂)\n| .app v₁ v₂ => .app (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .handle v c => .handle (instantiateValueLvl what level v) (instantiateComputationLvl what level c)\n| .join v₁ v₂ => .join (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .fst v => .fst (instantiateValueLvl what level v)\n| .snd v => .snd (instantiateValueLvl what level v)\n| .add v₁ v₂ => .add (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .sub v₁ v₂ => .sub (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .max v₁ v₂ => .max (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .lt v₁ v₂ => .lt (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .mul v₁ v₂ => .mul (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .eq v₁ v₂ => .eq (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\ntermination_by c => sizeOf c\ndecreasing_by\n  all_goals simp\n  all_goals grind"}, {"name": "instantiateValueLvl", "content": "def instantiateValueLvl (what : Value) (level : Nat) : Value → Value\n| var@(.var (.bvar bvarLevel)) => if bvarLevel = level then what else var\n| .lam c => .lam <| instantiateComputationLvl what (level + 1) c\n| .hdl h => .hdl <| instantiateHandlerLvl what level h\n| .pair v₁ v₂ => .pair (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)\n| .recfun c => .recfun <| instantiateComputationLvl what (level + 2) c\n| .string s => .string s\n| .bool b => .bool b\n| .unit => .unit\n| .var v => .var v\n| .num n => .num n\ntermination_by v => sizeOf v\ndecreasing_by\n  all_goals simp\n  all_goals grind"}, {"name": "instantiateHandlerLvl", "content": "def instantiateHandlerLvl (what : Value) (level : Nat) : Handler → Handler\n| ⟨ret, ops'⟩ =>\n    ⟨instantiateComputationLvl what (level+1) ret, instantiateOps what level ops'⟩\ntermination_by h => sizeOf h\ndecreasing_by\n  all_goals simp\n  all_goals grind"}, {"name": "instantiateOps", "content": "def instantiateOps (what : Value) (level : Nat) : List OpClause → List OpClause\n| [] => []\n| ⟨op, body⟩ :: ls =>\n  ⟨op, instantiateComputationLvl what (level+2) body⟩ :: instantiateOps what level ls\ntermination_by l => sizeOf l"}, {"name": "OpClause", "content": "structure OpClause where\n  op : Name\n   \n  body : Computation\nderiving Repr, BEq"}, {"name": "Computation", "content": "inductive Computation where\n| ret : Value → Computation\n \n| opCall : Name → Value → Computation → Computation\n \n| bind : Computation → Computation → Computation\n| ite (val : Value) (trueBranch falseBranch : Computation)\n| app : Value → Value → Computation\n \n| handle : Value → Computation → Computation\n| join : Value → Value → Computation\n| fst : Value → Computation\n| snd : Value → Computation\n| add : Value → Value → Computation\n| sub : Value → Value → Computation\n| mul : Value → Value → Computation\n| max : Value → Value → Computation\n| lt : Value → Value → Computation\n| eq : Value → Value → Computation\nderiving BEq"}, {"name": "Value", "content": "inductive Value where\n| var : Var → Value\n| bool : Bool → Value\n| string : String → Value\n| num : Int → Value\n| unit : Value\n| pair : Value → Value → Value\n \n| lam : Computation → Value\n \n| recfun : Computation → Value\n| hdl : Handler → Value\nderiving BEq"}, {"name": "Handler", "content": "structure Handler where\n   \n  ret : Computation\n  ops : List OpClause\nderiving Repr, BEq"}, {"name": "Var", "content": "inductive Var where\n| fvar : Name → Var\n| bvar : Nat → Var\nderiving Repr, DecidableEq"}, {"name": "Name", "content": "abbrev Name := String"}, {"name": "Step", "content": "@[grind cases]\ninductive Step : Computation → Computation → Prop\n \n| beta v body : Step (.app (.lam body) v) (instantiateComp v body)\n \n| recBeta v body : Step (.app (.recfun body) v) (instantiate2 (.recfun body) v body)\n \n| iteTrue c₁ c₂ : Step (.ite (.bool true) c₁ c₂) c₁\n \n| iteFalse c₁ c₂ : Step (.ite (.bool false) c₁ c₂) c₂\n \n| bindStep c₁ c₁' c₂ (h : Step c₁ c₁') : Step (.bind c₁ c₂) (.bind c₁' c₂)\n \n| bindReturn v c : Step (.bind (.ret v) c) (instantiateComp v c)\n \n| bindOp op v body c : Step (.bind (.opCall op v body) c) (.opCall op v (.bind body c))\n \n| handleInner h c₁ c₂ (h₁ : Step c₁ c₂) : Step (.handle (.hdl h) c₁) (.handle (.hdl h) c₂)\n \n| handleReturn h v :\n  Step (.handle (.hdl h) (.ret v)) (instantiateComp v h.ret)\n \n| handleOpHit h op v body c (hop : h.lookup op = some ⟨op, c⟩) :\n  Step (.handle (.hdl h) (.opCall op v body))\n    (instantiate2 v (.lam (.handle (.hdl h) body)) c)\n \n| handleOpMiss h op v body (hop : h.lookup op = none) :\n  Step (.handle (.hdl h) (.opCall op v body)) (.opCall op v (.handle (.hdl h) body))\n \n| join s₁ s₂ : Step (.join (.string s₁) (.string s₂)) (.ret (.string (strAppend s₁ s₂)))\n \n| fstStep v₁ v₂ : Step (.fst (.pair v₁ v₂)) (.ret v₁)\n \n| sndStep v₁ v₂ : Step (.snd (.pair v₁ v₂)) (.ret v₂)\n| add v₁ v₂ : Step (.add (.num v₁) (.num v₂)) (.ret (.num (v₁ + v₂)))\n| sub v₁ v₂ : Step (.sub (.num v₁) (.num v₂)) (.ret (.num (v₁ - v₂)))\n| max v₁ v₂ : Step (.max (.num v₁) (.num v₂)) (.ret (.num (Max.max v₁ v₂)))\n| lt v₁ v₂ : Step (.lt (.num v₁) (.num v₂)) (.ret (.bool (v₁ < v₂)))\n| mul v₁ v₂ : Step (.mul (.num v₁) (.num v₂)) (.ret (.num (v₁ * v₂)))\n| eq v₁ v₂ : Step (.eq v₁ v₂) (.ret (.bool (v₁ == v₂)))"}, {"name": "instantiate2", "content": "def instantiate2 (arg cont : Value) (body : Computation) : Computation :=\n  instantiateComputationLvl arg 1 (instantiateComputationLvl cont 0 body)"}, {"name": "strAppend", "content": "def strAppend : String → String → String\n| \"\", s => s\n| s, \"\" => s\n| s₁, s₂ => s₁ ++ \" \" ++ s₂"}, {"name": "StepStar", "content": "@[grind cases]\ninductive StepStar : Computation → Computation → Prop\n| refl (c) : StepStar c c\n| trans : Step c₁ c₂ → StepStar c₂ c₃ → StepStar c₁ c₃"}, {"name": "infix:50 \" ⤳ \" => Step", "content": "infix:50 \" ⤳ \" => Step"}, {"name": "infix:50 \" ⤳⋆ \" => StepStar", "content": "infix:50 \" ⤳⋆ \" => StepStar"}], "lib_lemmas": [{"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "evalSingleStep", "content": "def evalSingleStep : Computation → Option Computation\n \n| .app (.lam body) v => some <| instantiateComp v body\n \n| .app (.recfun body) v => some <| instantiate2 (.recfun body) v body\n| .ite (.bool true) c₁ _ => some c₁\n| .ite (.bool false) _ c₂ => some c₂\n \n| .bind (.ret v) c => some <| instantiateComp v c\n \n| .bind (.opCall op v body) c => some <| .opCall op v (.bind body c)\n| .bind c₁ c₂ => (evalSingleStep c₁).map (fun c₁' => .bind c₁' c₂)\n   \n| .handle (.hdl h) (.ret v) => some <| instantiateComp v h.ret\n| .handle (.hdl h) (.opCall op v body) =>\n  match h.lookup op with\n   \n  | some ⟨_, c⟩ =>\n    let cont := .lam (.handle (.hdl h) body)\n    some <| instantiate2 v cont c\n   \n  | none => some <| .opCall op v (.handle (.hdl h) body)\n| .handle (.hdl h) c =>\n  (evalSingleStep c).map (fun c' => .handle (.hdl h) c')\n| .join (.string s₁) (.string s₂) => some <| .ret (.string (strAppend s₁ s₂))\n| .fst (.pair v₁ _) => some <| .ret v₁\n| .snd (.pair _ v₂) => some <| .ret v₂\n| .add (.num v₁) (.num v₂) => some <| .ret (.num (v₁ + v₂))\n| .sub (.num v₁) (.num v₂) => some <| .ret (.num (v₁ - v₂))\n| .max (.num v₁) (.num v₂) => some <| .ret (.num (Max.max v₁ v₂))\n| .lt (.num v₁) (.num v₂) => some <| .ret (.bool (v₁ < v₂))\n| .mul (.num v₁) (.num v₂) => some <| .ret (.num (v₁ * v₂))\n| .eq v₁ v₂ => some <| .ret (.bool (v₁ == v₂))\n| _ => none"}, {"name": "evalFuel", "content": "def evalFuel : Nat → Computation → Option Computation\n| 0, _ => none\n| _+1, .ret v => some <| .ret v\n| _+1, .opCall n v c => some <| .opCall n v c\n| n+1, c =>\n  match evalSingleStep c with\n  | some c' => evalFuel n c'\n  | none => none"}], "used_local_lemmas": [{"name": "evalSingleStep_complete", "content": "theorem evalSingleStep_complete {c c' : Computation} :\n    c ⤳ c' → evalSingleStep c = (some c')"}, {"name": "evalFuel_step", "content": "theorem evalFuel_step (h : c ⤳ c') : evalFuel (n + 1) c = evalFuel n c'"}], "local_ctx": "import IntroEffects.SmallStep\n\ndef evalSingleStep : Computation → Option Computation\n \n| .app (.lam body) v => some <| instantiateComp v body\n \n| .app (.recfun body) v => some <| instantiate2 (.recfun body) v body\n| .ite (.bool true) c₁ _ => some c₁\n| .ite (.bool false) _ c₂ => some c₂\n \n| .bind (.ret v) c => some <| instantiateComp v c\n \n| .bind (.opCall op v body) c => some <| .opCall op v (.bind body c)\n| .bind c₁ c₂ => (evalSingleStep c₁).map (fun c₁' => .bind c₁' c₂)\n   \n| .handle (.hdl h) (.ret v) => some <| instantiateComp v h.ret\n| .handle (.hdl h) (.opCall op v body) =>\n  match h.lookup op with\n   \n  | some ⟨_, c⟩ =>\n    let cont := .lam (.handle (.hdl h) body)\n    some <| instantiate2 v cont c\n   \n  | none => some <| .opCall op v (.handle (.hdl h) body)\n| .handle (.hdl h) c =>\n  (evalSingleStep c).map (fun c' => .handle (.hdl h) c')\n| .join (.string s₁) (.string s₂) => some <| .ret (.string (strAppend s₁ s₂))\n| .fst (.pair v₁ _) => some <| .ret v₁\n| .snd (.pair _ v₂) => some <| .ret v₂\n| .add (.num v₁) (.num v₂) => some <| .ret (.num (v₁ + v₂))\n| .sub (.num v₁) (.num v₂) => some <| .ret (.num (v₁ - v₂))\n| .max (.num v₁) (.num v₂) => some <| .ret (.num (Max.max v₁ v₂))\n| .lt (.num v₁) (.num v₂) => some <| .ret (.bool (v₁ < v₂))\n| .mul (.num v₁) (.num v₂) => some <| .ret (.num (v₁ * v₂))\n| .eq v₁ v₂ => some <| .ret (.bool (v₁ == v₂))\n| _ => none\n\ndef evalFuel : Nat → Computation → Option Computation\n| 0, _ => none\n| _+1, .ret v => some <| .ret v\n| _+1, .opCall n v c => some <| .opCall n v c\n| n+1, c =>\n  match evalSingleStep c with\n  | some c' => evalFuel n c'\n  | none => none", "target_theorem": "theorem evalFuelOpCall_complete_aux (h : c ⤳⋆ r) :\n    ∀v, r = .opCall name v comp → ∃n, evalFuel n c = some (.opCall name v comp) :=", "ground_truth_proof": ":= by\n  induction h with\n  | refl c' =>\n    intro v hv\n    exact ⟨1, by grind [evalFuel]⟩\n  | @trans c1 c2 c3 hStep hTail ih =>\n    intro v hv\n    obtain ⟨n, ihFuel⟩ := ih v hv\n    refine ⟨n+1, ?_⟩\n    simp [evalFuel_step hStep, ihFuel]", "nesting_depth": 8, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Semantics"}
{"id": 221, "thm_name": "Iris.Agree.op_inv", "thm_stmt": "theorem Agree.op_inv {x y : Agree α} : (x.op y).valid → x ≡ y", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/Agree.lean", "imports": ["import Iris.Algebra.CMRA", "import src.Iris.Algebra.CMRA", "import src.Iris.Algebra.OFE", "import Iris.Algebra.OFE"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x", "content": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y"}, {"name": "Agree", "content": "@[ext]\nstructure Agree where\n  car : List α\n  not_nil : car ≠ []"}, {"name": "LeibnizO", "content": "@[ext] structure LeibnizO (α : Type _) where\n  car : α"}, {"name": "op", "content": "abbrev op (x y : α × β) : α × β :=\n  (x.1 • y.1, x.2 • y.2)"}, {"name": "valid", "content": "def valid : DFrac F → Prop\n  | .own f        => Proper f\n  | .discard      => True\n  | .ownDiscard f => Fractional f"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "Fractional", "content": "def Fractional [Fraction α] (a : α) : Prop := ∃ b, Proper (a + b)"}, {"name": "Fraction", "content": "class Fraction (α : Type _) extends Add α where\n   \n  Proper : α → Prop\n  add_comm : ∀ a b : α, a + b = b + a\n  add_assoc : ∀ a b c : α, a + (b + c) = (a + b) + c\n  add_left_cancel : ∀ {a b c : α}, a + b = a + c → b = c\n   \n  add_ne : ∀ {a b : α}, a ≠ b + a\n  proper_add_mono_left : ∀ {a b : α}, Proper (a + b) → Proper a"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Iso.symm", "content": "def Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where\n  hom := iso.inv\n  inv := iso.hom\n  hom_inv := by admit /- proof elided -/"}, {"name": "Iso", "content": "@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where\n  hom : α -n> β\n  inv : β -n> α\n  hom_inv : hom (inv x) ≡ x\n  inv_hom : inv (hom x) ≡ x"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}], "lib_lemmas": [{"name": "List.mem_append", "module": "Init.Data.List.Lemmas"}], "repo_lemmas": [{"name": "_root_.Iris.OFE.Dist.validN", "content": "theorem _root_.Iris.OFE.Dist.validN : (x : α) ≡{n}≡ y → (✓{n} x ↔ ✓{n} y)"}, {"name": "validN_iff", "content": "theorem validN_iff {x y : α} (e : x ≡{n}≡ y) : ✓{n} x ↔ ✓{n} y"}, {"name": "IncludedN.validN", "content": "theorem IncludedN.validN {n} {x y : α} : x ≼{n} y → ✓{n} y → ✓{n} x"}, {"name": "validN_of_incN", "content": "theorem validN_of_incN {n} {x y : α} : x ≼{n} y → ✓{n} y → ✓{n} x"}, {"name": "Included.validN", "content": "theorem Included.validN {n} {x y : α} : x ≼ y → ✓{n} y → ✓{n} x"}, {"name": "validN_of_inc", "content": "theorem validN_of_inc {n} {x y : α} : x ≼ y → ✓{n} y → ✓{n} x"}, {"name": "Valid.validN", "content": "theorem Valid.validN : ✓ (x : α) → ✓{n} x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "_root_.Iris.OFE.Dist.op", "content": "theorem _root_.Iris.OFE.Dist.op {x x' y y' : α}\n    (ex : x ≡{n}≡ x') (ey : y ≡{n}≡ y') : x • y ≡{n}≡ x' • y'"}, {"name": "Included.trans", "content": "theorem Included.trans : (x : α) ≼ y → y ≼ z → x ≼ z"}, {"name": "inc_trans", "content": "theorem inc_trans {x y z : α} : x ≼ y → y ≼ z → x ≼ z"}, {"name": "op_left_eqv", "content": "theorem op_left_eqv {x y : α} (z : α) (e : x ≡ y) : x • z ≡ y • z"}, {"name": "_root_.Iris.OFE.Dist.op_r", "content": "theorem _root_.Iris.OFE.Dist.op_r {x y z : α} : y ≡{n}≡ z → x • y ≡{n}≡ x • z"}, {"name": "op_right_dist", "content": "theorem op_right_dist (x : α) {y z : α} (e : y ≡{n}≡ z) : x • y ≡{n}≡ x • z"}, {"name": "_root_.Iris.OFE.Equiv.op_r", "content": "theorem _root_.Iris.OFE.Equiv.op_r {x y z : α} : y ≡ z → x • y ≡ x • z"}, {"name": "op_right_eqv", "content": "theorem op_right_eqv (x : α) {y z : α} (e : y ≡ z) : x • y ≡ x • z"}, {"name": "IncludedN.trans", "content": "theorem IncludedN.trans : (x : α) ≼{n} y → y ≼{n} z → x ≼{n} z"}, {"name": "incN_trans", "content": "theorem incN_trans {x y z : α} : x ≼{n} y → y ≼{n} z → x ≼{n} z"}, {"name": "op_left_dist", "content": "theorem op_left_dist {x y : α} (z : α) (e : x ≡{n}≡ y) : x • z ≡{n}≡ y • z"}, {"name": "_root_.Iris.OFE.Dist.op_l", "content": "theorem _root_.Iris.OFE.Dist.op_l {x y z : α} : x ≡{n}≡ y → x • z ≡{n}≡ y • z"}, {"name": "_root_.Iris.OFE.Equiv.op_l", "content": "theorem _root_.Iris.OFE.Equiv.op_l {x y z : α} : x ≡ y → x • z ≡ y • z"}, {"name": "_root_.Iris.OFE.Equiv.op", "content": "theorem _root_.Iris.OFE.Equiv.op : (x : α) ≡ x' → y ≡ y' → x • y ≡ x' • y'"}, {"name": "op_eqv", "content": "theorem op_eqv {x x' y y' : α} (ex : x ≡ x') (ey : y ≡ y') : x • y ≡ x' • y'"}], "used_local_defs": [{"name": "Iris.Agree", "content": "@[ext]\nstructure Agree where\n  car : List α\n  not_nil : car ≠ []"}, {"name": "Iris.Agree.dist", "content": "def Agree.dist (n : Nat) (x y : Agree α) : Prop :=\n  (∀ a ∈ x.car, ∃ b ∈ y.car, a ≡{n}≡ b) ∧\n  (∀ b ∈ y.car, ∃ a ∈ x.car, a ≡{n}≡ b)"}, {"name": "Iris.Agree.validN", "content": "def Agree.validN (n : Nat) (x : Agree α) : Prop :=\n  match x.car with\n  | [_] => True\n  | _ => ∀ a ∈ x.car, ∀ b ∈ x.car, a ≡{n}≡ b"}, {"name": "Iris.Agree.op", "content": "def Agree.op (x y : Agree α) : Agree α :=\n  ⟨x.car ++ y.car, by admit /- proof elided -/\n  ⟩"}], "used_local_lemmas": [{"name": "Iris.mem_of_agree", "content": "theorem mem_of_agree (x : Agree α) : ∃ a, a ∈ x.car"}, {"name": "Iris.Agree.equiv_def", "content": "theorem Agree.equiv_def {x y : Agree α} : x ≡ y ↔ ∀ n, Agree.dist n x y"}, {"name": "Iris.Agree.validN_iff", "content": "theorem Agree.validN_iff {x : Agree α} :\n    x.validN n ↔ ∀ a ∈ x.car, ∀ b ∈ x.car, a ≡{n}≡ b"}, {"name": "Iris.Agree.op_invN", "content": "theorem Agree.op_invN {x y : Agree α} : (x.op y).validN n → x ≡{n}≡ y"}], "local_ctx": "import Iris.Algebra.CMRA\n\nimport Iris.Algebra.OFE\n\nnamespace Iris\n\nsection agree\n\nvariable {α : Type u}\n\nvariable (α) in\n\n@[ext]\nstructure Agree where\n  car : List α\n  not_nil : car ≠ []\n\nvariable [OFE α]\n\ndef Agree.dist (n : Nat) (x y : Agree α) : Prop :=\n  (∀ a ∈ x.car, ∃ b ∈ y.car, a ≡{n}≡ b) ∧\n  (∀ b ∈ y.car, ∃ a ∈ x.car, a ≡{n}≡ b)\n\ndef Agree.validN (n : Nat) (x : Agree α) : Prop :=\n  match x.car with\n  | [_] => True\n  | _ => ∀ a ∈ x.car, ∀ b ∈ x.car, a ≡{n}≡ b\n\ndef Agree.valid (x : Agree α) : Prop := ∀ n, x.validN n\n\ndef Agree.op (x y : Agree α) : Agree α :=\n  ⟨x.car ++ y.car, by admit /- proof elided -/\n  ⟩", "target_theorem": "theorem Agree.op_inv {x y : Agree α} : (x.op y).valid → x ≡ y :=", "ground_truth_proof": ":= by\n  simp [valid, equiv_def]\n  intro h n\n  exact op_invN (h n)", "nesting_depth": 8, "transitive_dep_count": 58, "subset_aristotle": false, "category": "Framework"}
{"id": 222, "thm_name": "Iris.OFE.ContractiveHom.fixpoint_ind", "thm_stmt": "@[elab_as_elim]\ntheorem OFE.ContractiveHom.fixpoint_ind [COFE α] [Inhabited α] (f : α -c> α)\n    (P : α → Prop) (HProper : ∀ A B : α, A ≡ B → P A → P B) (x : α) (Hbase : P x)\n    (Hind : ∀ x, P x → P (f x)) (Hlim : LimitPreserving P) :\n    P f.fixpoint", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/OFE.lean", "imports": ["import src/Iris/Algebra/Excl.lean", "import src/Iris/Algebra/Agree.lean", "import src/Iris/Algebra/UPred.lean", "import src/Iris/Algebra/COFESolver.lean", "import src/Iris/Instances/UPred/Instance.lean", "import src/Iris/Algebra/Agree_task.lean", "import src/Iris/Algebra/CMRA.lean", "import src/Iris/Algebra/Heap.lean"], "used_lib_defs": [{"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Dist", "module": "Mathlib.Topology.MetricSpace.Pseudo.Defs"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "", "content": "instance : OFE.Discrete (DFrac F) := ⟨congrArg id⟩"}, {"name": "Agree.dist", "content": "def Agree.dist (n : Nat) (x y : Agree α) : Prop :=\n  (∀ a ∈ x.car, ∃ b ∈ y.car, a ≡{n}≡ b) ∧\n  (∀ b ∈ y.car, ∃ a ∈ x.car, a ≡{n}≡ b)"}, {"name": "", "content": "instance : OFE (Tower F) where\n  Equiv f g := ∀ k, f k ≡ g k\n  Dist n f g := ∀ k, f k ≡{n}≡ g k\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "", "content": "instance : OFE.NonExpansive (BUpd.bupd (PROP := UPred M)) := bupd_ne"}, {"name": "bupd_ne", "content": "instance bupd_ne : OFE.NonExpansive (bupd : UPred M → UPred M) where\n  ne n x1 x2 Hx m y Hm Hv := by admit /- proof elided -/"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "later_contractive", "content": "instance later_contractive : OFE.Contractive UPred.later (α := UPred M) where\n  distLater_dist {n x y} Hl :=\n    match n with\n    | 0 => by admit /- proof elided -/\n    | n + 1 => fun\n      | 0 => by admit /- proof elided -/\n      | n' + 1 => fun x' Hn' Hx' => Hl _ Hn' _ _ (Nat.le_refl _) (CMRA.validN_succ Hx')"}, {"name": "later", "content": "protected def later (P : UPred M) : UPred M where\n  holds n x := match n with | 0 => True | Nat.succ n' => P n' x\n  mono {n₁ n₂} := by admit /- proof elided -/"}, {"name": "BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "BIUpdate", "content": "class BIUpdate (PROP : Type _) [BI PROP] extends BUpd PROP where\n  [bupd_ne : OFE.NonExpansive (BUpd.bupd (PROP := PROP))]\n  intro {P : PROP} : iprop(P ⊢ |==> P)\n  mono {P Q : PROP} : iprop(P ⊢ Q) → iprop(|==> P ⊢ |==> Q)\n  trans {P : PROP} : iprop(|==> |==> P ⊢ |==> P)\n  frame_r {P R : PROP} : iprop((|==> P) ∗ R ⊢ |==> (P ∗ R))"}, {"name": "UPred", "content": "@[ext]\nstructure UPred (M : Type _) [UCMRA M] where\n  holds : Nat → M → Prop\n  mono {n1 n2 x1 x2} : holds n1 x1 → x1 ≼{n2} x2 → n2 ≤ n1 → holds n2 x2"}, {"name": "IsModal", "content": "class IsModal [BI PROP1] [BI PROP2] (M : PROP1 → PROP2)\n    (iaction saction : ModalityAction PROP1 PROP2) where\n  spec_intuitionistic : iaction.intuitionistic_action_spec M\n  spec_spatial : saction.spatial_action_spec M\n  emp : iprop(emp) ⊢ M iprop(emp)\n  mono : ∀ {P Q}, (P ⊢ Q) → M P ⊢ M Q\n  sep : ∀ {P Q}, iprop(M P ∗ M Q) ⊢ M iprop(P ∗ Q)"}, {"name": "UCMRA", "content": "class UCMRA (α : Type _) extends CMRA α where\n  unit : α\n  unit_valid : ✓ unit\n  unit_left_id : unit • x ≡ x\n  pcore_unit : pcore unit ≡ some unit"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "instOFE", "content": "instance instOFE [Store T K V] [OFE V] : OFE T where\n  Equiv s0 s1  := get s0 ≡ get s1\n  Dist n s0 s1 := get s0 ≡{n}≡ get s1\n  dist_eqv     := ⟨fun _ => .of_eq rfl, (·.symm), (·.trans ·)⟩\n  equiv_dist   := equiv_dist\n  dist_lt      := dist_lt"}, {"name": "Store.Equiv", "content": "@[simp] def Store.Equiv [Store T K V] (t1 t2 : T) : Prop := get t1 = get t2"}, {"name": "Store.Equiv_trans", "content": "instance Store.Equiv_trans [Store T K V] : Trans Equiv (Equiv (T := T)) Equiv := ⟨by admit /- proof elided -/\n⟩"}, {"name": "[CMRA", "content": "instance [CMRA β] : OFE (α -C> β) where\n  Equiv f g := f.toHom ≡ g.toHom\n  Dist n f g := f.toHom ≡{n}≡ g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "[OFE", "content": "instance [OFE α] [IsCOFE α] : IsCOFE (Excl α) where\n  compl c := (c 0).map fun x => IsCOFE.compl (exclChain c x)\n  conv_compl {n} c := by admit /- proof elided -/"}, {"name": "toAgree", "content": "def toAgree (a : α) : Agree α := ⟨[a], by admit /- proof elided -/\n⟩"}, {"name": "Agree", "content": "@[ext]\nstructure Agree where\n  car : List α\n  not_nil : car ≠ []"}, {"name": "(k", "content": "instance (k : Nat) : NonExpansive (fun X : Tower F => X.val k) := ⟨fun _ _ _ => (· _)⟩"}, {"name": "[OFE", "content": "instance [OFE α] : NonExpansive excl (α := α) where\n  ne _ _ _ a := a"}, {"name": "get_ne", "content": "instance get_ne [Store T K V] [OFE V] (k : K) : NonExpansive (get · k : T → V) where\n  ne {_ _ _} Ht := Ht k"}, {"name": "[Store", "content": "instance [Store T1 K V1] [Store T2 K V2] [OFE V1] [OFE V2] (f : K → V1 → V2)\n  [∀ k, NonExpansive (f k)] [HasStoreMap T1 T2 K V1 V2] : NonExpansive (dmap f : T1 → T2) where\n  ne _ {_ _} H k := by admit /- proof elided -/"}, {"name": "HasStoreMap", "content": "class HasStoreMap (T1 T2 : Type _) (K V1 V2 : outParam (Type _)) [Store T1 K V1] [Store T2 K V2] where\n   \n  dmap (f : K → V1 → V2) : T1 → T2\n  get_dmap : get (dmap f t) k = f k (get t k)"}, {"name": "", "content": "instance : NonExpansive (pcore (α := α)) where\n  ne n x {y} e := by admit /- proof elided -/"}, {"name": "[Heap", "content": "instance [Heap T K V] [OFE V] (op : V → V → V) [NonExpansive₂ op] :\n    NonExpansive₂ (merge (T := T) op) where\n  ne _ {_ _} Ht {_ _} Hs k := by admit /- proof elided -/"}, {"name": "Heap.instCOFE", "content": "instance Heap.instCOFE [Heap T K V] [COFE V] : COFE T where\n  compl c := hmap (fun _ => COFE.compl <| c.map ⟨_, Store.get_ne ·⟩) (c 0)\n  conv_compl {_ c} k := by admit /- proof elided -/"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -C> \" => Hom"}], "lib_lemmas": [{"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "Nat.le_of_lt_succ", "module": "Init.Prelude"}, {"name": "Nat.lt_succ_self", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Iris.OFE", "content": "class OFE (α : Type _) where\n  Equiv : α → α → Prop\n  Dist : Nat → α → α → Prop\n  dist_eqv : Equivalence (Dist n)\n  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y\n  dist_lt : Dist n x y → m < n → Dist m x y"}, {"name": "Iris.OFE.NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "Iris.OFE.id_ne", "content": "instance id_ne [OFE α] : NonExpansive (@id α) := ⟨fun _ _ _ h => h⟩"}, {"name": "Iris.OFE.NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "Iris.OFE.DistLater", "content": "def DistLater [OFE α] (n : Nat) (x y : α) : Prop := ∀ m, m < n → x ≡{m}≡ y"}, {"name": "Iris.OFE.Contractive", "content": "class Contractive [OFE α] [OFE β] (f : α → β) where\n  distLater_dist : DistLater n x y → f x ≡{n}≡ f y"}, {"name": "Iris.OFE.ne_of_contractive", "content": "instance ne_of_contractive [OFE α] [OFE β] (f : α → β) [Contractive f] : NonExpansive f where\n  ne := fun _ _ _ h => Contractive.distLater_dist (Dist.distLater h)"}, {"name": "Iris.OFE._inst_β", "content": "instance [OFE α] [OFE β] {x : β} : Contractive (fun _ : α => x) where\n  distLater_dist := fun _ => Dist.rfl"}, {"name": "Iris.OFE.Hom", "content": "@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f"}, {"name": "Iris.OFE._inst_α", "content": "instance [OFE α] [OFE β] (f : α -n> β) : NonExpansive f := f.ne"}, {"name": "Iris.OFE.ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "Iris.OFE._inst_α", "content": "instance [OFE α] [OFE β] (f : α -c> β) : Contractive f := f.contractive"}, {"name": "Iris.OFE._inst_OFE", "content": "instance : OFE Unit where\n  Equiv _ _ := True\n  Dist _ _ _ := True\n  dist_eqv := ⟨fun _ => ⟨⟩, id, fun _ => id⟩\n  equiv_dist := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] : OFE (ULift α) where\n  Equiv x y := x.down ≡ y.down\n  Dist n x y := x.down ≡{n}≡ y.down\n  dist_eqv := InvImage.equivalence dist_eqv\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] : OFE (Option α) where\n  Equiv := Option.Forall₂ Equiv\n  Dist n := Option.Forall₂ (Dist n)\n  dist_eqv := Option.Forall₂.equivalence dist_eqv\n  equiv_dist {x y} := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE.Discrete α] : OFE.Discrete (Option α) where\n  discrete_0 {mx my} e :=\n    match mx, my with\n    | none,   none   => e\n    | none,   some _ => e\n    | some _, none   => e\n    | some x, some y => show x ≡ y from discrete_0 e"}, {"name": "Iris.OFE.OFE", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Iris.OFE.Option", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_α", "content": "instance [OFEFun (β : α → _)] : OFE ((x : α) → β x) where\n  Equiv f g := ∀ x, f x ≡ g x\n  Dist n f g := ∀ x, f x ≡{n}≡ g x\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE β] : OFE (α -n> β) where\n  Equiv f g := f.f ≡ g.f\n  Dist n f g := f.f ≡{n}≡ g.f\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE β] : OFE (α -c> β) where\n  Equiv f g := Equiv f.toHom g.toHom\n  Dist n f g := Dist n f.toHom g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE β] : OFE (α × β) where\n  Equiv a b := a.1 ≡ b.1 ∧ a.2 ≡ b.2\n  Dist n a b := a.1 ≡{n}≡ b.1 ∧ a.2 ≡{n}≡ b.2\n  dist_eqv := {\n    refl _ := ⟨dist_eqv.refl _, dist_eqv.refl _⟩\n    symm h := ⟨dist_eqv.symm h.1, dist_eqv.symm h.2⟩\n    trans h1 h2 := ⟨dist_eqv.trans h1.1 h2.1, dist_eqv.trans h1.2 h2.2⟩\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "Iris.OFE.Iso", "content": "@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where\n  hom : α -n> β\n  inv : β -n> α\n  hom_inv : hom (inv x) ≡ x\n  inv_hom : inv (hom x) ≡ x"}, {"name": "Iris.OFE._inst_Iso", "content": "instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.hom := iso.hom.ne"}, {"name": "Iris.OFE._inst_Iso", "content": "instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.inv := iso.inv.ne"}, {"name": "Iris.OFE.Iso.symm", "content": "def Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where\n  hom := iso.inv\n  inv := iso.hom\n  hom_inv := by admit /- proof elided -/"}, {"name": "Iris.Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "Iris.IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Iris.LeibnizO", "content": "@[ext] structure LeibnizO (α : Type _) where\n  car : α"}, {"name": "Iris.optionChain", "content": "def optionChain (c : Chain (Option α)) (x : α) : Chain α :="}, {"name": "Iris.isCOFE_option", "content": "instance isCOFE_option [IsCOFE α] : IsCOFE (Option α) where\n  compl c := (c 0).map fun x => IsCOFE.compl (optionChain c x)\n  conv_compl {n} c := by admit /- proof elided -/"}, {"name": "Iris.LimitPreserving", "content": "def LimitPreserving [COFE α] (P : α → Prop) : Prop :=\n  ∀ (c : Chain α), (∀ n, P (c n)) → P (COFE.compl c)"}, {"name": "Iris.Fixpoint.chain", "content": "def Fixpoint.chain [OFE α] [Inhabited α] (f : α → α) [Contractive f] : Chain α where\n  chain n := Nat.repeat f (n + 1) default\n  cauchy {n} := by admit /- proof elided -/"}, {"name": "Iris.fixpoint", "content": "def fixpoint [COFE α] [Inhabited α] (f : α → α) [Contractive f] : α :=\n  COFE.compl <| Fixpoint.chain f\n\nnonrec abbrev OFE.ContractiveHom.fixpoint [COFE α] [Inhabited α] (f : α -c> α) : α := fixpoint f.f"}, {"name": "Iris.OFE", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "Iris.OFE.Dist.lt", "content": "theorem Dist.lt [OFE α] {m n} {x y : α} : x ≡{n}≡ y → m < n → x ≡{m}≡ y"}, {"name": "Iris.OFE.Dist.le", "content": "theorem Dist.le [OFE α] {m n} {x y : α} (h : x ≡{n}≡ y) (h' : m ≤ n) : x ≡{m}≡ y"}, {"name": "Iris.OFE.Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "Iris.OFE.Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Iris.OFE.Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "Iris.OFE.equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Iris.OFE.Equiv.symm", "content": "@[symm] theorem Equiv.symm [OFE α] {x : α} : x ≡ y → y ≡ x"}, {"name": "Iris.OFE.Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "Iris.OFE.Equiv.dist", "content": "theorem Equiv.dist [OFE α] {x : α} : x ≡ y → x ≡{n}≡ y"}, {"name": "Iris.OFE.DistLater.symm", "content": "@[symm] theorem DistLater.symm [OFE α] {n} {x : α} (h : DistLater n x y) : DistLater n y x"}, {"name": "Iris.OFE.DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}, {"name": "Iris.OFE.distLater_zero", "content": "@[simp] theorem distLater_zero [OFE α] {x y : α} : DistLater 0 x y"}, {"name": "Iris.OFE.distLater_succ", "content": "theorem distLater_succ [OFE α] {n} {x y : α} : DistLater n.succ x y ↔ x ≡{n}≡ y"}, {"name": "Iris.OFE.Contractive.zero", "content": "@[simp] theorem Contractive.zero [OFE α] [OFE β] (f : α → β) [Contractive f] {x y} :\n    f x ≡{0}≡ f y"}, {"name": "Iris.OFE.Contractive.succ", "content": "theorem Contractive.succ [OFE α] [OFE β] (f : α → β) [Contractive f] {n x y}\n    (h : x ≡{n}≡ y) : f x ≡{n.succ}≡ f y"}, {"name": "Iris.fixpoint_unfold", "content": "theorem fixpoint_unfold [COFE α] [Inhabited α] (f : α -c> α) :\n    fixpoint f ≡ f (fixpoint f)"}, {"name": "Iris.fixpoint_unique", "content": "theorem fixpoint_unique [COFE α] [Inhabited α] {f : α -c> α} {x : α} (H : x ≡ f x) :\n    x ≡ fixpoint f"}], "local_ctx": "namespace Iris\n\nclass OFE (α : Type _) where\n  Equiv : α → α → Prop\n  Dist : Nat → α → α → Prop\n  dist_eqv : Equivalence (Dist n)\n  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y\n  dist_lt : Dist n x y → m < n → Dist m x y\n\nopen OFE\n\nscoped infix:40 \" ≡ \" => OFE.Equiv\n\nscoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nnamespace OFE\n\nclass NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂\n\ninstance id_ne [OFE α] : NonExpansive (@id α) := ⟨fun _ _ _ h => h⟩\n\nclass NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂\n\ndef DistLater [OFE α] (n : Nat) (x y : α) : Prop := ∀ m, m < n → x ≡{m}≡ y\n\nclass Contractive [OFE α] [OFE β] (f : α → β) where\n  distLater_dist : DistLater n x y → f x ≡{n}≡ f y\n\ninstance ne_of_contractive [OFE α] [OFE β] (f : α → β) [Contractive f] : NonExpansive f where\n  ne := fun _ _ _ h => Contractive.distLater_dist (Dist.distLater h)\n\ninstance [OFE α] [OFE β] {x : β} : Contractive (fun _ : α => x) where\n  distLater_dist := fun _ => Dist.rfl\n\n@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f\n\n@[inherit_doc]\ninfixr:25 \" -n> \" => Hom\n\ninstance [OFE α] [OFE β] (f : α -n> β) : NonExpansive f := f.ne\n\n@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f\n\ninfixr:25 \" -c> \" => ContractiveHom\n\ninstance [OFE α] [OFE β] (f : α -c> β) : Contractive f := f.contractive\n\ninstance : OFE Unit where\n  Equiv _ _ := True\n  Dist _ _ _ := True\n  dist_eqv := ⟨fun _ => ⟨⟩, id, fun _ => id⟩\n  equiv_dist := by admit /- proof elided -/\n\ninstance [OFE α] : OFE (ULift α) where\n  Equiv x y := x.down ≡ y.down\n  Dist n x y := x.down ≡{n}≡ y.down\n  dist_eqv := InvImage.equivalence dist_eqv\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt\n\ninstance [OFE α] : OFE (Option α) where\n  Equiv := Option.Forall₂ Equiv\n  Dist n := Option.Forall₂ (Dist n)\n  dist_eqv := Option.Forall₂.equivalence dist_eqv\n  equiv_dist {x y} := by admit /- proof elided -/\n\ninstance [OFE α] [OFE.Discrete α] : OFE.Discrete (Option α) where\n  discrete_0 {mx my} e :=\n    match mx, my with\n    | none,   none   => e\n    | none,   some _ => e\n    | some _, none   => e\n    | some x, some y => show x ≡ y from discrete_0 e\n\ninstance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩\n\ninstance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/\n\ninstance [OFEFun (β : α → _)] : OFE ((x : α) → β x) where\n  Equiv f g := ∀ x, f x ≡ g x\n  Dist n f g := ∀ x, f x ≡{n}≡ g x\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/\n\ninstance [OFE α] [OFE β] : OFE (α -n> β) where\n  Equiv f g := f.f ≡ g.f\n  Dist n f g := f.f ≡{n}≡ g.f\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt\n\ninstance [OFE α] [OFE β] : OFE (α -c> β) where\n  Equiv f g := Equiv f.toHom g.toHom\n  Dist n f g := Dist n f.toHom g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt\n\ninstance [OFE α] [OFE β] : OFE (α × β) where\n  Equiv a b := a.1 ≡ b.1 ∧ a.2 ≡ b.2\n  Dist n a b := a.1 ≡{n}≡ b.1 ∧ a.2 ≡{n}≡ b.2\n  dist_eqv := {\n    refl _ := ⟨dist_eqv.refl _, dist_eqv.refl _⟩\n    symm h := ⟨dist_eqv.symm h.1, dist_eqv.symm h.2⟩\n    trans h1 h2 := ⟨dist_eqv.trans h1.1 h2.1, dist_eqv.trans h1.2 h2.2⟩\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/\n\n@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where\n  hom : α -n> β\n  inv : β -n> α\n  hom_inv : hom (inv x) ≡ x\n  inv_hom : inv (hom x) ≡ x\n\ninstance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.hom := iso.hom.ne\n\ninstance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.inv := iso.inv.ne\n\ndef Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where\n  hom := iso.inv\n  inv := iso.hom\n  hom_inv := by admit /- proof elided -/\n\nend OFE\n\nstructure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n\n\nnamespace Chain\n\nend Chain\n\nclass IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n\n\nnamespace COFE\n\nend COFE\n\n@[ext] structure LeibnizO (α : Type _) where\n  car : α\n\nsection DiscreteFunOF\n\nopen COFE\n\nend DiscreteFunOF\n\nsection Option\n\nvariable [OFE α]\n\ndef optionChain (c : Chain (Option α)) (x : α) : Chain α :=\n\ninstance isCOFE_option [IsCOFE α] : IsCOFE (Option α) where\n  compl c := (c 0).map fun x => IsCOFE.compl (optionChain c x)\n  conv_compl {n} c := by admit /- proof elided -/\n\nend Option\n\nsection OptionOF\n\nopen COFE\n\nvariable (F : OFunctorPre)\n\nend OptionOF\n\nsection Fixpoint\n\ndef LimitPreserving [COFE α] (P : α → Prop) : Prop :=\n  ∀ (c : Chain α), (∀ n, P (c n)) → P (COFE.compl c)\n\ndef Fixpoint.chain [OFE α] [Inhabited α] (f : α → α) [Contractive f] : Chain α where\n  chain n := Nat.repeat f (n + 1) default\n  cauchy {n} := by admit /- proof elided -/\n\ndef fixpoint [COFE α] [Inhabited α] (f : α → α) [Contractive f] : α :=\n  COFE.compl <| Fixpoint.chain f\n\nnonrec abbrev OFE.ContractiveHom.fixpoint [COFE α] [Inhabited α] (f : α -c> α) : α := fixpoint f.f\n\ninstance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/", "target_theorem": "@[elab_as_elim]\ntheorem OFE.ContractiveHom.fixpoint_ind [COFE α] [Inhabited α] (f : α -c> α)\n    (P : α → Prop) (HProper : ∀ A B : α, A ≡ B → P A → P B) (x : α) (Hbase : P x)\n    (Hind : ∀ x, P x → P (f x)) (Hlim : LimitPreserving P) :\n    P f.fixpoint :=", "ground_truth_proof": ":= by\n  let chain : Chain α := by\n    refine ⟨fun i => Nat.repeat f (i + 1) x, fun {n i} H => ?_⟩\n    induction n generalizing i with\n    | zero => simp [Nat.repeat]\n    | succ _ IH =>\n      cases i <;> simp at H\n      exact Contractive.succ _ <| IH H\n  refine HProper _ _ (fixpoint_unique (x := COFE.compl chain) ?_) ?_\n  · refine equiv_dist.mpr fun n => ?_\n    apply COFE.conv_compl.trans\n    refine .trans ?_ (f.ne.ne COFE.conv_compl).symm\n    induction n\n    · exact Contractive.zero f.f\n    · rename_i IH; apply Contractive.succ _ IH\n  · apply Hlim; intro n\n    induction n with\n    | zero => exact Hind (Nat.repeat f.f 0 x) Hbase\n    | succ _ IH => apply Hind (Nat.repeat f.f _ x) IH", "nesting_depth": 5, "transitive_dep_count": 47, "subset_aristotle": true, "category": "Framework"}
{"id": 223, "thm_name": "Iris.BI.wand_iff_exists_persistently", "thm_stmt": "theorem wand_iff_exists_persistently [BI PROP] [BIAffine PROP] {P Q : PROP} :\n    (P -∗ Q) ⊣⊢ ∃ R, R ∗ <pers> (P ∗ R → Q)", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/DerivedLaws.lean", "imports": ["import Iris.BI.Extensions", "import Iris.Std.TC", "import Iris.Std.Classes", "import Iris.BI.BI", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import Iris.BI.Classes", "import Iris.Std.Rewrite"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term\n\nsyntax:max \"<affine> \" term:40 : term\n\nsyntax:max \"□ \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "BIAffine", "content": "class BIAffine (PROP : Type _) [BI PROP] where\n  affine (P : PROP) : Affine P"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "Absorbing", "content": "class Absorbing [BI PROP] (P : PROP) where\n  absorbing : <absorb> P ⊢ P"}, {"name": "inductive", "content": "class inductive TCOr (T U : Sort _)\n  | l [t : T] : TCOr T U\n  | r [u : U] : TCOr T U"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Iris.BI.and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.imp_elim'", "content": "theorem imp_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P → R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.imp_elim_r", "content": "theorem imp_elim_r [BI PROP] {P Q : PROP} : P ∧ (P → Q) ⊢ Q"}, {"name": "Iris.BI.true_intro", "content": "theorem true_intro [BI PROP] {P : PROP} : P ⊢ True"}, {"name": "Iris.BI.and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "Iris.BI.and_mono_r", "content": "theorem and_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∧ Q ⊢ P ∧ Q'"}, {"name": "Iris.BI.exists_intro", "content": "theorem exists_intro [BI PROP] {Ψ : α → PROP} (a : α) : Ψ a ⊢ ∃ a, Ψ a"}, {"name": "Iris.BI.exists_elim", "content": "theorem exists_elim [BI PROP] {Φ : α → PROP} {Q : PROP} (h : ∀ a, Φ a ⊢ Q) : (∃ a, Φ a) ⊢ Q"}, {"name": "Iris.BI.sep_mono_l", "content": "theorem sep_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∗ Q ⊢ P' ∗ Q"}, {"name": "Iris.BI.sep_mono_r", "content": "theorem sep_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∗ Q ⊢ P ∗ Q'"}, {"name": "Iris.BI.sep_congr", "content": "@[rw_mono_rule]\ntheorem sep_congr [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊣⊢ Q) (h2 : P' ⊣⊢ Q') :\n    (P ∗ P') ⊣⊢ (Q ∗ Q')"}, {"name": "Iris.BI.sep_congr_l", "content": "theorem sep_congr_l [BI PROP] {P P' Q : PROP} (h : P ⊣⊢ P') : P ∗ Q ⊣⊢ P' ∗ Q"}, {"name": "Iris.BI.sep_congr_r", "content": "theorem sep_congr_r [BI PROP] {P Q Q' : PROP} (h : Q ⊣⊢ Q') : P ∗ Q ⊣⊢ P ∗ Q'"}, {"name": "Iris.BI.sep_comm", "content": "theorem sep_comm [BI PROP] {P Q : PROP} : P ∗ Q ⊣⊢ Q ∗ P"}, {"name": "Iris.BI.sep_assoc", "content": "theorem sep_assoc [BI PROP] {P Q R : PROP} : (P ∗ Q) ∗ R ⊣⊢ P ∗ Q ∗ R"}, {"name": "Iris.BI.sep_emp", "content": "theorem sep_emp [BI PROP] {P : PROP} : P ∗ emp ⊣⊢ P"}, {"name": "Iris.BI.true_sep_2", "content": "theorem true_sep_2 [BI PROP] {P : PROP} : P ⊢ True ∗ P"}, {"name": "Iris.BI.wand_intro'", "content": "theorem wand_intro' [BI PROP] {P Q R : PROP} (h : Q ∗ P ⊢ R) : P ⊢ Q -∗ R"}, {"name": "Iris.BI.wand_elim'", "content": "theorem wand_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P -∗ R) : P ∗ Q ⊢ R"}, {"name": "Iris.BI.wand_elim_r", "content": "theorem wand_elim_r [BI PROP] {P Q : PROP} : P ∗ (P -∗ Q) ⊢ Q"}, {"name": "Iris.BI.absorbingly_intro", "content": "theorem absorbingly_intro [BI PROP] {P : PROP} : P ⊢ <absorb> P"}, {"name": "Iris.BI.absorbingly_pure", "content": "theorem absorbingly_pure {φ : Prop} [BI PROP] : <absorb> ⌜φ⌝ ⊣⊢ (⌜φ⌝ : PROP)"}, {"name": "Iris.BI.sep_elim_l", "content": "theorem sep_elim_l [BI PROP] {P Q : PROP} : [TCOr (Affine Q) (Absorbing P)] → P ∗ Q ⊢ P\n  | TCOr.l => (sep_mono_r affine).trans sep_emp.1\n  | TCOr.r => (sep_mono_r true_intro).trans <| sep_comm.1.trans absorbing"}, {"name": "Iris.BI.sep_elim_r", "content": "theorem sep_elim_r [BI PROP] {P Q : PROP} [TCOr (Affine P) (Absorbing Q)] : P ∗ Q ⊢ Q"}, {"name": "Iris.BI.sep_and", "content": "theorem sep_and [BI PROP] {P Q : PROP}\n    [TCOr (Affine P) (Absorbing Q)] [TCOr (Affine Q) (Absorbing P)] : P ∗ Q ⊢ P ∧ Q"}, {"name": "Iris.BI.true_sep", "content": "theorem true_sep [BI PROP] {P : PROP} [Absorbing P] : True ∗ P ⊣⊢ P"}, {"name": "Iris.BI.sep_true", "content": "theorem sep_true [BI PROP] {P : PROP} [Absorbing P] : P ∗ True ⊣⊢ P"}, {"name": "Iris.BI.persistently_absorb_r", "content": "theorem persistently_absorb_r [BI PROP] {P Q : PROP} : P ∗ <pers> Q ⊢ <pers> Q"}, {"name": "Iris.BI.absorbingly_persistently", "content": "theorem absorbingly_persistently [BI PROP] {P : PROP} : <absorb> <pers> P ⊣⊢ <pers> P"}, {"name": "Iris.BI.persistently_emp_intro", "content": "theorem persistently_emp_intro [BI PROP] {P : PROP} : P ⊢ <pers> emp"}, {"name": "Iris.BI.persistently_emp", "content": "theorem persistently_emp [BI PROP] : <pers> (emp : PROP) ⊣⊢ True"}, {"name": "Iris.BI.persistently_true", "content": "theorem persistently_true [BI PROP] : <pers> (True : PROP) ⊣⊢ True"}, {"name": "Iris.BI.persistently_elim", "content": "theorem persistently_elim [BI PROP] {P : PROP} [Absorbing P] : <pers> P ⊢ P"}, {"name": "Iris.BI.persistently_idem", "content": "theorem persistently_idem [BI PROP] {P : PROP} : <pers> <pers> P ⊣⊢ <pers> P"}, {"name": "Iris.BI.persistently_intro'", "content": "theorem persistently_intro' [BI PROP] {P Q : PROP} (h : <pers> P ⊢ Q) : <pers> P ⊢ <pers> Q"}, {"name": "Iris.BI.persistently_pure", "content": "theorem persistently_pure {φ : Prop} [BI PROP] : <pers> ⌜φ⌝ ⊣⊢ (⌜φ⌝ : PROP)"}, {"name": "Iris.BI.and_persistently_imp_sep", "content": "theorem and_persistently_imp_sep [BI PROP] {P Q : PROP} : P ∧ <pers> Q ⊢ P ∗ <pers> Q"}, {"name": "Iris.BI.and_persistently_iff_sep", "content": "theorem and_persistently_iff_sep [BI PROP] [BIAffine PROP] {P Q : PROP} :\n    P ∧ <pers> Q ⊣⊢ P ∗ <pers> Q"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.Extensions\n\nimport Iris.BI.BI\n\nimport Iris.Std.Classes\n\nimport Iris.Std.Rewrite\n\nimport Iris.Std.TC\n\nnamespace Iris.BI\n\nopen Iris.Std BI", "target_theorem": "theorem wand_iff_exists_persistently [BI PROP] [BIAffine PROP] {P Q : PROP} :\n    (P -∗ Q) ⊣⊢ ∃ R, R ∗ <pers> (P ∗ R → Q) :=", "ground_truth_proof": ":= by\n  constructor\n  · refine (sep_true.2.trans ?_).trans (exists_intro iprop(P -∗ Q))\n    exact sep_mono_r <| persistently_pure.2.trans <| persistently_intro' <|\n      imp_intro <| (and_mono persistently_pure.1 wand_elim_r).trans and_elim_r\n  · exact exists_elim fun R => wand_intro' <| sep_assoc.2.trans <|\n      and_persistently_iff_sep.2.trans <| (and_mono_r persistently_elim).trans imp_elim_r", "nesting_depth": 6, "transitive_dep_count": 89, "subset_aristotle": false, "category": "Framework"}
{"id": 224, "thm_name": "Iris.COFE.OFunctor.Fix.Impl.Tower.embed_up", "thm_stmt": "theorem Tower.embed_up (x : A F k) :\n    Tower.embed (k+1) (up F k x) ≡ Tower.embed k x", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/COFESolver.lean", "imports": ["import Iris.Algebra.OFE"], "used_lib_defs": [{"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "ULift", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited.default", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}, {"name": "Nat.succ", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "Hom", "content": "@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "Hom.comp", "content": "protected def Hom.comp [OFE α] [OFE β] [OFE γ] (g : β -n> γ) (f : α -n> β) : α -n> γ where\n  f := g.f ∘ f.f\n  ne.1 _ _ _ h := g.ne.1 (f.ne.1 h)"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -n> \" => Hom"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -C> \" => Hom"}], "lib_lemmas": [{"name": "congrArg", "module": "Init.Prelude"}, {"name": "symm", "module": "Mathlib.Order.Defs.Unbundled"}, {"name": "Nat.add_assoc", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.add_left_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.add_left_cancel_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.add_right_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.le_antisymm", "module": "Init.Prelude"}, {"name": "Nat.le_of_succ_le", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.not_lt", "module": "Init.Data.Nat.Basic"}], "repo_lemmas": [{"name": "exact", "content": "theorem exact [BI PROP] (Q : PROP) : Q ⊢ Q"}], "used_local_defs": [{"name": "Iris.COFE.OFunctor.Fix.Impl.A'", "content": "def A' : Nat → Σ α : Type u, COFE α\n  | 0 => ⟨ULift Unit, inferInstance⟩\n  | n+1 => let ⟨A, _⟩ := A' n; ⟨F A A, inferInstance⟩"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.A", "content": "def A (n : Nat) : Type u := (A' F n).1"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.up", "content": "def up : ∀ k, A F k -n> A F (k+1)\n  | 0 => ⟨fun _ => inh.default, ⟨fun _ _ _ _ => .rfl⟩⟩\n  | k+1 => map (down k) (up k)"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.down", "content": "def down : ∀ k, A F (k+1) -n> A F k\n  | 0 => ⟨fun _ => ⟨()⟩, ⟨fun _ _ _ _ => .rfl⟩⟩\n  | k+1 => map (up k) (down k)"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.Tower", "content": "@[ext] structure Tower : Type u where\n  val k : A F k\n  protected down {k} : down F k (val (k+1)) ≡ val k"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.upN", "content": "def upN {k} : ∀ n, A F k -n> A F (k + n)\n  | 0 => .id\n  | n+1 => (up F (k + n)).comp (upN n)"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.downN", "content": "def downN {k} : ∀ n, A F (k + n) -n> A F k\n  | 0 => .id\n  | n+1 => (downN n).comp (down F (k + n))"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.eqToHom", "content": "def eqToHom (e : i = k) : A F i -n> A F k := e ▸ .id"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.embed", "content": "def embed : A F k -n> A F i :=\n  if h : k ≤ i then (eqToHom (Nat.add_sub_cancel' h)).comp (upN ..)\n  else (downN ..).comp (eqToHom (Nat.add_sub_cancel' (Nat.le_of_not_ge h)).symm)"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.Tower.embed", "content": "protected def Tower.embed (k) : A F k -n> Tower F :="}], "used_local_lemmas": [{"name": "Iris.COFE.OFunctor.Fix.Impl.down_up", "content": "theorem down_up : ∀ {k} x, down F k (up F k x) ≡ x\n  | 0, ⟨()⟩ => .rfl\n  | _+1, _ => (map_comp ..).symm.trans <|\n    (map_ne.eqv down_up down_up _).trans (map_id _)"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.eqToHom_up", "content": "theorem eqToHom_up {k k'} {x : A F k} (e : k = k') :\n    eqToHom (congrArg Nat.succ e) (up F k x) = up F k' (eqToHom e x)"}], "local_ctx": "import Iris.Algebra.OFE\n\nnamespace Iris.COFE.OFunctor\n\nopen OFE\n\nvariable {F : ∀ α β [OFE α] [OFE β], Type u} [OFunctorContractive F]\n\nvariable [∀ α [COFE α], IsCOFE (F α α)]\n\nvariable [inh : Inhabited (F (ULift Unit) (ULift Unit))]\n\nnamespace Fix.Impl\n\nvariable (F) in\n\ndef A' : Nat → Σ α : Type u, COFE α\n  | 0 => ⟨ULift Unit, inferInstance⟩\n  | n+1 => let ⟨A, _⟩ := A' n; ⟨F A A, inferInstance⟩\n\nvariable (F) in\n\ndef A (n : Nat) : Type u := (A' F n).1\n\nvariable (F) in\n\ndef up : ∀ k, A F k -n> A F (k+1)\n  | 0 => ⟨fun _ => inh.default, ⟨fun _ _ _ _ => .rfl⟩⟩\n  | k+1 => map (down k) (up k)\n\ndef down : ∀ k, A F (k+1) -n> A F k\n  | 0 => ⟨fun _ => ⟨()⟩, ⟨fun _ _ _ _ => .rfl⟩⟩\n  | k+1 => map (up k) (down k)\n\nend\n\n  open OFunctorContractive in exact match k with\n  | 0 => map_contractive.zero (x := (_, _)) (y := (_, _)) _ _\n  | k+1 => map_contractive.succ (x := (_, _)) (y := (_, _)) _ ⟨up_down, up_down⟩ _\n\nvariable (F) in\n\n@[ext] structure Tower : Type u where\n  val k : A F k\n  protected down {k} : down F k (val (k+1)) ≡ val k\n\nvariable (F) in\n\ndef upN {k} : ∀ n, A F k -n> A F (k + n)\n  | 0 => .id\n  | n+1 => (up F (k + n)).comp (upN n)\n\nvariable (F) in\n\ndef downN {k} : ∀ n, A F (k + n) -n> A F k\n  | 0 => .id\n  | n+1 => (downN n).comp (down F (k + n))\n\ndef eqToHom (e : i = k) : A F i -n> A F k := e ▸ .id\n\ndef embed : A F k -n> A F i :=\n  if h : k ≤ i then (eqToHom (Nat.add_sub_cancel' h)).comp (upN ..)\n  else (downN ..).comp (eqToHom (Nat.add_sub_cancel' (Nat.le_of_not_ge h)).symm)\n\nprotected def Tower.embed (k) : A F k -n> Tower F :=", "target_theorem": "theorem Tower.embed_up (x : A F k) :\n    Tower.embed (k+1) (up F k x) ≡ Tower.embed k x :=", "ground_truth_proof": ":= by\n  refine equiv_dist.2 fun n i => ?_\n  dsimp [Tower.embed, embed]; split <;> rename_i h₁\n  · simp [Nat.le_of_succ_le h₁]\n    suffices ∀ a b (e₁ : k + 1 + a = i) (e₂ : k+b = i),\n      eqToHom e₁ (upN F a (up F k x)) = eqToHom e₂ (upN F b x) from this .. ▸ .rfl\n    rintro a b eq rfl\n    rw [Nat.add_right_comm, Nat.add_assoc, Nat.add_left_cancel_iff] at eq; subst b\n    show _ = up F (k + a) (upN F a x); clear h₁\n    induction a with\n    | zero => rfl\n    | succ a ih =>\n      dsimp [upN, Hom.comp]\n      rw [eqToHom_up (by omega : k + 1 + a = k + (a + 1))]; congr 1; apply ih\n  · split <;> rename_i h₂\n    · cases Nat.le_antisymm h₂ (Nat.not_lt.1 h₁)\n      have {a b} {e₁ : k+1 = k+a} {e₂ : k+b = k+0} :\n        downN F a (eqToHom e₁ (up F k x)) ≡{n}≡ eqToHom e₂ (upN F b x) := by\n        cases Nat.add_left_cancel e₁; cases Nat.add_left_cancel e₂\n        exact (down_up ..).dist\n      exact this\n    · dsimp [Hom.comp]\n      suffices ∀ a b (e₁ : k + 1 = i + a) (e₂ : k = i + b),\n        downN F a (eqToHom e₁ (up F k x)) ≡{n}≡\n        downN F b (eqToHom e₂ x) from this ..\n      rintro a b eq rfl; cases Nat.add_left_cancel (m := b+1) eq\n      exact (downN ..).ne.1 (down_up x).dist", "nesting_depth": 8, "transitive_dep_count": 43, "subset_aristotle": false, "category": "Framework"}
{"id": 225, "thm_name": "Iris.not_add_le_self", "thm_stmt": "theorem not_add_le_self {a b : α} : ¬ a + b ≤ a", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/Frac.lean", "imports": ["import Iris.Algebra.CMRA", "import Iris.Algebra.OFE"], "used_lib_defs": [{"name": "structure BitVec (w : Nat) where", "module": ""}, {"name": "/-- Construct a `BitVec w` from a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "ofFin ::", "module": ""}, {"name": "/-- Interpret a bitvector as a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "toFin : Fin (hPow 2 w)", "module": ""}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "mt", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Iris.NumericFraction", "content": "class NumericFraction (α : Type _) extends One α, Add α, LE α, LT α where\n  add_comm : ∀ a b : α, a + b = b + a\n  add_assoc : ∀ a b c : α, a + (b + c) = (a + b) + c\n  add_left_cancel : ∀ {a b c : α}, a + b = a + c → b = c\n  le_def : ∀ {a b : α}, a ≤ b ↔ a = b ∨ a < b\n  lt_def : ∀ {a b : α}, a < b ↔ ∃ c : α, a + c = b\n  lt_irrefl : ∀ {a : α}, ¬a < a"}], "used_local_lemmas": [{"name": "Iris.add_ne_self", "content": "theorem add_ne_self {a b : α} : a + b ≠ a"}, {"name": "Iris.not_add_lt_self", "content": "theorem not_add_lt_self {a b : α} : ¬a + b < a"}], "local_ctx": "import Iris.Algebra.CMRA\n\nimport Iris.Algebra.OFE\n\nnamespace Iris\n\nnamespace Fraction\n\nvariable [Fraction α]\n\nend Fraction\n\nopen Fraction OFE CMRA\n\nsection NumericFraction\n\nclass NumericFraction (α : Type _) extends One α, Add α, LE α, LT α where\n  add_comm : ∀ a b : α, a + b = b + a\n  add_assoc : ∀ a b c : α, a + (b + c) = (a + b) + c\n  add_left_cancel : ∀ {a b c : α}, a + b = a + c → b = c\n  le_def : ∀ {a b : α}, a ≤ b ↔ a = b ∨ a < b\n  lt_def : ∀ {a b : α}, a < b ↔ ∃ c : α, a + c = b\n  lt_irrefl : ∀ {a : α}, ¬a < a\n\nvariable {α} [NumericFraction α]\n\nopen NumericFraction", "target_theorem": "theorem not_add_le_self {a b : α} : ¬ a + b ≤ a :=", "ground_truth_proof": ":= fun H => by\n  obtain H | H := le_def.mp H\n  · exact add_ne_self H\n  · exact not_add_lt_self H", "nesting_depth": 3, "transitive_dep_count": 4, "subset_aristotle": false, "category": "Framework"}
{"id": 226, "thm_name": "Iris.BI.imp_iff_exists_persistently", "thm_stmt": "theorem imp_iff_exists_persistently [BI PROP] [BIAffine PROP] {P Q : PROP} :\n    (P → Q) ⊣⊢ ∃ R, R ∧ <pers> (P ∧ R -∗ Q)", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/DerivedLaws.lean", "imports": ["import Iris.BI.Extensions", "import Iris.Std.TC", "import Iris.Std.Classes", "import Iris.BI.BI", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import Iris.BI.Classes", "import Iris.Std.Rewrite"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term\n\nsyntax:max \"<affine> \" term:40 : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term\n\nsyntax:max \"□ \" term:40 : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "BIAffine", "content": "class BIAffine (PROP : Type _) [BI PROP] where\n  affine (P : PROP) : Affine P"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Iris.BI.and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_symm", "content": "theorem and_symm [BI PROP] {P Q : PROP} : P ∧ Q ⊢ Q ∧ P"}, {"name": "Iris.BI.imp_intro'", "content": "theorem imp_intro' [BI PROP] {P Q R : PROP} (h : Q ∧ P ⊢ R) : P ⊢ Q → R"}, {"name": "Iris.BI.imp_elim'", "content": "theorem imp_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P → R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.imp_elim_r", "content": "theorem imp_elim_r [BI PROP] {P Q : PROP} : P ∧ (P → Q) ⊢ Q"}, {"name": "Iris.BI.true_intro", "content": "theorem true_intro [BI PROP] {P : PROP} : P ⊢ True"}, {"name": "Iris.BI.and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "Iris.BI.and_mono_l", "content": "theorem and_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∧ Q ⊢ P' ∧ Q"}, {"name": "Iris.BI.and_mono_r", "content": "theorem and_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∧ Q ⊢ P ∧ Q'"}, {"name": "Iris.BI.exists_intro", "content": "theorem exists_intro [BI PROP] {Ψ : α → PROP} (a : α) : Ψ a ⊢ ∃ a, Ψ a"}, {"name": "Iris.BI.exists_elim", "content": "theorem exists_elim [BI PROP] {Φ : α → PROP} {Q : PROP} (h : ∀ a, Φ a ⊢ Q) : (∃ a, Φ a) ⊢ Q"}, {"name": "Iris.BI.and_comm", "content": "theorem and_comm [BI PROP] {P Q : PROP} : P ∧ Q ⊣⊢ Q ∧ P"}, {"name": "Iris.BI.true_and", "content": "theorem true_and [BI PROP] {P : PROP} : True ∧ P ⊣⊢ P"}, {"name": "Iris.BI.and_true", "content": "theorem and_true [BI PROP] {P : PROP} : P ∧ True ⊣⊢ P"}, {"name": "Iris.BI.and_assoc", "content": "theorem and_assoc [BI PROP] {P Q R : PROP} : (P ∧ Q) ∧ R ⊣⊢ P ∧ Q ∧ R"}, {"name": "Iris.BI.sep_mono_l", "content": "theorem sep_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∗ Q ⊢ P' ∗ Q"}, {"name": "Iris.BI.sep_mono_r", "content": "theorem sep_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∗ Q ⊢ P ∗ Q'"}, {"name": "Iris.BI.sep_comm", "content": "theorem sep_comm [BI PROP] {P Q : PROP} : P ∗ Q ⊣⊢ Q ∗ P"}, {"name": "Iris.BI.sep_emp", "content": "theorem sep_emp [BI PROP] {P : PROP} : P ∗ emp ⊣⊢ P"}, {"name": "Iris.BI.wand_elim'", "content": "theorem wand_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P -∗ R) : P ∗ Q ⊢ R"}, {"name": "Iris.BI.wand_elim_r", "content": "theorem wand_elim_r [BI PROP] {P Q : PROP} : P ∗ (P -∗ Q) ⊢ Q"}, {"name": "Iris.BI.affinely_elim", "content": "theorem affinely_elim [BI PROP] {P : PROP} : <affine> P ⊢ P"}, {"name": "Iris.BI.persistently_emp_intro", "content": "theorem persistently_emp_intro [BI PROP] {P : PROP} : P ⊢ <pers> emp"}, {"name": "Iris.BI.persistently_emp", "content": "theorem persistently_emp [BI PROP] : <pers> (emp : PROP) ⊣⊢ True"}, {"name": "Iris.BI.intuitionistically_elim", "content": "theorem intuitionistically_elim [BI PROP] {P : PROP} : □ P ⊢ P"}, {"name": "Iris.BI.persistently_and_intuitionistically_sep_r", "content": "theorem persistently_and_intuitionistically_sep_r [BI PROP] {P Q : PROP} :\n    P ∧ <pers> Q ⊣⊢ P ∗ □ Q"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.Extensions\n\nimport Iris.BI.BI\n\nimport Iris.Std.Classes\n\nimport Iris.Std.Rewrite\n\nimport Iris.Std.TC\n\nnamespace Iris.BI\n\nopen Iris.Std BI", "target_theorem": "theorem imp_iff_exists_persistently [BI PROP] [BIAffine PROP] {P Q : PROP} :\n    (P → Q) ⊣⊢ ∃ R, R ∧ <pers> (P ∧ R -∗ Q) :=", "ground_truth_proof": ":= by\n  constructor\n  · refine (and_true.2.trans ?_).trans (exists_intro iprop(P → Q))\n    exact and_mono_r <| persistently_emp.2.trans <| persistently_mono <|\n      wand_intro <| emp_sep.1.trans imp_elim_r\n  · exact exists_elim fun R => imp_intro' <| and_assoc.2.trans <|\n      persistently_and_intuitionistically_sep_r.1.trans <|\n      (sep_mono_r intuitionistically_elim).trans wand_elim_r", "nesting_depth": 5, "transitive_dep_count": 75, "subset_aristotle": false, "category": "Framework"}
{"id": 227, "thm_name": "Iris.COFE.OFunctor.Fix.Impl.downN_upN", "thm_stmt": "theorem downN_upN {k} (x : A F k) : ∀ {i}, downN F i (upN F i x) ≡ x\n  | 0 => .rfl\n  | n+1 => ((downN F n).ne.eqv (down_up ..)).trans (downN_upN _)", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/COFESolver.lean", "imports": ["import Iris.Algebra.OFE"], "used_lib_defs": [{"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "ULift", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited.default", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "Hom", "content": "@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -n> \" => Hom"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -C> \" => Hom"}], "lib_lemmas": [{"name": "symm", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Iris.COFE.OFunctor.Fix.Impl.A'", "content": "def A' : Nat → Σ α : Type u, COFE α\n  | 0 => ⟨ULift Unit, inferInstance⟩\n  | n+1 => let ⟨A, _⟩ := A' n; ⟨F A A, inferInstance⟩"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.A", "content": "def A (n : Nat) : Type u := (A' F n).1"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.up", "content": "def up : ∀ k, A F k -n> A F (k+1)\n  | 0 => ⟨fun _ => inh.default, ⟨fun _ _ _ _ => .rfl⟩⟩\n  | k+1 => map (down k) (up k)"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.down", "content": "def down : ∀ k, A F (k+1) -n> A F k\n  | 0 => ⟨fun _ => ⟨()⟩, ⟨fun _ _ _ _ => .rfl⟩⟩\n  | k+1 => map (up k) (down k)"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.upN", "content": "def upN {k} : ∀ n, A F k -n> A F (k + n)\n  | 0 => .id\n  | n+1 => (up F (k + n)).comp (upN n)"}, {"name": "Iris.COFE.OFunctor.Fix.Impl.downN", "content": "def downN {k} : ∀ n, A F (k + n) -n> A F k\n  | 0 => .id\n  | n+1 => (downN n).comp (down F (k + n))"}], "used_local_lemmas": [{"name": "Iris.COFE.OFunctor.Fix.Impl.down_up", "content": "theorem down_up : ∀ {k} x, down F k (up F k x) ≡ x\n  | 0, ⟨()⟩ => .rfl\n  | _+1, _ => (map_comp ..).symm.trans <|\n    (map_ne.eqv down_up down_up _).trans (map_id _)"}], "local_ctx": "import Iris.Algebra.OFE\n\nnamespace Iris.COFE.OFunctor\n\nopen OFE\n\nvariable {F : ∀ α β [OFE α] [OFE β], Type u} [OFunctorContractive F]\n\nvariable [∀ α [COFE α], IsCOFE (F α α)]\n\nvariable [inh : Inhabited (F (ULift Unit) (ULift Unit))]\n\nnamespace Fix.Impl\n\nvariable (F) in\n\ndef A' : Nat → Σ α : Type u, COFE α\n  | 0 => ⟨ULift Unit, inferInstance⟩\n  | n+1 => let ⟨A, _⟩ := A' n; ⟨F A A, inferInstance⟩\n\nvariable (F) in\n\ndef A (n : Nat) : Type u := (A' F n).1\n\nvariable (F) in\n\ndef up : ∀ k, A F k -n> A F (k+1)\n  | 0 => ⟨fun _ => inh.default, ⟨fun _ _ _ _ => .rfl⟩⟩\n  | k+1 => map (down k) (up k)\n\ndef down : ∀ k, A F (k+1) -n> A F k\n  | 0 => ⟨fun _ => ⟨()⟩, ⟨fun _ _ _ _ => .rfl⟩⟩\n  | k+1 => map (up k) (down k)\n\nend\n\n  open OFunctorContractive in exact match k with\n  | 0 => map_contractive.zero (x := (_, _)) (y := (_, _)) _ _\n  | k+1 => map_contractive.succ (x := (_, _)) (y := (_, _)) _ ⟨up_down, up_down⟩ _\n\nvariable (F) in\n\nvariable (F) in\n\ndef upN {k} : ∀ n, A F k -n> A F (k + n)\n  | 0 => .id\n  | n+1 => (up F (k + n)).comp (upN n)\n\nvariable (F) in\n\ndef downN {k} : ∀ n, A F (k + n) -n> A F k\n  | 0 => .id\n  | n+1 => (downN n).comp (down F (k + n))", "target_theorem": "theorem downN_upN {k} (x : A F k) : ∀ {i}, downN F i (upN F i x) ≡ x :=", "ground_truth_proof": "| 0 => .rfl\n  | n+1 => ((downN F n).ne.eqv (down_up ..)).trans (downN_upN _)", "nesting_depth": 8, "transitive_dep_count": 27, "subset_aristotle": false, "category": "Framework"}
{"id": 228, "thm_name": "Iris.BI.persistently_sep", "thm_stmt": "theorem persistently_sep [BI PROP] [BIPositive PROP] {P Q : PROP} :\n    <pers> (P ∗ Q) ⊣⊢ <pers> P ∗ <pers> Q", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/DerivedLaws.lean", "imports": ["import Iris.BI.Extensions", "import Iris.Std.TC", "import Iris.Std.Classes", "import Iris.BI.BI", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import Iris.BI.Classes", "import Iris.Std.Rewrite"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term\n\nsyntax:max \"<affine> \" term:40 : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term\n\nsyntax:max \"□ \" term:40 : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "BIPositive", "content": "class BIPositive (PROP : Type _) [BI PROP] where\n  affinely_sep_l {P Q : PROP} : <affine> (P ∗ Q) ⊢ <affine> P ∗ Q"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Iris.BI.and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_symm", "content": "theorem and_symm [BI PROP] {P Q : PROP} : P ∧ Q ⊢ Q ∧ P"}, {"name": "Iris.BI.true_intro", "content": "theorem true_intro [BI PROP] {P : PROP} : P ⊢ True"}, {"name": "Iris.BI.and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "Iris.BI.and_mono_l", "content": "theorem and_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∧ Q ⊢ P' ∧ Q"}, {"name": "Iris.BI.and_mono_r", "content": "theorem and_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∧ Q ⊢ P ∧ Q'"}, {"name": "Iris.BI.and_congr", "content": "@[rw_mono_rule]\ntheorem and_congr [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊣⊢ Q) (h2 : P' ⊣⊢ Q') : P ∧ P' ⊣⊢ Q ∧ Q'"}, {"name": "Iris.BI.and_congr_l", "content": "theorem and_congr_l [BI PROP] {P P' Q : PROP} (h : P ⊣⊢ P') : P ∧ Q ⊣⊢ P' ∧ Q"}, {"name": "Iris.BI.and_self", "content": "theorem and_self [BI PROP] {P : PROP} : P ∧ P ⊣⊢ P"}, {"name": "Iris.BI.and_comm", "content": "theorem and_comm [BI PROP] {P Q : PROP} : P ∧ Q ⊣⊢ Q ∧ P"}, {"name": "Iris.BI.and_assoc", "content": "theorem and_assoc [BI PROP] {P Q R : PROP} : (P ∧ Q) ∧ R ⊣⊢ P ∧ Q ∧ R"}, {"name": "Iris.BI.sep_mono_l", "content": "theorem sep_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∗ Q ⊢ P' ∗ Q"}, {"name": "Iris.BI.sep_mono_r", "content": "theorem sep_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∗ Q ⊢ P ∗ Q'"}, {"name": "Iris.BI.sep_congr", "content": "@[rw_mono_rule]\ntheorem sep_congr [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊣⊢ Q) (h2 : P' ⊣⊢ Q') :\n    (P ∗ P') ⊣⊢ (Q ∗ Q')"}, {"name": "Iris.BI.sep_congr_l", "content": "theorem sep_congr_l [BI PROP] {P P' Q : PROP} (h : P ⊣⊢ P') : P ∗ Q ⊣⊢ P' ∗ Q"}, {"name": "Iris.BI.sep_congr_r", "content": "theorem sep_congr_r [BI PROP] {P Q Q' : PROP} (h : Q ⊣⊢ Q') : P ∗ Q ⊣⊢ P ∗ Q'"}, {"name": "Iris.BI.sep_comm", "content": "theorem sep_comm [BI PROP] {P Q : PROP} : P ∗ Q ⊣⊢ Q ∗ P"}, {"name": "Iris.BI.sep_assoc", "content": "theorem sep_assoc [BI PROP] {P Q R : PROP} : (P ∗ Q) ∗ R ⊣⊢ P ∗ Q ∗ R"}, {"name": "Iris.BI.sep_emp", "content": "theorem sep_emp [BI PROP] {P : PROP} : P ∗ emp ⊣⊢ P"}, {"name": "Iris.BI.true_sep_2", "content": "theorem true_sep_2 [BI PROP] {P : PROP} : P ⊢ True ∗ P"}, {"name": "Iris.BI.affinely_elim_emp", "content": "theorem affinely_elim_emp [BI PROP] {P : PROP} : <affine> P ⊢ emp"}, {"name": "Iris.BI.affinely_elim", "content": "theorem affinely_elim [BI PROP] {P : PROP} : <affine> P ⊢ P"}, {"name": "Iris.BI.affinely_mono", "content": "@[rw_mono_rule]\ntheorem affinely_mono [BI PROP] {P Q : PROP} : (P ⊢ Q) → <affine> P ⊢ <affine> Q"}, {"name": "Iris.BI.affinely_idem", "content": "theorem affinely_idem [BI PROP] {P : PROP} : <affine> <affine> P ⊣⊢ <affine> P"}, {"name": "Iris.BI.affinely_sep_2", "content": "theorem affinely_sep_2 [BI PROP] {P Q : PROP} : <affine> P ∗ <affine> Q ⊢ <affine> (P ∗ Q)"}, {"name": "Iris.BI.affinely_sep_r", "content": "theorem affinely_sep_r [BI PROP] [BIPositive PROP] {P Q : PROP} :\n    <affine> (P ∗ Q) ⊢ P ∗ <affine> Q"}, {"name": "Iris.BI.affinely_sep", "content": "theorem affinely_sep [BI PROP] [BIPositive PROP] {P Q : PROP} :\n    <affine> (P ∗ Q) ⊣⊢ <affine> P ∗ <affine> Q"}, {"name": "Iris.BI.absorbingly_intro", "content": "theorem absorbingly_intro [BI PROP] {P : PROP} : P ⊢ <absorb> P"}, {"name": "Iris.BI.persistently_absorb_r", "content": "theorem persistently_absorb_r [BI PROP] {P Q : PROP} : P ∗ <pers> Q ⊢ <pers> Q"}, {"name": "Iris.BI.absorbingly_persistently", "content": "theorem absorbingly_persistently [BI PROP] {P : PROP} : <absorb> <pers> P ⊣⊢ <pers> P"}, {"name": "Iris.BI.persistently_and", "content": "theorem persistently_and [BI PROP] {P Q : PROP} : <pers> (P ∧ Q) ⊣⊢ <pers> P ∧ <pers> Q"}, {"name": "Iris.BI.persistently_emp_intro", "content": "theorem persistently_emp_intro [BI PROP] {P : PROP} : P ⊢ <pers> emp"}, {"name": "Iris.BI.persistently_affinely", "content": "theorem persistently_affinely [BI PROP] {P : PROP} : <pers> <affine> P ⊣⊢ <pers> P"}, {"name": "Iris.BI.persistently_and_sep_assoc", "content": "theorem persistently_and_sep_assoc [BI PROP] {P Q R : PROP} :\n    <pers> P ∧ (Q ∗ R) ⊣⊢ (<pers> P ∧ Q) ∗ R"}, {"name": "Iris.BI.intuitionistically_elim", "content": "theorem intuitionistically_elim [BI PROP] {P : PROP} : □ P ⊢ P"}, {"name": "Iris.BI.persistently_idem", "content": "theorem persistently_idem [BI PROP] {P : PROP} : <pers> <pers> P ⊣⊢ <pers> P"}, {"name": "Iris.BI.persistently_and_imp_sep", "content": "theorem persistently_and_imp_sep [BI PROP] {P Q : PROP} : <pers> P ∧ Q ⊢ <pers> P ∗ Q"}, {"name": "Iris.BI.persistently_and_sep", "content": "theorem persistently_and_sep [BI PROP] {P Q : PROP} : <pers> (P ∧ Q) ⊢ <pers> (P ∗ Q)"}, {"name": "Iris.BI.persistently_and_persistently_sep", "content": "theorem persistently_and_persistently_sep [BI PROP] {P Q : PROP} :\n    <pers> P ∧ <pers> Q ⊣⊢ <pers> P ∗ <pers> Q"}, {"name": "Iris.BI.persistently_sep_2", "content": "theorem persistently_sep_2 [BI PROP] {P Q : PROP} : <pers> P ∗ <pers> Q ⊢ <pers> (P ∗ Q)"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.Extensions\n\nimport Iris.BI.BI\n\nimport Iris.Std.Classes\n\nimport Iris.Std.Rewrite\n\nimport Iris.Std.TC\n\nnamespace Iris.BI\n\nopen Iris.Std BI", "target_theorem": "theorem persistently_sep [BI PROP] [BIPositive PROP] {P Q : PROP} :\n    <pers> (P ∗ Q) ⊣⊢ <pers> P ∗ <pers> Q :=", "ground_truth_proof": ":= by\n  refine ⟨persistently_affinely.2.trans ?_, persistently_sep_2⟩\n  refine persistently_mono affinely_sep.1 |>.trans ?_ |>.trans persistently_and_persistently_sep.1\n  exact and_intro\n    (persistently_mono <| (sep_mono_r affinely_elim_emp).trans <| sep_emp.1.trans affinely_elim)\n    (persistently_mono <| (sep_mono_l affinely_elim_emp).trans <| emp_sep.1.trans affinely_elim)", "nesting_depth": 7, "transitive_dep_count": 89, "subset_aristotle": false, "category": "Framework"}
{"id": 229, "thm_name": "Iris.BI.loeb_wand_intuitionistically", "thm_stmt": "theorem loeb_wand_intuitionistically [BI PROP] [BILoeb PROP] {P : PROP} :\n    □ (□ ▷ P -∗ P) ⊢ P", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/DerivedLaws.lean", "imports": ["import Iris.BI.Extensions", "import Iris.Std.TC", "import Iris.Std.Classes", "import Iris.BI.BI", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import Iris.BI.Classes", "import Iris.Std.Rewrite"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Function.const", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term\n\nsyntax:max \"<affine> \" term:40 : term\n\nsyntax:max \"□ \" term:40 : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "BILoeb", "content": "class BILoeb (PROP : Type _) [BI PROP] where\n  loeb_weak {P : PROP} : (▷ P ⊢ P) → True ⊢ P"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Iris.BI.and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_symm", "content": "theorem and_symm [BI PROP] {P Q : PROP} : P ∧ Q ⊢ Q ∧ P"}, {"name": "Iris.BI.imp_intro'", "content": "theorem imp_intro' [BI PROP] {P Q R : PROP} (h : Q ∧ P ⊢ R) : P ⊢ Q → R"}, {"name": "Iris.BI.imp_elim'", "content": "theorem imp_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P → R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.imp_elim_l", "content": "theorem imp_elim_l [BI PROP] {P Q : PROP} : (P → Q) ∧ P ⊢ Q"}, {"name": "Iris.BI.imp_elim_r", "content": "theorem imp_elim_r [BI PROP] {P Q : PROP} : P ∧ (P → Q) ⊢ Q"}, {"name": "Iris.BI.true_intro", "content": "theorem true_intro [BI PROP] {P : PROP} : P ⊢ True"}, {"name": "Iris.BI.and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "Iris.BI.and_mono_l", "content": "theorem and_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∧ Q ⊢ P' ∧ Q"}, {"name": "Iris.BI.and_mono_r", "content": "theorem and_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∧ Q ⊢ P ∧ Q'"}, {"name": "Iris.BI.and_congr", "content": "@[rw_mono_rule]\ntheorem and_congr [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊣⊢ Q) (h2 : P' ⊣⊢ Q') : P ∧ P' ⊣⊢ Q ∧ Q'"}, {"name": "Iris.BI.and_congr_r", "content": "theorem and_congr_r [BI PROP] {P Q Q' : PROP} (h : Q ⊣⊢ Q') : P ∧ Q ⊣⊢ P ∧ Q'"}, {"name": "Iris.BI.and_self", "content": "theorem and_self [BI PROP] {P : PROP} : P ∧ P ⊣⊢ P"}, {"name": "Iris.BI.and_comm", "content": "theorem and_comm [BI PROP] {P Q : PROP} : P ∧ Q ⊣⊢ Q ∧ P"}, {"name": "Iris.BI.and_assoc", "content": "theorem and_assoc [BI PROP] {P Q R : PROP} : (P ∧ Q) ∧ R ⊣⊢ P ∧ Q ∧ R"}, {"name": "Iris.BI.sep_mono_l", "content": "theorem sep_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∗ Q ⊢ P' ∗ Q"}, {"name": "Iris.BI.sep_comm", "content": "theorem sep_comm [BI PROP] {P Q : PROP} : P ∗ Q ⊣⊢ Q ∗ P"}, {"name": "Iris.BI.sep_emp", "content": "theorem sep_emp [BI PROP] {P : PROP} : P ∗ emp ⊣⊢ P"}, {"name": "Iris.BI.true_sep_2", "content": "theorem true_sep_2 [BI PROP] {P : PROP} : P ⊢ True ∗ P"}, {"name": "Iris.BI.wand_elim'", "content": "theorem wand_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P -∗ R) : P ∗ Q ⊢ R"}, {"name": "Iris.BI.wand_elim_r", "content": "theorem wand_elim_r [BI PROP] {P Q : PROP} : P ∗ (P -∗ Q) ⊢ Q"}, {"name": "Iris.BI.pure_elim", "content": "theorem pure_elim [BI PROP] (φ : Prop) {Q R : PROP} (h1 : Q ⊢ ⌜φ⌝) (h2 : φ → Q ⊢ R) : Q ⊢ R"}, {"name": "Iris.BI.pure_elim_l", "content": "theorem pure_elim_l [BI PROP] {φ : Prop} {Q R : PROP} (h : φ → Q ⊢ R) : ⌜φ⌝ ∧ Q ⊢ R"}, {"name": "Iris.BI.pure_elim_r", "content": "theorem pure_elim_r [BI PROP] {φ : Prop} {Q R : PROP} (h : φ → Q ⊢ R) : Q ∧ ⌜φ⌝ ⊢ R"}, {"name": "Iris.BI.affinely_congr", "content": "@[rw_mono_rule]\ntheorem affinely_congr [BI PROP] {P P' : PROP} (h : P ⊣⊢ P') :\n    <affine> P ⊣⊢ <affine> P'"}, {"name": "Iris.BI.affinely_elim", "content": "theorem affinely_elim [BI PROP] {P : PROP} : <affine> P ⊢ P"}, {"name": "Iris.BI.affinely_mono", "content": "@[rw_mono_rule]\ntheorem affinely_mono [BI PROP] {P Q : PROP} : (P ⊢ Q) → <affine> P ⊢ <affine> Q"}, {"name": "Iris.BI.absorbingly_intro", "content": "theorem absorbingly_intro [BI PROP] {P : PROP} : P ⊢ <absorb> P"}, {"name": "Iris.BI.persistently_absorb_r", "content": "theorem persistently_absorb_r [BI PROP] {P Q : PROP} : P ∗ <pers> Q ⊢ <pers> Q"}, {"name": "Iris.BI.absorbingly_persistently", "content": "theorem absorbingly_persistently [BI PROP] {P : PROP} : <absorb> <pers> P ⊣⊢ <pers> P"}, {"name": "Iris.BI.persistently_and", "content": "theorem persistently_and [BI PROP] {P Q : PROP} : <pers> (P ∧ Q) ⊣⊢ <pers> P ∧ <pers> Q"}, {"name": "Iris.BI.persistently_emp_intro", "content": "theorem persistently_emp_intro [BI PROP] {P : PROP} : P ⊢ <pers> emp"}, {"name": "Iris.BI.persistently_affinely", "content": "theorem persistently_affinely [BI PROP] {P : PROP} : <pers> <affine> P ⊣⊢ <pers> P"}, {"name": "Iris.BI.intuitionistically_elim", "content": "theorem intuitionistically_elim [BI PROP] {P : PROP} : □ P ⊢ P"}, {"name": "Iris.BI.persistently_idem", "content": "theorem persistently_idem [BI PROP] {P : PROP} : <pers> <pers> P ⊣⊢ <pers> P"}, {"name": "Iris.BI.intuitionistically_mono", "content": "@[rw_mono_rule]\ntheorem intuitionistically_mono [BI PROP] {P Q : PROP} (h : P ⊢ Q) : □ P ⊢ □ Q"}, {"name": "Iris.BI.intuitionistically_idem", "content": "theorem intuitionistically_idem [BI PROP] {P : PROP} : □ □ P ⊣⊢ □ P"}, {"name": "Iris.BI.persistently_of_intuitionistically", "content": "theorem persistently_of_intuitionistically [BI PROP] {P : PROP} : □ P ⊢ <pers> P"}, {"name": "Iris.BI.later_impl", "content": "theorem later_impl [BI PROP] {P Q : PROP} : ▷ (P → Q) ⊢ ▷ P → ▷ Q"}, {"name": "Iris.BI.entails_impl_true", "content": "theorem entails_impl_true [BI PROP] {P Q : PROP} :\n    (P ⊢ Q) ↔ iprop((True : PROP) ⊢ (P → Q))"}, {"name": "Iris.BI.loeb", "content": "theorem loeb [BI PROP] [BILoeb PROP] {P : PROP} : (▷ P → P) ⊢ P"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.Extensions\n\nimport Iris.BI.BI\n\nimport Iris.Std.Classes\n\nimport Iris.Std.Rewrite\n\nimport Iris.Std.TC\n\nnamespace Iris.BI\n\nopen Iris.Std BI", "target_theorem": "theorem loeb_wand_intuitionistically [BI PROP] [BILoeb PROP] {P : PROP} :\n    □ (□ ▷ P -∗ P) ⊢ P :=", "ground_truth_proof": ":= by\n  refine .trans ?_ intuitionistically_elim\n  refine .trans ?_ loeb\n  refine imp_intro' ?_\n  refine (and_mono (later_mono persistently_of_intuitionistically) .rfl).trans ?_\n  refine (and_mono later_persistently.mp .rfl).trans ?_\n  refine persistently_and_intuitionistically_sep_l.mp.trans ?_\n  refine (sep_mono intuitionistically_idem.mpr .rfl).trans ?_\n  exact intuitionistically_sep_2.trans (intuitionistically_mono wand_elim_r)", "nesting_depth": 6, "transitive_dep_count": 91, "subset_aristotle": false, "category": "Framework"}
{"id": 230, "thm_name": "Iris.BI.plainly_persistently_elim", "thm_stmt": "theorem plainly_persistently_elim : ■ <pers> P ⊣⊢ ■ P", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/Plainly.lean", "imports": ["import Iris.Algebra", "import Iris.BI.DerivedLaws", "import src.Iris.Algebra.CMRA", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import src.Iris.BI.DerivedLaws", "import src/Iris/Instances/UPred/Instance.lean", "import Iris.BI.BI", "import Iris.BI.Classes"], "used_lib_defs": [{"name": "Lean.MonadQuotation", "module": "Init.Prelude"}, {"name": "Lean.MonadRef", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax term:26 \" ==∗ \" term:25 : term", "content": "syntax term:26 \" ==∗ \" term:25 : term\n\nsyntax term \"={ \" term \" , \" term \" }=∗ \" term : term\n\nsyntax term \"={ \" term \" }=∗ \" term : term\n\nsyntax term \"={ \" term \" }[ \" term \" ]▷=∗ \" term : term\n\nsyntax term \"={ \" term \" }▷=∗ \" term : term\n\nsyntax term \"={ \" term \" }[ \" term \" ]▷^\" term \"=∗ \" term : term\n\nsyntax term \"={ \" term \" }▷^\" term \"=∗ \" term : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term"}, {"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term\n\nsyntax \"■ \" term:40 : term"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y\n\nsyntax:max \"<affine> \" term:40 : term\n\nsyntax:max \"□ \" term:40 : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|==> $P))  => ``(BUpd.bupd iprop($P))\n  | `(iprop($P ==∗ $Q))  => ``(BIBase.wand iprop($P) (BUpd.bupd iprop($Q)))\n\ndelab_rule BUpd.bupd\n  | `($_ $P) => do ``(iprop(|==> $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 , $E2 }=> $P))  => ``(FUpd.fupd $E1 $E2 iprop($P))\n  | `(iprop($P ={ $E1 , $E2 }=∗ $Q))  => ``(BIBase.wand iprop($P) (FUpd.fupd $E1 $E2 iprop($Q)))\n  | `(iprop(|={ $E1 }=> $P))  => ``(FUpd.fupd $E1 $E1 iprop($P))\n  | `(iprop($P ={ $E1 }=∗ $Q))  => ``(BIBase.wand iprop($P) (FUpd.fupd $E1 $E1 iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 }[ $E2 ]▷=> $P))  => ``(iprop(|={$E1, $E2}=> ▷ (|={ $E2, $E1 }=> iprop($P))))\n  | `(iprop($P ={ $E1 }[ $E2 ]▷=∗ $Q))  => ``(iprop(iprop($P) -∗ |={$E1}[$E2]▷=> iprop($Q)))\n  | `(iprop(|={ $E1 }▷=> $P))  => ``(iprop(|={$E1}[$E1]▷=> iprop($P)))\n  | `(iprop($P ={ $E1 }▷=∗ $Q))  => ``(iprop(iprop($P) ={$E1}[$E1]▷=∗ iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 }[ $E2 ]▷^$n=> $P))  => ``(iprop(|={$E1, $E2}=> ▷^[$n] (|={ $E2, $E1 }=> iprop($P))))\n  | `(iprop($P ={ $E1 }[ $E2 ]▷^$n=∗ $Q))  => ``(iprop(iprop($P) -∗ |={$E1}[$E2]▷^$n=> iprop($Q)))\n  | `(iprop(|={ $E1 }▷^$n=> $P))  => ``(iprop(|={$E1}[$E1]▷^$n=> iprop($P)))\n  | `(iprop($P ={ $E1 }▷^$n=∗ $Q))  => ``(iprop(iprop($P) ={$E1}[$E1]▷^$n=∗ iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(■ $P))  => ``(Plainly.plainly iprop($P))\n\ndelab_rule Plainly.plainly\n  | `($_ $P) => do ``(iprop(■ $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(■? $p $P))  => ``(Plainly.plainlyIf $p iprop($P))\n\ndelab_rule Plainly.plainlyIf\n  | `($_ $p $P) => do ``(iprop(■? $p $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "unpackIprop", "content": "partial def unpackIprop [Monad m] [MonadRef m] [MonadQuotation m] : Term → m Term\n  | `(iprop($P))             => do `($P)\n  | `($P:ident)              => do `($P)\n  | `(?$P:ident)             => do `(?$P)\n  | `(($P))                  => do `(($(← unpackIprop P)))\n  | `($P $[ $Q]*)            => do ``($P $[ $Q]*)\n  | `(if $c then $t else $e) => do\n    let t ← unpackIprop t\n    let e ← unpackIprop e\n    `(if $c then $t else $e)\n  | `(($P : $t))             => do ``(($(← unpackIprop P) : $t))\n  | `($t)                    => `($t:term)"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "", "content": "instance : Plainly (UPred M) := ⟨UPred.plainly⟩"}, {"name": "plainly", "content": "protected def plainly (P : UPred M) : UPred M where\n  holds n _ := P n UCMRA.unit\n  mono H _ Hn := P.mono H (CMRA.incN_refl UCMRA.unit) Hn"}, {"name": "BIUpdate", "content": "class BIUpdate (PROP : Type _) [BI PROP] extends BUpd PROP where\n  [bupd_ne : OFE.NonExpansive (BUpd.bupd (PROP := PROP))]\n  intro {P : PROP} : iprop(P ⊢ |==> P)\n  mono {P Q : PROP} : iprop(P ⊢ Q) → iprop(|==> P ⊢ |==> Q)\n  trans {P : PROP} : iprop(|==> |==> P ⊢ |==> P)\n  frame_r {P R : PROP} : iprop((|==> P) ∗ R ⊢ |==> (P ∗ R))"}, {"name": "UPred", "content": "@[ext]\nstructure UPred (M : Type _) [UCMRA M] where\n  holds : Nat → M → Prop\n  mono {n1 n2 x1 x2} : holds n1 x1 → x1 ≼{n2} x2 → n2 ≤ n1 → holds n2 x2"}, {"name": "IsModal", "content": "class IsModal [BI PROP1] [BI PROP2] (M : PROP1 → PROP2)\n    (iaction saction : ModalityAction PROP1 PROP2) where\n  spec_intuitionistic : iaction.intuitionistic_action_spec M\n  spec_spatial : saction.spatial_action_spec M\n  emp : iprop(emp) ⊢ M iprop(emp)\n  mono : ∀ {P Q}, (P ⊢ Q) → M P ⊢ M Q\n  sep : ∀ {P Q}, iprop(M P ∗ M Q) ⊢ M iprop(P ∗ Q)"}, {"name": "UCMRA", "content": "class UCMRA (α : Type _) extends CMRA α where\n  unit : α\n  unit_valid : ✓ unit\n  unit_left_id : unit • x ≡ x\n  pcore_unit : pcore unit ≡ some unit"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "", "content": "instance : BIPlainly (UPred M) where\n  mono H _ _ _ := H _ _ CMRA.unit_validN\n  elim_persistently {P} n x Hx := by admit /- proof elided -/"}, {"name": "persistently", "content": "protected def persistently (P : UPred M) : UPred M where\n  holds n x := P n (CMRA.core x)\n  mono H Hx Hn := P.mono H (CMRA.core_incN_core Hx) Hn"}, {"name": "core", "content": "def core (x : α) := (pcore x).getD x"}, {"name": "BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "Idempotent", "content": "class Idempotent (R : Relation α) (f : α → α → α) where\n  idem {x : α} : R (f x x) x"}, {"name": "", "content": "instance : BIPlainlyExists (UPred M) where\n  plainly_sExists_1 _ _ _ := fun ⟨_, hp⟩ => ⟨_, ⟨_, rfl⟩, hp⟩"}, {"name": "BIPlainlyExists", "content": "class BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "imp_intro'", "content": "theorem imp_intro' [BI PROP] {P Q R : PROP} (h : Q ∧ P ⊢ R) : P ⊢ Q → R"}, {"name": "and_comm", "content": "theorem and_comm [BI PROP] {P Q : PROP} : P ∧ Q ⊣⊢ Q ∧ P"}, {"name": "and_symm", "content": "theorem and_symm [BI PROP] {P Q : PROP} : P ∧ Q ⊢ Q ∧ P"}, {"name": "imp_elim'", "content": "theorem imp_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P → R) : P ∧ Q ⊢ R"}, {"name": "pure_elim", "content": "theorem pure_elim [BI PROP] (φ : Prop) {Q R : PROP} (h1 : Q ⊢ ⌜φ⌝) (h2 : φ → Q ⊢ R) : Q ⊢ R"}, {"name": "and_self", "content": "theorem and_self [BI PROP] {P : PROP} : P ∧ P ⊣⊢ P"}, {"name": "forall_intro", "content": "theorem forall_intro [BI PROP] {P : PROP} {Ψ : α → PROP} (h : ∀ a, P ⊢ Ψ a) : P ⊢ ∀ a, Ψ a"}, {"name": "forall_elim", "content": "theorem forall_elim [BI PROP] {Ψ : α → PROP} (a : α) : (∀ a, Ψ a) ⊢ Ψ a"}, {"name": "Included.trans", "content": "theorem Included.trans : (x : α) ≼ y → y ≼ z → x ≼ z"}, {"name": "inc_trans", "content": "theorem inc_trans {x y z : α} : x ≼ y → y ≼ z → x ≼ z"}, {"name": "op_left_eqv", "content": "theorem op_left_eqv {x y : α} (z : α) (e : x ≡ y) : x • z ≡ y • z"}, {"name": "_root_.Iris.OFE.Dist.op_r", "content": "theorem _root_.Iris.OFE.Dist.op_r {x y z : α} : y ≡{n}≡ z → x • y ≡{n}≡ x • z"}, {"name": "op_right_dist", "content": "theorem op_right_dist (x : α) {y z : α} (e : y ≡{n}≡ z) : x • y ≡{n}≡ x • z"}, {"name": "_root_.Iris.OFE.Equiv.op_r", "content": "theorem _root_.Iris.OFE.Equiv.op_r {x y z : α} : y ≡ z → x • y ≡ x • z"}, {"name": "op_right_eqv", "content": "theorem op_right_eqv (x : α) {y z : α} (e : y ≡ z) : x • y ≡ x • z"}, {"name": "IncludedN.trans", "content": "theorem IncludedN.trans : (x : α) ≼{n} y → y ≼{n} z → x ≼{n} z"}, {"name": "incN_trans", "content": "theorem incN_trans {x y z : α} : x ≼{n} y → y ≼{n} z → x ≼{n} z"}, {"name": "op_left_dist", "content": "theorem op_left_dist {x y : α} (z : α) (e : x ≡{n}≡ y) : x • z ≡{n}≡ y • z"}, {"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "persistently_and_emp_elim", "content": "theorem persistently_and_emp_elim {P : PROP} [BI PROP] : emp ∧ <pers> P ⊢ P"}, {"name": "sep_emp", "content": "theorem sep_emp [BI PROP] {P : PROP} : P ∗ emp ⊣⊢ P"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "and_forall_bool", "content": "theorem and_forall_bool [BI PROP] {P Q : PROP} :\n    P ∧ Q ⊣⊢ «forall» (fun b : Bool => if b then P else Q)"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}, {"name": "true_and", "content": "theorem true_and [BI PROP] {P : PROP} : True ∧ P ⊣⊢ P"}, {"name": "forall_mono", "content": "@[rw_mono_rule]\ntheorem forall_mono [BI PROP] {Φ Ψ : α → PROP} (h : ∀ a, Φ a ⊢ Ψ a) : (∀ a, Φ a) ⊢ ∀ a, Ψ a"}, {"name": "and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}], "used_local_defs": [{"name": "Iris.Plainly", "content": "class Plainly (PROP : Type _) where\n  plainly : PROP → PROP"}, {"name": "Iris.Plainly.plainlyIf", "content": "def Plainly.plainlyIf [Iris.BI.BIBase PROP] [Plainly PROP] (p : Bool) (P : PROP) : PROP :=\n  iprop(if p then ■ P else P)"}, {"name": "Iris.BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "Iris.BIPlainlyExists", "content": "class BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p"}], "used_local_lemmas": [{"name": "Iris.BI.plainly_forall_2", "content": "theorem plainly_forall_2 {Ψ : α → PROP} : (∀ a, ■ (Ψ a)) ⊢ ■ (∀ a, Ψ a)"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.BI\n\nimport Iris.BI.DerivedLaws\n\nimport Iris.Algebra\n\nnamespace Iris\n\nopen BI\n\nclass Plainly (PROP : Type _) where\n  plainly : PROP → PROP\n\ndef Plainly.plainlyIf [Iris.BI.BIBase PROP] [Plainly PROP] (p : Bool) (P : PROP) : PROP :=\n  iprop(if p then ■ P else P)\n\nclass BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P\n\nclass BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p\n\nnamespace BI\n\nopen Iris.Std\n\nsection PlainlyLaws\n\nopen BIPlainly\n\nvariable [BI PROP] [BIPlainly PROP]\n\nvariable {P Q R : PROP}", "target_theorem": "theorem plainly_persistently_elim : ■ <pers> P ⊣⊢ ■ P :=", "ground_truth_proof": ":= by\n  constructor\n  · refine (true_and.2.trans <| and_mono emp_intro .rfl).trans ?_\n    refine .trans ?_ (mono <| and_forall_bool.2.trans persistently_and_emp_elim)\n    refine and_forall_bool.1.trans ?_\n    refine .trans ?_ plainly_forall_2\n    refine forall_mono ?_\n    exact (·.casesOn .rfl .rfl)\n  · exact idem.trans <| mono elim_persistently", "nesting_depth": 6, "transitive_dep_count": 75, "subset_aristotle": false, "category": "Framework"}
{"id": 231, "thm_name": "Iris.fixpoint_unique", "thm_stmt": "theorem fixpoint_unique [COFE α] [Inhabited α] {f : α -c> α} {x : α} (H : x ≡ f x) :\n    x ≡ fixpoint f", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/OFE.lean", "imports": ["import src/Iris/Algebra/Excl.lean", "import src/Iris/Instances/UPred/Instance.lean", "import src/Iris/Algebra/UPred.lean", "import src/Iris/Algebra/CMRA.lean", "import src/Iris/Algebra/Agree_task.lean", "import src/Iris/Algebra/COFESolver.lean", "import src/Iris/Algebra/Heap.lean"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Dist", "module": "Mathlib.Topology.MetricSpace.Pseudo.Defs"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "", "content": "instance : OFE.Discrete (DFrac F) := ⟨congrArg id⟩"}, {"name": "Agree.dist", "content": "def Agree.dist (n : Nat) (x y : Agree α) : Prop :=\n  (∀ a ∈ x.car, ∃ b ∈ y.car, a ≡{n}≡ b) ∧\n  (∀ b ∈ y.car, ∃ a ∈ x.car, a ≡{n}≡ b)"}, {"name": "", "content": "instance : OFE (Tower F) where\n  Equiv f g := ∀ k, f k ≡ g k\n  Dist n f g := ∀ k, f k ≡{n}≡ g k\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "", "content": "instance : OFE.NonExpansive (BUpd.bupd (PROP := UPred M)) := bupd_ne"}, {"name": "bupd_ne", "content": "instance bupd_ne : OFE.NonExpansive (bupd : UPred M → UPred M) where\n  ne n x1 x2 Hx m y Hm Hv := by admit /- proof elided -/"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "later_contractive", "content": "instance later_contractive : OFE.Contractive UPred.later (α := UPred M) where\n  distLater_dist {n x y} Hl :=\n    match n with\n    | 0 => by admit /- proof elided -/\n    | n + 1 => fun\n      | 0 => by admit /- proof elided -/\n      | n' + 1 => fun x' Hn' Hx' => Hl _ Hn' _ _ (Nat.le_refl _) (CMRA.validN_succ Hx')"}, {"name": "later", "content": "protected def later (P : UPred M) : UPred M where\n  holds n x := match n with | 0 => True | Nat.succ n' => P n' x\n  mono {n₁ n₂} := by admit /- proof elided -/"}, {"name": "BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "BIUpdate", "content": "class BIUpdate (PROP : Type _) [BI PROP] extends BUpd PROP where\n  [bupd_ne : OFE.NonExpansive (BUpd.bupd (PROP := PROP))]\n  intro {P : PROP} : iprop(P ⊢ |==> P)\n  mono {P Q : PROP} : iprop(P ⊢ Q) → iprop(|==> P ⊢ |==> Q)\n  trans {P : PROP} : iprop(|==> |==> P ⊢ |==> P)\n  frame_r {P R : PROP} : iprop((|==> P) ∗ R ⊢ |==> (P ∗ R))"}, {"name": "UPred", "content": "@[ext]\nstructure UPred (M : Type _) [UCMRA M] where\n  holds : Nat → M → Prop\n  mono {n1 n2 x1 x2} : holds n1 x1 → x1 ≼{n2} x2 → n2 ≤ n1 → holds n2 x2"}, {"name": "IsModal", "content": "class IsModal [BI PROP1] [BI PROP2] (M : PROP1 → PROP2)\n    (iaction saction : ModalityAction PROP1 PROP2) where\n  spec_intuitionistic : iaction.intuitionistic_action_spec M\n  spec_spatial : saction.spatial_action_spec M\n  emp : iprop(emp) ⊢ M iprop(emp)\n  mono : ∀ {P Q}, (P ⊢ Q) → M P ⊢ M Q\n  sep : ∀ {P Q}, iprop(M P ∗ M Q) ⊢ M iprop(P ∗ Q)"}, {"name": "UCMRA", "content": "class UCMRA (α : Type _) extends CMRA α where\n  unit : α\n  unit_valid : ✓ unit\n  unit_left_id : unit • x ≡ x\n  pcore_unit : pcore unit ≡ some unit"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "instOFE", "content": "instance instOFE [Store T K V] [OFE V] : OFE T where\n  Equiv s0 s1  := get s0 ≡ get s1\n  Dist n s0 s1 := get s0 ≡{n}≡ get s1\n  dist_eqv     := ⟨fun _ => .of_eq rfl, (·.symm), (·.trans ·)⟩\n  equiv_dist   := equiv_dist\n  dist_lt      := dist_lt"}, {"name": "Store.Equiv", "content": "@[simp] def Store.Equiv [Store T K V] (t1 t2 : T) : Prop := get t1 = get t2"}, {"name": "Store.Equiv_trans", "content": "instance Store.Equiv_trans [Store T K V] : Trans Equiv (Equiv (T := T)) Equiv := ⟨by admit /- proof elided -/\n⟩"}, {"name": "[CMRA", "content": "instance [CMRA β] : OFE (α -C> β) where\n  Equiv f g := f.toHom ≡ g.toHom\n  Dist n f g := f.toHom ≡{n}≡ g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "[OFE", "content": "instance [OFE α] [IsCOFE α] : IsCOFE (Excl α) where\n  compl c := (c 0).map fun x => IsCOFE.compl (exclChain c x)\n  conv_compl {n} c := by admit /- proof elided -/"}, {"name": "(k", "content": "instance (k : Nat) : NonExpansive (fun X : Tower F => X.val k) := ⟨fun _ _ _ => (· _)⟩"}, {"name": "[OFE", "content": "instance [OFE α] : NonExpansive excl (α := α) where\n  ne _ _ _ a := a"}, {"name": "get_ne", "content": "instance get_ne [Store T K V] [OFE V] (k : K) : NonExpansive (get · k : T → V) where\n  ne {_ _ _} Ht := Ht k"}, {"name": "[Store", "content": "instance [Store T1 K V1] [Store T2 K V2] [OFE V1] [OFE V2] (f : K → V1 → V2)\n  [∀ k, NonExpansive (f k)] [HasStoreMap T1 T2 K V1 V2] : NonExpansive (dmap f : T1 → T2) where\n  ne _ {_ _} H k := by admit /- proof elided -/"}, {"name": "HasStoreMap", "content": "class HasStoreMap (T1 T2 : Type _) (K V1 V2 : outParam (Type _)) [Store T1 K V1] [Store T2 K V2] where\n   \n  dmap (f : K → V1 → V2) : T1 → T2\n  get_dmap : get (dmap f t) k = f k (get t k)"}, {"name": "", "content": "instance : NonExpansive (pcore (α := α)) where\n  ne n x {y} e := by admit /- proof elided -/"}, {"name": "[Heap", "content": "instance [Heap T K V] [OFE V] (op : V → V → V) [NonExpansive₂ op] :\n    NonExpansive₂ (merge (T := T) op) where\n  ne _ {_ _} Ht {_ _} Hs k := by admit /- proof elided -/"}, {"name": "Heap.instCOFE", "content": "instance Heap.instCOFE [Heap T K V] [COFE V] : COFE T where\n  compl c := hmap (fun _ => COFE.compl <| c.map ⟨_, Store.get_ne ·⟩) (c 0)\n  conv_compl {_ c} k := by admit /- proof elided -/"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -C> \" => Hom"}], "lib_lemmas": [{"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "Nat.le_of_lt_succ", "module": "Init.Prelude"}, {"name": "Nat.lt_succ_self", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Iris.OFE", "content": "class OFE (α : Type _) where\n  Equiv : α → α → Prop\n  Dist : Nat → α → α → Prop\n  dist_eqv : Equivalence (Dist n)\n  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y\n  dist_lt : Dist n x y → m < n → Dist m x y"}, {"name": "Iris.OFE.NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "Iris.OFE.id_ne", "content": "instance id_ne [OFE α] : NonExpansive (@id α) := ⟨fun _ _ _ h => h⟩"}, {"name": "Iris.OFE.NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "Iris.OFE.DistLater", "content": "def DistLater [OFE α] (n : Nat) (x y : α) : Prop := ∀ m, m < n → x ≡{m}≡ y"}, {"name": "Iris.OFE.Contractive", "content": "class Contractive [OFE α] [OFE β] (f : α → β) where\n  distLater_dist : DistLater n x y → f x ≡{n}≡ f y"}, {"name": "Iris.OFE.ne_of_contractive", "content": "instance ne_of_contractive [OFE α] [OFE β] (f : α → β) [Contractive f] : NonExpansive f where\n  ne := fun _ _ _ h => Contractive.distLater_dist (Dist.distLater h)"}, {"name": "Iris.OFE._inst_β", "content": "instance [OFE α] [OFE β] {x : β} : Contractive (fun _ : α => x) where\n  distLater_dist := fun _ => Dist.rfl"}, {"name": "Iris.OFE.Hom", "content": "@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f"}, {"name": "Iris.OFE._inst_α", "content": "instance [OFE α] [OFE β] (f : α -n> β) : NonExpansive f := f.ne"}, {"name": "Iris.OFE.ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "Iris.OFE._inst_α", "content": "instance [OFE α] [OFE β] (f : α -c> β) : Contractive f := f.contractive"}, {"name": "Iris.OFE._inst_OFE", "content": "instance : OFE Unit where\n  Equiv _ _ := True\n  Dist _ _ _ := True\n  dist_eqv := ⟨fun _ => ⟨⟩, id, fun _ => id⟩\n  equiv_dist := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] : OFE (ULift α) where\n  Equiv x y := x.down ≡ y.down\n  Dist n x y := x.down ≡{n}≡ y.down\n  dist_eqv := InvImage.equivalence dist_eqv\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] : OFE (Option α) where\n  Equiv := Option.Forall₂ Equiv\n  Dist n := Option.Forall₂ (Dist n)\n  dist_eqv := Option.Forall₂.equivalence dist_eqv\n  equiv_dist {x y} := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE.Discrete α] : OFE.Discrete (Option α) where\n  discrete_0 {mx my} e :=\n    match mx, my with\n    | none,   none   => e\n    | none,   some _ => e\n    | some _, none   => e\n    | some x, some y => show x ≡ y from discrete_0 e"}, {"name": "Iris.OFE.OFE", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Iris.OFE.Option", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_α", "content": "instance [OFEFun (β : α → _)] : OFE ((x : α) → β x) where\n  Equiv f g := ∀ x, f x ≡ g x\n  Dist n f g := ∀ x, f x ≡{n}≡ g x\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE β] : OFE (α -n> β) where\n  Equiv f g := f.f ≡ g.f\n  Dist n f g := f.f ≡{n}≡ g.f\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE β] : OFE (α -c> β) where\n  Equiv f g := Equiv f.toHom g.toHom\n  Dist n f g := Dist n f.toHom g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE β] : OFE (α × β) where\n  Equiv a b := a.1 ≡ b.1 ∧ a.2 ≡ b.2\n  Dist n a b := a.1 ≡{n}≡ b.1 ∧ a.2 ≡{n}≡ b.2\n  dist_eqv := {\n    refl _ := ⟨dist_eqv.refl _, dist_eqv.refl _⟩\n    symm h := ⟨dist_eqv.symm h.1, dist_eqv.symm h.2⟩\n    trans h1 h2 := ⟨dist_eqv.trans h1.1 h2.1, dist_eqv.trans h1.2 h2.2⟩\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "Iris.OFE.Iso", "content": "@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where\n  hom : α -n> β\n  inv : β -n> α\n  hom_inv : hom (inv x) ≡ x\n  inv_hom : inv (hom x) ≡ x"}, {"name": "Iris.OFE._inst_Iso", "content": "instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.hom := iso.hom.ne"}, {"name": "Iris.OFE._inst_Iso", "content": "instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.inv := iso.inv.ne"}, {"name": "Iris.OFE.Iso.symm", "content": "def Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where\n  hom := iso.inv\n  inv := iso.hom\n  hom_inv := by admit /- proof elided -/"}, {"name": "Iris.Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "Iris.IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Iris.LeibnizO", "content": "@[ext] structure LeibnizO (α : Type _) where\n  car : α"}, {"name": "Iris.optionChain", "content": "def optionChain (c : Chain (Option α)) (x : α) : Chain α :="}, {"name": "Iris.isCOFE_option", "content": "instance isCOFE_option [IsCOFE α] : IsCOFE (Option α) where\n  compl c := (c 0).map fun x => IsCOFE.compl (optionChain c x)\n  conv_compl {n} c := by admit /- proof elided -/"}, {"name": "Iris.Fixpoint.chain", "content": "def Fixpoint.chain [OFE α] [Inhabited α] (f : α → α) [Contractive f] : Chain α where\n  chain n := Nat.repeat f (n + 1) default\n  cauchy {n} := by admit /- proof elided -/"}, {"name": "Iris.fixpoint", "content": "def fixpoint [COFE α] [Inhabited α] (f : α → α) [Contractive f] : α :=\n  COFE.compl <| Fixpoint.chain f\n\nnonrec abbrev OFE.ContractiveHom.fixpoint [COFE α] [Inhabited α] (f : α -c> α) : α := fixpoint f.f"}], "used_local_lemmas": [{"name": "Iris.OFE.Dist.lt", "content": "theorem Dist.lt [OFE α] {m n} {x y : α} : x ≡{n}≡ y → m < n → x ≡{m}≡ y"}, {"name": "Iris.OFE.Dist.le", "content": "theorem Dist.le [OFE α] {m n} {x y : α} (h : x ≡{n}≡ y) (h' : m ≤ n) : x ≡{m}≡ y"}, {"name": "Iris.OFE.Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "Iris.OFE.Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Iris.OFE.Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "Iris.OFE.equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Iris.OFE.Equiv.symm", "content": "@[symm] theorem Equiv.symm [OFE α] {x : α} : x ≡ y → y ≡ x"}, {"name": "Iris.OFE.Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "Iris.OFE.Equiv.dist", "content": "theorem Equiv.dist [OFE α] {x : α} : x ≡ y → x ≡{n}≡ y"}, {"name": "Iris.OFE.DistLater.symm", "content": "@[symm] theorem DistLater.symm [OFE α] {n} {x : α} (h : DistLater n x y) : DistLater n y x"}, {"name": "Iris.OFE.DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}, {"name": "Iris.OFE.distLater_zero", "content": "@[simp] theorem distLater_zero [OFE α] {x y : α} : DistLater 0 x y"}, {"name": "Iris.OFE.distLater_succ", "content": "theorem distLater_succ [OFE α] {n} {x y : α} : DistLater n.succ x y ↔ x ≡{n}≡ y"}, {"name": "Iris.OFE.Contractive.zero", "content": "@[simp] theorem Contractive.zero [OFE α] [OFE β] (f : α → β) [Contractive f] {x y} :\n    f x ≡{0}≡ f y"}, {"name": "Iris.OFE.Contractive.succ", "content": "theorem Contractive.succ [OFE α] [OFE β] (f : α → β) [Contractive f] {n x y}\n    (h : x ≡{n}≡ y) : f x ≡{n.succ}≡ f y"}, {"name": "Iris.fixpoint_unfold", "content": "theorem fixpoint_unfold [COFE α] [Inhabited α] (f : α -c> α) :\n    fixpoint f ≡ f (fixpoint f)"}], "local_ctx": "namespace Iris\n\nclass OFE (α : Type _) where\n  Equiv : α → α → Prop\n  Dist : Nat → α → α → Prop\n  dist_eqv : Equivalence (Dist n)\n  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y\n  dist_lt : Dist n x y → m < n → Dist m x y\n\nopen OFE\n\nscoped infix:40 \" ≡ \" => OFE.Equiv\n\nscoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nnamespace OFE\n\nclass NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂\n\ninstance id_ne [OFE α] : NonExpansive (@id α) := ⟨fun _ _ _ h => h⟩\n\nclass NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂\n\ndef DistLater [OFE α] (n : Nat) (x y : α) : Prop := ∀ m, m < n → x ≡{m}≡ y\n\nclass Contractive [OFE α] [OFE β] (f : α → β) where\n  distLater_dist : DistLater n x y → f x ≡{n}≡ f y\n\ninstance ne_of_contractive [OFE α] [OFE β] (f : α → β) [Contractive f] : NonExpansive f where\n  ne := fun _ _ _ h => Contractive.distLater_dist (Dist.distLater h)\n\ninstance [OFE α] [OFE β] {x : β} : Contractive (fun _ : α => x) where\n  distLater_dist := fun _ => Dist.rfl\n\n@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f\n\n@[inherit_doc]\ninfixr:25 \" -n> \" => Hom\n\ninstance [OFE α] [OFE β] (f : α -n> β) : NonExpansive f := f.ne\n\n@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f\n\ninfixr:25 \" -c> \" => ContractiveHom\n\ninstance [OFE α] [OFE β] (f : α -c> β) : Contractive f := f.contractive\n\ninstance : OFE Unit where\n  Equiv _ _ := True\n  Dist _ _ _ := True\n  dist_eqv := ⟨fun _ => ⟨⟩, id, fun _ => id⟩\n  equiv_dist := by admit /- proof elided -/\n\ninstance [OFE α] : OFE (ULift α) where\n  Equiv x y := x.down ≡ y.down\n  Dist n x y := x.down ≡{n}≡ y.down\n  dist_eqv := InvImage.equivalence dist_eqv\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt\n\ninstance [OFE α] : OFE (Option α) where\n  Equiv := Option.Forall₂ Equiv\n  Dist n := Option.Forall₂ (Dist n)\n  dist_eqv := Option.Forall₂.equivalence dist_eqv\n  equiv_dist {x y} := by admit /- proof elided -/\n\ninstance [OFE α] [OFE.Discrete α] : OFE.Discrete (Option α) where\n  discrete_0 {mx my} e :=\n    match mx, my with\n    | none,   none   => e\n    | none,   some _ => e\n    | some _, none   => e\n    | some x, some y => show x ≡ y from discrete_0 e\n\ninstance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩\n\ninstance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/\n\ninstance [OFEFun (β : α → _)] : OFE ((x : α) → β x) where\n  Equiv f g := ∀ x, f x ≡ g x\n  Dist n f g := ∀ x, f x ≡{n}≡ g x\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/\n\ninstance [OFE α] [OFE β] : OFE (α -n> β) where\n  Equiv f g := f.f ≡ g.f\n  Dist n f g := f.f ≡{n}≡ g.f\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt\n\ninstance [OFE α] [OFE β] : OFE (α -c> β) where\n  Equiv f g := Equiv f.toHom g.toHom\n  Dist n f g := Dist n f.toHom g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt\n\ninstance [OFE α] [OFE β] : OFE (α × β) where\n  Equiv a b := a.1 ≡ b.1 ∧ a.2 ≡ b.2\n  Dist n a b := a.1 ≡{n}≡ b.1 ∧ a.2 ≡{n}≡ b.2\n  dist_eqv := {\n    refl _ := ⟨dist_eqv.refl _, dist_eqv.refl _⟩\n    symm h := ⟨dist_eqv.symm h.1, dist_eqv.symm h.2⟩\n    trans h1 h2 := ⟨dist_eqv.trans h1.1 h2.1, dist_eqv.trans h1.2 h2.2⟩\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/\n\n@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where\n  hom : α -n> β\n  inv : β -n> α\n  hom_inv : hom (inv x) ≡ x\n  inv_hom : inv (hom x) ≡ x\n\ninstance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.hom := iso.hom.ne\n\ninstance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.inv := iso.inv.ne\n\ndef Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where\n  hom := iso.inv\n  inv := iso.hom\n  hom_inv := by admit /- proof elided -/\n\nend OFE\n\nstructure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n\n\nnamespace Chain\n\nend Chain\n\nclass IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n\n\nnamespace COFE\n\nend COFE\n\n@[ext] structure LeibnizO (α : Type _) where\n  car : α\n\nsection DiscreteFunOF\n\nopen COFE\n\nend DiscreteFunOF\n\nsection Option\n\nvariable [OFE α]\n\ndef optionChain (c : Chain (Option α)) (x : α) : Chain α :=\n\ninstance isCOFE_option [IsCOFE α] : IsCOFE (Option α) where\n  compl c := (c 0).map fun x => IsCOFE.compl (optionChain c x)\n  conv_compl {n} c := by admit /- proof elided -/\n\nend Option\n\nsection OptionOF\n\nopen COFE\n\nvariable (F : OFunctorPre)\n\nend OptionOF\n\nsection Fixpoint\n\ndef Fixpoint.chain [OFE α] [Inhabited α] (f : α → α) [Contractive f] : Chain α where\n  chain n := Nat.repeat f (n + 1) default\n  cauchy {n} := by admit /- proof elided -/\n\ndef fixpoint [COFE α] [Inhabited α] (f : α → α) [Contractive f] : α :=\n  COFE.compl <| Fixpoint.chain f\n\nnonrec abbrev OFE.ContractiveHom.fixpoint [COFE α] [Inhabited α] (f : α -c> α) : α := fixpoint f.f", "target_theorem": "theorem fixpoint_unique [COFE α] [Inhabited α] {f : α -c> α} {x : α} (H : x ≡ f x) :\n    x ≡ fixpoint f :=", "ground_truth_proof": ":= by\n  refine equiv_dist.mpr fun n => ?_\n  induction n with refine H.dist.trans <| .trans ?_ (fixpoint_unfold f).dist.symm\n  | zero => exact Contractive.zero f.f\n  | succ _ IH => exact Contractive.succ f.f IH", "nesting_depth": 6, "transitive_dep_count": 45, "subset_aristotle": false, "category": "Framework"}
{"id": 232, "thm_name": "Iris.fixpoint_unfold", "thm_stmt": "theorem fixpoint_unfold [COFE α] [Inhabited α] (f : α -c> α) :\n    fixpoint f ≡ f (fixpoint f)", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/OFE.lean", "imports": ["import src/Iris/Algebra/Excl.lean", "import src/Iris/Instances/UPred/Instance.lean", "import src/Iris/Algebra/UPred.lean", "import src/Iris/Algebra/CMRA.lean", "import src/Iris/Algebra/Agree_task.lean", "import src/Iris/Algebra/COFESolver.lean", "import src/Iris/Algebra/Heap.lean"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Dist", "module": "Mathlib.Topology.MetricSpace.Pseudo.Defs"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "", "content": "instance : OFE.Discrete (DFrac F) := ⟨congrArg id⟩"}, {"name": "Agree.dist", "content": "def Agree.dist (n : Nat) (x y : Agree α) : Prop :=\n  (∀ a ∈ x.car, ∃ b ∈ y.car, a ≡{n}≡ b) ∧\n  (∀ b ∈ y.car, ∃ a ∈ x.car, a ≡{n}≡ b)"}, {"name": "", "content": "instance : OFE (Tower F) where\n  Equiv f g := ∀ k, f k ≡ g k\n  Dist n f g := ∀ k, f k ≡{n}≡ g k\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "", "content": "instance : OFE.NonExpansive (BUpd.bupd (PROP := UPred M)) := bupd_ne"}, {"name": "bupd_ne", "content": "instance bupd_ne : OFE.NonExpansive (bupd : UPred M → UPred M) where\n  ne n x1 x2 Hx m y Hm Hv := by admit /- proof elided -/"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "later_contractive", "content": "instance later_contractive : OFE.Contractive UPred.later (α := UPred M) where\n  distLater_dist {n x y} Hl :=\n    match n with\n    | 0 => by admit /- proof elided -/\n    | n + 1 => fun\n      | 0 => by admit /- proof elided -/\n      | n' + 1 => fun x' Hn' Hx' => Hl _ Hn' _ _ (Nat.le_refl _) (CMRA.validN_succ Hx')"}, {"name": "later", "content": "protected def later (P : UPred M) : UPred M where\n  holds n x := match n with | 0 => True | Nat.succ n' => P n' x\n  mono {n₁ n₂} := by admit /- proof elided -/"}, {"name": "BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "BIUpdate", "content": "class BIUpdate (PROP : Type _) [BI PROP] extends BUpd PROP where\n  [bupd_ne : OFE.NonExpansive (BUpd.bupd (PROP := PROP))]\n  intro {P : PROP} : iprop(P ⊢ |==> P)\n  mono {P Q : PROP} : iprop(P ⊢ Q) → iprop(|==> P ⊢ |==> Q)\n  trans {P : PROP} : iprop(|==> |==> P ⊢ |==> P)\n  frame_r {P R : PROP} : iprop((|==> P) ∗ R ⊢ |==> (P ∗ R))"}, {"name": "UPred", "content": "@[ext]\nstructure UPred (M : Type _) [UCMRA M] where\n  holds : Nat → M → Prop\n  mono {n1 n2 x1 x2} : holds n1 x1 → x1 ≼{n2} x2 → n2 ≤ n1 → holds n2 x2"}, {"name": "IsModal", "content": "class IsModal [BI PROP1] [BI PROP2] (M : PROP1 → PROP2)\n    (iaction saction : ModalityAction PROP1 PROP2) where\n  spec_intuitionistic : iaction.intuitionistic_action_spec M\n  spec_spatial : saction.spatial_action_spec M\n  emp : iprop(emp) ⊢ M iprop(emp)\n  mono : ∀ {P Q}, (P ⊢ Q) → M P ⊢ M Q\n  sep : ∀ {P Q}, iprop(M P ∗ M Q) ⊢ M iprop(P ∗ Q)"}, {"name": "UCMRA", "content": "class UCMRA (α : Type _) extends CMRA α where\n  unit : α\n  unit_valid : ✓ unit\n  unit_left_id : unit • x ≡ x\n  pcore_unit : pcore unit ≡ some unit"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "instOFE", "content": "instance instOFE [Store T K V] [OFE V] : OFE T where\n  Equiv s0 s1  := get s0 ≡ get s1\n  Dist n s0 s1 := get s0 ≡{n}≡ get s1\n  dist_eqv     := ⟨fun _ => .of_eq rfl, (·.symm), (·.trans ·)⟩\n  equiv_dist   := equiv_dist\n  dist_lt      := dist_lt"}, {"name": "Store.Equiv", "content": "@[simp] def Store.Equiv [Store T K V] (t1 t2 : T) : Prop := get t1 = get t2"}, {"name": "Store.Equiv_trans", "content": "instance Store.Equiv_trans [Store T K V] : Trans Equiv (Equiv (T := T)) Equiv := ⟨by admit /- proof elided -/\n⟩"}, {"name": "[CMRA", "content": "instance [CMRA β] : OFE (α -C> β) where\n  Equiv f g := f.toHom ≡ g.toHom\n  Dist n f g := f.toHom ≡{n}≡ g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "[OFE", "content": "instance [OFE α] [IsCOFE α] : IsCOFE (Excl α) where\n  compl c := (c 0).map fun x => IsCOFE.compl (exclChain c x)\n  conv_compl {n} c := by admit /- proof elided -/"}, {"name": "(k", "content": "instance (k : Nat) : NonExpansive (fun X : Tower F => X.val k) := ⟨fun _ _ _ => (· _)⟩"}, {"name": "[OFE", "content": "instance [OFE α] : NonExpansive excl (α := α) where\n  ne _ _ _ a := a"}, {"name": "get_ne", "content": "instance get_ne [Store T K V] [OFE V] (k : K) : NonExpansive (get · k : T → V) where\n  ne {_ _ _} Ht := Ht k"}, {"name": "[Store", "content": "instance [Store T1 K V1] [Store T2 K V2] [OFE V1] [OFE V2] (f : K → V1 → V2)\n  [∀ k, NonExpansive (f k)] [HasStoreMap T1 T2 K V1 V2] : NonExpansive (dmap f : T1 → T2) where\n  ne _ {_ _} H k := by admit /- proof elided -/"}, {"name": "HasStoreMap", "content": "class HasStoreMap (T1 T2 : Type _) (K V1 V2 : outParam (Type _)) [Store T1 K V1] [Store T2 K V2] where\n   \n  dmap (f : K → V1 → V2) : T1 → T2\n  get_dmap : get (dmap f t) k = f k (get t k)"}, {"name": "", "content": "instance : NonExpansive (pcore (α := α)) where\n  ne n x {y} e := by admit /- proof elided -/"}, {"name": "[Heap", "content": "instance [Heap T K V] [OFE V] (op : V → V → V) [NonExpansive₂ op] :\n    NonExpansive₂ (merge (T := T) op) where\n  ne _ {_ _} Ht {_ _} Hs k := by admit /- proof elided -/"}, {"name": "Heap.instCOFE", "content": "instance Heap.instCOFE [Heap T K V] [COFE V] : COFE T where\n  compl c := hmap (fun _ => COFE.compl <| c.map ⟨_, Store.get_ne ·⟩) (c 0)\n  conv_compl {_ c} k := by admit /- proof elided -/"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -C> \" => Hom"}], "lib_lemmas": [{"name": "Nat.lt_of_le_of_ne", "module": "Init.Prelude"}, {"name": "Nat.le_of_lt_succ", "module": "Init.Prelude"}, {"name": "Nat.lt_succ_self", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Iris.OFE", "content": "class OFE (α : Type _) where\n  Equiv : α → α → Prop\n  Dist : Nat → α → α → Prop\n  dist_eqv : Equivalence (Dist n)\n  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y\n  dist_lt : Dist n x y → m < n → Dist m x y"}, {"name": "Iris.OFE.NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "Iris.OFE.id_ne", "content": "instance id_ne [OFE α] : NonExpansive (@id α) := ⟨fun _ _ _ h => h⟩"}, {"name": "Iris.OFE.NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "Iris.OFE.DistLater", "content": "def DistLater [OFE α] (n : Nat) (x y : α) : Prop := ∀ m, m < n → x ≡{m}≡ y"}, {"name": "Iris.OFE.Contractive", "content": "class Contractive [OFE α] [OFE β] (f : α → β) where\n  distLater_dist : DistLater n x y → f x ≡{n}≡ f y"}, {"name": "Iris.OFE.ne_of_contractive", "content": "instance ne_of_contractive [OFE α] [OFE β] (f : α → β) [Contractive f] : NonExpansive f where\n  ne := fun _ _ _ h => Contractive.distLater_dist (Dist.distLater h)"}, {"name": "Iris.OFE._inst_β", "content": "instance [OFE α] [OFE β] {x : β} : Contractive (fun _ : α => x) where\n  distLater_dist := fun _ => Dist.rfl"}, {"name": "Iris.OFE.Hom", "content": "@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f"}, {"name": "Iris.OFE._inst_α", "content": "instance [OFE α] [OFE β] (f : α -n> β) : NonExpansive f := f.ne"}, {"name": "Iris.OFE.ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "Iris.OFE._inst_α", "content": "instance [OFE α] [OFE β] (f : α -c> β) : Contractive f := f.contractive"}, {"name": "Iris.OFE._inst_OFE", "content": "instance : OFE Unit where\n  Equiv _ _ := True\n  Dist _ _ _ := True\n  dist_eqv := ⟨fun _ => ⟨⟩, id, fun _ => id⟩\n  equiv_dist := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] : OFE (ULift α) where\n  Equiv x y := x.down ≡ y.down\n  Dist n x y := x.down ≡{n}≡ y.down\n  dist_eqv := InvImage.equivalence dist_eqv\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] : OFE (Option α) where\n  Equiv := Option.Forall₂ Equiv\n  Dist n := Option.Forall₂ (Dist n)\n  dist_eqv := Option.Forall₂.equivalence dist_eqv\n  equiv_dist {x y} := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE.Discrete α] : OFE.Discrete (Option α) where\n  discrete_0 {mx my} e :=\n    match mx, my with\n    | none,   none   => e\n    | none,   some _ => e\n    | some _, none   => e\n    | some x, some y => show x ≡ y from discrete_0 e"}, {"name": "Iris.OFE.OFE", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Iris.OFE.Option", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_α", "content": "instance [OFEFun (β : α → _)] : OFE ((x : α) → β x) where\n  Equiv f g := ∀ x, f x ≡ g x\n  Dist n f g := ∀ x, f x ≡{n}≡ g x\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE β] : OFE (α -n> β) where\n  Equiv f g := f.f ≡ g.f\n  Dist n f g := f.f ≡{n}≡ g.f\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE β] : OFE (α -c> β) where\n  Equiv f g := Equiv f.toHom g.toHom\n  Dist n f g := Dist n f.toHom g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "Iris.OFE._inst_OFE", "content": "instance [OFE α] [OFE β] : OFE (α × β) where\n  Equiv a b := a.1 ≡ b.1 ∧ a.2 ≡ b.2\n  Dist n a b := a.1 ≡{n}≡ b.1 ∧ a.2 ≡{n}≡ b.2\n  dist_eqv := {\n    refl _ := ⟨dist_eqv.refl _, dist_eqv.refl _⟩\n    symm h := ⟨dist_eqv.symm h.1, dist_eqv.symm h.2⟩\n    trans h1 h2 := ⟨dist_eqv.trans h1.1 h2.1, dist_eqv.trans h1.2 h2.2⟩\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "Iris.OFE.Iso", "content": "@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where\n  hom : α -n> β\n  inv : β -n> α\n  hom_inv : hom (inv x) ≡ x\n  inv_hom : inv (hom x) ≡ x"}, {"name": "Iris.OFE._inst_Iso", "content": "instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.hom := iso.hom.ne"}, {"name": "Iris.OFE._inst_Iso", "content": "instance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.inv := iso.inv.ne"}, {"name": "Iris.OFE.Iso.symm", "content": "def Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where\n  hom := iso.inv\n  inv := iso.hom\n  hom_inv := by admit /- proof elided -/"}, {"name": "Iris.Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "Iris.IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Iris.LeibnizO", "content": "@[ext] structure LeibnizO (α : Type _) where\n  car : α"}, {"name": "Iris.optionChain", "content": "def optionChain (c : Chain (Option α)) (x : α) : Chain α :="}, {"name": "Iris.isCOFE_option", "content": "instance isCOFE_option [IsCOFE α] : IsCOFE (Option α) where\n  compl c := (c 0).map fun x => IsCOFE.compl (optionChain c x)\n  conv_compl {n} c := by admit /- proof elided -/"}, {"name": "Iris.Fixpoint.chain", "content": "def Fixpoint.chain [OFE α] [Inhabited α] (f : α → α) [Contractive f] : Chain α where\n  chain n := Nat.repeat f (n + 1) default\n  cauchy {n} := by admit /- proof elided -/"}, {"name": "Iris.fixpoint", "content": "def fixpoint [COFE α] [Inhabited α] (f : α → α) [Contractive f] : α :=\n  COFE.compl <| Fixpoint.chain f\n\nnonrec abbrev OFE.ContractiveHom.fixpoint [COFE α] [Inhabited α] (f : α -c> α) : α := fixpoint f.f"}], "used_local_lemmas": [{"name": "Iris.OFE.Dist.lt", "content": "theorem Dist.lt [OFE α] {m n} {x y : α} : x ≡{n}≡ y → m < n → x ≡{m}≡ y"}, {"name": "Iris.OFE.Dist.le", "content": "theorem Dist.le [OFE α] {m n} {x y : α} (h : x ≡{n}≡ y) (h' : m ≤ n) : x ≡{m}≡ y"}, {"name": "Iris.OFE.Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "Iris.OFE.Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Iris.OFE.Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "Iris.OFE.equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Iris.OFE.Equiv.symm", "content": "@[symm] theorem Equiv.symm [OFE α] {x : α} : x ≡ y → y ≡ x"}, {"name": "Iris.OFE.DistLater.symm", "content": "@[symm] theorem DistLater.symm [OFE α] {n} {x : α} (h : DistLater n x y) : DistLater n y x"}, {"name": "Iris.OFE.distLater_zero", "content": "@[simp] theorem distLater_zero [OFE α] {x y : α} : DistLater 0 x y"}, {"name": "Iris.OFE.distLater_succ", "content": "theorem distLater_succ [OFE α] {n} {x y : α} : DistLater n.succ x y ↔ x ≡{n}≡ y"}, {"name": "Iris.OFE.Contractive.zero", "content": "@[simp] theorem Contractive.zero [OFE α] [OFE β] (f : α → β) [Contractive f] {x y} :\n    f x ≡{0}≡ f y"}, {"name": "Iris.OFE.Contractive.succ", "content": "theorem Contractive.succ [OFE α] [OFE β] (f : α → β) [Contractive f] {n x y}\n    (h : x ≡{n}≡ y) : f x ≡{n.succ}≡ f y"}], "local_ctx": "namespace Iris\n\nclass OFE (α : Type _) where\n  Equiv : α → α → Prop\n  Dist : Nat → α → α → Prop\n  dist_eqv : Equivalence (Dist n)\n  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y\n  dist_lt : Dist n x y → m < n → Dist m x y\n\nopen OFE\n\nscoped infix:40 \" ≡ \" => OFE.Equiv\n\nscoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nnamespace OFE\n\nclass NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂\n\ninstance id_ne [OFE α] : NonExpansive (@id α) := ⟨fun _ _ _ h => h⟩\n\nclass NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂\n\ndef DistLater [OFE α] (n : Nat) (x y : α) : Prop := ∀ m, m < n → x ≡{m}≡ y\n\nclass Contractive [OFE α] [OFE β] (f : α → β) where\n  distLater_dist : DistLater n x y → f x ≡{n}≡ f y\n\ninstance ne_of_contractive [OFE α] [OFE β] (f : α → β) [Contractive f] : NonExpansive f where\n  ne := fun _ _ _ h => Contractive.distLater_dist (Dist.distLater h)\n\ninstance [OFE α] [OFE β] {x : β} : Contractive (fun _ : α => x) where\n  distLater_dist := fun _ => Dist.rfl\n\n@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f\n\n@[inherit_doc]\ninfixr:25 \" -n> \" => Hom\n\ninstance [OFE α] [OFE β] (f : α -n> β) : NonExpansive f := f.ne\n\n@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f\n\ninfixr:25 \" -c> \" => ContractiveHom\n\ninstance [OFE α] [OFE β] (f : α -c> β) : Contractive f := f.contractive\n\ninstance : OFE Unit where\n  Equiv _ _ := True\n  Dist _ _ _ := True\n  dist_eqv := ⟨fun _ => ⟨⟩, id, fun _ => id⟩\n  equiv_dist := by admit /- proof elided -/\n\ninstance [OFE α] : OFE (ULift α) where\n  Equiv x y := x.down ≡ y.down\n  Dist n x y := x.down ≡{n}≡ y.down\n  dist_eqv := InvImage.equivalence dist_eqv\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt\n\ninstance [OFE α] : OFE (Option α) where\n  Equiv := Option.Forall₂ Equiv\n  Dist n := Option.Forall₂ (Dist n)\n  dist_eqv := Option.Forall₂.equivalence dist_eqv\n  equiv_dist {x y} := by admit /- proof elided -/\n\ninstance [OFE α] [OFE.Discrete α] : OFE.Discrete (Option α) where\n  discrete_0 {mx my} e :=\n    match mx, my with\n    | none,   none   => e\n    | none,   some _ => e\n    | some _, none   => e\n    | some x, some y => show x ≡ y from discrete_0 e\n\ninstance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩\n\ninstance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/\n\ninstance [OFEFun (β : α → _)] : OFE ((x : α) → β x) where\n  Equiv f g := ∀ x, f x ≡ g x\n  Dist n f g := ∀ x, f x ≡{n}≡ g x\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/\n\ninstance [OFE α] [OFE β] : OFE (α -n> β) where\n  Equiv f g := f.f ≡ g.f\n  Dist n f g := f.f ≡{n}≡ g.f\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt\n\ninstance [OFE α] [OFE β] : OFE (α -c> β) where\n  Equiv f g := Equiv f.toHom g.toHom\n  Dist n f g := Dist n f.toHom g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt\n\ninstance [OFE α] [OFE β] : OFE (α × β) where\n  Equiv a b := a.1 ≡ b.1 ∧ a.2 ≡ b.2\n  Dist n a b := a.1 ≡{n}≡ b.1 ∧ a.2 ≡{n}≡ b.2\n  dist_eqv := {\n    refl _ := ⟨dist_eqv.refl _, dist_eqv.refl _⟩\n    symm h := ⟨dist_eqv.symm h.1, dist_eqv.symm h.2⟩\n    trans h1 h2 := ⟨dist_eqv.trans h1.1 h2.1, dist_eqv.trans h1.2 h2.2⟩\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/\n\n@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where\n  hom : α -n> β\n  inv : β -n> α\n  hom_inv : hom (inv x) ≡ x\n  inv_hom : inv (hom x) ≡ x\n\ninstance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.hom := iso.hom.ne\n\ninstance [OFE α] [OFE β] (iso : Iso α β) : NonExpansive iso.inv := iso.inv.ne\n\ndef Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where\n  hom := iso.inv\n  inv := iso.hom\n  hom_inv := by admit /- proof elided -/\n\nend OFE\n\nstructure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n\n\nnamespace Chain\n\nend Chain\n\nclass IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n\n\nnamespace COFE\n\nend COFE\n\n@[ext] structure LeibnizO (α : Type _) where\n  car : α\n\nsection DiscreteFunOF\n\nopen COFE\n\nend DiscreteFunOF\n\nsection Option\n\nvariable [OFE α]\n\ndef optionChain (c : Chain (Option α)) (x : α) : Chain α :=\n\ninstance isCOFE_option [IsCOFE α] : IsCOFE (Option α) where\n  compl c := (c 0).map fun x => IsCOFE.compl (optionChain c x)\n  conv_compl {n} c := by admit /- proof elided -/\n\nend Option\n\nsection OptionOF\n\nopen COFE\n\nvariable (F : OFunctorPre)\n\nend OptionOF\n\nsection Fixpoint\n\ndef Fixpoint.chain [OFE α] [Inhabited α] (f : α → α) [Contractive f] : Chain α where\n  chain n := Nat.repeat f (n + 1) default\n  cauchy {n} := by admit /- proof elided -/\n\ndef fixpoint [COFE α] [Inhabited α] (f : α → α) [Contractive f] : α :=\n  COFE.compl <| Fixpoint.chain f\n\nnonrec abbrev OFE.ContractiveHom.fixpoint [COFE α] [Inhabited α] (f : α -c> α) : α := fixpoint f.f", "target_theorem": "theorem fixpoint_unfold [COFE α] [Inhabited α] (f : α -c> α) :\n    fixpoint f ≡ f (fixpoint f) :=", "ground_truth_proof": ":= by\n  refine equiv_dist.mpr fun n => ?_\n  apply COFE.conv_compl.trans\n  refine .trans ?_ (NonExpansive.ne COFE.conv_compl.symm)\n  induction n with\n  | zero => exact Contractive.zero f.f\n  | succ _ IH => exact (Contractive.succ f.f IH.symm).symm", "nesting_depth": 4, "transitive_dep_count": 41, "subset_aristotle": false, "category": "Framework"}
{"id": 233, "thm_name": "Iris.BI.loeb", "thm_stmt": "theorem loeb [BI PROP] [BILoeb PROP] {P : PROP} : (▷ P → P) ⊢ P", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/DerivedLaws.lean", "imports": ["import Iris.BI.Extensions", "import Iris.Std.TC", "import Iris.Std.Classes", "import Iris.BI.BI", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import Iris.BI.Classes", "import Iris.Std.Rewrite"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Function.const", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term\n\nsyntax:max \"<affine> \" term:40 : term\n\nsyntax:max \"□ \" term:40 : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "BILoeb", "content": "class BILoeb (PROP : Type _) [BI PROP] where\n  loeb_weak {P : PROP} : (▷ P ⊢ P) → True ⊢ P"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Iris.BI.and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_symm", "content": "theorem and_symm [BI PROP] {P Q : PROP} : P ∧ Q ⊢ Q ∧ P"}, {"name": "Iris.BI.imp_intro'", "content": "theorem imp_intro' [BI PROP] {P Q R : PROP} (h : Q ∧ P ⊢ R) : P ⊢ Q → R"}, {"name": "Iris.BI.imp_elim'", "content": "theorem imp_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P → R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.imp_elim_l", "content": "theorem imp_elim_l [BI PROP] {P Q : PROP} : (P → Q) ∧ P ⊢ Q"}, {"name": "Iris.BI.imp_elim_r", "content": "theorem imp_elim_r [BI PROP] {P Q : PROP} : P ∧ (P → Q) ⊢ Q"}, {"name": "Iris.BI.true_intro", "content": "theorem true_intro [BI PROP] {P : PROP} : P ⊢ True"}, {"name": "Iris.BI.and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "Iris.BI.and_mono_l", "content": "theorem and_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∧ Q ⊢ P' ∧ Q"}, {"name": "Iris.BI.and_mono_r", "content": "theorem and_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∧ Q ⊢ P ∧ Q'"}, {"name": "Iris.BI.and_self", "content": "theorem and_self [BI PROP] {P : PROP} : P ∧ P ⊣⊢ P"}, {"name": "Iris.BI.and_comm", "content": "theorem and_comm [BI PROP] {P Q : PROP} : P ∧ Q ⊣⊢ Q ∧ P"}, {"name": "Iris.BI.and_assoc", "content": "theorem and_assoc [BI PROP] {P Q R : PROP} : (P ∧ Q) ∧ R ⊣⊢ P ∧ Q ∧ R"}, {"name": "Iris.BI.pure_elim", "content": "theorem pure_elim [BI PROP] (φ : Prop) {Q R : PROP} (h1 : Q ⊢ ⌜φ⌝) (h2 : φ → Q ⊢ R) : Q ⊢ R"}, {"name": "Iris.BI.pure_elim_l", "content": "theorem pure_elim_l [BI PROP] {φ : Prop} {Q R : PROP} (h : φ → Q ⊢ R) : ⌜φ⌝ ∧ Q ⊢ R"}, {"name": "Iris.BI.pure_elim_r", "content": "theorem pure_elim_r [BI PROP] {φ : Prop} {Q R : PROP} (h : φ → Q ⊢ R) : Q ∧ ⌜φ⌝ ⊢ R"}, {"name": "Iris.BI.later_impl", "content": "theorem later_impl [BI PROP] {P Q : PROP} : ▷ (P → Q) ⊢ ▷ P → ▷ Q"}, {"name": "Iris.BI.entails_impl_true", "content": "theorem entails_impl_true [BI PROP] {P Q : PROP} :\n    (P ⊢ Q) ↔ iprop((True : PROP) ⊢ (P → Q))"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.Extensions\n\nimport Iris.BI.BI\n\nimport Iris.Std.Classes\n\nimport Iris.Std.Rewrite\n\nimport Iris.Std.TC\n\nnamespace Iris.BI\n\nopen Iris.Std BI", "target_theorem": "theorem loeb [BI PROP] [BILoeb PROP] {P : PROP} : (▷ P → P) ⊢ P :=", "ground_truth_proof": ":= by\n  apply entails_impl_true.mpr\n  apply BILoeb.loeb_weak\n  apply imp_intro\n  apply (and_mono .rfl and_self.mpr).trans\n  apply (and_mono .rfl (and_mono later_intro .rfl)).trans\n  apply (and_mono later_impl .rfl).trans\n  apply and_assoc.mpr.trans\n  apply (and_mono imp_elim_l .rfl).trans\n  exact imp_elim_r", "nesting_depth": 7, "transitive_dep_count": 68, "subset_aristotle": false, "category": "Framework"}
{"id": 234, "thm_name": "Iris.BI.plainly_and_sep", "thm_stmt": "theorem plainly_and_sep : ■ (P ∧ Q) ⊢ ■ (P ∗ Q)", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/Plainly.lean", "imports": ["import Iris.Algebra", "import src.Iris.BI.DerivedLaws", "import src/Iris/Instances/UPred/Instance.lean", "import Iris.BI.DerivedLaws", "import Iris.BI.BI", "import src.Iris.BI.BI", "import Iris.BI.Classes"], "used_lib_defs": [{"name": "Lean.MonadQuotation", "module": "Init.Prelude"}, {"name": "Lean.MonadRef", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax term:26 \" ==∗ \" term:25 : term", "content": "syntax term:26 \" ==∗ \" term:25 : term\n\nsyntax term \"={ \" term \" , \" term \" }=∗ \" term : term\n\nsyntax term \"={ \" term \" }=∗ \" term : term\n\nsyntax term \"={ \" term \" }[ \" term \" ]▷=∗ \" term : term\n\nsyntax term \"={ \" term \" }▷=∗ \" term : term\n\nsyntax term \"={ \" term \" }[ \" term \" ]▷^\" term \"=∗ \" term : term\n\nsyntax term \"={ \" term \" }▷^\" term \"=∗ \" term : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term"}, {"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term\n\nsyntax \"■ \" term:40 : term"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|==> $P))  => ``(BUpd.bupd iprop($P))\n  | `(iprop($P ==∗ $Q))  => ``(BIBase.wand iprop($P) (BUpd.bupd iprop($Q)))\n\ndelab_rule BUpd.bupd\n  | `($_ $P) => do ``(iprop(|==> $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 , $E2 }=> $P))  => ``(FUpd.fupd $E1 $E2 iprop($P))\n  | `(iprop($P ={ $E1 , $E2 }=∗ $Q))  => ``(BIBase.wand iprop($P) (FUpd.fupd $E1 $E2 iprop($Q)))\n  | `(iprop(|={ $E1 }=> $P))  => ``(FUpd.fupd $E1 $E1 iprop($P))\n  | `(iprop($P ={ $E1 }=∗ $Q))  => ``(BIBase.wand iprop($P) (FUpd.fupd $E1 $E1 iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 }[ $E2 ]▷=> $P))  => ``(iprop(|={$E1, $E2}=> ▷ (|={ $E2, $E1 }=> iprop($P))))\n  | `(iprop($P ={ $E1 }[ $E2 ]▷=∗ $Q))  => ``(iprop(iprop($P) -∗ |={$E1}[$E2]▷=> iprop($Q)))\n  | `(iprop(|={ $E1 }▷=> $P))  => ``(iprop(|={$E1}[$E1]▷=> iprop($P)))\n  | `(iprop($P ={ $E1 }▷=∗ $Q))  => ``(iprop(iprop($P) ={$E1}[$E1]▷=∗ iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 }[ $E2 ]▷^$n=> $P))  => ``(iprop(|={$E1, $E2}=> ▷^[$n] (|={ $E2, $E1 }=> iprop($P))))\n  | `(iprop($P ={ $E1 }[ $E2 ]▷^$n=∗ $Q))  => ``(iprop(iprop($P) -∗ |={$E1}[$E2]▷^$n=> iprop($Q)))\n  | `(iprop(|={ $E1 }▷^$n=> $P))  => ``(iprop(|={$E1}[$E1]▷^$n=> iprop($P)))\n  | `(iprop($P ={ $E1 }▷^$n=∗ $Q))  => ``(iprop(iprop($P) ={$E1}[$E1]▷^$n=∗ iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(■ $P))  => ``(Plainly.plainly iprop($P))\n\ndelab_rule Plainly.plainly\n  | `($_ $P) => do ``(iprop(■ $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(■? $p $P))  => ``(Plainly.plainlyIf $p iprop($P))\n\ndelab_rule Plainly.plainlyIf\n  | `($_ $p $P) => do ``(iprop(■? $p $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "unpackIprop", "content": "partial def unpackIprop [Monad m] [MonadRef m] [MonadQuotation m] : Term → m Term\n  | `(iprop($P))             => do `($P)\n  | `($P:ident)              => do `($P)\n  | `(?$P:ident)             => do `(?$P)\n  | `(($P))                  => do `(($(← unpackIprop P)))\n  | `($P $[ $Q]*)            => do ``($P $[ $Q]*)\n  | `(if $c then $t else $e) => do\n    let t ← unpackIprop t\n    let e ← unpackIprop e\n    `(if $c then $t else $e)\n  | `(($P : $t))             => do ``(($(← unpackIprop P) : $t))\n  | `($t)                    => `($t:term)"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "Idempotent", "content": "class Idempotent (R : Relation α) (f : α → α → α) where\n  idem {x : α} : R (f x x) x"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "", "content": "instance : BIPlainly (UPred M) where\n  mono H _ _ _ := H _ _ CMRA.unit_validN\n  elim_persistently {P} n x Hx := by admit /- proof elided -/"}, {"name": "persistently", "content": "protected def persistently (P : UPred M) : UPred M where\n  holds n x := P n (CMRA.core x)\n  mono H Hx Hn := P.mono H (CMRA.core_incN_core Hx) Hn"}, {"name": "BIUpdate", "content": "class BIUpdate (PROP : Type _) [BI PROP] extends BUpd PROP where\n  [bupd_ne : OFE.NonExpansive (BUpd.bupd (PROP := PROP))]\n  intro {P : PROP} : iprop(P ⊢ |==> P)\n  mono {P Q : PROP} : iprop(P ⊢ Q) → iprop(|==> P ⊢ |==> Q)\n  trans {P : PROP} : iprop(|==> |==> P ⊢ |==> P)\n  frame_r {P R : PROP} : iprop((|==> P) ∗ R ⊢ |==> (P ∗ R))"}, {"name": "UPred", "content": "@[ext]\nstructure UPred (M : Type _) [UCMRA M] where\n  holds : Nat → M → Prop\n  mono {n1 n2 x1 x2} : holds n1 x1 → x1 ≼{n2} x2 → n2 ≤ n1 → holds n2 x2"}, {"name": "IsModal", "content": "class IsModal [BI PROP1] [BI PROP2] (M : PROP1 → PROP2)\n    (iaction saction : ModalityAction PROP1 PROP2) where\n  spec_intuitionistic : iaction.intuitionistic_action_spec M\n  spec_spatial : saction.spatial_action_spec M\n  emp : iprop(emp) ⊢ M iprop(emp)\n  mono : ∀ {P Q}, (P ⊢ Q) → M P ⊢ M Q\n  sep : ∀ {P Q}, iprop(M P ∗ M Q) ⊢ M iprop(P ∗ Q)"}, {"name": "core", "content": "def core (x : α) := (pcore x).getD x"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "UCMRA", "content": "class UCMRA (α : Type _) extends CMRA α where\n  unit : α\n  unit_valid : ✓ unit\n  unit_left_id : unit • x ≡ x\n  pcore_unit : pcore unit ≡ some unit"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "plainly", "content": "protected def plainly (P : UPred M) : UPred M where\n  holds n _ := P n UCMRA.unit\n  mono H _ Hn := P.mono H (CMRA.incN_refl UCMRA.unit) Hn"}, {"name": "BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "", "content": "instance : BIPlainlyExists (UPred M) where\n  plainly_sExists_1 _ _ _ := fun ⟨_, hp⟩ => ⟨_, ⟨_, rfl⟩, hp⟩"}, {"name": "BIPlainlyExists", "content": "class BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p"}, {"name": "", "content": "instance : Plainly (UPred M) := ⟨UPred.plainly⟩"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "persistently_and_sep_assoc", "content": "theorem persistently_and_sep_assoc [BI PROP] {P Q R : PROP} :\n    <pers> P ∧ (Q ∗ R) ⊣⊢ (<pers> P ∧ Q) ∗ R"}, {"name": "and_mono_l", "content": "theorem and_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∧ Q ⊢ P' ∧ Q"}, {"name": "and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}, {"name": "and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "sep_assoc", "content": "theorem sep_assoc [BI PROP] {P Q R : PROP} : (P ∗ Q) ∗ R ⊣⊢ P ∗ Q ∗ R"}, {"name": "sep_congr_l", "content": "theorem sep_congr_l [BI PROP] {P P' Q : PROP} (h : P ⊣⊢ P') : P ∗ Q ⊣⊢ P' ∗ Q"}, {"name": "sep_congr", "content": "@[rw_mono_rule]\ntheorem sep_congr [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊣⊢ Q) (h2 : P' ⊣⊢ Q') :\n    (P ∗ P') ⊣⊢ (Q ∗ Q')"}, {"name": "sep_congr_r", "content": "theorem sep_congr_r [BI PROP] {P Q Q' : PROP} (h : Q ⊣⊢ Q') : P ∗ Q ⊣⊢ P ∗ Q'"}, {"name": "sep_mono_l", "content": "theorem sep_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∗ Q ⊢ P' ∗ Q"}, {"name": "BIBase.Entails.rfl", "content": "@[simp] theorem BIBase.Entails.rfl [BI PROP] {P : PROP} : P ⊢ P"}, {"name": "imp_intro'", "content": "theorem imp_intro' [BI PROP] {P Q R : PROP} (h : Q ∧ P ⊢ R) : P ⊢ Q → R"}, {"name": "and_comm", "content": "theorem and_comm [BI PROP] {P Q : PROP} : P ∧ Q ⊣⊢ Q ∧ P"}, {"name": "and_symm", "content": "theorem and_symm [BI PROP] {P Q : PROP} : P ∧ Q ⊢ Q ∧ P"}, {"name": "imp_elim'", "content": "theorem imp_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P → R) : P ∧ Q ⊢ R"}, {"name": "pure_elim", "content": "theorem pure_elim [BI PROP] (φ : Prop) {Q R : PROP} (h1 : Q ⊢ ⌜φ⌝) (h2 : φ → Q ⊢ R) : Q ⊢ R"}, {"name": "and_self", "content": "theorem and_self [BI PROP] {P : PROP} : P ∧ P ⊣⊢ P"}, {"name": "forall_intro", "content": "theorem forall_intro [BI PROP] {P : PROP} {Ψ : α → PROP} (h : ∀ a, P ⊢ Ψ a) : P ⊢ ∀ a, Ψ a"}, {"name": "forall_elim", "content": "theorem forall_elim [BI PROP] {Ψ : α → PROP} (a : α) : (∀ a, Ψ a) ⊢ Ψ a"}, {"name": "forall_mono", "content": "@[rw_mono_rule]\ntheorem forall_mono [BI PROP] {Φ Ψ : α → PROP} (h : ∀ a, Φ a ⊢ Ψ a) : (∀ a, Φ a) ⊢ ∀ a, Ψ a"}, {"name": "and_forall_bool", "content": "theorem and_forall_bool [BI PROP] {P Q : PROP} :\n    P ∧ Q ⊣⊢ «forall» (fun b : Bool => if b then P else Q)"}, {"name": "persistently_and_emp_elim", "content": "theorem persistently_and_emp_elim {P : PROP} [BI PROP] : emp ∧ <pers> P ⊢ P"}, {"name": "sep_emp", "content": "theorem sep_emp [BI PROP] {P : PROP} : P ∗ emp ⊣⊢ P"}], "used_local_defs": [{"name": "Iris.Plainly", "content": "class Plainly (PROP : Type _) where\n  plainly : PROP → PROP"}, {"name": "Iris.Plainly.plainlyIf", "content": "def Plainly.plainlyIf [Iris.BI.BIBase PROP] [Plainly PROP] (p : Bool) (P : PROP) : PROP :=\n  iprop(if p then ■ P else P)"}, {"name": "Iris.BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "Iris.BIPlainlyExists", "content": "class BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p"}], "used_local_lemmas": [{"name": "Iris.BI.persistently_elim_plainly", "content": "theorem persistently_elim_plainly : <pers> ■ P ⊣⊢ ■ P"}, {"name": "Iris.BI.plainly_forall_2", "content": "theorem plainly_forall_2 {Ψ : α → PROP} : (∀ a, ■ (Ψ a)) ⊢ ■ (∀ a, Ψ a)"}, {"name": "Iris.BI.plainly_and_sep_assoc", "content": "theorem plainly_and_sep_assoc : ■ P ∧ (Q ∗ R) ⊣⊢ (■ P ∧ Q) ∗ R"}, {"name": "Iris.BI.plainly_and_emp_elim", "content": "theorem plainly_and_emp_elim : emp ∧ ■ P ⊢ P"}, {"name": "Iris.BI.plainly_forall", "content": "theorem plainly_forall {Ψ : α → PROP} : ■ (∀ a, Ψ a) ⊣⊢ ∀ a, ■ (Ψ a)"}, {"name": "Iris.BI.plainly_and", "content": "theorem plainly_and : ■ (P ∧ Q) ⊣⊢ ■ P ∧ ■ Q"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.BI\n\nimport Iris.BI.DerivedLaws\n\nimport Iris.Algebra\n\nnamespace Iris\n\nopen BI\n\nclass Plainly (PROP : Type _) where\n  plainly : PROP → PROP\n\ndef Plainly.plainlyIf [Iris.BI.BIBase PROP] [Plainly PROP] (p : Bool) (P : PROP) : PROP :=\n  iprop(if p then ■ P else P)\n\nclass BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P\n\nclass BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p\n\nnamespace BI\n\nopen Iris.Std\n\nsection PlainlyLaws\n\nopen BIPlainly\n\nvariable [BI PROP] [BIPlainly PROP]\n\nvariable {P Q R : PROP}", "target_theorem": "theorem plainly_and_sep : ■ (P ∧ Q) ⊢ ■ (P ∗ Q) :=", "ground_truth_proof": ":= by\n  refine (plainly_and.mp.trans <| (and_mono idem .rfl).trans plainly_and.mpr).trans ?_\n  refine (mono <| and_mono .rfl emp_sep.mpr).trans ?_\n  refine (mono <| plainly_and_sep_assoc.1).trans ?_\n  refine (mono <| sep_mono and_comm.mp .rfl).trans ?_\n  exact (mono <| sep_mono plainly_and_emp_elim .rfl).trans .rfl", "nesting_depth": 6, "transitive_dep_count": 55, "subset_aristotle": false, "category": "Framework"}
{"id": 235, "thm_name": "Iris.BI.later_exists_false", "thm_stmt": "theorem later_exists_false [BI PROP] {Φ : α → PROP} :\n    (▷ ∃ a, Φ a) ⊢ ▷ False ∨ ∃ a, ▷ Φ a", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/DerivedLaws.lean", "imports": ["import Iris.BI.Extensions", "import Iris.Std.TC", "import Iris.Std.Classes", "import Iris.BI.BI", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import Iris.BI.Classes", "import Iris.Std.Rewrite"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term\n\nsyntax:max \"<affine> \" term:40 : term\n\nsyntax:max \"□ \" term:40 : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Iris.BI.or_intro_r'", "content": "theorem or_intro_r' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ⊢ Q ∨ R"}, {"name": "Iris.BI.imp_intro'", "content": "theorem imp_intro' [BI PROP] {P Q R : PROP} (h : Q ∧ P ⊢ R) : P ⊢ Q → R"}, {"name": "Iris.BI.exists_intro", "content": "theorem exists_intro [BI PROP] {Ψ : α → PROP} (a : α) : Ψ a ⊢ ∃ a, Ψ a"}, {"name": "Iris.BI.exists_elim", "content": "theorem exists_elim [BI PROP] {Φ : α → PROP} {Q : PROP} (h : ∀ a, Φ a ⊢ Q) : (∃ a, Φ a) ⊢ Q"}, {"name": "Iris.BI.exists_intro'", "content": "theorem exists_intro' [BI PROP] {P : PROP} {Ψ : α → PROP} (a : α) (h : P ⊢ Ψ a) : P ⊢ ∃ a, Ψ a"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.Extensions\n\nimport Iris.BI.BI\n\nimport Iris.Std.Classes\n\nimport Iris.Std.Rewrite\n\nimport Iris.Std.TC\n\nnamespace Iris.BI\n\nopen Iris.Std BI", "target_theorem": "theorem later_exists_false [BI PROP] {Φ : α → PROP} :\n    (▷ ∃ a, Φ a) ⊢ ▷ False ∨ ∃ a, ▷ Φ a :=", "ground_truth_proof": ":= by\n  apply later_sExists_false.trans\n  apply or_elim\n  · apply or_intro_l\n  · refine or_intro_r' <| exists_elim ?_\n    intro P\n    refine imp_elim <| pure_elim' ?_\n    rintro ⟨a, rfl⟩\n    exact imp_intro' <| exists_intro' a and_elim_l", "nesting_depth": 4, "transitive_dep_count": 52, "subset_aristotle": false, "category": "Framework"}
{"id": 236, "thm_name": "Iris.DFrac.update_discard", "thm_stmt": "theorem DFrac.update_discard {dq : DFrac F} : dq ~~> .discard", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/DFrac.lean", "imports": ["import Iris.Algebra.CMRA", "import Iris.Algebra.Updates", "import src.Iris.Algebra.Frac", "import Iris.Algebra.OFE", "import Iris.Algebra.LocalUpdates", "import src.Iris.Algebra.CMRA", "import Iris.Algebra.Frac"], "used_lib_defs": [{"name": "Add", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x", "content": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x"}, {"name": "Fractional", "content": "def Fractional [Fraction α] (a : α) : Prop := ∃ b, Proper (a + b)"}, {"name": "Fraction", "content": "class Fraction (α : Type _) extends Add α where\n   \n  Proper : α → Prop\n  add_comm : ∀ a b : α, a + b = b + a\n  add_assoc : ∀ a b c : α, a + (b + c) = (a + b) + c\n  add_left_cancel : ∀ {a b c : α}, a + b = a + c → b = c\n   \n  add_ne : ∀ {a b : α}, a ≠ b + a\n  proper_add_mono_left : ∀ {a b : α}, Proper (a + b) → Proper a"}, {"name": "NumericFraction", "content": "class NumericFraction (α : Type _) extends One α, Add α, LE α, LT α where\n  add_comm : ∀ a b : α, a + b = b + a\n  add_assoc : ∀ a b c : α, a + (b + c) = (a + b) + c\n  add_left_cancel : ∀ {a b c : α}, a + b = a + c → b = c\n  le_def : ∀ {a b : α}, a ≤ b ↔ a = b ∨ a < b\n  lt_def : ∀ {a b : α}, a < b ↔ ∃ c : α, a + c = b\n  lt_irrefl : ∀ {a : α}, ¬a < a"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "Update", "content": "def Update [CMRA α] (x y : α) := ∀ n mz,\n  ✓{n} (x •? mz) → ✓{n} (y •? mz)"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "op?", "content": "def op? [CMRA α] (x : α) : Option α → α\n  | some y => x • y\n  | none => x"}, {"name": "Discrete", "content": "class Discrete (α : Type _) [CMRA α] extends OFE.Discrete α where\n  discrete_valid {x : α} : ✓{0} x → ✓ x"}, {"name": "Discrete", "content": "class Discrete (α : Type _) [OFE α] where\n  discrete_0 {x y : α} : x ≡{0}≡ y → x ≡ y"}, {"name": "infixr:50 \" ~~> \" => Update", "content": "infixr:50 \" ~~> \" => Update"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "infix:60 \" •? \" => op?", "content": "infix:60 \" •? \" => op?"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Fractional.of_add_right", "content": "theorem Fractional.of_add_right {a a' : α} (H : Fractional (a + a')) : Fractional a'"}, {"name": "valid_iff_validN'", "content": "theorem valid_iff_validN' [Discrete α] (n) {x : α} : ✓ x ↔ ✓{n} x"}, {"name": "validN_of_le", "content": "theorem validN_of_le {n n'} {x : α} : n' ≤ n → ✓{n} x → ✓{n'} x"}, {"name": "Valid.validN", "content": "theorem Valid.validN : ✓ (x : α) → ✓{n} x"}], "used_local_defs": [{"name": "Iris.DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "Iris.valid", "content": "def valid : DFrac F → Prop\n  | .own f        => Proper f\n  | .discard      => True\n  | .ownDiscard f => Fractional f"}, {"name": "Iris.pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "Iris.op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}], "used_local_lemmas": [{"name": "Iris.valid_op_own", "content": "theorem valid_op_own {dq : DFrac F} {q : F} : ✓ dq • own q → Fractional q"}], "local_ctx": "import Iris.Algebra.CMRA\n\nimport Iris.Algebra.OFE\n\nimport Iris.Algebra.Frac\n\nimport Iris.Algebra.Updates\n\nimport Iris.Algebra.LocalUpdates\n\nnamespace Iris\n\ninductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F\n\nsection dfrac\n\nopen DFrac Fraction OFE.Discrete\n\nvariable [UFraction F]\n\ndef valid : DFrac F → Prop\n  | .own f        => Proper f\n  | .discard      => True\n  | .ownDiscard f => Fractional f\n\ndef pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard\n\ndef op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')", "target_theorem": "theorem DFrac.update_discard {dq : DFrac F} : dq ~~> .discard :=", "ground_truth_proof": ":= by\n  intros n q H\n  apply (CMRA.valid_iff_validN' n).mp\n  have H' := (CMRA.valid_iff_validN' n).mpr H\n  simp [CMRA.op?] at H' ⊢\n  rcases q with (_|⟨q|_|q⟩) <;> simp [CMRA.Valid, valid, CMRA.op, op]\n  · cases dq <;> first | exact valid_op_own H | exact H\n  · cases dq <;> first | exact Fractional.of_add_right H | exact H", "nesting_depth": 3, "transitive_dep_count": 31, "subset_aristotle": false, "category": "Framework"}
{"id": 237, "thm_name": "Iris.local_update_unital_discrete", "thm_stmt": "theorem local_update_unital_discrete [CMRA.Discrete α] (x y x' y' : α) :\n    (x, y) ~l~> (x', y') ↔ ∀ z, ✓ x → x ≡ y • z → (✓ x' ∧ x' ≡ y' • z)", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/LocalUpdates.lean", "imports": ["import Iris.Algebra.CMRA", "import src.Iris.Algebra.CMRA", "import src.Iris.Algebra.OFE"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x", "content": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "Discrete", "content": "class Discrete (α : Type _) [CMRA α] extends OFE.Discrete α where\n  discrete_valid {x : α} : ✓{0} x → ✓ x"}, {"name": "Discrete", "content": "class Discrete (α : Type _) [OFE α] where\n  discrete_0 {x y : α} : x ≡{0}≡ y → x ≡ y"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "UCMRA", "content": "class UCMRA (α : Type _) extends CMRA α where\n  unit : α\n  unit_valid : ✓ unit\n  unit_left_id : unit • x ≡ x\n  pcore_unit : pcore unit ≡ some unit"}, {"name": "Iso.symm", "content": "def Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where\n  hom := iso.inv\n  inv := iso.hom\n  hom_inv := by admit /- proof elided -/"}, {"name": "Iso", "content": "@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where\n  hom : α -n> β\n  inv : β -n> α\n  hom_inv : hom (inv x) ≡ x\n  inv_hom : inv (hom x) ≡ x"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "Included.trans", "content": "theorem Included.trans : (x : α) ≼ y → y ≼ z → x ≼ z"}, {"name": "inc_trans", "content": "theorem inc_trans {x y z : α} : x ≼ y → y ≼ z → x ≼ z"}, {"name": "op_left_eqv", "content": "theorem op_left_eqv {x y : α} (z : α) (e : x ≡ y) : x • z ≡ y • z"}, {"name": "_root_.Iris.OFE.Dist.op_r", "content": "theorem _root_.Iris.OFE.Dist.op_r {x y z : α} : y ≡{n}≡ z → x • y ≡{n}≡ x • z"}, {"name": "op_right_dist", "content": "theorem op_right_dist (x : α) {y z : α} (e : y ≡{n}≡ z) : x • y ≡{n}≡ x • z"}, {"name": "_root_.Iris.OFE.Equiv.op_r", "content": "theorem _root_.Iris.OFE.Equiv.op_r {x y z : α} : y ≡ z → x • y ≡ x • z"}, {"name": "op_right_eqv", "content": "theorem op_right_eqv (x : α) {y z : α} (e : y ≡ z) : x • y ≡ x • z"}, {"name": "IncludedN.trans", "content": "theorem IncludedN.trans : (x : α) ≼{n} y → y ≼{n} z → x ≼{n} z"}, {"name": "incN_trans", "content": "theorem incN_trans {x y z : α} : x ≼{n} y → y ≼{n} z → x ≼{n} z"}, {"name": "op_left_dist", "content": "theorem op_left_dist {x y : α} (z : α) (e : x ≡{n}≡ y) : x • z ≡{n}≡ y • z"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}, {"name": "unit_right_id_dist", "content": "theorem unit_right_id_dist (x : α) : x • unit ≡{n}≡ x"}, {"name": "unit_left_id_dist", "content": "theorem unit_left_id_dist {n} (x : α) : unit • x ≡{n}≡ x"}, {"name": "Equiv.dist", "content": "theorem Equiv.dist [OFE α] {x : α} : x ≡ y → x ≡{n}≡ y"}, {"name": "valid_iff_validN'", "content": "theorem valid_iff_validN' [Discrete α] (n) {x : α} : ✓ x ↔ ✓{n} x"}, {"name": "validN_of_le", "content": "theorem validN_of_le {n n'} {x : α} : n' ≤ n → ✓{n} x → ✓{n'} x"}, {"name": "Valid.validN", "content": "theorem Valid.validN : ✓ (x : α) → ✓{n} x"}, {"name": "IncludedN.validN", "content": "theorem IncludedN.validN {n} {x y : α} : x ≼{n} y → ✓{n} y → ✓{n} x"}, {"name": "validN_of_incN", "content": "theorem validN_of_incN {n} {x y : α} : x ≼{n} y → ✓{n} y → ✓{n} x"}, {"name": "_root_.Iris.OFE.Dist.validN", "content": "theorem _root_.Iris.OFE.Dist.validN : (x : α) ≡{n}≡ y → (✓{n} x ↔ ✓{n} y)"}, {"name": "validN_iff", "content": "theorem validN_iff {x y : α} (e : x ≡{n}≡ y) : ✓{n} x ↔ ✓{n} y"}, {"name": "Included.validN", "content": "theorem Included.validN {n} {x y : α} : x ≼ y → ✓{n} y → ✓{n} x"}, {"name": "validN_of_inc", "content": "theorem validN_of_inc {n} {x y : α} : x ≼ y → ✓{n} y → ✓{n} x"}], "used_local_defs": [{"name": "Iris.LocalUpdate", "content": "def LocalUpdate [CMRA α] (x y : α × α) : Prop :=\n  ∀n mz, ✓{n} x.1 → x.1 ≡{n}≡ x.2 •? mz → ✓{n} y.1 ∧ y.1 ≡{n}≡ y.2 •? mz"}], "used_local_lemmas": [{"name": "Iris.local_update_unital", "content": "theorem local_update_unital {x y x' y' : α} :\n    (x, y) ~l~> (x', y') ↔ ∀ n z, ✓{n} x → x ≡{n}≡ y • z → (✓{n} x' ∧ x' ≡{n}≡ y' • z)"}], "local_ctx": "import Iris.Algebra.CMRA\n\nnamespace Iris\n\ndef LocalUpdate [CMRA α] (x y : α × α) : Prop :=\n  ∀n mz, ✓{n} x.1 → x.1 ≡{n}≡ x.2 •? mz → ✓{n} y.1 ∧ y.1 ≡{n}≡ y.2 •? mz\n\ninfixr:50 \" ~l~> \" => LocalUpdate\n\nsection localUpdate\n\nsection CMRA\n\nvariable [CMRA α]\n\nend CMRA\n\nsection UCMRA\n\nvariable [UCMRA α]", "target_theorem": "theorem local_update_unital_discrete [CMRA.Discrete α] (x y x' y' : α) :\n    (x, y) ~l~> (x', y') ↔ ∀ z, ✓ x → x ≡ y • z → (✓ x' ∧ x' ≡ y' • z) :=", "ground_truth_proof": "where\n  mp h z vx e :=\n    have ⟨vx', e'⟩ := h 0 (some z) (CMRA.Valid.validN vx) e.dist\n    ⟨CMRA.discrete_valid vx', OFE.discrete_0 e'⟩\n  mpr h := by\n    refine local_update_unital.mpr fun n z vnx e => ?_\n    have ⟨vx', e'⟩ := h z ((CMRA.valid_iff_validN' n).mpr vnx) (OFE.discrete e)\n    exact ⟨vx'.validN, e'.dist⟩", "nesting_depth": 5, "transitive_dep_count": 51, "subset_aristotle": false, "category": "Framework"}
{"id": 238, "thm_name": "Iris.BI.persistently_and_sep_assoc", "thm_stmt": "theorem persistently_and_sep_assoc [BI PROP] {P Q R : PROP} :\n    <pers> P ∧ (Q ∗ R) ⊣⊢ (<pers> P ∧ Q) ∗ R", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/DerivedLaws.lean", "imports": ["import Iris.BI.Extensions", "import Iris.Std.TC", "import Iris.Std.Classes", "import Iris.BI.BI", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import Iris.BI.Classes", "import Iris.Std.Rewrite"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term\n\nsyntax:max \"<affine> \" term:40 : term\n\nsyntax:max \"□ \" term:40 : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Iris.BI.and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "Iris.BI.and_mono_l", "content": "theorem and_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∧ Q ⊢ P' ∧ Q"}, {"name": "Iris.BI.sep_mono_l", "content": "theorem sep_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∗ Q ⊢ P' ∗ Q"}, {"name": "Iris.BI.sep_congr", "content": "@[rw_mono_rule]\ntheorem sep_congr [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊣⊢ Q) (h2 : P' ⊣⊢ Q') :\n    (P ∗ P') ⊣⊢ (Q ∗ Q')"}, {"name": "Iris.BI.sep_congr_l", "content": "theorem sep_congr_l [BI PROP] {P P' Q : PROP} (h : P ⊣⊢ P') : P ∗ Q ⊣⊢ P' ∗ Q"}, {"name": "Iris.BI.sep_congr_r", "content": "theorem sep_congr_r [BI PROP] {P Q Q' : PROP} (h : Q ⊣⊢ Q') : P ∗ Q ⊣⊢ P ∗ Q'"}, {"name": "Iris.BI.sep_assoc", "content": "theorem sep_assoc [BI PROP] {P Q R : PROP} : (P ∗ Q) ∗ R ⊣⊢ P ∗ Q ∗ R"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.Extensions\n\nimport Iris.BI.BI\n\nimport Iris.Std.Classes\n\nimport Iris.Std.Rewrite\n\nimport Iris.Std.TC\n\nnamespace Iris.BI\n\nopen Iris.Std BI", "target_theorem": "theorem persistently_and_sep_assoc [BI PROP] {P Q R : PROP} :\n    <pers> P ∧ (Q ∗ R) ⊣⊢ (<pers> P ∧ Q) ∗ R :=", "ground_truth_proof": ":= by\n  constructor\n  · refine (and_mono_l persistently_idem_2).trans <| persistently_and_affinely_sep.trans <|\n      sep_assoc.2.trans <| sep_mono_l <| and_intro ?_ ?_\n    · exact (sep_mono_l and_elim_r).trans persistently_absorb_l\n    · exact (sep_mono_l and_elim_l).trans emp_sep.1\n  · exact and_intro ((sep_mono_l and_elim_l).trans persistently_absorb_l) (sep_mono_l and_elim_r)", "nesting_depth": 4, "transitive_dep_count": 56, "subset_aristotle": false, "category": "Framework"}
{"id": 239, "thm_name": "Iris.UpdateP.iso", "thm_stmt": "theorem UpdateP.iso\n    (gf : ∀ x, g (f x) ≡ x)\n    (g_op : ∀ y1 y2, g (y1 • y2) ≡ g y1 • g y2)\n    (g_validN : ∀ n y, ✓{n} (g y) ↔ ✓{n} y)\n    (uyp : y ~~>: P)\n    (pq : ∀ y', P y' → Q (g y')) :\n    g y ~~>: Q", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/Updates.lean", "imports": ["import Iris.Algebra.CMRA", "import src.Iris.Algebra.CMRA", "import src.Iris.Algebra.OFE"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Option.map", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x", "content": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "map", "content": "def map [OFE α] [OFE β] (f : α -n> β) (c : Chain α) : Chain β where\n  chain n := f (c n)\n  cauchy h := f.ne.1 (c.cauchy h)"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "Hom", "content": "@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -n> \" => Hom"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -C> \" => Hom"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "op_right_eqv", "content": "theorem op_right_eqv (x : α) {y z : α} (e : y ≡ z) : x • y ≡ x • z"}, {"name": "Equiv.rfl", "content": "@[simp, refl] theorem Equiv.rfl [OFE α] {x : α} : x ≡ x"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}], "used_local_defs": [{"name": "Iris.UpdateP", "content": "def UpdateP [CMRA α] (x : α) (P : α → Prop) := ∀ n mz,\n  ✓{n} (x •? mz) → ∃ y, P y ∧ ✓{n} (y •? mz)"}, {"name": "Iris.Update", "content": "def Update [CMRA α] (x y : α) := ∀ n mz,\n  ✓{n} (x •? mz) → ✓{n} (y •? mz)"}], "used_local_lemmas": [], "local_ctx": "import Iris.Algebra.CMRA\n\nnamespace Iris\n\ndef UpdateP [CMRA α] (x : α) (P : α → Prop) := ∀ n mz,\n  ✓{n} (x •? mz) → ∃ y, P y ∧ ✓{n} (y •? mz)\n\ninfixr:50 \" ~~>: \" => UpdateP\n\ndef Update [CMRA α] (x y : α) := ∀ n mz,\n  ✓{n} (x •? mz) → ✓{n} (y •? mz)\n\ninfixr:50 \" ~~> \" => Update\n\nsection updates\n\nvariable [CMRA α] [CMRA β] (f : α → β) (g : β → α)", "target_theorem": "theorem UpdateP.iso\n    (gf : ∀ x, g (f x) ≡ x)\n    (g_op : ∀ y1 y2, g (y1 • y2) ≡ g y1 • g y2)\n    (g_validN : ∀ n y, ✓{n} (g y) ↔ ✓{n} y)\n    (uyp : y ~~>: P)\n    (pq : ∀ y', P y' → Q (g y')) :\n    g y ~~>: Q :=", "ground_truth_proof": ":= by\n  intro n mz v\n  have : ✓{n} y •? Option.map f mz :=\n    match mz with\n    | none => (g_validN n _).mp v\n    | some z =>\n      have : g y • z ≡ g (y • f z) :=\n        (CMRA.op_right_eqv _ (gf z).symm).trans (g_op y (f z)).symm\n      (g_validN n _).mp (CMRA.validN_ne this.dist v)\n  have ⟨x, px, vx⟩ := uyp n (mz.map f) this\n  have : g (x •? Option.map f mz) ≡ g x •? mz :=\n    match mz with\n    | none => OFE.Equiv.rfl\n    | some z => (g_op x (f z)).trans (CMRA.op_right_eqv (g x) (gf z))\n  exact ⟨g x, pq x px, CMRA.validN_ne this.dist ((g_validN n _).mpr vx)⟩", "nesting_depth": 4, "transitive_dep_count": 31, "subset_aristotle": false, "category": "Framework"}
{"id": 240, "thm_name": "Iris.OFE.equiv_eqv", "thm_stmt": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/OFE.lean", "imports": ["import src/Iris/Algebra/CMRA.lean", "import src/Iris/Instances/UPred/Instance.lean", "import src/Iris/Algebra/Agree_task.lean", "import src/Iris/Algebra/COFESolver.lean", "import src/Iris/Algebra/Heap.lean"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Dist", "module": "Mathlib.Topology.MetricSpace.Pseudo.Defs"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "", "content": "instance : OFE.Discrete (DFrac F) := ⟨congrArg id⟩"}, {"name": "Agree.dist", "content": "def Agree.dist (n : Nat) (x y : Agree α) : Prop :=\n  (∀ a ∈ x.car, ∃ b ∈ y.car, a ≡{n}≡ b) ∧\n  (∀ b ∈ y.car, ∃ a ∈ x.car, a ≡{n}≡ b)"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "", "content": "instance : OFE (Tower F) where\n  Equiv f g := ∀ k, f k ≡ g k\n  Dist n f g := ∀ k, f k ≡{n}≡ g k\n  dist_eqv := {\n    refl _ _ := dist_eqv.refl _\n    symm h _ := dist_eqv.symm (h _)\n    trans h1 h2 _ := dist_eqv.trans (h1 _) (h2 _)\n  }\n  equiv_dist {_ _} := by admit /- proof elided -/"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "", "content": "instance : OFE.NonExpansive (BUpd.bupd (PROP := UPred M)) := bupd_ne"}, {"name": "bupd_ne", "content": "instance bupd_ne : OFE.NonExpansive (bupd : UPred M → UPred M) where\n  ne n x1 x2 Hx m y Hm Hv := by admit /- proof elided -/"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "later_contractive", "content": "instance later_contractive : OFE.Contractive UPred.later (α := UPred M) where\n  distLater_dist {n x y} Hl :=\n    match n with\n    | 0 => by admit /- proof elided -/\n    | n + 1 => fun\n      | 0 => by admit /- proof elided -/\n      | n' + 1 => fun x' Hn' Hx' => Hl _ Hn' _ _ (Nat.le_refl _) (CMRA.validN_succ Hx')"}, {"name": "later", "content": "protected def later (P : UPred M) : UPred M where\n  holds n x := match n with | 0 => True | Nat.succ n' => P n' x\n  mono {n₁ n₂} := by admit /- proof elided -/"}, {"name": "BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "BIUpdate", "content": "class BIUpdate (PROP : Type _) [BI PROP] extends BUpd PROP where\n  [bupd_ne : OFE.NonExpansive (BUpd.bupd (PROP := PROP))]\n  intro {P : PROP} : iprop(P ⊢ |==> P)\n  mono {P Q : PROP} : iprop(P ⊢ Q) → iprop(|==> P ⊢ |==> Q)\n  trans {P : PROP} : iprop(|==> |==> P ⊢ |==> P)\n  frame_r {P R : PROP} : iprop((|==> P) ∗ R ⊢ |==> (P ∗ R))"}, {"name": "UPred", "content": "@[ext]\nstructure UPred (M : Type _) [UCMRA M] where\n  holds : Nat → M → Prop\n  mono {n1 n2 x1 x2} : holds n1 x1 → x1 ≼{n2} x2 → n2 ≤ n1 → holds n2 x2"}, {"name": "IsModal", "content": "class IsModal [BI PROP1] [BI PROP2] (M : PROP1 → PROP2)\n    (iaction saction : ModalityAction PROP1 PROP2) where\n  spec_intuitionistic : iaction.intuitionistic_action_spec M\n  spec_spatial : saction.spatial_action_spec M\n  emp : iprop(emp) ⊢ M iprop(emp)\n  mono : ∀ {P Q}, (P ⊢ Q) → M P ⊢ M Q\n  sep : ∀ {P Q}, iprop(M P ∗ M Q) ⊢ M iprop(P ∗ Q)"}, {"name": "UCMRA", "content": "class UCMRA (α : Type _) extends CMRA α where\n  unit : α\n  unit_valid : ✓ unit\n  unit_left_id : unit • x ≡ x\n  pcore_unit : pcore unit ≡ some unit"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "instOFE", "content": "instance instOFE [Store T K V] [OFE V] : OFE T where\n  Equiv s0 s1  := get s0 ≡ get s1\n  Dist n s0 s1 := get s0 ≡{n}≡ get s1\n  dist_eqv     := ⟨fun _ => .of_eq rfl, (·.symm), (·.trans ·)⟩\n  equiv_dist   := equiv_dist\n  dist_lt      := dist_lt"}, {"name": "Store.Equiv", "content": "@[simp] def Store.Equiv [Store T K V] (t1 t2 : T) : Prop := get t1 = get t2"}, {"name": "Store.Equiv_trans", "content": "instance Store.Equiv_trans [Store T K V] : Trans Equiv (Equiv (T := T)) Equiv := ⟨by admit /- proof elided -/\n⟩"}, {"name": "[CMRA", "content": "instance [CMRA β] : OFE (α -C> β) where\n  Equiv f g := f.toHom ≡ g.toHom\n  Dist n f g := f.toHom ≡{n}≡ g.toHom\n  dist_eqv := {\n    refl _ := dist_eqv.refl _\n    symm h := dist_eqv.symm h\n    trans h1 h2 := dist_eqv.trans h1 h2\n  }\n  equiv_dist := equiv_dist\n  dist_lt := dist_lt"}, {"name": "[Store", "content": "instance [Store T1 K V1] [Store T2 K V2] [OFE V1] [OFE V2] (f : K → V1 → V2)\n  [∀ k, NonExpansive (f k)] [HasStoreMap T1 T2 K V1 V2] : NonExpansive (dmap f : T1 → T2) where\n  ne _ {_ _} H k := by admit /- proof elided -/"}, {"name": "[Heap", "content": "instance [Heap T K V] [OFE V] (op : V → V → V) [NonExpansive₂ op] :\n    NonExpansive₂ (merge (T := T) op) where\n  ne _ {_ _} Ht {_ _} Hs k := by admit /- proof elided -/"}, {"name": "Heap.instCOFE", "content": "instance Heap.instCOFE [Heap T K V] [COFE V] : COFE T where\n  compl c := hmap (fun _ => COFE.compl <| c.map ⟨_, Store.get_ne ·⟩) (c 0)\n  conv_compl {_ c} k := by admit /- proof elided -/"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Iris.OFE", "content": "class OFE (α : Type _) where\n  Equiv : α → α → Prop\n  Dist : Nat → α → α → Prop\n  dist_eqv : Equivalence (Dist n)\n  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y\n  dist_lt : Dist n x y → m < n → Dist m x y"}], "used_local_lemmas": [{"name": "Iris.OFE.Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "Iris.OFE.Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Iris.OFE.Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}], "local_ctx": "namespace Iris\n\nclass OFE (α : Type _) where\n  Equiv : α → α → Prop\n  Dist : Nat → α → α → Prop\n  dist_eqv : Equivalence (Dist n)\n  equiv_dist : Equiv x y ↔ ∀ n, Dist n x y\n  dist_lt : Dist n x y → m < n → Dist m x y\n\nopen OFE\n\nscoped infix:40 \" ≡ \" => OFE.Equiv\n\nscoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nnamespace OFE", "target_theorem": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv :=", "ground_truth_proof": ":= by\n  constructor\n  · rintro x; rw [ofe.equiv_dist]; rintro n; exact Dist.rfl\n  · rintro x y; simp [ofe.equiv_dist]; rintro h n; exact Dist.symm (h n)\n  · rintro x y z; simp [ofe.equiv_dist]; rintro h₁ h₂ n; exact Dist.trans (h₁ n) (h₂ n)", "nesting_depth": 3, "transitive_dep_count": 14, "subset_aristotle": false, "category": "Framework"}
{"id": 241, "thm_name": "Iris.BI.plainly_sep", "thm_stmt": "theorem plainly_sep [BIPositive PROP] : ■ (P ∗ Q) ⊣⊢ ■ P ∗ ■ Q", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/Plainly.lean", "imports": ["import Iris.Algebra", "import src.Iris.BI.DerivedLaws", "import src/Iris/Instances/UPred/Instance.lean", "import Iris.BI.DerivedLaws", "import Iris.BI.BI", "import src.Iris.BI.BI", "import Iris.BI.Classes"], "used_lib_defs": [{"name": "Lean.MonadQuotation", "module": "Init.Prelude"}, {"name": "Lean.MonadRef", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax term:26 \" ==∗ \" term:25 : term", "content": "syntax term:26 \" ==∗ \" term:25 : term\n\nsyntax term \"={ \" term \" , \" term \" }=∗ \" term : term\n\nsyntax term \"={ \" term \" }=∗ \" term : term\n\nsyntax term \"={ \" term \" }[ \" term \" ]▷=∗ \" term : term\n\nsyntax term \"={ \" term \" }▷=∗ \" term : term\n\nsyntax term \"={ \" term \" }[ \" term \" ]▷^\" term \"=∗ \" term : term\n\nsyntax term \"={ \" term \" }▷^\" term \"=∗ \" term : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term"}, {"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term\n\nsyntax \"■ \" term:40 : term\n\nsyntax:max \"<affine> \" term:40 : term"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|==> $P))  => ``(BUpd.bupd iprop($P))\n  | `(iprop($P ==∗ $Q))  => ``(BIBase.wand iprop($P) (BUpd.bupd iprop($Q)))\n\ndelab_rule BUpd.bupd\n  | `($_ $P) => do ``(iprop(|==> $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 , $E2 }=> $P))  => ``(FUpd.fupd $E1 $E2 iprop($P))\n  | `(iprop($P ={ $E1 , $E2 }=∗ $Q))  => ``(BIBase.wand iprop($P) (FUpd.fupd $E1 $E2 iprop($Q)))\n  | `(iprop(|={ $E1 }=> $P))  => ``(FUpd.fupd $E1 $E1 iprop($P))\n  | `(iprop($P ={ $E1 }=∗ $Q))  => ``(BIBase.wand iprop($P) (FUpd.fupd $E1 $E1 iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 }[ $E2 ]▷=> $P))  => ``(iprop(|={$E1, $E2}=> ▷ (|={ $E2, $E1 }=> iprop($P))))\n  | `(iprop($P ={ $E1 }[ $E2 ]▷=∗ $Q))  => ``(iprop(iprop($P) -∗ |={$E1}[$E2]▷=> iprop($Q)))\n  | `(iprop(|={ $E1 }▷=> $P))  => ``(iprop(|={$E1}[$E1]▷=> iprop($P)))\n  | `(iprop($P ={ $E1 }▷=∗ $Q))  => ``(iprop(iprop($P) ={$E1}[$E1]▷=∗ iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 }[ $E2 ]▷^$n=> $P))  => ``(iprop(|={$E1, $E2}=> ▷^[$n] (|={ $E2, $E1 }=> iprop($P))))\n  | `(iprop($P ={ $E1 }[ $E2 ]▷^$n=∗ $Q))  => ``(iprop(iprop($P) -∗ |={$E1}[$E2]▷^$n=> iprop($Q)))\n  | `(iprop(|={ $E1 }▷^$n=> $P))  => ``(iprop(|={$E1}[$E1]▷^$n=> iprop($P)))\n  | `(iprop($P ={ $E1 }▷^$n=∗ $Q))  => ``(iprop(iprop($P) ={$E1}[$E1]▷^$n=∗ iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(■ $P))  => ``(Plainly.plainly iprop($P))\n\ndelab_rule Plainly.plainly\n  | `($_ $P) => do ``(iprop(■ $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(■? $p $P))  => ``(Plainly.plainlyIf $p iprop($P))\n\ndelab_rule Plainly.plainlyIf\n  | `($_ $p $P) => do ``(iprop(■? $p $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "unpackIprop", "content": "partial def unpackIprop [Monad m] [MonadRef m] [MonadQuotation m] : Term → m Term\n  | `(iprop($P))             => do `($P)\n  | `($P:ident)              => do `($P)\n  | `(?$P:ident)             => do `(?$P)\n  | `(($P))                  => do `(($(← unpackIprop P)))\n  | `($P $[ $Q]*)            => do ``($P $[ $Q]*)\n  | `(if $c then $t else $e) => do\n    let t ← unpackIprop t\n    let e ← unpackIprop e\n    `(if $c then $t else $e)\n  | `(($P : $t))             => do ``(($(← unpackIprop P) : $t))\n  | `($t)                    => `($t:term)"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "BIPositive", "content": "class BIPositive (PROP : Type _) [BI PROP] where\n  affinely_sep_l {P Q : PROP} : <affine> (P ∗ Q) ⊢ <affine> P ∗ Q"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "Idempotent", "content": "class Idempotent (R : Relation α) (f : α → α → α) where\n  idem {x : α} : R (f x x) x"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "", "content": "instance : BIPlainly (UPred M) where\n  mono H _ _ _ := H _ _ CMRA.unit_validN\n  elim_persistently {P} n x Hx := by admit /- proof elided -/"}, {"name": "persistently", "content": "protected def persistently (P : UPred M) : UPred M where\n  holds n x := P n (CMRA.core x)\n  mono H Hx Hn := P.mono H (CMRA.core_incN_core Hx) Hn"}, {"name": "BIUpdate", "content": "class BIUpdate (PROP : Type _) [BI PROP] extends BUpd PROP where\n  [bupd_ne : OFE.NonExpansive (BUpd.bupd (PROP := PROP))]\n  intro {P : PROP} : iprop(P ⊢ |==> P)\n  mono {P Q : PROP} : iprop(P ⊢ Q) → iprop(|==> P ⊢ |==> Q)\n  trans {P : PROP} : iprop(|==> |==> P ⊢ |==> P)\n  frame_r {P R : PROP} : iprop((|==> P) ∗ R ⊢ |==> (P ∗ R))"}, {"name": "UPred", "content": "@[ext]\nstructure UPred (M : Type _) [UCMRA M] where\n  holds : Nat → M → Prop\n  mono {n1 n2 x1 x2} : holds n1 x1 → x1 ≼{n2} x2 → n2 ≤ n1 → holds n2 x2"}, {"name": "IsModal", "content": "class IsModal [BI PROP1] [BI PROP2] (M : PROP1 → PROP2)\n    (iaction saction : ModalityAction PROP1 PROP2) where\n  spec_intuitionistic : iaction.intuitionistic_action_spec M\n  spec_spatial : saction.spatial_action_spec M\n  emp : iprop(emp) ⊢ M iprop(emp)\n  mono : ∀ {P Q}, (P ⊢ Q) → M P ⊢ M Q\n  sep : ∀ {P Q}, iprop(M P ∗ M Q) ⊢ M iprop(P ∗ Q)"}, {"name": "core", "content": "def core (x : α) := (pcore x).getD x"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "UCMRA", "content": "class UCMRA (α : Type _) extends CMRA α where\n  unit : α\n  unit_valid : ✓ unit\n  unit_left_id : unit • x ≡ x\n  pcore_unit : pcore unit ≡ some unit"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "plainly", "content": "protected def plainly (P : UPred M) : UPred M where\n  holds n _ := P n UCMRA.unit\n  mono H _ Hn := P.mono H (CMRA.incN_refl UCMRA.unit) Hn"}, {"name": "BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "", "content": "instance : BIPlainlyExists (UPred M) where\n  plainly_sExists_1 _ _ _ := fun ⟨_, hp⟩ => ⟨_, ⟨_, rfl⟩, hp⟩"}, {"name": "BIPlainlyExists", "content": "class BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p"}, {"name": "", "content": "instance : Plainly (UPred M) := ⟨UPred.plainly⟩"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "persistently_and_sep_assoc", "content": "theorem persistently_and_sep_assoc [BI PROP] {P Q R : PROP} :\n    <pers> P ∧ (Q ∗ R) ⊣⊢ (<pers> P ∧ Q) ∗ R"}, {"name": "and_mono_l", "content": "theorem and_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∧ Q ⊢ P' ∧ Q"}, {"name": "and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}, {"name": "and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "sep_assoc", "content": "theorem sep_assoc [BI PROP] {P Q R : PROP} : (P ∗ Q) ∗ R ⊣⊢ P ∗ Q ∗ R"}, {"name": "sep_congr_l", "content": "theorem sep_congr_l [BI PROP] {P P' Q : PROP} (h : P ⊣⊢ P') : P ∗ Q ⊣⊢ P' ∗ Q"}, {"name": "sep_congr", "content": "@[rw_mono_rule]\ntheorem sep_congr [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊣⊢ Q) (h2 : P' ⊣⊢ Q') :\n    (P ∗ P') ⊣⊢ (Q ∗ Q')"}, {"name": "sep_congr_r", "content": "theorem sep_congr_r [BI PROP] {P Q Q' : PROP} (h : Q ⊣⊢ Q') : P ∗ Q ⊣⊢ P ∗ Q'"}, {"name": "sep_mono_l", "content": "theorem sep_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∗ Q ⊢ P' ∗ Q"}, {"name": "BIBase.Entails.rfl", "content": "@[simp] theorem BIBase.Entails.rfl [BI PROP] {P : PROP} : P ⊢ P"}, {"name": "imp_intro'", "content": "theorem imp_intro' [BI PROP] {P Q R : PROP} (h : Q ∧ P ⊢ R) : P ⊢ Q → R"}, {"name": "and_comm", "content": "theorem and_comm [BI PROP] {P Q : PROP} : P ∧ Q ⊣⊢ Q ∧ P"}, {"name": "and_symm", "content": "theorem and_symm [BI PROP] {P Q : PROP} : P ∧ Q ⊢ Q ∧ P"}, {"name": "imp_elim'", "content": "theorem imp_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P → R) : P ∧ Q ⊢ R"}, {"name": "pure_elim", "content": "theorem pure_elim [BI PROP] (φ : Prop) {Q R : PROP} (h1 : Q ⊢ ⌜φ⌝) (h2 : φ → Q ⊢ R) : Q ⊢ R"}, {"name": "and_self", "content": "theorem and_self [BI PROP] {P : PROP} : P ∧ P ⊣⊢ P"}, {"name": "forall_intro", "content": "theorem forall_intro [BI PROP] {P : PROP} {Ψ : α → PROP} (h : ∀ a, P ⊢ Ψ a) : P ⊢ ∀ a, Ψ a"}, {"name": "forall_elim", "content": "theorem forall_elim [BI PROP] {Ψ : α → PROP} (a : α) : (∀ a, Ψ a) ⊢ Ψ a"}, {"name": "forall_mono", "content": "@[rw_mono_rule]\ntheorem forall_mono [BI PROP] {Φ Ψ : α → PROP} (h : ∀ a, Φ a ⊢ Ψ a) : (∀ a, Φ a) ⊢ ∀ a, Ψ a"}, {"name": "and_forall_bool", "content": "theorem and_forall_bool [BI PROP] {P Q : PROP} :\n    P ∧ Q ⊣⊢ «forall» (fun b : Bool => if b then P else Q)"}, {"name": "persistently_and_emp_elim", "content": "theorem persistently_and_emp_elim {P : PROP} [BI PROP] : emp ∧ <pers> P ⊢ P"}, {"name": "sep_emp", "content": "theorem sep_emp [BI PROP] {P : PROP} : P ∗ emp ⊣⊢ P"}, {"name": "affinely_elim_emp", "content": "theorem affinely_elim_emp [BI PROP] {P : PROP} : <affine> P ⊢ emp"}, {"name": "affinely_sep", "content": "theorem affinely_sep [BI PROP] [BIPositive PROP] {P Q : PROP} :\n    <affine> (P ∗ Q) ⊣⊢ <affine> P ∗ <affine> Q"}, {"name": "affinely_idem", "content": "theorem affinely_idem [BI PROP] {P : PROP} : <affine> <affine> P ⊣⊢ <affine> P"}, {"name": "and_assoc", "content": "theorem and_assoc [BI PROP] {P Q R : PROP} : (P ∧ Q) ∧ R ⊣⊢ P ∧ Q ∧ R"}, {"name": "and_mono_r", "content": "theorem and_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∧ Q ⊢ P ∧ Q'"}, {"name": "and_congr_l", "content": "theorem and_congr_l [BI PROP] {P P' Q : PROP} (h : P ⊣⊢ P') : P ∧ Q ⊣⊢ P' ∧ Q"}, {"name": "and_congr", "content": "@[rw_mono_rule]\ntheorem and_congr [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊣⊢ Q) (h2 : P' ⊣⊢ Q') : P ∧ P' ⊣⊢ Q ∧ Q'"}, {"name": "affinely_mono", "content": "@[rw_mono_rule]\ntheorem affinely_mono [BI PROP] {P Q : PROP} : (P ⊢ Q) → <affine> P ⊢ <affine> Q"}, {"name": "affinely_sep_2", "content": "theorem affinely_sep_2 [BI PROP] {P Q : PROP} : <affine> P ∗ <affine> Q ⊢ <affine> (P ∗ Q)"}, {"name": "affinely_elim", "content": "theorem affinely_elim [BI PROP] {P : PROP} : <affine> P ⊢ P"}, {"name": "affinely_sep_r", "content": "theorem affinely_sep_r [BI PROP] [BIPositive PROP] {P Q : PROP} :\n    <affine> (P ∗ Q) ⊢ P ∗ <affine> Q"}], "used_local_defs": [{"name": "Iris.Plainly", "content": "class Plainly (PROP : Type _) where\n  plainly : PROP → PROP"}, {"name": "Iris.Plainly.plainlyIf", "content": "def Plainly.plainlyIf [Iris.BI.BIBase PROP] [Plainly PROP] (p : Bool) (P : PROP) : PROP :=\n  iprop(if p then ■ P else P)"}, {"name": "Iris.BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "Iris.BIPlainlyExists", "content": "class BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p"}], "used_local_lemmas": [{"name": "Iris.BI.persistently_elim_plainly", "content": "theorem persistently_elim_plainly : <pers> ■ P ⊣⊢ ■ P"}, {"name": "Iris.BI.plainly_forall_2", "content": "theorem plainly_forall_2 {Ψ : α → PROP} : (∀ a, ■ (Ψ a)) ⊢ ■ (∀ a, Ψ a)"}, {"name": "Iris.BI.plainly_and_sep_assoc", "content": "theorem plainly_and_sep_assoc : ■ P ∧ (Q ∗ R) ⊣⊢ (■ P ∧ Q) ∗ R"}, {"name": "Iris.BI.plainly_and_emp_elim", "content": "theorem plainly_and_emp_elim : emp ∧ ■ P ⊢ P"}, {"name": "Iris.BI.plainly_forall", "content": "theorem plainly_forall {Ψ : α → PROP} : ■ (∀ a, Ψ a) ⊣⊢ ∀ a, ■ (Ψ a)"}, {"name": "Iris.BI.plainly_and", "content": "theorem plainly_and : ■ (P ∧ Q) ⊣⊢ ■ P ∧ ■ Q"}, {"name": "Iris.BI.plainly_and_sep_l_1", "content": "theorem plainly_and_sep_l_1 : ■ P ∧ Q ⊢ ■ P ∗ Q"}, {"name": "Iris.BI.plainly_and_sep", "content": "theorem plainly_and_sep : ■ (P ∧ Q) ⊢ ■ (P ∗ Q)"}, {"name": "Iris.BI.and_sep_plainly", "content": "theorem and_sep_plainly : ■ P ∧ ■ Q ⊣⊢ ■ P ∗ ■ Q"}, {"name": "Iris.BI.plainly_sep_2", "content": "theorem plainly_sep_2 : ■ P ∗ ■ Q ⊢ ■ (P ∗ Q)"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.BI\n\nimport Iris.BI.DerivedLaws\n\nimport Iris.Algebra\n\nnamespace Iris\n\nopen BI\n\nclass Plainly (PROP : Type _) where\n  plainly : PROP → PROP\n\ndef Plainly.plainlyIf [Iris.BI.BIBase PROP] [Plainly PROP] (p : Bool) (P : PROP) : PROP :=\n  iprop(if p then ■ P else P)\n\nclass BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P\n\nclass BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p\n\nnamespace BI\n\nopen Iris.Std\n\nsection PlainlyLaws\n\nopen BIPlainly\n\nvariable [BI PROP] [BIPlainly PROP]\n\nvariable {P Q R : PROP}", "target_theorem": "theorem plainly_sep [BIPositive PROP] : ■ (P ∗ Q) ⊣⊢ ■ P ∗ ■ Q :=", "ground_truth_proof": ":= by\n  refine ⟨?_, plainly_sep_2⟩\n  refine plainly_affinely_elim.mpr.trans ?_\n  refine (mono <| affinely_sep.mp).trans ?_\n  refine .trans ?_ and_sep_plainly.mp\n  refine and_intro (mono ?_) (mono ?_)\n  · exact ((sep_mono .rfl affinely_elim_emp).trans sep_emp.mp).trans affinely_elim\n  · exact ((sep_mono affinely_elim_emp .rfl).trans emp_sep.mp).trans affinely_elim", "nesting_depth": 7, "transitive_dep_count": 72, "subset_aristotle": false, "category": "Framework"}
{"id": 242, "thm_name": "Iris.Agree.toAgree_uninj", "thm_stmt": "theorem Agree.toAgree_uninj {x : Agree α} : ✓ x → ∃ a, toAgree a ≡ x", "lean_root": "iris-lean", "rel_path": "src/Iris/Algebra/Agree.lean", "imports": ["import Iris.Algebra.CMRA", "import src.Iris.Algebra.CMRA", "import src.Iris.Algebra.OFE", "import Iris.Algebra.OFE"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "m", "module": "QqTest.matching"}], "used_repo_defs": [{"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x", "content": "notation:50 \"✓{\" n \"} \" x:51 => ValidN n x"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "valid", "content": "def valid : DFrac F → Prop\n  | .own f        => Proper f\n  | .discard      => True\n  | .ownDiscard f => Fractional f"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "Fractional", "content": "def Fractional [Fraction α] (a : α) : Prop := ∃ b, Proper (a + b)"}, {"name": "Fraction", "content": "class Fraction (α : Type _) extends Add α where\n   \n  Proper : α → Prop\n  add_comm : ∀ a b : α, a + b = b + a\n  add_assoc : ∀ a b c : α, a + (b + c) = (a + b) + c\n  add_left_cancel : ∀ {a b c : α}, a + b = a + c → b = c\n   \n  add_ne : ∀ {a b : α}, a ≠ b + a\n  proper_add_mono_left : ∀ {a b : α}, Proper (a + b) → Proper a"}, {"name": "Agree", "content": "@[ext]\nstructure Agree where\n  car : List α\n  not_nil : car ≠ []"}, {"name": "LeibnizO", "content": "@[ext] structure LeibnizO (α : Type _) where\n  car : α"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "Iso.symm", "content": "def Iso.symm [OFE α] [OFE β] (iso : Iso α β) : Iso β α where\n  hom := iso.inv\n  inv := iso.hom\n  hom_inv := by admit /- proof elided -/"}, {"name": "Iso", "content": "@[ext] structure Iso (α β : Type _) [OFE α] [OFE β] where\n  hom : α -n> β\n  inv : β -n> α\n  hom_inv : hom (inv x) ≡ x\n  inv_hom : inv (hom x) ≡ x"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "Hom", "content": "@[ext] structure Hom (α β : Type _) [CMRA α] [CMRA β] extends OFE.Hom α β where\n  protected validN {n x} : ✓{n} x → ✓{n} (f x)\n  protected pcore x : (pcore x).map f ≡ pcore (f x)\n  protected op x y : f (x • y) ≡ f x • f y"}, {"name": "Hom", "content": "@[ext] structure Hom (α β : Type _) [OFE α] [OFE β] where\n  f : α → β\n  ne : NonExpansive f"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -n> \" => Hom"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}, {"name": "@[inherit_doc]", "content": "@[inherit_doc]\ninfixr:25 \" -C> \" => Hom"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "_root_.Iris.OFE.Dist.validN", "content": "theorem _root_.Iris.OFE.Dist.validN : (x : α) ≡{n}≡ y → (✓{n} x ↔ ✓{n} y)"}, {"name": "validN_iff", "content": "theorem validN_iff {x y : α} (e : x ≡{n}≡ y) : ✓{n} x ↔ ✓{n} y"}, {"name": "IncludedN.validN", "content": "theorem IncludedN.validN {n} {x y : α} : x ≼{n} y → ✓{n} y → ✓{n} x"}, {"name": "validN_of_incN", "content": "theorem validN_of_incN {n} {x y : α} : x ≼{n} y → ✓{n} y → ✓{n} x"}, {"name": "Included.validN", "content": "theorem Included.validN {n} {x y : α} : x ≼ y → ✓{n} y → ✓{n} x"}, {"name": "validN_of_inc", "content": "theorem validN_of_inc {n} {x y : α} : x ≼ y → ✓{n} y → ✓{n} x"}, {"name": "Valid.validN", "content": "theorem Valid.validN : ✓ (x : α) → ✓{n} x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "_root_.Iris.OFE.Equiv.valid", "content": "theorem _root_.Iris.OFE.Equiv.valid : (x : α) ≡ y → (✓ x ↔ ✓ y)"}, {"name": "valid_iff", "content": "theorem valid_iff {x y : α} (e : x ≡ y) : ✓ x ↔ ✓ y"}, {"name": "valid_of_eqv", "content": "theorem valid_of_eqv {x y : α} : x ≡ y → ✓ x → ✓ y"}, {"name": "valid_mapN", "content": "theorem valid_mapN {x y : α} (f : ∀ n, ✓{n} x → ✓{n} y) (v : ✓ x) : ✓ y"}, {"name": "validN_of_eqv", "content": "theorem validN_of_eqv {x y : α} : x ≡ y → ✓{n} x → ✓{n} y"}, {"name": "Hom.valid", "content": "protected theorem Hom.valid [CMRA β] (f : α -C> β) {x : α} (H : ✓ x) : ✓ f x"}], "used_local_defs": [{"name": "Iris.Agree", "content": "@[ext]\nstructure Agree where\n  car : List α\n  not_nil : car ≠ []"}, {"name": "Iris.toAgree", "content": "def toAgree (a : α) : Agree α := ⟨[a], by admit /- proof elided -/\n⟩"}, {"name": "Iris.Agree.dist", "content": "def Agree.dist (n : Nat) (x y : Agree α) : Prop :=\n  (∀ a ∈ x.car, ∃ b ∈ y.car, a ≡{n}≡ b) ∧\n  (∀ b ∈ y.car, ∃ a ∈ x.car, a ≡{n}≡ b)"}, {"name": "Iris.Agree.validN", "content": "def Agree.validN (n : Nat) (x : Agree α) : Prop :=\n  match x.car with\n  | [_] => True\n  | _ => ∀ a ∈ x.car, ∀ b ∈ x.car, a ≡{n}≡ b"}, {"name": "Iris.Agree.valid", "content": "def Agree.valid (x : Agree α) : Prop := ∀ n, x.validN n"}], "used_local_lemmas": [{"name": "Iris.mem_of_agree", "content": "theorem mem_of_agree (x : Agree α) : ∃ a, a ∈ x.car"}, {"name": "Iris.Agree.equiv_def", "content": "theorem Agree.equiv_def {x y : Agree α} : x ≡ y ↔ ∀ n, Agree.dist n x y"}, {"name": "Iris.Agree.validN_iff", "content": "theorem Agree.validN_iff {x : Agree α} :\n    x.validN n ↔ ∀ a ∈ x.car, ∀ b ∈ x.car, a ≡{n}≡ b"}, {"name": "Iris.Agree.valid_def", "content": "theorem Agree.valid_def {x : Agree α} : ✓ x ↔ x.valid"}], "local_ctx": "import Iris.Algebra.CMRA\n\nimport Iris.Algebra.OFE\n\nnamespace Iris\n\nsection agree\n\nvariable {α : Type u}\n\nvariable (α) in\n\n@[ext]\nstructure Agree where\n  car : List α\n  not_nil : car ≠ []\n\ndef toAgree (a : α) : Agree α := ⟨[a], by admit /- proof elided -/\n⟩\n\nvariable [OFE α]\n\ndef Agree.dist (n : Nat) (x y : Agree α) : Prop :=\n  (∀ a ∈ x.car, ∃ b ∈ y.car, a ≡{n}≡ b) ∧\n  (∀ b ∈ y.car, ∃ a ∈ x.car, a ≡{n}≡ b)\n\ndef Agree.validN (n : Nat) (x : Agree α) : Prop :=\n  match x.car with\n  | [_] => True\n  | _ => ∀ a ∈ x.car, ∀ b ∈ x.car, a ≡{n}≡ b\n\ndef Agree.valid (x : Agree α) : Prop := ∀ n, x.validN n", "target_theorem": "theorem Agree.toAgree_uninj {x : Agree α} : ✓ x → ∃ a, toAgree a ≡ x :=", "ground_truth_proof": ":= by\n  simp only [valid_def, valid, validN_iff, equiv_def]\n  obtain ⟨a, ha⟩ := mem_of_agree x\n  intro h; exists a; intro n\n  constructor <;> intros\n  · exists a; simp_all [toAgree]\n  · simp_all [toAgree]", "nesting_depth": 7, "transitive_dep_count": 53, "subset_aristotle": false, "category": "Framework"}
{"id": 243, "thm_name": "Iris.BI.plainly_sep_dup", "thm_stmt": "theorem plainly_sep_dup : ■ P ⊣⊢ ■ P ∗ ■ P", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/Plainly.lean", "imports": ["import Iris.Algebra", "import Iris.BI.DerivedLaws", "import src.Iris.Algebra.CMRA", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import src.Iris.BI.DerivedLaws", "import src/Iris/Instances/UPred/Instance.lean", "import Iris.BI.BI", "import Iris.BI.Classes"], "used_lib_defs": [{"name": "Lean.MonadQuotation", "module": "Init.Prelude"}, {"name": "Lean.MonadRef", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax term:26 \" ==∗ \" term:25 : term", "content": "syntax term:26 \" ==∗ \" term:25 : term\n\nsyntax term \"={ \" term \" , \" term \" }=∗ \" term : term\n\nsyntax term \"={ \" term \" }=∗ \" term : term\n\nsyntax term \"={ \" term \" }[ \" term \" ]▷=∗ \" term : term\n\nsyntax term \"={ \" term \" }▷=∗ \" term : term\n\nsyntax term \"={ \" term \" }[ \" term \" ]▷^\" term \"=∗ \" term : term\n\nsyntax term \"={ \" term \" }▷^\" term \"=∗ \" term : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term"}, {"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term\n\nsyntax \"■ \" term:40 : term"}, {"name": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y", "content": "notation:50 x \" ≼{\" n \"} \" y:51 => IncludedN n x y\n\nsyntax:max \"<affine> \" term:40 : term\n\nsyntax:max \"□ \" term:40 : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|==> $P))  => ``(BUpd.bupd iprop($P))\n  | `(iprop($P ==∗ $Q))  => ``(BIBase.wand iprop($P) (BUpd.bupd iprop($Q)))\n\ndelab_rule BUpd.bupd\n  | `($_ $P) => do ``(iprop(|==> $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 , $E2 }=> $P))  => ``(FUpd.fupd $E1 $E2 iprop($P))\n  | `(iprop($P ={ $E1 , $E2 }=∗ $Q))  => ``(BIBase.wand iprop($P) (FUpd.fupd $E1 $E2 iprop($Q)))\n  | `(iprop(|={ $E1 }=> $P))  => ``(FUpd.fupd $E1 $E1 iprop($P))\n  | `(iprop($P ={ $E1 }=∗ $Q))  => ``(BIBase.wand iprop($P) (FUpd.fupd $E1 $E1 iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 }[ $E2 ]▷=> $P))  => ``(iprop(|={$E1, $E2}=> ▷ (|={ $E2, $E1 }=> iprop($P))))\n  | `(iprop($P ={ $E1 }[ $E2 ]▷=∗ $Q))  => ``(iprop(iprop($P) -∗ |={$E1}[$E2]▷=> iprop($Q)))\n  | `(iprop(|={ $E1 }▷=> $P))  => ``(iprop(|={$E1}[$E1]▷=> iprop($P)))\n  | `(iprop($P ={ $E1 }▷=∗ $Q))  => ``(iprop(iprop($P) ={$E1}[$E1]▷=∗ iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(|={ $E1 }[ $E2 ]▷^$n=> $P))  => ``(iprop(|={$E1, $E2}=> ▷^[$n] (|={ $E2, $E1 }=> iprop($P))))\n  | `(iprop($P ={ $E1 }[ $E2 ]▷^$n=∗ $Q))  => ``(iprop(iprop($P) -∗ |={$E1}[$E2]▷^$n=> iprop($Q)))\n  | `(iprop(|={ $E1 }▷^$n=> $P))  => ``(iprop(|={$E1}[$E1]▷^$n=> iprop($P)))\n  | `(iprop($P ={ $E1 }▷^$n=∗ $Q))  => ``(iprop(iprop($P) ={$E1}[$E1]▷^$n=∗ iprop($Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(■ $P))  => ``(Plainly.plainly iprop($P))\n\ndelab_rule Plainly.plainly\n  | `($_ $P) => do ``(iprop(■ $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(■? $p $P))  => ``(Plainly.plainlyIf $p iprop($P))\n\ndelab_rule Plainly.plainlyIf\n  | `($_ $p $P) => do ``(iprop(■? $p $(← Iris.BI.unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "unpackIprop", "content": "partial def unpackIprop [Monad m] [MonadRef m] [MonadQuotation m] : Term → m Term\n  | `(iprop($P))             => do `($P)\n  | `($P:ident)              => do `($P)\n  | `(?$P:ident)             => do `(?$P)\n  | `(($P))                  => do `(($(← unpackIprop P)))\n  | `($P $[ $Q]*)            => do ``($P $[ $Q]*)\n  | `(if $c then $t else $e) => do\n    let t ← unpackIprop t\n    let e ← unpackIprop e\n    `(if $c then $t else $e)\n  | `(($P : $t))             => do ``(($(← unpackIprop P) : $t))\n  | `($t)                    => `($t:term)"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/\n\n  and_ne : OFE.NonExpansive₂ and\n  or_ne : OFE.NonExpansive₂ or\n  imp_ne : OFE.NonExpansive₂ imp\n  sForall_ne {P₁ P₂} : liftRel (· ≡{n}≡ ·) P₁ P₂ → sForall P₁ ≡{n}≡ sForall P₂\n  sExists_ne {P₁ P₂} : liftRel (· ≡{n}≡ ·) P₁ P₂ → sExists P₁ ≡{n}≡ sExists P₂\n  sep_ne : OFE.NonExpansive₂ sep\n  wand_ne : OFE.NonExpansive₂ wand\n  persistently_ne : OFE.NonExpansive persistently\n  later_ne : OFE.NonExpansive later\n\n  pure_intro {φ : Prop} {P : PROP} : φ → P ⊢ ⌜φ⌝\n  pure_elim' {φ : Prop} {P : PROP} : (φ → True ⊢ P) → ⌜φ⌝ ⊢ P\n\n  and_elim_l {P Q : PROP} : P ∧ Q ⊢ P\n  and_elim_r {P Q : PROP} : P ∧ Q ⊢ Q\n  and_intro {P Q R : PROP} : (P ⊢ Q) → (P ⊢ R) → P ⊢ Q ∧ R\n\n  or_intro_l {P Q : PROP} : P ⊢ P ∨ Q\n  or_intro_r {P Q : PROP} : Q ⊢ P ∨ Q\n  or_elim {P Q R : PROP} : (P ⊢ R) → (Q ⊢ R) → P ∨ Q ⊢ R\n\n  imp_intro {P Q R : PROP} : (P ∧ Q ⊢ R) → P ⊢ Q → R\n  imp_elim {P Q R : PROP} : (P ⊢ Q → R) → P ∧ Q ⊢ R\n\n  sForall_intro {P : PROP} {Ψ : PROP → Prop} : (∀ p, Ψ p → P ⊢ p) → P ⊢ sForall Ψ\n  sForall_elim {Ψ : PROP → Prop} {p : PROP} : Ψ p → sForall Ψ ⊢ p\n\n  sExists_intro {Ψ : PROP → Prop} {p : PROP} : Ψ p → p ⊢ sExists Ψ\n  sExists_elim {Φ : PROP → Prop} {Q : PROP} : (∀ p, Φ p → p ⊢ Q) → sExists Φ ⊢ Q\n\n  sep_mono {P P' Q Q' : PROP} : (P ⊢ Q) → (P' ⊢ Q') → P ∗ P' ⊢ Q ∗ Q'\n  emp_sep {P : PROP} : emp ∗ P ⊣⊢ P\n  sep_symm {P Q : PROP} : P ∗ Q ⊢ Q ∗ P\n  sep_assoc_l {P Q R : PROP} : (P ∗ Q) ∗ R ⊢ P ∗ (Q ∗ R)\n\n  wand_intro {P Q R : PROP} : (P ∗ Q ⊢ R) → P ⊢ Q -∗ R\n  wand_elim {P Q R : PROP} : (P ⊢ Q -∗ R) → P ∗ Q ⊢ R\n\n  persistently_mono {P Q : PROP} : (P ⊢ Q) → <pers> P ⊢ <pers> Q\n  persistently_idem_2 {P : PROP} : <pers> P ⊢ <pers> <pers> P\n  persistently_emp_2 : (emp : PROP) ⊢ <pers> emp\n  persistently_and_2 {P Q : PROP} : (<pers> P) ∧ (<pers> Q) ⊢ <pers> (P ∧ Q)\n  persistently_sExists_1 {Ψ : PROP → Prop} : <pers> (sExists Ψ) ⊢ ∃ p, ⌜Ψ p⌝ ∧ <pers> p\n  persistently_absorb_l {P Q : PROP} : <pers> P ∗ Q ⊢ <pers> P\n  persistently_and_l {P Q : PROP} : <pers> P ∧ Q ⊢ P ∗ Q\n\n  later_mono {P Q : PROP} : (P ⊢ Q) → ▷ P ⊢ ▷ Q\n  later_intro {P : PROP} : P ⊢ ▷ P\n\n  later_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ▷ p) ⊢ ▷ sForall Φ\n  later_sExists_false {Φ : PROP → Prop} : (▷ sExists Φ) ⊢ ▷ False ∨ ∃ p, ⌜Φ p⌝ ∧ ▷ p\n  later_sep {P Q : PROP} : ▷ (P ∗ Q) ⊣⊢ ▷ P ∗ ▷ Q\n  later_persistently {P : PROP} : ▷ <pers> P ⊣⊢ <pers> ▷ P\n  later_false_em {P : PROP} : ▷ P ⊢ ▷ False ∨ (▷ False → P)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "Included", "content": "def Included (x y : α) : Prop := ∃ z, y ≡ x • z"}, {"name": "CMRA", "content": "class CMRA (α : Type _) extends OFE α where\n  pcore : α → Option α\n  op : α → α → α\n  ValidN : Nat → α → Prop\n  Valid : α → Prop\n\n  op_ne : NonExpansive (op x)\n  pcore_ne : x ≡{n}≡ y → pcore x = some cx →\n    ∃ cy, pcore y = some cy ∧ cx ≡{n}≡ cy\n  validN_ne : x ≡{n}≡ y → ValidN n x → ValidN n y\n\n  valid_iff_validN : Valid x ↔ ∀ n, ValidN n x\n  validN_succ : ValidN n.succ x → ValidN n x\n  validN_op_left : ValidN n (op x y) → ValidN n x\n\n  assoc : op x (op y z) ≡ op (op x y) z\n  comm : op x y ≡ op y x\n\n  pcore_op_left : pcore x = some cx → op cx x ≡ x\n  pcore_idem : pcore x = some cx → pcore cx ≡ some cx\n  pcore_op_mono : pcore x = some cx → ∀ y, ∃ cy, pcore (op x y) ≡ some (op cx cy)\n\n  extend : ValidN n x → x ≡{n}≡ op y₁ y₂ →\n    Σ' z₁ z₂, x ≡ op z₁ z₂ ∧ z₁ ≡{n}≡ y₁ ∧ z₂ ≡{n}≡ y₂"}, {"name": "Commutative", "content": "class Commutative (R : Relation α) (f : β → β → α) where\n  comm {x y : β} : R (f x y) (f y x)"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "Absorbing", "content": "class Absorbing [BI PROP] (P : PROP) where\n  absorbing : <absorb> P ⊢ P"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "", "content": "instance : BIPlainly (UPred M) where\n  mono H _ _ _ := H _ _ CMRA.unit_validN\n  elim_persistently {P} n x Hx := by admit /- proof elided -/"}, {"name": "persistently", "content": "protected def persistently (P : UPred M) : UPred M where\n  holds n x := P n (CMRA.core x)\n  mono H Hx Hn := P.mono H (CMRA.core_incN_core Hx) Hn"}, {"name": "BIUpdate", "content": "class BIUpdate (PROP : Type _) [BI PROP] extends BUpd PROP where\n  [bupd_ne : OFE.NonExpansive (BUpd.bupd (PROP := PROP))]\n  intro {P : PROP} : iprop(P ⊢ |==> P)\n  mono {P Q : PROP} : iprop(P ⊢ Q) → iprop(|==> P ⊢ |==> Q)\n  trans {P : PROP} : iprop(|==> |==> P ⊢ |==> P)\n  frame_r {P R : PROP} : iprop((|==> P) ∗ R ⊢ |==> (P ∗ R))"}, {"name": "UPred", "content": "@[ext]\nstructure UPred (M : Type _) [UCMRA M] where\n  holds : Nat → M → Prop\n  mono {n1 n2 x1 x2} : holds n1 x1 → x1 ≼{n2} x2 → n2 ≤ n1 → holds n2 x2"}, {"name": "IsModal", "content": "class IsModal [BI PROP1] [BI PROP2] (M : PROP1 → PROP2)\n    (iaction saction : ModalityAction PROP1 PROP2) where\n  spec_intuitionistic : iaction.intuitionistic_action_spec M\n  spec_spatial : saction.spatial_action_spec M\n  emp : iprop(emp) ⊢ M iprop(emp)\n  mono : ∀ {P Q}, (P ⊢ Q) → M P ⊢ M Q\n  sep : ∀ {P Q}, iprop(M P ∗ M Q) ⊢ M iprop(P ∗ Q)"}, {"name": "core", "content": "def core (x : α) := (pcore x).getD x"}, {"name": "UCMRA", "content": "class UCMRA (α : Type _) extends CMRA α where\n  unit : α\n  unit_valid : ✓ unit\n  unit_left_id : unit • x ≡ x\n  pcore_unit : pcore unit ≡ some unit"}, {"name": "pcore", "content": "def pcore : DFrac F → Option (DFrac F)\n  | own _        => none\n  | .discard     => some discard\n  | ownDiscard _ => some discard"}, {"name": "DFrac", "content": "inductive DFrac (F : Type _) where\n \n| own (f : F) : DFrac F\n \n| discard : DFrac F\n \n| ownDiscard (f : F) : DFrac F"}, {"name": "op", "content": "def op : DFrac F → DFrac F → DFrac F\n  | .discard, .discard => discard\n  | own f, .discard\n  | ownDiscard f, .discard\n  | .discard, own f\n  | .discard, ownDiscard f => ownDiscard f\n  | own f, own f' => own (f + f')\n  | own f, ownDiscard f'\n  | ownDiscard f, own f'\n  | ownDiscard f, ownDiscard f' => ownDiscard (f + f')"}, {"name": "Associative", "content": "class Associative (R : Relation α) (f : α → α → α) where\n  assoc {x y z : α} : R (f (f x y) z) (f x (f y z))"}, {"name": "plainly", "content": "protected def plainly (P : UPred M) : UPred M where\n  holds n _ := P n UCMRA.unit\n  mono H _ Hn := P.mono H (CMRA.incN_refl UCMRA.unit) Hn"}, {"name": "BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "Idempotent", "content": "class Idempotent (R : Relation α) (f : α → α → α) where\n  idem {x : α} : R (f x x) x"}, {"name": "", "content": "instance : BIPlainlyExists (UPred M) where\n  plainly_sExists_1 _ _ _ := fun ⟨_, hp⟩ => ⟨_, ⟨_, rfl⟩, hp⟩"}, {"name": "BIPlainlyExists", "content": "class BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p"}, {"name": "", "content": "instance : Plainly (UPred M) := ⟨UPred.plainly⟩"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}, {"name": "infix:60 \" • \" => op", "content": "infix:60 \" • \" => op"}, {"name": "infix:50 \" ≼ \" => Included", "content": "infix:50 \" ≼ \" => Included"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "persistently_and_sep_assoc", "content": "theorem persistently_and_sep_assoc [BI PROP] {P Q R : PROP} :\n    <pers> P ∧ (Q ∗ R) ⊣⊢ (<pers> P ∧ Q) ∗ R"}, {"name": "and_mono_l", "content": "theorem and_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∧ Q ⊢ P' ∧ Q"}, {"name": "and_mono", "content": "@[rw_mono_rule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∧ P' ⊢ Q ∧ Q'"}, {"name": "and_elim_r'", "content": "theorem and_elim_r' [BI PROP] {P Q R : PROP} (h : Q ⊢ R) : P ∧ Q ⊢ R"}, {"name": "and_elim_l'", "content": "theorem and_elim_l' [BI PROP] {P Q R : PROP} (h : P ⊢ R) : P ∧ Q ⊢ R"}, {"name": "sep_assoc", "content": "theorem sep_assoc [BI PROP] {P Q R : PROP} : (P ∗ Q) ∗ R ⊣⊢ P ∗ Q ∗ R"}, {"name": "sep_congr_l", "content": "theorem sep_congr_l [BI PROP] {P P' Q : PROP} (h : P ⊣⊢ P') : P ∗ Q ⊣⊢ P' ∗ Q"}, {"name": "sep_congr", "content": "@[rw_mono_rule]\ntheorem sep_congr [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊣⊢ Q) (h2 : P' ⊣⊢ Q') :\n    (P ∗ P') ⊣⊢ (Q ∗ Q')"}, {"name": "sep_congr_r", "content": "theorem sep_congr_r [BI PROP] {P Q Q' : PROP} (h : Q ⊣⊢ Q') : P ∗ Q ⊣⊢ P ∗ Q'"}, {"name": "sep_mono_l", "content": "theorem sep_mono_l [BI PROP] {P P' Q : PROP} (h : P ⊢ P') : P ∗ Q ⊢ P' ∗ Q"}, {"name": "BIBase.Entails.rfl", "content": "@[simp] theorem BIBase.Entails.rfl [BI PROP] {P : PROP} : P ⊢ P"}, {"name": "Included.trans", "content": "theorem Included.trans : (x : α) ≼ y → y ≼ z → x ≼ z"}, {"name": "inc_trans", "content": "theorem inc_trans {x y z : α} : x ≼ y → y ≼ z → x ≼ z"}, {"name": "op_left_eqv", "content": "theorem op_left_eqv {x y : α} (z : α) (e : x ≡ y) : x • z ≡ y • z"}, {"name": "_root_.Iris.OFE.Dist.op_r", "content": "theorem _root_.Iris.OFE.Dist.op_r {x y z : α} : y ≡{n}≡ z → x • y ≡{n}≡ x • z"}, {"name": "op_right_dist", "content": "theorem op_right_dist (x : α) {y z : α} (e : y ≡{n}≡ z) : x • y ≡{n}≡ x • z"}, {"name": "_root_.Iris.OFE.Equiv.op_r", "content": "theorem _root_.Iris.OFE.Equiv.op_r {x y z : α} : y ≡ z → x • y ≡ x • z"}, {"name": "op_right_eqv", "content": "theorem op_right_eqv (x : α) {y z : α} (e : y ≡ z) : x • y ≡ x • z"}, {"name": "IncludedN.trans", "content": "theorem IncludedN.trans : (x : α) ≼{n} y → y ≼{n} z → x ≼{n} z"}, {"name": "incN_trans", "content": "theorem incN_trans {x y z : α} : x ≼{n} y → y ≼{n} z → x ≼{n} z"}, {"name": "op_left_dist", "content": "theorem op_left_dist {x y : α} (z : α) (e : x ≡{n}≡ y) : x • z ≡{n}≡ y • z"}, {"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}, {"name": "and_self", "content": "theorem and_self [BI PROP] {P : PROP} : P ∧ P ⊣⊢ P"}, {"name": "emp_sep", "content": "theorem emp_sep [BI PROP] {P : PROP} : emp ∗ P ⊣⊢ P"}, {"name": "sep_mono_r", "content": "theorem sep_mono_r [BI PROP] {P Q Q' : PROP} (h : Q ⊢ Q') : P ∗ Q ⊢ P ∗ Q'"}, {"name": "sep_mono", "content": "@[rw_mono_rule]\ntheorem sep_mono [BI PROP] {P P' Q Q' : PROP} (h1 : P ⊢ Q) (h2 : P' ⊢ Q') : P ∗ P' ⊢ Q ∗ Q'"}, {"name": "sep_comm", "content": "theorem sep_comm [BI PROP] {P Q : PROP} : P ∗ Q ⊣⊢ Q ∗ P"}, {"name": "sep_elim_r", "content": "theorem sep_elim_r [BI PROP] {P Q : PROP} [TCOr (Affine P) (Absorbing Q)] : P ∗ Q ⊢ Q"}, {"name": "absorbingly_of_persistently", "content": "theorem absorbingly_of_persistently [BI PROP] {P : PROP} : <pers> P ⊢ <absorb> P"}, {"name": "persistently_elim", "content": "theorem persistently_elim [BI PROP] {P : PROP} [Absorbing P] : <pers> P ⊢ P"}, {"name": "persistently_entails_r", "content": "theorem persistently_entails_r [BI PROP] {P Q : PROP} (h : P ⊢ <pers> Q) : P ⊢ P ∗ <pers> Q"}], "used_local_defs": [{"name": "Iris.Plainly", "content": "class Plainly (PROP : Type _) where\n  plainly : PROP → PROP"}, {"name": "Iris.Plainly.plainlyIf", "content": "def Plainly.plainlyIf [Iris.BI.BIBase PROP] [Plainly PROP] (p : Bool) (P : PROP) : PROP :=\n  iprop(if p then ■ P else P)"}, {"name": "Iris.BIPlainly", "content": "class BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P"}, {"name": "Iris.BIPlainlyExists", "content": "class BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p"}], "used_local_lemmas": [{"name": "Iris.BI.persistently_elim_plainly", "content": "theorem persistently_elim_plainly : <pers> ■ P ⊣⊢ ■ P"}, {"name": "Iris.BI.plainly_and_sep_assoc", "content": "theorem plainly_and_sep_assoc : ■ P ∧ (Q ∗ R) ⊣⊢ (■ P ∧ Q) ∗ R"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.BI\n\nimport Iris.BI.DerivedLaws\n\nimport Iris.Algebra\n\nnamespace Iris\n\nopen BI\n\nclass Plainly (PROP : Type _) where\n  plainly : PROP → PROP\n\ndef Plainly.plainlyIf [Iris.BI.BIBase PROP] [Plainly PROP] (p : Bool) (P : PROP) : PROP :=\n  iprop(if p then ■ P else P)\n\nclass BIPlainly (PROP : Type _) [Iris.BI PROP] extends Plainly PROP where\n  [ne : Iris.OFE.NonExpansive (Plainly.plainly (PROP := PROP))]\n  mono {P Q : PROP} : (P ⊢ Q) → ■ P ⊢ ■ Q\n  elim_persistently {P : PROP} : ■ P ⊢ <pers> P\n  idem {P : PROP} : ■ P ⊢ ■ ■ P\n  plainly_sForall_2 {Φ : PROP → Prop} : (∀ p, ⌜Φ p⌝ → ■ p) ⊢ ■ sForall Φ\n  plainly_impl_plainly {P Q : PROP} : (■ P → ■ Q) ⊢ ■ (■ P → Q)\n  emp_intro {P : PROP} : P ⊢ ■ emp\n  plainly_absorb {P Q : PROP} : ■ P ∗ Q ⊢ ■ P\n  later_plainly {P : PROP} : ▷ ■ P ⊣⊢ ■ ▷ P\n\nclass BIPlainlyExists (PROP : Type _) [Iris.BI PROP] [BIPlainly PROP] where\n  plainly_sExists_1 {Φ : PROP → Prop} : ■ sExists Φ ⊢ ∃ p, ⌜Φ p⌝ ∧ ■ p\n\nnamespace BI\n\nopen Iris.Std\n\nsection PlainlyLaws\n\nopen BIPlainly\n\nvariable [BI PROP] [BIPlainly PROP]\n\nvariable {P Q R : PROP}", "target_theorem": "theorem plainly_sep_dup : ■ P ⊣⊢ ■ P ∗ ■ P :=", "ground_truth_proof": ":= by\n  refine ⟨?_, plainly_absorb⟩\n  refine and_self.2.trans ?_\n  refine ((and_mono .rfl emp_sep.2).trans plainly_and_sep_assoc.1).trans ?_\n  exact (sep_mono and_elim_l .rfl).trans .rfl", "nesting_depth": 6, "transitive_dep_count": 72, "subset_aristotle": false, "category": "Framework"}
{"id": 244, "thm_name": "Iris.BI.later_and", "thm_stmt": "theorem later_and [BI PROP] {P Q : PROP} : ▷ (P ∧ Q) ⊣⊢ ▷ P ∧ ▷ Q", "lean_root": "iris-lean", "rel_path": "src/Iris/BI/DerivedLaws.lean", "imports": ["import Iris.BI.Extensions", "import Iris.Std.TC", "import Iris.Std.Classes", "import Iris.BI.BI", "import src.Iris.Algebra.OFE", "import src.Iris.BI.BI", "import Iris.BI.Classes", "import Iris.Std.Rewrite"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Equivalence", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "m", "module": "QqTest.matching"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do", "content": "macro \"∃\" xs:explicitBinders \", \" b:term : term => do\n  return ⟨← expandExplicitBinders ``BIBase.exists xs b⟩"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y\n\nsyntax:max \"<pers> \" term:40 : term"}, {"name": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entai", "content": "macro:25 P:term:29 \" ⊢ \" Q:term:25 : term => ``(BIBase.Entails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊢ $(← unpackIprop Q))"}, {"name": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails i", "content": "macro:25 P:term:29 \" ⊣⊢ \" Q:term:29 : term => ``(BiEntails iprop($P) iprop($Q))\n\ndelab_rule BIBase.Entails\n  | `($_ iprop(emp) $P) => do ``(⊢ $(← unpackIprop P))\n\ndelab_rule BIBase.BiEntails\n  | `($_ $P $Q) => do ``($(← unpackIprop P) ⊣⊢ $(← unpackIprop Q))\n\nsyntax \"⌜\" term \"⌝\" : term\n\nsyntax:max \"▷ \" term:40 : term"}, {"name": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((ipro", "content": "macro:max \"iprop(\" P:term \" : \" t:term \")\" : term => `((iprop($P) : $t))\n\nsyntax:max \"iprop(\" term \")\" : term\n\nsyntax:max \"<affine> \" term:40 : term\n\nsyntax:max \"□ \" term:40 : term\n\nsyntax:max \"<absorb> \" term:40 : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(term($t))) => pure t\n  | `(iprop($t))       => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(($P)))                  => ``((iprop($P)))\n  | `(iprop(if $c then $t else $e)) => ``(if $c then iprop($t) else iprop($e))\n  | `(iprop(($P : $t)))             => ``((iprop($P) : $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(emp))       => ``(BIBase.emp)\n  | `(iprop(⌜$φ⌝))      => ``(BIBase.pure $φ)\n  | `(iprop($P ∧ $Q))   => ``(BIBase.and iprop($P) iprop($Q))\n  | `(iprop($P ∨ $Q))   => ``(BIBase.or iprop($P) iprop($Q))\n  | `(iprop($P → $Q))   => ``(BIBase.imp iprop($P) iprop($Q))\n  | `(iprop(∃ $xs, $Ψ)) => do expandExplicitBinders ``BIBase.exists xs (← ``(iprop($Ψ)))\n  | `(iprop($P ∗ $Q))   => ``(BIBase.sep iprop($P) iprop($Q))\n  | `(iprop($P -∗ $Q))  => ``(BIBase.wand iprop($P) iprop($Q))\n  | `(iprop(<pers> $P)) => ``(BIBase.persistently iprop($P))\n  | `(iprop(▷ $P))      => ``(BIBase.later iprop($P))\n\ndelab_rule BIBase.emp\n  | `($_) => ``(iprop($(mkIdent `emp)))\ndelab_rule BIBase.pure\n  | `($_ $φ) => ``(iprop(⌜$φ⌝))\ndelab_rule BIBase.and\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∧ $(← unpackIprop Q)))\ndelab_rule BIBase.or\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∨ $(← unpackIprop Q)))\ndelab_rule BIBase.imp\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) → $(← unpackIprop Q)))\ndelab_rule BIBase.forall\n  | `($_ fun $x:ident => iprop(∀ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∀ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∀ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.exists\n  | `($_ fun $x:ident => iprop(∃ $y:ident $[$z:ident]*, $Ψ)) => do\n    ``(iprop(∃ $x:ident $y:ident $[$z:ident]*, $Ψ))\n  | `($_ fun $x:ident => $Ψ) => do ``(iprop(∃ $x:ident, $(← unpackIprop Ψ)))\ndelab_rule BIBase.sep\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗ $(← unpackIprop Q)))\ndelab_rule BIBase.wand\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) -∗ $(← unpackIprop Q)))\ndelab_rule BIBase.persistently\n  | `($_ $P) => do ``(iprop(<pers> $(← unpackIprop P)))\n\ndelab_rule BIBase.pure\n  | `($_ True) => ``(iprop($(mkIdent `True)))\n  | `($_ False) => ``(iprop($(mkIdent `False)))\ndelab_rule BIBase.imp\n  | `($_ $P iprop(False)) => do ``(iprop(¬$(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ _%$tk, $Ψ)) => ``(BIBase.forall (fun _%$tk => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x:ident, $Ψ)) => ``(BIBase.forall (fun $x => iprop($Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ (_%$tk : $t), $Ψ)) => ``(BIBase.forall (fun (_%$tk : $t) => iprop($Ψ)))\n  | `(iprop(∀ (_%$tk $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun (_%$tk : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ ($x:ident : $t), $Ψ)) => ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ ($x:ident $xs* : $t), $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ ($xs* : $t), $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {_%$tk : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop($Ψ)))\n  | `(iprop(∀ {_%$tk $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun {_%$tk : $t}  => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ {$x:ident : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop($Ψ)))\n  | `(iprop(∀ {$x:ident $xs* : $t}, $Ψ)) =>\n    ``(BIBase.forall (fun ($x : $t) => iprop(∀ {$xs* : $t}, $Ψ)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(∀ $x $y $xs*, $Ψ)) => ``(iprop(∀ $x, ∀ $y $xs*, $Ψ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(True))  => ``(BIBase.pure True)\n  | `(iprop(False)) => ``(BIBase.pure False)\n  | `(iprop(¬$P))   => ``(iprop($P → False))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop($P ↔ $Q))   => ``(iff iprop($P) iprop($Q))\n  | `(iprop($P ∗-∗ $Q)) => ``(wandIff iprop($P) iprop($Q))\n\ndelab_rule iff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ↔ $(← unpackIprop Q)))\ndelab_rule wandIff\n  | `($_ $P $Q) => do ``(iprop($(← unpackIprop P) ∗-∗ $(← unpackIprop Q)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<affine> $P)) => ``(affinely iprop($P))\n  | `(iprop(<absorb> $P)) => ``(absorbingly iprop($P))\n\ndelab_rule affinely\n  | `($_ $P) => do ``(iprop(<affine> $(← unpackIprop P)))\ndelab_rule absorbingly\n  | `($_ $P) => do ``(iprop(<absorb> $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(□ $P)) => ``(intuitionistically iprop($P))\n\ndelab_rule intuitionistically\n  | `($_ $P) => do ``(iprop(□ $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(<pers>?$p $P))   => ``(persistentlyIf $p iprop($P))\n  | `(iprop(<affine>?$p $P)) => ``(affinelyIf $p iprop($P))\n  | `(iprop(<absorb>?$p $P)) => ``(absorbinglyIf $p iprop($P))\n  | `(iprop(□?$p $P))        => ``(intuitionisticallyIf $p iprop($P))\n\ndelab_rule persistentlyIf\n  | `($_ $p $P) => do ``(iprop(<pers>?$p $(← unpackIprop P)))\ndelab_rule affinelyIf\n  | `($_ $p $P) => do ``(iprop(<affine>?$p $(← unpackIprop P)))\ndelab_rule absorbinglyIf\n  | `($_ $p $P) => do ``(iprop(<absorb>?$p $(← unpackIprop P)))\ndelab_rule intuitionisticallyIf\n  | `($_ $p $P) => do ``(iprop(□?$p $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(▷^[$n] $P))   => ``(laterN $n iprop($P))\n\ndelab_rule laterN\n  | `($_ $n $P) => do ``(iprop(▷^[$n] $(← unpackIprop P)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(iprop(◇ $P)) => ``(except0 iprop($P))\n\ndelab_rule except0\n  | `($_ $P) => do ``(iprop(◇ $(← unpackIprop P)))"}, {"name": "BIBase", "content": "class BIBase (PROP : Type u) where\n  Entails : PROP → PROP → Prop\n  emp : PROP\n  pure : Prop → PROP\n  and : PROP → PROP → PROP\n  or : PROP → PROP → PROP\n  imp : PROP → PROP → PROP\n  sForall : (PROP → Prop) → PROP\n  sExists : (PROP → Prop) → PROP\n  sep : PROP → PROP → PROP\n  wand : PROP → PROP → PROP\n  persistently : PROP → PROP\n  later : PROP → PROP"}, {"name": "BI", "content": "class BI (PROP : Type _) extends COFE PROP, BI.BIBase PROP where\n  Equiv P Q := P ⊣⊢ Q\n\n  entails_preorder : Preorder Entails\n  equiv_iff {P Q : PROP} : (P ≡ Q) ↔ P ⊣⊢ Q := by admit /- proof elided -/"}, {"name": "liftRel", "content": "def liftRel (R : α → β → Prop) (A : α → Prop) (B : β → Prop) : Prop :=\n  (∀ a, A a → ∃ b, B b ∧ R a b) ∧ (∀ b, B b → ∃ a, A a ∧ R a b)"}, {"name": "Preorder", "content": "class Preorder (R : Relation α) extends Reflexive R, Transitive R"}, {"name": "Reflexive", "content": "class Reflexive (R : Relation α) where\n  refl {x : α} : R x x"}, {"name": "Relation", "content": "abbrev Relation (α : Type _) := α → α → Prop"}, {"name": "Transitive", "content": "class Transitive (R : Relation α) where\n  trans {x y z : α} : R x y → R y z → R x z"}, {"name": "NonExpansive₂", "content": "class NonExpansive₂ [OFE α] [OFE β] [OFE γ] (f : α → β → γ) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → ∀ ⦃y₁ y₂⦄, y₁ ≡{n}≡ y₂ → f x₁ y₁ ≡{n}≡ f x₂ y₂"}, {"name": "OFE.ContractiveHom.fixpoint_ne", "content": "instance OFE.ContractiveHom.fixpoint_ne [COFE α] [Inhabited α] :\n    NonExpansive (ContractiveHom.fixpoint (α := α)) where\n  ne n f1 f2 H := by admit /- proof elided -/"}, {"name": "ContractiveHom", "content": "@[ext] structure ContractiveHom (α β : Type _) [OFE α] [OFE β] extends Hom α β where\n  [contractive : Contractive f]\n  ne := ne_of_contractive f"}, {"name": "OFE.Option.some.ne", "content": "instance OFE.Option.some.ne [OFE α] : OFE.NonExpansive (some : α → Option α) := ⟨fun _ _ _ => id⟩"}, {"name": "Option.merge_ne", "content": "instance Option.merge_ne [OFE α] {op : α → α → α} [NonExpansive₂ op] :\n    NonExpansive₂ (Option.merge op) where\n  ne n x1 x2 Hx y1 y2 Hy := by admit /- proof elided -/"}, {"name": "NonExpansive", "content": "class NonExpansive [OFE α] [OFE β] (f : α → β) where\n  ne : ∀ ⦃n x₁ x₂⦄, x₁ ≡{n}≡ x₂ → f x₁ ≡{n}≡ f x₂"}, {"name": "COFE.OFunctor.constOF_RFunctor", "content": "instance COFE.OFunctor.constOF_RFunctor [CMRA B] : RFunctor (constOF B) where\n  map f g := by admit /- proof elided -/"}, {"name": "IsCOFE", "content": "class IsCOFE (α : Type _) [OFE α] where\n  compl : Chain α → α\n  conv_compl {c : Chain α} : compl c ≡{n}≡ c n"}, {"name": "Chain", "content": "structure Chain (α : Type _) [OFE α] where\n  chain : Nat → α\n  cauchy : n ≤ i → chain i ≡{n}≡ chain n"}, {"name": "LawfulBigOp", "content": "class LawfulBigOp (f : PROP → PROP → PROP) (unit : outParam PROP)\n    (eq : outParam (PROP → PROP → Prop)) where\n  refl : eq a a\n  symm : eq a b → eq b a\n  trans : eq a b → eq b c → eq a c\n  comm : eq (f a b) (f b a)\n  assoc : eq (f (f a b) c) (f a (f b c))\n  left_id : eq (f unit a) a\n  congr_l : eq a a' → eq (f a b) (f a' b)"}, {"name": "intuitionistically", "content": "def intuitionistically [BIBase PROP] (P : PROP) : PROP := iprop(<affine> <pers> P)"}, {"name": "Affine", "content": "class Affine [BI PROP] (P : PROP) where\n  affine : P ⊢ emp"}, {"name": "bigAnd", "content": "def bigAnd [BIBase PROP] (Ps : List PROP) : PROP := bigOp and iprop(True) Ps"}, {"name": "bigOp", "content": "def bigOp (f : PROP → PROP → PROP) (unit : PROP) : List PROP → PROP\n  | []      => unit\n  | [P]     => P\n  | P :: Ps => f P (bigOp f unit Ps)"}, {"name": "absorbingly", "content": "def absorbingly [BIBase PROP] (P : PROP) : PROP := iprop(True ∗ P)"}, {"name": "BiEntails", "content": "structure BiEntails [BIBase PROP] (P Q : PROP) where\n  mp : P ⊢ Q\n  mpr : Q ⊢ P"}, {"name": "persistentlyIf", "content": "def persistentlyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <pers> P else P)"}, {"name": "intuitionisticallyIf", "content": "def intuitionisticallyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then □ P else P)"}, {"name": "bigSep", "content": "def bigSep [BIBase PROP] (Ps : List PROP) : PROP := bigOp sep iprop(emp) Ps"}, {"name": "affinely", "content": "def affinely    [BIBase PROP] (P : PROP) : PROP := iprop(emp ∧ P)"}, {"name": "bigOr", "content": "def bigOr [BIBase PROP] (Ps : List PROP) : PROP := bigOp or iprop(False) Ps"}, {"name": "absorbinglyIf", "content": "def absorbinglyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <absorb> P else P)"}, {"name": "affinelyIf", "content": "def affinelyIf [BIBase PROP] (p : Bool) (P : PROP) : PROP := iprop(if p then <affine> P else P)"}, {"name": "wandIff", "content": "def wandIff [BIBase PROP] (P Q : PROP) : PROP := iprop((P -∗ Q) ∧ (Q -∗ P))"}, {"name": "scoped infix:40 \" ≡ \" => OFE.Equiv", "content": "scoped infix:40 \" ≡ \" => OFE.Equiv"}, {"name": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y", "content": "scoped notation:40 x \" ≡{\" n \"}≡ \" y:41 => OFE.Dist n x y"}, {"name": "infixr:25 \" -c> \" => ContractiveHom", "content": "infixr:25 \" -c> \" => ContractiveHom"}, {"name": "notation:40 \"[∧] \" Ps:max => bigAnd Ps", "content": "notation:40 \"[∧] \" Ps:max => bigAnd Ps"}, {"name": "notation:40 \"[∨] \" Ps:max => bigOr Ps", "content": "notation:40 \"[∨] \" Ps:max => bigOr Ps"}, {"name": "notation:40 \"[∗] \" Ps:max => bigSep Ps", "content": "notation:40 \"[∗] \" Ps:max => bigSep Ps"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Dist.trans", "content": "theorem Dist.trans [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ z → x ≡{n}≡ z"}, {"name": "BIBase.BiEntails.trans", "content": "theorem BIBase.BiEntails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊣⊢ Q) (h2 : Q ⊣⊢ R) : P ⊣⊢ R"}, {"name": "BIBase.Entails.trans", "content": "theorem BIBase.Entails.trans [BI PROP] {P Q R : PROP} (h1 : P ⊢ Q) (h2 : Q ⊢ R) : P ⊢ R"}, {"name": "Equiv.trans", "content": "theorem Equiv.trans [OFE α] {x : α} : x ≡ y → y ≡ z → x ≡ z"}, {"name": "equiv_eqv", "content": "theorem equiv_eqv [ofe : OFE α] : Equivalence ofe.Equiv"}, {"name": "Dist.symm", "content": "@[symm] theorem Dist.symm [OFE α] {n} {x : α} : x ≡{n}≡ y → y ≡{n}≡ x"}, {"name": "Dist.rfl", "content": "@[simp, refl] theorem Dist.rfl [OFE α] {n} {x : α} : x ≡{n}≡ x"}, {"name": "DistLater.trans", "content": "theorem DistLater.trans [OFE α] {n} {x : α} (h1 : DistLater n x y) (h2 : DistLater n y z) :\n    DistLater n x z"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Iris.BI.and_symm", "content": "theorem and_symm [BI PROP] {P Q : PROP} : P ∧ Q ⊢ Q ∧ P"}, {"name": "Iris.BI.imp_intro'", "content": "theorem imp_intro' [BI PROP] {P Q R : PROP} (h : Q ∧ P ⊢ R) : P ⊢ Q → R"}, {"name": "Iris.BI.imp_elim'", "content": "theorem imp_elim' [BI PROP] {P Q R : PROP} (h : Q ⊢ P → R) : P ∧ Q ⊢ R"}, {"name": "Iris.BI.forall_intro", "content": "theorem forall_intro [BI PROP] {P : PROP} {Ψ : α → PROP} (h : ∀ a, P ⊢ Ψ a) : P ⊢ ∀ a, Ψ a"}, {"name": "Iris.BI.forall_elim", "content": "theorem forall_elim [BI PROP] {Ψ : α → PROP} (a : α) : (∀ a, Ψ a) ⊢ Ψ a"}, {"name": "Iris.BI.forall_mono", "content": "@[rw_mono_rule]\ntheorem forall_mono [BI PROP] {Φ Ψ : α → PROP} (h : ∀ a, Φ a ⊢ Ψ a) : (∀ a, Φ a) ⊢ ∀ a, Ψ a"}, {"name": "Iris.BI.and_self", "content": "theorem and_self [BI PROP] {P : PROP} : P ∧ P ⊣⊢ P"}, {"name": "Iris.BI.and_comm", "content": "theorem and_comm [BI PROP] {P Q : PROP} : P ∧ Q ⊣⊢ Q ∧ P"}, {"name": "Iris.BI.pure_elim", "content": "theorem pure_elim [BI PROP] (φ : Prop) {Q R : PROP} (h1 : Q ⊢ ⌜φ⌝) (h2 : φ → Q ⊢ R) : Q ⊢ R"}, {"name": "Iris.BI.and_forall_bool", "content": "theorem and_forall_bool [BI PROP] {P Q : PROP} :\n    P ∧ Q ⊣⊢ «forall» (fun b : Bool => if b then P else Q)"}, {"name": "Iris.BI.later_forall_2", "content": "theorem later_forall_2 [BI PROP] {α} {Φ : α → PROP} : (∀ a, ▷ Φ a) ⊢ ▷ ∀ a, Φ a"}, {"name": "Iris.BI.later_forall", "content": "theorem later_forall [BI PROP] {Φ : α → PROP} :\n    ▷ (∀ a, Φ a) ⊣⊢ (∀ a, ▷ Φ a)"}], "local_ctx": "import Iris.BI.Classes\n\nimport Iris.BI.Extensions\n\nimport Iris.BI.BI\n\nimport Iris.Std.Classes\n\nimport Iris.Std.Rewrite\n\nimport Iris.Std.TC\n\nnamespace Iris.BI\n\nopen Iris.Std BI", "target_theorem": "theorem later_and [BI PROP] {P Q : PROP} : ▷ (P ∧ Q) ⊣⊢ ▷ P ∧ ▷ Q :=", "ground_truth_proof": ":= by\n  constructor\n  · refine (later_mono and_forall_bool.mp).trans ?_\n    refine .trans ?_ and_forall_bool.mpr\n    refine (later_forall).mp.trans (forall_mono ?_)\n    exact (·.casesOn .rfl .rfl)\n  · refine .trans ?_ (later_mono and_forall_bool.mpr)\n    refine and_forall_bool.mp.trans ?_\n    refine .trans (forall_mono ?_) later_forall.mpr\n    exact (·.casesOn .rfl .rfl)", "nesting_depth": 6, "transitive_dep_count": 59, "subset_aristotle": false, "category": "Framework"}
{"id": 245, "thm_name": "Juvix.Core.Main.Value.Approx.Indexed.preserved_step", "thm_stmt": "lemma Value.Approx.Indexed.preserved_step {k} :\n  (∀ k' < k, Preservation k') → Preservation k", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx/Indexed.lean", "imports": ["import Juvix.Utils", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Mathlib.Tactic.Linarith", "import Mathlib.Data.List.Forall2", "import Juvix.Core.Main.Semantics.Eval", "import Aesop"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "List.Forall₂.cons", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.nil", "module": "Batteries.Data.List.Basic"}], "used_repo_defs": [{"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\nsyntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))\n\nsyntax \"let \" ident \" := \" expr \" in \" expr : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}, {"name": "Expr.Approx.Param", "content": "def Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)"}, {"name": "Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "Eval.Indexed", "content": "inductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "Context", "content": "inductive Context : Type where\n  | hole : Context\n  | app_left : Context → Expr → Context\n  | app_right : Expr → Context → Context\n  | constr_app_left : Context → Expr → Context\n  | constr_app_right : Expr → Context → Context\n  | binop_left : (oper : BinaryOp) → (arg₁ : Context) → (arg₂ : Expr) → Context\n  | binop_right : (oper : BinaryOp) → (arg₁ : Expr) → (arg₂ : Context) → Context\n  | lambda : (var_name : String) → (body : Context) → Context\n  | save_left : (var_name : String) → (value : Context) → (body : Expr) → Context\n  | save_right : (var_name : String) → (value : Expr) → (body : Context) → Context\n  | branch_left : (constr : Name) → (var_names : List Name) → (body : Context) → (next : Expr) → Context\n  | branch_right : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Context) → Context\n  | recur : (var_name : Name) → (ctx : Context) → Context\n  deriving Inhabited, BEq"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "List.Forall₂.get", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.Forall₂.length_eq", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.getElem?_eq_some_iff", "module": "Init.Data.List.Lemmas"}, {"name": "List.get_eq_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "forall_true_left", "module": "Mathlib.Logic.Basic"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "exists_and_left", "module": "Init.PropLemmas"}, {"name": "exists_const", "module": "Init.PropLemmas"}, {"name": "exists_eq_left'", "module": "Init.PropLemmas"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}, {"name": "Juvix.Core.Main.Expr.Approx.Param.Indexed", "content": "def Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Juvix.Core.Main.Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}, {"name": "Juvix.Core.Main.Expr.Approx.Indexed'", "content": "def Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.Preservation", "content": "def Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Env.Approx.Indexed'.get", "content": "lemma Env.Approx.Indexed'.get {n i : Nat} {env env' o₁}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some o₁) :\n  ∃ o₂, env'[i]? = some o₂ ∧ o₁ ≲ₒ'(n) o₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.value", "content": "lemma Env.Approx.Indexed'.value {n i : Nat} {env env' v}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some (Object.value v)) :\n  ∃ v', env'[i]? = some (Object.value v') ∧ v ≲ᵥ(n) v'"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.delayed", "content": "lemma Env.Approx.Indexed'.delayed {n i : Nat} {env₁ env₂ env e}\n  (h₁ : env₁ ≲ₑ'(n) env₂)\n  (h₂ : env₁[i]? = some (Object.delayed env e)) :\n  (∃ env' e', e ≲(n)⟨env, env'⟩ e' ∧ env₂[i]? = some (Object.delayed env' e')) ∨\n  ∃ env', env ≲ₑ'(n) env' ∧ env₂[i]? = some (Object.delayed env' e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.from_value", "content": "lemma Env.Approx.Indexed'.from_value {n l₁ l₂} (h : l₁ ≲ₐ(n) l₂) :\n  List.map Object.value l₁ ≲ₑ'(n) List.map Object.value l₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.concat", "content": "lemma Env.Approx.Indexed'.concat {n env₁ env₂ env₁' env₂'}\n  (h₁ : env₁ ≲ₑ'(n) env₁')\n  (h₂ : env₂ ≲ₑ'(n) env₂') :\n  env₁ ++ env₂ ≲ₑ'(n) env₁' ++ env₂'"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.cons", "content": "lemma Env.Approx.Indexed'.cons {n o₁ o₂ env₁ env₂}\n  (h₁ : o₁ ≲ₒ'(n) o₂)\n  (h₂ : env₁ ≲ₑ'(n) env₂) :\n  o₁ :: env₁ ≲ₑ'(n) o₂ :: env₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.anti_monotone", "content": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂"}, {"name": "Juvix.Core.Main.Expr.Approx.Param.Indexed.anti_monotone", "content": "lemma Expr.Approx.Param.Indexed.anti_monotone {n n' env₁ env₂ e₁ e₂}\n  (h : e₁ ≲(n)⟨env₁, env₂⟩ e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲(n')⟨env₁, env₂⟩ e₂"}, {"name": "Juvix.Core.Main.Expr.Approx.Indexed'.anti_monotone", "content": "lemma Expr.Approx.Indexed'.anti_monotone {n n' e₁ e₂}\n  (h : e₁ ≲'(n) e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲'(n') e₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.anti_monotone", "content": "  lemma Env.Approx.Indexed'.anti_monotone {n n' env₁ env₂}\n    (h : env₁ ≲ₑ'(n) env₂)\n    (h' : n' ≤ n)\n    : env₁ ≲ₑ'(n') env₂"}, {"name": "Juvix.Core.Main.Object.Approx.Indexed'.anti_monotone", "content": "  lemma Object.Approx.Indexed'.anti_monotone {n n' o₁ o₂} (h : o₁ ≲ₒ'(n) o₂) (h' : n' ≤ n) : o₁ ≲ₒ'(n') o₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.Preservation.anti_monotone", "content": "lemma Value.Approx.Indexed.Preservation.anti_monotone {k k'} (h : Value.Approx.Indexed.Preservation k) (h' : k' ≤ k) : Value.Approx.Indexed.Preservation k'"}], "local_ctx": "import Juvix.Core.Main.Semantics.Eval\n\nimport Juvix.Core.Main.Semantics.Eval.Indexed\n\nimport Juvix.Utils\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Data.List.Forall2\n\nimport Aesop\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))\n\nnotation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nnotation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\ndef Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)\n\nnotation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\ninductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)\n\ndef Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)\n\nnotation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'\n\nnotation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\ndef Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)\n\nnotation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'\n\n@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)\n\nend\n\ndef Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'", "target_theorem": "lemma Value.Approx.Indexed.preserved_step {k} :\n  (∀ k' < k, Preservation k') → Preservation k :=", "ground_truth_proof": ":= by\n  intro ihk\n  intro m n env env' e v h₀ h₁ h₂\n  induction h₂ generalizing m env'\n  case var env i v h =>\n    obtain ⟨v', hget, happrox⟩ := Env.Approx.Indexed'.value h₁ h\n    exists v'\n    constructor\n    · apply Eval.var\n      aesop\n    · apply Value.Approx.Indexed.anti_monotone\n      · assumption\n      · linarith\n  case var_rec n' env₁ name i env₁' e₁' v' hget heval ih =>\n    have hget_or :\n        (∃ env₂' e₂', e₁' ≲(m + n')⟨env₁', env₂'⟩ e₂' ∧\n          env'[i]? = some (Object.delayed env₂' e₂')) ∨\n        (∃ env₂', env₁' ≲ₑ'(m + n') env₂' ∧\n          env'[i]? = some (Object.delayed env₂' e₁')) := by\n      apply Env.Approx.Indexed'.delayed h₁; assumption\n    cases hget_or\n    case inl hget' =>\n      obtain ⟨env₂', e₂', h₂, hget'⟩ := hget'\n      simp [Expr.Approx.Param.Indexed] at h₂\n      obtain ⟨v'', heval', happrox'⟩ := h₂ n' m v' (by linarith) heval\n      exists v''\n      constructor\n      · apply Eval.var_rec\n        · exact hget'\n        · assumption\n      · assumption\n    case inr hget' =>\n      obtain ⟨env₂, happrox, hget'⟩ := hget'\n      obtain ⟨v'', heval', happrox'⟩ := ih m env₂ (by linarith) happrox\n      exists v''\n      constructor\n      · apply Eval.var_rec\n        · exact hget'\n        · assumption\n      · assumption\n  case unit =>\n    exists Value.unit\n    constructor\n    · apply Eval.unit\n    · apply Value.Approx.Indexed.unit\n  case const c =>\n    exists Value.const c\n    constructor\n    · apply Eval.const\n    · apply Value.Approx.Indexed.const\n  case constr ctr_name =>\n    exists Value.constr_app ctr_name []\n    constructor\n    · apply Eval.constr\n    · apply Value.Approx.Indexed.constr_app\n      intro k hk\n      simp\n  case app n n₁ n₂ env env₀ f body arg val val' hn hcl h₂ h₃ ih_f ih_arg ih_body =>\n    have : env ≲ₑ'(m + n₂ + 1 + n₁) env' := by\n      apply Env.Approx.Indexed'.anti_monotone\n      · assumption\n      · linarith\n    obtain ⟨v_f, heval_f, happrox_f⟩ := ih_f (m + n₂ + 1) env' (by linarith) this\n    clear this\n    have : env ≲ₑ'(m + n₂ + n₁ + 1) env' := by\n      apply Env.Approx.Indexed'.anti_monotone\n      · assumption\n      · linarith\n    obtain ⟨v_arg, heval_arg, happrox_arg⟩ := ih_arg (m + n₂) env' (by nlinarith) this\n    clear this\n    invert happrox_f\n    case closure cenv cbody ch =>\n      have happrox_arg' : val ≲ᵥ(n₂ + m) v_arg := by\n        have : m + n₂ = n₂ + m := by linarith\n        rw [this] at happrox_arg\n        exact happrox_arg\n      obtain ⟨v_body, heval_body, happrox_body⟩ := ch n₂ m (by linarith) val v_arg val' happrox_arg' h₃\n      exists v_body\n      constructor\n      · apply Eval.app\n        · exact heval_f\n        · exact heval_arg\n        · exact heval_body\n      · assumption\n  case constr_app n₁ n₂ env₁ e₁ cname args arg v' hlt _ h₂ h₃ h₄ =>\n    obtain ⟨v₁, heval_e₁, happrox_e₁⟩ := h₃ m env' (by linarith) h₁\n    have : env₁ ≲ₑ'(m + n₂) env' := by\n      apply Env.Approx.Indexed'.anti_monotone\n      · assumption\n      · linarith\n    obtain ⟨v₂, heval_arg, happrox_arg⟩ := h₄ m env' (by linarith) this\n    invert happrox_e₁\n    case constr_app args_rev hargs =>\n      exists (Value.constr_app cname (v₂ :: args_rev))\n      constructor\n      · apply Eval.constr_app\n        · exact heval_e₁\n        · exact heval_arg\n      · apply Value.Approx.Indexed.constr_app\n        intro k hk\n        constructor\n        · apply Value.Approx.Indexed.anti_monotone\n          · exact happrox_arg\n          · linarith\n        · apply hargs\n          linarith\n  case binop n' env₁ op arg₁ arg₂ i₁ i₂ v₁ v₂ hv₁ hv₂ =>\n    obtain ⟨v₁', heval₁, happrox₁⟩ := hv₁ m env' (by linarith) h₁\n    obtain ⟨v₂', heval₂, happrox₂⟩ := hv₂ m env' (by linarith) h₁\n    invert happrox₁\n    invert happrox₂\n    exists (Value.const (Constant.int (eval_binop_int op i₁ i₂)))\n    constructor\n    · apply Eval.binop\n      · exact heval₁\n      · exact heval₂\n    · apply Value.Approx.Indexed.const\n  case lambda n' env₁ name body =>\n    exists (Value.closure env' body)\n    constructor\n    · apply Eval.lambda\n    · apply Value.Approx.Indexed.closure\n      intro n₁ n₂ hlt a₁ a₂ v₁ happrox heval\n      have happrox_env : a₁ ∷ env₁ ≲ₑ'(n₁ + n₂) a₂ ∷ env' := by\n        constructor\n        · constructor\n          exact happrox\n        · apply Env.Approx.Indexed'.anti_monotone (n := m + n')\n          · assumption\n          · linarith\n      apply ihk (n₁ + n₂ + 1) (by linarith) (m := n₂) (n := n₁)\n      · linarith\n      · have : n₁ + n₂ = n₂ + n₁ := by linarith\n        rw [<- this]\n        exact happrox_env\n      · exact heval\n  case save n' n₁ n₂ env₁ name e₁ e₂ val val' hle ha₁ ha₂ ih₁ ih₂ =>\n    have : env₁ ≲ₑ'(m + n₂ + n₁) env' := by\n      apply Env.Approx.Indexed'.anti_monotone\n      · assumption\n      · linarith\n    obtain ⟨v₁, heval₁, happrox₁⟩ := ih₁ (m + n₂) env' (by linarith) this\n    clear this\n    have : val ∷ env₁ ≲ₑ'(m + n₂) v₁ ∷ env' := by\n      constructor\n      · apply Object.Approx.Indexed'.value\n        · apply Value.Approx.Indexed.anti_monotone\n          · exact happrox₁\n          · linarith\n      apply Env.Approx.Indexed'.anti_monotone\n      · assumption\n      · linarith\n    obtain ⟨v₂, heval₂, happrox₂⟩ := ih₂ m (v₁ ∷ env') (by linarith) this\n    exists v₂\n    constructor\n    · apply Eval.save\n      · exact heval₁\n      · exact heval₂\n    · assumption\n  case branch_matches e' n₁ n₁' env₁ name args_rev body val hlt heval_body ih =>\n    rcases env' with\n      ⟨⟩ | ⟨⟨hd⟩, env'⟩\n    case nil =>\n      cases h₁\n    case cons.value =>\n      cases h₁\n      case cons h₁ h₂ =>\n        cases h₁\n        case value h₁ =>\n          invert h₁\n          case constr_app args_rev' happrox =>\n            let args := List.map Object.value args_rev\n            let args' := List.map Object.value args_rev'\n            have : args ++ env₁ ≲ₑ'(m + n₁') args' ++ env' := by\n              apply Env.Approx.Indexed'.concat\n              · apply Env.Approx.Indexed'.from_value\n                apply happrox\n                linarith\n              · apply Env.Approx.Indexed'.anti_monotone\n                · assumption\n                · linarith\n            obtain ⟨v', heval', happrox'⟩ := ih m (args' ++ env') (by linarith) this\n            exists v'\n            constructor\n            · apply Eval.branch_matches\n              · exact heval'\n            · assumption\n    case cons.delayed =>\n      cases h₁\n      case cons h _ =>\n        cases h\n  case branch_fails e' n' env₁ name name' args e₂ v₂ hname heval ih =>\n    rcases env' with\n      ⟨⟩ | ⟨⟨hd⟩, env'⟩\n    case nil =>\n      cases h₁\n    case cons.value =>\n      cases h₁\n      case cons h₁ h₂ =>\n        cases h₁\n        case value h₁ =>\n          invert h₁\n          case constr_app args_rev h =>\n            have h₃ : Value.constr_app name args ∷ env₁ ≲ₑ'(m + n') Value.constr_app name args_rev ∷ env' := by\n              constructor\n              · apply Object.Approx.Indexed'.value\n                apply Value.Approx.Indexed.constr_app\n                intro k hk\n                apply h\n                exact hk\n              · apply Env.Approx.Indexed'.anti_monotone\n                · assumption\n                · linarith\n            obtain ⟨v', heval', happrox'⟩ := ih m (Value.constr_app name args_rev ∷ env') (by linarith) h₃\n            exists v'\n            constructor\n            · apply Eval.branch_fails\n              · exact hname\n              · exact heval'\n            · assumption\n    case cons.delayed =>\n      cases h₁\n      case cons h _ =>\n        cases h\n  case recur n₁ n₂ env₁ name body val hn ha₁ ih =>\n    let d := Object.delayed env₁ (Expr.recur name body)\n    let d' := Object.delayed env' (Expr.recur name body)\n    have : d ≲ₒ'(m + n₂) d' := by\n      simp [d, d']\n      apply Object.Approx.Indexed'.delayed_eq\n      apply Env.Approx.Indexed'.anti_monotone\n      · assumption\n      · linarith\n    have : d :: env₁ ≲ₑ'(m + n₂) d' :: env' := by\n      apply Env.Approx.Indexed'.cons\n      · assumption\n      · apply Env.Approx.Indexed'.anti_monotone\n        · assumption\n        · linarith\n    obtain ⟨v, heval, happrox⟩ := ih m (d' :: env') (by linarith) this\n    exists v\n    constructor\n    · apply Eval.recur\n      exact heval\n    · assumption", "nesting_depth": 4, "transitive_dep_count": 68, "subset_aristotle": true, "category": "Semantics"}
{"id": 246, "thm_name": "Juvix.Core.Main.Value.Approx.Indexed.refl", "thm_stmt": "@[refl]\nlemma Value.Approx.Indexed.refl {n} v : v ≲ᵥ(n) v", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx/Indexed.lean", "imports": ["import Juvix.Utils", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Mathlib.Tactic.Linarith", "import Mathlib.Data.List.Forall2", "import Juvix.Core.Main.Semantics.Eval", "import Aesop"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.cons", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.nil", "module": "Batteries.Data.List.Basic"}, {"name": "List.cons", "module": "Init.Prelude"}, {"name": "List.nil", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "List.map", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nsyntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))\n\nsyntax \"let \" ident \" := \" expr \" in \" expr : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}, {"name": "Expr.Approx.Param", "content": "def Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)"}, {"name": "Eval.Indexed", "content": "inductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "Context", "content": "inductive Context : Type where\n  | hole : Context\n  | app_left : Context → Expr → Context\n  | app_right : Expr → Context → Context\n  | constr_app_left : Context → Expr → Context\n  | constr_app_right : Expr → Context → Context\n  | binop_left : (oper : BinaryOp) → (arg₁ : Context) → (arg₂ : Expr) → Context\n  | binop_right : (oper : BinaryOp) → (arg₁ : Expr) → (arg₂ : Context) → Context\n  | lambda : (var_name : String) → (body : Context) → Context\n  | save_left : (var_name : String) → (value : Context) → (body : Expr) → Context\n  | save_right : (var_name : String) → (value : Expr) → (body : Context) → Context\n  | branch_left : (constr : Name) → (var_names : List Name) → (body : Context) → (next : Expr) → Context\n  | branch_right : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Context) → Context\n  | recur : (var_name : Name) → (ctx : Context) → Context\n  deriving Inhabited, BEq"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "exists_and_left", "module": "Init.PropLemmas"}, {"name": "exists_const", "module": "Init.PropLemmas"}, {"name": "exists_eq_left'", "module": "Init.PropLemmas"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "List.forall₂_same", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.Forall₂.get", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.Forall₂.length_eq", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.getElem?_eq_some_iff", "module": "Init.Data.List.Lemmas"}, {"name": "List.get_eq_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "forall_true_left", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.le_eq", "module": "Init.Data.Nat.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}, {"name": "Juvix.Core.Main.Expr.Approx.Param.Indexed", "content": "def Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Juvix.Core.Main.Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}, {"name": "Juvix.Core.Main.Expr.Approx.Indexed'", "content": "def Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.Preservation", "content": "def Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Env.Approx.Indexed'.get", "content": "lemma Env.Approx.Indexed'.get {n i : Nat} {env env' o₁}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some o₁) :\n  ∃ o₂, env'[i]? = some o₂ ∧ o₁ ≲ₒ'(n) o₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.value", "content": "lemma Env.Approx.Indexed'.value {n i : Nat} {env env' v}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some (Object.value v)) :\n  ∃ v', env'[i]? = some (Object.value v') ∧ v ≲ᵥ(n) v'"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.delayed", "content": "lemma Env.Approx.Indexed'.delayed {n i : Nat} {env₁ env₂ env e}\n  (h₁ : env₁ ≲ₑ'(n) env₂)\n  (h₂ : env₁[i]? = some (Object.delayed env e)) :\n  (∃ env' e', e ≲(n)⟨env, env'⟩ e' ∧ env₂[i]? = some (Object.delayed env' e')) ∨\n  ∃ env', env ≲ₑ'(n) env' ∧ env₂[i]? = some (Object.delayed env' e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.from_value", "content": "lemma Env.Approx.Indexed'.from_value {n l₁ l₂} (h : l₁ ≲ₐ(n) l₂) :\n  List.map Object.value l₁ ≲ₑ'(n) List.map Object.value l₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.concat", "content": "lemma Env.Approx.Indexed'.concat {n env₁ env₂ env₁' env₂'}\n  (h₁ : env₁ ≲ₑ'(n) env₁')\n  (h₂ : env₂ ≲ₑ'(n) env₂') :\n  env₁ ++ env₂ ≲ₑ'(n) env₁' ++ env₂'"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.cons", "content": "lemma Env.Approx.Indexed'.cons {n o₁ o₂ env₁ env₂}\n  (h₁ : o₁ ≲ₒ'(n) o₂)\n  (h₂ : env₁ ≲ₑ'(n) env₂) :\n  o₁ :: env₁ ≲ₑ'(n) o₂ :: env₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.anti_monotone", "content": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂"}, {"name": "Juvix.Core.Main.Expr.Approx.Param.Indexed.anti_monotone", "content": "lemma Expr.Approx.Param.Indexed.anti_monotone {n n' env₁ env₂ e₁ e₂}\n  (h : e₁ ≲(n)⟨env₁, env₂⟩ e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲(n')⟨env₁, env₂⟩ e₂"}, {"name": "Juvix.Core.Main.Expr.Approx.Indexed'.anti_monotone", "content": "lemma Expr.Approx.Indexed'.anti_monotone {n n' e₁ e₂}\n  (h : e₁ ≲'(n) e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲'(n') e₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.anti_monotone", "content": "  lemma Env.Approx.Indexed'.anti_monotone {n n' env₁ env₂}\n    (h : env₁ ≲ₑ'(n) env₂)\n    (h' : n' ≤ n)\n    : env₁ ≲ₑ'(n') env₂"}, {"name": "Juvix.Core.Main.Object.Approx.Indexed'.anti_monotone", "content": "  lemma Object.Approx.Indexed'.anti_monotone {n n' o₁ o₂} (h : o₁ ≲ₒ'(n) o₂) (h' : n' ≤ n) : o₁ ≲ₒ'(n') o₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.Preservation.anti_monotone", "content": "lemma Value.Approx.Indexed.Preservation.anti_monotone {k k'} (h : Value.Approx.Indexed.Preservation k) (h' : k' ≤ k) : Value.Approx.Indexed.Preservation k'"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.preserved_step", "content": "lemma Value.Approx.Indexed.preserved_step {k} :\n  (∀ k' < k, Preservation k') → Preservation k"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.preserved'", "content": "lemma Value.Approx.Indexed.preserved' {k} : Preservation k"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.preserved", "content": "theorem Value.Approx.Indexed.preserved :\n  ∀ m n env env' e v,\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.refl'", "content": "  lemma Env.Approx.Indexed'.refl' {n env} (h : ∀ v, v ≲ᵥ(n) v) : env ≲ₑ'(n) env"}, {"name": "Juvix.Core.Main.Object.Approx.Indexed'.refl'", "content": "  lemma Object.Approx.Indexed'.refl' {n o} (h : ∀ v, v ≲ᵥ(n) v) : o ≲ₒ'(n) o"}], "local_ctx": "import Juvix.Core.Main.Semantics.Eval\n\nimport Juvix.Core.Main.Semantics.Eval.Indexed\n\nimport Juvix.Utils\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Data.List.Forall2\n\nimport Aesop\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))\n\nnotation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nnotation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\ndef Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)\n\nnotation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\ninductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)\n\ndef Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)\n\nnotation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'\n\nnotation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\ndef Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)\n\nnotation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'\n\n@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)\n\nend\n\ndef Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'\n\nend", "target_theorem": "@[refl]\nlemma Value.Approx.Indexed.refl {n} v : v ≲ᵥ(n) v :=", "ground_truth_proof": ":= by\n  revert n\n  suffices ∀ n, ∀ k ≤ n, v ≲ᵥ(k) v by\n    intro k\n    exact this k k k.le_refl\n  intro n\n  induction n generalizing v with\n  | zero =>\n    intros k hk\n    cases v\n    case unit =>\n      exact Value.Approx.Indexed.unit\n    case const c =>\n      exact Value.Approx.Indexed.const\n    case constr_app ctr_name args_rev =>\n      apply Value.Approx.Indexed.constr_app\n      · intros\n        have : k = 0 := by linarith\n        subst k\n        contradiction\n    case closure env body =>\n      apply Value.Approx.Indexed.closure\n      · intros\n        have : k = 0 := by linarith\n        subst k\n        contradiction\n  | succ n ih =>\n    intros k hk\n    cases v\n    case unit =>\n      exact Value.Approx.Indexed.unit\n    case const c =>\n      exact Value.Approx.Indexed.const\n    case constr_app ctr_name args_rev =>\n      apply Value.Approx.Indexed.constr_app\n      · intros k' hk'\n        have : k' ≤ n := by linarith\n        rw [List.forall₂_same]\n        intros\n        apply ih\n        · assumption\n    case closure env body =>\n      apply Value.Approx.Indexed.closure\n      · intros n₁ n₂ hn a₁ a₂ r₁ happrox heval\n        have henv : a₁ ∷ env ≲ₑ'(n₁ + n₂) a₂ ∷ env := by\n          simp [Env.Approx.Indexed']\n          constructor\n          · constructor; exact happrox\n          · intro x _\n            cases x\n            case value v₁ v₂ =>\n              apply Object.Approx.Indexed'.value\n              · apply ih\n                · linarith\n            case delayed =>\n              apply Object.Approx.Indexed'.refl'\n              intro v\n              apply ih\n              linarith\n        apply Value.Approx.Indexed.preserved (n := n₁) (m := n₂)\n        · have : n₁ + n₂ = n₂ + n₁ := by linarith\n          rw [<- this]\n          exact henv\n        · assumption", "nesting_depth": 7, "transitive_dep_count": 80, "subset_aristotle": true, "category": "Semantics"}
{"id": 247, "thm_name": "Juvix.Core.Main.Eval.toIndexed", "thm_stmt": "lemma Eval.toIndexed {env e v} (h : env ⊢ e ↦ v) : ∃ n, env ⊢ e ↦(n) v", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Eval/Indexed.lean", "imports": ["import Mathlib.Tactic.Linarith", "import Juvix.Core.Main.Semantics.Eval"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}], "used_repo_defs": [{"name": "syntax:100 expr:100 ppSpace expr:101 : expr", "content": "syntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Context", "content": "inductive Context : Type where\n  | hole : Context\n  | app_left : Context → Expr → Context\n  | app_right : Expr → Context → Context\n  | constr_app_left : Context → Expr → Context\n  | constr_app_right : Expr → Context → Context\n  | binop_left : (oper : BinaryOp) → (arg₁ : Context) → (arg₂ : Expr) → Context\n  | binop_right : (oper : BinaryOp) → (arg₁ : Expr) → (arg₂ : Context) → Context\n  | lambda : (var_name : String) → (body : Context) → Context\n  | save_left : (var_name : String) → (value : Context) → (body : Expr) → Context\n  | save_right : (var_name : String) → (value : Expr) → (body : Context) → Context\n  | branch_left : (constr : Name) → (var_names : List Name) → (body : Context) → (next : Expr) → Context\n  | branch_right : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Context) → Context\n  | recur : (var_name : Name) → (ctx : Context) → Context\n  deriving Inhabited, BEq"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Termination.var", "content": "lemma Termination.var {env name i v} :\n  env[i]? = some (Object.value v) →\n  env ⊢ Expr.var name i ↓"}, {"name": "Termination.var_rec", "content": "lemma Termination.var_rec {env name i env' e} :\n    env[i]? = some (Object.delayed env' e) →\n    env' ⊢ e ↓ →\n    env ⊢ (Expr.var name i) ↓"}, {"name": "Termination.unit", "content": "lemma Termination.unit {env} :\n  env ⊢ Expr.unit ↓"}, {"name": "Termination.const", "content": "lemma Termination.const {env c} :\n  env ⊢ Expr.const c ↓"}, {"name": "Termination.constr", "content": "lemma Termination.constr {env nm} :\n  env ⊢ Expr.constr nm ↓"}, {"name": "Termination.app", "content": "lemma Termination.app {env e₁ e₂ v env' body} :\n  env ⊢ e₁ ↦ Value.closure env' body →\n  env ⊢ e₂ ↦ v →\n  v ∷ env' ⊢ body ↓ →\n  env ⊢ Expr.app e₁ e₂ ↓"}, {"name": "Termination.constr_app", "content": "lemma Termination.constr_app {env e₁ e₂ e₁₁ e₁₂} :\n  env ⊢ e₁ ↦ Value.constr_app e₁₁ e₁₂ →\n  env ⊢ e₂ ↓ →\n  env ⊢ Expr.constr_app e₁ e₂ ↓"}, {"name": "Termination.binop", "content": "lemma Termination.binop {env op e₁ e₂ c₁ c₂} :\n  env ⊢ e₁ ↦ Value.const (Constant.int c₁) →\n  env ⊢ e₂ ↦ Value.const (Constant.int c₂) →\n  env ⊢ Expr.binop op e₁ e₂ ↓"}, {"name": "Termination.lambda", "content": "lemma Termination.lambda {env name e} : env ⊢ Expr.lambda name e ↓"}, {"name": "Termination.save", "content": "lemma Termination.save {env name e₁ e₂ v} :\n  env ⊢ e₁ ↦ v →\n  v ∷ env ⊢ e₂ ↓ →\n  env ⊢ Expr.save name e₁ e₂ ↓"}, {"name": "Termination.branch_matches", "content": "lemma Termination.branch_matches {env name vnames args_rev e e'} :\n  args_rev.map Object.value ++ env ⊢ e ↓ →\n  Value.constr_app name args_rev ∷ env ⊢ Expr.branch name vnames e e' ↓"}, {"name": "Termination.branch_fails", "content": "lemma Termination.branch_fails {env name name' vnames args_rev e e'} :\n  name ≠ name' →\n  Value.constr_app name args_rev ∷ env ⊢ e' ↓ →\n  Value.constr_app name args_rev ∷ env ⊢ Expr.branch name' vnames e e' ↓"}, {"name": "Termination.recur", "content": "lemma Termination.recur {env name e} :\n  Object.delayed env (Expr.recur name e) :: env ⊢ e ↓ →\n  env ⊢ Expr.recur name e ↓"}], "used_local_defs": [{"name": "Juvix.Core.Main.Eval.Indexed", "content": "inductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Eval.Indexed.monotone", "content": "lemma Eval.Indexed.monotone {n n' env e v} (h : env ⊢ e ↦(n) v) (h' : n ≤ n') : env ⊢ e ↦(n') v"}], "local_ctx": "import Juvix.Core.Main.Semantics.Eval\n\nimport Mathlib.Tactic.Linarith\n\nnamespace Juvix.Core.Main\n\ninductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val\n\nnotation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v", "target_theorem": "lemma Eval.toIndexed {env e v} (h : env ⊢ e ↦ v) : ∃ n, env ⊢ e ↦(n) v :=", "ground_truth_proof": ":= by\n  induction h\n  case var =>\n    exists 0; apply Eval.Indexed.var; assumption\n  case var_rec ih =>\n    obtain ⟨n, ih⟩ := ih\n    exists n\n    apply Eval.Indexed.var_rec <;> assumption\n  case unit =>\n    exists 0; apply Eval.Indexed.unit\n  case const =>\n    exists 0; apply Eval.Indexed.const\n  case constr =>\n    exists 0; apply Eval.Indexed.constr\n  case app env env' f body arg val val' _ _ _ ih1 ih2 ih3 =>\n    obtain ⟨n₁, ih₁⟩ := ih1\n    obtain ⟨n₂, ih₂⟩ := ih2\n    obtain ⟨n₃, ih₃⟩ := ih3\n    have ih₁' : env ⊢ f ↦(n₁ + n₂) Value.closure env' body := by\n      apply Eval.Indexed.monotone ih₁; linarith\n    have ih₂' : env ⊢ arg ↦(n₁ + n₂ + 1) val := by\n      apply Eval.Indexed.monotone ih₂; linarith\n    exists (n₁ + n₂ + n₃ + 1)\n    apply Eval.Indexed.app\n    · rfl\n    · exact ih₁'\n    · exact ih₂'\n    · exact ih₃\n  case constr_app ih₁ ih₂ =>\n    obtain ⟨n₁, ih₁⟩ := ih₁\n    obtain ⟨n₂, ih₂⟩ := ih₂\n    exists (n₁ + n₂ + 1)\n    apply Eval.Indexed.constr_app (n' := n₁ + n₂)\n    · linarith\n    · apply Eval.Indexed.monotone ih₁; linarith\n    · apply Eval.Indexed.monotone ih₂; linarith\n  case binop ih₁ ih₂ =>\n    obtain ⟨n₁, ih₁⟩ := ih₁\n    obtain ⟨n₂, ih₂⟩ := ih₂\n    exists (n₁ + n₂)\n    apply Eval.Indexed.binop (n := n₁ + n₂)\n    · apply Eval.Indexed.monotone ih₁; linarith\n    · apply Eval.Indexed.monotone ih₂; linarith\n  case lambda =>\n    exists 0; apply Eval.Indexed.lambda\n  case save ih₁ ih₂ =>\n    obtain ⟨n₁, ih₁⟩ := ih₁\n    obtain ⟨n₂, ih₂⟩ := ih₂\n    exists (n₁ + n₂)\n    apply Eval.Indexed.save (n := n₁ + n₂)\n    · rfl\n    · apply Eval.Indexed.monotone ih₁; linarith\n    · apply Eval.Indexed.monotone ih₂; linarith\n  case branch_matches ih =>\n    obtain ⟨n, ih⟩ := ih\n    exists (n + 1)\n    apply Eval.Indexed.branch_matches (n' := n)\n    · linarith\n    · assumption\n  case branch_fails ih =>\n    obtain ⟨n, ih⟩ := ih\n    exists n\n    apply Eval.Indexed.branch_fails <;> assumption\n  case recur ih =>\n    obtain ⟨n, ih⟩ := ih\n    exists n.succ\n    apply Eval.Indexed.recur (n' := n)\n    · linarith\n    · assumption", "nesting_depth": 3, "transitive_dep_count": 24, "subset_aristotle": true, "category": "Semantics"}
{"id": 248, "thm_name": "Juvix.Core.Main.Termination.recur", "thm_stmt": "lemma Termination.recur {env name e} :\n  Object.delayed env (Expr.recur name e) :: env ⊢ e ↓ →\n  env ⊢ Expr.recur name e ↓", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Tactics/Termination.lean", "imports": ["import Juvix.Core.Main.Tactics.Base"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}], "used_repo_defs": [{"name": "syntax:100 expr:100 ppSpace expr:101 : expr", "content": "syntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Eval.Defined", "content": "def Eval.Defined (env : Env) (e : Expr) : Prop :=\n  ∃ v, env ⊢ e ↦ v"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Juvix.Core.Main.Tactics.Base\n\nnamespace Juvix.Core.Main", "target_theorem": "lemma Termination.recur {env name e} :\n  Object.delayed env (Expr.recur name e) :: env ⊢ e ↓ →\n  env ⊢ Expr.recur name e ↓ :=", "ground_truth_proof": ":= by\n  unfold Eval.Defined\n  intro h\n  obtain ⟨w, h⟩ := h\n  exists w\n  · apply Juvix.Core.Main.Eval.recur\n    exact h", "nesting_depth": 4, "transitive_dep_count": 19, "subset_aristotle": false, "category": "Semantics"}
{"id": 249, "thm_name": "Juvix.Core.Main.Value.Approx.Indexed.trans", "thm_stmt": "@[trans]\nlemma Value.Approx.Indexed.trans {n v₁ v₂ v₃} : v₁ ≲ᵥ(n) v₂ → v₂ ≲ᵥ v₃ → v₁ ≲ᵥ(n) v₃", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx.lean", "imports": ["import Juvix.Core.Main.Semantics.Approx.Indexed", "import Juvix.Utils"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}], "used_repo_defs": [{"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.A", "content": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Value.Approx.Indexed.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "IsEmpty.forall_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}], "repo_lemmas": [{"name": "Value.Approx.Indexed.invert", "content": "@[aesop unsafe destruct]\nlemma Value.Approx.Indexed.invert {n v v'} :\n  v ≲ᵥ(n) v' →\n  Value.Approx.Indexed.Inversion n v v'"}, {"name": "Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "forall₂_trans'", "content": "theorem forall₂_trans' {α} {P Q R : α → α → Prop} {l₁ l₂ l₃}\n  (h : ∀ x y z, P x y → Q y z → R x z)\n  (h₁ : List.Forall₂ P l₁ l₂)\n  (h₂ : List.Forall₂ Q l₂ l₃)\n  : List.Forall₂ R l₁ l₃"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Juvix.Core.Main.Value.Approx.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Inversion : Value -> Value -> Prop where\n  | unit : Value.Approx.Inversion Value.unit Value.unit\n  | const {c} : Value.Approx.Inversion (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    args_rev ≲ₐ args_rev' →\n    Value.Approx.Inversion (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ a₁ a₂, a₁ ≲ᵥ a₂ → body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n    Value.Approx.Inversion (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Value.Approx.invert", "content": "@[aesop unsafe destruct]\nlemma Value.Approx.invert {v v'} :\n  v ≲ᵥ v' →\n  Value.Approx.Inversion v v'"}], "local_ctx": "import Juvix.Core.Main.Semantics.Approx.Indexed\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'\n\nnotation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'\n\nnotation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂\n\nnotation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'\n\nnotation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'\n\nnotation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂\n\nnotation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂\n\n@[aesop safe cases]\ninductive Value.Approx.Inversion : Value -> Value -> Prop where\n  | unit : Value.Approx.Inversion Value.unit Value.unit\n  | const {c} : Value.Approx.Inversion (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    args_rev ≲ₐ args_rev' →\n    Value.Approx.Inversion (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ a₁ a₂, a₁ ≲ᵥ a₂ → body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n    Value.Approx.Inversion (Value.closure env₁ body₁) (Value.closure env₂ body₂)", "target_theorem": "@[trans]\nlemma Value.Approx.Indexed.trans {n v₁ v₂ v₃} : v₁ ≲ᵥ(n) v₂ → v₂ ≲ᵥ v₃ → v₁ ≲ᵥ(n) v₃ :=", "ground_truth_proof": ":= by\n  revert n\n  suffices ∀ n, ∀ k ≤ n, v₁ ≲ᵥ(k) v₂ → v₂ ≲ᵥ v₃ → v₁ ≲ᵥ(k) v₃ by\n    intro k\n    exact this k k k.le_refl\n  intros n\n  induction n generalizing v₁ v₂ v₃ with\n  | zero =>\n    intros k hk h₁ h₂\n    invert h₁\n    case unit =>\n      invert h₂\n      case unit =>\n        exact Value.Approx.Indexed.unit\n    case const =>\n      invert h₂\n      case const =>\n        exact Value.Approx.Indexed.const\n    case constr_app ctr_name args_rev₁ args_rev₁' ch₁ =>\n      cases h₂.invert\n      case constr_app args_rev₂ ch₂ =>\n        apply Value.Approx.Indexed.constr_app;\n        intros\n        simp_all only [nonpos_iff_eq_zero, not_lt_zero', IsEmpty.forall_iff, implies_true]\n    case closure env₁ body₁ env₁' body₁' ch₁ =>\n      cases h₂.invert\n      case closure env₂ body₂ ch₂ =>\n        apply Value.Approx.Indexed.closure\n        · intro k' hk' v v₁ h\n          have : k = 0 := by linarith\n          subst k\n          contradiction\n  | succ n ih =>\n    intros k hk h₁ h₂\n    invert h₁\n    case unit =>\n      invert h₂\n      case unit =>\n        exact Value.Approx.Indexed.unit\n    case const =>\n      invert h₂\n      case const =>\n        exact Value.Approx.Indexed.const\n    case constr_app ctr_name args_rev args_rev' ch₁ =>\n      invert h₂\n      case constr_app args_rev'' ch₂ =>\n        apply Value.Approx.Indexed.constr_app\n        · intro k' hk'\n          have hk' : k' ≤ n := by linarith\n          apply Utils.forall₂_trans' (P := fun v₁ v₂ => v₁ ≲ᵥ(k') v₂) (Q := fun v₁ v₂ => v₁ ≲ᵥ v₂)\n          · intros v₁ v₂ v₃ h₁ h₂\n            apply ih <;> assumption\n          · apply ch₁\n            simp_all only\n          · apply ch₂\n    case closure env₁ body₁ env₂ body₂ ch₁ =>\n      invert h₂\n      case closure env₃ body₃ ch₂ =>\n        apply Value.Approx.Indexed.closure\n        · intro n₁ n₂ hn a₁ a₃ v₁ happrox heval₁\n          have ah₁ : ∃ v₂, (a₃ ∷ env₂) ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂ := by\n            apply ch₁\n            · assumption\n            · assumption\n            · assumption\n          obtain ⟨v₂, heval₂, h₂⟩ := ah₁\n          have ah₂ : ∃ v₃, (a₃ ∷ env₃) ⊢ body₃ ↦ v₃ ∧ v₂ ≲ᵥ v₃ := by\n            apply ch₂\n            · rfl\n            · assumption\n          obtain ⟨v₃, _, h₃⟩ := ah₂\n          have : n₂ ≤ n := by linarith\n          exists v₃\n          aesop", "nesting_depth": 3, "transitive_dep_count": 33, "subset_aristotle": true, "category": "Semantics"}
{"id": 250, "thm_name": "Juvix.Core.Main.Value.Approx.Indexed.anti_monotone", "thm_stmt": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx/Indexed.lean", "imports": ["import Juvix.Utils", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Mathlib.Tactic.Linarith", "import Mathlib.Data.List.Forall2", "import Juvix.Core.Main.Semantics.Eval", "import Aesop"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}], "used_repo_defs": [{"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "exists_and_left", "module": "Init.PropLemmas"}, {"name": "exists_const", "module": "Init.PropLemmas"}, {"name": "exists_eq_left'", "module": "Init.PropLemmas"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}], "local_ctx": "import Juvix.Core.Main.Semantics.Eval\n\nimport Juvix.Core.Main.Semantics.Eval.Indexed\n\nimport Juvix.Utils\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Data.List.Forall2\n\nimport Aesop\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))\n\nnotation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nnotation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nnotation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\nnotation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'\n\nnotation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\nnotation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'", "target_theorem": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂ :=", "ground_truth_proof": ":= by\n  revert n n'\n  suffices ∀ n, v₁ ≲ᵥ(n) v₂ → ∀ k ≤ n, v₁ ≲ᵥ(k) v₂ by\n    intro n n' h hn\n    exact this n h n' hn\n  intro n\n  induction n generalizing v₁ v₂ with\n  | zero =>\n    intros h k hk\n    invert h\n    case unit =>\n      exact Value.Approx.Indexed.unit\n    case const =>\n      exact Value.Approx.Indexed.const\n    case constr_app ctr_name args_rev args_rev' hargs =>\n      apply Value.Approx.Indexed.constr_app\n      · intros\n        have : k = 0 := by linarith\n        subst k\n        contradiction\n    case closure env₁ body₁ env₂ body₂ h =>\n      apply Value.Approx.Indexed.closure\n      · intros\n        have : k = 0 := by linarith\n        subst k\n        contradiction\n  | succ n ih =>\n    intros h k hk\n    invert h\n    case unit =>\n      exact Value.Approx.Indexed.unit\n    case const =>\n      exact Value.Approx.Indexed.const\n    case constr_app ctr_name args_rev args_rev' hargs =>\n      apply Value.Approx.Indexed.constr_app\n      · intros k' hk'\n        have : k' < n + 1 := by linarith\n        simp_all only\n    case closure env₁ body₁ env₂ body₂ ch =>\n      apply Value.Approx.Indexed.closure\n      · intro n₁ n₂ hn a₁ a₂ v₁ happrox heval\n        apply ch n₁ n₂\n        · linarith\n        · assumption\n        · assumption", "nesting_depth": 3, "transitive_dep_count": 34, "subset_aristotle": true, "category": "Semantics"}
{"id": 251, "thm_name": "Juvix.Core.Main.Expr.Approx.Context.preserved", "thm_stmt": "lemma Expr.Approx.Context.preserved (e₁ e₂ : Expr) :\n  e₁ ≲ e₂ →\n  ∀ (C : Context) env₁ env₂ v₁,\n  env₁ ≲ₑ env₂ → env₁ ⊢ C.subst e₁ ↦ v₁ →\n  ∃ v₂, env₂ ⊢ C.subst e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx/Soundness.lean", "imports": ["import Juvix.Core.Main.Semantics.Approx.Indexed", "import Juvix.Core.Main.Semantics.Approx", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Juvix.Core.Main.Semantics.Approx.Contextual", "import Juvix.Core.Main.Semantics.Eval"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.map", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\nsyntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr"}, {"name": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.A", "content": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nsyntax \"let \" ident \" := \" expr \" in \" expr : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)"}, {"name": "Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}, {"name": "Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Context.subst", "content": "@[simp]\ndef Context.subst (C : Context) (e : Expr) : Expr :=\n  match C with\n  | Context.hole => e\n  | Context.app_left C' e' => Expr.app (C'.subst e) e'\n  | Context.app_right e' C' => Expr.app e' (C'.subst e)\n  | Context.constr_app_left C' e' => Expr.constr_app (C'.subst e) e'\n  | Context.constr_app_right e' C' => Expr.constr_app e' (C'.subst e)\n  | Context.binop_left oper C₁ e₂ => Expr.binop oper (C₁.subst e) e₂\n  | Context.binop_right oper e₁ C₂ => Expr.binop oper e₁ (C₂.subst e)\n  | Context.lambda s C' => Expr.lambda s (C'.subst e)\n  | Context.save_left s C' e' => Expr.save s (C'.subst e) e'\n  | Context.save_right s e' C' => Expr.save s e' (C'.subst e)\n  | Context.branch_left constr names C' next => Expr.branch constr names (C'.subst e) next\n  | Context.branch_right constr names body C' => Expr.branch constr names body (C'.subst e)\n  | Context.recur name C' => Expr.recur name (C'.subst e)"}, {"name": "Context", "content": "inductive Context : Type where\n  | hole : Context\n  | app_left : Context → Expr → Context\n  | app_right : Expr → Context → Context\n  | constr_app_left : Context → Expr → Context\n  | constr_app_right : Expr → Context → Context\n  | binop_left : (oper : BinaryOp) → (arg₁ : Context) → (arg₂ : Expr) → Context\n  | binop_right : (oper : BinaryOp) → (arg₁ : Expr) → (arg₂ : Context) → Context\n  | lambda : (var_name : String) → (body : Context) → Context\n  | save_left : (var_name : String) → (value : Context) → (body : Expr) → Context\n  | save_right : (var_name : String) → (value : Expr) → (body : Context) → Context\n  | branch_left : (constr : Name) → (var_names : List Name) → (body : Context) → (next : Expr) → Context\n  | branch_right : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Context) → Context\n  | recur : (var_name : Name) → (ctx : Context) → Context\n  deriving Inhabited, BEq"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Value.Approx.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Inversion : Value -> Value -> Prop where\n  | unit : Value.Approx.Inversion Value.unit Value.unit\n  | const {c} : Value.Approx.Inversion (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    args_rev ≲ₐ args_rev' →\n    Value.Approx.Inversion (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ a₁ a₂, a₁ ≲ᵥ a₂ → body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n    Value.Approx.Inversion (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Eval.Indexed", "content": "inductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "Expr.Approx.Param.Indexed", "content": "def Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Env", "content": "abbrev Env : Type := List Object\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Expr.Approx.Param", "content": "def Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)"}, {"name": "Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Value.Approx.Indexed.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env"}, {"name": "Expr.Approx.Indexed'", "content": "def Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}, {"name": "Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}, {"name": "Value.Approx.Indexed.Preservation", "content": "def Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "List.forall₂_cons", "module": "Batteries.Data.List.Basic"}, {"name": "List.forall₂_same", "module": "Mathlib.Data.List.Forall2"}], "repo_lemmas": [{"name": "Expr.Approx.toIndexed", "content": "lemma Expr.Approx.toIndexed {e₁ e₂} : e₁ ≲ e₂ → ∀ n, e₁ ≲'(n) e₂"}, {"name": "Value.Approx.Indexed.trans", "content": "@[trans]\nlemma Value.Approx.Indexed.trans {n v₁ v₂ v₃} : v₁ ≲ᵥ(n) v₂ → v₂ ≲ᵥ v₃ → v₁ ≲ᵥ(n) v₃"}, {"name": "Value.Approx.invert", "content": "@[aesop unsafe destruct]\nlemma Value.Approx.invert {v v'} :\n  v ≲ᵥ v' →\n  Value.Approx.Inversion v v'"}, {"name": "Env.Approx.toIndexed", "content": "lemma Env.Approx.toIndexed {env₁ env₂} : env₁ ≲ₑ env₂ → ∀ n, env₁ ≲ₑ'(n) env₂"}, {"name": "Env.Approx.cons", "content": "lemma Env.Approx.cons {env₁ env₂ o₁ o₂} :\n  o₁ ≲ₒ o₂ → env₁ ≲ₑ env₂ → (o₁ :: env₁) ≲ₑ (o₂ :: env₂)"}, {"name": "Object.Approx.toIndexed", "content": "lemma Object.Approx.toIndexed {o₁ o₂} : o₁ ≲ₒ o₂ → ∀ n, o₁ ≲ₒ'(n) o₂"}, {"name": "Eval.deterministic", "content": "theorem Eval.deterministic {env e v₁ v₂} (h₁ : env ⊢ e ↦ v₁) (h₂ : env ⊢ e ↦ v₂) : v₁ = v₂"}, {"name": "Eval.toIndexed", "content": "lemma Eval.toIndexed {env e v} (h : env ⊢ e ↦ v) : ∃ n, env ⊢ e ↦(n) v"}, {"name": "Eval.Indexed.monotone", "content": "lemma Eval.Indexed.monotone {n n' env e v} (h : env ⊢ e ↦(n) v) (h' : n ≤ n') : env ⊢ e ↦(n') v"}, {"name": "Value.Approx.Indexed.preserved", "content": "theorem Value.Approx.Indexed.preserved :\n  ∀ m n env env' e v,\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}, {"name": "Value.Approx.Indexed.preserved'", "content": "lemma Value.Approx.Indexed.preserved' {k} : Preservation k"}, {"name": "Env.Approx.Indexed'.refl", "content": "@[refl]\nlemma Env.Approx.Indexed'.refl {n env} : env ≲ₑ'(n) env"}, {"name": "Value.Approx.Indexed.refl", "content": "@[refl]\nlemma Value.Approx.Indexed.refl {n} v : v ≲ᵥ(n) v"}, {"name": "Object.Approx.Indexed'.refl'", "content": "lemma Object.Approx.Indexed'.refl' {n o} (h : ∀ v, v ≲ᵥ(n) v) : o ≲ₒ'(n) o"}, {"name": "Env.Approx.Indexed'.refl'", "content": "lemma Env.Approx.Indexed'.refl' {n env} (h : ∀ v, v ≲ᵥ(n) v) : env ≲ₑ'(n) env"}, {"name": "Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "Object.Approx.Indexed'.refl", "content": "@[refl]\nlemma Object.Approx.Indexed'.refl {n o} : o ≲ₒ'(n) o"}, {"name": "Value.Approx.Indexed.preserved_step", "content": "lemma Value.Approx.Indexed.preserved_step {k} :\n  (∀ k' < k, Preservation k') → Preservation k"}, {"name": "Expr.Approx.Param.Indexed.anti_monotone", "content": "lemma Expr.Approx.Param.Indexed.anti_monotone {n n' env₁ env₂ e₁ e₂}\n  (h : e₁ ≲(n)⟨env₁, env₂⟩ e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲(n')⟨env₁, env₂⟩ e₂"}, {"name": "Expr.Approx.Indexed'.anti_monotone", "content": "lemma Expr.Approx.Indexed'.anti_monotone {n n' e₁ e₂}\n  (h : e₁ ≲'(n) e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲'(n') e₂"}, {"name": "Env.Approx.Indexed'.delayed", "content": "lemma Env.Approx.Indexed'.delayed {n i : Nat} {env₁ env₂ env e}\n  (h₁ : env₁ ≲ₑ'(n) env₂)\n  (h₂ : env₁[i]? = some (Object.delayed env e)) :\n  (∃ env' e', e ≲(n)⟨env, env'⟩ e' ∧ env₂[i]? = some (Object.delayed env' e')) ∨\n  ∃ env', env ≲ₑ'(n) env' ∧ env₂[i]? = some (Object.delayed env' e)"}, {"name": "Env.Approx.Indexed'.value", "content": "lemma Env.Approx.Indexed'.value {n i : Nat} {env env' v}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some (Object.value v)) :\n  ∃ v', env'[i]? = some (Object.value v') ∧ v ≲ᵥ(n) v'"}, {"name": "Env.Approx.Indexed'.get", "content": "lemma Env.Approx.Indexed'.get {n i : Nat} {env env' o₁}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some o₁) :\n  ∃ o₂, env'[i]? = some o₂ ∧ o₁ ≲ₒ'(n) o₂"}, {"name": "Value.Approx.Indexed.anti_monotone", "content": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂"}, {"name": "Value.Approx.Indexed.Preservation.anti_monotone", "content": "lemma Value.Approx.Indexed.Preservation.anti_monotone {k k'} (h : Value.Approx.Indexed.Preservation k) (h' : k' ≤ k) : Value.Approx.Indexed.Preservation k'"}, {"name": "Env.Approx.Indexed'.anti_monotone", "content": "lemma Env.Approx.Indexed'.anti_monotone {n n' env₁ env₂}\n    (h : env₁ ≲ₑ'(n) env₂)\n    (h' : n' ≤ n)\n    : env₁ ≲ₑ'(n') env₂"}, {"name": "Object.Approx.Indexed'.anti_monotone", "content": "lemma Object.Approx.Indexed'.anti_monotone {n n' o₁ o₂} (h : o₁ ≲ₒ'(n) o₂) (h' : n' ≤ n) : o₁ ≲ₒ'(n') o₂"}, {"name": "Env.Approx.Indexed'.cons", "content": "lemma Env.Approx.Indexed'.cons {n o₁ o₂ env₁ env₂}\n  (h₁ : o₁ ≲ₒ'(n) o₂)\n  (h₂ : env₁ ≲ₑ'(n) env₂) :\n  o₁ :: env₁ ≲ₑ'(n) o₂ :: env₂"}, {"name": "Env.Approx.Indexed'.concat", "content": "lemma Env.Approx.Indexed'.concat {n env₁ env₂ env₁' env₂'}\n  (h₁ : env₁ ≲ₑ'(n) env₁')\n  (h₂ : env₂ ≲ₑ'(n) env₂') :\n  env₁ ++ env₂ ≲ₑ'(n) env₁' ++ env₂'"}, {"name": "Env.Approx.Indexed'.from_value", "content": "lemma Env.Approx.Indexed'.from_value {n l₁ l₂} (h : l₁ ≲ₐ(n) l₂) :\n  List.map Object.value l₁ ≲ₑ'(n) List.map Object.value l₂"}, {"name": "Value.Approx.preserved", "content": "theorem Value.Approx.preserved :\n  ∀ env env' e v,\n    env ≲ₑ env' →\n    env ⊢ e ↦ v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ v'"}, {"name": "Value.Approx.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.constr_app {ctr_name args_rev args_rev'} :\n  args_rev ≲ₐ args_rev' →\n  Value.constr_app ctr_name args_rev ≲ᵥ Value.constr_app ctr_name args_rev'"}, {"name": "Value.Approx.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.closure {env₁ body₁ env₂ body₂} :\n  (∀ a₁ a₂,\n    a₁ ≲ᵥ a₂ →\n    body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n  Value.closure env₁ body₁ ≲ᵥ Value.closure env₂ body₂"}, {"name": "Expr.Approx.Param.preserved", "content": "lemma Expr.Approx.Param.preserved {e e' env₁ env₂ env₃} :\n  e ≲⟨env₁, env₂⟩ e' →\n  env₂ ≲ₑ env₃ →\n  e ≲⟨env₁, env₃⟩ e'"}, {"name": "Expr.Approx.const", "content": "lemma Expr.Approx.const {c₁ c₂} :\n  Expr.const c₁ ≲ Expr.const c₂ →\n  c₁ = c₂"}, {"name": "Expr.Approx.preservation", "content": "lemma Expr.Approx.preservation {e₁ e₂} :\n  e₁ ≲ e₂ ↔\n  ∀ env v₁, env ⊢ e₁ ↦ v₁ → ∃ v₂, env ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂"}, {"name": "Expr.Approx.to_preservation", "content": "lemma Expr.Approx.to_preservation {e₁ e₂} :\n  e₁ ≲ e₂ →\n  ∀ env v₁, env ⊢ e₁ ↦ v₁ → ∃ v₂, env ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂"}, {"name": "Expr.Approx.from_preservation", "content": "lemma Expr.Approx.from_preservation {e₁ e₂}\n  (h : ∀ env v₁, env ⊢ e₁ ↦ v₁ → ∃ v₂, env ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)\n  : e₁ ≲ e₂"}], "used_local_defs": [{"name": "Juvix.Core.Main.Context.Indexed.Preserved", "content": "def Context.Indexed.Preserved (k : Nat) : Prop :=\n  ∀ n m,\n    n + m < k →\n    ∀ e₁ e₂,\n      e₁ ≲'(n + m) e₂ → ∀ (C : Context) env₁ env₂ v₁,\n      env₁ ≲ₑ'(n + m) env₂ →\n      env₁ ⊢ C.subst e₁ ↦(n) v₁ →\n      ∃ v₂, env₂ ⊢ C.subst e₂ ↦ v₂ ∧ v₁ ≲ᵥ(m) v₂"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Expr.Approx.Context.Indexed.preserved_step", "content": "lemma Expr.Approx.Context.Indexed.preserved_step (k : Nat) :\n  Context.Indexed.Preserved k → Context.Indexed.Preserved (k + 1)"}, {"name": "Juvix.Core.Main.Expr.Approx.Context.Indexed.preserved", "content": "lemma Expr.Approx.Context.Indexed.preserved (k : Nat) :\n  Context.Indexed.Preserved k"}], "local_ctx": "import Juvix.Core.Main.Semantics.Approx\n\nimport Juvix.Core.Main.Semantics.Approx.Contextual\n\nnamespace Juvix.Core.Main\n\ndef Context.Indexed.Preserved (k : Nat) : Prop :=\n  ∀ n m,\n    n + m < k →\n    ∀ e₁ e₂,\n      e₁ ≲'(n + m) e₂ → ∀ (C : Context) env₁ env₂ v₁,\n      env₁ ≲ₑ'(n + m) env₂ →\n      env₁ ⊢ C.subst e₁ ↦(n) v₁ →\n      ∃ v₂, env₂ ⊢ C.subst e₂ ↦ v₂ ∧ v₁ ≲ᵥ(m) v₂", "target_theorem": "lemma Expr.Approx.Context.preserved (e₁ e₂ : Expr) :\n  e₁ ≲ e₂ →\n  ∀ (C : Context) env₁ env₂ v₁,\n  env₁ ≲ₑ env₂ → env₁ ⊢ C.subst e₁ ↦ v₁ →\n  ∃ v₂, env₂ ⊢ C.subst e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂ :=", "ground_truth_proof": ":= by\n  intro happrox C env₁ env₂ v₁ henv heval\n  obtain ⟨n, heval'⟩ := Eval.toIndexed heval\n  have happrox' : e₁ ≲'(n + 0) e₂ := by apply Expr.Approx.toIndexed happrox\n  have henv' : env₁ ≲ₑ'(n + 0) env₂ := by apply Env.Approx.toIndexed henv\n  obtain ⟨v₂, heval₂, happrox₂⟩ := Expr.Approx.Context.Indexed.preserved (n + 1) n 0 (by linarith) e₁ e₂ happrox' C env₁ env₂ v₁ henv' heval'\n  have : ∀ m, v₁ ≲ᵥ(m) v₂ := by\n    intro m\n    have happrox_m : e₁ ≲'(n + m) e₂ := by apply Expr.Approx.toIndexed happrox\n    have henv_m : env₁ ≲ₑ'(n + m) env₂ := by apply Env.Approx.toIndexed henv\n    obtain ⟨v₂', heval₂', happrox₂'⟩ := Expr.Approx.Context.Indexed.preserved (n + m + 1) n m (by linarith) e₁ e₂ happrox_m C env₁ env₂ v₁ henv_m heval'\n    have : v₂ = v₂' := Eval.deterministic heval₂ heval₂'\n    rw [this]\n    assumption\n  exists v₂", "nesting_depth": 5, "transitive_dep_count": 84, "subset_aristotle": false, "category": "Semantics"}
{"id": 252, "thm_name": "Juvix.Core.Main.Value.Approx.closure", "thm_stmt": "@[aesop unsafe apply]\nlemma Value.Approx.closure {env₁ body₁ env₂ body₂} :\n  (∀ a₁ a₂,\n    a₁ ≲ᵥ a₂ →\n    body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n  Value.closure env₁ body₁ ≲ᵥ Value.closure env₂ body₂", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx.lean", "imports": ["import Juvix.Core.Main.Semantics.Approx.Indexed", "import Juvix.Utils"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}, {"name": "List.map", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.A", "content": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂\n\nsyntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'\n\nsyntax \"let \" ident \" := \" expr \" in \" expr : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}, {"name": "Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}, {"name": "Value.Approx.Indexed.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "Value.Approx.Indexed.Preservation", "content": "def Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}, {"name": "Expr.Approx.Param.Indexed", "content": "def Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Eval.Indexed", "content": "inductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "Context", "content": "inductive Context : Type where\n  | hole : Context\n  | app_left : Context → Expr → Context\n  | app_right : Expr → Context → Context\n  | constr_app_left : Context → Expr → Context\n  | constr_app_right : Expr → Context → Context\n  | binop_left : (oper : BinaryOp) → (arg₁ : Context) → (arg₂ : Expr) → Context\n  | binop_right : (oper : BinaryOp) → (arg₁ : Expr) → (arg₂ : Context) → Context\n  | lambda : (var_name : String) → (body : Context) → Context\n  | save_left : (var_name : String) → (value : Context) → (body : Expr) → Context\n  | save_right : (var_name : String) → (value : Expr) → (body : Context) → Context\n  | branch_left : (constr : Name) → (var_names : List Name) → (body : Context) → (next : Expr) → Context\n  | branch_right : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Context) → Context\n  | recur : (var_name : Name) → (ctx : Context) → Context\n  deriving Inhabited, BEq"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "Expr.Approx.Indexed'", "content": "def Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "IsEmpty.forall_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}], "repo_lemmas": [{"name": "Value.Approx.Indexed.invert", "content": "@[aesop unsafe destruct]\nlemma Value.Approx.Indexed.invert {n v v'} :\n  v ≲ᵥ(n) v' →\n  Value.Approx.Indexed.Inversion n v v'"}, {"name": "Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "forall₂_trans'", "content": "theorem forall₂_trans' {α} {P Q R : α → α → Prop} {l₁ l₂ l₃}\n  (h : ∀ x y z, P x y → Q y z → R x z)\n  (h₁ : List.Forall₂ P l₁ l₂)\n  (h₂ : List.Forall₂ Q l₂ l₃)\n  : List.Forall₂ R l₁ l₃"}, {"name": "Value.Approx.Indexed.preserved", "content": "theorem Value.Approx.Indexed.preserved :\n  ∀ m n env env' e v,\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}, {"name": "Value.Approx.Indexed.preserved'", "content": "lemma Value.Approx.Indexed.preserved' {k} : Preservation k"}, {"name": "Env.Approx.Indexed'.refl", "content": "@[refl]\nlemma Env.Approx.Indexed'.refl {n env} : env ≲ₑ'(n) env"}, {"name": "Value.Approx.Indexed.refl", "content": "@[refl]\nlemma Value.Approx.Indexed.refl {n} v : v ≲ᵥ(n) v"}, {"name": "Object.Approx.Indexed'.refl'", "content": "lemma Object.Approx.Indexed'.refl' {n o} (h : ∀ v, v ≲ᵥ(n) v) : o ≲ₒ'(n) o"}, {"name": "Env.Approx.Indexed'.refl'", "content": "lemma Env.Approx.Indexed'.refl' {n env} (h : ∀ v, v ≲ᵥ(n) v) : env ≲ₑ'(n) env"}, {"name": "Object.Approx.Indexed'.refl", "content": "@[refl]\nlemma Object.Approx.Indexed'.refl {n o} : o ≲ₒ'(n) o"}, {"name": "Value.Approx.Indexed.preserved_step", "content": "lemma Value.Approx.Indexed.preserved_step {k} :\n  (∀ k' < k, Preservation k') → Preservation k"}, {"name": "Expr.Approx.Param.Indexed.anti_monotone", "content": "lemma Expr.Approx.Param.Indexed.anti_monotone {n n' env₁ env₂ e₁ e₂}\n  (h : e₁ ≲(n)⟨env₁, env₂⟩ e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲(n')⟨env₁, env₂⟩ e₂"}, {"name": "Expr.Approx.Indexed'.anti_monotone", "content": "lemma Expr.Approx.Indexed'.anti_monotone {n n' e₁ e₂}\n  (h : e₁ ≲'(n) e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲'(n') e₂"}, {"name": "Env.Approx.Indexed'.delayed", "content": "lemma Env.Approx.Indexed'.delayed {n i : Nat} {env₁ env₂ env e}\n  (h₁ : env₁ ≲ₑ'(n) env₂)\n  (h₂ : env₁[i]? = some (Object.delayed env e)) :\n  (∃ env' e', e ≲(n)⟨env, env'⟩ e' ∧ env₂[i]? = some (Object.delayed env' e')) ∨\n  ∃ env', env ≲ₑ'(n) env' ∧ env₂[i]? = some (Object.delayed env' e)"}, {"name": "Env.Approx.Indexed'.value", "content": "lemma Env.Approx.Indexed'.value {n i : Nat} {env env' v}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some (Object.value v)) :\n  ∃ v', env'[i]? = some (Object.value v') ∧ v ≲ᵥ(n) v'"}, {"name": "Env.Approx.Indexed'.get", "content": "lemma Env.Approx.Indexed'.get {n i : Nat} {env env' o₁}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some o₁) :\n  ∃ o₂, env'[i]? = some o₂ ∧ o₁ ≲ₒ'(n) o₂"}, {"name": "Value.Approx.Indexed.anti_monotone", "content": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂"}, {"name": "Value.Approx.Indexed.Preservation.anti_monotone", "content": "lemma Value.Approx.Indexed.Preservation.anti_monotone {k k'} (h : Value.Approx.Indexed.Preservation k) (h' : k' ≤ k) : Value.Approx.Indexed.Preservation k'"}, {"name": "Env.Approx.Indexed'.anti_monotone", "content": "lemma Env.Approx.Indexed'.anti_monotone {n n' env₁ env₂}\n    (h : env₁ ≲ₑ'(n) env₂)\n    (h' : n' ≤ n)\n    : env₁ ≲ₑ'(n') env₂"}, {"name": "Object.Approx.Indexed'.anti_monotone", "content": "lemma Object.Approx.Indexed'.anti_monotone {n n' o₁ o₂} (h : o₁ ≲ₒ'(n) o₂) (h' : n' ≤ n) : o₁ ≲ₒ'(n') o₂"}, {"name": "Env.Approx.Indexed'.cons", "content": "lemma Env.Approx.Indexed'.cons {n o₁ o₂ env₁ env₂}\n  (h₁ : o₁ ≲ₒ'(n) o₂)\n  (h₂ : env₁ ≲ₑ'(n) env₂) :\n  o₁ :: env₁ ≲ₑ'(n) o₂ :: env₂"}, {"name": "Env.Approx.Indexed'.concat", "content": "lemma Env.Approx.Indexed'.concat {n env₁ env₂ env₁' env₂'}\n  (h₁ : env₁ ≲ₑ'(n) env₁')\n  (h₂ : env₂ ≲ₑ'(n) env₂') :\n  env₁ ++ env₂ ≲ₑ'(n) env₁' ++ env₂'"}, {"name": "Env.Approx.Indexed'.from_value", "content": "lemma Env.Approx.Indexed'.from_value {n l₁ l₂} (h : l₁ ≲ₐ(n) l₂) :\n  List.map Object.value l₁ ≲ₑ'(n) List.map Object.value l₂"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Juvix.Core.Main.Expr.Approx.Param", "content": "def Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)"}, {"name": "Juvix.Core.Main.Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Juvix.Core.Main.Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "Juvix.Core.Main.Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}, {"name": "Juvix.Core.Main.Value.Approx.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Inversion : Value -> Value -> Prop where\n  | unit : Value.Approx.Inversion Value.unit Value.unit\n  | const {c} : Value.Approx.Inversion (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    args_rev ≲ₐ args_rev' →\n    Value.Approx.Inversion (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ a₁ a₂, a₁ ≲ᵥ a₂ → body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n    Value.Approx.Inversion (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Value.Approx.invert", "content": "@[aesop unsafe destruct]\nlemma Value.Approx.invert {v v'} :\n  v ≲ᵥ v' →\n  Value.Approx.Inversion v v'"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.trans", "content": "@[trans]\nlemma Value.Approx.Indexed.trans {n v₁ v₂ v₃} : v₁ ≲ᵥ(n) v₂ → v₂ ≲ᵥ v₃ → v₁ ≲ᵥ(n) v₃"}], "local_ctx": "import Juvix.Core.Main.Semantics.Approx.Indexed\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'\n\nnotation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'\n\nnotation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂\n\ndef Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)\n\nnotation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'\n\ninductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n\ndef Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx\n\nnotation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'\n\nnotation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂\n\ndef Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂\n\nnotation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂\n\n@[aesop safe cases]\ninductive Value.Approx.Inversion : Value -> Value -> Prop where\n  | unit : Value.Approx.Inversion Value.unit Value.unit\n  | const {c} : Value.Approx.Inversion (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    args_rev ≲ₐ args_rev' →\n    Value.Approx.Inversion (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ a₁ a₂, a₁ ≲ᵥ a₂ → body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n    Value.Approx.Inversion (Value.closure env₁ body₁) (Value.closure env₂ body₂)", "target_theorem": "@[aesop unsafe apply]\nlemma Value.Approx.closure {env₁ body₁ env₂ body₂} :\n  (∀ a₁ a₂,\n    a₁ ≲ᵥ a₂ →\n    body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n  Value.closure env₁ body₁ ≲ᵥ Value.closure env₂ body₂ :=", "ground_truth_proof": ":= by\n  intro h n\n  apply Value.Approx.Indexed.closure\n  intro n₁ n₂ hn a₁ a₂ v₁ happrox heval\n  have h₁ : ∃ v₁', a₂ ∷ env₁ ⊢ body₁ ↦ v₁' ∧ v₁ ≲ᵥ(n₂) v₁' := by\n    apply Indexed.preserved n₂ n₁ (a₁ ∷ env₁) (a₂ ∷ env₁) body₁ v₁\n    · simp [Env.Approx.Indexed']\n      constructor\n      · constructor\n        have : n₂ + n₁ = n₁ + n₂ := by linarith\n        rw [this]\n        exact happrox\n      · intros\n        rfl\n    · assumption\n  obtain ⟨v₁', heval', happrox'⟩ := h₁\n  obtain ⟨v₂, heval₂, happrox₂⟩ := h a₂ a₂ (by rfl) v₁' heval'\n  exists v₂\n  constructor\n  · assumption\n  · apply Value.Approx.Indexed.trans <;> assumption", "nesting_depth": 4, "transitive_dep_count": 68, "subset_aristotle": false, "category": "Semantics"}
{"id": 253, "thm_name": "Juvix.Core.Main.Object.Approx.Indexed'.anti_monotone", "thm_stmt": "lemma Object.Approx.Indexed'.anti_monotone {n n' o₁ o₂} (h : o₁ ≲ₒ'(n) o₂) (h' : n' ≤ n) : o₁ ≲ₒ'(n') o₂ :=\n    match h with\n    | @Object.Approx.Indexed'.value n v₁ v₂ happrox => by\n      apply Object.Approx.Indexed'.value\n      · apply Value.Approx.Indexed.anti_monotone\n        · assumption\n        · assumption\n    | @Object.Approx.Indexed'.delayed n env₁ env₂ e₁ e₂ happrox => by\n      apply Object.Approx.Indexed'.delayed\n      apply Expr.Approx.Param.Indexed.anti_monotone\n      · assumption\n      · assumption\n    | @Object.Approx.Indexed'.delayed_eq n env₁ env₂ e happrox =>\n      Object.Approx.Indexed'.delayed_eq (Env.Approx.Indexed'.anti_monotone happrox h')", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx/Indexed.lean", "imports": ["import Juvix.Utils", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Mathlib.Tactic.Linarith", "import Mathlib.Data.List.Forall2", "import Juvix.Core.Main.Semantics.Eval", "import Aesop"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "List.Forall₂.cons", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.nil", "module": "Batteries.Data.List.Basic"}], "used_repo_defs": [{"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}, {"name": "Expr.Approx.Param", "content": "def Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "exists_and_left", "module": "Init.PropLemmas"}, {"name": "exists_const", "module": "Init.PropLemmas"}, {"name": "exists_eq_left'", "module": "Init.PropLemmas"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}, {"name": "Juvix.Core.Main.Expr.Approx.Param.Indexed", "content": "def Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Juvix.Core.Main.Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.anti_monotone", "content": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂"}, {"name": "Juvix.Core.Main.Expr.Approx.Param.Indexed.anti_monotone", "content": "lemma Expr.Approx.Param.Indexed.anti_monotone {n n' env₁ env₂ e₁ e₂}\n  (h : e₁ ≲(n)⟨env₁, env₂⟩ e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲(n')⟨env₁, env₂⟩ e₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.anti_monotone", "content": "  lemma Env.Approx.Indexed'.anti_monotone {n n' env₁ env₂}\n    (h : env₁ ≲ₑ'(n) env₂)\n    (h' : n' ≤ n)\n    : env₁ ≲ₑ'(n') env₂"}], "local_ctx": "import Juvix.Core.Main.Semantics.Eval\n\nimport Juvix.Core.Main.Semantics.Eval.Indexed\n\nimport Juvix.Utils\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Data.List.Forall2\n\nimport Aesop\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))\n\nnotation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nnotation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\ndef Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)\n\nnotation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\ninductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)\n\ndef Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)\n\nnotation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'\n\nnotation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\nnotation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'", "target_theorem": "lemma Object.Approx.Indexed'.anti_monotone {n n' o₁ o₂} (h : o₁ ≲ₒ'(n) o₂) (h' : n' ≤ n) : o₁ ≲ₒ'(n') o₂ :=", "ground_truth_proof": ":=\n    match h with\n    | @Object.Approx.Indexed'.value n v₁ v₂ happrox => by\n      apply Object.Approx.Indexed'.value\n      · apply Value.Approx.Indexed.anti_monotone\n        · assumption\n        · assumption\n    | @Object.Approx.Indexed'.delayed n env₁ env₂ e₁ e₂ happrox => by\n      apply Object.Approx.Indexed'.delayed\n      apply Expr.Approx.Param.Indexed.anti_monotone\n      · assumption\n      · assumption\n    | @Object.Approx.Indexed'.delayed_eq n env₁ env₂ e happrox =>\n      Object.Approx.Indexed'.delayed_eq (Env.Approx.Indexed'.anti_monotone happrox h')", "nesting_depth": 3, "transitive_dep_count": 47, "subset_aristotle": false, "category": "Semantics"}
{"id": 254, "thm_name": "Juvix.Core.Main.Value.Approx.closure_inv", "thm_stmt": "lemma Value.Approx.closure_inv {env₁ body₁ env₂ body₂}\n  (h : Value.closure env₁ body₁ ≲ᵥ Value.closure env₂ body₂) :\n  ∀ a₁ a₂, a₁ ≲ᵥ a₂ → body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx.lean", "imports": ["import Juvix.Core.Main.Semantics.Approx.Indexed", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Juvix.Utils", "import Juvix.Core.Main.Semantics.Eval"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}, {"name": "List.map", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.A", "content": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂\n\nsyntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'\n\nsyntax \"let \" ident \" := \" expr \" in \" expr : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}, {"name": "Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}, {"name": "Value.Approx.Indexed.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "Value.Approx.Indexed.Preservation", "content": "def Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}, {"name": "Expr.Approx.Param.Indexed", "content": "def Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Eval.Indexed", "content": "inductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "Context", "content": "inductive Context : Type where\n  | hole : Context\n  | app_left : Context → Expr → Context\n  | app_right : Expr → Context → Context\n  | constr_app_left : Context → Expr → Context\n  | constr_app_right : Expr → Context → Context\n  | binop_left : (oper : BinaryOp) → (arg₁ : Context) → (arg₂ : Expr) → Context\n  | binop_right : (oper : BinaryOp) → (arg₁ : Expr) → (arg₂ : Context) → Context\n  | lambda : (var_name : String) → (body : Context) → Context\n  | save_left : (var_name : String) → (value : Context) → (body : Expr) → Context\n  | save_right : (var_name : String) → (value : Expr) → (body : Context) → Context\n  | branch_left : (constr : Name) → (var_names : List Name) → (body : Context) → (next : Expr) → Context\n  | branch_right : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Context) → Context\n  | recur : (var_name : Name) → (ctx : Context) → Context\n  deriving Inhabited, BEq"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "Expr.Approx.Indexed'", "content": "def Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "IsEmpty.forall_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}], "repo_lemmas": [{"name": "Value.Approx.Indexed.invert", "content": "@[aesop unsafe destruct]\nlemma Value.Approx.Indexed.invert {n v v'} :\n  v ≲ᵥ(n) v' →\n  Value.Approx.Indexed.Inversion n v v'"}, {"name": "Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "forall₂_trans'", "content": "theorem forall₂_trans' {α} {P Q R : α → α → Prop} {l₁ l₂ l₃}\n  (h : ∀ x y z, P x y → Q y z → R x z)\n  (h₁ : List.Forall₂ P l₁ l₂)\n  (h₂ : List.Forall₂ Q l₂ l₃)\n  : List.Forall₂ R l₁ l₃"}, {"name": "Value.Approx.Indexed.preserved", "content": "theorem Value.Approx.Indexed.preserved :\n  ∀ m n env env' e v,\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}, {"name": "Value.Approx.Indexed.preserved'", "content": "lemma Value.Approx.Indexed.preserved' {k} : Preservation k"}, {"name": "Env.Approx.Indexed'.refl", "content": "@[refl]\nlemma Env.Approx.Indexed'.refl {n env} : env ≲ₑ'(n) env"}, {"name": "Value.Approx.Indexed.refl", "content": "@[refl]\nlemma Value.Approx.Indexed.refl {n} v : v ≲ᵥ(n) v"}, {"name": "Object.Approx.Indexed'.refl'", "content": "lemma Object.Approx.Indexed'.refl' {n o} (h : ∀ v, v ≲ᵥ(n) v) : o ≲ₒ'(n) o"}, {"name": "Env.Approx.Indexed'.refl'", "content": "lemma Env.Approx.Indexed'.refl' {n env} (h : ∀ v, v ≲ᵥ(n) v) : env ≲ₑ'(n) env"}, {"name": "Object.Approx.Indexed'.refl", "content": "@[refl]\nlemma Object.Approx.Indexed'.refl {n o} : o ≲ₒ'(n) o"}, {"name": "Value.Approx.Indexed.preserved_step", "content": "lemma Value.Approx.Indexed.preserved_step {k} :\n  (∀ k' < k, Preservation k') → Preservation k"}, {"name": "Expr.Approx.Param.Indexed.anti_monotone", "content": "lemma Expr.Approx.Param.Indexed.anti_monotone {n n' env₁ env₂ e₁ e₂}\n  (h : e₁ ≲(n)⟨env₁, env₂⟩ e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲(n')⟨env₁, env₂⟩ e₂"}, {"name": "Expr.Approx.Indexed'.anti_monotone", "content": "lemma Expr.Approx.Indexed'.anti_monotone {n n' e₁ e₂}\n  (h : e₁ ≲'(n) e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲'(n') e₂"}, {"name": "Env.Approx.Indexed'.delayed", "content": "lemma Env.Approx.Indexed'.delayed {n i : Nat} {env₁ env₂ env e}\n  (h₁ : env₁ ≲ₑ'(n) env₂)\n  (h₂ : env₁[i]? = some (Object.delayed env e)) :\n  (∃ env' e', e ≲(n)⟨env, env'⟩ e' ∧ env₂[i]? = some (Object.delayed env' e')) ∨\n  ∃ env', env ≲ₑ'(n) env' ∧ env₂[i]? = some (Object.delayed env' e)"}, {"name": "Env.Approx.Indexed'.value", "content": "lemma Env.Approx.Indexed'.value {n i : Nat} {env env' v}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some (Object.value v)) :\n  ∃ v', env'[i]? = some (Object.value v') ∧ v ≲ᵥ(n) v'"}, {"name": "Env.Approx.Indexed'.get", "content": "lemma Env.Approx.Indexed'.get {n i : Nat} {env env' o₁}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some o₁) :\n  ∃ o₂, env'[i]? = some o₂ ∧ o₁ ≲ₒ'(n) o₂"}, {"name": "Value.Approx.Indexed.anti_monotone", "content": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂"}, {"name": "Value.Approx.Indexed.Preservation.anti_monotone", "content": "lemma Value.Approx.Indexed.Preservation.anti_monotone {k k'} (h : Value.Approx.Indexed.Preservation k) (h' : k' ≤ k) : Value.Approx.Indexed.Preservation k'"}, {"name": "Env.Approx.Indexed'.anti_monotone", "content": "lemma Env.Approx.Indexed'.anti_monotone {n n' env₁ env₂}\n    (h : env₁ ≲ₑ'(n) env₂)\n    (h' : n' ≤ n)\n    : env₁ ≲ₑ'(n') env₂"}, {"name": "Object.Approx.Indexed'.anti_monotone", "content": "lemma Object.Approx.Indexed'.anti_monotone {n n' o₁ o₂} (h : o₁ ≲ₒ'(n) o₂) (h' : n' ≤ n) : o₁ ≲ₒ'(n') o₂"}, {"name": "Env.Approx.Indexed'.cons", "content": "lemma Env.Approx.Indexed'.cons {n o₁ o₂ env₁ env₂}\n  (h₁ : o₁ ≲ₒ'(n) o₂)\n  (h₂ : env₁ ≲ₑ'(n) env₂) :\n  o₁ :: env₁ ≲ₑ'(n) o₂ :: env₂"}, {"name": "Env.Approx.Indexed'.concat", "content": "lemma Env.Approx.Indexed'.concat {n env₁ env₂ env₁' env₂'}\n  (h₁ : env₁ ≲ₑ'(n) env₁')\n  (h₂ : env₂ ≲ₑ'(n) env₂') :\n  env₁ ++ env₂ ≲ₑ'(n) env₁' ++ env₂'"}, {"name": "Env.Approx.Indexed'.from_value", "content": "lemma Env.Approx.Indexed'.from_value {n l₁ l₂} (h : l₁ ≲ₐ(n) l₂) :\n  List.map Object.value l₁ ≲ₑ'(n) List.map Object.value l₂"}, {"name": "Eval.deterministic", "content": "theorem Eval.deterministic {env e v₁ v₂} (h₁ : env ⊢ e ↦ v₁) (h₂ : env ⊢ e ↦ v₂) : v₁ = v₂"}, {"name": "Eval.toIndexed", "content": "lemma Eval.toIndexed {env e v} (h : env ⊢ e ↦ v) : ∃ n, env ⊢ e ↦(n) v"}, {"name": "Eval.Indexed.monotone", "content": "lemma Eval.Indexed.monotone {n n' env e v} (h : env ⊢ e ↦(n) v) (h' : n ≤ n') : env ⊢ e ↦(n') v"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Juvix.Core.Main.Expr.Approx.Param", "content": "def Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)"}, {"name": "Juvix.Core.Main.Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Juvix.Core.Main.Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "Juvix.Core.Main.Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}], "used_local_lemmas": [], "local_ctx": "import Juvix.Core.Main.Semantics.Approx.Indexed\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'\n\nnotation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'\n\nnotation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂\n\ndef Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)\n\nnotation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'\n\ninductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n\ndef Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx\n\nnotation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'\n\nnotation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂\n\ndef Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂\n\nnotation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "target_theorem": "lemma Value.Approx.closure_inv {env₁ body₁ env₂ body₂}\n  (h : Value.closure env₁ body₁ ≲ᵥ Value.closure env₂ body₂) :\n  ∀ a₁ a₂, a₁ ≲ᵥ a₂ → body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂ :=", "ground_truth_proof": ":= by\n  intro a₁ a₂ ha v₁ h'\n  suffices ∀ n, ∃ v₂, (a₂ ∷ env₂) ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n) v₂ by\n    obtain ⟨v₂, _, _⟩ := this 0\n    exists v₂\n    constructor\n    · assumption\n    · intro n\n      obtain ⟨v₂', _, _⟩ := this n\n      have : v₂ = v₂' := by\n        apply Eval.deterministic <;> assumption\n      subst this\n      simp_all only\n  intro n₂\n  obtain ⟨n₁, h''⟩ := Eval.toIndexed h'\n  invert (h (n₁ + n₂ + 1))\n  case closure ch =>\n    apply ch (n₁ := n₁) (n₂ := n₂)\n    · linarith\n    · apply ha\n    · assumption", "nesting_depth": 5, "transitive_dep_count": 72, "subset_aristotle": false, "category": "Semantics"}
{"id": 255, "thm_name": "Juvix.Core.Main.Value.Approx.preserved", "thm_stmt": "theorem Value.Approx.preserved :\n  ∀ env env' e v,\n    env ≲ₑ env' →\n    env ⊢ e ↦ v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ v'", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx.lean", "imports": ["import Juvix.Core.Main.Semantics.Approx.Indexed", "import Juvix.Core.Main.Semantics.Eval", "import Juvix.Core.Main.Semantics.Eval.Indexed"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂.cons", "module": "Batteries.Data.List.Basic"}, {"name": "List.map", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\nsyntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))\n\nsyntax \"let \" ident \" := \" expr \" in \" expr : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}, {"name": "Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Eval.Indexed", "content": "inductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "Value.Approx.Indexed.Preservation", "content": "def Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}, {"name": "Value.Approx.Indexed.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "Expr.Approx.Param.Indexed", "content": "def Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Context", "content": "inductive Context : Type where\n  | hole : Context\n  | app_left : Context → Expr → Context\n  | app_right : Expr → Context → Context\n  | constr_app_left : Context → Expr → Context\n  | constr_app_right : Expr → Context → Context\n  | binop_left : (oper : BinaryOp) → (arg₁ : Context) → (arg₂ : Expr) → Context\n  | binop_right : (oper : BinaryOp) → (arg₁ : Expr) → (arg₂ : Context) → Context\n  | lambda : (var_name : String) → (body : Context) → Context\n  | save_left : (var_name : String) → (value : Context) → (body : Expr) → Context\n  | save_right : (var_name : String) → (value : Expr) → (body : Context) → Context\n  | branch_left : (constr : Name) → (var_names : List Name) → (body : Context) → (next : Expr) → Context\n  | branch_right : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Context) → Context\n  | recur : (var_name : Name) → (ctx : Context) → Context\n  deriving Inhabited, BEq"}, {"name": "Expr.Approx.Indexed'", "content": "def Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Eval.Indexed.toEval", "content": "lemma Eval.Indexed.toEval {n env e v} (h : env ⊢ e ↦(n) v) : env ⊢ e ↦ v"}, {"name": "Value.Approx.Indexed.preserved", "content": "theorem Value.Approx.Indexed.preserved :\n  ∀ m n env env' e v,\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}, {"name": "Value.Approx.Indexed.preserved'", "content": "lemma Value.Approx.Indexed.preserved' {k} : Preservation k"}, {"name": "Env.Approx.Indexed'.refl", "content": "@[refl]\nlemma Env.Approx.Indexed'.refl {n env} : env ≲ₑ'(n) env"}, {"name": "Value.Approx.Indexed.refl", "content": "@[refl]\nlemma Value.Approx.Indexed.refl {n} v : v ≲ᵥ(n) v"}, {"name": "Object.Approx.Indexed'.refl'", "content": "lemma Object.Approx.Indexed'.refl' {n o} (h : ∀ v, v ≲ᵥ(n) v) : o ≲ₒ'(n) o"}, {"name": "Env.Approx.Indexed'.refl'", "content": "lemma Env.Approx.Indexed'.refl' {n env} (h : ∀ v, v ≲ᵥ(n) v) : env ≲ₑ'(n) env"}, {"name": "Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "Object.Approx.Indexed'.refl", "content": "@[refl]\nlemma Object.Approx.Indexed'.refl {n o} : o ≲ₒ'(n) o"}, {"name": "Value.Approx.Indexed.preserved_step", "content": "lemma Value.Approx.Indexed.preserved_step {k} :\n  (∀ k' < k, Preservation k') → Preservation k"}, {"name": "Expr.Approx.Param.Indexed.anti_monotone", "content": "lemma Expr.Approx.Param.Indexed.anti_monotone {n n' env₁ env₂ e₁ e₂}\n  (h : e₁ ≲(n)⟨env₁, env₂⟩ e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲(n')⟨env₁, env₂⟩ e₂"}, {"name": "Expr.Approx.Indexed'.anti_monotone", "content": "lemma Expr.Approx.Indexed'.anti_monotone {n n' e₁ e₂}\n  (h : e₁ ≲'(n) e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲'(n') e₂"}, {"name": "Env.Approx.Indexed'.delayed", "content": "lemma Env.Approx.Indexed'.delayed {n i : Nat} {env₁ env₂ env e}\n  (h₁ : env₁ ≲ₑ'(n) env₂)\n  (h₂ : env₁[i]? = some (Object.delayed env e)) :\n  (∃ env' e', e ≲(n)⟨env, env'⟩ e' ∧ env₂[i]? = some (Object.delayed env' e')) ∨\n  ∃ env', env ≲ₑ'(n) env' ∧ env₂[i]? = some (Object.delayed env' e)"}, {"name": "Env.Approx.Indexed'.value", "content": "lemma Env.Approx.Indexed'.value {n i : Nat} {env env' v}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some (Object.value v)) :\n  ∃ v', env'[i]? = some (Object.value v') ∧ v ≲ᵥ(n) v'"}, {"name": "Env.Approx.Indexed'.get", "content": "lemma Env.Approx.Indexed'.get {n i : Nat} {env env' o₁}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some o₁) :\n  ∃ o₂, env'[i]? = some o₂ ∧ o₁ ≲ₒ'(n) o₂"}, {"name": "Value.Approx.Indexed.anti_monotone", "content": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂"}, {"name": "Value.Approx.Indexed.Preservation.anti_monotone", "content": "lemma Value.Approx.Indexed.Preservation.anti_monotone {k k'} (h : Value.Approx.Indexed.Preservation k) (h' : k' ≤ k) : Value.Approx.Indexed.Preservation k'"}, {"name": "Env.Approx.Indexed'.anti_monotone", "content": "lemma Env.Approx.Indexed'.anti_monotone {n n' env₁ env₂}\n    (h : env₁ ≲ₑ'(n) env₂)\n    (h' : n' ≤ n)\n    : env₁ ≲ₑ'(n') env₂"}, {"name": "Object.Approx.Indexed'.anti_monotone", "content": "lemma Object.Approx.Indexed'.anti_monotone {n n' o₁ o₂} (h : o₁ ≲ₒ'(n) o₂) (h' : n' ≤ n) : o₁ ≲ₒ'(n') o₂"}, {"name": "Env.Approx.Indexed'.cons", "content": "lemma Env.Approx.Indexed'.cons {n o₁ o₂ env₁ env₂}\n  (h₁ : o₁ ≲ₒ'(n) o₂)\n  (h₂ : env₁ ≲ₑ'(n) env₂) :\n  o₁ :: env₁ ≲ₑ'(n) o₂ :: env₂"}, {"name": "Env.Approx.Indexed'.concat", "content": "lemma Env.Approx.Indexed'.concat {n env₁ env₂ env₁' env₂'}\n  (h₁ : env₁ ≲ₑ'(n) env₁')\n  (h₂ : env₂ ≲ₑ'(n) env₂') :\n  env₁ ++ env₂ ≲ₑ'(n) env₁' ++ env₂'"}, {"name": "Env.Approx.Indexed'.from_value", "content": "lemma Env.Approx.Indexed'.from_value {n l₁ l₂} (h : l₁ ≲ₐ(n) l₂) :\n  List.map Object.value l₁ ≲ₑ'(n) List.map Object.value l₂"}, {"name": "Eval.deterministic", "content": "theorem Eval.deterministic {env e v₁ v₂} (h₁ : env ⊢ e ↦ v₁) (h₂ : env ⊢ e ↦ v₂) : v₁ = v₂"}, {"name": "Eval.toIndexed", "content": "lemma Eval.toIndexed {env e v} (h : env ⊢ e ↦ v) : ∃ n, env ⊢ e ↦(n) v"}, {"name": "Eval.Indexed.monotone", "content": "lemma Eval.Indexed.monotone {n n' env e v} (h : env ⊢ e ↦(n) v) (h' : n ≤ n') : env ⊢ e ↦(n') v"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Juvix.Core.Main.Expr.Approx.Param", "content": "def Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)"}, {"name": "Juvix.Core.Main.Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Juvix.Core.Main.Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "Juvix.Core.Main.Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Env.Approx.cons", "content": "lemma Env.Approx.cons {env₁ env₂ o₁ o₂} :\n  o₁ ≲ₒ o₂ → env₁ ≲ₑ env₂ → (o₁ :: env₁) ≲ₑ (o₂ :: env₂)"}, {"name": "Juvix.Core.Main.Object.Approx.toIndexed", "content": "lemma Object.Approx.toIndexed {o₁ o₂} : o₁ ≲ₒ o₂ → ∀ n, o₁ ≲ₒ'(n) o₂"}, {"name": "Juvix.Core.Main.Env.Approx.toIndexed", "content": "lemma Env.Approx.toIndexed {env₁ env₂} : env₁ ≲ₑ env₂ → ∀ n, env₁ ≲ₑ'(n) env₂"}], "local_ctx": "import Juvix.Core.Main.Semantics.Approx.Indexed\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'\n\nnotation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'\n\nnotation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂\n\ndef Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)\n\nnotation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'\n\ninductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n\ndef Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx\n\nnotation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'\n\nnotation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂\n\ndef Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂\n\nnotation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "target_theorem": "theorem Value.Approx.preserved :\n  ∀ env env' e v,\n    env ≲ₑ env' →\n    env ⊢ e ↦ v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ v' :=", "ground_truth_proof": ":= by\n  intro env env' e v h₁ h₂\n  suffices ∀ n, ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(n) v' by\n    obtain ⟨v', heval', happrox'⟩ := this 0\n    exists v'\n    constructor\n    · assumption\n    · intro n\n      obtain ⟨v'', heval'', happrox''⟩ := this n\n      have : v' = v'' := by\n        apply Eval.deterministic <;> assumption\n      subst this\n      exact happrox''\n  intro m\n  obtain ⟨n, h₂'⟩ := Eval.toIndexed h₂\n  have h₁' : env ≲ₑ'(m + n) env' := by\n    apply Env.Approx.toIndexed\n    assumption\n  obtain ⟨v', heval', happrox'⟩ := Value.Approx.Indexed.preserved (env := env) (env' := env') m n e v h₁' h₂'\n  aesop", "nesting_depth": 4, "transitive_dep_count": 66, "subset_aristotle": false, "category": "Semantics"}
{"id": 256, "thm_name": "Juvix.Core.Main.Env.Approx.Indexed'.delayed", "thm_stmt": "lemma Env.Approx.Indexed'.delayed {n i : Nat} {env₁ env₂ env e}\n  (h₁ : env₁ ≲ₑ'(n) env₂)\n  (h₂ : env₁[i]? = some (Object.delayed env e)) :\n  (∃ env' e', e ≲(n)⟨env, env'⟩ e' ∧ env₂[i]? = some (Object.delayed env' e')) ∨\n  ∃ env', env ≲ₑ'(n) env' ∧ env₂[i]? = some (Object.delayed env' e)", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx/Indexed.lean", "imports": ["import Juvix.Utils", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Mathlib.Tactic.Linarith", "import Mathlib.Data.List.Forall2", "import Juvix.Core.Main.Semantics.Eval", "import Aesop"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}], "used_repo_defs": [{"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "List.Forall₂.get", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.Forall₂.length_eq", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.getElem?_eq_some_iff", "module": "Init.Data.List.Lemmas"}, {"name": "List.get_eq_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "forall_true_left", "module": "Mathlib.Logic.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Juvix.Core.Main.Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Env.Approx.Indexed'.get", "content": "lemma Env.Approx.Indexed'.get {n i : Nat} {env env' o₁}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some o₁) :\n  ∃ o₂, env'[i]? = some o₂ ∧ o₁ ≲ₒ'(n) o₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.value", "content": "lemma Env.Approx.Indexed'.value {n i : Nat} {env env' v}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some (Object.value v)) :\n  ∃ v', env'[i]? = some (Object.value v') ∧ v ≲ᵥ(n) v'"}], "local_ctx": "import Juvix.Core.Main.Semantics.Eval\n\nimport Juvix.Core.Main.Semantics.Eval.Indexed\n\nimport Juvix.Utils\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Data.List.Forall2\n\nimport Aesop\n\nnamespace Juvix.Core.Main\n\nnotation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nnotation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nnotation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\ninductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)\n\ndef Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)\n\nnotation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'\n\nnotation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\nnotation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'", "target_theorem": "lemma Env.Approx.Indexed'.delayed {n i : Nat} {env₁ env₂ env e}\n  (h₁ : env₁ ≲ₑ'(n) env₂)\n  (h₂ : env₁[i]? = some (Object.delayed env e)) :\n  (∃ env' e', e ≲(n)⟨env, env'⟩ e' ∧ env₂[i]? = some (Object.delayed env' e')) ∨\n  ∃ env', env ≲ₑ'(n) env' ∧ env₂[i]? = some (Object.delayed env' e) :=", "ground_truth_proof": ":= by\n  obtain ⟨o, h, happrox⟩ := Env.Approx.Indexed'.get h₁ h₂\n  cases h₃ : o\n  case value v =>\n    rw [h₃] at happrox\n    contradiction\n  case delayed env' e' =>\n    rw [h₃] at happrox\n    cases happrox\n    case delayed h =>\n      left\n      subst h₃\n      exists env'\n      exists e'\n    case delayed_eq =>\n      right\n      subst h₃\n      exists env'", "nesting_depth": 3, "transitive_dep_count": 15, "subset_aristotle": false, "category": "Semantics"}
{"id": 257, "thm_name": "Juvix.Core.Main.Value.Approx.Indexed.preserved'", "thm_stmt": "lemma Value.Approx.Indexed.preserved' {k} : Preservation k", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx/Indexed.lean", "imports": ["import Juvix.Utils", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Mathlib.Tactic.Linarith", "import Mathlib.Data.List.Forall2", "import Juvix.Core.Main.Semantics.Eval", "import Aesop"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.cons", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.nil", "module": "Batteries.Data.List.Basic"}, {"name": "List.cons", "module": "Init.Prelude"}, {"name": "List.nil", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "List.map", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nsyntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr"}, {"name": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Index", "content": "notation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))\n\nsyntax \"let \" ident \" := \" expr \" in \" expr : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}, {"name": "Expr.Approx.Param", "content": "def Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)"}, {"name": "Eval.Indexed", "content": "inductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "Context", "content": "inductive Context : Type where\n  | hole : Context\n  | app_left : Context → Expr → Context\n  | app_right : Expr → Context → Context\n  | constr_app_left : Context → Expr → Context\n  | constr_app_right : Expr → Context → Context\n  | binop_left : (oper : BinaryOp) → (arg₁ : Context) → (arg₂ : Expr) → Context\n  | binop_right : (oper : BinaryOp) → (arg₁ : Expr) → (arg₂ : Context) → Context\n  | lambda : (var_name : String) → (body : Context) → Context\n  | save_left : (var_name : String) → (value : Context) → (body : Expr) → Context\n  | save_right : (var_name : String) → (value : Expr) → (body : Context) → Context\n  | branch_left : (constr : Name) → (var_names : List Name) → (body : Context) → (next : Expr) → Context\n  | branch_right : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Context) → Context\n  | recur : (var_name : Name) → (ctx : Context) → Context\n  deriving Inhabited, BEq"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "exists_and_left", "module": "Init.PropLemmas"}, {"name": "exists_const", "module": "Init.PropLemmas"}, {"name": "exists_eq_left'", "module": "Init.PropLemmas"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "List.forall₂_same", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.Forall₂.get", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.Forall₂.length_eq", "module": "Mathlib.Data.List.Forall2"}, {"name": "List.getElem?_eq_some_iff", "module": "Init.Data.List.Lemmas"}, {"name": "List.get_eq_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "forall_true_left", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.le_eq", "module": "Init.Data.Nat.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}, {"name": "Juvix.Core.Main.Expr.Approx.Param.Indexed", "content": "def Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Juvix.Core.Main.Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}, {"name": "Juvix.Core.Main.Expr.Approx.Indexed'", "content": "def Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.Preservation", "content": "def Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Env.Approx.Indexed'.get", "content": "lemma Env.Approx.Indexed'.get {n i : Nat} {env env' o₁}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some o₁) :\n  ∃ o₂, env'[i]? = some o₂ ∧ o₁ ≲ₒ'(n) o₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.value", "content": "lemma Env.Approx.Indexed'.value {n i : Nat} {env env' v}\n  (h₁ : env ≲ₑ'(n) env')\n  (h₂ : env[i]? = some (Object.value v)) :\n  ∃ v', env'[i]? = some (Object.value v') ∧ v ≲ᵥ(n) v'"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.delayed", "content": "lemma Env.Approx.Indexed'.delayed {n i : Nat} {env₁ env₂ env e}\n  (h₁ : env₁ ≲ₑ'(n) env₂)\n  (h₂ : env₁[i]? = some (Object.delayed env e)) :\n  (∃ env' e', e ≲(n)⟨env, env'⟩ e' ∧ env₂[i]? = some (Object.delayed env' e')) ∨\n  ∃ env', env ≲ₑ'(n) env' ∧ env₂[i]? = some (Object.delayed env' e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.from_value", "content": "lemma Env.Approx.Indexed'.from_value {n l₁ l₂} (h : l₁ ≲ₐ(n) l₂) :\n  List.map Object.value l₁ ≲ₑ'(n) List.map Object.value l₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.concat", "content": "lemma Env.Approx.Indexed'.concat {n env₁ env₂ env₁' env₂'}\n  (h₁ : env₁ ≲ₑ'(n) env₁')\n  (h₂ : env₂ ≲ₑ'(n) env₂') :\n  env₁ ++ env₂ ≲ₑ'(n) env₁' ++ env₂'"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.cons", "content": "lemma Env.Approx.Indexed'.cons {n o₁ o₂ env₁ env₂}\n  (h₁ : o₁ ≲ₒ'(n) o₂)\n  (h₂ : env₁ ≲ₑ'(n) env₂) :\n  o₁ :: env₁ ≲ₑ'(n) o₂ :: env₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.anti_monotone", "content": "lemma Value.Approx.Indexed.anti_monotone {n n' v₁ v₂} (h : v₁ ≲ᵥ(n) v₂) (h' : n' ≤ n) : v₁ ≲ᵥ(n') v₂"}, {"name": "Juvix.Core.Main.Expr.Approx.Param.Indexed.anti_monotone", "content": "lemma Expr.Approx.Param.Indexed.anti_monotone {n n' env₁ env₂ e₁ e₂}\n  (h : e₁ ≲(n)⟨env₁, env₂⟩ e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲(n')⟨env₁, env₂⟩ e₂"}, {"name": "Juvix.Core.Main.Expr.Approx.Indexed'.anti_monotone", "content": "lemma Expr.Approx.Indexed'.anti_monotone {n n' e₁ e₂}\n  (h : e₁ ≲'(n) e₂)\n  (h' : n' ≤ n)\n  : e₁ ≲'(n') e₂"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'.anti_monotone", "content": "  lemma Env.Approx.Indexed'.anti_monotone {n n' env₁ env₂}\n    (h : env₁ ≲ₑ'(n) env₂)\n    (h' : n' ≤ n)\n    : env₁ ≲ₑ'(n') env₂"}, {"name": "Juvix.Core.Main.Object.Approx.Indexed'.anti_monotone", "content": "  lemma Object.Approx.Indexed'.anti_monotone {n n' o₁ o₂} (h : o₁ ≲ₒ'(n) o₂) (h' : n' ≤ n) : o₁ ≲ₒ'(n') o₂"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.Preservation.anti_monotone", "content": "lemma Value.Approx.Indexed.Preservation.anti_monotone {k k'} (h : Value.Approx.Indexed.Preservation k) (h' : k' ≤ k) : Value.Approx.Indexed.Preservation k'"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.preserved_step", "content": "lemma Value.Approx.Indexed.preserved_step {k} :\n  (∀ k' < k, Preservation k') → Preservation k"}], "local_ctx": "import Juvix.Core.Main.Semantics.Eval\n\nimport Juvix.Core.Main.Semantics.Eval.Indexed\n\nimport Juvix.Utils\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Data.List.Forall2\n\nimport Aesop\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))\n\nnotation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nnotation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\ndef Expr.Approx.Param.Indexed (n : Nat) (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)\n\nnotation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\ninductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)\n\ndef Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)\n\nnotation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'\n\nnotation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\ndef Expr.Approx.Indexed' (n : Nat) (e₁ e₂ : Expr) : Prop :=\n  (∀ n₁ n₂ v₁, n₁ + n₂ ≤ n →\n    ∀ env₁ env₂, env₁ ≲ₑ'(n₁ + n₂) env₂ → env₁ ⊢ e₁ ↦(n₁) v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂)\n\nnotation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'\n\n@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)\n\nend\n\ndef Value.Approx.Indexed.Preservation (k : Nat) : Prop :=\n  ∀ m n env env' e v,\n    m + n < k →\n    env ≲ₑ'(m + n) env' →\n    env ⊢ e ↦(n) v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ(m) v'", "target_theorem": "lemma Value.Approx.Indexed.preserved' {k} : Preservation k :=", "ground_truth_proof": ":= by\n  suffices ∀ k' ≤ k, Preservation k' by\n    aesop\n  induction k with\n  | zero =>\n    intro k' hk'\n    refine preserved_step ?_\n    intro k hk\n    linarith\n  | succ k ih =>\n    intro k' hk'\n    cases hk'\n    case succ.refl =>\n      refine preserved_step ?_\n      intro k' hk'\n      apply ih\n      linarith\n    case succ.step =>\n      apply ih\n      simp_all only [Nat.le_eq]", "nesting_depth": 9, "transitive_dep_count": 80, "subset_aristotle": false, "category": "Semantics"}
{"id": 258, "thm_name": "Juvix.Core.Main.Object.Approx.Indexed'.refl'", "thm_stmt": "lemma Object.Approx.Indexed'.refl' {n o} (h : ∀ v, v ≲ᵥ(n) v) : o ≲ₒ'(n) o", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx/Indexed.lean", "imports": ["import Juvix.Utils", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Mathlib.Tactic.Linarith", "import Mathlib.Data.List.Forall2", "import Juvix.Core.Main.Semantics.Eval", "import Aesop"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.cons", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.nil", "module": "Batteries.Data.List.Basic"}, {"name": "List.cons", "module": "Init.Prelude"}, {"name": "List.nil", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Juvix.Core.Main.Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Env.Approx.Indexed'.refl'", "content": "  lemma Env.Approx.Indexed'.refl' {n env} (h : ∀ v, v ≲ᵥ(n) v) : env ≲ₑ'(n) env"}], "local_ctx": "import Juvix.Core.Main.Semantics.Eval\n\nimport Juvix.Core.Main.Semantics.Eval.Indexed\n\nimport Juvix.Utils\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Data.List.Forall2\n\nimport Aesop\n\nnamespace Juvix.Core.Main\n\nnotation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nnotation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nnotation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\ninductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)\n\ndef Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)\n\nnotation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'\n\nnotation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\nnotation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'\n\nend", "target_theorem": "lemma Object.Approx.Indexed'.refl' {n o} (h : ∀ v, v ≲ᵥ(n) v) : o ≲ₒ'(n) o :=", "ground_truth_proof": ":= by\n    cases o\n    case value v =>\n      apply Object.Approx.Indexed'.value\n      apply h\n    case delayed env e =>\n      apply Object.Approx.Indexed'.delayed_eq\n      exact Env.Approx.Indexed'.refl' h", "nesting_depth": 3, "transitive_dep_count": 14, "subset_aristotle": false, "category": "Semantics"}
{"id": 259, "thm_name": "Juvix.Core.Main.Expr.Equiv.eval_const", "thm_stmt": "lemma Expr.Equiv.eval_const {op a₁ a₂ i₁ i₂ i₃} :\n  a₁ ≈ Expr.const (Constant.int i₁) →\n  a₂ ≈ Expr.const (Constant.int i₂) →\n  i₃ = eval_binop_int op i₁ i₂ →\n  Expr.binop op a₁ a₂ ≈\n    Expr.const (Constant.int i₃)", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Equiv.lean", "imports": ["import Juvix.Core.Main.Semantics.Approx"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}], "used_repo_defs": [{"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\nsyntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))"}, {"name": "notation:40 e₁:40 \" ≈ \" e₂:40 => Expr.Equiv e₁ e₂", "content": "notation:40 e₁:40 \" ≈ \" e₂:40 => Expr.Equiv e₁ e₂"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}, {"name": "Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Eval.Indexed", "content": "inductive Eval.Indexed : Nat → Env → Expr → Value → Prop where\n  | var {n env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval.Indexed n env (Expr.var name idx) val\n  | var_rec {n env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval.Indexed n env' expr val →\n    Eval.Indexed n env (Expr.var name idx) val\n  | unit {n env} :\n    Eval.Indexed n env Expr.unit Value.unit\n  | const {n env c} :\n    Eval.Indexed n env (Expr.const c) (Value.const c)\n  | constr {n env name} :\n    Eval.Indexed n env (Expr.constr name) (Value.constr_app name [])\n  | app {n n₁ n₂ env env' f body arg val val'} :\n    n₁ + n₂ + 1 ≤ n →\n    Eval.Indexed n₁ env f (Value.closure env' body) →\n    Eval.Indexed (n₁ + 1) env arg val →\n    Eval.Indexed n₂ (val ∷ env') body val' →\n    Eval.Indexed n env (Expr.app f arg) val'\n  | constr_app {n n' env ctr ctr_name ctr_args_rev arg val} :\n    n' < n →\n    Eval.Indexed n env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval.Indexed n' env arg val →\n    Eval.Indexed n env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {n env op arg₁ arg₂ val₁ val₂} :\n    Eval.Indexed n env arg₁ (Value.const (Constant.int val₁)) →\n    Eval.Indexed n env arg₂ (Value.const (Constant.int val₂)) →\n    Eval.Indexed n env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {n env name body} :\n    Eval.Indexed n env (Expr.lambda name body) (Value.closure env body)\n  | save {n n₁ n₂ env name value body val val'} :\n    n₁ + n₂ ≤ n →\n    Eval.Indexed n₁ env value val →\n    Eval.Indexed n₂ (val ∷ env) body val' →\n    Eval.Indexed n env (Expr.save name value body) val'\n  | branch_matches {n n' env name args_rev body val} :\n    n' < n →\n    Eval.Indexed n' (args_rev.map Object.value ++ env) body val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {n env name name' args_rev next val} :\n    name ≠ name' →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) next val →\n    Eval.Indexed n (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {n n' env name body val} :\n    n' < n →\n    Eval.Indexed n' (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval.Indexed n env (Expr.recur name body) val"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Expr.Approx.eval_const₁", "content": "lemma Expr.Approx.eval_const₁ {op a₁ a₂ i₁ i₂ i₃} :\n  a₁ ≲ Expr.const (Constant.int i₁) →\n  a₂ ≲ Expr.const (Constant.int i₂) →\n  i₃ = eval_binop_int op i₁ i₂ →\n  Expr.binop op a₁ a₂ ≲\n    Expr.const (Constant.int i₃)"}, {"name": "Expr.Approx.const_right", "content": "lemma Expr.Approx.const_right {c e env v} :\n  e ≲ Expr.const c →\n  env ⊢ e ↦ v →\n  v = Value.const c"}, {"name": "Expr.Approx.preservation", "content": "lemma Expr.Approx.preservation {e₁ e₂} :\n  e₁ ≲ e₂ ↔\n  ∀ env v₁, env ⊢ e₁ ↦ v₁ → ∃ v₂, env ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂"}, {"name": "Expr.Approx.to_preservation", "content": "lemma Expr.Approx.to_preservation {e₁ e₂} :\n  e₁ ≲ e₂ →\n  ∀ env v₁, env ⊢ e₁ ↦ v₁ → ∃ v₂, env ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂"}, {"name": "Expr.Approx.from_preservation", "content": "lemma Expr.Approx.from_preservation {e₁ e₂}\n  (h : ∀ env v₁, env ⊢ e₁ ↦ v₁ → ∃ v₂, env ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)\n  : e₁ ≲ e₂"}, {"name": "Value.Approx.preserved", "content": "theorem Value.Approx.preserved :\n  ∀ env env' e v,\n    env ≲ₑ env' →\n    env ⊢ e ↦ v →\n    ∃ v', env' ⊢ e ↦ v' ∧ v ≲ᵥ v'"}, {"name": "Env.Approx.toIndexed", "content": "lemma Env.Approx.toIndexed {env₁ env₂} : env₁ ≲ₑ env₂ → ∀ n, env₁ ≲ₑ'(n) env₂"}, {"name": "Object.Approx.toIndexed", "content": "lemma Object.Approx.toIndexed {o₁ o₂} : o₁ ≲ₒ o₂ → ∀ n, o₁ ≲ₒ'(n) o₂"}, {"name": "Env.Approx.cons", "content": "lemma Env.Approx.cons {env₁ env₂ o₁ o₂} :\n  o₁ ≲ₒ o₂ → env₁ ≲ₑ env₂ → (o₁ :: env₁) ≲ₑ (o₂ :: env₂)"}, {"name": "Expr.Approx.eval_const₂", "content": "lemma Expr.Approx.eval_const₂ {op a₁ a₂ i₁ i₂ i₃} :\n  Expr.const (Constant.int i₁) ≲ a₁ →\n  Expr.const (Constant.int i₂) ≲ a₂ →\n  i₃ = eval_binop_int op i₁ i₂ →\n  Expr.const (Constant.int i₃) ≲\n  Expr.binop op a₁ a₂"}, {"name": "Expr.Approx.const_left", "content": "lemma Expr.Approx.const_left {c e env} :\n  Expr.const c ≲ e →\n  env ⊢ e ↦ Value.const c"}, {"name": "Expr.Approx.const", "content": "lemma Expr.Approx.const {c₁ c₂} :\n  Expr.const c₁ ≲ Expr.const c₂ →\n  c₁ = c₂"}], "used_local_defs": [{"name": "Juvix.Core.Main.Expr.Equiv", "content": "def Expr.Equiv (e₁ e₂ : Expr) : Prop :=\n  e₁ ≲ e₂ ∧ e₂ ≲ e₁"}], "used_local_lemmas": [], "local_ctx": "import Juvix.Core.Main.Semantics.Approx\n\nnamespace Juvix.Core.Main\n\nnotation:40 v:40 \" ≈ᵥ \" v':40 => Value.Equiv v v'\n\nnotation:40 args₁:40 \" ≈ₐ \" args₂:40 => List.Forall₂ Value.Equiv args₁ args₂\n\nnotation:40 v:40 \" ≈ₒ \" v':40 => Object.Equiv v v'\n\nnotation:40 env₁:40 \" ≈ₑ \" env₂:40 => Env.Equiv env₁ env₂\n\nnotation:40 e₁:40 \" ≈⟨\" env₁:40 \", \" env₂:40 \"⟩ \" e₂:40 => Expr.Equiv.Param env₁ env₂ e₁ e₂\n\ndef Expr.Equiv (e₁ e₂ : Expr) : Prop :=\n  e₁ ≲ e₂ ∧ e₂ ≲ e₁\n\nnotation:40 e₁:40 \" ≈ \" e₂:40 => Expr.Equiv e₁ e₂", "target_theorem": "lemma Expr.Equiv.eval_const {op a₁ a₂ i₁ i₂ i₃} :\n  a₁ ≈ Expr.const (Constant.int i₁) →\n  a₂ ≈ Expr.const (Constant.int i₂) →\n  i₃ = eval_binop_int op i₁ i₂ →\n  Expr.binop op a₁ a₂ ≈\n    Expr.const (Constant.int i₃) :=", "ground_truth_proof": ":= by\n  simp only [Expr.Equiv]\n  intro h₁ h₂ h₃\n  constructor\n  · apply Expr.Approx.eval_const₁ (op := op) (i₁ := i₁) (i₂ := i₂) <;>\n      simp_all only\n  · apply Expr.Approx.eval_const₂ (op := op) (i₁ := i₁) (i₂ := i₂) <;>\n      simp_all only", "nesting_depth": 3, "transitive_dep_count": 40, "subset_aristotle": false, "category": "Semantics"}
{"id": 260, "thm_name": "Juvix.Core.Main.Env.Approx.Indexed'.from_delayed", "thm_stmt": "lemma Env.Approx.Indexed'.from_delayed {n env₁ env₂ l} (h : env₁ ≲ₑ'(n) env₂) :\n  List.map (Object.delayed env₁) l ≲ₑ'(n) List.map (Object.delayed env₂) l", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx/Indexed.lean", "imports": ["import Juvix.Utils", "import Juvix.Core.Main.Semantics.Eval.Indexed", "import Mathlib.Tactic.Linarith", "import Mathlib.Data.List.Forall2", "import Juvix.Core.Main.Semantics.Eval", "import Aesop"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "List.Forall₂.cons", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.nil", "module": "Batteries.Data.List.Basic"}], "used_repo_defs": [{"name": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e'", "content": "notation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.In", "content": "notation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.", "content": "notation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Juvix.Core.Main.Object.Approx.Indexed'", "content": "inductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)"}, {"name": "Juvix.Core.Main.Env.Approx.Indexed'", "content": "def Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)"}], "used_local_lemmas": [], "local_ctx": "import Juvix.Core.Main.Semantics.Eval\n\nimport Juvix.Core.Main.Semantics.Eval.Indexed\n\nimport Juvix.Utils\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Data.List.Forall2\n\nimport Aesop\n\nnamespace Juvix.Core.Main\n\nnotation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nnotation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nnotation:40 e:40 \" ≲(\" n:40 \")⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param.Indexed n env env' e e'\n\ninductive Object.Approx.Indexed' (n : Nat) : Object → Object → Prop where\n  | value {v₁ v₂} :\n    v₁ ≲ᵥ(n) v₂ →\n    Object.Approx.Indexed' n (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲(n)⟨env₁, env₂⟩ e₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n  | delayed_eq {env₁ env₂ e} :\n    List.Forall₂ (Object.Approx.Indexed' n) env₁ env₂ →\n    Object.Approx.Indexed' n (Object.delayed env₁ e) (Object.delayed env₂ e)\n\ndef Env.Approx.Indexed' (n : Nat) : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ (Object.Approx.Indexed' n)\n\nnotation:40 v:40 \" ≲ₒ'(\" n:40 \") \" v':40 => Object.Approx.Indexed' n v v'\n\nnotation:40 env₁:40 \" ≲ₑ'(\" n:40 \") \" env₂:40 => Env.Approx.Indexed' n env₁ env₂\n\nnotation:40 e:40 \" ≲'(\" n:40 \") \" e':40 => Expr.Approx.Indexed' n e e'", "target_theorem": "lemma Env.Approx.Indexed'.from_delayed {n env₁ env₂ l} (h : env₁ ≲ₑ'(n) env₂) :\n  List.map (Object.delayed env₁) l ≲ₑ'(n) List.map (Object.delayed env₂) l :=", "ground_truth_proof": ":= by\n  induction l\n  case nil =>\n    exact List.Forall₂.nil\n  case cons e l' h' =>\n    simp\n    apply List.Forall₂.cons\n    · apply Object.Approx.Indexed'.delayed_eq\n      assumption\n    · assumption", "nesting_depth": 3, "transitive_dep_count": 12, "subset_aristotle": false, "category": "Semantics"}
{"id": 261, "thm_name": "Juvix.Core.Main.Termination.binop", "thm_stmt": "lemma Termination.binop {env op e₁ e₂ c₁ c₂} :\n  env ⊢ e₁ ↦ Value.const (Constant.int c₁) →\n  env ⊢ e₂ ↦ Value.const (Constant.int c₂) →\n  env ⊢ Expr.binop op e₁ e₂ ↓", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Tactics/Termination.lean", "imports": ["import Juvix.Core.Main.Tactics.Base"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}], "used_repo_defs": [{"name": "syntax:100 expr:100 ppSpace expr:101 : expr", "content": "syntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Juvix.Core.Main.Tactics.Base\n\nnamespace Juvix.Core.Main", "target_theorem": "lemma Termination.binop {env op e₁ e₂ c₁ c₂} :\n  env ⊢ e₁ ↦ Value.const (Constant.int c₁) →\n  env ⊢ e₂ ↦ Value.const (Constant.int c₂) →\n  env ⊢ Expr.binop op e₁ e₂ ↓ :=", "ground_truth_proof": ":= by\n  intro h₁ h₂\n  constructor\n  apply Juvix.Core.Main.Eval.binop\n  · assumption\n  · assumption", "nesting_depth": 4, "transitive_dep_count": 18, "subset_aristotle": false, "category": "Semantics"}
{"id": 262, "thm_name": "Juvix.Core.Main.Object.Approx.trans", "thm_stmt": "@[trans]\nlemma Object.Approx.trans {o₁ o₂ o₃} : o₁ ≲ₒ o₂ → o₂ ≲ₒ o₃ → o₁ ≲ₒ o₃", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx.lean", "imports": ["import Juvix.Core.Main.Semantics.Approx.Indexed", "import Juvix.Utils"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}], "used_repo_defs": [{"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.A", "content": "notation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂"}, {"name": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).inve", "content": "macro \"invert\" h:term : tactic => `(tactic| (cases ($h).invert <;> try clear $h))"}, {"name": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Foral", "content": "notation:40 args₁:40 \" ≲ₐ(\" n:40 \") \" args₂:40 => List.Forall₂ (Value.Approx.Indexed n) args₁ args₂\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "Env", "content": "abbrev Env : Type := List Object\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Value.Approx.Indexed", "content": "def Value.Approx.Indexed (n : Nat) (v₁ v₂ : Value) : Prop :=\n  (v₁ = Value.unit ∧ v₂ = Value.unit) ∨\n  (∃ c, v₁ = Value.const c ∧ v₂ = Value.const c) ∨\n  (∃ ctr_name args_rev args_rev',\n    v₁ = Value.constr_app ctr_name args_rev ∧\n    v₂ = Value.constr_app ctr_name args_rev' ∧\n    (∀ k < n, List.Forall₂ (Value.Approx.Indexed k) args_rev args_rev')) ∨\n  (∃ env₁ body₁ env₂ body₂,\n    v₁ = Value.closure env₁ body₁ ∧\n    v₂ = Value.closure env₂ body₂ ∧\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ r₁,\n        Value.Approx.Indexed (n₁ + n₂) a₁ a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) r₁ →\n        ∃ r₂,\n          a₂ ∷ env₂ ⊢ body₂ ↦ r₂ ∧\n          Value.Approx.Indexed n₂ r₁ r₂))"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env"}, {"name": "Value.Approx.Indexed.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Indexed.Inversion (n : Nat) : Value → Value → Prop where\n  | unit : Value.Approx.Indexed.Inversion n Value.unit Value.unit\n  | const {c} : Value.Approx.Indexed.Inversion n (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.Approx.Indexed.Inversion n (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.Approx.Indexed.Inversion n (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "IsEmpty.forall_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "nonpos_iff_eq_zero", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}], "repo_lemmas": [{"name": "Value.Approx.Indexed.invert", "content": "@[aesop unsafe destruct]\nlemma Value.Approx.Indexed.invert {n v v'} :\n  v ≲ᵥ(n) v' →\n  Value.Approx.Indexed.Inversion n v v'"}, {"name": "Value.Approx.Indexed.constr_app", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.constr_app {n ctr_name args_rev args_rev'} :\n    (∀ k < n, args_rev ≲ₐ(k) args_rev') →\n    Value.constr_app ctr_name args_rev ≲ᵥ(n) Value.constr_app ctr_name args_rev'"}, {"name": "Value.Approx.Indexed.unit", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.unit {n} : Value.unit ≲ᵥ(n) Value.unit"}, {"name": "Value.Approx.Indexed.const", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.const {n c} : Value.const c ≲ᵥ(n) Value.const c"}, {"name": "Value.Approx.Indexed.closure", "content": "@[aesop unsafe apply]\nlemma Value.Approx.Indexed.closure {n env₁ body₁ env₂ body₂} :\n    (∀ n₁ n₂, n₁ + n₂ < n →\n      ∀ a₁ a₂ v₁,\n        a₁ ≲ᵥ(n₁ + n₂) a₂ →\n        a₁ ∷ env₁ ⊢ body₁ ↦(n₁) v₁ →\n        ∃ v₂, a₂ ∷ env₂ ⊢ body₂ ↦ v₂ ∧ v₁ ≲ᵥ(n₂) v₂) →\n    Value.closure env₁ body₁ ≲ᵥ(n) Value.closure env₂ body₂"}, {"name": "forall₂_trans'", "content": "theorem forall₂_trans' {α} {P Q R : α → α → Prop} {l₁ l₂ l₃}\n  (h : ∀ x y z, P x y → Q y z → R x z)\n  (h₁ : List.Forall₂ P l₁ l₂)\n  (h₂ : List.Forall₂ Q l₂ l₃)\n  : List.Forall₂ R l₁ l₃"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Juvix.Core.Main.Expr.Approx.Param", "content": "def Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)"}, {"name": "Juvix.Core.Main.Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Juvix.Core.Main.Expr.Approx", "content": "def Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂"}, {"name": "Juvix.Core.Main.Value.Approx.Inversion", "content": "@[aesop safe cases]\ninductive Value.Approx.Inversion : Value -> Value -> Prop where\n  | unit : Value.Approx.Inversion Value.unit Value.unit\n  | const {c} : Value.Approx.Inversion (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    args_rev ≲ₐ args_rev' →\n    Value.Approx.Inversion (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ a₁ a₂, a₁ ≲ᵥ a₂ → body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n    Value.Approx.Inversion (Value.closure env₁ body₁) (Value.closure env₂ body₂)"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Value.Approx.invert", "content": "@[aesop unsafe destruct]\nlemma Value.Approx.invert {v v'} :\n  v ≲ᵥ v' →\n  Value.Approx.Inversion v v'"}, {"name": "Juvix.Core.Main.Value.Approx.Indexed.trans", "content": "@[trans]\nlemma Value.Approx.Indexed.trans {n v₁ v₂ v₃} : v₁ ≲ᵥ(n) v₂ → v₂ ≲ᵥ v₃ → v₁ ≲ᵥ(n) v₃"}, {"name": "Juvix.Core.Main.Value.Approx.trans", "content": "@[trans]\nlemma Value.Approx.trans {v₁ v₂ v₃} : v₁ ≲ᵥ v₂ → v₂ ≲ᵥ v₃ → v₁ ≲ᵥ v₃"}, {"name": "Juvix.Core.Main.Expr.Approx.Param.trans", "content": "lemma Expr.Approx.Param.trans {env₁ env₂ env₃ e₁ e₂ e₃} :\n  e₁ ≲⟨env₁, env₂⟩ e₂ → e₂ ≲⟨env₂, env₃⟩ e₃ → e₁ ≲⟨env₁, env₃⟩ e₃"}], "local_ctx": "import Juvix.Core.Main.Semantics.Approx.Indexed\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'\n\nnotation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'\n\nnotation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂\n\ndef Expr.Approx.Param (env₁ env₂ : Env) (e₁ e₂ : Expr) : Prop :=\n  (∀ v₁, env₁ ⊢ e₁ ↦ v₁ → ∃ v₂, env₂ ⊢ e₂ ↦ v₂ ∧ v₁ ≲ᵥ v₂)\n\nnotation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'\n\ninductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n\nnotation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'\n\nnotation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂\n\ndef Expr.Approx (e₁ e₂ : Expr) : Prop :=\n  ∀ env₁ env₂, env₁ ≲ₑ env₂ → e₁ ≲⟨env₁, env₂⟩ e₂\n\nnotation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂\n\n@[aesop safe cases]\ninductive Value.Approx.Inversion : Value -> Value -> Prop where\n  | unit : Value.Approx.Inversion Value.unit Value.unit\n  | const {c} : Value.Approx.Inversion (Value.const c) (Value.const c)\n  | constr_app {ctr_name args_rev args_rev'} :\n    args_rev ≲ₐ args_rev' →\n    Value.Approx.Inversion (Value.constr_app ctr_name args_rev) (Value.constr_app ctr_name args_rev')\n  | closure {env₁ body₁ env₂ body₂} :\n    (∀ a₁ a₂, a₁ ≲ᵥ a₂ → body₁ ≲⟨a₁ ∷ env₁, a₂ ∷ env₂⟩ body₂) →\n    Value.Approx.Inversion (Value.closure env₁ body₁) (Value.closure env₂ body₂)", "target_theorem": "@[trans]\nlemma Object.Approx.trans {o₁ o₂ o₃} : o₁ ≲ₒ o₂ → o₂ ≲ₒ o₃ → o₁ ≲ₒ o₃ :=", "ground_truth_proof": ":= by\n  intros h₁ h₂\n  cases h₁\n  case value v₁ v₂ happrox =>\n    cases h₂\n    case value v₃ happrox' =>\n      apply Object.Approx.value\n      trans v₂ <;> assumption\n  case delayed env₁ env₂ e₁ e₂ happrox =>\n    cases h₂\n    case delayed env₂' e₂' happrox' =>\n      apply Object.Approx.delayed\n      exact Expr.Approx.Param.trans happrox happrox'", "nesting_depth": 6, "transitive_dep_count": 39, "subset_aristotle": false, "category": "Semantics"}
{"id": 263, "thm_name": "Juvix.Core.Main.Env.Approx.cons_value", "thm_stmt": "lemma Env.Approx.cons_value {v₁ v₂ env} :\n  v₁ ≲ᵥ v₂ → v₁ ∷ env ≲ₑ v₂ ∷ env", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Semantics/Approx.lean", "imports": ["import Juvix.Core.Main.Semantics.Approx.Indexed"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "List.Forall₂", "module": "Batteries.Data.List.Basic"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "List.Forall₂.cons", "module": "Batteries.Data.List.Basic"}, {"name": "List.Forall₂.nil", "module": "Batteries.Data.List.Basic"}], "used_repo_defs": [{"name": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Exp", "content": "notation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'"}, {"name": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'", "content": "notation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'"}, {"name": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "content": "notation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂"}, {"name": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Inde", "content": "notation:40 v:40 \" ≲ᵥ(\" n:40 \") \" v':40 => Value.Approx.Indexed n v v'\n\nsyntax \"case \" expr \" of \" cases \" end\" : expr"}, {"name": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'", "content": "notation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'"}, {"name": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂", "content": "notation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Juvix.Core.Main.Value.Approx", "content": "def Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'"}, {"name": "Juvix.Core.Main.Object.Approx", "content": "inductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)"}, {"name": "Juvix.Core.Main.Env.Approx", "content": "def Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx"}], "used_local_lemmas": [{"name": "Juvix.Core.Main.Value.Approx.refl", "content": "@[refl]\nlemma Value.Approx.refl v : v ≲ᵥ v"}, {"name": "Juvix.Core.Main.Object.Approx.refl", "content": "@[refl]\nlemma Object.Approx.refl {o} : o ≲ₒ o"}, {"name": "Juvix.Core.Main.Env.Approx.refl", "content": "@[refl]\nlemma Env.Approx.refl {env} : env ≲ₑ env"}, {"name": "Juvix.Core.Main.Env.Approx.cons", "content": "lemma Env.Approx.cons {env₁ env₂ o₁ o₂} :\n  o₁ ≲ₒ o₂ → env₁ ≲ₑ env₂ → (o₁ :: env₁) ≲ₑ (o₂ :: env₂)"}], "local_ctx": "import Juvix.Core.Main.Semantics.Approx.Indexed\n\nnamespace Juvix.Core.Main\n\ndef Value.Approx (v v' : Value) : Prop :=\n  ∀ n, v ≲ᵥ(n) v'\n\nnotation:40 v:40 \" ≲ᵥ \" v':40 => Value.Approx v v'\n\nnotation:40 args₁:40 \" ≲ₐ \" args₂:40 => List.Forall₂ Value.Approx args₁ args₂\n\nnotation:40 e:40 \" ≲⟨\" env:40 \", \" env':40 \"⟩ \" e':40 => Expr.Approx.Param env env' e e'\n\ninductive Object.Approx : Object → Object → Prop where\n  | value {v₁ v₂} : v₁ ≲ᵥ v₂ → Object.Approx (Object.value v₁) (Object.value v₂)\n  | delayed {env₁ env₂ e₁ e₂} :\n    e₁ ≲⟨env₁, env₂⟩ e₂ →\n    Object.Approx (Object.delayed env₁ e₁) (Object.delayed env₂ e₂)\n\ndef Env.Approx : (env₁ env₂ : Env) → Prop :=\n  List.Forall₂ Object.Approx\n\nnotation:40 v:40 \" ≲ₒ \" v':40 => Object.Approx v v'\n\nnotation:40 env₁:40 \" ≲ₑ \" env₂:40 => Env.Approx env₁ env₂\n\nnotation:40 e₁:40 \" ≲ \" e₂:40 => Expr.Approx e₁ e₂", "target_theorem": "lemma Env.Approx.cons_value {v₁ v₂ env} :\n  v₁ ≲ᵥ v₂ → v₁ ∷ env ≲ₑ v₂ ∷ env :=", "ground_truth_proof": ":= by\n  intro h\n  apply Env.Approx.cons\n  · apply Object.Approx.value\n    assumption\n  · apply Env.Approx.refl", "nesting_depth": 4, "transitive_dep_count": 26, "subset_aristotle": false, "category": "Semantics"}
{"id": 264, "thm_name": "Juvix.Core.Main.Termination.branch_matches", "thm_stmt": "lemma Termination.branch_matches {env name vnames args_rev e e'} :\n  args_rev.map Object.value ++ env ⊢ e ↓ →\n  Value.constr_app name args_rev ∷ env ⊢ Expr.branch name vnames e e' ↓", "lean_root": "juvix-lean", "rel_path": "Juvix/Core/Main/Tactics/Termination.lean", "imports": ["import Juvix.Core.Main.Tactics.Base"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "closure", "module": "Mathlib.Topology.Defs.Basic"}], "used_repo_defs": [{"name": "syntax:100 expr:100 ppSpace expr:101 : expr", "content": "syntax:100 expr:100 ppSpace expr:101 : expr\n\nsyntax:50 expr:50 \" + \" expr:51 : expr\n\nsyntax:50 expr:50 \" - \" expr:50 : expr\n\nsyntax:60 expr:60 \" * \" expr:61 : expr\n\nsyntax:60 expr:60 \" / \" expr:60 : expr"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↓\" => Eval.Defined env e"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Inde", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦(\" n \") \" v:40 => Eval.Indexed n env e v"}, {"name": "macro_rules", "content": "macro_rules\n  | `(⟪$s:ident ♯ $i:num⟫) => `(Expr.var $(Lean.Syntax.mkStrLit s.getId.toString) $i)\n  | `(⟪$num:num⟫) => `(Expr.const (Constant.int $num))\n  | `(⟪$s:str⟫) => `(Expr.const (Constant.string $s))\n  | `(⟪υ⟫) => `(Expr.unit)\n  | `(⟪$e₁:expr $e₂:expr⟫) => `(Expr.app ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪λ $s:ident $e:expr⟫) => `(Expr.lambda $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e⟫)\n  | `(⟪λ $ss:ident* . $e:expr⟫) => mkLambdas ss e\n  | `(⟪ $s:ident ⟫) => `(Expr.constr $(Lean.Syntax.mkStrLit s.getId.toString))\n  | `(⟪ $s:ident $es:expr* ⟫) => mkConstrApp s es\n  | `(⟪$e₁ + $e₂⟫) => `(Expr.binop BinaryOp.add_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ - $e₂⟫) => `(Expr.binop BinaryOp.sub_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ * $e₂⟫) => `(Expr.binop BinaryOp.mul_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪$e₁ / $e₂⟫) => `(Expr.binop BinaryOp.div_int ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪let $s:ident := $e₁:expr in $e₂:expr⟫) => `(Expr.save $(Lean.Syntax.mkStrLit s.getId.toString) ⟪$e₁⟫ ⟪$e₂⟫)\n  | `(⟪letrec $s:ident := $e₁:expr in $e₂:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.save $name (Expr.recur $name ⟪$e₁⟫) ⟪$e₂⟫)\n  | `(⟪rec $s:ident $e:expr⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.recur $name ⟪$e⟫)\n  | `(⟪⊥⟫) => `(Expr.fail)\n  | `(⟪cases| | $s:ident => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr $cs:cases ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪cases| | $s:ident => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    `(Expr.branch $name [] ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | $s:ident $ss:ident* => $e:expr ⟫) =>\n    let name := Lean.Syntax.mkStrLit s.getId.toString\n    let vnames := ss.toList.map fun s => s.getId.toString\n    `(Expr.branch $name $(quote vnames) ⟪$e⟫ Expr.fail)\n  | `(⟪cases| | _ => $e:expr ⟫) =>\n    `(⟪$e⟫)\n  | `(⟪case $e:expr of $cs:cases end⟫) => do\n    `(Expr.save \"_case_\" ⟪$e⟫ ⟪cases|$cs⟫)\n  | `(⟪($e)⟫) => `(⟪$e⟫)"}, {"name": "Eval", "content": "@[aesop unsafe constructors]\ninductive Eval : Env → Expr → Value → Prop where\n  | var {env name idx val} :\n    env[idx]? = some (Object.value val) →\n    Eval env (Expr.var name idx) val\n  | var_rec {env name idx env' expr val} :\n    env[idx]? = some (Object.delayed env' expr) →\n    Eval env' expr val →\n    Eval env (Expr.var name idx) val\n  | unit {env} :\n    Eval env Expr.unit Value.unit\n  | const {env c} :\n    Eval env (Expr.const c) (Value.const c)\n  | constr {env name} :\n    Eval env (Expr.constr name) (Value.constr_app name [])\n  | app {env env' f body arg val val'} :\n    Eval env f (Value.closure env' body) →\n    Eval env arg val →\n    Eval (val ∷ env') body val' →\n    Eval env (Expr.app f arg) val'\n  | constr_app {env ctr ctr_name ctr_args_rev arg val} :\n    Eval env ctr (Value.constr_app ctr_name ctr_args_rev) →\n    Eval env arg val →\n    Eval env (Expr.constr_app ctr arg) (Value.constr_app ctr_name (val :: ctr_args_rev))\n  | binop {env op arg₁ arg₂ val₁ val₂} :\n    Eval env arg₁ (Value.const (Constant.int val₁)) →\n    Eval env arg₂ (Value.const (Constant.int val₂)) →\n    Eval env (Expr.binop op arg₁ arg₂) (Value.const (Constant.int (eval_binop_int op val₁ val₂)))\n  | lambda {env name body} :\n    Eval env (Expr.lambda name body) (Value.closure env body)\n  | save {env name value body val val'} :\n    Eval env value val →\n    Eval (val ∷ env) body val' →\n    Eval env (Expr.save name value body) val'\n  | branch_matches {env name args_rev body val} :\n    Eval (args_rev.map Object.value ++ env) body val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name _ body _) val\n  | branch_fails {env name name' args_rev next val} :\n    name ≠ name' →\n    Eval (Value.constr_app name args_rev ∷ env) next val →\n    Eval (Value.constr_app name args_rev ∷ env) (Expr.branch name' _ _ next) val\n  | recur {env name body val} :\n    Eval (Object.delayed env (Expr.recur name body) :: env) body val →\n    Eval env (Expr.recur name body) val"}, {"name": "eval_binop_int", "content": "def eval_binop_int (op : BinaryOp) (i₁ i₂ : Int) : Int :=\n  match op with\n  | BinaryOp.add_int => i₁ + i₂\n  | BinaryOp.sub_int => i₁ - i₂\n  | BinaryOp.mul_int => i₁ * i₂\n  | BinaryOp.div_int => i₁ / i₂"}, {"name": "BinaryOp", "content": "inductive BinaryOp : Type where\n  | add_int : BinaryOp\n  | sub_int : BinaryOp\n  | mul_int : BinaryOp\n  | div_int : BinaryOp\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Expr", "content": "inductive Expr : Type where\n  | var : (name : String) → (index : Nat) → Expr\n  | unit : Expr\n  | const : Constant → Expr\n  | constr : Name → Expr\n  | app : Expr → Expr → Expr\n  | constr_app : Expr → Expr → Expr\n  | binop : (oper : BinaryOp) → (arg₁ arg₂ : Expr) → Expr\n  | lambda : (var_name : String) → (body : Expr) → Expr\n  | save : (var_name : String) → (value : Expr) → (body : Expr) → Expr\n  | branch : (constr : Name) → (var_names : List Name) → (body : Expr) → (next : Expr) → Expr\n  | recur : (var_name : Name) → (body : Expr) → Expr\n  | fail : Expr\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "cons_value", "content": "abbrev cons_value (v : Value) (env : Env) : Env := Object.value v :: env\n\n  inductive Object : Type where\n    | value : Value → Object\n    | delayed : (env : List Object) → Expr → Object\n    deriving Inhabited\n\n  inductive Value : Type where\n    | unit : Value\n    | const : Constant → Value\n    | constr_app : (constr : Name) → (args_rev : List Value) → Value\n    | closure : (env : List Object) → (value : Expr) → Value\n    deriving Inhabited"}, {"name": "Constant", "content": "inductive Constant : Type where\n  | int : Int → Constant\n  | string : String → Constant\n  deriving Inhabited, BEq, DecidableEq"}, {"name": "Name", "content": "abbrev Name : Type := String"}, {"name": "Env", "content": "abbrev Env : Type := List Object"}, {"name": "infixr:50 \" ∷ \" => cons_value", "content": "infixr:50 \" ∷ \" => cons_value"}, {"name": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v", "content": "notation:40 env:40 \" ⊢ \" e:40 \" ↦ \" v:40 => Eval env e v"}, {"name": "infixl:100 \" @@ \" => Expr.app", "content": "infixl:100 \" @@ \" => Expr.app"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Juvix.Core.Main.Tactics.Base\n\nnamespace Juvix.Core.Main", "target_theorem": "lemma Termination.branch_matches {env name vnames args_rev e e'} :\n  args_rev.map Object.value ++ env ⊢ e ↓ →\n  Value.constr_app name args_rev ∷ env ⊢ Expr.branch name vnames e e' ↓ :=", "ground_truth_proof": ":= by\n  intro h\n  obtain ⟨w, h⟩ := h\n  constructor\n  apply Juvix.Core.Main.Eval.branch_matches\n  assumption", "nesting_depth": 4, "transitive_dep_count": 18, "subset_aristotle": false, "category": "Semantics"}
{"id": 265, "thm_name": "eval_bool_completeness", "thm_stmt": "theorem eval_bool_completeness (e: BoolExpr V): BoolStepStar V val e (BoolExpr.Const (eval_bool V val e))", "lean_root": "LeanExprEvaluator", "rel_path": "ExprEval/Basic.lean", "imports": ["import ExprEval.Steps", "import ExprEval.Lemmas", "import ExprEval.Expr"], "used_lib_defs": [{"name": "And", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Not", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Add", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Mul", "module": "Init.Prelude"}, {"name": "Sub", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "BoolStepStar", "content": "def BoolStepStar (V: Type) := StepStar (Step := BoolStep V) V"}, {"name": "BoolStep", "content": "inductive BoolStep: (val: V -> Int) -> (BoolExpr V) -> (BoolExpr V) -> Prop where\n    | notIsBoolNot (b: Bool): BoolStep val\n        (BoolExpr.Not (BoolExpr.Const b))\n        (BoolExpr.Const !b)\n    | andLeftTrue (e: BoolExpr V): BoolStep val\n        (BoolExpr.And (BoolExpr.Const true) e)\n        e\n    | orLeftFalse (e: BoolExpr V): BoolStep val\n        (BoolExpr.Or (BoolExpr.Const false) e)\n        e\n    | andLeftShortCircuit e : BoolStep val\n        (BoolExpr.And (BoolExpr.Const false) e)\n        (BoolExpr.Const false)\n    | orLeftShortCircuit e : BoolStep val\n        (BoolExpr.Or (BoolExpr.Const true) e)\n        (BoolExpr.Const true)\n    | lessConstConstTrue n₁ n₂ : n₁ < n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | lessConstConstFalse n₁ n₂ : n₁ >= n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | eqConstConstTrue n₁ n₂ : n₁ = n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | eqConstConstFalse n₁ n₂ : n₁ != n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | lessArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁' e₂)\n    | eqArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁' e₂)\n     | lessArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁ e₂')\n    | eqArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁ e₂')\n    | orStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Or e₁ e₂)\n                (BoolExpr.Or e₁' e₂)\n    | andStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.And e₁ e₂)\n                (BoolExpr.And e₁' e₂)\n    | notStep (e e' : BoolExpr V):\n        BoolStep val e e' -> BoolStep val\n            (BoolExpr.Not e)\n            (BoolExpr.Not e')"}, {"name": "ArStep", "content": "inductive ArStep: (val: V -> Int) -> (ArExpr V) -> (ArExpr V) -> Prop where\n    | getVarValue (var: V) :\n        ArStep\n            val\n            (ArExpr.Var var)\n            (ArExpr.Const (val var))\n    | addConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Add (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ + n₂))\n    | subConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Sub (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ - n₂))\n    | mulConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Mul (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ * n₂))\n    | addLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Add e₁ e₂) (ArExpr.Add e₁' e₂)\n\n    | subLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Sub e₁ e₂) (ArExpr.Sub e₁' e₂)\n\n    | mulLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Mul e₁ e₂) (ArExpr.Mul e₁' e₂)\n\n    | addRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Add (ArExpr.Const n) e₂)  (ArExpr.Add (ArExpr.Const n) e₂')\n\n    | subRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Sub (ArExpr.Const n) e₂)  (ArExpr.Sub (ArExpr.Const n) e₂')\n    | mulRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Mul (ArExpr.Const n) e₂)  (ArExpr.Mul (ArExpr.Const n) e₂')\n\n    | ifStep (e e': BoolExpr V) (a b : ArExpr V): BoolStep val e e' ->\n        ArStep val (ArExpr.If e a b) (ArExpr.If e' a b)\n    | ifCondTrue (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const true) a b) a\n    | ifCondFalse (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const false) a b) b\n\n    inductive ArExpr: Type\n        | Const: Int -> ArExpr\n        | Add: ArExpr -> ArExpr -> ArExpr\n        | Sub: ArExpr -> ArExpr -> ArExpr\n        | Mul: ArExpr -> ArExpr -> ArExpr\n        | Var: V -> ArExpr\n        | If : BoolExpr -> ArExpr -> ArExpr -> ArExpr\n\n    inductive BoolExpr : Type\n        | Const: Bool -> BoolExpr\n        | Less: ArExpr -> ArExpr -> BoolExpr\n        | Eq: ArExpr -> ArExpr -> BoolExpr\n        | Not : BoolExpr -> BoolExpr\n        | And : BoolExpr -> BoolExpr -> BoolExpr\n        | Or : BoolExpr -> BoolExpr -> BoolExpr"}, {"name": "StepStar", "content": "inductive StepStar {ExprType: Type} {Step: StepKind V ExprType} : (val: V -> Int) -> ExprType -> ExprType -> Prop where\n        | refl val e : StepStar val e e\n        | trans e₁ e₂ e₃ : Step val e₁ e₂ -> StepStar val e₂ e₃ -> StepStar val e₁ e₃"}, {"name": "BoolStepStar.refl", "content": "def BoolStepStar.refl {V: Type} (val: V -> Int) := StepStar.refl (Step := BoolStep V) val\n\n    def eval (val: V -> Int) (e: ArExpr V) : Int :=\n        match e with\n            | ArExpr.Const x => x\n            | ArExpr.Add lhs rhs => (eval val lhs) + (eval val rhs)\n            | ArExpr.Sub lhs rhs => (eval val lhs) - (eval val rhs)\n            | ArExpr.Mul lhs rhs => (eval val lhs) * (eval val rhs)\n            | ArExpr.Var v => val v\n            | ArExpr.If cond then_e else_e => if eval_bool val cond then eval val then_e else eval val else_e\n\n    def eval_bool (val: V -> Int) (e: BoolExpr V): Bool :=\n        match e with\n        | BoolExpr.Const b => b\n        | BoolExpr.Less l r =>  (eval val l) < (eval val r)\n        | BoolExpr.Eq l r => (eval val l) == (eval val r)\n        | BoolExpr.Not e => not (eval_bool val e)\n        | BoolExpr.And l r => if eval_bool val l then eval_bool val r else false\n        | BoolExpr.Or l r => if eval_bool val l then true else eval_bool val r"}, {"name": "ArStepStar.trans", "content": "def ArStepStar.trans {V: Type} {val: V -> Int} e₁ e₂ e₃ := StepStar.trans e₁ e₂ e₃ (Step := ArStep V) (val := val)"}, {"name": "ArStepStar", "content": "def ArStepStar (V: Type) := StepStar (Step := ArStep V) V"}, {"name": "ArStepStar.refl", "content": "def ArStepStar.refl {V: Type} (val: V -> Int) := StepStar.refl (Step := ArStep V) val"}, {"name": "BoolStepStar.trans", "content": "def BoolStepStar.trans {V: Type} {val: V -> Int} e₁ e₂ e₃ := StepStar.trans e₁ e₂ e₃ (Step := BoolStep V) (val := val)"}, {"name": "StepKind", "content": "def StepKind (V: Type) (ExprType: Type):= (V -> Int) -> ExprType -> ExprType -> Prop"}], "lib_lemmas": [{"name": "Int.not_lt", "module": "Init.Data.Int.Order"}, {"name": "bne_iff_ne", "module": "Init.SimpLemmas"}, {"name": "decide_eq_false", "module": "Init.Prelude"}, {"name": "decide_eq_true", "module": "Init.Prelude"}, {"name": "eq_false_of_ne_true", "module": "Init.Prelude"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "ne_of_beq_false", "module": "Init.Core"}], "repo_lemmas": [{"name": "BoolStepStar.andStepLeft", "content": "theorem BoolStepStar.andStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStepStar V val e₁ e₁' -> BoolStepStar V val\n            (BoolExpr.And e₁ e₂)\n            (BoolExpr.And e₁' e₂)"}, {"name": "chasles_step_star", "content": "theorem chasles_step_star {ExprKind: Type} {Step: StepKind V ExprKind} {e₁ e₂ e₃: ExprKind}:\n    StepStar (Step := Step) V val e₁ e₂  ->\n        StepStar (Step := Step) V val e₂ e₃ ->\n            StepStar (Step := Step) V val e₁ e₃"}, {"name": "BoolStepStar.lessArStepStarLeft", "content": "theorem BoolStepStar.lessArStepStarLeft (e₁ e₁' e₂: ArExpr V):\n        ArStepStar V val e₁ e₁' ->\n            BoolStepStar V val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁' e₂)"}, {"name": "BoolStepStar.notStep", "content": "theorem BoolStepStar.notStep {V: Type} {val: V -> Int}\n        (e e': BoolExpr V):\n            BoolStepStar V val e e' -> BoolStepStar V val (BoolExpr.Not e) (BoolExpr.Not e')"}, {"name": "BoolStepStar.lessArStepStarRight", "content": "theorem BoolStepStar.lessArStepStarRight (e₁ e₂ e₂': ArExpr V):\n        ArStepStar V val e₂ e₂' ->\n            BoolStepStar V val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁ e₂')"}, {"name": "step_to_stepstar", "content": "theorem step_to_stepstar {ExprKind: Type} {Step: StepKind V ExprKind} {e e': ExprKind}:\n    Step val e e' -> StepStar (Step := Step) V val e e'"}, {"name": "BoolStepStar.eqArStepStarRight", "content": "theorem BoolStepStar.eqArStepStarRight (e₁ e₂ e₂': ArExpr V):\n        ArStepStar V val e₂ e₂' ->\n            BoolStepStar V val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁ e₂')"}, {"name": "BoolStepStar.eqArStepStarLeft", "content": "theorem BoolStepStar.eqArStepStarLeft (e₁ e₁' e₂: ArExpr V):\n        ArStepStar V val e₁ e₁' ->\n            BoolStepStar V val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁' e₂)"}, {"name": "BoolStepStar.orStepLeft", "content": "theorem BoolStepStar.orStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStepStar V val e₁ e₁' -> BoolStepStar V val\n            (BoolExpr.Or e₁ e₂)\n            (BoolExpr.Or e₁' e₂)"}, {"name": "arstepstar_add_right", "content": "theorem arstepstar_add_right (n: Int) (e₂ e₂': ArExpr V) :\n            ArStepStar V val e₂ e₂' ->\n                    ArStepStar V val\n                            (ArExpr.Add (ArExpr.Const n) e₂)\n                            (ArExpr.Add (ArExpr.Const n) e₂')"}, {"name": "arstepstar_sub_left", "content": "theorem arstepstar_sub_left (e₁ e₁' e₂: ArExpr V) :\n            ArStepStar V val e₁ e₁' ->\n                    ArStepStar V val\n                            (ArExpr.Sub e₁ e₂)\n                            (ArExpr.Sub e₁' e₂)"}, {"name": "arstepstar_add_left", "content": "theorem arstepstar_add_left (e₁ e₁' e₂: ArExpr V) :\n            ArStepStar V val e₁ e₁' -> ArStepStar V val (ArExpr.Add e₁ e₂) (ArExpr.Add e₁' e₂)"}, {"name": "arstepstar_sub_right", "content": "theorem arstepstar_sub_right (n: Int) (e₂ e₂': ArExpr V) :\n            ArStepStar V val e₂ e₂' ->\n                    ArStepStar V val\n                            (ArExpr.Sub (ArExpr.Const n) e₂)\n                            (ArExpr.Sub (ArExpr.Const n) e₂')"}, {"name": "ArStepStar.ifStep", "content": "theorem ArStepStar.ifStep (e e': BoolExpr V) (a b: ArExpr V): BoolStepStar V val e e' ->\n        ArStepStar V val (ArExpr.If e a b) (ArExpr.If e' a b)"}, {"name": "arstepstar_mul_right", "content": "theorem arstepstar_mul_right (n: Int) (e₂ e₂': ArExpr V) :\n            ArStepStar V val e₂ e₂' ->\n                    ArStepStar V val\n                            (ArExpr.Mul (ArExpr.Const n) e₂)\n                            (ArExpr.Mul (ArExpr.Const n) e₂')"}, {"name": "arstepstar_mul_left", "content": "theorem arstepstar_mul_left (e₁ e₁' e₂: ArExpr V) :\n            ArStepStar V val e₁ e₁' ->\n                    ArStepStar V val\n                            (ArExpr.Mul e₁ e₂)\n                            (ArExpr.Mul e₁' e₂)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "eval_completeness", "content": "    theorem eval_completeness (e: ArExpr V) : ArStepStar V val e (ArExpr.Const (eval V val e))"}], "local_ctx": "import ExprEval.Steps\n\nimport ExprEval.Lemmas\n\nvariable (V: Type)", "target_theorem": "theorem eval_bool_completeness (e: BoolExpr V): BoolStepStar V val e (BoolExpr.Const (eval_bool V val e)) :=", "ground_truth_proof": ":= by\n        cases e with\n        | Const b =>\n            simp [eval_bool]\n            apply BoolStepStar.refl\n        | Not e =>\n            simp [eval_bool]\n            have ihn := eval_bool_completeness e (val := val)\n            have h := BoolStepStar.notStep e _ ihn\n            have h1 := BoolStep.notIsBoolNot (eval_bool V val e) (val := val)\n            have h2 := (step_to_stepstar V h1)\n            apply chasles_step_star V h h2\n        | And l r =>\n            have l_step := eval_bool_completeness l (val := val)\n            have r_step := eval_bool_completeness r (val := val)\n            simp [eval_bool]\n            by_cases h: (eval_bool V val l)\n            .   simp [and]\n                rw [h]\n                simp\n                rw [h] at l_step\n                have h1 := BoolStep.andLeftTrue r (val := val)\n                have h2 := StepStar.trans _ _ _ h1 r_step\n                have h3 := BoolStepStar.andStepLeft _ _ r l_step\n                exact chasles_step_star V h3 h2\n\n            .   replace h := eq_false_of_ne_true h\n                rw [h]\n                rw [h] at l_step\n                simp [and]\n                have h1 := BoolStepStar.andStepLeft _ _ r l_step\n                have h2 := BoolStep.andLeftShortCircuit r (val := val)\n                have h3 := step_to_stepstar V h2\n                exact chasles_step_star V h1 h3\n        | Or l r =>\n            have l_step := eval_bool_completeness l (val := val)\n            have r_step := eval_bool_completeness r (val := val)\n            simp [eval_bool]\n            by_cases h: (eval_bool V val l)\n            .   rw [h]\n                rw [h] at l_step\n                simp [or]\n                have h1 := BoolStepStar.orStepLeft _ _ r l_step\n                have h2 := BoolStep.orLeftShortCircuit r (val := val)\n                have h3 := step_to_stepstar V h2\n                exact chasles_step_star V h1 h3\n\n            .   replace h := eq_false_of_ne_true h\n                rw [h]\n                rw [h] at l_step\n                simp\n                have h1 := BoolStep.orLeftFalse r (val := val)\n                have h2 := StepStar.trans _ _ _ h1 r_step\n                have h3 := BoolStepStar.orStepLeft _ _ r l_step\n                exact chasles_step_star V h3 h2\n        | Eq l r =>\n            simp [eval_bool]\n\n            by_cases h: (eval V val l == eval V val r)\n\n            .   replace h := eq_of_beq h\n                rw [h]\n                simp\n                have h1 := BoolStep.eqConstConstTrue _ _ h (val := val)\n                have hl := eval_completeness l (val := val)\n                have hr := eval_completeness r (val := val)\n                have h2 := BoolStepStar.eqArStepStarLeft l (ArExpr.Const (eval V val l)) r hl\n                have h3 := BoolStepStar.eqArStepStarRight (ArExpr.Const (eval V val l)) r (ArExpr.Const (eval V val r)) hr\n                have h4 := chasles_step_star V h2 h3\n                exact chasles_step_star V h4 (step_to_stepstar V h1)\n\n            .   replace h := eq_false_of_ne_true h\n                rw [h]\n                have h' := ne_of_beq_false h\n                rw [<- bne_iff_ne] at h'\n                have h1 := BoolStep.eqConstConstFalse _ _ h' (val := val)\n                have hl := eval_completeness l (val := val)\n                have hr := eval_completeness r (val := val)\n                have h2 := BoolStepStar.eqArStepStarLeft l (ArExpr.Const (eval V val l)) r hl\n                have h3 := BoolStepStar.eqArStepStarRight (ArExpr.Const (eval V val l)) r (ArExpr.Const (eval V val r)) hr\n                have h4 := chasles_step_star V h2 h3\n                exact chasles_step_star V h4 (step_to_stepstar V h1)\n        | Less l r =>\n            simp [eval_bool]\n\n            by_cases h: (eval V val l < eval V val r)\n\n            .   rw [decide_eq_true]\n                have h1 := BoolStep.lessConstConstTrue _ _ h (val := val)\n                have hr := eval_completeness r (val := val)\n                have hl := eval_completeness l (val := val)\n                have hr := eval_completeness r (val := val)\n                have h2 := BoolStepStar.lessArStepStarLeft l (ArExpr.Const (eval V val l)) r hl\n                have h3 := BoolStepStar.lessArStepStarRight (ArExpr.Const (eval V val l)) r (ArExpr.Const (eval V val r)) hr\n                have h4 := chasles_step_star V h2 h3\n                apply chasles_step_star V h4 (step_to_stepstar V h1)\n                exact h\n\n            .   rw [decide_eq_false]\n                have h' := h\n                rw [Int.not_lt] at h'\n                rw [<- ge_iff_le] at h'\n                have h1 := BoolStep.lessConstConstFalse _ _ h' (val := val)\n                have hl := eval_completeness l (val := val)\n                have hr := eval_completeness r (val := val)\n                have h2 := BoolStepStar.lessArStepStarLeft l (ArExpr.Const (eval V val l)) r hl\n                have h3 := BoolStepStar.lessArStepStarRight (ArExpr.Const (eval V val l)) r (ArExpr.Const (eval V val r)) hr\n                have h4 := chasles_step_star V h2 h3\n                exact chasles_step_star V h4 (step_to_stepstar V h1)\n                exact h", "nesting_depth": 4, "transitive_dep_count": 60, "subset_aristotle": true, "category": "Semantics"}
{"id": 266, "thm_name": "eval_completeness", "thm_stmt": "theorem eval_completeness (e: ArExpr V) : ArStepStar V val e (ArExpr.Const (eval V val e))", "lean_root": "LeanExprEvaluator", "rel_path": "ExprEval/Basic.lean", "imports": ["import ExprEval.Steps", "import ExprEval.Lemmas", "import ExprEval.Expr"], "used_lib_defs": [{"name": "Add", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Mul", "module": "Init.Prelude"}, {"name": "Sub", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Not", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "ArStep", "content": "inductive ArStep: (val: V -> Int) -> (ArExpr V) -> (ArExpr V) -> Prop where\n    | getVarValue (var: V) :\n        ArStep\n            val\n            (ArExpr.Var var)\n            (ArExpr.Const (val var))\n    | addConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Add (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ + n₂))\n    | subConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Sub (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ - n₂))\n    | mulConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Mul (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ * n₂))\n    | addLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Add e₁ e₂) (ArExpr.Add e₁' e₂)\n\n    | subLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Sub e₁ e₂) (ArExpr.Sub e₁' e₂)\n\n    | mulLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Mul e₁ e₂) (ArExpr.Mul e₁' e₂)\n\n    | addRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Add (ArExpr.Const n) e₂)  (ArExpr.Add (ArExpr.Const n) e₂')\n\n    | subRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Sub (ArExpr.Const n) e₂)  (ArExpr.Sub (ArExpr.Const n) e₂')\n    | mulRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Mul (ArExpr.Const n) e₂)  (ArExpr.Mul (ArExpr.Const n) e₂')\n\n    | ifStep (e e': BoolExpr V) (a b : ArExpr V): BoolStep val e e' ->\n        ArStep val (ArExpr.If e a b) (ArExpr.If e' a b)\n    | ifCondTrue (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const true) a b) a\n    | ifCondFalse (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const false) a b) b"}, {"name": "ArStepStar.trans", "content": "def ArStepStar.trans {V: Type} {val: V -> Int} e₁ e₂ e₃ := StepStar.trans e₁ e₂ e₃ (Step := ArStep V) (val := val)"}, {"name": "BoolStep", "content": "inductive BoolStep: (val: V -> Int) -> (BoolExpr V) -> (BoolExpr V) -> Prop where\n    | notIsBoolNot (b: Bool): BoolStep val\n        (BoolExpr.Not (BoolExpr.Const b))\n        (BoolExpr.Const !b)\n    | andLeftTrue (e: BoolExpr V): BoolStep val\n        (BoolExpr.And (BoolExpr.Const true) e)\n        e\n    | orLeftFalse (e: BoolExpr V): BoolStep val\n        (BoolExpr.Or (BoolExpr.Const false) e)\n        e\n    | andLeftShortCircuit e : BoolStep val\n        (BoolExpr.And (BoolExpr.Const false) e)\n        (BoolExpr.Const false)\n    | orLeftShortCircuit e : BoolStep val\n        (BoolExpr.Or (BoolExpr.Const true) e)\n        (BoolExpr.Const true)\n    | lessConstConstTrue n₁ n₂ : n₁ < n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | lessConstConstFalse n₁ n₂ : n₁ >= n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | eqConstConstTrue n₁ n₂ : n₁ = n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | eqConstConstFalse n₁ n₂ : n₁ != n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | lessArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁' e₂)\n    | eqArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁' e₂)\n     | lessArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁ e₂')\n    | eqArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁ e₂')\n    | orStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Or e₁ e₂)\n                (BoolExpr.Or e₁' e₂)\n    | andStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.And e₁ e₂)\n                (BoolExpr.And e₁' e₂)\n    | notStep (e e' : BoolExpr V):\n        BoolStep val e e' -> BoolStep val\n            (BoolExpr.Not e)\n            (BoolExpr.Not e')\n\n    inductive BoolExpr : Type\n        | Const: Bool -> BoolExpr\n        | Less: ArExpr -> ArExpr -> BoolExpr\n        | Eq: ArExpr -> ArExpr -> BoolExpr\n        | Not : BoolExpr -> BoolExpr\n        | And : BoolExpr -> BoolExpr -> BoolExpr\n        | Or : BoolExpr -> BoolExpr -> BoolExpr\n\n    inductive ArExpr: Type\n        | Const: Int -> ArExpr\n        | Add: ArExpr -> ArExpr -> ArExpr\n        | Sub: ArExpr -> ArExpr -> ArExpr\n        | Mul: ArExpr -> ArExpr -> ArExpr\n        | Var: V -> ArExpr\n        | If : BoolExpr -> ArExpr -> ArExpr -> ArExpr"}, {"name": "StepStar", "content": "inductive StepStar {ExprType: Type} {Step: StepKind V ExprType} : (val: V -> Int) -> ExprType -> ExprType -> Prop where\n        | refl val e : StepStar val e e\n        | trans e₁ e₂ e₃ : Step val e₁ e₂ -> StepStar val e₂ e₃ -> StepStar val e₁ e₃"}, {"name": "ArStepStar", "content": "def ArStepStar (V: Type) := StepStar (Step := ArStep V) V\n\n    def eval (val: V -> Int) (e: ArExpr V) : Int :=\n        match e with\n            | ArExpr.Const x => x\n            | ArExpr.Add lhs rhs => (eval val lhs) + (eval val rhs)\n            | ArExpr.Sub lhs rhs => (eval val lhs) - (eval val rhs)\n            | ArExpr.Mul lhs rhs => (eval val lhs) * (eval val rhs)\n            | ArExpr.Var v => val v\n            | ArExpr.If cond then_e else_e => if eval_bool val cond then eval val then_e else eval val else_e\n\n    def eval_bool (val: V -> Int) (e: BoolExpr V): Bool :=\n        match e with\n        | BoolExpr.Const b => b\n        | BoolExpr.Less l r =>  (eval val l) < (eval val r)\n        | BoolExpr.Eq l r => (eval val l) == (eval val r)\n        | BoolExpr.Not e => not (eval_bool val e)\n        | BoolExpr.And l r => if eval_bool val l then eval_bool val r else false\n        | BoolExpr.Or l r => if eval_bool val l then true else eval_bool val r"}, {"name": "ArStepStar.refl", "content": "def ArStepStar.refl {V: Type} (val: V -> Int) := StepStar.refl (Step := ArStep V) val"}, {"name": "BoolStepStar", "content": "def BoolStepStar (V: Type) := StepStar (Step := BoolStep V) V"}, {"name": "BoolStepStar.refl", "content": "def BoolStepStar.refl {V: Type} (val: V -> Int) := StepStar.refl (Step := BoolStep V) val"}, {"name": "StepKind", "content": "def StepKind (V: Type) (ExprType: Type):= (V -> Int) -> ExprType -> ExprType -> Prop"}, {"name": "BoolStepStar.trans", "content": "def BoolStepStar.trans {V: Type} {val: V -> Int} e₁ e₂ e₃ := StepStar.trans e₁ e₂ e₃ (Step := BoolStep V) (val := val)"}], "lib_lemmas": [{"name": "Int.not_lt", "module": "Init.Data.Int.Order"}, {"name": "bne_iff_ne", "module": "Init.SimpLemmas"}, {"name": "decide_eq_false", "module": "Init.Prelude"}, {"name": "decide_eq_true", "module": "Init.Prelude"}, {"name": "eq_false_of_ne_true", "module": "Init.Prelude"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "ne_of_beq_false", "module": "Init.Core"}], "repo_lemmas": [{"name": "arstepstar_add_right", "content": "theorem arstepstar_add_right (n: Int) (e₂ e₂': ArExpr V) :\n            ArStepStar V val e₂ e₂' ->\n                    ArStepStar V val\n                            (ArExpr.Add (ArExpr.Const n) e₂)\n                            (ArExpr.Add (ArExpr.Const n) e₂')"}, {"name": "arstepstar_sub_left", "content": "theorem arstepstar_sub_left (e₁ e₁' e₂: ArExpr V) :\n            ArStepStar V val e₁ e₁' ->\n                    ArStepStar V val\n                            (ArExpr.Sub e₁ e₂)\n                            (ArExpr.Sub e₁' e₂)"}, {"name": "chasles_step_star", "content": "theorem chasles_step_star {ExprKind: Type} {Step: StepKind V ExprKind} {e₁ e₂ e₃: ExprKind}:\n    StepStar (Step := Step) V val e₁ e₂  ->\n        StepStar (Step := Step) V val e₂ e₃ ->\n            StepStar (Step := Step) V val e₁ e₃"}, {"name": "arstepstar_add_left", "content": "theorem arstepstar_add_left (e₁ e₁' e₂: ArExpr V) :\n            ArStepStar V val e₁ e₁' -> ArStepStar V val (ArExpr.Add e₁ e₂) (ArExpr.Add e₁' e₂)"}, {"name": "arstepstar_sub_right", "content": "theorem arstepstar_sub_right (n: Int) (e₂ e₂': ArExpr V) :\n            ArStepStar V val e₂ e₂' ->\n                    ArStepStar V val\n                            (ArExpr.Sub (ArExpr.Const n) e₂)\n                            (ArExpr.Sub (ArExpr.Const n) e₂')"}, {"name": "ArStepStar.ifStep", "content": "theorem ArStepStar.ifStep (e e': BoolExpr V) (a b: ArExpr V): BoolStepStar V val e e' ->\n        ArStepStar V val (ArExpr.If e a b) (ArExpr.If e' a b)"}, {"name": "arstepstar_mul_right", "content": "theorem arstepstar_mul_right (n: Int) (e₂ e₂': ArExpr V) :\n            ArStepStar V val e₂ e₂' ->\n                    ArStepStar V val\n                            (ArExpr.Mul (ArExpr.Const n) e₂)\n                            (ArExpr.Mul (ArExpr.Const n) e₂')"}, {"name": "arstepstar_mul_left", "content": "theorem arstepstar_mul_left (e₁ e₁' e₂: ArExpr V) :\n            ArStepStar V val e₁ e₁' ->\n                    ArStepStar V val\n                            (ArExpr.Mul e₁ e₂)\n                            (ArExpr.Mul e₁' e₂)"}, {"name": "BoolStepStar.andStepLeft", "content": "theorem BoolStepStar.andStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStepStar V val e₁ e₁' -> BoolStepStar V val\n            (BoolExpr.And e₁ e₂)\n            (BoolExpr.And e₁' e₂)"}, {"name": "BoolStepStar.lessArStepStarLeft", "content": "theorem BoolStepStar.lessArStepStarLeft (e₁ e₁' e₂: ArExpr V):\n        ArStepStar V val e₁ e₁' ->\n            BoolStepStar V val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁' e₂)"}, {"name": "BoolStepStar.notStep", "content": "theorem BoolStepStar.notStep {V: Type} {val: V -> Int}\n        (e e': BoolExpr V):\n            BoolStepStar V val e e' -> BoolStepStar V val (BoolExpr.Not e) (BoolExpr.Not e')"}, {"name": "BoolStepStar.lessArStepStarRight", "content": "theorem BoolStepStar.lessArStepStarRight (e₁ e₂ e₂': ArExpr V):\n        ArStepStar V val e₂ e₂' ->\n            BoolStepStar V val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁ e₂')"}, {"name": "step_to_stepstar", "content": "theorem step_to_stepstar {ExprKind: Type} {Step: StepKind V ExprKind} {e e': ExprKind}:\n    Step val e e' -> StepStar (Step := Step) V val e e'"}, {"name": "BoolStepStar.eqArStepStarRight", "content": "theorem BoolStepStar.eqArStepStarRight (e₁ e₂ e₂': ArExpr V):\n        ArStepStar V val e₂ e₂' ->\n            BoolStepStar V val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁ e₂')"}, {"name": "BoolStepStar.eqArStepStarLeft", "content": "theorem BoolStepStar.eqArStepStarLeft (e₁ e₁' e₂: ArExpr V):\n        ArStepStar V val e₁ e₁' ->\n            BoolStepStar V val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁' e₂)"}, {"name": "BoolStepStar.orStepLeft", "content": "theorem BoolStepStar.orStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStepStar V val e₁ e₁' -> BoolStepStar V val\n            (BoolExpr.Or e₁ e₂)\n            (BoolExpr.Or e₁' e₂)"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import ExprEval.Steps\n\nimport ExprEval.Lemmas\n\nvariable (V: Type)", "target_theorem": "theorem eval_completeness (e: ArExpr V) : ArStepStar V val e (ArExpr.Const (eval V val e)) :=", "ground_truth_proof": ":= by\n        cases e with\n        | Const x =>\n            simp [eval]\n            exact ArStepStar.refl _ _\n        | Add e₁ e₂ =>\n            have ih1 := eval_completeness e₁ (val := val)\n            have ih2 := eval_completeness e₂ (val := val)\n            simp [eval]\n            have a := arstepstar_add_left _ _ e₂ ih1\n            have b := arstepstar_add_right (eval V val e₁) e₂ _ ih2\n            have c := ArStep.addConstConst (val := val) (eval V val e₁) (eval V val e₂)\n            have d := ArStepStar.trans _ _ _ c (ArStepStar.refl _ (ArExpr.Const (eval V val e₁ + eval V val e₂)))\n            have e := chasles_step_star _ a (chasles_step_star _ b d)\n            exact e\n        | Sub e₁ e₂ =>\n            have ih1 := eval_completeness e₁ (val := val)\n            have ih2 := eval_completeness e₂ (val := val)\n            simp [eval]\n            have a := arstepstar_sub_left _ _ e₂ ih1\n            have b := arstepstar_sub_right (eval V val e₁) e₂ _ ih2\n            have c := ArStep.subConstConst (val := val) (eval V val e₁) (eval V val e₂)\n            have d := ArStepStar.trans _ _ _ c (ArStepStar.refl _ (ArExpr.Const  (eval V val e₁ - eval V val e₂)))\n            have e := chasles_step_star _ a (chasles_step_star _ b d)\n            exact e\n        | Mul e₁ e₂ =>\n            have ih1 := eval_completeness e₁ (val := val)\n            have ih2 := eval_completeness e₂ (val := val)\n            simp [eval]\n            have a := arstepstar_mul_left _ _ e₂ ih1\n            have b := arstepstar_mul_right (eval V val e₁) e₂ _ ih2\n            have c := ArStep.mulConstConst (val := val) (eval V val e₁) (eval V val e₂)\n            have d := ArStepStar.trans _ _ _ c (ArStepStar.refl _ (ArExpr.Const (eval  V val e₁ * eval V val e₂)))\n            have e := chasles_step_star _ a (chasles_step_star _ b d)\n            exact e\n        | Var var =>\n            simp [eval]\n            have a := ArStep.getVarValue var (val := val)\n            have b := ArStepStar.refl val (ArExpr.Const (val var))\n            exact ArStepStar.trans (val := val) _ _ _ a b\n        | If cond a b =>\n            simp [eval]\n            cases h: (eval_bool V val cond)\n            .   simp\n                have h1 := ArStep.ifCondFalse a b (val := val)\n                have h2 := eval_completeness b (val := val)\n                have h3 := StepStar.trans _ _ _ h1 h2\n                have h4 := eval_bool_completeness cond (val := val)\n                rw [h] at h4\n                have h5 := ArStepStar.ifStep _ _ a b h4\n                exact chasles_step_star V h5 h3\n            .   simp\n                have h1 := ArStep.ifCondTrue a b (val := val)\n                have h2 := eval_completeness a (val := val)\n                have h3 := StepStar.trans _ _ _ h1 h2\n                have h4 := eval_bool_completeness cond (val := val)\n                rw [h] at h4\n                have h5 := ArStepStar.ifStep _ _ a b h4\n                exact chasles_step_star V h5 h3", "nesting_depth": 4, "transitive_dep_count": 60, "subset_aristotle": true, "category": "Semantics"}
{"id": 267, "thm_name": "boolstep_preserves_eval_bool", "thm_stmt": "theorem boolstep_preserves_eval_bool (e e': BoolExpr V):\n        BoolStep V val e e' -> eval_bool V val e = eval_bool V val e'", "lean_root": "LeanExprEvaluator", "rel_path": "ExprEval/Lemmas.lean", "imports": ["import ExprEval.Steps", "import ExprEval.Expr"], "used_lib_defs": [{"name": "Add", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Mul", "module": "Init.Prelude"}, {"name": "Sub", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Not", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "BoolExpr", "content": "inductive BoolExpr : Type\n        | Const: Bool -> BoolExpr\n        | Less: ArExpr -> ArExpr -> BoolExpr\n        | Eq: ArExpr -> ArExpr -> BoolExpr\n        | Not : BoolExpr -> BoolExpr\n        | And : BoolExpr -> BoolExpr -> BoolExpr\n        | Or : BoolExpr -> BoolExpr -> BoolExpr\n\n    inductive ArExpr: Type\n        | Const: Int -> ArExpr\n        | Add: ArExpr -> ArExpr -> ArExpr\n        | Sub: ArExpr -> ArExpr -> ArExpr\n        | Mul: ArExpr -> ArExpr -> ArExpr\n        | Var: V -> ArExpr\n        | If : BoolExpr -> ArExpr -> ArExpr -> ArExpr"}, {"name": "BoolStep", "content": "inductive BoolStep: (val: V -> Int) -> (BoolExpr V) -> (BoolExpr V) -> Prop where\n    | notIsBoolNot (b: Bool): BoolStep val\n        (BoolExpr.Not (BoolExpr.Const b))\n        (BoolExpr.Const !b)\n    | andLeftTrue (e: BoolExpr V): BoolStep val\n        (BoolExpr.And (BoolExpr.Const true) e)\n        e\n    | orLeftFalse (e: BoolExpr V): BoolStep val\n        (BoolExpr.Or (BoolExpr.Const false) e)\n        e\n    | andLeftShortCircuit e : BoolStep val\n        (BoolExpr.And (BoolExpr.Const false) e)\n        (BoolExpr.Const false)\n    | orLeftShortCircuit e : BoolStep val\n        (BoolExpr.Or (BoolExpr.Const true) e)\n        (BoolExpr.Const true)\n    | lessConstConstTrue n₁ n₂ : n₁ < n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | lessConstConstFalse n₁ n₂ : n₁ >= n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | eqConstConstTrue n₁ n₂ : n₁ = n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | eqConstConstFalse n₁ n₂ : n₁ != n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | lessArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁' e₂)\n    | eqArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁' e₂)\n     | lessArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁ e₂')\n    | eqArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁ e₂')\n    | orStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Or e₁ e₂)\n                (BoolExpr.Or e₁' e₂)\n    | andStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.And e₁ e₂)\n                (BoolExpr.And e₁' e₂)\n    | notStep (e e' : BoolExpr V):\n        BoolStep val e e' -> BoolStep val\n            (BoolExpr.Not e)\n            (BoolExpr.Not e')\n\n    def eval (val: V -> Int) (e: ArExpr V) : Int :=\n        match e with\n            | ArExpr.Const x => x\n            | ArExpr.Add lhs rhs => (eval val lhs) + (eval val rhs)\n            | ArExpr.Sub lhs rhs => (eval val lhs) - (eval val rhs)\n            | ArExpr.Mul lhs rhs => (eval val lhs) * (eval val rhs)\n            | ArExpr.Var v => val v\n            | ArExpr.If cond then_e else_e => if eval_bool val cond then eval val then_e else eval val else_e\n\n    def eval_bool (val: V -> Int) (e: BoolExpr V): Bool :=\n        match e with\n        | BoolExpr.Const b => b\n        | BoolExpr.Less l r =>  (eval val l) < (eval val r)\n        | BoolExpr.Eq l r => (eval val l) == (eval val r)\n        | BoolExpr.Not e => not (eval_bool val e)\n        | BoolExpr.And l r => if eval_bool val l then eval_bool val r else false\n        | BoolExpr.Or l r => if eval_bool val l then true else eval_bool val r"}, {"name": "ArStep", "content": "inductive ArStep: (val: V -> Int) -> (ArExpr V) -> (ArExpr V) -> Prop where\n    | getVarValue (var: V) :\n        ArStep\n            val\n            (ArExpr.Var var)\n            (ArExpr.Const (val var))\n    | addConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Add (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ + n₂))\n    | subConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Sub (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ - n₂))\n    | mulConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Mul (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ * n₂))\n    | addLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Add e₁ e₂) (ArExpr.Add e₁' e₂)\n\n    | subLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Sub e₁ e₂) (ArExpr.Sub e₁' e₂)\n\n    | mulLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Mul e₁ e₂) (ArExpr.Mul e₁' e₂)\n\n    | addRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Add (ArExpr.Const n) e₂)  (ArExpr.Add (ArExpr.Const n) e₂')\n\n    | subRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Sub (ArExpr.Const n) e₂)  (ArExpr.Sub (ArExpr.Const n) e₂')\n    | mulRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Mul (ArExpr.Const n) e₂)  (ArExpr.Mul (ArExpr.Const n) e₂')\n\n    | ifStep (e e': BoolExpr V) (a b : ArExpr V): BoolStep val e e' ->\n        ArStep val (ArExpr.If e a b) (ArExpr.If e' a b)\n    | ifCondTrue (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const true) a b) a\n    | ifCondFalse (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const false) a b) b"}], "lib_lemmas": [{"name": "bne_iff_ne", "module": "Init.SimpLemmas"}, {"name": "Int.mul_eq_mul_right_iff", "module": "Init.Data.Int.Lemmas"}, {"name": "beq_eq_false_iff_ne", "module": "Init.SimpLemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "arstep_preserve_eval", "content": "    theorem arstep_preserve_eval (e e': ArExpr V): (ArStep V val) e e' -> eval V val e = eval V val e'"}], "local_ctx": "import ExprEval.Expr\n\nimport ExprEval.Steps", "target_theorem": "theorem boolstep_preserves_eval_bool (e e': BoolExpr V):\n        BoolStep V val e e' -> eval_bool V val e = eval_bool V val e' :=", "ground_truth_proof": ":= by\n        intro h\n        cases h with\n        | notIsBoolNot b => simp [eval_bool]\n        | andLeftTrue e => simp [eval_bool]\n        | orLeftFalse e => simp [eval_bool]\n        | andLeftShortCircuit e => simp [eval_bool]\n        | orLeftShortCircuit e => simp [eval_bool]\n        | lessConstConstTrue n1 n2 h =>\n            simp [eval_bool]\n            simp [eval]\n            exact h\n        | lessConstConstFalse n1 n2 h =>\n            simp [eval_bool]\n            simp [eval]\n            exact h\n        | eqConstConstTrue n1 n2 h =>\n            simp [eval_bool]\n            simp [eval]\n            exact h\n        | eqConstConstFalse n1 n2 h =>\n            simp [eval_bool]\n            simp [eval]\n            simp [bne_iff_ne] at h\n            exact h\n        | lessArStepLeft e₁ e₁' e₂ arstep\n        | eqArStepLeft e₁ e₁' e₂ arstep =>\n            simp [eval_bool]\n            have h' := arstep_preserve_eval e₁ e₁' arstep\n            rw [h']\n        | lessArStepRight e₁ e₂ e₂' arstep\n        | eqArStepRight e₁ e₂ e₂' arstep =>\n            simp [eval_bool]\n            have h' := arstep_preserve_eval e₂ e₂' arstep\n            rw [h']\n        | orStepLeft e₁ e₁' e₂ step\n        | andStepLeft e₁ e₁' e₂ step =>\n            have ihn := boolstep_preserves_eval_bool e₁ e₁' step\n            simp [eval_bool]\n            rw [ihn]\n        | notStep e e' step =>\n            simp[eval_bool]\n            have ihn := boolstep_preserves_eval_bool e e' step\n            rw [ihn]", "nesting_depth": 3, "transitive_dep_count": 33, "subset_aristotle": false, "category": "Semantics"}
{"id": 268, "thm_name": "arstep_preserve_eval", "thm_stmt": "theorem arstep_preserve_eval (e e': ArExpr V): (ArStep V val) e e' -> eval V val e = eval V val e'", "lean_root": "LeanExprEvaluator", "rel_path": "ExprEval/Lemmas.lean", "imports": ["import ExprEval.Steps", "import ExprEval.Expr"], "used_lib_defs": [{"name": "Add", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Mul", "module": "Init.Prelude"}, {"name": "Sub", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Not", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "ArStep", "content": "inductive ArStep: (val: V -> Int) -> (ArExpr V) -> (ArExpr V) -> Prop where\n    | getVarValue (var: V) :\n        ArStep\n            val\n            (ArExpr.Var var)\n            (ArExpr.Const (val var))\n    | addConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Add (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ + n₂))\n    | subConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Sub (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ - n₂))\n    | mulConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Mul (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ * n₂))\n    | addLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Add e₁ e₂) (ArExpr.Add e₁' e₂)\n\n    | subLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Sub e₁ e₂) (ArExpr.Sub e₁' e₂)\n\n    | mulLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Mul e₁ e₂) (ArExpr.Mul e₁' e₂)\n\n    | addRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Add (ArExpr.Const n) e₂)  (ArExpr.Add (ArExpr.Const n) e₂')\n\n    | subRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Sub (ArExpr.Const n) e₂)  (ArExpr.Sub (ArExpr.Const n) e₂')\n    | mulRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Mul (ArExpr.Const n) e₂)  (ArExpr.Mul (ArExpr.Const n) e₂')\n\n    | ifStep (e e': BoolExpr V) (a b : ArExpr V): BoolStep val e e' ->\n        ArStep val (ArExpr.If e a b) (ArExpr.If e' a b)\n    | ifCondTrue (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const true) a b) a\n    | ifCondFalse (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const false) a b) b"}, {"name": "BoolStep", "content": "inductive BoolStep: (val: V -> Int) -> (BoolExpr V) -> (BoolExpr V) -> Prop where\n    | notIsBoolNot (b: Bool): BoolStep val\n        (BoolExpr.Not (BoolExpr.Const b))\n        (BoolExpr.Const !b)\n    | andLeftTrue (e: BoolExpr V): BoolStep val\n        (BoolExpr.And (BoolExpr.Const true) e)\n        e\n    | orLeftFalse (e: BoolExpr V): BoolStep val\n        (BoolExpr.Or (BoolExpr.Const false) e)\n        e\n    | andLeftShortCircuit e : BoolStep val\n        (BoolExpr.And (BoolExpr.Const false) e)\n        (BoolExpr.Const false)\n    | orLeftShortCircuit e : BoolStep val\n        (BoolExpr.Or (BoolExpr.Const true) e)\n        (BoolExpr.Const true)\n    | lessConstConstTrue n₁ n₂ : n₁ < n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | lessConstConstFalse n₁ n₂ : n₁ >= n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | eqConstConstTrue n₁ n₂ : n₁ = n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | eqConstConstFalse n₁ n₂ : n₁ != n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | lessArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁' e₂)\n    | eqArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁' e₂)\n     | lessArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁ e₂')\n    | eqArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁ e₂')\n    | orStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Or e₁ e₂)\n                (BoolExpr.Or e₁' e₂)\n    | andStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.And e₁ e₂)\n                (BoolExpr.And e₁' e₂)\n    | notStep (e e' : BoolExpr V):\n        BoolStep val e e' -> BoolStep val\n            (BoolExpr.Not e)\n            (BoolExpr.Not e')\n\n    inductive BoolExpr : Type\n        | Const: Bool -> BoolExpr\n        | Less: ArExpr -> ArExpr -> BoolExpr\n        | Eq: ArExpr -> ArExpr -> BoolExpr\n        | Not : BoolExpr -> BoolExpr\n        | And : BoolExpr -> BoolExpr -> BoolExpr\n        | Or : BoolExpr -> BoolExpr -> BoolExpr\n\n    inductive ArExpr: Type\n        | Const: Int -> ArExpr\n        | Add: ArExpr -> ArExpr -> ArExpr\n        | Sub: ArExpr -> ArExpr -> ArExpr\n        | Mul: ArExpr -> ArExpr -> ArExpr\n        | Var: V -> ArExpr\n        | If : BoolExpr -> ArExpr -> ArExpr -> ArExpr\n\n    def eval (val: V -> Int) (e: ArExpr V) : Int :=\n        match e with\n            | ArExpr.Const x => x\n            | ArExpr.Add lhs rhs => (eval val lhs) + (eval val rhs)\n            | ArExpr.Sub lhs rhs => (eval val lhs) - (eval val rhs)\n            | ArExpr.Mul lhs rhs => (eval val lhs) * (eval val rhs)\n            | ArExpr.Var v => val v\n            | ArExpr.If cond then_e else_e => if eval_bool val cond then eval val then_e else eval val else_e\n\n    def eval_bool (val: V -> Int) (e: BoolExpr V): Bool :=\n        match e with\n        | BoolExpr.Const b => b\n        | BoolExpr.Less l r =>  (eval val l) < (eval val r)\n        | BoolExpr.Eq l r => (eval val l) == (eval val r)\n        | BoolExpr.Not e => not (eval_bool val e)\n        | BoolExpr.And l r => if eval_bool val l then eval_bool val r else false\n        | BoolExpr.Or l r => if eval_bool val l then true else eval_bool val r"}], "lib_lemmas": [{"name": "Int.mul_eq_mul_right_iff", "module": "Init.Data.Int.Lemmas"}, {"name": "beq_eq_false_iff_ne", "module": "Init.SimpLemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "bne_iff_ne", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import ExprEval.Expr\n\nimport ExprEval.Steps", "target_theorem": "theorem arstep_preserve_eval (e e': ArExpr V): (ArStep V val) e e' -> eval V val e = eval V val e' :=", "ground_truth_proof": ":= by\n            intro h\n            cases h with\n\n            | addConstConst n₁ n₂\n            | subConstConst n₁ n₂\n            | mulConstConst n₁ n₂ => simp [eval]\n            | subLeft e₁ e₁' e₂ step\n            | addLeft e₁ e₁' e₂ step =>\n                simp [eval]\n                exact arstep_preserve_eval e₁ e₁' step\n            | mulLeft e₁ e₁' e₂ step =>\n                simp [eval]\n                cases h : (eval V val e₂ == 0)\n                .   rw [beq_eq_false_iff_ne] at h\n                    rw [Int.mul_eq_mul_right_iff]\n                    exact arstep_preserve_eval e₁ e₁' step\n                    exact h\n                .   rw [beq_iff_eq] at h\n                    rw [h]\n                    simp\n            | addRight n e₂ e₁' step\n            | subRight n e₂ e₁' step\n            | mulRight n e₂ e₁' step =>\n                -- have ih := arstep_preserve_eval e₁ e₁' step\n                have ihn := arstep_preserve_eval e₂ e₁' step\n                simp [eval, ihn]\n\n            | getVarValue var => simp [eval]\n            | ifCondTrue a b =>\n                simp [eval]\n                simp [eval_bool]\n            | ifCondFalse a b =>\n                simp [eval]\n                simp [eval_bool]\n            | ifStep e e' a b bstep =>\n                have ihn := boolstep_preserves_eval_bool e e' bstep\n                cases h: (eval_bool V val e)\n                .   simp [eval]\n                    rw [h]\n                    rw [<- ihn]\n                    simp\n                    intro noth\n                    rw [h] at noth\n                    nomatch noth\n                .   simp [eval]\n                    rw [h]\n                    rw [<- ihn]\n                    simp\n                    intro noth\n                    rw [h] at noth\n                    nomatch noth", "nesting_depth": 4, "transitive_dep_count": 33, "subset_aristotle": true, "category": "Semantics"}
{"id": 269, "thm_name": "boolstepstar_preserves_eval_bool", "thm_stmt": "theorem boolstepstar_preserves_eval_bool (e e': BoolExpr V):\n        BoolStepStar V val e e' -> eval_bool V val e = eval_bool V val e'", "lean_root": "LeanExprEvaluator", "rel_path": "ExprEval/Lemmas.lean", "imports": ["import ExprEval.Steps", "import ExprEval.Expr"], "used_lib_defs": [{"name": "Add", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Mul", "module": "Init.Prelude"}, {"name": "Sub", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Not", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "BoolExpr", "content": "inductive BoolExpr : Type\n        | Const: Bool -> BoolExpr\n        | Less: ArExpr -> ArExpr -> BoolExpr\n        | Eq: ArExpr -> ArExpr -> BoolExpr\n        | Not : BoolExpr -> BoolExpr\n        | And : BoolExpr -> BoolExpr -> BoolExpr\n        | Or : BoolExpr -> BoolExpr -> BoolExpr\n\n    inductive ArExpr: Type\n        | Const: Int -> ArExpr\n        | Add: ArExpr -> ArExpr -> ArExpr\n        | Sub: ArExpr -> ArExpr -> ArExpr\n        | Mul: ArExpr -> ArExpr -> ArExpr\n        | Var: V -> ArExpr\n        | If : BoolExpr -> ArExpr -> ArExpr -> ArExpr"}, {"name": "BoolStepStar", "content": "def BoolStepStar (V: Type) := StepStar (Step := BoolStep V) V"}, {"name": "BoolStep", "content": "inductive BoolStep: (val: V -> Int) -> (BoolExpr V) -> (BoolExpr V) -> Prop where\n    | notIsBoolNot (b: Bool): BoolStep val\n        (BoolExpr.Not (BoolExpr.Const b))\n        (BoolExpr.Const !b)\n    | andLeftTrue (e: BoolExpr V): BoolStep val\n        (BoolExpr.And (BoolExpr.Const true) e)\n        e\n    | orLeftFalse (e: BoolExpr V): BoolStep val\n        (BoolExpr.Or (BoolExpr.Const false) e)\n        e\n    | andLeftShortCircuit e : BoolStep val\n        (BoolExpr.And (BoolExpr.Const false) e)\n        (BoolExpr.Const false)\n    | orLeftShortCircuit e : BoolStep val\n        (BoolExpr.Or (BoolExpr.Const true) e)\n        (BoolExpr.Const true)\n    | lessConstConstTrue n₁ n₂ : n₁ < n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | lessConstConstFalse n₁ n₂ : n₁ >= n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | eqConstConstTrue n₁ n₂ : n₁ = n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | eqConstConstFalse n₁ n₂ : n₁ != n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | lessArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁' e₂)\n    | eqArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁' e₂)\n     | lessArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁ e₂')\n    | eqArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁ e₂')\n    | orStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Or e₁ e₂)\n                (BoolExpr.Or e₁' e₂)\n    | andStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.And e₁ e₂)\n                (BoolExpr.And e₁' e₂)\n    | notStep (e e' : BoolExpr V):\n        BoolStep val e e' -> BoolStep val\n            (BoolExpr.Not e)\n            (BoolExpr.Not e')"}, {"name": "ArStep", "content": "inductive ArStep: (val: V -> Int) -> (ArExpr V) -> (ArExpr V) -> Prop where\n    | getVarValue (var: V) :\n        ArStep\n            val\n            (ArExpr.Var var)\n            (ArExpr.Const (val var))\n    | addConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Add (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ + n₂))\n    | subConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Sub (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ - n₂))\n    | mulConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Mul (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ * n₂))\n    | addLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Add e₁ e₂) (ArExpr.Add e₁' e₂)\n\n    | subLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Sub e₁ e₂) (ArExpr.Sub e₁' e₂)\n\n    | mulLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Mul e₁ e₂) (ArExpr.Mul e₁' e₂)\n\n    | addRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Add (ArExpr.Const n) e₂)  (ArExpr.Add (ArExpr.Const n) e₂')\n\n    | subRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Sub (ArExpr.Const n) e₂)  (ArExpr.Sub (ArExpr.Const n) e₂')\n    | mulRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Mul (ArExpr.Const n) e₂)  (ArExpr.Mul (ArExpr.Const n) e₂')\n\n    | ifStep (e e': BoolExpr V) (a b : ArExpr V): BoolStep val e e' ->\n        ArStep val (ArExpr.If e a b) (ArExpr.If e' a b)\n    | ifCondTrue (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const true) a b) a\n    | ifCondFalse (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const false) a b) b"}, {"name": "StepStar", "content": "inductive StepStar {ExprType: Type} {Step: StepKind V ExprType} : (val: V -> Int) -> ExprType -> ExprType -> Prop where\n        | refl val e : StepStar val e e\n        | trans e₁ e₂ e₃ : Step val e₁ e₂ -> StepStar val e₂ e₃ -> StepStar val e₁ e₃\n\n    def eval_bool (val: V -> Int) (e: BoolExpr V): Bool :=\n        match e with\n        | BoolExpr.Const b => b\n        | BoolExpr.Less l r =>  (eval val l) < (eval val r)\n        | BoolExpr.Eq l r => (eval val l) == (eval val r)\n        | BoolExpr.Not e => not (eval_bool val e)\n        | BoolExpr.And l r => if eval_bool val l then eval_bool val r else false\n        | BoolExpr.Or l r => if eval_bool val l then true else eval_bool val r\n\n    def eval (val: V -> Int) (e: ArExpr V) : Int :=\n        match e with\n            | ArExpr.Const x => x\n            | ArExpr.Add lhs rhs => (eval val lhs) + (eval val rhs)\n            | ArExpr.Sub lhs rhs => (eval val lhs) - (eval val rhs)\n            | ArExpr.Mul lhs rhs => (eval val lhs) * (eval val rhs)\n            | ArExpr.Var v => val v\n            | ArExpr.If cond then_e else_e => if eval_bool val cond then eval val then_e else eval val else_e"}], "lib_lemmas": [{"name": "Int.mul_eq_mul_right_iff", "module": "Init.Data.Int.Lemmas"}, {"name": "beq_eq_false_iff_ne", "module": "Init.SimpLemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "bne_iff_ne", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "arstep_preserve_eval", "content": "    theorem arstep_preserve_eval (e e': ArExpr V): (ArStep V val) e e' -> eval V val e = eval V val e'"}, {"name": "boolstep_preserves_eval_bool", "content": "    theorem boolstep_preserves_eval_bool (e e': BoolExpr V):\n        BoolStep V val e e' -> eval_bool V val e = eval_bool V val e'"}], "local_ctx": "import ExprEval.Expr\n\nimport ExprEval.Steps", "target_theorem": "theorem boolstepstar_preserves_eval_bool (e e': BoolExpr V):\n        BoolStepStar V val e e' -> eval_bool V val e = eval_bool V val e' :=", "ground_truth_proof": ":= by\n        intro h\n        induction h with\n        | refl => rfl\n        | trans _ _ _ h_step _ ihn =>\n            rw [<- ihn]\n            exact boolstep_preserves_eval_bool _ _ h_step", "nesting_depth": 4, "transitive_dep_count": 35, "subset_aristotle": false, "category": "Semantics"}
{"id": 270, "thm_name": "arstepstar_preserves_eval", "thm_stmt": "theorem arstepstar_preserves_eval (e e': ArExpr V) :\n            ArStepStar V val e e' -> eval V val e = eval V val e'", "lean_root": "LeanExprEvaluator", "rel_path": "ExprEval/Lemmas.lean", "imports": ["import ExprEval.Steps", "import ExprEval.Expr"], "used_lib_defs": [{"name": "And", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Not", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Add", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Mul", "module": "Init.Prelude"}, {"name": "Sub", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "ArExpr", "content": "inductive ArExpr: Type\n        | Const: Int -> ArExpr\n        | Add: ArExpr -> ArExpr -> ArExpr\n        | Sub: ArExpr -> ArExpr -> ArExpr\n        | Mul: ArExpr -> ArExpr -> ArExpr\n        | Var: V -> ArExpr\n        | If : BoolExpr -> ArExpr -> ArExpr -> ArExpr\n\n    inductive BoolExpr : Type\n        | Const: Bool -> BoolExpr\n        | Less: ArExpr -> ArExpr -> BoolExpr\n        | Eq: ArExpr -> ArExpr -> BoolExpr\n        | Not : BoolExpr -> BoolExpr\n        | And : BoolExpr -> BoolExpr -> BoolExpr\n        | Or : BoolExpr -> BoolExpr -> BoolExpr"}, {"name": "ArStepStar", "content": "def ArStepStar (V: Type) := StepStar (Step := ArStep V) V"}, {"name": "ArStep", "content": "inductive ArStep: (val: V -> Int) -> (ArExpr V) -> (ArExpr V) -> Prop where\n    | getVarValue (var: V) :\n        ArStep\n            val\n            (ArExpr.Var var)\n            (ArExpr.Const (val var))\n    | addConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Add (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ + n₂))\n    | subConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Sub (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ - n₂))\n    | mulConstConst(n₁ n₂: Int) :\n        ArStep\n            val\n            (ArExpr.Mul (ArExpr.Const n₁) (ArExpr.Const n₂))\n            (ArExpr.Const (n₁ * n₂))\n    | addLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Add e₁ e₂) (ArExpr.Add e₁' e₂)\n\n    | subLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Sub e₁ e₂) (ArExpr.Sub e₁' e₂)\n\n    | mulLeft (e₁ e₁' e₂) : ArStep val e₁ e₁' -> ArStep val (ArExpr.Mul e₁ e₂) (ArExpr.Mul e₁' e₂)\n\n    | addRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Add (ArExpr.Const n) e₂)  (ArExpr.Add (ArExpr.Const n) e₂')\n\n    | subRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Sub (ArExpr.Const n) e₂)  (ArExpr.Sub (ArExpr.Const n) e₂')\n    | mulRight (n: Int) (e₂ e₂': ArExpr V) : ArStep val e₂ e₂' -> ArStep val (ArExpr.Mul (ArExpr.Const n) e₂)  (ArExpr.Mul (ArExpr.Const n) e₂')\n\n    | ifStep (e e': BoolExpr V) (a b : ArExpr V): BoolStep val e e' ->\n        ArStep val (ArExpr.If e a b) (ArExpr.If e' a b)\n    | ifCondTrue (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const true) a b) a\n    | ifCondFalse (a b: ArExpr V) : ArStep val (ArExpr.If (BoolExpr.Const false) a b) b"}, {"name": "BoolStep", "content": "inductive BoolStep: (val: V -> Int) -> (BoolExpr V) -> (BoolExpr V) -> Prop where\n    | notIsBoolNot (b: Bool): BoolStep val\n        (BoolExpr.Not (BoolExpr.Const b))\n        (BoolExpr.Const !b)\n    | andLeftTrue (e: BoolExpr V): BoolStep val\n        (BoolExpr.And (BoolExpr.Const true) e)\n        e\n    | orLeftFalse (e: BoolExpr V): BoolStep val\n        (BoolExpr.Or (BoolExpr.Const false) e)\n        e\n    | andLeftShortCircuit e : BoolStep val\n        (BoolExpr.And (BoolExpr.Const false) e)\n        (BoolExpr.Const false)\n    | orLeftShortCircuit e : BoolStep val\n        (BoolExpr.Or (BoolExpr.Const true) e)\n        (BoolExpr.Const true)\n    | lessConstConstTrue n₁ n₂ : n₁ < n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | lessConstConstFalse n₁ n₂ : n₁ >= n₂ -> BoolStep val\n        (BoolExpr.Less (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | eqConstConstTrue n₁ n₂ : n₁ = n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const true)\n    | eqConstConstFalse n₁ n₂ : n₁ != n₂ -> BoolStep val\n        (BoolExpr.Eq (ArExpr.Const n₁) (ArExpr.Const n₂))\n        (BoolExpr.Const false)\n    | lessArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁' e₂)\n    | eqArStepLeft (e₁ e₁' e₂: ArExpr V):\n        ArStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁' e₂)\n     | lessArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Less e₁ e₂)\n                (BoolExpr.Less e₁ e₂')\n    | eqArStepRight (e₁ e₂ e₂': ArExpr V):\n        ArStep val e₂ e₂' ->\n            BoolStep val\n                (BoolExpr.Eq e₁ e₂)\n                (BoolExpr.Eq e₁ e₂')\n    | orStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.Or e₁ e₂)\n                (BoolExpr.Or e₁' e₂)\n    | andStepLeft (e₁ e₁' e₂: BoolExpr V):\n        BoolStep val e₁ e₁' ->\n            BoolStep val\n                (BoolExpr.And e₁ e₂)\n                (BoolExpr.And e₁' e₂)\n    | notStep (e e' : BoolExpr V):\n        BoolStep val e e' -> BoolStep val\n            (BoolExpr.Not e)\n            (BoolExpr.Not e')"}, {"name": "StepStar", "content": "inductive StepStar {ExprType: Type} {Step: StepKind V ExprType} : (val: V -> Int) -> ExprType -> ExprType -> Prop where\n        | refl val e : StepStar val e e\n        | trans e₁ e₂ e₃ : Step val e₁ e₂ -> StepStar val e₂ e₃ -> StepStar val e₁ e₃\n\n    def eval (val: V -> Int) (e: ArExpr V) : Int :=\n        match e with\n            | ArExpr.Const x => x\n            | ArExpr.Add lhs rhs => (eval val lhs) + (eval val rhs)\n            | ArExpr.Sub lhs rhs => (eval val lhs) - (eval val rhs)\n            | ArExpr.Mul lhs rhs => (eval val lhs) * (eval val rhs)\n            | ArExpr.Var v => val v\n            | ArExpr.If cond then_e else_e => if eval_bool val cond then eval val then_e else eval val else_e\n\n    def eval_bool (val: V -> Int) (e: BoolExpr V): Bool :=\n        match e with\n        | BoolExpr.Const b => b\n        | BoolExpr.Less l r =>  (eval val l) < (eval val r)\n        | BoolExpr.Eq l r => (eval val l) == (eval val r)\n        | BoolExpr.Not e => not (eval_bool val e)\n        | BoolExpr.And l r => if eval_bool val l then eval_bool val r else false\n        | BoolExpr.Or l r => if eval_bool val l then true else eval_bool val r"}], "lib_lemmas": [{"name": "bne_iff_ne", "module": "Init.SimpLemmas"}, {"name": "Int.mul_eq_mul_right_iff", "module": "Init.Data.Int.Lemmas"}, {"name": "beq_eq_false_iff_ne", "module": "Init.SimpLemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "arstep_preserve_eval", "content": "    theorem arstep_preserve_eval (e e': ArExpr V): (ArStep V val) e e' -> eval V val e = eval V val e'"}], "local_ctx": "import ExprEval.Expr\n\nimport ExprEval.Steps", "target_theorem": "theorem arstepstar_preserves_eval (e e': ArExpr V) :\n            ArStepStar V val e e' -> eval V val e = eval V val e' :=", "ground_truth_proof": ":= by\n            intro h\n            induction h with\n            | refl _ => rfl\n            | trans _ _ _ h1 _ ih =>\n                rw [arstep_preserve_eval _ _ h1, ih]", "nesting_depth": 3, "transitive_dep_count": 35, "subset_aristotle": false, "category": "Semantics"}
{"id": 271, "thm_name": "TM.progress", "thm_stmt": "theorem progress t T\n    : HasType t T → value t ∨ ∃ t', Step t t'", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Types.lean", "imports": ["import Frap.SmallStep"], "used_lib_defs": [{"name": "structure BitVec (w : Nat) where", "module": ""}, {"name": "/-- Construct a `BitVec w` from a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "ofFin ::", "module": ""}, {"name": "/-- Interpret a bitvector as a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "toFin : Fin (hPow 2 w)", "module": ""}], "used_repo_defs": [{"name": "syntax term \"==>\" term : term", "content": "syntax term \"==>\" term : term\n\nsyntax term \"~~>\" term : term\n\nsyntax term \"~~>*\" term : term\n\nsyntax:30 term \" =[ \" imp \" ]=> \" term : term\n\nsyntax term \"!->\" term \"; \" term : term\n\nsyntax:36 term \"<<->>\" term : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$t:term ==> $n:term) => `(Eval $t $n)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$t₁:term ~~> $t₂:term) => `(Step $t₁ $t₂)\n\nexample :\n    p\n      (p (c 1) (c 3))\n      (p (c 2) (c 4))\n    ~~>\n    p\n      (c 4)\n      (p (c 2) (c 4)) := by admit /- proof elided -/"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$t₁:term ~~>* $t₂:term) => `(Multi Step $t₁ $t₂)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$p <<->> $q) => `($p ->> $q ∧ $q ->> $p)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|{*$p*} $c {*$q*}) => `(valid_hoare_triple $p $c $q)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$st =[ $c ]=> $st') => `(CEval <{$c}> $st $st')"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$x:term !-> $a:term ; $st) => `(update $st $x $a)\n\nexample : empty =[\n      x := 2;\n      if (x <= 1) then y := 3 else z := 4 end\n    ]=> (z !-> 4; x !-> 2; empty) := by admit /- proof elided -/"}, {"name": "Step", "content": "inductive Step : Tm → Tm → Prop :=\n  | st_ifTrue t₁ t₂ : Step (ite tru t₁ t₂) t₁\n  | st_ifFalse t₁ t₂ : Step (ite fls t₁ t₂) t₂\n  | st_if c c' t₁ t₂ :\n      Step c c' → Step (ite c t₁ t₂) (ite c' t₁ t₂)\n  | st_succ t₁ t₁' : Step t₁ t₁' → Step (scc t₁) (scc t₁')\n  | st_pred0 : Step (prd zro) zro\n  | st_predSucc v : NValue v → Step (prd (scc v)) v\n  | st_pred t₁ t₁' : Step t₁ t₁' → Step (prd t₁) (prd t₁')\n  | st_iszero0 : Step (iszero zro) tru\n  | st_iszeroSucc v : NValue v → Step (iszero (scc v)) fls\n  | st_iszero t₁ t₁' :\n      Step t₁ t₁' → Step (iszero t₁) (iszero t₁')"}, {"name": "Tm", "content": "inductive Tm : Type :=\n  | tru : Tm\n  | fls : Tm\n  | ite : Tm → Tm → Tm → Tm\n  | zro : Tm\n  | scc : Tm → Tm\n  | prd : Tm → Tm\n  | iszero : Tm → Tm"}, {"name": "NValue", "content": "inductive NValue : Tm → Prop :=\n  | nv_0 : NValue zro\n  | nv_succ t : NValue t → NValue (scc t)"}, {"name": "Ty", "content": "inductive Ty : Type :=\n  | bool : Ty\n  | nat : Ty"}, {"name": "HasType", "content": "inductive HasType : Tm → Ty → Prop :=\n  | t_true : HasType tru bool\n  | t_false : HasType fls bool\n  | t_if t₁ t₂ t₃ T :\n      HasType t₁ bool → HasType t₂ T → HasType t₃ T\n      → HasType (ite t₁ t₂ t₃) T\n  | t_0 : HasType zro nat\n  | t_succ t₁ : HasType t₁ nat → HasType (scc t₁) nat\n  | t_pred t₁ : HasType t₁ nat → HasType (prd t₁) nat\n  | t_iszero t₁ : HasType t₁ nat → HasType (iszero t₁) bool"}, {"name": "Tm", "content": "inductive Tm : Type :=\n  | c : Nat → Tm       \n  | p : Tm → Tm → Tm   "}, {"name": "BValue", "content": "inductive BValue : Tm → Prop :=\n  | bv_true : BValue tru\n  | bv_false : BValue fls"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "TM.Tm", "content": "inductive Tm : Type :=\n  | tru : Tm\n  | fls : Tm\n  | ite : Tm → Tm → Tm → Tm\n  | zro : Tm\n  | scc : Tm → Tm\n  | prd : Tm → Tm\n  | iszero : Tm → Tm"}, {"name": "TM.BValue", "content": "inductive BValue : Tm → Prop :=\n  | bv_true : BValue tru\n  | bv_false : BValue fls"}, {"name": "TM.NValue", "content": "inductive NValue : Tm → Prop :=\n  | nv_0 : NValue zro\n  | nv_succ t : NValue t → NValue (scc t)"}, {"name": "TM.value", "content": "abbrev value (t : Tm) := BValue t ∨ NValue t"}, {"name": "TM.Step", "content": "inductive Step : Tm → Tm → Prop :=\n  | st_ifTrue t₁ t₂ : Step (ite tru t₁ t₂) t₁\n  | st_ifFalse t₁ t₂ : Step (ite fls t₁ t₂) t₂\n  | st_if c c' t₁ t₂ :\n      Step c c' → Step (ite c t₁ t₂) (ite c' t₁ t₂)\n  | st_succ t₁ t₁' : Step t₁ t₁' → Step (scc t₁) (scc t₁')\n  | st_pred0 : Step (prd zro) zro\n  | st_predSucc v : NValue v → Step (prd (scc v)) v\n  | st_pred t₁ t₁' : Step t₁ t₁' → Step (prd t₁) (prd t₁')\n  | st_iszero0 : Step (iszero zro) tru\n  | st_iszeroSucc v : NValue v → Step (iszero (scc v)) fls\n  | st_iszero t₁ t₁' :\n      Step t₁ t₁' → Step (iszero t₁) (iszero t₁')"}, {"name": "TM.Ty", "content": "inductive Ty : Type :=\n  | bool : Ty\n  | nat : Ty"}, {"name": "TM.HasType", "content": "inductive HasType : Tm → Ty → Prop :=\n  | t_true : HasType tru bool\n  | t_false : HasType fls bool\n  | t_if t₁ t₂ t₃ T :\n      HasType t₁ bool → HasType t₂ T → HasType t₃ T\n      → HasType (ite t₁ t₂ t₃) T\n  | t_0 : HasType zro nat\n  | t_succ t₁ : HasType t₁ nat → HasType (scc t₁) nat\n  | t_pred t₁ : HasType t₁ nat → HasType (prd t₁) nat\n  | t_iszero t₁ : HasType t₁ nat → HasType (iszero t₁) bool"}], "used_local_lemmas": [{"name": "TM.bool_canonical", "content": "theorem bool_canonical t : HasType t bool → value t → BValue t"}, {"name": "TM.nat_canonical", "content": "theorem nat_canonical t : HasType t nat → value t → NValue t"}], "local_ctx": "import Frap.SmallStep\n\nnamespace TM\n\ninductive Tm : Type :=\n  | tru : Tm\n  | fls : Tm\n  | ite : Tm → Tm → Tm → Tm\n  | zro : Tm\n  | scc : Tm → Tm\n  | prd : Tm → Tm\n  | iszero : Tm → Tm\n\nopen Tm\n\ninductive BValue : Tm → Prop :=\n  | bv_true : BValue tru\n  | bv_false : BValue fls\n\ninductive NValue : Tm → Prop :=\n  | nv_0 : NValue zro\n  | nv_succ t : NValue t → NValue (scc t)\n\nopen BValue\n\nopen NValue\n\nabbrev value (t : Tm) := BValue t ∨ NValue t\n\ninductive Step : Tm → Tm → Prop :=\n  | st_ifTrue t₁ t₂ : Step (ite tru t₁ t₂) t₁\n  | st_ifFalse t₁ t₂ : Step (ite fls t₁ t₂) t₂\n  | st_if c c' t₁ t₂ :\n      Step c c' → Step (ite c t₁ t₂) (ite c' t₁ t₂)\n  | st_succ t₁ t₁' : Step t₁ t₁' → Step (scc t₁) (scc t₁')\n  | st_pred0 : Step (prd zro) zro\n  | st_predSucc v : NValue v → Step (prd (scc v)) v\n  | st_pred t₁ t₁' : Step t₁ t₁' → Step (prd t₁) (prd t₁')\n  | st_iszero0 : Step (iszero zro) tru\n  | st_iszeroSucc v : NValue v → Step (iszero (scc v)) fls\n  | st_iszero t₁ t₁' :\n      Step t₁ t₁' → Step (iszero t₁) (iszero t₁')\n\nopen Step\n\ninductive Ty : Type :=\n  | bool : Ty\n  | nat : Ty\n\nopen Ty\n\ninductive HasType : Tm → Ty → Prop :=\n  | t_true : HasType tru bool\n  | t_false : HasType fls bool\n  | t_if t₁ t₂ t₃ T :\n      HasType t₁ bool → HasType t₂ T → HasType t₃ T\n      → HasType (ite t₁ t₂ t₃) T\n  | t_0 : HasType zro nat\n  | t_succ t₁ : HasType t₁ nat → HasType (scc t₁) nat\n  | t_pred t₁ : HasType t₁ nat → HasType (prd t₁) nat\n  | t_iszero t₁ : HasType t₁ nat → HasType (iszero t₁) bool\n\nopen HasType\n\nexample  -- `⊢ if false then 0 else (succ 0) ∈ nat`\n    : HasType (ite fls zro (scc zro)) nat := by\n  apply t_if\n  . apply t_false\n  . apply t_0\n  . apply t_succ\n    apply t_0\n\n/-\nIt's important to realize that the typing relation is a _conservative_ (or _static_) approximation: it does not consider what happens when the term is reduced.\nIn particular, it does not calculate the type of its normal form.\n-/\n\nexample  -- `⊢ if false then zero else true ∉ bool`\n    : ¬ HasType (ite fls zro tru) bool := by\n  intro contra\n  cases contra\n  rename_i contra _\n  cases contra\n\n/-\nexercise (1-star)\n-/\nexample t : HasType (scc t) nat → HasType t nat := by\n  intro ht\n  cases ht\n  assumption", "target_theorem": "theorem progress t T\n    : HasType t T → value t ∨ ∃ t', Step t t' :=", "ground_truth_proof": ":= by\n  intro ht\n  induction ht with\n  | t_if t₁ t₂ t₃ T =>\n    rename_i ih₁ ih₂ ih₃\n    right; cases ih₁\n    . -- t₁ is a value\n      have h : BValue t₁ := by\n        apply bool_canonical <;> assumption\n      cases h\n      . exists t₂; apply st_ifTrue\n      . exists t₃; apply st_ifFalse\n    . -- t₁ can take a step\n      rename_i h\n      obtain ⟨t₁', h₁⟩ := h\n      exists ite t₁' t₂ t₃\n      apply st_if; exact h₁\n  | t_true =>\n    left;unfold value; left; apply bv_true\n  | t_false =>\n    left; unfold value; left; apply bv_false\n  | t_0 =>\n    left; unfold value; right; apply nv_0\n  | t_succ t =>\n    rename_i ih\n    cases ih\n    . -- t is a value\n      left\n      have h : NValue t := by\n        apply nat_canonical <;> assumption\n      cases h\n      . unfold value; right; apply nv_succ; apply nv_0\n      . unfold value; right; apply nv_succ; apply nv_succ; assumption\n    . -- t can take a step\n      right\n      rename_i ih\n      obtain ⟨t', h⟩ := ih\n      exists (scc t')\n      apply st_succ; exact h\n  | t_pred t =>\n    rename_i ih\n    right; cases ih\n    . -- t is a value\n      have h : NValue t := by\n        apply nat_canonical <;> assumption\n      cases h\n      . exists zro; apply st_pred0\n      . constructor; apply st_predSucc; assumption\n    . -- t can take a step\n      rename_i h\n      obtain ⟨t', h⟩ := h\n      exists (prd t')\n      apply st_pred; exact h\n  | t_iszero t =>\n    rename_i ih\n    right; cases ih\n    . -- t is a value\n      have h : NValue t := by\n        apply nat_canonical <;> assumption\n      cases h\n      . exists tru; apply st_iszero0\n      . exists fls; apply st_iszeroSucc; assumption\n    . -- t can take a step\n      rename_i h\n      obtain ⟨t', h⟩ := h\n      exists (iszero t')\n      apply st_iszero; exact h\n\n/-\n### Type preservation\n\nThe second critical property of typing is that, when a well-typed term takes a step, the result is a well-typed term (of the same type).\n-/\n\n/-\nexercise (3-star)\nComplete the following informal proof:\n\n_Theorem_: If `⊢ t ∈ T` and `t ~~> t'`, then `⊢ t' ∈ T`.\n_Proof_: By induction on a derivation of `⊢ t ∈ T`.\n\n* If the last rule in the derivation is `t_if`, then `t = if t₁ then t₂ else t₃`, with `⊢ t₁ ∈ bool`, `⊢ t₂ ∈ T`, and `⊢ t₃ ∈ T`.\n  Inspecting the rules for the small-step reduction relation and remembering that `t` is the form `if ...`, we see that the only ones that could have been used to prove `t ~~> t'` are `st_ifTrue`, `st_ifFalse`, or `st_if`.\n * If the last rule was `st_ifTrue`, then `t' = t₂`.\n   But we know that `⊢ t₂ ∈ T`, so we are done.\n * If the last rule was `st_ifFalse`, then `t' = t₃`.\n   But we know that `⊢ t₃ ∈ T`, so we are done.\n * If the last rule was `st_if`, then `t' = if t₁' then t₂ else t₃`, where `t₁ ~~> t₁'`.\n   We know `⊢ t₁ ∈ bool`, so by the IH, `⊢ t₁' ∈ bool`.\n   The `t_if` rule then gives us `⊢ if if t₁' then t₂ else t₃ ∈ T`, as required.\n/- **FILL IN HERE** -/\n-/", "nesting_depth": 2, "transitive_dep_count": 16, "subset_aristotle": true, "category": "Semantics"}
{"id": 272, "thm_name": "Imp.fold_constants_com_sound", "thm_stmt": "theorem fold_constants_com_sound : ctrans_sound fold_constants_com", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Trans.lean", "imports": ["import Frap.Exercises.Equiv", "import Frap.Equiv"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"<[\" term \"]>\" : imp", "content": "syntax \"<[\" term \"]>\" : imp\n\nsyntax \"<{\" imp \"}>\" : term\n\nsyntax \"false\" : imp\n\nsyntax \"false\" : term\n\nsyntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : imp\n\nsyntax \"true\" : imp\n\nsyntax \"true\" : term\n\nsyntax \"skip\" : imp\n\nsyntax:21 \"while\" imp:20 \"do\" imp:20 \"end\" : imp"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|true) => `(Bool.true)\n  | `(term|false) => `(Bool.false)\n  | `(term|<{$x}>) => `(imp|$x)\n  | `(imp|$n:num) => `(a_num $n)\n  | `(imp|$s:str) => `(a_id $s)\n  | `(imp|$x + $y) => `(a_plus <{$x}> <{$y}>)\n  | `(imp|$x - $y) => `(a_minus <{$x}> <{$y}>)\n  | `(imp|$x * $y) => `(a_mult <{$x}> <{$y}>)\n  | `(imp|true) => `(b_true)\n  | `(imp|false) => `(b_false)\n  | `(imp|$x = $y) => `(b_eq <{$x}> <{$y}>)\n  | `(imp|$x != $y) => `(b_neq <{$x}> <{$y}>)\n  | `(imp|$x <= $y) => `(b_le <{$x}> <{$y}>)\n  | `(imp|!$x) => `(b_not <{$x}>)\n  | `(imp|$x && $y) => `(b_and <{$x}> <{$y}>)\n  | `(imp|$x || $y) => `(b_or <{$x}> <{$y}>)\n  | `(imp|($x)) => `(<{$x}>)\n  | `(imp|$x:ident) => `(a_id $(Lean.quote (toString x.getId)))\n  | `(imp|<[$t:term]>) => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(imp|skip) => `(c_skip)\n  | `(imp|$x:str := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$x:ident := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$c1 ; $c2) => `(c_seq <{$c1}> <{$c2}>)\n  | `(imp|if $b then $c1 else $c2 end) => `(c_if <{$b}> <{$c1}> <{$c2}>)\n  | `(imp|while $b do $c end) => `(c_while <{$b}> <{$c}>)"}, {"name": "cequiv", "content": "def cequiv (c₁ c₂ : Com) : Prop :=\n  ∀ st st' : State, (st =[<[c₁]>]=> st') ↔ (st =[<[c₂]>]=> st')"}, {"name": "State", "content": "abbrev State := String → Nat"}, {"name": "Com", "content": "inductive Com :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_id : String → AExp  \n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "Com", "content": "inductive Com : Type :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com\n  | c_par : Com → Com → Com   "}, {"name": "BExp", "content": "inductive BExp where\n  | b_true : BExp\n  | b_false : BExp\n  | b_eq : AExp → AExp → BExp\n  | b_neq : AExp → AExp → BExp\n  | b_le : AExp → AExp → BExp\n  | b_not : BExp → BExp\n  | b_and : BExp → BExp → BExp\n  | b_or : BExp → BExp → BExp"}, {"name": "x", "content": "abbrev x := \"x\""}, {"name": "y", "content": "abbrev y := \"y\""}, {"name": "loop", "content": "def loop : Com :=\n  <{ while true do\n       skip\n     end }>"}, {"name": "bequiv", "content": "def bequiv (b₁ b₂ : BExp) : Prop :=\n  ∀ (st : State), beval st b₁ = beval st b₂"}, {"name": "beval", "content": "def beval (st : State) (b : BExp) : Bool :=\n  match b with\n  | b_true => true\n  | b_false => false\n  | b_eq a₁ a₂ => (aeval st a₁) == (aeval st a₂)\n  | b_neq a₁ a₂ => (aeval st a₁) != (aeval st a₂)\n  | b_le a₁ a₂ => (aeval st a₁) <= (aeval st a₂)\n  | b_not b₁ => not (beval st b₁)\n  | b_and b₁ b₂ => and (beval st b₁) (beval st b₂)\n  | b_or b₁ b₂ => or (beval st b₁) (beval st b₂)\n\nexample : aeval (update empty x 5)\n    <{3 + x * 2}>\n    \n    = 13 := by admit /- proof elided -/"}, {"name": "empty", "content": "def empty : State := fun _ => 0"}, {"name": "z", "content": "abbrev z := \"z\""}, {"name": "aeval", "content": "def aeval (st : State) (a : AExp) : Nat :=\n  match a with\n  | a_num n => n\n  | a_id x => st x\n  | a_plus a₁ a₂ => (aeval st a₁) + (aeval st a₂)\n  | a_minus a₁ a₂ => (aeval st a₁) - (aeval st a₂)\n  | a_mult a₁ a₂ => (aeval st a₁) * (aeval st a₂)"}, {"name": "update", "content": "def update (st : State) (k : String) (v : Nat) : State :=\n  fun x => if x == k then v else st x"}, {"name": "aequiv", "content": "def aequiv (a₁ a₂ : AExp) : Prop :=\n  ∀ (st : State), aeval st a₁ = aeval st a₂"}, {"name": "CEval", "content": "inductive CEval : Com → State → State → Prop :=\n  | e_skip : ∀ st,\n      CEval c_skip st st\n  | e_asgn : ∀ a n x st,\n      aeval st a = n\n      → CEval (c_asgn x a) st (update st x n)\n  | e_seq : ∀ c₁ c₂ st st' st'',\n      CEval c₁ st st' → CEval c₂ st' st''\n      → CEval (c_seq c₁ c₂) st st''\n  | e_ifTrue : ∀ b c₁ c₂ st st',\n      beval st b = true → CEval c₁ st st'\n      → CEval (c_if b c₁ c₂) st st'\n  | e_ifFalse : ∀ b c₁ c₂ st st',\n      beval st b = false → CEval c₂ st st'\n      → CEval (c_if b c₁ c₂) st st'\n  | e_whileFalse : ∀ b c st,\n      beval st b = false\n      → CEval (c_while b c) st st\n  | e_whileTrue : ∀ b c st st' st'',\n      beval st b = true\n      → CEval c st st'\n      → CEval (c_while b c) st' st''\n      → CEval (c_while b c) st st''"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "if_true", "content": "theorem if_true b c₁ c₂\n    : bequiv b <{true}> → cequiv <{if <[b]> then <[c₁]> else <[c₂]> end}> c₁"}, {"name": "if_false", "content": "theorem if_false b c₁ c₂\n    : bequiv b <{false}> → cequiv <{if <[b]> then <[c₁]> else <[c₂]> end}> c₂"}, {"name": "while_false", "content": "theorem while_false b c : bequiv b <{false}> →\n    cequiv <{while <[b]> do <[c]> end}> <{skip}>"}], "used_local_defs": [{"name": "Imp.atrans_sound", "content": "def atrans_sound (atrans : AExp → AExp) :=\n  ∀ (a : AExp), aequiv a (atrans a)"}, {"name": "Imp.btrans_sound", "content": "def btrans_sound (btrans : BExp → BExp) :=\n  ∀ (b : BExp), bequiv b (btrans b)"}, {"name": "Imp.ctrans_sound", "content": "def ctrans_sound (ctrans : Com → Com) :=\n  ∀ (c : Com), cequiv c (ctrans c)"}, {"name": "Imp.fold_constants_aexp", "content": "def fold_constants_aexp (a : AExp) : AExp :=\n  match a with\n  | a_num _\n  | a_id _\n    => a\n  | a_plus a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 + n2)\n    | (a1', a2') => a_plus a1' a2'\n  | a_minus a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 - n2)\n    | (a1', a2') => a_minus a1' a2'\n  | a_mult a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 * n2)\n    | (a1', a2') => a_mult a1' a2'\n\nexample : fold_constants_aexp <{(1 + 2) * x}> = <{3 * x}> := by admit /- proof elided -/"}, {"name": "Imp.fold_constants_bexp", "content": "def fold_constants_bexp (b : BExp) : BExp :=\n  match b with\n  | b_true\n  | b_false\n    => b\n  | b_eq a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 == n2 then b_true else b_false\n    | (a1', a2') => b_eq a1' a2'\n  | b_neq a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 != n2 then b_true else b_false\n    | (a1', a2') => b_neq a1' a2'\n  | b_le a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 <= n2 then b_true else b_false\n    | (a1', a2') => b_le a1' a2'\n  | b_not b1 =>\n    match fold_constants_bexp b1 with\n    | b_true => b_false\n    | b_false => b_true\n    | b1' => b_not b1'\n  | b_and b1 b2 =>\n    match (fold_constants_bexp b1, fold_constants_bexp b2) with\n    | (b_true, b_true) => b_true\n    | (b_true, b_false) => b_false\n    | (b_false, b_true) => b_false\n    | (b_false, b_false) => b_false\n    | (b1', b2') => b_and b1' b2'\n  | b_or b1 b2 =>\n    match (fold_constants_bexp b1, fold_constants_bexp b2) with\n    | (b_true, b_true) => b_true\n    | (b_true, b_false) => b_true\n    | (b_false, b_true) => b_true\n    | (b_false, b_false) => b_false\n    | (b1', b2') => b_or b1' b2'\n\nexample : fold_constants_bexp <{true && !(false && true)}> = <{true}> := by admit /- proof elided -/"}, {"name": "Imp.fold_constants_com", "content": "def fold_constants_com (c : Com) : Com :=\n  match c with\n  | c_skip => c\n  | c_asgn x a => c_asgn x (fold_constants_aexp a)\n  | c_seq c1 c2 => c_seq (fold_constants_com c1) (fold_constants_com c2)\n  | c_if b c1 c2 =>\n    match fold_constants_bexp b with\n    | b_true => fold_constants_com c1\n    | b_false => fold_constants_com c2\n    | b' => c_if b' (fold_constants_com c1) (fold_constants_com c2)\n  | c_while b c1 =>\n    match fold_constants_bexp b with\n    | b_true => loop\n    | b_false => c_skip\n    | b' => c_while b' (fold_constants_com c1)\n\nexample : fold_constants_com\n    \n    <{ x := 4 + 5;\n       y := x - 3;\n       if (x - y) = (2 + 4) then skip\n       else y := 0 end;\n       if 0 <= (4 - (2 + 1)) then y := 0\n       else skip end;\n       while y = 0 do\n         x := x + 1\n       end }>\n    = \n    <{ x := 9;\n       y := x - 3;\n       if (x - y) = 6 then skip\n       else y := 0 end;\n       y := 0;\n       while y = 0 do\n         x := x + 1\n       end }> := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "Imp.while_true", "content": "theorem while_true b c :\n    bequiv b <{true}> → cequiv <{while <[b]> do <[c]> end}> loop"}, {"name": "Imp.sym_bequiv", "content": "theorem sym_bequiv b₁ b₂ : bequiv b₁ b₂ → bequiv b₂ b₁"}, {"name": "Imp.sym_cequiv", "content": "theorem sym_cequiv c₁ c₂ : cequiv c₁ c₂ → cequiv c₂ c₁"}, {"name": "Imp.trans_cequiv", "content": "theorem trans_cequiv c₁ c₂ c₃ : cequiv c₁ c₂ → cequiv c₂ c₃ → cequiv c₁ c₃"}, {"name": "Imp.c_asgn_congruence", "content": "theorem c_asgn_congruence x a a'\n    : aequiv a a' → cequiv <{x := <[a]>}> <{x := <[a']>}>"}, {"name": "Imp.c_while_congruence", "content": "theorem c_while_congruence b b' c c'\n    : bequiv b b' → cequiv c c'\n      → cequiv <{while <[b]> do <[c]> end}> <{while <[b']> do <[c']> end}>"}, {"name": "Imp.c_seq_congruence", "content": "theorem c_seq_congruence c₁ c₁' c₂ c₂'\n    : cequiv c₁ c₁' → cequiv c₂ c₂' → cequiv (c_seq c₁ c₂) (c_seq c₁' c₂')"}, {"name": "Imp.c_if_congruence", "content": "theorem c_if_congruence b b' c₁ c₁' c₂ c₂'\n    : bequiv b b' → cequiv c₁ c₁' → cequiv c₂ c₂'\n      → cequiv (c_if b c₁ c₂) (c_if b' c₁' c₂')"}, {"name": "Imp.fold_constants_aexp_sound", "content": "theorem fold_constants_aexp_sound : atrans_sound fold_constants_aexp"}, {"name": "Imp.fold_constants_bexp_sound", "content": "theorem fold_constants_bexp_sound : btrans_sound fold_constants_bexp"}], "local_ctx": "import Frap.Equiv\n\nimport Frap.Exercises.Equiv\n\nnamespace Imp\n\nopen AExp\n\nopen BExp\n\nopen Com\n\nopen CEval\n\ndef atrans_sound (atrans : AExp → AExp) :=\n  ∀ (a : AExp), aequiv a (atrans a)\n\ndef btrans_sound (btrans : BExp → BExp) :=\n  ∀ (b : BExp), bequiv b (btrans b)\n\ndef ctrans_sound (ctrans : Com → Com) :=\n  ∀ (c : Com), cequiv c (ctrans c)\n\ndef fold_constants_aexp (a : AExp) : AExp :=\n  match a with\n  | a_num _\n  | a_id _\n    => a\n  | a_plus a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 + n2)\n    | (a1', a2') => a_plus a1' a2'\n  | a_minus a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 - n2)\n    | (a1', a2') => a_minus a1' a2'\n  | a_mult a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 * n2)\n    | (a1', a2') => a_mult a1' a2'\n\nexample : fold_constants_aexp <{(1 + 2) * x}> = <{3 * x}> := by admit /- proof elided -/\n\ndef fold_constants_bexp (b : BExp) : BExp :=\n  match b with\n  | b_true\n  | b_false\n    => b\n  | b_eq a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 == n2 then b_true else b_false\n    | (a1', a2') => b_eq a1' a2'\n  | b_neq a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 != n2 then b_true else b_false\n    | (a1', a2') => b_neq a1' a2'\n  | b_le a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 <= n2 then b_true else b_false\n    | (a1', a2') => b_le a1' a2'\n  | b_not b1 =>\n    match fold_constants_bexp b1 with\n    | b_true => b_false\n    | b_false => b_true\n    | b1' => b_not b1'\n  | b_and b1 b2 =>\n    match (fold_constants_bexp b1, fold_constants_bexp b2) with\n    | (b_true, b_true) => b_true\n    | (b_true, b_false) => b_false\n    | (b_false, b_true) => b_false\n    | (b_false, b_false) => b_false\n    | (b1', b2') => b_and b1' b2'\n  | b_or b1 b2 =>\n    match (fold_constants_bexp b1, fold_constants_bexp b2) with\n    | (b_true, b_true) => b_true\n    | (b_true, b_false) => b_true\n    | (b_false, b_true) => b_true\n    | (b_false, b_false) => b_false\n    | (b1', b2') => b_or b1' b2'\n\nexample : fold_constants_bexp <{true && !(false && true)}> = <{true}> := by admit /- proof elided -/\n\ndef fold_constants_com (c : Com) : Com :=\n  match c with\n  | c_skip => c\n  | c_asgn x a => c_asgn x (fold_constants_aexp a)\n  | c_seq c1 c2 => c_seq (fold_constants_com c1) (fold_constants_com c2)\n  | c_if b c1 c2 =>\n    match fold_constants_bexp b with\n    | b_true => fold_constants_com c1\n    | b_false => fold_constants_com c2\n    | b' => c_if b' (fold_constants_com c1) (fold_constants_com c2)\n  | c_while b c1 =>\n    match fold_constants_bexp b with\n    | b_true => loop\n    | b_false => c_skip\n    | b' => c_while b' (fold_constants_com c1)\n\nexample : fold_constants_com\n    \n    <{ x := 4 + 5;\n       y := x - 3;\n       if (x - y) = (2 + 4) then skip\n       else y := 0 end;\n       if 0 <= (4 - (2 + 1)) then y := 0\n       else skip end;\n       while y = 0 do\n         x := x + 1\n       end }>\n    = \n    <{ x := 9;\n       y := x - 3;\n       if (x - y) = 6 then skip\n       else y := 0 end;\n       y := 0;\n       while y = 0 do\n         x := x + 1\n       end }> := by admit /- proof elided -/", "target_theorem": "theorem fold_constants_com_sound : ctrans_sound fold_constants_com :=", "ground_truth_proof": ":= by\n  intro c; induction c with simp [fold_constants_com]\n  | c_asgn =>\n    apply c_asgn_congruence\n    apply fold_constants_aexp_sound\n  | c_seq =>\n    apply c_seq_congruence <;> assumption\n  | c_if b c₁ c₂ ih1 ih2 =>\n    split\n    . apply trans_cequiv _ c₁\n      . apply if_true\n        rename_i heq; rw [← heq]\n        apply fold_constants_bexp_sound\n      . assumption\n    . apply trans_cequiv _ c₂\n      . apply if_false\n        rename_i heq; rw [← heq]\n        apply fold_constants_bexp_sound\n      . assumption\n    . apply c_if_congruence\n      . apply fold_constants_bexp_sound\n      . assumption\n      . assumption\n  | c_while b c' ih =>\n    split\n    . apply while_true\n      rename_i heq; rw [← heq]\n      apply fold_constants_bexp_sound\n    . apply while_false\n      rename_i heq; rw [← heq]\n      apply fold_constants_bexp_sound\n    . apply c_while_congruence\n      . apply fold_constants_bexp_sound\n      . assumption", "nesting_depth": 5, "transitive_dep_count": 40, "subset_aristotle": true, "category": "Semantics"}
{"id": 273, "thm_name": "CImp.par_loop_any_x", "thm_stmt": "theorem par_loop_any_x n\n    : ∃ st', Multi CStep (par_loop, empty) (c_skip, st')\n        ∧ st' x = n", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/SmallStepImp.lean", "imports": ["import Frap.SmallStep", "import Frap.Equiv"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : im", "content": "syntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : imp"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|true) => `(Bool.true)\n  | `(term|false) => `(Bool.false)\n  | `(term|<{$x}>) => `(imp|$x)\n  | `(imp|$n:num) => `(a_num $n)\n  | `(imp|$s:str) => `(a_id $s)\n  | `(imp|$x + $y) => `(a_plus <{$x}> <{$y}>)\n  | `(imp|$x - $y) => `(a_minus <{$x}> <{$y}>)\n  | `(imp|$x * $y) => `(a_mult <{$x}> <{$y}>)\n  | `(imp|true) => `(b_true)\n  | `(imp|false) => `(b_false)\n  | `(imp|$x = $y) => `(b_eq <{$x}> <{$y}>)\n  | `(imp|$x != $y) => `(b_neq <{$x}> <{$y}>)\n  | `(imp|$x <= $y) => `(b_le <{$x}> <{$y}>)\n  | `(imp|!$x) => `(b_not <{$x}>)\n  | `(imp|$x && $y) => `(b_and <{$x}> <{$y}>)\n  | `(imp|$x || $y) => `(b_or <{$x}> <{$y}>)\n  | `(imp|($x)) => `(<{$x}>)\n  | `(imp|$x:ident) => `(a_id $(Lean.quote (toString x.getId)))\n  | `(imp|<[$t:term]>) => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(imp|skip) => `(c_skip)\n  | `(imp|$x:str := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$x:ident := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$c1 ; $c2) => `(c_seq <{$c1}> <{$c2}>)\n  | `(imp|if $b then $c1 else $c2 end) => `(c_if <{$b}> <{$c1}> <{$c2}>)\n  | `(imp|while $b do $c end) => `(c_while <{$b}> <{$c}>)"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_id : String → AExp  \n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "State", "content": "abbrev State := String → Nat"}, {"name": "BExp", "content": "inductive BExp where\n  | b_true : BExp\n  | b_false : BExp\n  | b_eq : AExp → AExp → BExp\n  | b_neq : AExp → AExp → BExp\n  | b_le : AExp → AExp → BExp\n  | b_not : BExp → BExp\n  | b_and : BExp → BExp → BExp\n  | b_or : BExp → BExp → BExp"}, {"name": "x", "content": "abbrev x := \"x\""}, {"name": "Com", "content": "inductive Com :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com"}, {"name": "empty", "content": "def empty : State := fun _ => 0"}, {"name": "Multi", "content": "inductive Multi {X : Type} (R : relation X) : relation X :=\n  | multi_refl x : Multi R x x\n  | multi_step x y z : R x y → Multi R y z → Multi R x z"}, {"name": "relation", "content": "def relation (X : Type) := X → X → Prop"}, {"name": "update", "content": "def update (st : State) (k : String) (v : Nat) : State :=\n  fun x => if x == k then v else st x"}, {"name": "y", "content": "abbrev y := \"y\""}, {"name": "And", "content": "inductive And : Prop → Prop → Prop where\n  | intro : a → b → And a b"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "multi_R", "content": "theorem multi_R (X : Type) (R : relation X) x y : R x y → Multi R x y"}, {"name": "multi_trans", "content": "theorem multi_trans (X : Type) (R : relation X) x y z\n    : Multi R x y → Multi R y z → Multi R x z"}], "used_local_defs": [{"name": "AVal", "content": "inductive AVal : AExp → Prop :=\n  | av_num : ∀ n, AVal (a_num n)"}, {"name": "AStep", "content": "inductive AStep (st : State) : AExp → AExp → Prop :=\n  | as_id : ∀ v, AStep st (a_id v) (a_num (st v))\n  | as_plus1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_plus a₁ a₂) (a_plus a₁' a₂)\n  | as_plus2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_plus v₁ a₂) (a_plus v₁ a₂')\n  | as_plus : ∀ (v₁ v₂ : Nat),\n      AStep st (a_plus (a_num v₁) (a_num v₂)) (a_num (v₁ + v₂))\n  | as_minus1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_minus a₁ a₂) (a_minus a₁' a₂)\n  | as_minus2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_minus v₁ a₂) (a_minus v₁ a₂')\n  | as_minus : ∀ (v₁ v₂ : Nat),\n      AStep st (a_minus (a_num v₁) (a_num v₂)) (a_num (v₁ - v₂))\n  | as_mult1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_mult a₁ a₂) (a_mult a₁' a₂)\n  | as_mult2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_mult v₁ a₂) (a_mult v₁ a₂')\n  | as_mult : ∀ (v₁ v₂ : Nat),\n      AStep st (a_mult (a_num v₁) (a_num v₂)) (a_num (v₁ * v₂))"}, {"name": "BStep", "content": "inductive BStep (st : State) : BExp → BExp → Prop :=\n  | bs_eq1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_eq a₁ a₂) (b_eq a₁' a₂)\n  | bs_eq2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_eq v₁ a₂) (b_eq v₁ a₂')\n  | bs_eq : ∀ (v₁ v₂ : Nat),\n      BStep st (b_eq (a_num v₁) (a_num v₂))\n          (if v₁ == v₂ then b_true else b_false)\n  | bs_neq1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_neq a₁ a₂) (b_neq a₁' a₂)\n  | bs_neq2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_neq v₁ a₂) (b_neq v₁ a₂')\n  | bs_neq : ∀ (v₁ v₂ : Nat),\n      BStep st (b_neq (a_num v₁) (a_num v₂))\n          (if v₁ != v₂ then b_true else b_false)\n  | bs_le1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_le a₁ a₂) (b_le a₁' a₂)\n  | bs_le2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_le v₁ a₂) (b_le v₁ a₂')\n  | bs_le : ∀ (v₁ v₂ : Nat),\n      BStep st (b_le (a_num v₁) (a_num v₂))\n          (if v₁ <= v₂ then b_true else b_false)\n  | bs_notStep : ∀ b₁ b₁',\n      BStep st b₁ b₁'\n      → BStep st (b_not b₁) (b_not b₁')\n  | bs_notTrue : BStep st (b_not b_true) b_false\n  | bs_notFalse : BStep st (b_not b_false) b_true\n  | bs_andStep : ∀ b₁ b₁' b₂,\n      BStep st b₁ b₁'\n      → BStep st (b_and b₁ b₂) (b_and b₁' b₂)\n  | bs_andFalse : ∀ b₂,\n      BStep st (b_and b_false b₂) b_false\n  | bs_andTrueStep : ∀ b₂ b₂',\n      BStep st b₂ b₂'\n      → BStep st (b_and b_true b₂) (b_and b_true b₂')\n  | bs_andTrueTrue :\n      BStep st (b_and b_true b_true) b_true\n  | bs_andTrueFalse :\n      BStep st (b_and b_true b_false) b_false\n  | bs_orStep : ∀ b₁ b₁' b₂,\n      BStep st b₁ b₁'\n      → BStep st (b_or b₁ b₂) (b_or b₁' b₂)\n  | bs_orTrue : ∀ b₂,\n      BStep st (b_or b_true b₂) b_true\n  | bs_orFalseStep : ∀ b₂ b₂',\n      BStep st b₂ b₂'\n      → BStep st (b_or b_false b₂) (b_or b_false b₂')\n  | bs_orFalseTrue :\n      BStep st (b_or b_false b_true) b_true\n  | bs_orFalseFalse :\n      BStep st (b_or b_false b_false) b_false"}, {"name": "CStep", "content": "inductive CStep : (Com × State) → (Com × State) → Prop :=\n  | cs_asgnStep : ∀ st v a₁ a₁',\n      AStep st a₁ a₁'\n      → CStep (c_asgn v a₁, st) (c_asgn v a₁', st)\n  | cs_asgn : ∀ st v (n : Nat),\n      CStep (c_asgn v (a_num n), st) (c_skip, x !-> n; st)\n  | cs_seqStep : ∀ st c₁ c₁' st' c₂,\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_seq c₁ c₂, st) (c_seq c₁' c₂, st')\n  | cs_seqFinish : ∀ st c₂,\n      CStep (c_seq c_skip c₂, st) (c₂, st)\n  | cs_ifStep : ∀ st b₁ b₁' c₁ c₂,\n      BStep st b₁ b₁'\n      → CStep (c_if b₁ c₁ c₂, st) (c_if b₁' c₁ c₂, st)\n  | cs_ifTrue : ∀ st c₁ c₂,\n      CStep (c_if b_true c₁ c₂, st) (c₁, st)\n  | cs_ifFalse : ∀ st c₁ c₂,\n      CStep (c_if b_false c₁ c₂, st) (c₂, st)\n  | cs_while : ∀ st b₁ c₁,\n      CStep (c_while b₁ c₁, st)\n        (c_if b₁ (c_seq c₁ (c_while b₁ c₁)) c_skip, st)"}, {"name": "CImp.Com", "content": "inductive Com : Type :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com\n  | c_par : Com → Com → Com"}, {"name": "CImp.CStep", "content": "inductive CStep : (Com × State) → (Com × State) → Prop :=\n   \n  | cs_asgnStep : ∀ st v a₁ a₁',\n      AStep st a₁ a₁'\n      → CStep (c_asgn v a₁, st) (c_asgn v a₁', st)\n  | cs_asgn : ∀ st v (n : Nat),\n      CStep (c_asgn v (a_num n), st) (c_skip, v !-> n; st)\n  | cs_seqStep : ∀ st c₁ c₁' st' c₂,\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_seq c₁ c₂, st) (c_seq c₁' c₂, st')\n  | cs_seqFinish : ∀ st c₂,\n      CStep (c_seq c_skip c₂, st) (c₂, st)\n  | cs_ifStep : ∀ st b₁ b₁' c₁ c₂,\n      BStep st b₁ b₁'\n      → CStep (c_if b₁ c₁ c₂, st) (c_if b₁' c₁ c₂, st)\n  | cs_ifTrue : ∀ st c₁ c₂,\n      CStep (c_if b_true c₁ c₂, st) (c₁, st)\n  | cs_ifFalse : ∀ st c₁ c₂,\n      CStep (c_if b_false c₁ c₂, st) (c₂, st)\n  | cs_while : ∀ st b₁ c₁,\n      CStep (c_while b₁ c₁, st)\n        (c_if b₁ (c_seq c₁ (c_while b₁ c₁)) c_skip, st)\n   \n  | cs_par1 : ∀ st c₁ c₁' c₂ st',\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_par c₁ c₂, st) (c_par c₁' c₂, st')\n  | cs_par2 : ∀ st c₁ c₂ c₂' st',\n      CStep (c₂, st) (c₂', st')\n      → CStep (c_par c₁ c₂, st) (c_par c₁ c₂', st')\n  | cs_parDone : ∀ st,\n      CStep (c_par c_skip c_skip, st) (c_skip, st)"}, {"name": "CImp.par_loop", "content": "def par_loop : Com :=\n  c_par\n    (c_asgn y (a_num 1))\n    (c_while (b_eq (a_id y) (a_num 0))\n      (c_asgn x (a_plus (a_id x) (a_num 1))))\n\n/-\nIn particular, it can terminate with `x` set to `0`.\n-/\n\nexample : ∃ st',\n    Multi CStep (par_loop, empty) (c_skip, st')\n    ∧ st' x = 0 :="}], "used_local_lemmas": [{"name": "CImp.par_body_n__Sn", "content": "theorem par_body_n__Sn n st\n    : st x = n ∧ st y = 0\n      → Multi CStep (par_loop, st) (par_loop, x !-> n + 1; st)"}, {"name": "CImp.par_body_n", "content": "theorem par_body_n n st\n    : st x = 0 ∧ st y = 0\n      → ∃ st', Multi CStep (par_loop, st) (par_loop, st')\n          ∧ st' x = n ∧ st' y = 0"}], "local_ctx": "import Frap.Equiv\n\nimport Frap.SmallStep\n\nopen Imp\n\nopen AExp\n\nopen BExp\n\nopen Com\n\nopen Multi\n\ninductive AVal : AExp → Prop :=\n  | av_num : ∀ n, AVal (a_num n)\n\ninductive AStep (st : State) : AExp → AExp → Prop :=\n  | as_id : ∀ v, AStep st (a_id v) (a_num (st v))\n  | as_plus1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_plus a₁ a₂) (a_plus a₁' a₂)\n  | as_plus2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_plus v₁ a₂) (a_plus v₁ a₂')\n  | as_plus : ∀ (v₁ v₂ : Nat),\n      AStep st (a_plus (a_num v₁) (a_num v₂)) (a_num (v₁ + v₂))\n  | as_minus1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_minus a₁ a₂) (a_minus a₁' a₂)\n  | as_minus2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_minus v₁ a₂) (a_minus v₁ a₂')\n  | as_minus : ∀ (v₁ v₂ : Nat),\n      AStep st (a_minus (a_num v₁) (a_num v₂)) (a_num (v₁ - v₂))\n  | as_mult1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_mult a₁ a₂) (a_mult a₁' a₂)\n  | as_mult2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_mult v₁ a₂) (a_mult v₁ a₂')\n  | as_mult : ∀ (v₁ v₂ : Nat),\n      AStep st (a_mult (a_num v₁) (a_num v₂)) (a_num (v₁ * v₂))\n\nopen AStep\n\ninductive BStep (st : State) : BExp → BExp → Prop :=\n  | bs_eq1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_eq a₁ a₂) (b_eq a₁' a₂)\n  | bs_eq2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_eq v₁ a₂) (b_eq v₁ a₂')\n  | bs_eq : ∀ (v₁ v₂ : Nat),\n      BStep st (b_eq (a_num v₁) (a_num v₂))\n          (if v₁ == v₂ then b_true else b_false)\n  | bs_neq1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_neq a₁ a₂) (b_neq a₁' a₂)\n  | bs_neq2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_neq v₁ a₂) (b_neq v₁ a₂')\n  | bs_neq : ∀ (v₁ v₂ : Nat),\n      BStep st (b_neq (a_num v₁) (a_num v₂))\n          (if v₁ != v₂ then b_true else b_false)\n  | bs_le1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_le a₁ a₂) (b_le a₁' a₂)\n  | bs_le2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_le v₁ a₂) (b_le v₁ a₂')\n  | bs_le : ∀ (v₁ v₂ : Nat),\n      BStep st (b_le (a_num v₁) (a_num v₂))\n          (if v₁ <= v₂ then b_true else b_false)\n  | bs_notStep : ∀ b₁ b₁',\n      BStep st b₁ b₁'\n      → BStep st (b_not b₁) (b_not b₁')\n  | bs_notTrue : BStep st (b_not b_true) b_false\n  | bs_notFalse : BStep st (b_not b_false) b_true\n  | bs_andStep : ∀ b₁ b₁' b₂,\n      BStep st b₁ b₁'\n      → BStep st (b_and b₁ b₂) (b_and b₁' b₂)\n  | bs_andFalse : ∀ b₂,\n      BStep st (b_and b_false b₂) b_false\n  | bs_andTrueStep : ∀ b₂ b₂',\n      BStep st b₂ b₂'\n      → BStep st (b_and b_true b₂) (b_and b_true b₂')\n  | bs_andTrueTrue :\n      BStep st (b_and b_true b_true) b_true\n  | bs_andTrueFalse :\n      BStep st (b_and b_true b_false) b_false\n  | bs_orStep : ∀ b₁ b₁' b₂,\n      BStep st b₁ b₁'\n      → BStep st (b_or b₁ b₂) (b_or b₁' b₂)\n  | bs_orTrue : ∀ b₂,\n      BStep st (b_or b_true b₂) b_true\n  | bs_orFalseStep : ∀ b₂ b₂',\n      BStep st b₂ b₂'\n      → BStep st (b_or b_false b₂) (b_or b_false b₂')\n  | bs_orFalseTrue :\n      BStep st (b_or b_false b_true) b_true\n  | bs_orFalseFalse :\n      BStep st (b_or b_false b_false) b_false\n\nopen BStep\n\ninductive CStep : (Com × State) → (Com × State) → Prop :=\n  | cs_asgnStep : ∀ st v a₁ a₁',\n      AStep st a₁ a₁'\n      → CStep (c_asgn v a₁, st) (c_asgn v a₁', st)\n  | cs_asgn : ∀ st v (n : Nat),\n      CStep (c_asgn v (a_num n), st) (c_skip, x !-> n; st)\n  | cs_seqStep : ∀ st c₁ c₁' st' c₂,\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_seq c₁ c₂, st) (c_seq c₁' c₂, st')\n  | cs_seqFinish : ∀ st c₂,\n      CStep (c_seq c_skip c₂, st) (c₂, st)\n  | cs_ifStep : ∀ st b₁ b₁' c₁ c₂,\n      BStep st b₁ b₁'\n      → CStep (c_if b₁ c₁ c₂, st) (c_if b₁' c₁ c₂, st)\n  | cs_ifTrue : ∀ st c₁ c₂,\n      CStep (c_if b_true c₁ c₂, st) (c₁, st)\n  | cs_ifFalse : ∀ st c₁ c₂,\n      CStep (c_if b_false c₁ c₂, st) (c₂, st)\n  | cs_while : ∀ st b₁ c₁,\n      CStep (c_while b₁ c₁, st)\n        (c_if b₁ (c_seq c₁ (c_while b₁ c₁)) c_skip, st)\n\nnamespace CImp\n\ninductive Com : Type :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com\n  | c_par : Com → Com → Com   \n\nopen Com\n\ninductive CStep : (Com × State) → (Com × State) → Prop :=\n   \n  | cs_asgnStep : ∀ st v a₁ a₁',\n      AStep st a₁ a₁'\n      → CStep (c_asgn v a₁, st) (c_asgn v a₁', st)\n  | cs_asgn : ∀ st v (n : Nat),\n      CStep (c_asgn v (a_num n), st) (c_skip, v !-> n; st)\n  | cs_seqStep : ∀ st c₁ c₁' st' c₂,\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_seq c₁ c₂, st) (c_seq c₁' c₂, st')\n  | cs_seqFinish : ∀ st c₂,\n      CStep (c_seq c_skip c₂, st) (c₂, st)\n  | cs_ifStep : ∀ st b₁ b₁' c₁ c₂,\n      BStep st b₁ b₁'\n      → CStep (c_if b₁ c₁ c₂, st) (c_if b₁' c₁ c₂, st)\n  | cs_ifTrue : ∀ st c₁ c₂,\n      CStep (c_if b_true c₁ c₂, st) (c₁, st)\n  | cs_ifFalse : ∀ st c₁ c₂,\n      CStep (c_if b_false c₁ c₂, st) (c₂, st)\n  | cs_while : ∀ st b₁ c₁,\n      CStep (c_while b₁ c₁, st)\n        (c_if b₁ (c_seq c₁ (c_while b₁ c₁)) c_skip, st)\n   \n  | cs_par1 : ∀ st c₁ c₁' c₂ st',\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_par c₁ c₂, st) (c_par c₁' c₂, st')\n  | cs_par2 : ∀ st c₁ c₂ c₂' st',\n      CStep (c₂, st) (c₂', st')\n      → CStep (c_par c₁ c₂, st) (c_par c₁ c₂', st')\n  | cs_parDone : ∀ st,\n      CStep (c_par c_skip c_skip, st) (c_skip, st)\n\nopen CStep\n\ndef par_loop : Com :=\n  c_par\n    (c_asgn y (a_num 1))\n    (c_while (b_eq (a_id y) (a_num 0))\n      (c_asgn x (a_plus (a_id x) (a_num 1))))\n\n/-\nIn particular, it can terminate with `x` set to `0`.\n-/\n\nexample : ∃ st',\n    Multi CStep (par_loop, empty) (c_skip, st')\n    ∧ st' x = 0 :=", "target_theorem": "theorem par_loop_any_x n\n    : ∃ st', Multi CStep (par_loop, empty) (c_skip, st')\n        ∧ st' x = n :=", "ground_truth_proof": ":= by\n  let h := par_body_n n empty\n  simp [empty] at h\n  obtain ⟨st', ⟨h', ⟨hx, _⟩⟩⟩ := h\n  exists y !-> 1; st'\n  constructor\n  . apply multi_trans\n    . apply h'\n    . apply multi_step\n      . apply cs_par1; apply cs_asgn\n      . apply multi_step\n        . apply cs_par2; apply cs_while\n        . apply multi_step\n          . apply cs_par2; apply cs_ifStep\n            apply bs_eq1; apply as_id\n          . apply multi_step\n            . apply cs_par2; apply cs_ifStep\n              apply bs_eq\n            . simp [update]; apply multi_step\n              . apply cs_par2; apply cs_ifFalse\n              . apply multi_R\n                apply cs_parDone\n  . simp [update]; assumption", "nesting_depth": 3, "transitive_dep_count": 27, "subset_aristotle": true, "category": "Semantics"}
{"id": 274, "thm_name": "Imp.fold_constants_bexp_sound", "thm_stmt": "theorem fold_constants_bexp_sound : btrans_sound fold_constants_bexp", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Trans.lean", "imports": ["import Frap.Exercises.Equiv", "import Frap.Equiv"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"<{\" imp \"}>\" : term", "content": "syntax \"<{\" imp \"}>\" : term\n\nsyntax \"false\" : imp\n\nsyntax \"false\" : term\n\nsyntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : imp\n\nsyntax \"true\" : imp\n\nsyntax \"true\" : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|true) => `(Bool.true)\n  | `(term|false) => `(Bool.false)\n  | `(term|<{$x}>) => `(imp|$x)\n  | `(imp|$n:num) => `(a_num $n)\n  | `(imp|$s:str) => `(a_id $s)\n  | `(imp|$x + $y) => `(a_plus <{$x}> <{$y}>)\n  | `(imp|$x - $y) => `(a_minus <{$x}> <{$y}>)\n  | `(imp|$x * $y) => `(a_mult <{$x}> <{$y}>)\n  | `(imp|true) => `(b_true)\n  | `(imp|false) => `(b_false)\n  | `(imp|$x = $y) => `(b_eq <{$x}> <{$y}>)\n  | `(imp|$x != $y) => `(b_neq <{$x}> <{$y}>)\n  | `(imp|$x <= $y) => `(b_le <{$x}> <{$y}>)\n  | `(imp|!$x) => `(b_not <{$x}>)\n  | `(imp|$x && $y) => `(b_and <{$x}> <{$y}>)\n  | `(imp|$x || $y) => `(b_or <{$x}> <{$y}>)\n  | `(imp|($x)) => `(<{$x}>)\n  | `(imp|$x:ident) => `(a_id $(Lean.quote (toString x.getId)))\n  | `(imp|<[$t:term]>) => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(imp|skip) => `(c_skip)\n  | `(imp|$x:str := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$x:ident := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$c1 ; $c2) => `(c_seq <{$c1}> <{$c2}>)\n  | `(imp|if $b then $c1 else $c2 end) => `(c_if <{$b}> <{$c1}> <{$c2}>)\n  | `(imp|while $b do $c end) => `(c_while <{$b}> <{$c}>)"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_id : String → AExp  \n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "x", "content": "abbrev x := \"x\""}, {"name": "y", "content": "abbrev y := \"y\""}, {"name": "BExp", "content": "inductive BExp where\n  | b_true : BExp\n  | b_false : BExp\n  | b_eq : AExp → AExp → BExp\n  | b_neq : AExp → AExp → BExp\n  | b_le : AExp → AExp → BExp\n  | b_not : BExp → BExp\n  | b_and : BExp → BExp → BExp\n  | b_or : BExp → BExp → BExp"}, {"name": "bequiv", "content": "def bequiv (b₁ b₂ : BExp) : Prop :=\n  ∀ (st : State), beval st b₁ = beval st b₂"}, {"name": "beval", "content": "def beval (st : State) (b : BExp) : Bool :=\n  match b with\n  | b_true => true\n  | b_false => false\n  | b_eq a₁ a₂ => (aeval st a₁) == (aeval st a₂)\n  | b_neq a₁ a₂ => (aeval st a₁) != (aeval st a₂)\n  | b_le a₁ a₂ => (aeval st a₁) <= (aeval st a₂)\n  | b_not b₁ => not (beval st b₁)\n  | b_and b₁ b₂ => and (beval st b₁) (beval st b₂)\n  | b_or b₁ b₂ => or (beval st b₁) (beval st b₂)\n\nexample : aeval (update empty x 5)\n    <{3 + x * 2}>\n    \n    = 13 := by admit /- proof elided -/"}, {"name": "empty", "content": "def empty : State := fun _ => 0"}, {"name": "State", "content": "abbrev State := String → Nat"}, {"name": "z", "content": "abbrev z := \"z\""}, {"name": "aeval", "content": "def aeval (st : State) (a : AExp) : Nat :=\n  match a with\n  | a_num n => n\n  | a_id x => st x\n  | a_plus a₁ a₂ => (aeval st a₁) + (aeval st a₂)\n  | a_minus a₁ a₂ => (aeval st a₁) - (aeval st a₂)\n  | a_mult a₁ a₂ => (aeval st a₁) * (aeval st a₂)"}, {"name": "update", "content": "def update (st : State) (k : String) (v : Nat) : State :=\n  fun x => if x == k then v else st x"}, {"name": "aequiv", "content": "def aequiv (a₁ a₂ : AExp) : Prop :=\n  ∀ (st : State), aeval st a₁ = aeval st a₂"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Imp.atrans_sound", "content": "def atrans_sound (atrans : AExp → AExp) :=\n  ∀ (a : AExp), aequiv a (atrans a)"}, {"name": "Imp.btrans_sound", "content": "def btrans_sound (btrans : BExp → BExp) :=\n  ∀ (b : BExp), bequiv b (btrans b)"}, {"name": "Imp.fold_constants_aexp", "content": "def fold_constants_aexp (a : AExp) : AExp :=\n  match a with\n  | a_num _\n  | a_id _\n    => a\n  | a_plus a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 + n2)\n    | (a1', a2') => a_plus a1' a2'\n  | a_minus a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 - n2)\n    | (a1', a2') => a_minus a1' a2'\n  | a_mult a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 * n2)\n    | (a1', a2') => a_mult a1' a2'\n\nexample : fold_constants_aexp <{(1 + 2) * x}> = <{3 * x}> := by admit /- proof elided -/"}, {"name": "Imp.fold_constants_bexp", "content": "def fold_constants_bexp (b : BExp) : BExp :=\n  match b with\n  | b_true\n  | b_false\n    => b\n  | b_eq a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 == n2 then b_true else b_false\n    | (a1', a2') => b_eq a1' a2'\n  | b_neq a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 != n2 then b_true else b_false\n    | (a1', a2') => b_neq a1' a2'\n  | b_le a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 <= n2 then b_true else b_false\n    | (a1', a2') => b_le a1' a2'\n  | b_not b1 =>\n    match fold_constants_bexp b1 with\n    | b_true => b_false\n    | b_false => b_true\n    | b1' => b_not b1'\n  | b_and b1 b2 =>\n    match (fold_constants_bexp b1, fold_constants_bexp b2) with\n    | (b_true, b_true) => b_true\n    | (b_true, b_false) => b_false\n    | (b_false, b_true) => b_false\n    | (b_false, b_false) => b_false\n    | (b1', b2') => b_and b1' b2'\n  | b_or b1 b2 =>\n    match (fold_constants_bexp b1, fold_constants_bexp b2) with\n    | (b_true, b_true) => b_true\n    | (b_true, b_false) => b_true\n    | (b_false, b_true) => b_true\n    | (b_false, b_false) => b_false\n    | (b1', b2') => b_or b1' b2'\n\nexample : fold_constants_bexp <{true && !(false && true)}> = <{true}> := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "Imp.fold_constants_aexp_sound", "content": "theorem fold_constants_aexp_sound : atrans_sound fold_constants_aexp"}], "local_ctx": "import Frap.Equiv\n\nimport Frap.Exercises.Equiv\n\nnamespace Imp\n\nopen AExp\n\nopen BExp\n\nopen Com\n\nopen CEval\n\ndef atrans_sound (atrans : AExp → AExp) :=\n  ∀ (a : AExp), aequiv a (atrans a)\n\ndef btrans_sound (btrans : BExp → BExp) :=\n  ∀ (b : BExp), bequiv b (btrans b)\n\ndef fold_constants_aexp (a : AExp) : AExp :=\n  match a with\n  | a_num _\n  | a_id _\n    => a\n  | a_plus a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 + n2)\n    | (a1', a2') => a_plus a1' a2'\n  | a_minus a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 - n2)\n    | (a1', a2') => a_minus a1' a2'\n  | a_mult a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => a_num (n1 * n2)\n    | (a1', a2') => a_mult a1' a2'\n\nexample : fold_constants_aexp <{(1 + 2) * x}> = <{3 * x}> := by admit /- proof elided -/\n\ndef fold_constants_bexp (b : BExp) : BExp :=\n  match b with\n  | b_true\n  | b_false\n    => b\n  | b_eq a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 == n2 then b_true else b_false\n    | (a1', a2') => b_eq a1' a2'\n  | b_neq a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 != n2 then b_true else b_false\n    | (a1', a2') => b_neq a1' a2'\n  | b_le a1 a2 =>\n    match (fold_constants_aexp a1, fold_constants_aexp a2) with\n    | (a_num n1, a_num n2) => if n1 <= n2 then b_true else b_false\n    | (a1', a2') => b_le a1' a2'\n  | b_not b1 =>\n    match fold_constants_bexp b1 with\n    | b_true => b_false\n    | b_false => b_true\n    | b1' => b_not b1'\n  | b_and b1 b2 =>\n    match (fold_constants_bexp b1, fold_constants_bexp b2) with\n    | (b_true, b_true) => b_true\n    | (b_true, b_false) => b_false\n    | (b_false, b_true) => b_false\n    | (b_false, b_false) => b_false\n    | (b1', b2') => b_and b1' b2'\n  | b_or b1 b2 =>\n    match (fold_constants_bexp b1, fold_constants_bexp b2) with\n    | (b_true, b_true) => b_true\n    | (b_true, b_false) => b_true\n    | (b_false, b_true) => b_true\n    | (b_false, b_false) => b_false\n    | (b1', b2') => b_or b1' b2'\n\nexample : fold_constants_bexp <{true && !(false && true)}> = <{true}> := by admit /- proof elided -/", "target_theorem": "theorem fold_constants_bexp_sound : btrans_sound fold_constants_bexp :=", "ground_truth_proof": ":= by\n  intro b st; induction b with simp [fold_constants_bexp]\n  | b_eq a1 a2 =>\n    rw [fold_constants_aexp_sound a1, fold_constants_aexp_sound a2]\n    split\n    . rename_i hm\n      simp at hm\n      obtain ⟨⟩ := hm\n      split <;> simp [*]\n    . rename_i hm\n      simp at hm\n      obtain ⟨⟩ := hm\n      simp [*]\n  | b_neq a1 a2 =>\n    rw [fold_constants_aexp_sound a1, fold_constants_aexp_sound a2]\n    split\n    . rename_i hm\n      simp at hm\n      obtain ⟨⟩ := hm\n      split <;> simp [*]\n    . rename_i hm\n      simp at hm\n      obtain ⟨⟩ := hm\n      simp [*]\n  | b_le a1 a2 =>\n    rw [fold_constants_aexp_sound a1, fold_constants_aexp_sound a2]\n    split\n    . rename_i hm\n      simp at hm\n      obtain ⟨⟩ := hm\n      split <;> simp [*]\n    . rename_i hm\n      simp at hm\n      obtain ⟨⟩ := hm\n      simp [*]\n  | b_not =>\n    split <;> simp [*]\n  | b_and =>\n    simp [*]\n    split <;> (\n      rename_i hm\n      simp at hm\n      obtain ⟨⟩ := hm\n      simp [*]\n    )\n  | b_or =>\n    simp [*]\n    split <;> (\n      rename_i hm\n      simp at hm\n      obtain ⟨⟩ := hm\n      simp [*]\n    )", "nesting_depth": 5, "transitive_dep_count": 21, "subset_aristotle": true, "category": "Semantics"}
{"id": 275, "thm_name": "step_normalizing", "thm_stmt": "theorem step_normalizing : normalizing Step", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/SmallStep.lean", "imports": [], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Tm", "content": "inductive Tm : Type :=\n  | c : Nat → Tm       \n  | p : Tm → Tm → Tm"}, {"name": "SimpleArith1.Step", "content": "inductive Step : Tm → Tm → Prop :=\n  | st_plusConstConst n₁ n₂ : Step (p (c n₁) (c n₂)) (c (n₁ + n₂))\n  | st_plus1 t₁ t₁' t₂ : Step t₁ t₁' → Step (p t₁ t₂) (p t₁' t₂)\n  | st_plus2 n₁ t₂ t₂' : Step t₂ t₂' → Step (p (c n₁) t₂) (p (c n₁) t₂')"}, {"name": "relation", "content": "def relation (X : Type) := X → X → Prop"}, {"name": "Value", "content": "inductive Value : Tm → Prop :=\n  | v_const n : Value (c n)"}, {"name": "Step", "content": "inductive Step : Tm → Tm → Prop :=\n  | st_plusConstConst n₁ n₂ : Step (p (c n₁) (c n₂)) (c (n₁ + n₂))\n  | st_plus1 t₁ t₁' t₂ : Step t₁ t₁' → Step (p t₁ t₂) (p t₁' t₂)\n  | st_plus2 v₁ t₂ t₂' : Value v₁ → Step t₂ t₂' → Step (p v₁ t₂) (p v₁ t₂')"}, {"name": "normal_form", "content": "def normal_form {X : Type} (R : relation X) (t : X) : Prop := ¬∃t', R t t'"}, {"name": "Multi", "content": "inductive Multi {X : Type} (R : relation X) : relation X :=\n  | multi_refl x : Multi R x x\n  | multi_step x y z : R x y → Multi R y z → Multi R x z"}, {"name": "normalizing", "content": "def normalizing {X : Type} (R : relation X) :=\n  ∀t, ∃t', (Multi R) t t' ∧ normal_form R t'"}], "used_local_lemmas": [{"name": "strong_progress", "content": "theorem strong_progress t : Value t ∨ ∃t', Step t t'"}, {"name": "value_is_nf", "content": "theorem value_is_nf v : Value v → normal_form Step v"}, {"name": "nf_is_value", "content": "theorem nf_is_value t : normal_form Step t → Value t"}, {"name": "nf_same_as_value", "content": "theorem nf_same_as_value t : normal_form Step t ↔ Value t"}, {"name": "multi_R", "content": "theorem multi_R (X : Type) (R : relation X) x y : R x y → Multi R x y"}, {"name": "multi_trans", "content": "theorem multi_trans (X : Type) (R : relation X) x y z\n    : Multi R x y → Multi R y z → Multi R x z"}, {"name": "multistep_congr_1", "content": "theorem multistep_congr_1 t₁ t₁' t₂\n    : (t₁ ~~>* t₁') → (p t₁ t₂ ~~>* p t₁' t₂)"}, {"name": "multistep_congr_2", "content": "theorem multistep_congr_2 v₁ t₂ t₂'\n    : (t₂ ~~>* t₂') → (p v₁ t₂ ~~>* p v₁ t₂')"}], "local_ctx": "inductive Tm : Type :=\n  | c : Nat → Tm       \n  | p : Tm → Tm → Tm   \n\nopen Tm\n\nopen Eval\n\nnamespace SimpleArith1\n\ninductive Step : Tm → Tm → Prop :=\n  | st_plusConstConst n₁ n₂ : Step (p (c n₁) (c n₂)) (c (n₁ + n₂))\n  | st_plus1 t₁ t₁' t₂ : Step t₁ t₁' → Step (p t₁ t₂) (p t₁' t₂)\n  | st_plus2 n₁ t₂ t₂' : Step t₂ t₂' → Step (p (c n₁) t₂) (p (c n₁) t₂')\n\nopen Step\n\nend SimpleArith1\n\ndef relation (X : Type) := X → X → Prop\n\nnamespace SimpleArith2\n\nopen SimpleArith1\n\nend SimpleArith2\n\ninductive Value : Tm → Prop :=\n  | v_const n : Value (c n)\n\nopen Value\n\ninductive Step : Tm → Tm → Prop :=\n  | st_plusConstConst n₁ n₂ : Step (p (c n₁) (c n₂)) (c (n₁ + n₂))\n  | st_plus1 t₁ t₁' t₂ : Step t₁ t₁' → Step (p t₁ t₂) (p t₁' t₂)\n  | st_plus2 v₁ t₂ t₂' : Value v₁ → Step t₂ t₂' → Step (p v₁ t₂) (p v₁ t₂')\n\nopen Step\n\ndef normal_form {X : Type} (R : relation X) (t : X) : Prop := ¬∃t', R t t'\n\ninductive Multi {X : Type} (R : relation X) : relation X :=\n  | multi_refl x : Multi R x x\n  | multi_step x y z : R x y → Multi R y z → Multi R x z\n\nopen Multi\n\ndef normalizing {X : Type} (R : relation X) :=\n  ∀t, ∃t', (Multi R) t t' ∧ normal_form R t'", "target_theorem": "theorem step_normalizing : normalizing Step :=", "ground_truth_proof": ":= by\n  unfold normalizing\n  intro t\n  induction t with\n  | c n =>\n    exists c n\n    constructor\n    . apply multi_refl\n    . rw [nf_same_as_value]; apply v_const\n  | p t₁ t₂ ih₁ ih₂ =>\n    obtain ⟨t₁', ⟨hs₁, hn₁⟩⟩ := ih₁\n    obtain ⟨t₂', ⟨hs₂, hn₂⟩⟩ := ih₂\n    rw [nf_same_as_value] at hn₁\n    rw [nf_same_as_value] at hn₂\n    cases hn₁; cases hn₂\n    rename_i n₁ n₂\n    exists c (n₁ + n₂)\n    constructor\n    . apply multi_trans\n      . apply multistep_congr_1\n        apply hs₁\n      . apply multi_trans\n        . apply multistep_congr_2\n          apply hs₂\n        . apply multi_R\n          apply st_plusConstConst\n    . rw [nf_same_as_value]; apply v_const", "nesting_depth": 3, "transitive_dep_count": 17, "subset_aristotle": true, "category": "Semantics"}
{"id": 276, "thm_name": "Imp.optimize_0plus_com_sound", "thm_stmt": "theorem optimize_0plus_com_sound : ctrans_sound optimize_0plus_com", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Exercises/Trans.lean", "imports": ["import Frap.Trans"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"<{\" imp \"}>\" : term", "content": "syntax \"<{\" imp \"}>\" : term\n\nsyntax:21 \"while\" imp:20 \"do\" imp:20 \"end\" : imp\n\nsyntax \"<[\" term \"]>\" : imp\n\nsyntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : imp\n\nsyntax \"false\" : imp\n\nsyntax \"false\" : term\n\nsyntax \"true\" : imp\n\nsyntax \"true\" : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|true) => `(Bool.true)\n  | `(term|false) => `(Bool.false)\n  | `(term|<{$x}>) => `(imp|$x)\n  | `(imp|$n:num) => `(a_num $n)\n  | `(imp|$s:str) => `(a_id $s)\n  | `(imp|$x + $y) => `(a_plus <{$x}> <{$y}>)\n  | `(imp|$x - $y) => `(a_minus <{$x}> <{$y}>)\n  | `(imp|$x * $y) => `(a_mult <{$x}> <{$y}>)\n  | `(imp|true) => `(b_true)\n  | `(imp|false) => `(b_false)\n  | `(imp|$x = $y) => `(b_eq <{$x}> <{$y}>)\n  | `(imp|$x != $y) => `(b_neq <{$x}> <{$y}>)\n  | `(imp|$x <= $y) => `(b_le <{$x}> <{$y}>)\n  | `(imp|!$x) => `(b_not <{$x}>)\n  | `(imp|$x && $y) => `(b_and <{$x}> <{$y}>)\n  | `(imp|$x || $y) => `(b_or <{$x}> <{$y}>)\n  | `(imp|($x)) => `(<{$x}>)\n  | `(imp|$x:ident) => `(a_id $(Lean.quote (toString x.getId)))\n  | `(imp|<[$t:term]>) => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(imp|skip) => `(c_skip)\n  | `(imp|$x:str := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$x:ident := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$c1 ; $c2) => `(c_seq <{$c1}> <{$c2}>)\n  | `(imp|if $b then $c1 else $c2 end) => `(c_if <{$b}> <{$c1}> <{$c2}>)\n  | `(imp|while $b do $c end) => `(c_while <{$b}> <{$c}>)"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_id : String → AExp  \n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "BExp", "content": "inductive BExp where\n  | b_true : BExp\n  | b_false : BExp\n  | b_eq : AExp → AExp → BExp\n  | b_neq : AExp → AExp → BExp\n  | b_le : AExp → AExp → BExp\n  | b_not : BExp → BExp\n  | b_and : BExp → BExp → BExp\n  | b_or : BExp → BExp → BExp"}, {"name": "Com", "content": "inductive Com :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com"}, {"name": "Com", "content": "inductive Com : Type :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com\n  | c_par : Com → Com → Com   "}, {"name": "ctrans_sound", "content": "def ctrans_sound (ctrans : Com → Com) :=\n  ∀ (c : Com), cequiv c (ctrans c)"}, {"name": "cequiv", "content": "def cequiv (c₁ c₂ : Com) : Prop :=\n  ∀ st st' : State, (st =[<[c₁]>]=> st') ↔ (st =[<[c₂]>]=> st')"}, {"name": "State", "content": "abbrev State := String → Nat"}, {"name": "btrans_sound", "content": "def btrans_sound (btrans : BExp → BExp) :=\n  ∀ (b : BExp), bequiv b (btrans b)"}, {"name": "bequiv", "content": "def bequiv (b₁ b₂ : BExp) : Prop :=\n  ∀ (st : State), beval st b₁ = beval st b₂"}, {"name": "beval", "content": "def beval (st : State) (b : BExp) : Bool :=\n  match b with\n  | b_true => true\n  | b_false => false\n  | b_eq a₁ a₂ => (aeval st a₁) == (aeval st a₂)\n  | b_neq a₁ a₂ => (aeval st a₁) != (aeval st a₂)\n  | b_le a₁ a₂ => (aeval st a₁) <= (aeval st a₂)\n  | b_not b₁ => not (beval st b₁)\n  | b_and b₁ b₂ => and (beval st b₁) (beval st b₂)\n  | b_or b₁ b₂ => or (beval st b₁) (beval st b₂)\n\nexample : aeval (update empty x 5)\n    <{3 + x * 2}>\n    \n    = 13 := by admit /- proof elided -/"}, {"name": "empty", "content": "def empty : State := fun _ => 0"}, {"name": "y", "content": "abbrev y := \"y\""}, {"name": "x", "content": "abbrev x := \"x\""}, {"name": "z", "content": "abbrev z := \"z\""}, {"name": "aeval", "content": "def aeval (st : State) (a : AExp) : Nat :=\n  match a with\n  | a_num n => n\n  | a_id x => st x\n  | a_plus a₁ a₂ => (aeval st a₁) + (aeval st a₂)\n  | a_minus a₁ a₂ => (aeval st a₁) - (aeval st a₂)\n  | a_mult a₁ a₂ => (aeval st a₁) * (aeval st a₂)"}, {"name": "update", "content": "def update (st : State) (k : String) (v : Nat) : State :=\n  fun x => if x == k then v else st x"}, {"name": "atrans_sound", "content": "def atrans_sound (atrans : AExp → AExp) :=\n  ∀ (a : AExp), aequiv a (atrans a)"}, {"name": "aequiv", "content": "def aequiv (a₁ a₂ : AExp) : Prop :=\n  ∀ (st : State), aeval st a₁ = aeval st a₂"}, {"name": "CEval", "content": "inductive CEval : Com → State → State → Prop :=\n  | e_skip : ∀ st,\n      CEval c_skip st st\n  | e_asgn : ∀ a n x st,\n      aeval st a = n\n      → CEval (c_asgn x a) st (update st x n)\n  | e_seq : ∀ c₁ c₂ st st' st'',\n      CEval c₁ st st' → CEval c₂ st' st''\n      → CEval (c_seq c₁ c₂) st st''\n  | e_ifTrue : ∀ b c₁ c₂ st st',\n      beval st b = true → CEval c₁ st st'\n      → CEval (c_if b c₁ c₂) st st'\n  | e_ifFalse : ∀ b c₁ c₂ st st',\n      beval st b = false → CEval c₂ st st'\n      → CEval (c_if b c₁ c₂) st st'\n  | e_whileFalse : ∀ b c st,\n      beval st b = false\n      → CEval (c_while b c) st st\n  | e_whileTrue : ∀ b c st st' st'',\n      beval st b = true\n      → CEval c st st'\n      → CEval (c_while b c) st' st''\n      → CEval (c_while b c) st st''"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "c_while_congruence", "content": "theorem c_while_congruence b b' c c'\n    : bequiv b b' → cequiv c c'\n      → cequiv <{while <[b]> do <[c]> end}> <{while <[b']> do <[c']> end}>"}, {"name": "sym_bequiv", "content": "theorem sym_bequiv b₁ b₂ : bequiv b₁ b₂ → bequiv b₂ b₁"}, {"name": "sym_cequiv", "content": "theorem sym_cequiv c₁ c₂ : cequiv c₁ c₂ → cequiv c₂ c₁"}, {"name": "c_if_congruence", "content": "theorem c_if_congruence b b' c₁ c₁' c₂ c₂'\n    : bequiv b b' → cequiv c₁ c₁' → cequiv c₂ c₂'\n      → cequiv (c_if b c₁ c₂) (c_if b' c₁' c₂')"}, {"name": "c_asgn_congruence", "content": "theorem c_asgn_congruence x a a'\n    : aequiv a a' → cequiv <{x := <[a]>}> <{x := <[a']>}>"}, {"name": "c_seq_congruence", "content": "theorem c_seq_congruence c₁ c₁' c₂ c₂'\n    : cequiv c₁ c₁' → cequiv c₂ c₂' → cequiv (c_seq c₁ c₂) (c_seq c₁' c₂')"}, {"name": "refl_cequiv", "content": "theorem refl_cequiv c : cequiv c c"}], "used_local_defs": [{"name": "Imp.optimize_0plus_aexp", "content": "def optimize_0plus_aexp (a : AExp) : AExp :=\n  match a with\n  | a_num _ => a\n  | a_id _ => a\n  | a_plus (a_num 0) a₂ => optimize_0plus_aexp a₂\n  | a_plus a₁ a₂ => a_plus (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | a_minus a₁ a₂ => a_minus (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | a_mult a₁ a₂ => a_mult (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)"}, {"name": "Imp.optimize_0plus_bexp", "content": "def optimize_0plus_bexp (b : BExp) : BExp :=\n  match b with\n  | b_true => b_true\n  | b_false => b_false\n  | b_eq a₁ a₂ => b_eq (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | b_neq a₁ a₂ => b_neq (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | b_le a₁ a₂ => b_le (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | b_not b₁ => b_not (optimize_0plus_bexp b₁)\n  | b_and b₁ b₂ => b_and (optimize_0plus_bexp b₁) (optimize_0plus_bexp b₂)\n  | b_or b₁ b₂ => b_or (optimize_0plus_bexp b₁) (optimize_0plus_bexp b₂)"}, {"name": "Imp.optimize_0plus_com", "content": "def optimize_0plus_com (c : Com) : Com :=\n  match c with\n  | c_skip => c_skip\n  | c_asgn x a => c_asgn x (optimize_0plus_aexp a)\n  | c_seq c₁ c₂ => c_seq (optimize_0plus_com c₁) (optimize_0plus_com c₂)\n  | c_if b c₁ c₂ => c_if (optimize_0plus_bexp b) (optimize_0plus_com c₁) (optimize_0plus_com c₂)\n  | c_while b c => c_while (optimize_0plus_bexp b) (optimize_0plus_com c)\n\n \n\nexample :\n    optimize_0plus_com <{ while x != 0 do x := 0 + x - 1 end }>\n    = <{ while x != 0 do x := x - 1 end }> := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "Imp.optimize_0plus_aexp_sound", "content": "theorem optimize_0plus_aexp_sound : atrans_sound optimize_0plus_aexp"}, {"name": "Imp.optimize_0plus_bexp_sound", "content": "theorem optimize_0plus_bexp_sound : btrans_sound optimize_0plus_bexp"}], "local_ctx": "import Frap.Trans\n\nnamespace Hidden.AExp\n\nopen AExp\n\nend Hidden.AExp\n\nnamespace Imp\n\nopen AExp\n\nopen BExp\n\nopen Com\n\nopen CEval\n\ndef optimize_0plus_aexp (a : AExp) : AExp :=\n  match a with\n  | a_num _ => a\n  | a_id _ => a\n  | a_plus (a_num 0) a₂ => optimize_0plus_aexp a₂\n  | a_plus a₁ a₂ => a_plus (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | a_minus a₁ a₂ => a_minus (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | a_mult a₁ a₂ => a_mult (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n\ndef optimize_0plus_bexp (b : BExp) : BExp :=\n  match b with\n  | b_true => b_true\n  | b_false => b_false\n  | b_eq a₁ a₂ => b_eq (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | b_neq a₁ a₂ => b_neq (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | b_le a₁ a₂ => b_le (optimize_0plus_aexp a₁) (optimize_0plus_aexp a₂)\n  | b_not b₁ => b_not (optimize_0plus_bexp b₁)\n  | b_and b₁ b₂ => b_and (optimize_0plus_bexp b₁) (optimize_0plus_bexp b₂)\n  | b_or b₁ b₂ => b_or (optimize_0plus_bexp b₁) (optimize_0plus_bexp b₂)\n\ndef optimize_0plus_com (c : Com) : Com :=\n  match c with\n  | c_skip => c_skip\n  | c_asgn x a => c_asgn x (optimize_0plus_aexp a)\n  | c_seq c₁ c₂ => c_seq (optimize_0plus_com c₁) (optimize_0plus_com c₂)\n  | c_if b c₁ c₂ => c_if (optimize_0plus_bexp b) (optimize_0plus_com c₁) (optimize_0plus_com c₂)\n  | c_while b c => c_while (optimize_0plus_bexp b) (optimize_0plus_com c)\n\n \n\nexample :\n    optimize_0plus_com <{ while x != 0 do x := 0 + x - 1 end }>\n    = <{ while x != 0 do x := x - 1 end }> := by admit /- proof elided -/", "target_theorem": "theorem optimize_0plus_com_sound : ctrans_sound optimize_0plus_com :=", "ground_truth_proof": ":= by\n  intros c\n  induction c with simp [*] at *\n  | c_skip => apply refl_cequiv\n  | c_asgn => apply c_asgn_congruence; apply optimize_0plus_aexp_sound\n  | c_seq =>\n    apply c_seq_congruence\n    . assumption\n    . assumption\n  | c_if =>\n    apply c_if_congruence\n    . apply optimize_0plus_bexp_sound\n    . assumption\n    . assumption\n  | c_while =>\n    apply c_while_congruence\n    . apply optimize_0plus_bexp_sound\n    . assumption", "nesting_depth": 6, "transitive_dep_count": 35, "subset_aristotle": false, "category": "Semantics"}
{"id": 277, "thm_name": "Tree.balance_BST", "thm_stmt": "theorem balance_BST {α : Type u} c (l : Tree α) k vk (r : Tree α)\n    : ForallTree (fun x _ => x < k) l\n      -> ForallTree (fun x _ => x > k) r\n      -> BST l\n      -> BST r\n      -> BST (balance c l k vk r)", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/RedBlack.lean", "imports": [], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Color", "content": "inductive Color where\n  | red\n  | black"}, {"name": "Tree", "content": "inductive Tree (α : Type u) where\n  | empty  : Tree α\n  | tree (c: Color) (l : Tree α) (k : Nat) (v : α) (r : Tree α) : Tree α"}, {"name": "Tree.ex_tree", "content": "def ex_tree : Tree String :=\n  tree black (tree red empty 2 \"two\" empty) 4 \"four\" (tree red empty 5 \"five\" empty)"}, {"name": "Tree.balance", "content": "def balance {α : Type u} (c : Color) (l : Tree α) (k : Nat) (vk : α) (r : Tree α) : Tree α :=\n  match c with\n  | red => tree red l k vk r\n  | black =>\n    match (l, k, vk, r) with\n    | (tree red (tree red a x vx b) y vy c, z, vz, d)\n    | (tree red a x vx (tree red b y vy c), z, vz, d)\n    | (a, x, vx, tree red (tree red b y vy c) z vz d)\n    | (a, x, vx, tree red b y vy (tree red c z vz d))\n        => tree red (tree black a x vx b) y vy (tree black c z vz d)\n    | _ => tree black l k vk r"}, {"name": "Tree.ForallTree", "content": "inductive ForallTree {α : Type u} (p : Nat → α → Prop) : Tree α → Prop where\n  | empty : ForallTree p empty\n  | tree : ∀ c l k v r,\n      p k v → ForallTree p l → ForallTree p r\n      → ForallTree p (tree c l k v r)"}, {"name": "Tree.BST", "content": "inductive BST {α : Type u} : Tree α → Prop where\n  | empty : BST empty\n  | tree : ∀ c l k v r,\n      ForallTree (fun x _ => x < k) l\n      → ForallTree (fun x _ => x > k) r\n      → BST l\n      → BST r\n      → BST (tree c l k v r)\n\n \n\nexample : BST ex_tree := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "Tree.forallTree_imp", "content": "theorem forallTree_imp {α : Type u} (P Q : Nat → α → Prop) t\n    : ForallTree P t → (∀ k v, P k v → Q k v) → ForallTree Q t"}, {"name": "Tree.forallTree_lt", "content": "theorem forallTree_lt {α : Type u} (t : Tree α) k k'\n    : ForallTree (fun x _ => x < k) t → k < k'\n      → ForallTree (fun x _ => x < k') t"}, {"name": "Tree.forallTree_gt", "content": "theorem forallTree_gt {α : Type u} (t : Tree α) k k'\n    : ForallTree (fun x _ => x > k) t → k > k'\n      → ForallTree (fun x _ => x > k') t"}], "local_ctx": "inductive Color where\n  | red\n  | black\n\ninductive Tree (α : Type u) where\n  | empty  : Tree α\n  | tree (c: Color) (l : Tree α) (k : Nat) (v : α) (r : Tree α) : Tree α\n\nnamespace Tree\n\nopen Color\n\ndef ex_tree : Tree String :=\n  tree black (tree red empty 2 \"two\" empty) 4 \"four\" (tree red empty 5 \"five\" empty)\n\ndef balance {α : Type u} (c : Color) (l : Tree α) (k : Nat) (vk : α) (r : Tree α) : Tree α :=\n  match c with\n  | red => tree red l k vk r\n  | black =>\n    match (l, k, vk, r) with\n    | (tree red (tree red a x vx b) y vy c, z, vz, d)\n    | (tree red a x vx (tree red b y vy c), z, vz, d)\n    | (a, x, vx, tree red (tree red b y vy c) z vz d)\n    | (a, x, vx, tree red b y vy (tree red c z vz d))\n        => tree red (tree black a x vx b) y vy (tree black c z vz d)\n    | _ => tree black l k vk r\n\ninductive ForallTree {α : Type u} (p : Nat → α → Prop) : Tree α → Prop where\n  | empty : ForallTree p empty\n  | tree : ∀ c l k v r,\n      p k v → ForallTree p l → ForallTree p r\n      → ForallTree p (tree c l k v r)\n\ninductive BST {α : Type u} : Tree α → Prop where\n  | empty : BST empty\n  | tree : ∀ c l k v r,\n      ForallTree (fun x _ => x < k) l\n      → ForallTree (fun x _ => x > k) r\n      → BST l\n      → BST r\n      → BST (tree c l k v r)\n\n \n\nexample : BST ex_tree := by admit /- proof elided -/", "target_theorem": "theorem balance_BST {α : Type u} c (l : Tree α) k vk (r : Tree α)\n    : ForallTree (fun x _ => x < k) l\n      -> ForallTree (fun x _ => x > k) r\n      -> BST l\n      -> BST r\n      -> BST (balance c l k vk r) :=", "ground_truth_proof": ":= by\n  intro hlk hkr hbl hbr; simp\n  split\n  . constructor <;> assumption\n  . split\n    . -- we are in the case where `x` is left child of `y`\n      cases l <;> repeat simp [*] at *\n      rcases hbl with ⟨⟩ | ⟨_, _, _, _, _, hxy, hyc, hbx, hbc⟩\n      rcases hlk with ⟨⟩ | ⟨_, _, _, _, _, hyz, hxz, hzc⟩\n      rcases hbx with ⟨⟩ | ⟨_, _, _, _, _, hax, hxb, hba, hbb⟩\n      rcases hxy with ⟨⟩ | ⟨_, _, _, _, _, hxy, hay, hby⟩\n      rcases hxz with ⟨⟩ | ⟨_, _, _, _, _, haz, hxz, hbz⟩\n      /-\n      Here, we have destructed all the assumptions to the atomic level.\n      The goal remains to show that the resulting tree is a BST.\n      We apply the constructor tactic repeatedly.\n      Most of the resulting subgoals are trivial by assumption.\n      -/\n      (repeat' constructor) <;> try assumption\n      /-\n      The only interesting subgoal is to show that each node in the right subtree of the original root is greater than the left child of the original root.\n      This should follow from assumptions, although not trivially.\n      To facilitate reasoning at this point, we prove lemmas `forallTree_imp`, `forallTree_lt`, and `forallTree_gt` above.\n      -/\n      apply forallTree_gt <;> assumption\n    . -- we are in the case where `y` is right child of `x`\n      cases l <;> repeat simp [*] at *\n      rcases hbl with ⟨⟩ | ⟨_, _, _, _, _, hax, hxy, hba, hby⟩\n      rcases hlk with ⟨⟩ | ⟨_, _, _, _, _, hxz, haz, hyz⟩\n      rcases hby with ⟨⟩ | ⟨_, _, _, _, _, hby, hyc, hbb, hbc⟩\n      rcases hxy with ⟨⟩ | ⟨_, _, _, _, _, hxy, hxb, hxc⟩\n      rcases hyz with ⟨⟩ | ⟨_, _, _, _, _, hyz, hbz, hcz⟩\n      (repeat' constructor) <;> try assumption\n      . apply forallTree_lt\n        . exact hax\n        . assumption\n      . apply forallTree_gt <;> assumption\n    . -- we are in the case where `y` is left child of `z`\n      cases r <;> repeat simp [*] at *\n      rcases hbr with ⟨⟩ | ⟨_, _, _, _, _, hyz, hzd, hby, hbd⟩\n      rcases hkr with ⟨⟩ | ⟨_, _, _, _, _, hxz, hxy, hxd⟩\n      rcases hby with ⟨⟩ | ⟨_, _, _, _, _, hby, hyc, hbb, hbc⟩\n      rcases hxy with ⟨⟩ | ⟨_, _, _, _, _, hxy, hxb, hxc⟩\n      rcases hyz with ⟨⟩ | ⟨_, _, _, _, _, hyz, hbz, hcz⟩\n      (repeat' constructor) <;> try assumption\n      . apply forallTree_lt <;> assumption\n      . apply forallTree_gt\n        . exact hzd\n        . assumption\n    . -- we are in the case where `z` is right child of `y`\n      cases r <;> repeat simp [*] at *\n      rcases hbr with ⟨⟩ | ⟨_, _, _, _, _, hby, hyz, hbb, hbz⟩\n      rcases hkr with ⟨⟩ | ⟨_, _, _, _, _, hxy, hxb, hxz⟩\n      rcases hbz with ⟨⟩ | ⟨_, _, _, _, _, hcz, hzd, hbc, hbd⟩\n      rcases hxz with ⟨⟩ | ⟨_, _, _, _, _, hxz, hxc, hxd⟩\n      rcases hyz with ⟨⟩ | ⟨_, _, _, _, _, hyz, hyc, hyd⟩\n      (repeat' constructor) <;> try assumption\n      apply forallTree_lt <;> assumption\n    . constructor <;> assumption", "nesting_depth": 2, "transitive_dep_count": 11, "subset_aristotle": true, "category": "Semantics"}
{"id": 278, "thm_name": "permutation_app_comm", "thm_stmt": "theorem permutation_app_comm (l l' : List α)\n    : Permutation (l ++ l') (l' ++ l)", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Sort.lean", "imports": [], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "append_assoc", "content": "theorem append_assoc (as bs cs : List α)\n        : (as ++ bs) ++ cs = as ++ (bs ++ cs)"}], "used_local_defs": [{"name": "Permutation", "content": "inductive Permutation {α : Type} : List α → List α → Prop :=\n  | perm_nil : Permutation [] []\n  | perm_skip x l l'\n      : Permutation l l' → Permutation (x::l) (x::l')\n  | perm_swap x y l : Permutation (y::x::l) (x::y::l)\n  | perm_trans l l' l''\n      : Permutation l l' → Permutation l' l''\n        → Permutation l l''"}], "used_local_lemmas": [{"name": "permutation_refl", "content": "theorem permutation_refl (l : List α) : Permutation l l"}, {"name": "permutation_symm", "content": "theorem permutation_symm (l l' : List α)\n    : Permutation l l' → Permutation l' l"}, {"name": "permutation_app_tail", "content": "theorem permutation_app_tail (l l' tl : List α)\n    : Permutation l l' → Permutation (l++tl) (l'++tl)"}, {"name": "permutation_app_head", "content": "theorem permutation_app_head (l tl tl' : List α)\n    : Permutation tl tl' → Permutation (l++tl) (l++tl')"}, {"name": "permutation_cons_append", "content": "theorem permutation_cons_append (l : List α) x\n    : Permutation (x :: l) (l ++ [x])"}], "local_ctx": "inductive Permutation {α : Type} : List α → List α → Prop :=\n  | perm_nil : Permutation [] []\n  | perm_skip x l l'\n      : Permutation l l' → Permutation (x::l) (x::l')\n  | perm_swap x y l : Permutation (y::x::l) (x::y::l)\n  | perm_trans l l' l''\n      : Permutation l l' → Permutation l' l''\n        → Permutation l l''\n\nopen Permutation", "target_theorem": "theorem permutation_app_comm (l l' : List α)\n    : Permutation (l ++ l') (l' ++ l) :=", "ground_truth_proof": ":= by\n  induction l with simp\n  | nil => apply permutation_refl\n  | cons a al ih =>\n    -- a :: (al ++ l')\n    -- al ++ l' ++ [a]\n    -- l' ++ al ++ [a]\n    -- l' ++ (al ++ [a])\n    -- l' ++ a :: al\n    apply perm_trans\n    . apply permutation_cons_append\n    . apply perm_trans\n      . apply permutation_app_tail; exact ih\n      . rw [List.append_assoc]\n        apply permutation_app_head\n        apply permutation_symm\n        apply permutation_cons_append", "nesting_depth": 2, "transitive_dep_count": 8, "subset_aristotle": false, "category": "Semantics"}
{"id": 279, "thm_name": "Hidden.Nat.add_comm", "thm_stmt": "theorem add_comm (m n : Nat) : m + n = n + m", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Inductive.lean", "imports": [], "used_lib_defs": [{"name": "structure BitVec (w : Nat) where", "module": ""}, {"name": "/-- Construct a `BitVec w` from a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "ofFin ::", "module": ""}, {"name": "/-- Interpret a bitvector as a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "toFin : Fin (hPow 2 w)", "module": ""}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Hidden.Nat", "content": "inductive Nat where\n  | zero : Nat\n  | succ : Nat → Nat"}], "used_local_lemmas": [{"name": "Hidden.Nat.add_zero", "content": "theorem add_zero (m : Nat) : m + zero = m"}, {"name": "Hidden.Nat.add_succ", "content": "theorem add_succ (m n : Nat) : m + succ n = succ (m + n)"}, {"name": "Hidden.Nat.zero_add", "content": "theorem zero_add (n : Nat) : zero + n = n"}, {"name": "Hidden.Nat.succ_add", "content": "theorem succ_add (m n : Nat) : succ (m + n) = succ m + n"}], "local_ctx": "section rewriting\n\nvariable (α : Type) (a b c d : α)  -- a, b, c, d are elements of some type\n\n/-\n## Rewriting\n\nRecall our equality proof from last time.\n-/\n\nexample : a = b → c = b → c = d → a = d := by\n  intro hab hcb hcd\n  apply Eq.trans\n  . exact hab\n  . apply Eq.trans\n    . apply Eq.symm\n      exact hcb\n    . exact hcd\n\n/-\nNotice that in each subgoal, we are attempting to use a hypothesis by way of an equality property.\nFor example, we invoke the transitive property on `a = d` so that we can use the assumption `a = b` and be left with `b = d` to prove.\nIn effect, we are substituting `b` for `a` in the goal.\n\nThis substitution operation can be seen as rewriting some term in the goal with another, via an equality in the hypothesis.\nThis rewriting operation is so common that Lean provides a `rw` tactic to do so.\nIt replaces equals for equals, which could be an equality (`=`) or equivalence (`↔`).\n\nBe default, it rewrites the left-hand side with the right-hand side.\nFor right-to-left rewriting, a left arrow needs to be indicated before the hypothesis (`←`, typed with `\\<-` shortcut) .\n-/\n\nexample : a = b → c = b → c = d → a = d := by\n  intro hab hcb hcd\n  rw [hab]\n  rw [← hcd]\n  rw [hcb]\n\nend rewriting\n\n/-\n## reflexivity tactic\n\nOne of the most powerful tactics is the _reflexivity_ tactic, written `rfl` in Lean.\nIt will attempt to conclude that both sides of an equality goal is indeed equal, up to some trivial computation, such as expansion of definitions and function applications.\n-/\n\nexample : 2 + 5 = 7 := by\n  rfl\n\nexample : ¬ Even = 3 := by\n  intro h\n  sorry\n\nnamespace Hidden\n\ninductive Nat where\n  | zero : Nat\n  | succ : Nat → Nat\n\nnamespace Nat\n\nend Nat\n\nend Hidden\n\nsection opendemo\n\nopen Nat\n\nend opendemo\n\nnamespace Hidden\n\nnamespace Nat\n\nend Nat\n\nend Hidden\n\nnamespace Hidden\n\nnamespace Nat", "target_theorem": "theorem add_comm (m n : Nat) : m + n = n + m :=", "ground_truth_proof": ":= by\n  induction n with\n  | zero =>\n      rw [add_zero]\n      rw [zero_add]\n  | succ n' ih =>\n      rw [add_succ]\n      rw [ih]\n      rw [succ_add]\n\n/-\nAlternatively, if the supporting fact is only useful within a single theorem, we can embed the proof within, using the\n-/\n\nexample (m n : Nat) : m + n = n + m := by\n  induction n with\n  | zero =>\n      rw [add_zero]\n      rw [zero_add]\n  | succ n' ih =>\n      rw [add_succ]\n      rw [ih]\n      have h (m n : Nat) : succ (m + n) = succ m + n := by\n        induction n with\n        | zero =>\n            rw [add_zero]\n            rw [add_zero]\n        | succ n' ih =>\n            rw [add_succ]\n            rw [add_succ]\n            rw [ih]\n      rw [h]", "nesting_depth": 1, "transitive_dep_count": 5, "subset_aristotle": false, "category": "Semantics"}
{"id": 280, "thm_name": "Hidden.List.Palindrome.Palindrome_rev", "thm_stmt": "theorem Palindrome_rev {α : Type} (l : List α)\n    : Palindrome l → l = reverse l", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Exercises/IndProp.lean", "imports": [], "used_lib_defs": [{"name": "structure BitVec (w : Nat) where", "module": ""}, {"name": "/-- Construct a `BitVec w` from a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "ofFin ::", "module": ""}, {"name": "/-- Interpret a bitvector as a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "toFin : Fin (hPow 2 w)", "module": ""}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Hidden.List", "content": "inductive List (α : Type u) where\n  | nil  : List α\n  | cons : α → List α → List α"}, {"name": "Hidden.List.reverse", "content": "def reverse {α : Type u} (as : List α) : List α :=\n  match as with\n  | nil        => nil\n  | cons a as' => reverse as' ++ cons a nil"}, {"name": "Hidden.List.Palindrome", "content": "inductive Palindrome {α : Type} : List α → Prop where\n  | nil : Palindrome nil\n  | single (x : α) : Palindrome (cons x nil)\n  | sandwich (x : α) (xs : List α)\n      : Palindrome xs → Palindrome (cons x xs ++ cons x nil)"}], "used_local_lemmas": [{"name": "Hidden.List.nil_append", "content": "theorem nil_append (as : List α) : nil ++ as = as"}, {"name": "Hidden.List.cons_append", "content": "theorem cons_append (a : α) (as bs : List α)\n        : (cons a as) ++ bs = cons a (as ++ bs)"}, {"name": "Hidden.List.append_nil", "content": "theorem append_nil (as : List α) : as ++ nil = as"}, {"name": "Hidden.List.append_assoc", "content": "theorem append_assoc (as bs cs : List α)\n        : (as ++ bs) ++ cs = as ++ (bs ++ cs)"}, {"name": "Hidden.List.reverse_append", "content": "theorem reverse_append {α : Type u} (as bs : List α)\n    : reverse (as ++ bs) = reverse bs ++ reverse as"}], "local_ctx": "namespace Hidden\n\ninductive List (α : Type u) where\n  | nil  : List α\n  | cons : α → List α → List α\n\nnamespace List\n\ndef reverse {α : Type u} (as : List α) : List α :=\n  match as with\n  | nil        => nil\n  | cons a as' => reverse as' ++ cons a nil\n\ninductive Palindrome {α : Type} : List α → Prop where\n  | nil : Palindrome nil\n  | single (x : α) : Palindrome (cons x nil)\n  | sandwich (x : α) (xs : List α)\n      : Palindrome xs → Palindrome (cons x xs ++ cons x nil)\n\nnamespace Palindrome", "target_theorem": "theorem Palindrome_rev {α : Type} (l : List α)\n    : Palindrome l → l = reverse l :=", "ground_truth_proof": ":= by\n  intro lPal\n  induction lPal with\n  | nil => rfl\n  | single x => rfl\n  | sandwich x xs xsPal xsRev =>\n    rw [reverse_append, reverse, reverse, nil_append, reverse, ←xsRev]\n    rw [cons_append, ←append_assoc, cons_append, nil_append, cons_append]", "nesting_depth": 2, "transitive_dep_count": 8, "subset_aristotle": false, "category": "Semantics"}
{"id": 281, "thm_name": "Imp.Hoare.hoare_while", "thm_stmt": "theorem hoare_while P b c :\n    {* fun st => P st ∧ beval st b *} c {* P *}\n    → {* P *} c_while b c {* fun st => P st ∧ ¬(beval st b) *}", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Hoare.lean", "imports": ["import Frap.Trans"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : im", "content": "syntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : imp\n\nsyntax \"false\" : imp\n\nsyntax \"false\" : term\n\nsyntax \"true\" : imp\n\nsyntax \"true\" : term\n\nsyntax \"<{\" imp \"}>\" : term\n\nsyntax:21 \"while\" imp:20 \"do\" imp:20 \"end\" : imp\n\nsyntax:30 \"{*\" term \"*}\" term \"{*\" term \"*}\" : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|true) => `(Bool.true)\n  | `(term|false) => `(Bool.false)\n  | `(term|<{$x}>) => `(imp|$x)\n  | `(imp|$n:num) => `(a_num $n)\n  | `(imp|$s:str) => `(a_id $s)\n  | `(imp|$x + $y) => `(a_plus <{$x}> <{$y}>)\n  | `(imp|$x - $y) => `(a_minus <{$x}> <{$y}>)\n  | `(imp|$x * $y) => `(a_mult <{$x}> <{$y}>)\n  | `(imp|true) => `(b_true)\n  | `(imp|false) => `(b_false)\n  | `(imp|$x = $y) => `(b_eq <{$x}> <{$y}>)\n  | `(imp|$x != $y) => `(b_neq <{$x}> <{$y}>)\n  | `(imp|$x <= $y) => `(b_le <{$x}> <{$y}>)\n  | `(imp|!$x) => `(b_not <{$x}>)\n  | `(imp|$x && $y) => `(b_and <{$x}> <{$y}>)\n  | `(imp|$x || $y) => `(b_or <{$x}> <{$y}>)\n  | `(imp|($x)) => `(<{$x}>)\n  | `(imp|$x:ident) => `(a_id $(Lean.quote (toString x.getId)))\n  | `(imp|<[$t:term]>) => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(imp|skip) => `(c_skip)\n  | `(imp|$x:str := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$x:ident := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$c1 ; $c2) => `(c_seq <{$c1}> <{$c2}>)\n  | `(imp|if $b then $c1 else $c2 end) => `(c_if <{$b}> <{$c1}> <{$c2}>)\n  | `(imp|while $b do $c end) => `(c_while <{$b}> <{$c}>)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$p <<->> $q) => `($p ->> $q ∧ $q ->> $p)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|{*$p*} $c {*$q*}) => `(valid_hoare_triple $p $c $q)"}, {"name": "x", "content": "abbrev x := \"x\""}, {"name": "beval", "content": "def beval (st : State) (b : BExp) : Bool :=\n  match b with\n  | b_true => true\n  | b_false => false\n  | b_eq a₁ a₂ => (aeval st a₁) == (aeval st a₂)\n  | b_neq a₁ a₂ => (aeval st a₁) != (aeval st a₂)\n  | b_le a₁ a₂ => (aeval st a₁) <= (aeval st a₂)\n  | b_not b₁ => not (beval st b₁)\n  | b_and b₁ b₂ => and (beval st b₁) (beval st b₂)\n  | b_or b₁ b₂ => or (beval st b₁) (beval st b₂)\n\nexample : aeval (update empty x 5)\n    <{3 + x * 2}>\n    \n    = 13 := by admit /- proof elided -/"}, {"name": "empty", "content": "def empty : State := fun _ => 0"}, {"name": "State", "content": "abbrev State := String → Nat"}, {"name": "y", "content": "abbrev y := \"y\""}, {"name": "z", "content": "abbrev z := \"z\""}, {"name": "aeval", "content": "def aeval (st : State) (a : AExp) : Nat :=\n  match a with\n  | a_num n => n\n  | a_id x => st x\n  | a_plus a₁ a₂ => (aeval st a₁) + (aeval st a₂)\n  | a_minus a₁ a₂ => (aeval st a₁) - (aeval st a₂)\n  | a_mult a₁ a₂ => (aeval st a₁) * (aeval st a₂)"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_id : String → AExp  \n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "update", "content": "def update (st : State) (k : String) (v : Nat) : State :=\n  fun x => if x == k then v else st x"}, {"name": "BExp", "content": "inductive BExp where\n  | b_true : BExp\n  | b_false : BExp\n  | b_eq : AExp → AExp → BExp\n  | b_neq : AExp → AExp → BExp\n  | b_le : AExp → AExp → BExp\n  | b_not : BExp → BExp\n  | b_and : BExp → BExp → BExp\n  | b_or : BExp → BExp → BExp"}, {"name": "Com", "content": "inductive Com : Type :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com\n  | c_par : Com → Com → Com"}, {"name": "CEval", "content": "inductive CEval : Com → State → State → Prop :=\n  | e_skip : ∀ st,\n      CEval c_skip st st\n  | e_asgn : ∀ a n x st,\n      aeval st a = n\n      → CEval (c_asgn x a) st (update st x n)\n  | e_seq : ∀ c₁ c₂ st st' st'',\n      CEval c₁ st st' → CEval c₂ st' st''\n      → CEval (c_seq c₁ c₂) st st''\n  | e_ifTrue : ∀ b c₁ c₂ st st',\n      beval st b = true → CEval c₁ st st'\n      → CEval (c_if b c₁ c₂) st st'\n  | e_ifFalse : ∀ b c₁ c₂ st st',\n      beval st b = false → CEval c₂ st st'\n      → CEval (c_if b c₁ c₂) st st'\n  | e_whileFalse : ∀ b c st,\n      beval st b = false\n      → CEval (c_while b c) st st\n  | e_whileTrue : ∀ b c st st' st'',\n      beval st b = true\n      → CEval c st st'\n      → CEval (c_while b c) st' st''\n      → CEval (c_while b c) st st''\n\nsyntax:30 term \" =[ \" imp \" ]=> \" term : term\nmacro_rules\n  | `(term|$st =[ $c ]=> $st') => `(CEval <{$c}> $st $st')"}, {"name": "Assertion", "content": "abbrev Assertion := State → Prop"}, {"name": "assert_implies", "content": "def assert_implies (P Q : Assertion) : Prop :=\n  ∀ st, P st → Q st"}, {"name": "valid_hoare_triple", "content": "def valid_hoare_triple (P : Assertion) (c : Com) (Q : Assertion) : Prop :=\n  ∀ st st', P st → (st =[<[c]>]=> st') → Q st'"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Frap.Trans\n\nnamespace Imp\n\nopen AExp\n\nopen BExp\n\nopen Com\n\nopen CEval\n\nnamespace Hoare\n\ninfix:36 \" ->> \" => assert_implies", "target_theorem": "theorem hoare_while P b c :\n    {* fun st => P st ∧ beval st b *} c {* P *}\n    → {* P *} c_while b c {* fun st => P st ∧ ¬(beval st b) *} :=", "ground_truth_proof": ":= by\n  intro hHoare st st' hPre hEval\n  -- we proceed by induction on `hEval`\n  generalize hloop : c_while b c = cmd at *\n  induction hEval with simp at *\n  | e_whileFalse =>\n    constructor\n    . exact hPre\n    . simp [*]\n  | e_whileTrue =>\n    simp [*] at *\n    rename_i ih\n    apply ih\n    apply hHoare\n    . constructor\n      . apply hPre\n      . assumption\n    . assumption\n\n/-\nWe call `P` a _loop invariant_ of `while b do c` if\n  `{P ∧ b} c {P}`\nis a valid Hoare triple.\n\nThis means that `P` will be true at the end of the loop body whenever the loop body executes.\nIf `P` contradicts `b`, this holds trivially since the precondition is false.\n\nFor instance, `X = 0` is a loop invariant of\n  `while X = 2 do X := 1 end`\nsince the program will never enter the loop.\n\nThe program\n  `while Y > 10 do Y := Y - 1; Z := Z + 1 end`\nadmits an interesting loop invariant:\n  `X = Y + Z`\nNote that this doesn't contradict the loop guard but neither is it a command invariant of\n  `Y := Y - 1; Z := Z + 1`\nsince, if `X = 5`, `Y = 0` and `Z = 5`, running the command will set `Y + Z` to 6, because `Y` remains 0.\nThe loop guard `Y > 10` guarantees that this will not be the case.\nWe will see many such loop invariants in the following lecture.\n-/\n\nexample :\n    -- { x ≤ 3 }\n    {* fun st => st x ≤ 3 *}\n    <{\n      while x <= 2 do\n        x := x + 1", "nesting_depth": 4, "transitive_dep_count": 15, "subset_aristotle": true, "category": "Semantics"}
{"id": 282, "thm_name": "Tree.bst_insert_of_bst", "thm_stmt": "theorem bst_insert_of_bst {α : Type u} (k : Nat) (v : α) (t : Tree α)\n    : BST t → BST (insert k v t)", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/ADT.lean", "imports": [], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.le_antisymm", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Tree", "content": "inductive Tree (α : Type u) where\n  | empty  : Tree α\n  | tree (l : Tree α) (k : Nat) (v : α) (r : Tree α) : Tree α"}, {"name": "Tree.ex_tree", "content": "def ex_tree : Tree String :=\n  tree (tree empty 2 \"two\" empty) 4 \"four\" (tree empty 5 \"five\" empty)"}, {"name": "Tree.insert", "content": "def insert {α : Type u} (x : Nat) (v : α) (t : Tree α) : Tree α :=\n  match t with\n  | empty => tree empty x v empty\n  | tree l k v' r =>\n      if x < k then tree (insert x v l) k v' r\n      else if x > k then tree l k v' (insert x v r)\n      else tree l x v r"}, {"name": "Tree.ForallTree", "content": "inductive ForallTree {α : Type u} (p : Nat → α → Prop) : Tree α → Prop where\n  | empty : ForallTree p empty\n  | tree : ∀ l k v r,\n      p k v → ForallTree p l → ForallTree p r\n      → ForallTree p (tree l k v r)"}, {"name": "Tree.BST", "content": "inductive BST {α : Type u} : Tree α → Prop where\n  | empty : BST empty\n  | tree : ∀ l k v r,\n      ForallTree (fun x _ => x < k) l\n      → ForallTree (fun x _ => x > k) r\n      → BST l\n      → BST r\n      → BST (tree l k v r)\n\nexample : BST ex_tree := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "Tree.forall_insert_of_forall", "content": "theorem forall_insert_of_forall {α : Type u} (P : Nat → α → Prop) (t : Tree α)\n    : ForallTree P t → ∀ k v, P k v → ForallTree P (insert k v t)"}], "local_ctx": "inductive Tree (α : Type u) where\n  | empty  : Tree α\n  | tree (l : Tree α) (k : Nat) (v : α) (r : Tree α) : Tree α\n\nnamespace Tree\n\ndef ex_tree : Tree String :=\n  tree (tree empty 2 \"two\" empty) 4 \"four\" (tree empty 5 \"five\" empty)\n\ndef insert {α : Type u} (x : Nat) (v : α) (t : Tree α) : Tree α :=\n  match t with\n  | empty => tree empty x v empty\n  | tree l k v' r =>\n      if x < k then tree (insert x v l) k v' r\n      else if x > k then tree l k v' (insert x v r)\n      else tree l x v r  \n\ninductive ForallTree {α : Type u} (p : Nat → α → Prop) : Tree α → Prop where\n  | empty : ForallTree p empty\n  | tree : ∀ l k v r,\n      p k v → ForallTree p l → ForallTree p r\n      → ForallTree p (tree l k v r)\n\ninductive BST {α : Type u} : Tree α → Prop where\n  | empty : BST empty\n  | tree : ∀ l k v r,\n      ForallTree (fun x _ => x < k) l\n      → ForallTree (fun x _ => x > k) r\n      → BST l\n      → BST r\n      → BST (tree l k v r)\n\nexample : BST ex_tree := by admit /- proof elided -/", "target_theorem": "theorem bst_insert_of_bst {α : Type u} (k : Nat) (v : α) (t : Tree α)\n    : BST t → BST (insert k v t) :=", "ground_truth_proof": ":= by\n  intro hbst\n  induction hbst with\n  | empty => constructor <;> constructor\n  | tree l k' v' r hlt hgt hbstl hbstr =>\n    unfold insert\n    split\n    . constructor\n      . apply forall_insert_of_forall\n        . assumption\n        . assumption\n      . assumption\n      . assumption\n      . assumption\n    . split\n      . constructor\n        . assumption\n        . apply forall_insert_of_forall\n          . assumption\n          . assumption\n        . assumption\n        . assumption\n      . have heq : k = k' := by\n          simp [*] at *\n          apply Nat.le_antisymm <;> assumption\n        constructor\n        . rw [heq]; assumption\n        . rw [heq]; assumption\n        . assumption\n        . assumption", "nesting_depth": 4, "transitive_dep_count": 11, "subset_aristotle": false, "category": "Semantics"}
{"id": 283, "thm_name": "Hidden.List.Palindrome.Palindrome_app_rev", "thm_stmt": "theorem Palindrome_app_rev {α : Type} (l : List α) : Palindrome (l ++ reverse l)", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Exercises/IndProp.lean", "imports": [], "used_lib_defs": [{"name": "structure BitVec (w : Nat) where", "module": ""}, {"name": "/-- Construct a `BitVec w` from a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "ofFin ::", "module": ""}, {"name": "/-- Interpret a bitvector as a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "toFin : Fin (hPow 2 w)", "module": ""}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Hidden.List", "content": "inductive List (α : Type u) where\n  | nil  : List α\n  | cons : α → List α → List α"}, {"name": "Hidden.List.reverse", "content": "def reverse {α : Type u} (as : List α) : List α :=\n  match as with\n  | nil        => nil\n  | cons a as' => reverse as' ++ cons a nil"}, {"name": "Hidden.List.Palindrome", "content": "inductive Palindrome {α : Type} : List α → Prop where\n  | nil : Palindrome nil\n  | single (x : α) : Palindrome (cons x nil)\n  | sandwich (x : α) (xs : List α)\n      : Palindrome xs → Palindrome (cons x xs ++ cons x nil)"}], "used_local_lemmas": [{"name": "Hidden.List.nil_append", "content": "theorem nil_append (as : List α) : nil ++ as = as"}, {"name": "Hidden.List.cons_append", "content": "theorem cons_append (a : α) (as bs : List α)\n        : (cons a as) ++ bs = cons a (as ++ bs)"}, {"name": "Hidden.List.append_assoc", "content": "theorem append_assoc (as bs cs : List α)\n        : (as ++ bs) ++ cs = as ++ (bs ++ cs)"}], "local_ctx": "namespace Hidden\n\ninductive List (α : Type u) where\n  | nil  : List α\n  | cons : α → List α → List α\n\nnamespace List\n\ndef reverse {α : Type u} (as : List α) : List α :=\n  match as with\n  | nil        => nil\n  | cons a as' => reverse as' ++ cons a nil\n\ninductive Palindrome {α : Type} : List α → Prop where\n  | nil : Palindrome nil\n  | single (x : α) : Palindrome (cons x nil)\n  | sandwich (x : α) (xs : List α)\n      : Palindrome xs → Palindrome (cons x xs ++ cons x nil)\n\nnamespace Palindrome", "target_theorem": "theorem Palindrome_app_rev {α : Type} (l : List α) : Palindrome (l ++ reverse l) :=", "ground_truth_proof": ":= by\n  induction l with\n  | nil => rw [nil_append, reverse]; exact nil\n  | cons x xs ih =>\n    rw [cons_append, reverse, ←append_assoc, ←cons_append]\n    apply sandwich x\n    exact ih", "nesting_depth": 2, "transitive_dep_count": 6, "subset_aristotle": false, "category": "Semantics"}
{"id": 284, "thm_name": "Hidden.Nat.mul_assoc", "thm_stmt": "theorem mul_assoc (m n k : Nat) : m * n * k = m * (n * k)", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Inductive.lean", "imports": [], "used_lib_defs": [{"name": "structure BitVec (w : Nat) where", "module": ""}, {"name": "/-- Construct a `BitVec w` from a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "ofFin ::", "module": ""}, {"name": "/-- Interpret a bitvector as a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "toFin : Fin (hPow 2 w)", "module": ""}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Hidden.Nat", "content": "inductive Nat where\n  | zero : Nat\n  | succ : Nat → Nat"}, {"name": "Hidden.Nat.add", "content": "def add (m n : Nat) : Nat :=\n  match n with\n  | zero    => m\n  | succ n' => succ (add m n')"}], "used_local_lemmas": [{"name": "Hidden.Nat.add_succ", "content": "theorem add_succ (m n : Nat) : m + succ n = succ (m + n)"}, {"name": "Hidden.Nat.add_assoc", "content": "theorem add_assoc (m n k : Nat) : m + n + k = m + (n + k)"}, {"name": "Hidden.Nat.mul_succ", "content": "theorem mul_succ (m n : Nat) : m * succ n = add (m * n) m"}, {"name": "Hidden.Nat.add_infix", "content": "theorem add_infix (m n : Nat) : m.add n = m + n"}], "local_ctx": "section rewriting\n\nvariable (α : Type) (a b c d : α)  -- a, b, c, d are elements of some type\n\n/-\n## Rewriting\n\nRecall our equality proof from last time.\n-/\n\nexample : a = b → c = b → c = d → a = d := by\n  intro hab hcb hcd\n  apply Eq.trans\n  . exact hab\n  . apply Eq.trans\n    . apply Eq.symm\n      exact hcb\n    . exact hcd\n\n/-\nNotice that in each subgoal, we are attempting to use a hypothesis by way of an equality property.\nFor example, we invoke the transitive property on `a = d` so that we can use the assumption `a = b` and be left with `b = d` to prove.\nIn effect, we are substituting `b` for `a` in the goal.\n\nThis substitution operation can be seen as rewriting some term in the goal with another, via an equality in the hypothesis.\nThis rewriting operation is so common that Lean provides a `rw` tactic to do so.\nIt replaces equals for equals, which could be an equality (`=`) or equivalence (`↔`).\n\nBe default, it rewrites the left-hand side with the right-hand side.\nFor right-to-left rewriting, a left arrow needs to be indicated before the hypothesis (`←`, typed with `\\<-` shortcut) .\n-/\n\nexample : a = b → c = b → c = d → a = d := by\n  intro hab hcb hcd\n  rw [hab]\n  rw [← hcd]\n  rw [hcb]\n\nend rewriting\n\n/-\n## reflexivity tactic\n\nOne of the most powerful tactics is the _reflexivity_ tactic, written `rfl` in Lean.\nIt will attempt to conclude that both sides of an equality goal is indeed equal, up to some trivial computation, such as expansion of definitions and function applications.\n-/\n\nexample : 2 + 5 = 7 := by\n  rfl\n\nexample : ¬ Even = 3 := by\n  intro h\n  sorry\n\nnamespace Hidden\n\ninductive Nat where\n  | zero : Nat\n  | succ : Nat → Nat\n\nnamespace Nat\n\ndef add (m n : Nat) : Nat :=\n  match n with\n  | zero    => m\n  | succ n' => succ (add m n')\n\ndef mul (m n : Nat) : Nat :=\n  match n with\n  | zero    => zero\n  | succ n' => add (mul m n') m\n\ninstance : Add Nat where\n  add := add\n\ninstance : Mul Nat where\n  mul := mul\n\nend Nat\n\nend Hidden\n\nsection opendemo\n\nopen Nat\n\nend opendemo\n\nnamespace Hidden\n\nnamespace Nat\n\nend Nat\n\nend Hidden\n\nnamespace Hidden\n\nnamespace Nat", "target_theorem": "theorem mul_assoc (m n k : Nat) : m * n * k = m * (n * k) :=", "ground_truth_proof": ":= by\n  induction k with\n  | zero => rfl\n  | succ k' ih =>\n      rw [mul_succ, mul_succ, ih, add_infix, add_infix]\n      have nat_distrib (m n k : Nat)\n            : m * n + m * k = m * (n + k) := by\n        -- distributive property from left\n        induction k with\n        | zero => rfl\n        | succ k' ih =>\n            rw [mul_succ, add_succ, mul_succ, add_infix, add_infix, ← add_assoc, ih]\n      rw [nat_distrib]", "nesting_depth": 2, "transitive_dep_count": 6, "subset_aristotle": false, "category": "Semantics"}
{"id": 285, "thm_name": "permutation_cons_app", "thm_stmt": "theorem permutation_cons_app (l l₁ l₂ : List α)\n    : Permutation l (l₁ ++ l₂)\n      → Permutation (a :: l) (l₁ ++ a :: l₂)", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Sort.lean", "imports": [], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "append_assoc", "content": "theorem append_assoc (as bs cs : List α)\n        : (as ++ bs) ++ cs = as ++ (bs ++ cs)"}], "used_local_defs": [{"name": "Permutation", "content": "inductive Permutation {α : Type} : List α → List α → Prop :=\n  | perm_nil : Permutation [] []\n  | perm_skip x l l'\n      : Permutation l l' → Permutation (x::l) (x::l')\n  | perm_swap x y l : Permutation (y::x::l) (x::y::l)\n  | perm_trans l l' l''\n      : Permutation l l' → Permutation l' l''\n        → Permutation l l''"}], "used_local_lemmas": [{"name": "permutation_refl", "content": "theorem permutation_refl (l : List α) : Permutation l l"}, {"name": "permutation_symm", "content": "theorem permutation_symm (l l' : List α)\n    : Permutation l l' → Permutation l' l"}, {"name": "permutation_app_head", "content": "theorem permutation_app_head (l tl tl' : List α)\n    : Permutation tl tl' → Permutation (l++tl) (l++tl')"}, {"name": "permutation_cons_append", "content": "theorem permutation_cons_append (l : List α) x\n    : Permutation (x :: l) (l ++ [x])"}], "local_ctx": "inductive Permutation {α : Type} : List α → List α → Prop :=\n  | perm_nil : Permutation [] []\n  | perm_skip x l l'\n      : Permutation l l' → Permutation (x::l) (x::l')\n  | perm_swap x y l : Permutation (y::x::l) (x::y::l)\n  | perm_trans l l' l''\n      : Permutation l l' → Permutation l' l''\n        → Permutation l l''\n\nopen Permutation", "target_theorem": "theorem permutation_cons_app (l l₁ l₂ : List α)\n    : Permutation l (l₁ ++ l₂)\n      → Permutation (a :: l) (l₁ ++ a :: l₂) :=", "ground_truth_proof": ":= by\n  intro h\n  -- a :: l\n  -- a :: (l₁ ++ l₂)\n  -- l₁ ++ l₂ ++ [a]\n  -- l₁ ++ (l₂ ++ [a])\n  -- l₁ ++ a :: l₂\n  apply perm_trans\n  . apply perm_skip; exact h\n  . apply perm_trans\n    . apply permutation_cons_append\n    . rw [List.append_assoc]\n      apply permutation_app_head\n      apply permutation_symm\n      apply permutation_cons_append", "nesting_depth": 2, "transitive_dep_count": 7, "subset_aristotle": false, "category": "Semantics"}
{"id": 286, "thm_name": "TM.soundness", "thm_stmt": "theorem soundness t t' T\n    : HasType t T → multistep t t' → ¬ stuck t'", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Types.lean", "imports": ["import Frap.SmallStep"], "used_lib_defs": [{"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax term \"==>\" term : term", "content": "syntax term \"==>\" term : term\n\nsyntax term \"~~>\" term : term\n\nsyntax term \"~~>*\" term : term\n\nsyntax:30 term \" =[ \" imp \" ]=> \" term : term\n\nsyntax term \"!->\" term \"; \" term : term\n\nsyntax:36 term \"<<->>\" term : term"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$t:term ==> $n:term) => `(Eval $t $n)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$t₁:term ~~> $t₂:term) => `(Step $t₁ $t₂)\n\nexample :\n    p\n      (p (c 1) (c 3))\n      (p (c 2) (c 4))\n    ~~>\n    p\n      (c 4)\n      (p (c 2) (c 4)) := by admit /- proof elided -/"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$t₁:term ~~>* $t₂:term) => `(Multi Step $t₁ $t₂)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$p <<->> $q) => `($p ->> $q ∧ $q ->> $p)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|{*$p*} $c {*$q*}) => `(valid_hoare_triple $p $c $q)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$st =[ $c ]=> $st') => `(CEval <{$c}> $st $st')"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$x:term !-> $a:term ; $st) => `(update $st $x $a)\n\nexample : empty =[\n      x := 2;\n      if (x <= 1) then y := 3 else z := 4 end\n    ]=> (z !-> 4; x !-> 2; empty) := by admit /- proof elided -/"}, {"name": "Tm", "content": "inductive Tm : Type :=\n  | tru : Tm\n  | fls : Tm\n  | ite : Tm → Tm → Tm → Tm\n  | zro : Tm\n  | scc : Tm → Tm\n  | prd : Tm → Tm\n  | iszero : Tm → Tm"}, {"name": "Step", "content": "inductive Step : Tm → Tm → Prop :=\n  | st_ifTrue t₁ t₂ : Step (ite tru t₁ t₂) t₁\n  | st_ifFalse t₁ t₂ : Step (ite fls t₁ t₂) t₂\n  | st_if c c' t₁ t₂ :\n      Step c c' → Step (ite c t₁ t₂) (ite c' t₁ t₂)\n  | st_succ t₁ t₁' : Step t₁ t₁' → Step (scc t₁) (scc t₁')\n  | st_pred0 : Step (prd zro) zro\n  | st_predSucc v : NValue v → Step (prd (scc v)) v\n  | st_pred t₁ t₁' : Step t₁ t₁' → Step (prd t₁) (prd t₁')\n  | st_iszero0 : Step (iszero zro) tru\n  | st_iszeroSucc v : NValue v → Step (iszero (scc v)) fls\n  | st_iszero t₁ t₁' :\n      Step t₁ t₁' → Step (iszero t₁) (iszero t₁')"}, {"name": "NValue", "content": "inductive NValue : Tm → Prop :=\n  | nv_0 : NValue zro\n  | nv_succ t : NValue t → NValue (scc t)"}, {"name": "Tm", "content": "inductive Tm : Type :=\n  | c : Nat → Tm       \n  | p : Tm → Tm → Tm   "}, {"name": "normal_form", "content": "def normal_form {X : Type} (R : relation X) (t : X) : Prop := ¬∃t', R t t'"}, {"name": "relation", "content": "def relation (X : Type) := X → X → Prop"}, {"name": "BValue", "content": "inductive BValue : Tm → Prop :=\n  | bv_true : BValue tru\n  | bv_false : BValue fls"}, {"name": "And", "content": "inductive And : Prop → Prop → Prop where\n  | intro : a → b → And a b"}, {"name": "Ty", "content": "inductive Ty : Type :=\n  | bool : Ty\n  | nat : Ty"}, {"name": "HasType", "content": "inductive HasType : Tm → Ty → Prop :=\n  | t_true : HasType tru bool\n  | t_false : HasType fls bool\n  | t_if t₁ t₂ t₃ T :\n      HasType t₁ bool → HasType t₂ T → HasType t₃ T\n      → HasType (ite t₁ t₂ t₃) T\n  | t_0 : HasType zro nat\n  | t_succ t₁ : HasType t₁ nat → HasType (scc t₁) nat\n  | t_pred t₁ : HasType t₁ nat → HasType (prd t₁) nat\n  | t_iszero t₁ : HasType t₁ nat → HasType (iszero t₁) bool"}, {"name": "Multi", "content": "inductive Multi {X : Type} (R : relation X) : relation X :=\n  | multi_refl x : Multi R x x\n  | multi_step x y z : R x y → Multi R y z → Multi R x z"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "TM.Tm", "content": "inductive Tm : Type :=\n  | tru : Tm\n  | fls : Tm\n  | ite : Tm → Tm → Tm → Tm\n  | zro : Tm\n  | scc : Tm → Tm\n  | prd : Tm → Tm\n  | iszero : Tm → Tm"}, {"name": "TM.BValue", "content": "inductive BValue : Tm → Prop :=\n  | bv_true : BValue tru\n  | bv_false : BValue fls"}, {"name": "TM.NValue", "content": "inductive NValue : Tm → Prop :=\n  | nv_0 : NValue zro\n  | nv_succ t : NValue t → NValue (scc t)"}, {"name": "TM.value", "content": "abbrev value (t : Tm) := BValue t ∨ NValue t"}, {"name": "TM.Step", "content": "inductive Step : Tm → Tm → Prop :=\n  | st_ifTrue t₁ t₂ : Step (ite tru t₁ t₂) t₁\n  | st_ifFalse t₁ t₂ : Step (ite fls t₁ t₂) t₂\n  | st_if c c' t₁ t₂ :\n      Step c c' → Step (ite c t₁ t₂) (ite c' t₁ t₂)\n  | st_succ t₁ t₁' : Step t₁ t₁' → Step (scc t₁) (scc t₁')\n  | st_pred0 : Step (prd zro) zro\n  | st_predSucc v : NValue v → Step (prd (scc v)) v\n  | st_pred t₁ t₁' : Step t₁ t₁' → Step (prd t₁) (prd t₁')\n  | st_iszero0 : Step (iszero zro) tru\n  | st_iszeroSucc v : NValue v → Step (iszero (scc v)) fls\n  | st_iszero t₁ t₁' :\n      Step t₁ t₁' → Step (iszero t₁) (iszero t₁')"}, {"name": "TM.step_normal_form", "content": "abbrev step_normal_form := normal_form Step"}, {"name": "TM.stuck", "content": "def stuck (t: Tm) : Prop :=\n  step_normal_form t ∧ ¬ value t\n\n/-\nexercise (2-star)\n-/\nexample : ∃ t, stuck t :="}, {"name": "TM.Ty", "content": "inductive Ty : Type :=\n  | bool : Ty\n  | nat : Ty"}, {"name": "TM.HasType", "content": "inductive HasType : Tm → Ty → Prop :=\n  | t_true : HasType tru bool\n  | t_false : HasType fls bool\n  | t_if t₁ t₂ t₃ T :\n      HasType t₁ bool → HasType t₂ T → HasType t₃ T\n      → HasType (ite t₁ t₂ t₃) T\n  | t_0 : HasType zro nat\n  | t_succ t₁ : HasType t₁ nat → HasType (scc t₁) nat\n  | t_pred t₁ : HasType t₁ nat → HasType (prd t₁) nat\n  | t_iszero t₁ : HasType t₁ nat → HasType (iszero t₁) bool"}, {"name": "TM.multistep", "content": "abbrev multistep := Multi Step"}], "used_local_lemmas": [{"name": "TM.bool_canonical", "content": "theorem bool_canonical t : HasType t bool → value t → BValue t"}, {"name": "TM.nat_canonical", "content": "theorem nat_canonical t : HasType t nat → value t → NValue t"}, {"name": "TM.progress", "content": "theorem progress t T\n    : HasType t T → value t ∨ ∃ t', Step t t'"}, {"name": "TM.preservation", "content": "theorem preservation t t' T\n    : HasType t T → Step t t' → HasType t' T"}], "local_ctx": "import Frap.SmallStep\n\nnamespace TM\n\ninductive Tm : Type :=\n  | tru : Tm\n  | fls : Tm\n  | ite : Tm → Tm → Tm → Tm\n  | zro : Tm\n  | scc : Tm → Tm\n  | prd : Tm → Tm\n  | iszero : Tm → Tm\n\nopen Tm\n\ninductive BValue : Tm → Prop :=\n  | bv_true : BValue tru\n  | bv_false : BValue fls\n\ninductive NValue : Tm → Prop :=\n  | nv_0 : NValue zro\n  | nv_succ t : NValue t → NValue (scc t)\n\nopen BValue\n\nopen NValue\n\nabbrev value (t : Tm) := BValue t ∨ NValue t\n\ninductive Step : Tm → Tm → Prop :=\n  | st_ifTrue t₁ t₂ : Step (ite tru t₁ t₂) t₁\n  | st_ifFalse t₁ t₂ : Step (ite fls t₁ t₂) t₂\n  | st_if c c' t₁ t₂ :\n      Step c c' → Step (ite c t₁ t₂) (ite c' t₁ t₂)\n  | st_succ t₁ t₁' : Step t₁ t₁' → Step (scc t₁) (scc t₁')\n  | st_pred0 : Step (prd zro) zro\n  | st_predSucc v : NValue v → Step (prd (scc v)) v\n  | st_pred t₁ t₁' : Step t₁ t₁' → Step (prd t₁) (prd t₁')\n  | st_iszero0 : Step (iszero zro) tru\n  | st_iszeroSucc v : NValue v → Step (iszero (scc v)) fls\n  | st_iszero t₁ t₁' :\n      Step t₁ t₁' → Step (iszero t₁) (iszero t₁')\n\nopen Step\n\nabbrev step_normal_form := normal_form Step\n\ndef stuck (t: Tm) : Prop :=\n  step_normal_form t ∧ ¬ value t\n\n/-\nexercise (2-star)\n-/\nexample : ∃ t, stuck t :=\n\ninductive Ty : Type :=\n  | bool : Ty\n  | nat : Ty\n\nopen Ty\n\ninductive HasType : Tm → Ty → Prop :=\n  | t_true : HasType tru bool\n  | t_false : HasType fls bool\n  | t_if t₁ t₂ t₃ T :\n      HasType t₁ bool → HasType t₂ T → HasType t₃ T\n      → HasType (ite t₁ t₂ t₃) T\n  | t_0 : HasType zro nat\n  | t_succ t₁ : HasType t₁ nat → HasType (scc t₁) nat\n  | t_pred t₁ : HasType t₁ nat → HasType (prd t₁) nat\n  | t_iszero t₁ : HasType t₁ nat → HasType (iszero t₁) bool\n\nopen HasType\n\nexample  -- `⊢ if false then 0 else (succ 0) ∈ nat`\n    : HasType (ite fls zro (scc zro)) nat := by\n  apply t_if\n  . apply t_false\n  . apply t_0\n  . apply t_succ\n    apply t_0\n\n/-\nIt's important to realize that the typing relation is a _conservative_ (or _static_) approximation: it does not consider what happens when the term is reduced.\nIn particular, it does not calculate the type of its normal form.\n-/\n\nexample  -- `⊢ if false then zero else true ∉ bool`\n    : ¬ HasType (ite fls zro tru) bool := by\n  intro contra\n  cases contra\n  rename_i contra _\n  cases contra\n\n/-\nexercise (1-star)\n-/\nexample t : HasType (scc t) nat → HasType t nat := by\n  intro ht\n  cases ht\n  assumption\n\nabbrev multistep := Multi Step", "target_theorem": "theorem soundness t t' T\n    : HasType t T → multistep t t' → ¬ stuck t' :=", "ground_truth_proof": ":= by\n  intro hT P\n  induction P with (intro contra; obtain ⟨R, S⟩ := contra)\n  | multi_refl =>\n    cases (progress _ _ hT)\n    . -- value\n      simp [*] at *\n    . -- step\n      simp [step_normal_form, normal_form, *] at *\n  | multi_step =>\n    rename_i ih; apply ih\n    . apply preservation\n      . apply hT\n      . assumption\n    . unfold stuck; constructor <;> assumption", "nesting_depth": 4, "transitive_dep_count": 26, "subset_aristotle": false, "category": "Semantics"}
{"id": 287, "thm_name": "CImp.par_body_n", "thm_stmt": "theorem par_body_n n st\n    : st x = 0 ∧ st y = 0\n      → ∃ st', Multi CStep (par_loop, st) (par_loop, st')\n          ∧ st' x = n ∧ st' y = 0", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/SmallStepImp.lean", "imports": ["import Frap.SmallStep", "import Frap.Equiv"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : im", "content": "syntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : imp"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|true) => `(Bool.true)\n  | `(term|false) => `(Bool.false)\n  | `(term|<{$x}>) => `(imp|$x)\n  | `(imp|$n:num) => `(a_num $n)\n  | `(imp|$s:str) => `(a_id $s)\n  | `(imp|$x + $y) => `(a_plus <{$x}> <{$y}>)\n  | `(imp|$x - $y) => `(a_minus <{$x}> <{$y}>)\n  | `(imp|$x * $y) => `(a_mult <{$x}> <{$y}>)\n  | `(imp|true) => `(b_true)\n  | `(imp|false) => `(b_false)\n  | `(imp|$x = $y) => `(b_eq <{$x}> <{$y}>)\n  | `(imp|$x != $y) => `(b_neq <{$x}> <{$y}>)\n  | `(imp|$x <= $y) => `(b_le <{$x}> <{$y}>)\n  | `(imp|!$x) => `(b_not <{$x}>)\n  | `(imp|$x && $y) => `(b_and <{$x}> <{$y}>)\n  | `(imp|$x || $y) => `(b_or <{$x}> <{$y}>)\n  | `(imp|($x)) => `(<{$x}>)\n  | `(imp|$x:ident) => `(a_id $(Lean.quote (toString x.getId)))\n  | `(imp|<[$t:term]>) => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(imp|skip) => `(c_skip)\n  | `(imp|$x:str := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$x:ident := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$c1 ; $c2) => `(c_seq <{$c1}> <{$c2}>)\n  | `(imp|if $b then $c1 else $c2 end) => `(c_if <{$b}> <{$c1}> <{$c2}>)\n  | `(imp|while $b do $c end) => `(c_while <{$b}> <{$c}>)"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_id : String → AExp  \n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "State", "content": "abbrev State := String → Nat"}, {"name": "BExp", "content": "inductive BExp where\n  | b_true : BExp\n  | b_false : BExp\n  | b_eq : AExp → AExp → BExp\n  | b_neq : AExp → AExp → BExp\n  | b_le : AExp → AExp → BExp\n  | b_not : BExp → BExp\n  | b_and : BExp → BExp → BExp\n  | b_or : BExp → BExp → BExp"}, {"name": "x", "content": "abbrev x := \"x\""}, {"name": "Com", "content": "inductive Com :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com"}, {"name": "empty", "content": "def empty : State := fun _ => 0"}, {"name": "Multi", "content": "inductive Multi {X : Type} (R : relation X) : relation X :=\n  | multi_refl x : Multi R x x\n  | multi_step x y z : R x y → Multi R y z → Multi R x z"}, {"name": "relation", "content": "def relation (X : Type) := X → X → Prop"}, {"name": "update", "content": "def update (st : State) (k : String) (v : Nat) : State :=\n  fun x => if x == k then v else st x"}, {"name": "y", "content": "abbrev y := \"y\""}, {"name": "And", "content": "inductive And : Prop → Prop → Prop where\n  | intro : a → b → And a b"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "AVal", "content": "inductive AVal : AExp → Prop :=\n  | av_num : ∀ n, AVal (a_num n)"}, {"name": "AStep", "content": "inductive AStep (st : State) : AExp → AExp → Prop :=\n  | as_id : ∀ v, AStep st (a_id v) (a_num (st v))\n  | as_plus1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_plus a₁ a₂) (a_plus a₁' a₂)\n  | as_plus2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_plus v₁ a₂) (a_plus v₁ a₂')\n  | as_plus : ∀ (v₁ v₂ : Nat),\n      AStep st (a_plus (a_num v₁) (a_num v₂)) (a_num (v₁ + v₂))\n  | as_minus1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_minus a₁ a₂) (a_minus a₁' a₂)\n  | as_minus2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_minus v₁ a₂) (a_minus v₁ a₂')\n  | as_minus : ∀ (v₁ v₂ : Nat),\n      AStep st (a_minus (a_num v₁) (a_num v₂)) (a_num (v₁ - v₂))\n  | as_mult1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_mult a₁ a₂) (a_mult a₁' a₂)\n  | as_mult2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_mult v₁ a₂) (a_mult v₁ a₂')\n  | as_mult : ∀ (v₁ v₂ : Nat),\n      AStep st (a_mult (a_num v₁) (a_num v₂)) (a_num (v₁ * v₂))"}, {"name": "BStep", "content": "inductive BStep (st : State) : BExp → BExp → Prop :=\n  | bs_eq1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_eq a₁ a₂) (b_eq a₁' a₂)\n  | bs_eq2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_eq v₁ a₂) (b_eq v₁ a₂')\n  | bs_eq : ∀ (v₁ v₂ : Nat),\n      BStep st (b_eq (a_num v₁) (a_num v₂))\n          (if v₁ == v₂ then b_true else b_false)\n  | bs_neq1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_neq a₁ a₂) (b_neq a₁' a₂)\n  | bs_neq2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_neq v₁ a₂) (b_neq v₁ a₂')\n  | bs_neq : ∀ (v₁ v₂ : Nat),\n      BStep st (b_neq (a_num v₁) (a_num v₂))\n          (if v₁ != v₂ then b_true else b_false)\n  | bs_le1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_le a₁ a₂) (b_le a₁' a₂)\n  | bs_le2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_le v₁ a₂) (b_le v₁ a₂')\n  | bs_le : ∀ (v₁ v₂ : Nat),\n      BStep st (b_le (a_num v₁) (a_num v₂))\n          (if v₁ <= v₂ then b_true else b_false)\n  | bs_notStep : ∀ b₁ b₁',\n      BStep st b₁ b₁'\n      → BStep st (b_not b₁) (b_not b₁')\n  | bs_notTrue : BStep st (b_not b_true) b_false\n  | bs_notFalse : BStep st (b_not b_false) b_true\n  | bs_andStep : ∀ b₁ b₁' b₂,\n      BStep st b₁ b₁'\n      → BStep st (b_and b₁ b₂) (b_and b₁' b₂)\n  | bs_andFalse : ∀ b₂,\n      BStep st (b_and b_false b₂) b_false\n  | bs_andTrueStep : ∀ b₂ b₂',\n      BStep st b₂ b₂'\n      → BStep st (b_and b_true b₂) (b_and b_true b₂')\n  | bs_andTrueTrue :\n      BStep st (b_and b_true b_true) b_true\n  | bs_andTrueFalse :\n      BStep st (b_and b_true b_false) b_false\n  | bs_orStep : ∀ b₁ b₁' b₂,\n      BStep st b₁ b₁'\n      → BStep st (b_or b₁ b₂) (b_or b₁' b₂)\n  | bs_orTrue : ∀ b₂,\n      BStep st (b_or b_true b₂) b_true\n  | bs_orFalseStep : ∀ b₂ b₂',\n      BStep st b₂ b₂'\n      → BStep st (b_or b_false b₂) (b_or b_false b₂')\n  | bs_orFalseTrue :\n      BStep st (b_or b_false b_true) b_true\n  | bs_orFalseFalse :\n      BStep st (b_or b_false b_false) b_false"}, {"name": "CStep", "content": "inductive CStep : (Com × State) → (Com × State) → Prop :=\n  | cs_asgnStep : ∀ st v a₁ a₁',\n      AStep st a₁ a₁'\n      → CStep (c_asgn v a₁, st) (c_asgn v a₁', st)\n  | cs_asgn : ∀ st v (n : Nat),\n      CStep (c_asgn v (a_num n), st) (c_skip, x !-> n; st)\n  | cs_seqStep : ∀ st c₁ c₁' st' c₂,\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_seq c₁ c₂, st) (c_seq c₁' c₂, st')\n  | cs_seqFinish : ∀ st c₂,\n      CStep (c_seq c_skip c₂, st) (c₂, st)\n  | cs_ifStep : ∀ st b₁ b₁' c₁ c₂,\n      BStep st b₁ b₁'\n      → CStep (c_if b₁ c₁ c₂, st) (c_if b₁' c₁ c₂, st)\n  | cs_ifTrue : ∀ st c₁ c₂,\n      CStep (c_if b_true c₁ c₂, st) (c₁, st)\n  | cs_ifFalse : ∀ st c₁ c₂,\n      CStep (c_if b_false c₁ c₂, st) (c₂, st)\n  | cs_while : ∀ st b₁ c₁,\n      CStep (c_while b₁ c₁, st)\n        (c_if b₁ (c_seq c₁ (c_while b₁ c₁)) c_skip, st)"}, {"name": "CImp.Com", "content": "inductive Com : Type :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com\n  | c_par : Com → Com → Com"}, {"name": "CImp.CStep", "content": "inductive CStep : (Com × State) → (Com × State) → Prop :=\n   \n  | cs_asgnStep : ∀ st v a₁ a₁',\n      AStep st a₁ a₁'\n      → CStep (c_asgn v a₁, st) (c_asgn v a₁', st)\n  | cs_asgn : ∀ st v (n : Nat),\n      CStep (c_asgn v (a_num n), st) (c_skip, v !-> n; st)\n  | cs_seqStep : ∀ st c₁ c₁' st' c₂,\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_seq c₁ c₂, st) (c_seq c₁' c₂, st')\n  | cs_seqFinish : ∀ st c₂,\n      CStep (c_seq c_skip c₂, st) (c₂, st)\n  | cs_ifStep : ∀ st b₁ b₁' c₁ c₂,\n      BStep st b₁ b₁'\n      → CStep (c_if b₁ c₁ c₂, st) (c_if b₁' c₁ c₂, st)\n  | cs_ifTrue : ∀ st c₁ c₂,\n      CStep (c_if b_true c₁ c₂, st) (c₁, st)\n  | cs_ifFalse : ∀ st c₁ c₂,\n      CStep (c_if b_false c₁ c₂, st) (c₂, st)\n  | cs_while : ∀ st b₁ c₁,\n      CStep (c_while b₁ c₁, st)\n        (c_if b₁ (c_seq c₁ (c_while b₁ c₁)) c_skip, st)\n   \n  | cs_par1 : ∀ st c₁ c₁' c₂ st',\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_par c₁ c₂, st) (c_par c₁' c₂, st')\n  | cs_par2 : ∀ st c₁ c₂ c₂' st',\n      CStep (c₂, st) (c₂', st')\n      → CStep (c_par c₁ c₂, st) (c_par c₁ c₂', st')\n  | cs_parDone : ∀ st,\n      CStep (c_par c_skip c_skip, st) (c_skip, st)"}, {"name": "CImp.par_loop", "content": "def par_loop : Com :=\n  c_par\n    (c_asgn y (a_num 1))\n    (c_while (b_eq (a_id y) (a_num 0))\n      (c_asgn x (a_plus (a_id x) (a_num 1))))\n\n/-\nIn particular, it can terminate with `x` set to `0`.\n-/\n\nexample : ∃ st',\n    Multi CStep (par_loop, empty) (c_skip, st')\n    ∧ st' x = 0 :="}], "used_local_lemmas": [{"name": "CImp.par_body_n__Sn", "content": "theorem par_body_n__Sn n st\n    : st x = n ∧ st y = 0\n      → Multi CStep (par_loop, st) (par_loop, x !-> n + 1; st)"}], "local_ctx": "import Frap.Equiv\n\nimport Frap.SmallStep\n\nopen Imp\n\nopen AExp\n\nopen BExp\n\nopen Com\n\nopen Multi\n\ninductive AVal : AExp → Prop :=\n  | av_num : ∀ n, AVal (a_num n)\n\ninductive AStep (st : State) : AExp → AExp → Prop :=\n  | as_id : ∀ v, AStep st (a_id v) (a_num (st v))\n  | as_plus1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_plus a₁ a₂) (a_plus a₁' a₂)\n  | as_plus2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_plus v₁ a₂) (a_plus v₁ a₂')\n  | as_plus : ∀ (v₁ v₂ : Nat),\n      AStep st (a_plus (a_num v₁) (a_num v₂)) (a_num (v₁ + v₂))\n  | as_minus1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_minus a₁ a₂) (a_minus a₁' a₂)\n  | as_minus2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_minus v₁ a₂) (a_minus v₁ a₂')\n  | as_minus : ∀ (v₁ v₂ : Nat),\n      AStep st (a_minus (a_num v₁) (a_num v₂)) (a_num (v₁ - v₂))\n  | as_mult1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → AStep st (a_mult a₁ a₂) (a_mult a₁' a₂)\n  | as_mult2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → AStep st (a_mult v₁ a₂) (a_mult v₁ a₂')\n  | as_mult : ∀ (v₁ v₂ : Nat),\n      AStep st (a_mult (a_num v₁) (a_num v₂)) (a_num (v₁ * v₂))\n\nopen AStep\n\ninductive BStep (st : State) : BExp → BExp → Prop :=\n  | bs_eq1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_eq a₁ a₂) (b_eq a₁' a₂)\n  | bs_eq2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_eq v₁ a₂) (b_eq v₁ a₂')\n  | bs_eq : ∀ (v₁ v₂ : Nat),\n      BStep st (b_eq (a_num v₁) (a_num v₂))\n          (if v₁ == v₂ then b_true else b_false)\n  | bs_neq1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_neq a₁ a₂) (b_neq a₁' a₂)\n  | bs_neq2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_neq v₁ a₂) (b_neq v₁ a₂')\n  | bs_neq : ∀ (v₁ v₂ : Nat),\n      BStep st (b_neq (a_num v₁) (a_num v₂))\n          (if v₁ != v₂ then b_true else b_false)\n  | bs_le1 : ∀ a₁ a₁' a₂,\n      AStep st a₁ a₁'\n      → BStep st (b_le a₁ a₂) (b_le a₁' a₂)\n  | bs_le2 : ∀ v₁ a₂ a₂',\n      AVal v₁ → AStep st a₂ a₂'\n      → BStep st (b_le v₁ a₂) (b_le v₁ a₂')\n  | bs_le : ∀ (v₁ v₂ : Nat),\n      BStep st (b_le (a_num v₁) (a_num v₂))\n          (if v₁ <= v₂ then b_true else b_false)\n  | bs_notStep : ∀ b₁ b₁',\n      BStep st b₁ b₁'\n      → BStep st (b_not b₁) (b_not b₁')\n  | bs_notTrue : BStep st (b_not b_true) b_false\n  | bs_notFalse : BStep st (b_not b_false) b_true\n  | bs_andStep : ∀ b₁ b₁' b₂,\n      BStep st b₁ b₁'\n      → BStep st (b_and b₁ b₂) (b_and b₁' b₂)\n  | bs_andFalse : ∀ b₂,\n      BStep st (b_and b_false b₂) b_false\n  | bs_andTrueStep : ∀ b₂ b₂',\n      BStep st b₂ b₂'\n      → BStep st (b_and b_true b₂) (b_and b_true b₂')\n  | bs_andTrueTrue :\n      BStep st (b_and b_true b_true) b_true\n  | bs_andTrueFalse :\n      BStep st (b_and b_true b_false) b_false\n  | bs_orStep : ∀ b₁ b₁' b₂,\n      BStep st b₁ b₁'\n      → BStep st (b_or b₁ b₂) (b_or b₁' b₂)\n  | bs_orTrue : ∀ b₂,\n      BStep st (b_or b_true b₂) b_true\n  | bs_orFalseStep : ∀ b₂ b₂',\n      BStep st b₂ b₂'\n      → BStep st (b_or b_false b₂) (b_or b_false b₂')\n  | bs_orFalseTrue :\n      BStep st (b_or b_false b_true) b_true\n  | bs_orFalseFalse :\n      BStep st (b_or b_false b_false) b_false\n\nopen BStep\n\ninductive CStep : (Com × State) → (Com × State) → Prop :=\n  | cs_asgnStep : ∀ st v a₁ a₁',\n      AStep st a₁ a₁'\n      → CStep (c_asgn v a₁, st) (c_asgn v a₁', st)\n  | cs_asgn : ∀ st v (n : Nat),\n      CStep (c_asgn v (a_num n), st) (c_skip, x !-> n; st)\n  | cs_seqStep : ∀ st c₁ c₁' st' c₂,\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_seq c₁ c₂, st) (c_seq c₁' c₂, st')\n  | cs_seqFinish : ∀ st c₂,\n      CStep (c_seq c_skip c₂, st) (c₂, st)\n  | cs_ifStep : ∀ st b₁ b₁' c₁ c₂,\n      BStep st b₁ b₁'\n      → CStep (c_if b₁ c₁ c₂, st) (c_if b₁' c₁ c₂, st)\n  | cs_ifTrue : ∀ st c₁ c₂,\n      CStep (c_if b_true c₁ c₂, st) (c₁, st)\n  | cs_ifFalse : ∀ st c₁ c₂,\n      CStep (c_if b_false c₁ c₂, st) (c₂, st)\n  | cs_while : ∀ st b₁ c₁,\n      CStep (c_while b₁ c₁, st)\n        (c_if b₁ (c_seq c₁ (c_while b₁ c₁)) c_skip, st)\n\nnamespace CImp\n\ninductive Com : Type :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com\n  | c_par : Com → Com → Com   \n\nopen Com\n\ninductive CStep : (Com × State) → (Com × State) → Prop :=\n   \n  | cs_asgnStep : ∀ st v a₁ a₁',\n      AStep st a₁ a₁'\n      → CStep (c_asgn v a₁, st) (c_asgn v a₁', st)\n  | cs_asgn : ∀ st v (n : Nat),\n      CStep (c_asgn v (a_num n), st) (c_skip, v !-> n; st)\n  | cs_seqStep : ∀ st c₁ c₁' st' c₂,\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_seq c₁ c₂, st) (c_seq c₁' c₂, st')\n  | cs_seqFinish : ∀ st c₂,\n      CStep (c_seq c_skip c₂, st) (c₂, st)\n  | cs_ifStep : ∀ st b₁ b₁' c₁ c₂,\n      BStep st b₁ b₁'\n      → CStep (c_if b₁ c₁ c₂, st) (c_if b₁' c₁ c₂, st)\n  | cs_ifTrue : ∀ st c₁ c₂,\n      CStep (c_if b_true c₁ c₂, st) (c₁, st)\n  | cs_ifFalse : ∀ st c₁ c₂,\n      CStep (c_if b_false c₁ c₂, st) (c₂, st)\n  | cs_while : ∀ st b₁ c₁,\n      CStep (c_while b₁ c₁, st)\n        (c_if b₁ (c_seq c₁ (c_while b₁ c₁)) c_skip, st)\n   \n  | cs_par1 : ∀ st c₁ c₁' c₂ st',\n      CStep (c₁, st) (c₁', st')\n      → CStep (c_par c₁ c₂, st) (c_par c₁' c₂, st')\n  | cs_par2 : ∀ st c₁ c₂ c₂' st',\n      CStep (c₂, st) (c₂', st')\n      → CStep (c_par c₁ c₂, st) (c_par c₁ c₂', st')\n  | cs_parDone : ∀ st,\n      CStep (c_par c_skip c_skip, st) (c_skip, st)\n\nopen CStep\n\ndef par_loop : Com :=\n  c_par\n    (c_asgn y (a_num 1))\n    (c_while (b_eq (a_id y) (a_num 0))\n      (c_asgn x (a_plus (a_id x) (a_num 1))))\n\n/-\nIn particular, it can terminate with `x` set to `0`.\n-/\n\nexample : ∃ st',\n    Multi CStep (par_loop, empty) (c_skip, st')\n    ∧ st' x = 0 :=", "target_theorem": "theorem par_body_n n st\n    : st x = 0 ∧ st y = 0\n      → ∃ st', Multi CStep (par_loop, st) (par_loop, st')\n          ∧ st' x = n ∧ st' y = 0 :=", "ground_truth_proof": ":= by\n  intro h\n  obtain ⟨hx, hy⟩ := h\n  induction n with\n  | zero =>\n      exists st\n      apply And.intro\n      . apply multi_refl\n      . apply And.intro\n        . apply hx\n        . apply hy\n  | succ n' hn' =>\n      obtain ⟨st', ⟨h', ⟨hx', hy'⟩⟩⟩ := hn'\n      constructor\n      apply And.intro\n      . apply multi_trans\n        . apply h'\n        . apply par_body_n__Sn\n          . apply And.intro\n            . exact hx'\n            . exact hy'\n      . apply And.intro\n        . rfl\n        . exact hy'", "nesting_depth": 3, "transitive_dep_count": 24, "subset_aristotle": false, "category": "Semantics"}
{"id": 288, "thm_name": "Imp.Hoare.hoare_if", "thm_stmt": "theorem hoare_if P Q b c₁ c₂ :\n    {* fun st => P st ∧ beval st b *} c₁ {* Q *}\n    → {* fun st => P st ∧ ¬(beval st b) *} c₂ {* Q *}\n    → {* P *} c_if b c₁ c₂ {* Q *}", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Hoare.lean", "imports": ["import Frap.Trans"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : im", "content": "syntax:21 \"if\" imp:20 \"then\" imp:20 \"else\" imp:20 \"end\" : imp\n\nsyntax \"false\" : imp\n\nsyntax \"false\" : term\n\nsyntax \"true\" : imp\n\nsyntax \"true\" : term\n\nsyntax \"<{\" imp \"}>\" : term\n\nsyntax:30 \"{*\" term \"*}\" term \"{*\" term \"*}\" : term\n\nsyntax \"<[\" term \"]>\" : imp"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|true) => `(Bool.true)\n  | `(term|false) => `(Bool.false)\n  | `(term|<{$x}>) => `(imp|$x)\n  | `(imp|$n:num) => `(a_num $n)\n  | `(imp|$s:str) => `(a_id $s)\n  | `(imp|$x + $y) => `(a_plus <{$x}> <{$y}>)\n  | `(imp|$x - $y) => `(a_minus <{$x}> <{$y}>)\n  | `(imp|$x * $y) => `(a_mult <{$x}> <{$y}>)\n  | `(imp|true) => `(b_true)\n  | `(imp|false) => `(b_false)\n  | `(imp|$x = $y) => `(b_eq <{$x}> <{$y}>)\n  | `(imp|$x != $y) => `(b_neq <{$x}> <{$y}>)\n  | `(imp|$x <= $y) => `(b_le <{$x}> <{$y}>)\n  | `(imp|!$x) => `(b_not <{$x}>)\n  | `(imp|$x && $y) => `(b_and <{$x}> <{$y}>)\n  | `(imp|$x || $y) => `(b_or <{$x}> <{$y}>)\n  | `(imp|($x)) => `(<{$x}>)\n  | `(imp|$x:ident) => `(a_id $(Lean.quote (toString x.getId)))\n  | `(imp|<[$t:term]>) => pure t"}, {"name": "macro_rules", "content": "macro_rules\n  | `(imp|skip) => `(c_skip)\n  | `(imp|$x:str := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$x:ident := $y) => `(c_asgn $x <{$y}>)\n  | `(imp|$c1 ; $c2) => `(c_seq <{$c1}> <{$c2}>)\n  | `(imp|if $b then $c1 else $c2 end) => `(c_if <{$b}> <{$c1}> <{$c2}>)\n  | `(imp|while $b do $c end) => `(c_while <{$b}> <{$c}>)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|$p <<->> $q) => `($p ->> $q ∧ $q ->> $p)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term|{*$p*} $c {*$q*}) => `(valid_hoare_triple $p $c $q)"}, {"name": "x", "content": "abbrev x := \"x\""}, {"name": "Com", "content": "inductive Com : Type :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com\n  | c_par : Com → Com → Com   "}, {"name": "beval", "content": "def beval (st : State) (b : BExp) : Bool :=\n  match b with\n  | b_true => true\n  | b_false => false\n  | b_eq a₁ a₂ => (aeval st a₁) == (aeval st a₂)\n  | b_neq a₁ a₂ => (aeval st a₁) != (aeval st a₂)\n  | b_le a₁ a₂ => (aeval st a₁) <= (aeval st a₂)\n  | b_not b₁ => not (beval st b₁)\n  | b_and b₁ b₂ => and (beval st b₁) (beval st b₂)\n  | b_or b₁ b₂ => or (beval st b₁) (beval st b₂)\n\nexample : aeval (update empty x 5)\n    <{3 + x * 2}>\n    \n    = 13 := by admit /- proof elided -/"}, {"name": "empty", "content": "def empty : State := fun _ => 0"}, {"name": "State", "content": "abbrev State := String → Nat"}, {"name": "y", "content": "abbrev y := \"y\""}, {"name": "z", "content": "abbrev z := \"z\""}, {"name": "aeval", "content": "def aeval (st : State) (a : AExp) : Nat :=\n  match a with\n  | a_num n => n\n  | a_id x => st x\n  | a_plus a₁ a₂ => (aeval st a₁) + (aeval st a₂)\n  | a_minus a₁ a₂ => (aeval st a₁) - (aeval st a₂)\n  | a_mult a₁ a₂ => (aeval st a₁) * (aeval st a₂)"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_id : String → AExp  \n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "AExp", "content": "inductive AExp where\n  | a_num : Nat → AExp\n  | a_plus : AExp → AExp → AExp\n  | a_minus : AExp → AExp → AExp\n  | a_mult : AExp → AExp → AExp"}, {"name": "update", "content": "def update (st : State) (k : String) (v : Nat) : State :=\n  fun x => if x == k then v else st x"}, {"name": "BExp", "content": "inductive BExp where\n  | b_true : BExp\n  | b_false : BExp\n  | b_eq : AExp → AExp → BExp\n  | b_neq : AExp → AExp → BExp\n  | b_le : AExp → AExp → BExp\n  | b_not : BExp → BExp\n  | b_and : BExp → BExp → BExp\n  | b_or : BExp → BExp → BExp"}, {"name": "Com", "content": "inductive Com :=\n  | c_skip : Com\n  | c_asgn : String → AExp → Com\n  | c_seq : Com → Com → Com\n  | c_if : BExp → Com → Com → Com\n  | c_while : BExp → Com → Com"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Imp.Hoare.Assertion", "content": "abbrev Assertion := State → Prop"}, {"name": "Imp.Hoare.assert_implies", "content": "def assert_implies (P Q : Assertion) : Prop :=\n  ∀ st, P st → Q st"}, {"name": "Imp.Hoare.valid_hoare_triple", "content": "def valid_hoare_triple (P : Assertion) (c : Com) (Q : Assertion) : Prop :=\n  ∀ st st', P st → (st =[<[c]>]=> st') → Q st'"}], "used_local_lemmas": [{"name": "Imp.Hoare.hoare_asgn", "content": "theorem hoare_asgn Q x a :\n    {* fun st => Q (st[x ↦ aeval st a]) *} c_asgn x a {* Q *}"}, {"name": "Imp.Hoare.hoare_consequence_pre", "content": "theorem hoare_consequence_pre P P' Q c :\n    {* P' *} c {* Q *}\n    → P ->> P'\n    → {* P *} c {* Q *}"}], "local_ctx": "import Frap.Trans\n\nnamespace Imp\n\nopen AExp\n\nopen BExp\n\nopen Com\n\nopen CEval\n\nnamespace Hoare\n\nabbrev Assertion := State → Prop\n\ndef assert_implies (P Q : Assertion) : Prop :=\n  ∀ st, P st → Q st\n\ninfix:36 \" ->> \" => assert_implies\n\ndef valid_hoare_triple (P : Assertion) (c : Com) (Q : Assertion) : Prop :=\n  ∀ st st', P st → (st =[<[c]>]=> st') → Q st'", "target_theorem": "theorem hoare_if P Q b c₁ c₂ :\n    {* fun st => P st ∧ beval st b *} c₁ {* Q *}\n    → {* fun st => P st ∧ ¬(beval st b) *} c₂ {* Q *}\n    → {* P *} c_if b c₁ c₂ {* Q *} :=", "ground_truth_proof": ":= by\n  intro hTrue hFalse st st' hPre hEval\n  cases hEval\n  . -- e_ifTrue\n    rename_i hb hc₁\n    apply hTrue\n    . constructor\n      . exact hPre\n      . exact hb\n    . exact hc₁\n  . -- e_ifFalse\n    rename_i hb hc₂\n    apply hFalse\n    . constructor\n      . exact hPre\n      . simp [hb]\n    . exact hc₂\n\nexample :\n    -- { True }\n    {* fun _ => True *}\n      <{if x = 0 then y := 2 else y := x + 1 end}>\n    -- { x ≤ y }\n    {* fun st => st x <= st y *} := by\n  apply hoare_if\n  . apply hoare_consequence_pre\n    . apply hoare_asgn\n    . intro st _\n      simp [update] at *\n      split <;> simp [*]\n  . apply hoare_consequence_pre\n    . apply hoare_asgn\n    . intro st _\n      simp [update]\n      split\n      . simp [*]\n      . unfold x\n        omega", "nesting_depth": 4, "transitive_dep_count": 21, "subset_aristotle": false, "category": "Semantics"}
{"id": 289, "thm_name": "Hidden.List.reverse_append", "thm_stmt": "theorem reverse_append {α : Type u} (as bs : List α)\n    : reverse (as ++ bs) = reverse bs ++ reverse as", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Exercises/IndProp.lean", "imports": [], "used_lib_defs": [{"name": "structure BitVec (w : Nat) where", "module": ""}, {"name": "/-- Construct a `BitVec w` from a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "ofFin ::", "module": ""}, {"name": "/-- Interpret a bitvector as a number less than `2^w`.", "module": ""}, {"name": "O(1), because we use `Fin` as the internal representation of a bitvector. -/", "module": ""}, {"name": "toFin : Fin (hPow 2 w)", "module": ""}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Hidden.List", "content": "inductive List (α : Type u) where\n  | nil  : List α\n  | cons : α → List α → List α"}, {"name": "Hidden.List.reverse", "content": "def reverse {α : Type u} (as : List α) : List α :=\n  match as with\n  | nil        => nil\n  | cons a as' => reverse as' ++ cons a nil"}], "used_local_lemmas": [{"name": "Hidden.List.nil_append", "content": "theorem nil_append (as : List α) : nil ++ as = as"}, {"name": "Hidden.List.cons_append", "content": "theorem cons_append (a : α) (as bs : List α)\n        : (cons a as) ++ bs = cons a (as ++ bs)"}, {"name": "Hidden.List.append_nil", "content": "theorem append_nil (as : List α) : as ++ nil = as"}, {"name": "Hidden.List.append_assoc", "content": "theorem append_assoc (as bs cs : List α)\n        : (as ++ bs) ++ cs = as ++ (bs ++ cs)"}], "local_ctx": "namespace Hidden\n\ninductive List (α : Type u) where\n  | nil  : List α\n  | cons : α → List α → List α\n\nnamespace List\n\ndef reverse {α : Type u} (as : List α) : List α :=\n  match as with\n  | nil        => nil\n  | cons a as' => reverse as' ++ cons a nil", "target_theorem": "theorem reverse_append {α : Type u} (as bs : List α)\n    : reverse (as ++ bs) = reverse bs ++ reverse as :=", "ground_truth_proof": ":= by\n  induction as with\n  | nil =>\n      rw [nil_append, reverse, append_nil]\n  | cons a as' ih =>\n      rw [cons_append, reverse, reverse, ih, append_assoc]", "nesting_depth": 2, "transitive_dep_count": 7, "subset_aristotle": false, "category": "Semantics"}
{"id": 290, "thm_name": "and_associative", "thm_stmt": "theorem and_associative (p q r : Prop) : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r)", "lean_root": "lean-formal-reasoning-program", "rel_path": "Frap/Propositional.lean", "imports": [], "used_lib_defs": [{"name": "Iff", "module": "Init.Core"}], "used_repo_defs": [{"name": "Iff", "content": "inductive Iff : Prop → Prop → Prop where\n  | intro : (a → b) → (b → a) → Iff a b"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "and_commutative", "content": "theorem and_commutative (p q : Prop) : p ∧ q ↔ q ∧ p"}], "local_ctx": "import Mathlib.Computability.NFA\nimport Mathlib.Data.FinEnum\nimport Mathlib.Data.Rel\nimport Mathlib.Data.Vector.Basic\nimport Blase.AutoStructs.ForLean\n\nopen Set\nopen Mathlib\nopen SetRel\n\n-- this is better because card is defeq to n\ninstance (α : Type) : Inter (Language α) := ⟨Set.inter⟩\ninstance (α : Type) : Union (Language α) := ⟨Set.union⟩\n\n/--\nThe set of `n`-tuples of bit vectors of an arbitrary width.\n-/\nstructure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n\n\ndef BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=\n  { w := bvs.w, bvs := h ▸ bvs.bvs }\n\nabbrev BitVecs.empty : BitVecs n := ⟨0, List.Vector.replicate n .nil⟩\nabbrev BitVecs.singleton {w : Nat} (bv : BitVec w) : BitVecs 1 := ⟨w, bv ::ᵥ .nil⟩\nabbrev BitVecs.pair {w : Nat} (bv1 bv2 : BitVec w) : BitVecs 2 := ⟨w, bv1 ::ᵥ bv2 ::ᵥ .nil⟩\n\n/--\nThe set of `n`-tuples of bit vectors of an arbitrary width, encoded as a list of\nbit vectors of width `n`. The width of the encoded bit vectors is the length of\nthe list.\n-/\nabbrev BitVecs' (n : Nat) := List (BitVec n)\n\n@[simps]\ndef dec (bvs' : BitVecs' n) : BitVecs n where\n  w := bvs'.length\n  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k\n\n@[simp]", "target_theorem": "theorem and_associative (p q r : Prop) : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) :=", "ground_truth_proof": ":= by\n  constructor\n  . intro h\n    cases h with\n    | intro hpq hr =>\n      cases hpq with\n      | intro hp hq =>\n        apply And.intro\n        . exact hp\n        . apply And.intro\n          . exact hq\n          . exact hr\n  . intro h\n    cases h with\n    | intro hp hqr =>\n      cases hqr with\n      | intro hq hr =>\n        apply And.intro\n        . apply And.intro\n          . exact hp\n          . exact hq\n        . exact hr\n\n/-\n## Rewriting proof terms using previously proven results\n\nOften in our proof, we'll need to use the same tactic sequence as in a prior proof.\nLike in software development, duplicating the sequence results in duplicate code and can lead to maintenance problems.\nWe can refer to an earlier propeosition by simply writing the name of the theorem or lemma of interest.\nThis is similar to breaking down a large program into functions.\n\nFirst, if we have a previously proven result, we can apply that result directly in the `apply` tactic.\n-/\nexample (p q : Prop) : p ∧ q ↔ q ∧ p := by\n  apply and_commutative\n\n/-\nWe can even refer to asserted but unproven propositions.\n-/\nexample (p q r : Prop) : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) := by\n  apply and_associative\n\n/-\nFor nested proof terms, we can use the `rewrite` tactic, along with a list of identities (equivalences) to apply to the proof goal or a hypothesis.\n\nThe syntax here is a bit complicated.\nWe need to pass a configuration stating which occurrence(s) of the proof term we want to rewrite.\nAn occurrence is a preorder position within the AST of the proof term, with 1 as the root (the entire proof term).\nYou might need to do some trial-and-error to get the desired occurrence.\nRefer to the pop-up contextual guide for more information.\n-/\nexample (p q r : Prop) : (p ∧ q) ∧ r ↔ (q ∧ p) ∧ r := by\n  apply Iff.intro\n  . intro h\n    rewrite (config := {occs := .pos [2]}) [and_commutative] at h\n    assumption\n  . intro h\n    rewrite (config := {occs := .pos [2]}) [and_commutative] at h\n    assumption", "nesting_depth": 2, "transitive_dep_count": 3, "subset_aristotle": false, "category": "Semantics"}
{"id": 291, "thm_name": "While.WellTyped.some_ty", "thm_stmt": "theorem WellTyped.some_ty {e ty} : e.ty = some ty → WellTyped e ty", "lean_root": "lean-hoare", "rel_path": "Hoare/While/Types.lean", "imports": ["import Hoare.While.Syntax"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.isSome", "module": "Init.Data.Option.Basic"}], "used_repo_defs": [{"name": "syntax num : nexpr", "content": "syntax num : nexpr\n\nsyntax \"if \" bexpr \" then \" com \" else \" com \" fi\" : com"}, {"name": "macro_rules", "content": "macro_rules\n| `([nexpr| $n:num]) => `(Expr.num $n)\n| `([nexpr| $x:ident]) => `(Expr.var $(Lean.quote x.getId.toString))\n| `([bexpr| true]) => `(Expr.bool «true»)\n| `([bexpr| false]) => `(Expr.bool «false»)\n| `([nexpr| $e1 + $e2]) => `(Expr.add [nexpr| $e1] [nexpr| $e2])\n| `([nexpr| $e1 - $e2]) => `(Expr.sub [nexpr| $e1] [nexpr| $e2])\n| `([nexpr| $e1 * $e2]) => `(Expr.mul [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr == $e2]) => `(Expr.eq [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr < $e2]) => `(Expr.lt [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr > $e2]) => `(Expr.gt [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr <= $e2]) => `(Expr.le [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr >= $e2]) => `(Expr.ge [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:bexpr && $e2]) => `(Expr.and [bexpr| $e1] [bexpr| $e2])\n| `([bexpr| $e1:bexpr || $e2]) => `(Expr.or [bexpr| $e1] [bexpr| $e2])\n| `([bexpr| ($e)]) => `([bexpr| $e])\n| `([nexpr| ($e)]) => `([nexpr| $e])\n| `([com| skip]) => `(Com.skip)\n| `([com| let $x:ident := $e]) => `(Com.assign $(Lean.quote x.getId.toString) [nexpr| $e])\n| `([com| $x:ident := $e]) => `(Com.assign $(Lean.quote x.getId.toString) [nexpr| $e])\n| `([com| $c1; $c2]) => `(Com.seq [com| $c1] [com| $c2])\n| `([com| if $e then $c1 else $c2 fi]) => `(Com.cond [bexpr| $e] [com| $c1] [com| $c2])\n| `([com| while $e do $c od]) => `(Com.while [bexpr| $e] [com| $c])"}, {"name": "macro_rules", "content": "macro_rules\n| `([bexpr| $($t:term)]) => `($t)\n| `([nexpr| $($t:term)]) => `($t)\n| `([com| $($t:term)]) => `($t)"}, {"name": "Expr.ty", "content": "def Expr.ty : Expr → Option Ty\n  | Expr.num _ => Ty.num\n  | Expr.bool _ => Ty.bool\n  | Expr.add e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.sub e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.mul e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.eq e1 e2 => if e1.ty = e2.ty ∧ e1.ty.isSome then some .bool else none\n  | Expr.lt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.gt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.le e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.ge e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.and e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.or e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.var _ => none "}, {"name": "Com", "content": "inductive Com\n| assign : String → Expr → Com\n| seq : Com → Com → Com\n| cond : Expr → Com → Com → Com\n| while : Expr → Com → Com\n| skip : Com"}, {"name": "Expr", "content": "inductive Expr\n| num : Nat → Expr\n| bool : Bool → Expr\n| var : String → Expr\n| add : Expr → Expr → Expr\n| sub : Expr → Expr → Expr\n| mul : Expr → Expr → Expr\n| eq : Expr → Expr → Expr\n| lt : Expr → Expr → Expr\n| gt : Expr → Expr → Expr\n| le : Expr → Expr → Expr\n| ge : Expr → Expr → Expr\n| and : Expr → Expr → Expr\n| or : Expr → Expr → Expr\nderiving Repr, DecidableEq"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "While.Ty", "content": "inductive Ty\n| num : Ty\n| bool : Ty\nderiving Repr, DecidableEq"}, {"name": "While.WellTyped", "content": "inductive WellTyped : Expr → Ty → Prop\n| num : WellTyped (Expr.num _) Ty.num\n| bool : WellTyped (Expr.bool _) Ty.bool\n| add : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.add e1 e2) Ty.num\n| sub : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.sub e1 e2) Ty.num\n| mul : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.mul e1 e2) Ty.num\n| eq : ∀ e1 e2 t, WellTyped e1 t → WellTyped e2 t →\n    WellTyped (Expr.eq e1 e2) Ty.bool\n| lt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.lt e1 e2) Ty.bool\n| gt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.gt e1 e2) Ty.bool\n| le : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.le e1 e2) Ty.bool\n| ge : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.ge e1 e2) Ty.bool\n| and : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.and e1 e2) Ty.bool\n| or : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.or e1 e2) Ty.bool"}, {"name": "While.Expr.ty", "content": "def Expr.ty : Expr → Option Ty\n  | Expr.num _ => Ty.num\n  | Expr.bool _ => Ty.bool\n  | Expr.add e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.sub e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.mul e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.eq e1 e2 => if e1.ty = e2.ty ∧ e1.ty.isSome then some .bool else none\n  | Expr.lt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.gt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.le e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.ge e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.and e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.or e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.var _ => none"}], "used_local_lemmas": [{"name": "While.WellTyped.ty_some", "content": "theorem WellTyped.ty_some {e ty} : WellTyped e ty → e.ty = some ty"}], "local_ctx": "import Hoare.While.Syntax\n\nnamespace While\n\ninductive Ty\n| num : Ty\n| bool : Ty\nderiving Repr, DecidableEq\n\ninductive WellTyped : Expr → Ty → Prop\n| num : WellTyped (Expr.num _) Ty.num\n| bool : WellTyped (Expr.bool _) Ty.bool\n| add : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.add e1 e2) Ty.num\n| sub : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.sub e1 e2) Ty.num\n| mul : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.mul e1 e2) Ty.num\n| eq : ∀ e1 e2 t, WellTyped e1 t → WellTyped e2 t →\n    WellTyped (Expr.eq e1 e2) Ty.bool\n| lt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.lt e1 e2) Ty.bool\n| gt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.gt e1 e2) Ty.bool\n| le : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.le e1 e2) Ty.bool\n| ge : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.ge e1 e2) Ty.bool\n| and : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.and e1 e2) Ty.bool\n| or : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.or e1 e2) Ty.bool\n\ndef Expr.ty : Expr → Option Ty\n  | Expr.num _ => Ty.num\n  | Expr.bool _ => Ty.bool\n  | Expr.add e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.sub e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.mul e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.eq e1 e2 => if e1.ty = e2.ty ∧ e1.ty.isSome then some .bool else none\n  | Expr.lt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.gt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.le e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.ge e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.and e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.or e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.var _ => none", "target_theorem": "theorem WellTyped.some_ty {e ty} : e.ty = some ty → WellTyped e ty :=", "ground_truth_proof": ":= by\n  intro h\n  induction e generalizing ty <;> simp_all [Expr.ty]\n  case num _ => rw [← h]; exact WellTyped.num\n  case bool _ => rw [← h]; exact WellTyped.bool\n  case add e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'; exact WellTyped.add e1 e2 ih_e1 ih_e2\n  case sub e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'; exact WellTyped.sub e1 e2 ih_e1 ih_e2\n  case mul e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'; exact WellTyped.mul e1 e2 ih_e1 ih_e2\n  case eq e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'\n    match hty : e1.ty with\n    | some .num => have hty' := h.1.1.symm; rw [hty] at hty'; exact WellTyped.eq e1 e2 .num (ih_e1 hty') (ih_e2 hty')\n    | some .bool => have hty' := h.1.1.symm; rw [hty] at hty'; exact WellTyped.eq e1 e2 .bool (ih_e1 hty') (ih_e2 hty')\n    | none => rw [hty] at h; have hc := h.1.2; simp [Option.isSome] at hc\n  case lt e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'; exact WellTyped.lt e1 e2 ih_e1 ih_e2\n  case gt e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'; exact WellTyped.gt e1 e2 ih_e1 ih_e2\n  case le e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'; exact WellTyped.le e1 e2 ih_e1 ih_e2\n  case ge e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'; exact WellTyped.ge e1 e2 ih_e1 ih_e2\n  case and e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'; exact WellTyped.and e1 e2 ih_e1 ih_e2\n  case or e1 e2 ih_e1 ih_e2 =>\n    have h' := h.2.symm; subst h'; exact WellTyped.or e1 e2 ih_e1 ih_e2", "nesting_depth": 3, "transitive_dep_count": 13, "subset_aristotle": false, "category": "Semantics"}
{"id": 292, "thm_name": "While.WellTyped.ty_some", "thm_stmt": "theorem WellTyped.ty_some {e ty} : WellTyped e ty → e.ty = some ty", "lean_root": "lean-hoare", "rel_path": "Hoare/While/Types.lean", "imports": ["import Hoare.While.Syntax"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.isSome", "module": "Init.Data.Option.Basic"}], "used_repo_defs": [{"name": "syntax num : nexpr", "content": "syntax num : nexpr\n\nsyntax \"if \" bexpr \" then \" com \" else \" com \" fi\" : com"}, {"name": "macro_rules", "content": "macro_rules\n| `([nexpr| $n:num]) => `(Expr.num $n)\n| `([nexpr| $x:ident]) => `(Expr.var $(Lean.quote x.getId.toString))\n| `([bexpr| true]) => `(Expr.bool «true»)\n| `([bexpr| false]) => `(Expr.bool «false»)\n| `([nexpr| $e1 + $e2]) => `(Expr.add [nexpr| $e1] [nexpr| $e2])\n| `([nexpr| $e1 - $e2]) => `(Expr.sub [nexpr| $e1] [nexpr| $e2])\n| `([nexpr| $e1 * $e2]) => `(Expr.mul [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr == $e2]) => `(Expr.eq [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr < $e2]) => `(Expr.lt [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr > $e2]) => `(Expr.gt [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr <= $e2]) => `(Expr.le [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:nexpr >= $e2]) => `(Expr.ge [nexpr| $e1] [nexpr| $e2])\n| `([bexpr| $e1:bexpr && $e2]) => `(Expr.and [bexpr| $e1] [bexpr| $e2])\n| `([bexpr| $e1:bexpr || $e2]) => `(Expr.or [bexpr| $e1] [bexpr| $e2])\n| `([bexpr| ($e)]) => `([bexpr| $e])\n| `([nexpr| ($e)]) => `([nexpr| $e])\n| `([com| skip]) => `(Com.skip)\n| `([com| let $x:ident := $e]) => `(Com.assign $(Lean.quote x.getId.toString) [nexpr| $e])\n| `([com| $x:ident := $e]) => `(Com.assign $(Lean.quote x.getId.toString) [nexpr| $e])\n| `([com| $c1; $c2]) => `(Com.seq [com| $c1] [com| $c2])\n| `([com| if $e then $c1 else $c2 fi]) => `(Com.cond [bexpr| $e] [com| $c1] [com| $c2])\n| `([com| while $e do $c od]) => `(Com.while [bexpr| $e] [com| $c])"}, {"name": "macro_rules", "content": "macro_rules\n| `([bexpr| $($t:term)]) => `($t)\n| `([nexpr| $($t:term)]) => `($t)\n| `([com| $($t:term)]) => `($t)"}, {"name": "Expr.ty", "content": "def Expr.ty : Expr → Option Ty\n  | Expr.num _ => Ty.num\n  | Expr.bool _ => Ty.bool\n  | Expr.add e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.sub e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.mul e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.eq e1 e2 => if e1.ty = e2.ty ∧ e1.ty.isSome then some .bool else none\n  | Expr.lt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.gt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.le e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.ge e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.and e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.or e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.var _ => none "}, {"name": "Com", "content": "inductive Com\n| assign : String → Expr → Com\n| seq : Com → Com → Com\n| cond : Expr → Com → Com → Com\n| while : Expr → Com → Com\n| skip : Com"}, {"name": "Expr", "content": "inductive Expr\n| num : Nat → Expr\n| bool : Bool → Expr\n| var : String → Expr\n| add : Expr → Expr → Expr\n| sub : Expr → Expr → Expr\n| mul : Expr → Expr → Expr\n| eq : Expr → Expr → Expr\n| lt : Expr → Expr → Expr\n| gt : Expr → Expr → Expr\n| le : Expr → Expr → Expr\n| ge : Expr → Expr → Expr\n| and : Expr → Expr → Expr\n| or : Expr → Expr → Expr\nderiving Repr, DecidableEq"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "While.Ty", "content": "inductive Ty\n| num : Ty\n| bool : Ty\nderiving Repr, DecidableEq"}, {"name": "While.WellTyped", "content": "inductive WellTyped : Expr → Ty → Prop\n| num : WellTyped (Expr.num _) Ty.num\n| bool : WellTyped (Expr.bool _) Ty.bool\n| add : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.add e1 e2) Ty.num\n| sub : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.sub e1 e2) Ty.num\n| mul : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.mul e1 e2) Ty.num\n| eq : ∀ e1 e2 t, WellTyped e1 t → WellTyped e2 t →\n    WellTyped (Expr.eq e1 e2) Ty.bool\n| lt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.lt e1 e2) Ty.bool\n| gt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.gt e1 e2) Ty.bool\n| le : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.le e1 e2) Ty.bool\n| ge : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.ge e1 e2) Ty.bool\n| and : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.and e1 e2) Ty.bool\n| or : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.or e1 e2) Ty.bool"}, {"name": "While.Expr.ty", "content": "def Expr.ty : Expr → Option Ty\n  | Expr.num _ => Ty.num\n  | Expr.bool _ => Ty.bool\n  | Expr.add e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.sub e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.mul e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.eq e1 e2 => if e1.ty = e2.ty ∧ e1.ty.isSome then some .bool else none\n  | Expr.lt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.gt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.le e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.ge e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.and e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.or e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.var _ => none"}], "used_local_lemmas": [], "local_ctx": "import Hoare.While.Syntax\n\nnamespace While\n\ninductive Ty\n| num : Ty\n| bool : Ty\nderiving Repr, DecidableEq\n\ninductive WellTyped : Expr → Ty → Prop\n| num : WellTyped (Expr.num _) Ty.num\n| bool : WellTyped (Expr.bool _) Ty.bool\n| add : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.add e1 e2) Ty.num\n| sub : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.sub e1 e2) Ty.num\n| mul : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.mul e1 e2) Ty.num\n| eq : ∀ e1 e2 t, WellTyped e1 t → WellTyped e2 t →\n    WellTyped (Expr.eq e1 e2) Ty.bool\n| lt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.lt e1 e2) Ty.bool\n| gt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.gt e1 e2) Ty.bool\n| le : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.le e1 e2) Ty.bool\n| ge : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.ge e1 e2) Ty.bool\n| and : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.and e1 e2) Ty.bool\n| or : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.or e1 e2) Ty.bool\n\ndef Expr.ty : Expr → Option Ty\n  | Expr.num _ => Ty.num\n  | Expr.bool _ => Ty.bool\n  | Expr.add e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.sub e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.mul e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .num else none\n  | Expr.eq e1 e2 => if e1.ty = e2.ty ∧ e1.ty.isSome then some .bool else none\n  | Expr.lt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.gt e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.le e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.ge e1 e2 => if e1.ty = some .num ∧ e2.ty = some .num then some .bool else none\n  | Expr.and e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.or e1 e2 => if e1.ty = some .bool ∧ e2.ty = some .bool then some .bool else none\n  | Expr.var _ => none", "target_theorem": "theorem WellTyped.ty_some {e ty} : WellTyped e ty → e.ty = some ty :=", "ground_truth_proof": ":=\n  by\n    intro h\n    induction h\n    case num => rfl\n    case bool => rfl\n    case add _ _ h1 h2 => simp [Expr.ty, h1, h2]\n    case sub _ _ h1 h2 => simp [Expr.ty, h1, h2]\n    case mul _ _ h1 h2 => simp [Expr.ty, h1, h2]\n    case eq e1 _ _ h1 h2 => simp [Expr.ty, h1, h2]\n    case lt _ _ h1 h2 => simp [Expr.ty, h1, h2]\n    case gt _ _ h1 h2 => simp [Expr.ty, h1, h2]\n    case le _ _ h1 h2 => simp [Expr.ty, h1, h2]\n    case ge _ _ h1 h2 => simp [Expr.ty, h1, h2]\n    case and _ _ h1 h2 => simp [Expr.ty, h1, h2]\n    case or _ _ h1 h2 => simp [Expr.ty, h1, h2]", "nesting_depth": 4, "transitive_dep_count": 13, "subset_aristotle": false, "category": "Semantics"}
{"id": 293, "thm_name": "While.WellTyped.not_eq_not_eq_ty", "thm_stmt": "theorem WellTyped.not_eq_not_eq_ty {e1 e2 : Expr} {t1 t2 : Ty} :\n  WellTyped e1 t1 → WellTyped e2 t2 → t1 ≠ t2 → ∀ t, ¬ (WellTyped (Expr.eq e1 e2) t)", "lean_root": "lean-hoare", "rel_path": "Hoare/While/Types.lean", "imports": ["import Hoare.While.Syntax"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "bool", "module": "Init.Control.Basic"}], "used_repo_defs": [{"name": "Expr", "content": "inductive Expr\n| num : Nat → Expr\n| bool : Bool → Expr\n| var : String → Expr\n| add : Expr → Expr → Expr\n| sub : Expr → Expr → Expr\n| mul : Expr → Expr → Expr\n| eq : Expr → Expr → Expr\n| lt : Expr → Expr → Expr\n| gt : Expr → Expr → Expr\n| le : Expr → Expr → Expr\n| ge : Expr → Expr → Expr\n| and : Expr → Expr → Expr\n| or : Expr → Expr → Expr\nderiving Repr, DecidableEq"}, {"name": "WellTyped", "content": "inductive WellTyped : Expr → Ty → Prop\n| num : WellTyped (Expr.num _) Ty.num\n| bool : WellTyped (Expr.bool _) Ty.bool\n| add : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.add e1 e2) Ty.num\n| sub : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.sub e1 e2) Ty.num\n| mul : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.mul e1 e2) Ty.num\n| eq : ∀ e1 e2 t, WellTyped e1 t → WellTyped e2 t →\n    WellTyped (Expr.eq e1 e2) Ty.bool\n| lt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.lt e1 e2) Ty.bool\n| gt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.gt e1 e2) Ty.bool\n| le : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.le e1 e2) Ty.bool\n| ge : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.ge e1 e2) Ty.bool\n| and : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.and e1 e2) Ty.bool\n| or : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.or e1 e2) Ty.bool"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "While.Ty", "content": "inductive Ty\n| num : Ty\n| bool : Ty\nderiving Repr, DecidableEq"}, {"name": "While.WellTyped", "content": "inductive WellTyped : Expr → Ty → Prop\n| num : WellTyped (Expr.num _) Ty.num\n| bool : WellTyped (Expr.bool _) Ty.bool\n| add : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.add e1 e2) Ty.num\n| sub : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.sub e1 e2) Ty.num\n| mul : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.mul e1 e2) Ty.num\n| eq : ∀ e1 e2 t, WellTyped e1 t → WellTyped e2 t →\n    WellTyped (Expr.eq e1 e2) Ty.bool\n| lt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.lt e1 e2) Ty.bool\n| gt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.gt e1 e2) Ty.bool\n| le : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.le e1 e2) Ty.bool\n| ge : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.ge e1 e2) Ty.bool\n| and : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.and e1 e2) Ty.bool\n| or : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.or e1 e2) Ty.bool"}], "used_local_lemmas": [{"name": "While.WellTyped.unique", "content": "theorem WellTyped.unique : ∀ {e t1 t2}, WellTyped e t1 → WellTyped e t2 → t1 = t2"}], "local_ctx": "import Hoare.While.Syntax\n\nnamespace While\n\ninductive Ty\n| num : Ty\n| bool : Ty\nderiving Repr, DecidableEq", "target_theorem": "theorem WellTyped.not_eq_not_eq_ty {e1 e2 : Expr} {t1 t2 : Ty} :\n  WellTyped e1 t1 → WellTyped e2 t2 → t1 ≠ t2 → ∀ t, ¬ (WellTyped (Expr.eq e1 e2) t) :=", "ground_truth_proof": ":= by\n  intro h1 h2 h3 t h4\n  cases h4\n  · case eq t h1' h2' =>\n      have ht1 : t1 = t := WellTyped.unique h1 h1'\n      have ht2 : t2 = t := WellTyped.unique h2 h2'\n      rw [ht1, ht2] at h3\n      contradiction", "nesting_depth": 2, "transitive_dep_count": 6, "subset_aristotle": false, "category": "Semantics"}
{"id": 294, "thm_name": "While.WellTyped.not_welltyped_not_eq_ty", "thm_stmt": "theorem WellTyped.not_welltyped_not_eq_ty {e1 e2 : Expr} {t : Ty} :\n  WellTyped e1 t → ¬ WellTyped e2 t → ∀ t', ¬ (WellTyped (Expr.eq e1 e2) t')", "lean_root": "lean-hoare", "rel_path": "Hoare/While/Types.lean", "imports": ["import Hoare.While.Syntax"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "bool", "module": "Init.Control.Basic"}], "used_repo_defs": [{"name": "Expr", "content": "inductive Expr\n| num : Nat → Expr\n| bool : Bool → Expr\n| var : String → Expr\n| add : Expr → Expr → Expr\n| sub : Expr → Expr → Expr\n| mul : Expr → Expr → Expr\n| eq : Expr → Expr → Expr\n| lt : Expr → Expr → Expr\n| gt : Expr → Expr → Expr\n| le : Expr → Expr → Expr\n| ge : Expr → Expr → Expr\n| and : Expr → Expr → Expr\n| or : Expr → Expr → Expr\nderiving Repr, DecidableEq"}, {"name": "WellTyped", "content": "inductive WellTyped : Expr → Ty → Prop\n| num : WellTyped (Expr.num _) Ty.num\n| bool : WellTyped (Expr.bool _) Ty.bool\n| add : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.add e1 e2) Ty.num\n| sub : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.sub e1 e2) Ty.num\n| mul : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.mul e1 e2) Ty.num\n| eq : ∀ e1 e2 t, WellTyped e1 t → WellTyped e2 t →\n    WellTyped (Expr.eq e1 e2) Ty.bool\n| lt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.lt e1 e2) Ty.bool\n| gt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.gt e1 e2) Ty.bool\n| le : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.le e1 e2) Ty.bool\n| ge : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.ge e1 e2) Ty.bool\n| and : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.and e1 e2) Ty.bool\n| or : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.or e1 e2) Ty.bool"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "While.Ty", "content": "inductive Ty\n| num : Ty\n| bool : Ty\nderiving Repr, DecidableEq"}, {"name": "While.WellTyped", "content": "inductive WellTyped : Expr → Ty → Prop\n| num : WellTyped (Expr.num _) Ty.num\n| bool : WellTyped (Expr.bool _) Ty.bool\n| add : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.add e1 e2) Ty.num\n| sub : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.sub e1 e2) Ty.num\n| mul : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.mul e1 e2) Ty.num\n| eq : ∀ e1 e2 t, WellTyped e1 t → WellTyped e2 t →\n    WellTyped (Expr.eq e1 e2) Ty.bool\n| lt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.lt e1 e2) Ty.bool\n| gt : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.gt e1 e2) Ty.bool\n| le : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.le e1 e2) Ty.bool\n| ge : ∀ e1 e2, WellTyped e1 Ty.num → WellTyped e2 Ty.num →\n    WellTyped (Expr.ge e1 e2) Ty.bool\n| and : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.and e1 e2) Ty.bool\n| or : ∀ e1 e2, WellTyped e1 Ty.bool → WellTyped e2 Ty.bool →\n    WellTyped (Expr.or e1 e2) Ty.bool"}], "used_local_lemmas": [{"name": "While.WellTyped.unique", "content": "theorem WellTyped.unique : ∀ {e t1 t2}, WellTyped e t1 → WellTyped e t2 → t1 = t2"}], "local_ctx": "import Hoare.While.Syntax\n\nnamespace While\n\ninductive Ty\n| num : Ty\n| bool : Ty\nderiving Repr, DecidableEq", "target_theorem": "theorem WellTyped.not_welltyped_not_eq_ty {e1 e2 : Expr} {t : Ty} :\n  WellTyped e1 t → ¬ WellTyped e2 t → ∀ t', ¬ (WellTyped (Expr.eq e1 e2) t') :=", "ground_truth_proof": ":= by\n  intro h1 h2 t' h3\n  cases h3\n  · case eq t' h1' h2' =>\n      have ht : t = t' := WellTyped.unique h1 h1'\n      apply h2\n      rw [ht]\n      exact h2'", "nesting_depth": 2, "transitive_dep_count": 6, "subset_aristotle": false, "category": "Semantics"}
{"id": 295, "thm_name": "processOneElem_spec", "thm_stmt": "omit [Fintype S] in\nlemma processOneElem_spec {st : worklist.St A S} (s : State) (sa : S) (k : ℕ) :\n    ∀ a sa', (f sa)[k]? = some (a, sa') →\n    processOneElem_mot inits final f s sa k st →\n    processOneElem_mot inits final f s sa (k+1) (processOneElem A S final s st (a, sa'))", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/Worklist.lean", "imports": ["import Blase.Blase.AutoStructs.Basic", "import Blase.Blase.AutoStructs.ForLean", "import Blase.AutoStructs.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "SetRel", "module": "Mathlib.Data.Rel"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "SetRel.inv", "module": "Mathlib.Data.Rel"}, {"name": "LawfulBEq", "module": "Init.Core"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}, {"name": "LawfulHashable", "module": "Init.Data.LawfulHashable"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "List.insert", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "State", "content": "abbrev State := Nat"}, {"name": "RawCNFA.WF", "content": "structure RawCNFA.WF (m : RawCNFA A) where\n  initials_lt : ∀ {s}, s ∈ m.initials → s ∈ m.states\n  finals_lt : ∀ {s}, s ∈ m.finals → s ∈ m.states\n  trans_src_lt : ∀ s_a ∈ m.trans, s_a.1 ∈ m.states\n  trans_tgt_lt : s' ∈ m.tr s a → s' ∈ m.states"}, {"name": "RawCNFA.tr", "content": "@[inline]\ndef RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅"}, {"name": "RawCNFA.states", "content": "def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax"}, {"name": "RawCNFA.addFinal", "content": "def RawCNFA.addFinal (m : RawCNFA A) (s : State) : RawCNFA A :=\n  { m with finals := m.finals.insert s }"}, {"name": "RawCNFA.newState", "content": "def RawCNFA.newState (m : RawCNFA A) : State × RawCNFA A :=\n  let old := m.stateMax\n  let m := { m with stateMax := old + 1 }\n  (old, m)"}, {"name": "RawCNFA.addTrans", "content": "def RawCNFA.addTrans (m : RawCNFA A) (a : A) (s s' : State) : RawCNFA A :=\n  let ns := m.trans.getD (s, a) ∅\n  let ns := ns.insert s'\n  { m with trans :=  m.trans.insert (s, a) ns }"}], "lib_lemmas": [{"name": "Std.HashMap.getElem?_insert", "module": "Std.Data.HashMap.Lemmas"}, {"name": "getElem?_eq_none_iff", "module": "Init.GetElem"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "Array.mem_of_getElem?", "module": "Init.Data.Array.Lemmas"}, {"name": "Prod.mk.eta", "module": "Mathlib.Data.Prod.Basic"}, {"name": "Std.HashSet.mem_insert", "module": "Std.Data.HashSet.Lemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "not_and", "module": "Init.SimpLemmas"}, {"name": "not_exists", "module": "Init.PropLemmas"}, {"name": "or_true", "module": "Init.SimpLemmas"}, {"name": "true_or", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "Std.HashMap.get?_none_not_mem", "content": "@[aesop 50% unsafe]\ntheorem Std.HashMap.get?_none_not_mem [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] {m : Std.HashMap K V} {k : K} : m.get? k = none → k ∉ m"}, {"name": "Std.HashMap.mem_of_getElem?", "content": "@[aesop 50% unsafe]\ntheorem Std.HashMap.mem_of_getElem? [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] {m : Std.HashMap K V} {k : K} :\n    m[k]? = some v → k ∈ m"}, {"name": "Std.HashMap.mem_iff_getElem?", "content": "theorem Std.HashMap.mem_iff_getElem? [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] [Inhabited V] {m : Std.HashMap K V} {k : K} :\n     k ∈ m ↔ ∃ v, m[k]? = some v"}, {"name": "addTrans_tr_eq", "content": "@[grind =, simp]\nlemma addTrans_tr_eq (m : RawCNFA A) [LawfulBEq A] (a : A) (s₁ s₂ : State) :\n    (m.addTrans a s₁ s₂).tr s₁ a = (m.tr s₁ a).insert s₂"}, {"name": "@[grind =] -- TODO: should I?", "content": "@[grind =] -- TODO: should I?\nlemma addTrans_tr (m : RawCNFA A) [LawfulBEq A] {a b : A} {s₁ s₁' s₂ : State} :\n    (m.addTrans a s₁ s₂).tr s₁' b = if s₁ = s₁' ∧ a = b then (m.tr s₁ a).insert s₂ else m.tr s₁' b"}, {"name": "RawCNFA.WF.trans_src_lt''", "content": "@[grind ., simp, aesop 50% unsafe]\nlemma RawCNFA.WF.trans_src_lt'' [LawfulBEq A] {m : RawCNFA A} (hwf : m.WF) :\n    ∀ {s a s'}, s' ∈ m.tr s a → s ∈ m.states"}, {"name": "RawCNFA.WF.trans_src_lt'", "content": "@[grind ., simp, aesop 50% unsafe]\nlemma RawCNFA.WF.trans_src_lt' {m : RawCNFA A} (hwf : m.WF) :\n    ∀ {s a}, (s, a) ∈ m.trans → s ∈ m.states"}, {"name": "Std.HashMap.getElem?_none_not_mem", "content": "@[aesop 50% unsafe]\ntheorem Std.HashMap.getElem?_none_not_mem [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] {m : Std.HashMap K V} {k : K} :\n    m[k]? = none → k ∉ m"}], "used_local_defs": [{"name": "nfa", "content": "def nfa : NFA A S where\n  start := { sa | sa ∈ inits }\n  accept := { sa | final sa }\n  step sa a := { sa' | (a, sa') ∈ f sa }"}, {"name": "worklist.St", "content": "structure worklist.St where\n  m : RawCNFA A\n  map : Std.HashMap S State := ∅\n  worklist : Array S := ∅\n  worklist_nodup : worklist.toList.Nodup\n  worklist_incl : ∀ sa ∈ worklist, sa ∈ map"}, {"name": "worklist.St.addOrCreateState", "content": "def worklist.St.addOrCreateState (st : worklist.St A S) (final? : Bool) (sa : S) : State × worklist.St A S :=\n  match heq : st.map[sa]? with\n  | some s => (s, st)\n  | none =>\n    let (s, m) := st.m.newState\n    let m := if final? then m.addFinal s else m\n    let map := st.map.insert sa s\n    let worklist := st.worklist.push sa\n    have worklist_nodup : worklist.toList.Nodup := by admit /- proof elided -/"}, {"name": "processOneElem", "content": "def processOneElem (final : S → Bool) (s : State) (st : worklist.St A S) : A × S → worklist.St A S :=\n  fun (a', sa') =>\n    let (s', st') := st.addOrCreateState _ _ (final sa') sa'\n    let m := st'.m.addTrans a' s s'\n    { st' with m }"}, {"name": "worklist.St.visited", "content": "def worklist.St.visited (st : worklist.St A S) : Set S := { s : S | s ∈ st.map ∧ s ∉ st.worklist }"}, {"name": "StInv", "content": "structure StInv (m : RawCNFA A) (map : Std.HashMap S State) where\n  wf : m.WF\n  map_states : ∀ (sa : S) s, map[sa]? = some s → s ∈ m.states\n  map_surj : ∀ s : m.states, ∃ (sa : S), map[sa]? = some s.val\n  map_inj : ∀ {s} {sa sa' : S}, map[sa]? = some s → map[sa']? = some s → sa = sa'"}, {"name": "worklist.St.rel", "content": "def worklist.St.rel (st : worklist.St A S) : SetRel State S := {(s, sa) | st.map[sa]? = some s }"}, {"name": "worklist.St.D", "content": "def worklist.St.D (st : worklist.St A S) : Set S := st.visited"}, {"name": "worklist.St.sim", "content": "abbrev worklist.St.sim {st : worklist.St A S} (T : Set (S × A × S)) :=\n  st.m.Simul (nfa inits final f) st.rel st.D T"}, {"name": "processOneElem_mot", "content": "def processOneElem_mot (s : State) (sa : S) (n : ℕ) (st : worklist.St A S) : Prop :=\n  st.map[sa]? = some s ∧\n  sa ∈ st.visited ∧\n  StInv A S st.m st.map ∧\n  st.sim inits final f  {(sa1, a, sa') | sa1 = sa ∧ ∃ k ≥ n, (f sa)[k]? = some (a, sa') }"}, {"name": "processOneElem_inv", "content": "def processOneElem_inv {st : worklist.St A S} (s : State) (sa : S) (k : ℕ) :\n    ∀ a sa', (f sa)[k]? = some (a, sa') →\n    processOneElem_mot inits final f s sa k st →\n    let st' := processOneElem A S final s st (a, sa')\n    StInv A S st'.m st'.map :="}], "used_local_lemmas": [{"name": "addOrCreate_preserves_map", "content": "omit [Fintype S] [LawfulBEq A] in\nlemma addOrCreate_preserves_map (st : worklist.St A S) (final? : Bool) (sa sa' : S) :\n    let (_, st') := st.addOrCreateState _ _  final? sa'\n    st.map[sa]? = some s →\n    st'.map[sa]? = some s"}, {"name": "processOneElem_preserves_map", "content": "omit [Fintype S] [LawfulBEq A] in\nlemma processOneElem_preserves_map (st : worklist.St A S) (final : S → Bool) (a : A) (sa sa' : S) (s s' : State) :\n    let st' := processOneElem _ _  final s st (a, sa')\n    st.map[sa]? = some s' →\n    st'.map[sa]? = some s'"}, {"name": "addOrCreateElem_visited", "content": "omit [LawfulBEq A] [Fintype S] [DecidableEq S] in\nlemma addOrCreateElem_visited final? (st : worklist.St A S) sa :\n    st.addOrCreateState _ _ final? sa |>.2.visited = st.visited"}, {"name": "processOneElem_visited", "content": "omit [LawfulBEq A] [Fintype S] [DecidableEq S] in\nlemma processOneElem_visited (st : worklist.St A S) :\n    let st' := processOneElem _ _  final s st (a, sa')\n    st'.visited = st.visited"}, {"name": "processOneElem_map", "content": "omit [LawfulBEq A] [Fintype S] in\nlemma processOneElem_map (st : worklist.St A S) (final : S → Bool) (a : A) (sa sa' : S) (s : State) :\n  (processOneElem A S final s st (a, sa)).map[sa']? =\n    match st.map[sa']? with\n    | some s => some s\n    | none => if sa = sa' then some st.m.stateMax else none"}, {"name": "processOneElem_initials", "content": "omit [LawfulBEq A] [Fintype S] [DecidableEq S] in\n@[simp]\nlemma processOneElem_initials (st : worklist.St A S) (final : S → Bool) (a : A) (sa : S) (s : State) :\n    (processOneElem A S final s st (a, sa)).m.initials = st.m.initials"}, {"name": "processOneElem_finals", "content": "omit [LawfulBEq A] [Fintype S] [DecidableEq S] in\nlemma processOneElem_finals (st : worklist.St A S) (final : S → Bool) (a : A) (sa : S) (s : State) :\n    (processOneElem A S final s st (a, sa)).m.finals =\n      if sa ∉ st.map ∧ final sa then st.m.finals.insert st.m.stateMax else st.m.finals"}, {"name": "processOneElem_tr", "content": "omit [Fintype S] [DecidableEq S] in\nlemma processOneElem_tr (st : worklist.St A S) (final : S → Bool) (a b : A) (sa : S) (s s' : State) :\n    if a = b ∧ s = s' then\n      ∃ ssa, (processOneElem A S final s st (a, sa)).map[sa]? = some ssa ∧\n        (processOneElem A S final s st (a, sa)).m.tr s' b =\n        (st.m.tr s a |>.insert ssa)\n    else\n      (processOneElem A S final s st (a, sa)).m.tr s' b = st.m.tr s' b"}, {"name": "processOneElem_rel", "content": "omit [LawfulBEq A] [Fintype S] in\nlemma processOneElem_rel {s₁ s₂ : State} :\n    s₂ ~[(processOneElem A S final s₁ st (a, sa)).rel] sa' ↔\n      (s₂ ~[st.rel] sa' ∨ (s₂ = st.m.stateMax ∧ sa' = sa ∧ st.map[sa']? = none))"}, {"name": "rel_in_states", "content": "omit [LawfulBEq A] [Fintype S] [LawfulBEq S] [DecidableEq S] in\nlemma rel_in_states {st : worklist.St A S} (hinv : StInv A S st.m st.map) :\n    s ~[st.rel] sa → s ∈ st.m.states"}, {"name": "processOneElem_rel_preserve", "content": "omit [LawfulBEq A] [Fintype S] in\nlemma processOneElem_rel_preserve :\n    s₂ ~[st.rel] sa' →\n    s₂ ~[(processOneElem A S final s₁ st (a, sa)).rel] sa'"}, {"name": "processOneElem_rel_preserve_olds", "content": "omit [LawfulBEq A] [Fintype S] in\nlemma processOneElem_rel_preserve_olds :\n    s₂ ~[(processOneElem A S final s₁ st (a, sa)).rel] sa' →\n    s₂ ∈ st.m.states → s₂ ~[st.rel] sa'"}], "local_ctx": "import Blase.AutoStructs.Basic\n\nopen SetRel\n\nsection nfa\n\nvariable {A : Type} [BEq A] [LawfulBEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nvariable {S : Type} [Fintype S] [BEq S] [LawfulBEq S] [Hashable S] [DecidableEq S]\n\nvariable (inits : Array S) (final : S → Bool) (f : S → Array (A × S))\n\ndef nfa : NFA A S where\n  start := { sa | sa ∈ inits }\n  accept := { sa | final sa }\n  step sa a := { sa' | (a, sa') ∈ f sa }\n\nend nfa\n\nsection nfa'\n\nvariable {S : Type} [Fintype S] [BEq S] [LawfulBEq S] [Hashable S] [DecidableEq S]\n\nvariable (inits : Array S) (final : S → Bool) (f : S → Array (BitVec n × S))\n\nend nfa'\n\nsection worklist\n\nvariable (A : Type) [BEq A] [LawfulBEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nvariable (S : Type) [Fintype S] [BEq S] [LawfulBEq S] [Hashable S] [DecidableEq S]\n\nstructure worklist.St where\n  m : RawCNFA A\n  map : Std.HashMap S State := ∅\n  worklist : Array S := ∅\n  worklist_nodup : worklist.toList.Nodup\n  worklist_incl : ∀ sa ∈ worklist, sa ∈ map\n\nopen List in\n\ndef worklist.St.addOrCreateState (st : worklist.St A S) (final? : Bool) (sa : S) : State × worklist.St A S :=\n  match heq : st.map[sa]? with\n  | some s => (s, st)\n  | none =>\n    let (s, m) := st.m.newState\n    let m := if final? then m.addFinal s else m\n    let map := st.map.insert sa s\n    let worklist := st.worklist.push sa\n    have worklist_nodup : worklist.toList.Nodup := by admit /- proof elided -/\n\ndef processOneElem (final : S → Bool) (s : State) (st : worklist.St A S) : A × S → worklist.St A S :=\n  fun (a', sa') =>\n    let (s', st') := st.addOrCreateState _ _ (final sa') sa'\n    let m := st'.m.addTrans a' s s'\n    { st' with m }\n\n      open List in\n      have hgrow : ∃ sas, st2.map.keys ~ (sas ++ st1.map.keys) ∧ st2.worklist.toList = st1.worklist.toList ++ sas := by\n        rcases a with ⟨al⟩\n        unfold st2\n        generalize hst1 : st1 = x; clear hst1; revert x\n        induction al with\n        | nil => simp\n        | cons asa al ih =>\n          simp; simp at ih; intros st\n          let wl' := processOneElem A S final s st asa\n          rcases ih wl' with ⟨sas', h1', h2'⟩; clear ih\n          rcases processOneElem_grow _ _ st final asa.1 asa.2 s with ⟨sas, h1, h2⟩\n          use (sas ++ sas')\n          constructor\n          { simp [wl'] at h1'; apply list_perm_trick; exact h1'; exact h1 }\n          { simp [wl', h2] at h2'; aesop }\n      have hincl : ∀ k, k ∈ st1.map → k ∈ st2.map := by\n        intros k; rcases hgrow with ⟨sas, hkeys, -⟩;\n        have := @(List.perm_subset_iff_right hkeys st1.map.keys).mpr (by aesop) k; aesop\n      have : st1.meas < st0.meas := by\n        rcases heq' : sa? with ⟨⟩ | ⟨sa⟩\n        { simp_all }\n        apply Finset.card_lt_card\n        simp [Finset.ssubset_iff, Finset.subset_iff]\n        use sa\n        simp [sa?] at heq'\n        constructor\n        { constructor\n          { apply Array.mem_of_back? at heq'; apply st0.worklist_incl; assumption }\n          { apply Array.not_elem_back_pop at heq' <;> simp_all +zetaDelta [Array.pop] } }\n        constructor\n        { right; apply Array.mem_of_back? at heq'; assumption }\n        rintro sa hh; rcases hh with hnin | hin\n        { simp +zetaDelta [hnin] }\n        right\n        exact Array.mem_of_mem_pop st0.worklist sa hin\n      have : st2.meas ≤ st1.meas := by\n        apply Finset.card_le_card\n        simp +zetaDelta [Finset.subset_iff]\n        intros sa' h\n        rcases h with hnin | hin\n        { left; simp [st1] at hincl; intros hc; apply hnin; apply hincl; assumption }\n        by_cases hnew : sa' ∈ st0.map\n        all_goals try (left; trivial)\n        right\n        simp [st1] at hgrow\n        rcases hgrow with ⟨sas, hkeys2, hwl2⟩\n        have hnin : sa'∉ sas := by\n          intros hc\n          have hdisj : st0.map.keys.Disjoint sas := by\n            have : (sas ++ st0.map.keys).Nodup := by\n              apply List.Perm.nodup\n              assumption\n              apply st2.map.keys_nodup\n            simp [List.nodup_append_comm, List.disjoint_of_nodup_append, this]\n          apply hdisj\n          { simp_all [Std.HashMap.mem_keys]; apply hnew }\n          { apply hc }\n        rcases hin with ⟨hin⟩; simp_all +zetaDelta\n      have : st2.meas < st0.meas := by omega\n      go st2\n    else\n      st0.m -- never happens\n  | none => st0.m -- never happens\n  termination_by st0.meas\n\ndef worklist.St.visited (st : worklist.St A S) : Set S := { s : S | s ∈ st.map ∧ s ∉ st.worklist }\n\nstructure StInv (m : RawCNFA A) (map : Std.HashMap S State) where\n  wf : m.WF\n  map_states : ∀ (sa : S) s, map[sa]? = some s → s ∈ m.states\n  map_surj : ∀ s : m.states, ∃ (sa : S), map[sa]? = some s.val\n  map_inj : ∀ {s} {sa sa' : S}, map[sa]? = some s → map[sa']? = some s → sa = sa'\n\nend worklist\n\nsection worklist_correct\n\nvariable {A : Type} [BEq A] [LawfulBEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nvariable {S : Type} [Fintype S] [BEq S] [LawfulBEq S] [Hashable S] [DecidableEq S]\n\nvariable (inits : Array S) (final : S → Bool) (f : S → Array (A × S))\n\ndef worklist.St.rel (st : worklist.St A S) : SetRel State S := {(s, sa) | st.map[sa]? = some s }\n\ndef worklist.St.D (st : worklist.St A S) : Set S := st.visited\n\nabbrev worklist.St.sim {st : worklist.St A S} (T : Set (S × A × S)) :=\n  st.m.Simul (nfa inits final f) st.rel st.D T\n\ndef processOneElem_mot (s : State) (sa : S) (n : ℕ) (st : worklist.St A S) : Prop :=\n  st.map[sa]? = some s ∧\n  sa ∈ st.visited ∧\n  StInv A S st.m st.map ∧\n  st.sim inits final f  {(sa1, a, sa') | sa1 = sa ∧ ∃ k ≥ n, (f sa)[k]? = some (a, sa') }\n\ndef processOneElem_inv {st : worklist.St A S} (s : State) (sa : S) (k : ℕ) :\n    ∀ a sa', (f sa)[k]? = some (a, sa') →\n    processOneElem_mot inits final f s sa k st →\n    let st' := processOneElem A S final s st (a, sa')\n    StInv A S st'.m st'.map :=", "target_theorem": "omit [Fintype S] in\nlemma processOneElem_spec {st : worklist.St A S} (s : State) (sa : S) (k : ℕ) :\n    ∀ a sa', (f sa)[k]? = some (a, sa') →\n    processOneElem_mot inits final f s sa k st →\n    processOneElem_mot inits final f s sa (k+1) (processOneElem A S final s st (a, sa')) :=", "ground_truth_proof": ":= by\n  intro a sa' hf ⟨hmap, hvisited, inv, hsim⟩\n  have hmem : ∀ s (sa : S), st.map[sa]? = some s → s ∈ st.m.states := by intros; apply inv.map_states; assumption\n  have hwf : st.m.WF := by apply inv.wf\n  have inv' := processOneElem_inv inits final f s sa k a sa' hf ⟨hmap, hvisited, inv, hsim⟩\n  unfold processOneElem_mot\n  constructor\n  (rw [processOneElem_preserves_map]; assumption)\n  constructor\n  (rw [processOneElem_visited]; exact hvisited)\n  use inv'; constructor\n  { rw [processOneElem_finals]\n    rintro s' q hR\n    have hs' : s ∈ st.m.states := by apply hmem <;> assumption\n    rw [processOneElem_rel] at hR\n    rcases hR with hR | ⟨rfl, rfl, heq⟩\n    · have heq := rel_in_states inv hR\n      split_ifs with hcond\n      · have hneq : st.m.stateMax ≠ s' := by rintro rfl; simp [RawCNFA.states] at heq\n        simp [hneq]\n        apply hsim.accept; assumption\n      · apply hsim.accept; assumption\n    · split_ifs with h\n      · rcases h with ⟨_, hfin⟩\n        simp [nfa, hfin]\n      · simp [nfa]\n        suffices hnin : q ∉ st.map by\n          push_neg at h; specialize h hnin; simp_all\n          rintro hc; apply hwf.finals_lt at hc; simp at hc\n        exact Std.HashMap.getElem?_none_not_mem heq }\n  { rintro s₁ hs₁; rw [processOneElem_initials] at hs₁\n    obtain ⟨q, hq, hR⟩ := hsim.initial₁ hs₁\n    use q, hq, processOneElem_rel_preserve final hR }\n  { intros q hq; obtain ⟨s, hs, hR⟩ := hsim.initial₂ hq\n    simp only [processOneElem_initials]\n    use s, hs, (by exact processOneElem_rel_preserve final hR) }\n  { rintro s₁ s₂ b q₁ hR htr\n    have h := processOneElem_tr st final a b sa' s s₁\n    split_ifs at h with hcond\n    on_goal 2 => {\n      rw [h] at htr\n      apply processOneElem_rel_preserve_olds at hR\n      specialize hR (RawCNFA.WF.trans_src_lt'' hwf htr)\n      obtain ⟨q₂, hst, hrel⟩ := hsim.trans_match₁ hR htr\n      use q₂; simp only [hst, true_and]\n      exact processOneElem_rel_preserve final hrel }\n    rcases hcond with ⟨rfl, rfl⟩\n    rcases h with ⟨sₙ, hmap', htr'⟩\n    rw [htr'] at htr; clear htr'\n    simp only [Std.HashSet.mem_insert, beq_iff_eq] at htr\n    rcases htr with rfl | htr\n    on_goal 2 =>\n      have hold := RawCNFA.WF.trans_src_lt'' hwf htr\n      obtain ⟨q₂, hst, hrel⟩ := hsim.trans_match₁\n        (by apply processOneElem_rel_preserve_olds final hR hold) htr\n      use q₂, hst, processOneElem_rel_preserve final hrel\n    use sa'; constructor\n    · suffices heq : q₁ = sa by\n        subst heq; apply Array.mem_of_getElem? hf\n      apply processOneElem_preserves_map at hmap\n      unfold worklist.St.rel at hR\n      apply inv'.map_inj hR hmap\n    · exact hmap' }\n  { rintro s₁ b q₁ q₂ hR hs hD hnT\n    simp only [ge_iff_le, Prod.mk.eta, Set.mem_setOf_eq, not_and, not_exists] at hnT\n    have h := processOneElem_tr st final a b sa' s s₁\n    split_ifs at h with hcond\n    on_goal 2 =>\n      have hR' : s₁ ~[st.rel] q₁ := by\n        rw [processOneElem_rel] at hR; rcases hR with hR' | ⟨_, _, hnone⟩; exact hR'\n        unfold worklist.St.D at hD; rw [processOneElem_visited] at hD\n        rcases hD with ⟨hnin, -⟩\n        apply Std.HashMap.getElem?_none_not_mem at hnone\n        contradiction\n      obtain ⟨s₂, hs', hR⟩ := hsim.trans_match₂ hR' hs\n        (by simp_all [worklist.St.D, processOneElem_visited])\n        (by simp; rintro rfl i hi hc\n            by_cases heq: k = i\n            · subst heq; apply hcond; constructor\n              · simp [hc] at hf; simp [hf]\n              · rw [hR'] at hmap; simp at hmap; exact hmap.symm\n            · apply hnT rfl _ (by omega) hc)\n      use s₂; simp only [h, hs', true_and]\n      apply processOneElem_rel_preserve; assumption\n    rcases hcond with ⟨rfl, rfl⟩\n    rcases h with ⟨sₙ, hmap', htr'⟩\n    obtain rfl : sa = q₁ := by\n      apply processOneElem_preserves_map at hmap\n      apply inv'.map_inj hmap hR\n    simp only [htr', Std.HashSet.mem_insert, beq_iff_eq]\n    by_cases heq : sa' = q₂\n    · subst heq; use sₙ; simp only [true_or, true_and]; exact hmap'\n    · have hold := hmem _ _ hmap\n      apply processOneElem_rel_preserve_olds at hR\n      obtain ⟨s₂, hs', hR⟩ := hsim.trans_match₂ (hR hold) hs\n        (by simp_all [worklist.St.D, processOneElem_visited])\n        (by simp; rintro i hi hc\n            have _ : i ≠ k := by rintro rfl; simp_all\n            apply hnT rfl i (by omega) hc)\n      use s₂; simp only [hs', or_true, true_and]\n      apply processOneElem_rel_preserve; assumption }", "nesting_depth": 5, "transitive_dep_count": 79, "subset_aristotle": false, "category": "Compiler"}
{"id": 296, "thm_name": "formula_language_case_atom", "thm_stmt": "lemma formula_language_case_atom :\n    let φ := Formula.atom rel t1 t2\n    φ.language = λ (bvs : BitVecs φ.arity) => φ.sat (fun k => bvs.bvs.get k)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/Defs.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.ForMathlib"], "used_lib_defs": [{"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.card", "module": "Mathlib.Data.FinEnum"}, {"name": "Fin.isValue", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Set.Mem", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.instMembership", "module": "Mathlib.Data.Set.Defs"}, {"name": "List.Vector.get", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}], "used_repo_defs": [{"name": "syntax \"max\" : MLIR.Pretty.uniform_op", "content": "syntax \"max\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "Term.arity", "content": "@[simp] def Term.arity : Term → Nat\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| ofNat _ => 0\n| Term.and t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.or t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.xor t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.not t => arity t\n| add t₁ t₂ => max (arity t₁) (arity t₂)\n| sub t₁ t₂ => max (arity t₁) (arity t₂)\n| neg t => arity t\n\n\n| shiftL t .. => arity t"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "WidthPredicate", "content": "inductive WidthPredicate\n| eq\n| neq\n| lt\n| le\n| gt\n| ge\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "BitVecs.transport", "content": "def BitVecs.transport (f : Fin n → Fin m) (bvs : BitVecs m) : BitVecs n :=\n  { w := bvs.w, bvs := bvs.bvs.transport f }"}, {"name": "BitVec.transport", "content": "def BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=\n  BitVec.ofFn fun i => bv.getLsbD (f i)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "List.Vector.transport", "content": "def List.Vector.transport (v : Vector α m) (f : Fin n → Fin m) : Vector α n :=\n  Vector.ofFn fun i => v.get (f i)"}, {"name": "BitVecs'.transport", "content": "def BitVecs'.transport (f : Fin n → Fin m) (bvs' : BitVecs' m): BitVecs' n :=\n  bvs'.map fun bv => bv.transport f"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "BitVecs.cast", "content": "def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=\n  { w := bvs.w, bvs := h ▸ bvs.bvs }"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.val_last", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "le_add_iff_nonneg_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "zero_le", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}], "repo_lemmas": [{"name": "List.Vector.append_get_ge", "content": "@[simp]\nlemma List.Vector.append_get_ge {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: n ≤ i) :\n    (x ++ y).get i = y.get ((i.cast (Nat.add_comm n m) |>.subNat n hlt))"}, {"name": "List.Vector.append_get_lt", "content": "@[simp]\nlemma List.Vector.append_get_lt {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: i < n) :\n    (x ++ y).get i = x.get (i.castLT hlt)"}, {"name": "BitVecs.transport_getElem", "content": "@[simp]\nlemma BitVecs.transport_getElem {bvs : BitVecs m} (f : Fin n → Fin m) (i : Fin n) :\n    (bvs.transport f).bvs.get i = bvs.bvs.get (f i)"}], "used_local_defs": [{"name": "liftMaxSucc1", "content": "def liftMaxSucc1 (n m : Nat) : Fin (n + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = n then Fin.last (max n m) else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMaxSucc2", "content": "def liftMaxSucc2 (n m : Nat) : Fin (m + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = m then Fin.last (max n m + 1) else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftLast2", "content": "def liftLast2 n : Fin 2 → Fin (n + 2)\n| 0 => n\n| 1 => Fin.last (n + 1)"}, {"name": "liftExcept2", "content": "def liftExcept2 n : Fin n → Fin (n + 2) :=\n  fun k => Fin.castLE (by admit /- proof elided -/\n  ) k"}, {"name": "liftMax1", "content": "def liftMax1 (n m : Nat) : Fin n → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMax2", "content": "def liftMax2 (n m : Nat) : Fin m → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )"}, {"name": "Term.evalFinBV", "content": "@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n"}, {"name": "Term.language", "content": "def Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }"}, {"name": "RelationOrdering", "content": "inductive RelationOrdering\n| lt | le | gt | ge\nderiving Repr, Fintype"}, {"name": "Relation", "content": "inductive Relation\n| eq\n| signed (ord : RelationOrdering)\n| unsigned (ord : RelationOrdering)\nderiving Repr"}, {"name": "evalRelation", "content": "def evalRelation (rel : Relation) {w} (bv1 bv2 : BitVec w) : Prop :=\n  match rel with\n  | .eq => bv1 = bv2\n  | .signed .lt => bv1.slt bv2\n  | .signed .le => bv1.sle  bv2\n  | .signed .gt => bv2.slt bv1\n  | .signed .ge => bv2.sle bv1\n  | .unsigned .lt => bv1.ult bv2\n  | .unsigned .le => bv1.ule bv2\n  | .unsigned .gt => bv2.ult bv1\n  | .unsigned .ge => bv2.ule bv1"}, {"name": "Relation.language", "content": "@[simp]\ndef Relation.language (rel : Relation) : Set (BitVecs 2) :=\n  { bvs | evalRelation rel (bvs.bvs.get 0) (bvs.bvs.get 1) }"}, {"name": "Binop", "content": "inductive Binop\n| and | or | impl | equiv\nderiving Repr"}, {"name": "evalBinop", "content": "def evalBinop (op : Binop) (b1 b2 : Prop) : Prop :=\n  match op with\n  | .and => b1 ∧ b2\n  | .or => b1 ∨ b2\n  | .impl => b1 → b2\n  | .equiv => b1 ↔ b2"}, {"name": "langBinop", "content": "def langBinop (op : Binop) (l1 l2 : Set (BitVecs n)) : Set (BitVecs n) :=\n  match op with\n  | .and => l1 ∩ l2\n  | .or => l1 ∪ l2\n  | .impl => l1ᶜ ∪ l2\n  | .equiv => (l1ᶜ ∪ l2) ∩ (l2ᶜ ∪ l1)"}, {"name": "Unop", "content": "inductive Unop\n| neg\nderiving Repr"}, {"name": "Formula", "content": "inductive Formula : Type\n| width : WidthPredicate → Nat → Formula\n| atom : Relation → Term → Term → Formula\n| msbSet : Term → Formula\n| unop : Unop → Formula → Formula\n| binop : Binop → Formula → Formula → Formula\nderiving Repr"}, {"name": "Formula.arity", "content": "@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity"}, {"name": "WidthPredicate.sat", "content": "@[simp]\ndef WidthPredicate.sat (wp : WidthPredicate) (w n : Nat) : Bool :=\n  match wp with\n  | .eq => w = n\n  | .neq => w ≠ n\n  | .lt => w < n\n  | .le => w ≤ n\n  | .gt => w > n\n  | .ge => w ≥ n"}, {"name": "Formula.sat", "content": "@[simp]\ndef Formula.sat {w : Nat} (φ : Formula) (ρ : Fin φ.arity → BitVec w) : Prop :=\n  match φ with\n  | .width wp n => wp.sat w n\n  | .atom rel t1 t2 =>\n    let bv1 := t1.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let bv2 := t2.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalRelation rel bv1 bv2\n  | .unop .neg φ => ¬ φ.sat ρ\n  | .binop op φ1 φ2 =>\n    let b1 := φ1.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let b2 := φ2.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalBinop op b1 b2\n  | .msbSet t => (t.evalFinBV ρ).msb"}, {"name": "_root_.Set.lift", "content": "@[simp]\ndef _root_.Set.lift (f : Fin n → Fin m) (bvs : Set (BitVecs n)) : Set (BitVecs m) :=\n  BitVecs.transport f ⁻¹' bvs"}, {"name": "_root_.Set.proj", "content": "@[simp]\ndef _root_.Set.proj (f : Fin n → Fin m) (bvs : Set (BitVecs m)) : Set (BitVecs n) :=\n  BitVecs.transport f '' bvs"}, {"name": "Formula.language", "content": "@[simp]\ndef Formula.language (φ : Formula) : Set (BitVecs φ.arity) :=\n  match φ with\n  | .width wp n => { bvs | wp.sat bvs.w n }\n  | .atom rel t1 t2 =>\n    let l1 := t1.language.lift (liftMaxSucc1 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let l2 := t2.language.lift (liftMaxSucc2 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let lrel := rel.language.lift $ liftLast2 (max (FinEnum.card (Fin t1.arity)) (FinEnum.card (Fin t2.arity)))\n    let l := lrel ∩ l1 ∩ l2\n    l.proj (liftExcept2 _)\n  | .unop .neg φ => φ.languageᶜ\n  | .binop op φ1 φ2 =>\n    let l1 := φ1.language.lift $ liftMax1 φ1.arity φ2.arity\n    let l2 := φ2.language.lift $ liftMax2 φ1.arity φ2.arity\n    langBinop op l1 l2\n  | .msbSet t =>\n    let lmsb := langMsb.lift $ fun _ => Fin.last t.arity\n    let l' := t.language ∩ lmsb\n    l'.proj fun n => n.castLE (by admit /- proof elided -/\n    )"}], "used_local_lemmas": [{"name": "evalFin_eq", "content": "lemma evalFin_eq {t : Term} {vars1 : Fin t.arity → BitVec w1} {vars2 : Fin t.arity → BitVec w2} :\n    ∀ (heq : w1 = w2),\n    (∀ n, vars1 n = heq ▸ vars2 n) →\n    t.evalFinBV vars1 = heq ▸ t.evalFinBV vars2"}, {"name": "evalRelation_coe", "content": "@[simp]\nlemma evalRelation_coe (rel : Relation) (bv1 bv2 : BitVec w1) (heq : w1 = w2) :\n    evalRelation rel (heq ▸ bv1) (heq ▸ bv2) = evalRelation rel bv1 bv2"}, {"name": "helper1", "content": "lemma helper1 : (k = 0) → (x ::ᵥ vs).get k = x"}, {"name": "helper2", "content": "lemma helper2 : (k = 1) → (x ::ᵥ y ::ᵥ vs).get k = y"}], "local_ctx": "import Blase.AutoStructs.ForMathlib\n\nimport Blase.SingleWidth.Defs\n\nopen Fin.NatCast\n\ndef liftMaxSucc1 (n m : Nat) : Fin (n + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = n then Fin.last (max n m) else k.castLE (by admit /- proof elided -/\n  )\n\ndef liftMaxSucc2 (n m : Nat) : Fin (m + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = m then Fin.last (max n m + 1) else k.castLE (by admit /- proof elided -/\n  )\n\ndef liftLast2 n : Fin 2 → Fin (n + 2)\n| 0 => n\n| 1 => Fin.last (n + 1)\n\ndef liftExcept2 n : Fin n → Fin (n + 2) :=\n  fun k => Fin.castLE (by admit /- proof elided -/\n  ) k\n\ndef liftMax1 (n m : Nat) : Fin n → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )\n\ndef liftMax2 (n m : Nat) : Fin m → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )\n\n@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n\n\ndef Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }\n\ninductive RelationOrdering\n| lt | le | gt | ge\nderiving Repr, Fintype\n\ninductive Relation\n| eq\n| signed (ord : RelationOrdering)\n| unsigned (ord : RelationOrdering)\nderiving Repr\n\ndef evalRelation (rel : Relation) {w} (bv1 bv2 : BitVec w) : Prop :=\n  match rel with\n  | .eq => bv1 = bv2\n  | .signed .lt => bv1.slt bv2\n  | .signed .le => bv1.sle  bv2\n  | .signed .gt => bv2.slt bv1\n  | .signed .ge => bv2.sle bv1\n  | .unsigned .lt => bv1.ult bv2\n  | .unsigned .le => bv1.ule bv2\n  | .unsigned .gt => bv2.ult bv1\n  | .unsigned .ge => bv2.ule bv1\n\n@[simp]\ndef Relation.language (rel : Relation) : Set (BitVecs 2) :=\n  { bvs | evalRelation rel (bvs.bvs.get 0) (bvs.bvs.get 1) }\n\ninductive Binop\n| and | or | impl | equiv\nderiving Repr\n\ndef evalBinop (op : Binop) (b1 b2 : Prop) : Prop :=\n  match op with\n  | .and => b1 ∧ b2\n  | .or => b1 ∨ b2\n  | .impl => b1 → b2\n  | .equiv => b1 ↔ b2\n\ndef langBinop (op : Binop) (l1 l2 : Set (BitVecs n)) : Set (BitVecs n) :=\n  match op with\n  | .and => l1 ∩ l2\n  | .or => l1 ∪ l2\n  | .impl => l1ᶜ ∪ l2\n  | .equiv => (l1ᶜ ∪ l2) ∩ (l2ᶜ ∪ l1)\n\ninductive Unop\n| neg\nderiving Repr\n\ninductive Formula : Type\n| width : WidthPredicate → Nat → Formula\n| atom : Relation → Term → Term → Formula\n| msbSet : Term → Formula\n| unop : Unop → Formula → Formula\n| binop : Binop → Formula → Formula → Formula\nderiving Repr\n\n@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity\n\n@[simp]\ndef WidthPredicate.sat (wp : WidthPredicate) (w n : Nat) : Bool :=\n  match wp with\n  | .eq => w = n\n  | .neq => w ≠ n\n  | .lt => w < n\n  | .le => w ≤ n\n  | .gt => w > n\n  | .ge => w ≥ n\n\n@[simp]\ndef Formula.sat {w : Nat} (φ : Formula) (ρ : Fin φ.arity → BitVec w) : Prop :=\n  match φ with\n  | .width wp n => wp.sat w n\n  | .atom rel t1 t2 =>\n    let bv1 := t1.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let bv2 := t2.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalRelation rel bv1 bv2\n  | .unop .neg φ => ¬ φ.sat ρ\n  | .binop op φ1 φ2 =>\n    let b1 := φ1.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let b2 := φ2.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalBinop op b1 b2\n  | .msbSet t => (t.evalFinBV ρ).msb\n\n@[simp]\ndef _root_.Set.lift (f : Fin n → Fin m) (bvs : Set (BitVecs n)) : Set (BitVecs m) :=\n  BitVecs.transport f ⁻¹' bvs\n\n@[simp]\ndef _root_.Set.proj (f : Fin n → Fin m) (bvs : Set (BitVecs m)) : Set (BitVecs n) :=\n  BitVecs.transport f '' bvs\n\n@[simp]\ndef Formula.language (φ : Formula) : Set (BitVecs φ.arity) :=\n  match φ with\n  | .width wp n => { bvs | wp.sat bvs.w n }\n  | .atom rel t1 t2 =>\n    let l1 := t1.language.lift (liftMaxSucc1 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let l2 := t2.language.lift (liftMaxSucc2 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let lrel := rel.language.lift $ liftLast2 (max (FinEnum.card (Fin t1.arity)) (FinEnum.card (Fin t2.arity)))\n    let l := lrel ∩ l1 ∩ l2\n    l.proj (liftExcept2 _)\n  | .unop .neg φ => φ.languageᶜ\n  | .binop op φ1 φ2 =>\n    let l1 := φ1.language.lift $ liftMax1 φ1.arity φ2.arity\n    let l2 := φ2.language.lift $ liftMax2 φ1.arity φ2.arity\n    langBinop op l1 l2\n  | .msbSet t =>\n    let lmsb := langMsb.lift $ fun _ => Fin.last t.arity\n    let l' := t.language ∩ lmsb\n    l'.proj fun n => n.castLE (by admit /- proof elided -/\n    )", "target_theorem": "lemma formula_language_case_atom :\n    let φ :=", "ground_truth_proof": ":= Formula.atom rel t1 t2\n    φ.language = λ (bvs : BitVecs φ.arity) => φ.sat (fun k => bvs.bvs.get k) := by\n  unfold Formula.language\n  rintro φ\n  let n := φ.arity\n  unfold φ\n  dsimp (config := { zeta := false })\n  lift_lets\n  intros l1 l2 lrel l\n  ext bvs\n  constructor\n  · intros h; simp at h\n    obtain ⟨bvsb, h, heqb⟩ := h\n    unfold l at h\n    simp at h\n    unfold lrel l1 l2 at h\n    obtain ⟨⟨hrel, h1⟩, h2⟩ := h\n    have _ : n+1 < bvsb.bvs.length := by simp +zetaDelta\n    have _ : n < bvsb.bvs.length := by simp +zetaDelta\n    have hrel : evalRelation rel (bvsb.bvs.get n) (bvsb.bvs.get (Fin.last (n + 1))) := by\n      simp at hrel\n      apply hrel\n    have ht1 : bvsb.bvs.get n = t1.evalFinBV fun n => bvsb.bvs.get n := by\n      unfold Term.language at h1\n      simp [liftMaxSucc1] at h1\n      unfold n; simp +zetaDelta; rw [←h1]\n      congr; ext1 k\n      congr; ext; simp; rw [Nat.mod_eq_of_lt]; omega\n    have ht2 : bvsb.bvs.get (Fin.last (n+1)) = t2.evalFinBV fun n => bvsb.bvs.get n := by\n      unfold Term.language at h2\n      simp [liftMaxSucc2] at h2\n      unfold n; simp +zetaDelta only [Formula.arity]; rw [←h2]\n      congr; ext1 k\n      congr; ext; simp; rw [Nat.mod_eq_of_lt]; omega\n    have hw : bvsb.w = bvs.w := by rw [←heqb]; simp\n    have heq1 : (t1.evalFinBV fun n => bvsb.bvs.get n) =\n        hw ▸ t1.evalFinBV fun n => bvs.bvs.get $ n.castLE (by simp) := by\n      apply evalFin_eq hw; intros k\n      rcases bvs with ⟨w, bvs⟩; rcases hw\n      injection heqb with _ heqb; rw [←heqb]\n      simp [List.Vector.transport, liftExcept2]\n      congr; ext; simp; omega\n    have heq2 : (t2.evalFinBV fun n => bvsb.bvs.get n) =\n        hw ▸ t2.evalFinBV fun n => bvs.bvs.get $ n.castLE (by simp) := by\n      apply evalFin_eq hw; intros k\n      rcases bvs with ⟨w, bvs⟩; rcases hw\n      injection heqb with _ heqb; rw [←heqb]\n      simp [List.Vector.transport, liftExcept2]\n      congr; ext; simp; omega\n    rw [ht1, ht2, heq1, heq2, evalRelation_coe] at hrel\n    dsimp only [Set.instMembership, Set.Mem]\n    simp_all\n  · intros h\n    simp\n    let bv1 := t1.evalFinBV fun k => bvs.bvs.get $ k.castLE (by simp)\n    let bv2 := t2.evalFinBV fun k => bvs.bvs.get $ k.castLE (by simp)\n    use ⟨bvs.w, bvs.bvs ++ bv1 ::ᵥ bv2 ::ᵥ List.Vector.nil⟩\n    rcases bvs with ⟨w, bvs⟩\n    simp\n    constructor\n    · unfold l; simp; split_ands\n      · unfold lrel; simp only [Fin.isValue, BitVecs.transport_getElem,\n          liftLast2, Set.mem_setOf_eq, Fin.val_last, le_add_iff_nonneg_right, zero_le,\n          List.Vector.append_get_ge]\n        rw [List.Vector.append_get_ge (by dsimp; rw [Nat.mod_eq_of_lt]; omega)]\n        simp [Set.instMembership, Set.Mem] at h\n        convert h using 2\n        · apply helper1; ext; simp; rw [Nat.mod_eq_of_lt] <;> omega\n        · apply helper2; ext; simp\n      · unfold l1 Term.language; simp [liftMaxSucc1]\n        rw [List.Vector.append_get_ge (by dsimp; rw [Nat.mod_eq_of_lt]; omega)]\n        rw [helper1 (by ext; simp; rw [Nat.mod_eq_of_lt] <;> omega)]\n        unfold bv1\n        congr;\n      · unfold l2 Term.language; simp [liftMaxSucc2]\n        rw [helper2 (by ext; simp)]\n        unfold bv2\n        congr\n    · ext1; simp\n      next i =>\n        simp [liftExcept2]\n        rw [List.Vector.append_get_lt i.isLt]\n        congr 1", "nesting_depth": 5, "transitive_dep_count": 73, "subset_aristotle": false, "category": "Compiler"}
{"id": 297, "thm_name": "TermBinop.alt_lang", "thm_stmt": "lemma TermBinop.alt_lang {t₁ t₂ : Term} (op : TermBinop) :\n  (op.subst_arity' ▸ (op.subst t₁ t₂).language) =\n    let lop : Set (BitVecs 3) := op.openTerm_arity ▸ op.openTerm.language\n    let lop' : Set (BitVecs ((t₁.arity ⊔ t₂.arity) + 3)) := lop.lift (liftLast3 (max t₁.arity t₂.arity))\n    let l₁ := t₁.language.lift (liftMaxSuccSucc1 t₁.arity t₂.arity)\n    let l₂ := t₂.language.lift (liftMaxSuccSucc2 t₁.arity t₂.arity)\n    let l := l₁ ∩ l₂ ∩ lop'\n    l.proj (liftOp _)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.mk", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Fin.cast", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.castLT", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.subNat", "module": "Init.Data.Fin.Basic"}, {"name": "List.Vector.get", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}], "used_repo_defs": [{"name": "syntax \"xor\" : MLIR.Pretty.uniform_op", "content": "syntax \"xor\" : MLIR.Pretty.uniform_op\n\nsyntax \"max\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "neg", "content": "def neg (x : BitStream) : BitStream :=\n  fun n => (negAux x n).1"}, {"name": "negAux", "content": "def negAux (x : BitStream) : Nat → Bool × Bool\n  | 0 => (x 0, !(x 0))\n  | n+1 =>\n    let borrow := (negAux x n).2\n    let a := x (n + 1)\n    (xor (!a) borrow, !a && borrow)"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "sub", "content": "def sub (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).1"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "add", "content": "def add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1"}, {"name": "addAux", "content": "def addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "liftMaxSuccSucc2", "content": "def liftMaxSuccSucc2 (n m : Nat) : Fin (m + 1) → Fin (max n m + 3) :=\n  fun k => if _ : k = Fin.last m then max n m + 1 else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMaxSuccSucc1", "content": "def liftMaxSuccSucc1 (n m : Nat) : Fin (n + 1) → Fin (max n m + 3) :=\n  fun k => if _ : k = Fin.last n then (max n m).cast else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftLast3", "content": "def liftLast3 n : Fin 3 → Fin (n + 3)\n| 0 => n\n| 1 => n + 1\n| 2 => Fin.last (n + 2)"}, {"name": "BitVecs.cast", "content": "def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=\n  { w := bvs.w, bvs := h ▸ bvs.bvs }"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "Term.language", "content": "def Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }"}, {"name": "Formula.arity", "content": "@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity"}, {"name": "Term.evalFinBV", "content": "@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n"}, {"name": "Term.arity", "content": "@[simp] def Term.arity : Term → Nat\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| ofNat _ => 0\n| Term.and t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.or t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.xor t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.not t => arity t\n| add t₁ t₂ => max (arity t₁) (arity t₂)\n| sub t₁ t₂ => max (arity t₁) (arity t₂)\n| neg t => arity t\n\n\n| shiftL t .. => arity t"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.add_def", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.castLE_castLE", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.le_of_eq", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}], "repo_lemmas": [{"name": "List.Vector.append_get_ge", "content": "@[simp]\nlemma List.Vector.append_get_ge {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: n ≤ i) :\n    (x ++ y).get i = y.get ((i.cast (Nat.add_comm n m) |>.subNat n hlt))"}, {"name": "List.Vector.append_get_lt", "content": "@[simp]\nlemma List.Vector.append_get_lt {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: i < n) :\n    (x ++ y).get i = x.get (i.castLT hlt)"}, {"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}], "used_local_defs": [{"name": "liftOp", "content": "def liftOp n : Fin (n + 1) → Fin (n + 3) :=\n  fun k =>\n    if k = n then Fin.last (n+2) else k.castLE (by admit /- proof elided -/\n    )"}, {"name": "liftOp_unchanged", "content": "@[simp]\ndef liftOp_unchanged (k : Fin n) : liftOp n k.castSucc = k.castLE (by simp) :="}, {"name": "TermBinop", "content": "inductive TermBinop where\n| and | or | xor | add | sub"}, {"name": "TermBinop.subst", "content": "def TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂"}, {"name": "TermBinop.openTerm", "content": "def TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)"}, {"name": "TermBinop.openTerm_arity", "content": "@[simp]\ndef TermBinop.openTerm_arity (op : TermBinop) : op.openTerm.arity + 1 = 3 :="}, {"name": "swapLastTwoBlock", "content": "def swapLastTwoBlock (x : Fin (n + 3)) : Fin (n + 3) :=\n  if x = Fin.last (n+2) then n\n  else if x = n+1 then Fin.last (n + 2)\n  else if x = n then n + 1\n  else x"}], "used_local_lemmas": [{"name": "TermBinop.subst_arity'", "content": "lemma TermBinop.subst_arity' {op : TermBinop} : (op.subst t₁ t₂).arity + 1= t₁.arity ⊔ t₂.arity + 1"}, {"name": "BitVecs.cast_eq", "content": "@[simp]\nlemma BitVecs.cast_eq  (x : BitVecs n) (h : n = n') : h ▸ x = x.cast h"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\nsection fsm\n\nvariable {arity : Type} [FinEnum arity]\n\nopen BitStream in\n\nend fsm\n\nsection nfas_relations\n\nend nfas_relations\n\ndef liftOp n : Fin (n + 1) → Fin (n + 3) :=\n  fun k =>\n    if k = n then Fin.last (n+2) else k.castLE (by admit /- proof elided -/\n    )\n\n@[simp]\ndef liftOp_unchanged (k : Fin n) : liftOp n k.castSucc = k.castLE (by simp) :=\n\ninductive TermBinop where\n| and | or | xor | add | sub\n\ndef TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂\n\ndef TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)\n\n@[simp]\ndef TermBinop.openTerm_arity (op : TermBinop) : op.openTerm.arity + 1 = 3 :=\n\ndef swapLastTwoBlock (x : Fin (n + 3)) : Fin (n + 3) :=\n  if x = Fin.last (n+2) then n\n  else if x = n+1 then Fin.last (n + 2)\n  else if x = n then n + 1\n  else x", "target_theorem": "lemma TermBinop.alt_lang {t₁ t₂ : Term} (op : TermBinop) :\n  (op.subst_arity' ▸ (op.subst t₁ t₂).language) =\n    let lop : Set (BitVecs 3) :=", "ground_truth_proof": ":= op.openTerm_arity ▸ op.openTerm.language\n    let lop' : Set (BitVecs ((t₁.arity ⊔ t₂.arity) + 3)) := lop.lift (liftLast3 (max t₁.arity t₂.arity))\n    let l₁ := t₁.language.lift (liftMaxSuccSucc1 t₁.arity t₂.arity)\n    let l₂ := t₂.language.lift (liftMaxSuccSucc2 t₁.arity t₂.arity)\n    let l := l₁ ∩ l₂ ∩ lop'\n    l.proj (liftOp _)\n    := by\n  simp [Term.language]\n  ext bvs\n  simp\n  constructor\n  · rintro heq\n    let bvs' := bvs.bvs ++\n                  (t₁.evalFinBV (λ n ↦ bvs.bvs.get n) ::ᵥ t₂.evalFinBV (λ n ↦ bvs.bvs.get n) ::ᵥ List.Vector.nil)\n                |>.transport swapLastTwoBlock\n    use ⟨_, bvs'⟩\n    simp [bvs']\n    split_ands\n    · rw [liftMaxSuccSucc1]; simp\n      conv =>\n        enter [1, 2, n]\n        rw [List.Vector.append_get_lt (by rcases n with ⟨n, hn⟩; simp_all; rw [Nat.mod_eq_of_lt (by omega)]; omega)]\n        rfl\n\n      rw [List.Vector.append_get_ge]\n      on_goal 2 => apply Nat.le_of_eq; simp [Fin.add_def]; rw [Nat.mod_eq_of_lt (by omega)]\n\n      generalize_proofs h₁ h₂ h₃ h₄\n      have heq h : Fin.subNat (t₁.arity ⊔ t₂.arity + 1) (Fin.cast h₄ (↑(t₁.arity ⊔ t₂.arity) + 1)) h = 0 := by\n         ext; simp [Fin.add_def]; rw [Nat.mod_eq_of_lt (by omega)]; omega\n      simp [heq]\n      congr; ext1 i; congr 1\n      rcases i with ⟨i, hi⟩; simp [Fin.castLT]; ext; simp; repeat rw [Nat.mod_eq_of_lt (by omega)]\n    · rw [liftMaxSuccSucc2]; simp\n      conv =>\n        enter [1, 2, n]\n        rw [List.Vector.append_get_lt (by rcases n with ⟨n, hn⟩; simp_all; rw [Nat.mod_eq_of_lt (by omega)]; omega)]\n        rfl\n      generalize_proofs h₁ h₂ h₃ h₄\n      have heq h : Fin.subNat (t₁.arity ⊔ t₂.arity + 1) (Fin.last (2 + t₁.arity ⊔ t₂.arity)) h = 1 := by\n         ext; simp; omega\n      simp [heq]\n      congr; ext1 i; congr 1\n      rcases i with ⟨i, hi⟩; simp [Fin.castLT]; ext; simp; repeat rw [Nat.mod_eq_of_lt (by omega)]\n    · rw [BitVecs.cast_eq] at *\n      simp [BitVecs.cast] at *\n      rw [liftLast3]\n      simp\n      rw [List.Vector.append_get_lt (by simp; rw [Nat.mod_eq_of_lt (by omega)]; omega)]\n      convert heq using 1\n      have h n : n + 1 < n + 3 := by omega\n      have h' n m k : n < m → n < (k ⊔ m) + 1 := by omega\n      · cases op <;>\n        · simp [openTerm, subst, liftLast3]; congr\n          · rw [List.Vector.append_get_ge]\n            · simp [Fin.subNat, Fin.add_def]; simp [Nat.mod_eq_of_lt, h]; congr! with ⟨n, hn⟩; ext; simp; omega\n            · rw [Fin.add_def]; simp; rw [Nat.mod_eq_of_lt (by omega)]\n          · have heq h : (Fin.subNat (t₁.arity ⊔ t₂.arity + 1) (Fin.last (2 + t₁.arity ⊔ t₂.arity)) h) = 1 := by\n              simp [Fin.subNat]; omega\n            simp [heq, List.Vector.get]; congr; ext1 ⟨i, hi⟩; simp; simp [Nat.mod_eq_of_lt (h' _ _ t₁.arity hi)]\n      · congr!; simp [Fin.castLT, Fin.last]; omega\n    · ext1\n      · simp\n      next i =>\n        simp [liftOp]\n        split_ifs with h\n        · subst h\n          simp\n          rw [List.Vector.append_get_lt]\n          on_goal 2 =>\n            simp +arith\n            repeat (rw [Nat.mod_eq_of_lt (by omega)])\n            omega\n          congr; ext; simp +arith\n          omega\n        · simp [h]\n          rw [List.Vector.append_get_lt (by rcases i; simp_all [Fin.last]; rw [Nat.mod_eq_of_lt (by omega)]; omega)]\n          rcases i with ⟨i, hi⟩; congr!; simp_all; omega\n  · rintro ⟨bvs', ⟨⟨⟨heq₁, heq₂⟩, heq₃⟩, heq₄⟩⟩\n    rw [BitVecs.cast_eq] at *\n    simp [BitVecs.cast] at *\n    rw [←heq₄]\n    conv_rhs =>\n      simp\n      simp [liftOp]\n      rfl\n    simp [liftMaxSuccSucc1, liftMaxSuccSucc2] at heq₁ heq₂\n    rw [liftLast3] at heq₃\n    convert heq₃ using 1\n    have h₁ : (t₁.evalFinBV fun i => bvs'.bvs.get (liftOp (t₁.arity ⊔ t₂.arity) (Fin.castLE (by omega) i).castSucc)) =\n                  bvs'.bvs.get (liftLast3 (t₁.arity ⊔ t₂.arity) 0) := by\n      simp only [liftOp_unchanged, Fin.castLE_castLE, liftLast3]; convert heq₁ using 1\n    have h₂ : (t₂.evalFinBV fun i => bvs'.bvs.get (liftOp (t₁.arity ⊔ t₂.arity) (Fin.castLE (by omega) i).castSucc)) =\n                  bvs'.bvs.get (liftLast3 (t₁.arity ⊔ t₂.arity) 1) := by\n      simp only [liftOp_unchanged, Fin.castLE_castLE, liftLast3]; convert heq₂\n    rcases op with _ | _ | _ <;>\n    . simp [subst, openTerm] at *; congr", "nesting_depth": 4, "transitive_dep_count": 57, "subset_aristotle": false, "category": "Compiler"}
{"id": 298, "thm_name": "Predicate.eval_eq_denote", "thm_stmt": "theorem Predicate.eval_eq_denote (w : Nat) (p : Predicate) (vars : List (BitVec w)) :\n    (p.eval (vars.map .ofBitVecSext) w = false) ↔ p.denote w vars", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/Fast/Defs.lean", "imports": ["import Blase.SingleWidth.Defs", "import Mathlib.Data.Fin.Basic", "import Mathlib.Data.Bool.Basic", "import Blase.Fast.BitStream", "import Blase.Blase.Fast.BitStream"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "BitVec.ult", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.carry", "module": "Init.Data.BitVec.Bitblast"}, {"name": "BitVec.sle", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.slt", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ule", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"slt\" : MLIR.Pretty.uniform_op", "content": "syntax \"slt\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "borrow", "content": "def borrow (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).2"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "nxor", "content": "def nxor (a b : BitStream) : BitStream := fun i => a i == b i"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "falseIffNeq", "content": "abbrev falseIffNeq (n : Nat) : BitStream := fun i => decide (i == n)"}, {"name": "falseIffLt", "content": "abbrev falseIffLt (n : Nat) : BitStream := fun i => decide (i ≥ n)"}, {"name": "falseIffGe", "content": "abbrev falseIffGe (n : Nat) : BitStream := fun i => decide (i < n)"}, {"name": "falseIffEq", "content": "abbrev falseIffEq (n : Nat) : BitStream := fun i => decide (i != n)"}, {"name": "falseIffGt", "content": "abbrev falseIffGt (n : Nat) : BitStream := fun i => decide (i ≤ n)"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "falseIffLe", "content": "abbrev falseIffLe (n : Nat) : BitStream := fun i => decide (i > n)"}, {"name": "negOne", "content": "abbrev negOne : BitStream := fun _ => true"}, {"name": "shiftLeft", "content": "def shiftLeft (x : BitStream) (k : Nat) : BitStream :=\n  fun i => if i < k then false else x (i - k) "}, {"name": "ofNat", "content": "def ofNat (x : Nat) : BitStream :=\n  Nat.testBit x"}, {"name": "one", "content": "abbrev one    : BitStream := (· == 0)"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "Predicate", "content": "inductive Predicate : Type where\n \n| width (wp : WidthPredicate) (n : Nat) : Predicate\n| binary (p : BinaryPredicate) (t₁ t₂ : Term)\n| land  (p q : Predicate) : Predicate\n| lor (p q : Predicate) : Predicate\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "neg", "content": "def neg (x : BitStream) : BitStream :=\n  fun n => (negAux x n).1"}, {"name": "negAux", "content": "def negAux (x : BitStream) : Nat → Bool × Bool\n  | 0 => (x 0, !(x 0))\n  | n+1 =>\n    let borrow := (negAux x n).2\n    let a := x (n + 1)\n    (xor (!a) borrow, !a && borrow)"}, {"name": "map", "content": "abbrev map (f : Bool → Bool) : BitStream → BitStream :=\n  fun x i => f (x i)"}, {"name": "ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}, {"name": "Term.denote", "content": "def Term.denote (w : Nat) (t : Term) (vars : List (BitVec w)) : BitVec w :=\n  match t with\n  | ofNat n => BitVec.ofNat w n\n  | var n => vars.getD n default\n  | zero => 0#w\n  | negOne => -1#w\n  | one  => 1#w\n  | and a b => (a.denote w vars) &&& (b.denote w vars)\n  | or a b => (a.denote w vars) ||| (b.denote w vars)\n  | xor a b => (a.denote w vars) ^^^ (b.denote w vars)\n  | not a => ~~~ (a.denote w vars)\n  | add a b => (a.denote w vars) + (b.denote w vars)\n  | sub a b => (a.denote w vars) - (b.denote w vars)\n  | neg a => - (a.denote w vars)\n  \n  \n  | shiftL a n => (a.denote w vars) <<< n"}, {"name": "Predicate.denote", "content": "def Predicate.denote (p : Predicate) (w : Nat) (vars : List (BitVec w)) : Prop :=\n  match p with\n  | .width .ge k => k ≤ w \n  | .width .gt k => k < w \n  | .width .le k => w ≤ k\n  | .width .lt k => w < k\n  | .width .neq k => w ≠ k\n  | .width .eq k => w = k\n  | .binary .eq t₁ t₂ => t₁.denote w vars = t₂.denote w vars\n  | .binary .neq t₁ t₂ => t₁.denote w vars ≠ t₂.denote w vars\n  | .binary .sle  t₁ t₂ => ((t₁.denote w vars).sle (t₂.denote w vars)) = true\n  | .binary .slt  t₁ t₂ => ((t₁.denote w vars).slt (t₂.denote w vars)) = true\n  | .binary .ule  t₁ t₂ => ((t₁.denote w vars).ule (t₂.denote w vars)) = true\n  | .binary .ult  t₁ t₂ => (t₁.denote w vars).ult (t₂.denote w vars) = true\n  | .land  p q => p.denote w vars ∧ q.denote w vars\n  | .lor  p q => p.denote w vars ∨ q.denote w vars"}, {"name": "WidthPredicate", "content": "inductive WidthPredicate\n| eq\n| neq\n| lt\n| le\n| gt\n| ge\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "toBitVec", "content": "def toBitVec (w : Nat) (x : BitStream) : BitVec w :=\n  match w with\n  | 0   => 0#0\n  | w+1 => (x.toBitVec w).cons (x w)"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "concat", "content": "def concat (b : Bool) (x : BitStream) : BitStream\n  | 0   => b\n  | i+1 => x i"}, {"name": "sub", "content": "def sub (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).1"}, {"name": "add", "content": "def add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1"}, {"name": "addAux", "content": "def addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}], "lib_lemmas": [{"name": "BitVec.lt_def", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.of_length_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.ult_eq_not_carry", "module": "Init.Data.BitVec.Bitblast"}, {"name": "BitVec.eq_of_getLsbD_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "Bool.not_eq_false", "module": "Init.SimpLemmas"}, {"name": "BitVec.eq_of_toInt_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "Bool.not_eq_true", "module": "Init.SimpLemmas"}, {"name": "Bool.not_false", "module": "Init.SimpLemmas"}, {"name": "Bool.not_true", "module": "Init.SimpLemmas"}, {"name": "decide_false", "module": "Init.Core"}, {"name": "decide_true", "module": "Init.Core"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "BitVec.neg_one_eq_allOnes", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Bool.true_and", "module": "Init.SimpLemmas"}, {"name": "BitVec.msb_eq_getLsbD_last", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.le_def", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.slt_eq_ult", "module": "Init.Data.BitVec.Lemmas"}], "repo_lemmas": [{"name": "subAux_eq_BitVec_carry", "content": "@[simp] theorem subAux_eq_BitVec_carry (a b : BitStream) (w i : Nat) (hi : i < w) :\n    (a.subAux b i).2 = !(BitVec.carry (i + 1) (a.toBitVec w) ((~~~b).toBitVec w) true)"}, {"name": "scanOr_true_iff", "content": "theorem scanOr_true_iff (s : BitStream) (n : Nat)\n    : s.scanOr n = true ↔ ∃ (i : Nat), (i ≤ n) ∧ s i = true"}, {"name": "scanOr_false_iff", "content": "theorem scanOr_false_iff (s : BitStream) (n : Nat) : s.scanOr n = false ↔ ∀ (i : Nat), (hi : i ≤ n) → s i = false"}, {"name": "scanOr_succ", "content": "@[simp]\ntheorem scanOr_succ (s : BitStream) : scanOr s (n+1) = ((s.scanOr n) || s (n + 1))"}, {"name": "scanAnd_true_iff", "content": "theorem scanAnd_true_iff (s : BitStream) (n : Nat) :\n    s.scanAnd n = true ↔ ∀ (i : Nat), (hi : i ≤ n) → s i = true"}, {"name": "scanAnd_succ", "content": "@[simp] theorem scanAnd_succ (s : BitStream) : scanAnd s (n+1) = ((s.scanAnd n) && s (n + 1))"}, {"name": "scanAnd_false_iff", "content": "theorem scanAnd_false_iff (s : BitStream) (n : Nat)\n    : s.scanAnd n = false ↔ ∃ (i : Nat), (i ≤ n) ∧ s i = false"}, {"name": "and_eq", "content": "@[simp] theorem and_eq : (x &&& y) i = (x i && y i)"}, {"name": "getLsbD_toBitVec", "content": "@[simp] theorem getLsbD_toBitVec (w : Nat) (x : BitStream) :\n    (x.toBitVec w).getLsbD i = ((decide (i < w)) && x i)"}, {"name": "xor_eq", "content": "@[simp] theorem xor_eq : (x ^^^ y) i = (xor (x i) (y i))"}], "used_local_defs": [{"name": "Term.eval", "content": "def Term.eval (t : Term) (vars : List BitStream) : BitStream :=\n  match t with\n  | var n       => vars.getD n default\n  | zero        => BitStream.zero\n  | one         => BitStream.one\n  | negOne      => BitStream.negOne\n  | ofNat n     => BitStream.ofNat n\n  | and t₁ t₂   => (t₁.eval vars) &&& (t₂.eval vars)\n  | or t₁ t₂    => (t₁.eval vars) ||| (t₂.eval vars)\n  | xor t₁ t₂   => (t₁.eval vars) ^^^ (t₂.eval vars)\n  | not t       => ~~~(t.eval vars)\n  | add t₁ t₂   => (Term.eval t₁ vars) + (Term.eval t₂ vars)\n  | sub t₁ t₂   => (Term.eval t₁ vars) - (Term.eval t₂ vars)\n  | neg t       => -(Term.eval t vars)\n\n\n  | shiftL t n  => BitStream.shiftLeft (Term.eval t vars) n"}, {"name": "Predicate.evalEq", "content": "def Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr"}, {"name": "Predicate.evalNeq", "content": "def Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd"}, {"name": "Predicate.evalLor", "content": "def Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)"}, {"name": "Predicate.evalLand", "content": "def Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)"}, {"name": "Predicate.evalUlt", "content": "def Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true"}, {"name": "Predicate.evalMsbEq", "content": "def Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false"}, {"name": "Predicate.evalSlt", "content": "def Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))"}, {"name": "Predicate.eval", "content": "def Predicate.eval (p : Predicate) (vars : List BitStream) : BitStream :=\n  match p with\n  | .width .eq n => BitStream.falseIffEq n\n  | .width .neq n => BitStream.falseIffNeq n\n  | .width .lt n => BitStream.falseIffLt n\n  | .width .le n => BitStream.falseIffLe n\n  | .width .gt n => BitStream.falseIffGt n\n  | .width .ge n => BitStream.falseIffGe n\n  | lor p q => Predicate.evalLor (p.eval vars) (q.eval vars)\n  | land p q => Predicate.evalLand (p.eval vars) (q.eval vars)\n  | binary .eq t₁ t₂ => Predicate.evalEq (t₁.eval vars) (t₂.eval vars)\n   \n  | binary .neq t1 t2 => Predicate.evalNeq (t1.eval vars) (t2.eval vars)\n  | binary .ult t₁ t₂ => Predicate.evalUlt (t₁.eval vars) (t₂.eval vars)\n  | binary .ule t₁ t₂ =>\n     Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalUlt (t₁.eval vars) (t₂.eval vars))\n  | binary .slt t₁ t₂ => Predicate.evalSlt (t₁.eval vars) (t₂.eval vars)\n  | binary .sle t₁ t₂ => Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalSlt (t₁.eval vars) (t₂.eval vars))"}], "used_local_lemmas": [{"name": "Term.eval_eq_denote", "content": "@[simp] theorem Term.eval_eq_denote (t : Term) (w : Nat) (vars : List (BitVec w)) :\n    (t.eval (vars.map BitStream.ofBitVecSext)).toBitVec w = t.denote w vars"}, {"name": "Term.eval_eq_denote_apply", "content": "theorem Term.eval_eq_denote_apply (t : Term) {w : Nat} {vars : List (BitVec w)}\n    {i : Nat} (hi : i < w) :\n    (t.eval (vars.map BitStream.ofBitVecSext)) i = (t.denote w vars).getLsbD i"}, {"name": "Predicate.evalEq_denote_false_iff", "content": "theorem Predicate.evalEq_denote_false_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalEq (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = false ↔\n    Term.denote w a vars = Term.denote w b vars"}, {"name": "Predicate.evalEq_iff_not_evalNeq", "content": "theorem Predicate.evalEq_iff_not_evalNeq (a b : BitStream) :\n    ∀ (w : Nat), evalEq a b w ↔ ¬ (evalNeq a b w)"}, {"name": "Predicate.evalNeq_denote", "content": "theorem Predicate.evalNeq_denote {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalNeq (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = false ↔\n    Term.denote w a vars ≠ Term.denote w b vars"}, {"name": "Predicate.evalEq_denote_true_iff", "content": "theorem Predicate.evalEq_denote_true_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalEq (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = true ↔\n    Term.denote w a vars ≠ Term.denote w b vars"}, {"name": "BitVec.lt_eq_decide_ult", "content": "private theorem BitVec.lt_eq_decide_ult {x y : BitVec w} : (x < y) = decide (x.ult y)"}, {"name": "Predicate.evalUlt_denote_false_iff", "content": "theorem Predicate.evalUlt_denote_false_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalUlt (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = false ↔\n    (Term.denote w a vars < Term.denote w b vars)"}, {"name": "Predicate.evalUlt_denote_true_iff", "content": "theorem Predicate.evalUlt_denote_true_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalUlt (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = true ↔\n      (Term.denote w b vars) ≤ (Term.denote w a vars)"}, {"name": "evalMsbEq_denote_false_iff", "content": "private theorem evalMsbEq_denote_false_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    Predicate.evalMsbEq (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = false ↔\n   ((Term.denote w a vars).msb = (Term.denote w b vars).msb)"}, {"name": "evalMsbEq_denote_true_iff", "content": "private theorem evalMsbEq_denote_true_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    Predicate.evalMsbEq (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = true ↔\n   ((Term.denote w a vars).msb ≠ (Term.denote w b vars).msb)"}, {"name": "eq_true_iff_of_eq_false_iff", "content": "theorem eq_true_iff_of_eq_false_iff (b : Bool) (rhs : Prop) (h : (b = false) ↔ rhs) :\n    (b = true) ↔ ¬ rhs"}, {"name": "Predicate.evalSlt_denote_false_iff", "content": "private theorem Predicate.evalSlt_denote_false_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalSlt (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = false ↔\n    (Term.denote w a vars).slt (Term.denote w b vars)"}, {"name": "Predicate.evalSlt_denote_true_iff", "content": "private theorem Predicate.evalSlt_denote_true_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalSlt (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = true ↔\n    ¬ (Term.denote w a vars).slt (Term.denote w b vars)"}, {"name": "BitVec.ForLean.ule_iff_ult_or_eq'", "content": "private theorem BitVec.ForLean.ule_iff_ult_or_eq' (x y : BitVec w) : (x.ule y) = (decide (x = y ∨ x < y))"}, {"name": "BitVec.sle_iff_slt_or_eq", "content": "private theorem BitVec.sle_iff_slt_or_eq (x y : BitVec w) : x.sle y ↔ (decide (x = y) ∨ x.slt y)"}, {"name": "BitVec.ult_notation_eq_decide_ult", "content": "theorem BitVec.ult_notation_eq_decide_ult (x y : BitVec w) : (x.ult y) = decide (x < y)"}], "local_ctx": "import Mathlib.Data.Bool.Basic\n\nimport Mathlib.Data.Fin.Basic\n\nimport Blase.Fast.BitStream\n\nimport Blase.SingleWidth.Defs\n\nopen Term\n\nopen BitStream in\n\ndef Term.eval (t : Term) (vars : List BitStream) : BitStream :=\n  match t with\n  | var n       => vars.getD n default\n  | zero        => BitStream.zero\n  | one         => BitStream.one\n  | negOne      => BitStream.negOne\n  | ofNat n     => BitStream.ofNat n\n  | and t₁ t₂   => (t₁.eval vars) &&& (t₂.eval vars)\n  | or t₁ t₂    => (t₁.eval vars) ||| (t₂.eval vars)\n  | xor t₁ t₂   => (t₁.eval vars) ^^^ (t₂.eval vars)\n  | not t       => ~~~(t.eval vars)\n  | add t₁ t₂   => (Term.eval t₁ vars) + (Term.eval t₂ vars)\n  | sub t₁ t₂   => (Term.eval t₁ vars) - (Term.eval t₂ vars)\n  | neg t       => -(Term.eval t vars)\n\n\n  | shiftL t n  => BitStream.shiftLeft (Term.eval t vars) n\n\ndef Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr\n\ndef Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd\n\ndef Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)\n\ndef Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)\n\ndef Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true\n\ndef Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false\n\ndef Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))\n\nopen BitStream in\n\ndef Predicate.eval (p : Predicate) (vars : List BitStream) : BitStream :=\n  match p with\n  | .width .eq n => BitStream.falseIffEq n\n  | .width .neq n => BitStream.falseIffNeq n\n  | .width .lt n => BitStream.falseIffLt n\n  | .width .le n => BitStream.falseIffLe n\n  | .width .gt n => BitStream.falseIffGt n\n  | .width .ge n => BitStream.falseIffGe n\n  | lor p q => Predicate.evalLor (p.eval vars) (q.eval vars)\n  | land p q => Predicate.evalLand (p.eval vars) (q.eval vars)\n  | binary .eq t₁ t₂ => Predicate.evalEq (t₁.eval vars) (t₂.eval vars)\n   \n  | binary .neq t1 t2 => Predicate.evalNeq (t1.eval vars) (t2.eval vars)\n  | binary .ult t₁ t₂ => Predicate.evalUlt (t₁.eval vars) (t₂.eval vars)\n  | binary .ule t₁ t₂ =>\n     Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalUlt (t₁.eval vars) (t₂.eval vars))\n  | binary .slt t₁ t₂ => Predicate.evalSlt (t₁.eval vars) (t₂.eval vars)\n  | binary .sle t₁ t₂ => Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalSlt (t₁.eval vars) (t₂.eval vars))\n\nsection Predicate\n\nend Predicate", "target_theorem": "theorem Predicate.eval_eq_denote (w : Nat) (p : Predicate) (vars : List (BitVec w)) :\n    (p.eval (vars.map .ofBitVecSext) w = false) ↔ p.denote w vars :=", "ground_truth_proof": ":= by\n  induction p generalizing vars w\n  case width wp n => cases wp <;> simp [eval, denote]\n  case binary p a b =>\n    cases p with\n    | eq => simp [eval, denote]; apply evalEq_denote_false_iff\n    | neq => simp [eval, denote]; apply evalNeq_denote\n    | ult =>\n      simp [eval, denote]\n      rw [BitVec.ult_notation_eq_decide_ult]\n      by_cases h: Term.denote w a vars < Term.denote w b vars\n      case pos => simp only [h, decide_true, Bool.not_true]; rw [evalUlt_denote_false_iff]; exact h\n      case neg => simp only [h, decide_false, Bool.not_false]; rw [evalUlt_denote_true_iff]; simpa using h\n    | slt =>\n      simp [eval, denote];\n      by_cases h : (Term.denote w a vars).slt (Term.denote w b vars)\n      · rw [h]\n        simp [Predicate.evalSlt_denote_false_iff, h]\n      · simp at h\n        rw [h]\n        simp only [Bool.not_false]\n        by_contra h'\n        simp only [Bool.not_eq_true] at h'\n        rw [Predicate.evalSlt_denote_false_iff] at h'\n        simp only [h, Bool.false_eq_true] at h'\n    | ule =>\n      simp [eval, denote];\n      simp only [evalLor, BitStream.and_eq]\n      rw [BitVec.ForLean.ule_iff_ult_or_eq' (Term.denote w a vars) (Term.denote w b vars)]\n      by_cases heq : Term.denote w a vars = Term.denote w b vars\n      · rw [heq]\n        simp [evalEq_denote_false_iff a b vars |>.mpr heq]\n      · simp [heq]\n        by_cases hlt : Term.denote w a vars < Term.denote w b vars\n        · simp [hlt]\n          simp [evalUlt_denote_false_iff a b vars |>.mpr hlt]\n        · simp [hlt]\n          have := evalEq_denote_false_iff a b vars |>.not |>.mpr heq\n          simp only [this, true_and]\n          have := evalUlt_denote_false_iff a b vars |>.not |>.mpr hlt\n          simp only [this]\n    | sle =>\n      simp [eval, denote]\n      simp only [evalLor, BitStream.and_eq]\n      have h := BitVec.sle_iff_slt_or_eq (Term.denote w a vars) (Term.denote w b vars) |>.eq\n      rcases hSle : (Term.denote w a vars).sle (Term.denote w b vars)\n      · simp [hSle] at h ⊢\n        obtain ⟨h₁, h₂⟩ := h\n        simp [evalEq_denote_true_iff .. |>.mpr h₁]\n        rw [evalSlt_denote_true_iff .. |>.mpr]\n        simp [h₂]\n      · simp [hSle] at h ⊢\n        intros hEq\n        simp [evalEq_denote_true_iff .. |>.mp hEq] at h\n        apply evalSlt_denote_false_iff .. |>.mpr h\n  case land p q hp hq => simp [eval, denote, hp, hq, evalLand]\n  case lor p q hp hq =>\n    simp [eval, denote]\n    simp only [evalLor, BitStream.and_eq]\n    constructor\n    · intros heval\n      by_cases hp' : p.denote w vars\n      · simp [hp']\n      · by_cases hq' : q.denote w vars\n        · simp [hq']\n        · have := hp .. |>.not |>.mpr hp'\n          simp [this] at heval\n          have := hq .. |>.not |>.mpr hq'\n          simp [this] at heval\n    · intros hdenote\n      rcases hdenote with hp' | hq'\n      · have := hp .. |>.mpr hp'\n        simp [this]\n      · have := hq .. |>.mpr hq'\n        simp [this]", "nesting_depth": 9, "transitive_dep_count": 96, "subset_aristotle": false, "category": "Compiler"}
{"id": 299, "thm_name": "eq_true_iff_of_eq_false_iff", "thm_stmt": "theorem eq_true_iff_of_eq_false_iff (b : Bool) (rhs : Prop) (h : (b = false) ↔ rhs) :\n    (b = true) ↔ ¬ rhs", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/Fast/Defs.lean", "imports": ["import Blase.SingleWidth.Defs", "import Mathlib.Data.Fin.Basic", "import Blase.Fast.BitStream", "import Mathlib.Data.Bool.Basic"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Mathlib.Data.Bool.Basic\n\nimport Mathlib.Data.Fin.Basic\n\nimport Blase.Fast.BitStream\n\nimport Blase.SingleWidth.Defs\n\nopen Term\n\nopen BitStream in\n\nopen BitStream in\n\nsection Predicate\n\nend Predicate", "target_theorem": "theorem eq_true_iff_of_eq_false_iff (b : Bool) (rhs : Prop) (h : (b = false) ↔ rhs) :\n    (b = true) ↔ ¬ rhs :=", "ground_truth_proof": ":= by\nconstructor\n· intros h'\n  apply h.not.mp\n  simp [h']\n· intros h'\n  by_contra hcontra\n  simp at hcontra\n  have := h.mp hcontra\n  exact h' this", "nesting_depth": 1, "transitive_dep_count": 2, "subset_aristotle": false, "category": "Compiler"}
{"id": 300, "thm_name": "ReflectVerif.BvDecide.KInductionCircuits.FSM.carryWith_congrEnv_envBitstream_set_of_le", "thm_stmt": "theorem FSM.carryWith_congrEnv_envBitstream_set_of_le (fsm : FSM arity)\n    (s0 : fsm.α → Bool) (env : arity → BitStream) (n : Nat) (v : arity → Bool)\n    (k : Nat) (hk : k ≤ n) :\n  fsm.carryWith s0 (envBitstream_set env n v) k =\n  fsm.carryWith s0 env k", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/KInduction/KInduction.lean", "imports": ["import Blase.Fast.Defs", "import Blase.Vars", "import Blase.Fast.ForLean", "import Lean.Meta.ForEachExpr", "import Mathlib.Data.Bool.Basic", "import Mathlib.Data.Finset.Defs", "import Blase.Fast.FiniteStateMachine", "import Blase.SingleWidth.Syntax", "import Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "import Blase.Fast.BitStream", "import Blase.EnvBitstream", "import Blase.Blase.Fast.FiniteStateMachine", "import Mathlib.Data.Finset.Basic", "import Mathlib.Data.Fin.Basic", "import Blase.Fast.Decide", "import Lean", "import Mathlib.Data.Multiset.FinsetOps"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Empty", "module": "Init.Prelude"}, {"name": "Empty.elim", "module": "Init.Core"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}], "used_repo_defs": [{"name": "syntax \"llvm.and\"     : MLIR.Pretty.uniform_op", "content": "syntax \"llvm.and\"     : MLIR.Pretty.uniform_op\n\nsyntax \"llvm.ashr\"    : MLIR.Pretty.exact_op\n\nsyntax \"llvm.add\"     : MLIR.Pretty.overflow_op\n\nsyntax \"llvm.return\"  : MLIR.Pretty.uniform_op\n\nsyntax \"return\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  \n  | `(mlir_op| $v:mlir_op_operand = mod_arith.constant $x:neg_num : $t) =>\n    `(mlir_op| $v:mlir_op_operand = \"mod_arith.constant\" () {value = $x:neg_num} : () -> ($t))\n\n  \n  | `(mlir_op| $v:mlir_op_operand = mod_arith.constant ${ $x:term } : $t) => do\n      let ctor := mkIdent ``MLIR.AST.AttrValue.int\n      let x ← `($ctor $x [mlir_type| i64])\n      `(mlir_op| $v:mlir_op_operand = \"mod_arith.constant\" () {value = $$($x)} : () -> ($t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $resName:mlir_op_operand = $name:InstCombine.cmp_op_name $x, $y $[: $t]?) => do\n    let some opName := extractOpName name.raw\n      | Macro.throwUnsupported\n    let t ← t.getDM `(mlir_type| _)\n    `(mlir_op| $resName:mlir_op_operand = $opName ($x, $y) : ($t, $t) -> (i1) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $resName:mlir_op_operand = $name:InstCombine.int_cast_op $x : $t to $t') => do\n    let some opName := extractOpName name.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $resName:mlir_op_operand = $opName ($x) : ($t) -> $t')"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant( $x $[: $inner_type]?)\n      $[: $outer_type]? ) => do\n       \n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      let inner_type := inner_type.getD outer_type\n      `(mlir_op| $res:mlir_op_operand = \"llvm.mlir.constant\"()\n          {value = $x:neg_num : $inner_type} : () -> ($outer_type) )\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant( ${ $x:term }) $[: $t]?) => do\n      let t ← t.getDM `(mlir_type| _)\n      let x ← `(MLIR.AST.AttrValue.int $x [mlir_type| $t])\n      `(mlir_op| $res:mlir_op_operand = \"llvm.mlir.constant\"() {value = $$($x) } : () -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (true) $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (1 : i1) : i1)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (false) $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (0 : i1) : i1)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant $x $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant($x $[: $t]?) $[: $t]?)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant ${ $x:term } $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant($$($x) $[: $t]?) $[: $t]?)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.icmp $p $x, $y $[: $t]?) => do\n    let t ← t.getDM `(mlir_type| _)\n    match p.getString with\n      | \"eq\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.eq\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ne\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ne\" ($x, $y) : ($t, $t) -> (i1))\n      | \"slt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.slt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sle\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sle\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sgt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sgt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sge\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sge\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ult\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ult\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ule\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ule\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ugt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ugt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"uge\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.uge\" ($x, $y) : ($t, $t) -> (i1))\n      | _ => Macro.throwErrorAt p s!\"unexpected predicate {p.getString}\""}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.select $c, $x, $y $[: $t]?) => do\n    let t ← t.getDM `(mlir_type| _)\n    `(mlir_op| $res:mlir_op_operand = \"llvm.select\" ($c, $x, $y) : (i1, $t, $t) -> ($t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $v:mlir_op_operand = arith.const $x:neg_num : $t) =>\n      `(mlir_op| $v:mlir_op_operand = \"arith.const\" () {value = $x:neg_num } : () -> ($t))\n  | `(mlir_op| $v:mlir_op_operand = arith.const ${ $x:term } : $t) => do\n      let ctor := mkIdent ``MLIR.AST.AttrValue.int\n      let x ← `($ctor $x [mlir_type| i64])\n      \n      \n      \n      \n      `(mlir_op| $v:mlir_op_operand = \"arith.const\" () {value = $$($x) } : () -> ($t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $v:mlir_op_operand = poly.const $x:neg_num : $t) =>\n      `(mlir_op| $v:mlir_op_operand = \"poly.const\" () {value = $x:neg_num } : () -> ($t))\n  | `(mlir_op| $v:mlir_op_operand = poly.const ${ $x:term } : $t) => do\n      let ctor := mkIdent ``MLIR.AST.AttrValue.int\n      let x ← `($ctor $x [mlir_type| i64])\n      \n      \n      \n      \n      `(mlir_op| $v:mlir_op_operand = \"poly.const\" () {value = $$($x) } : () -> ($t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $v:mlir_op_operand = poly.monomial $xs,* : ($ts,*) -> $t) =>\n      `(mlir_op| $v:mlir_op_operand = \"poly.monomial\" ($xs,*) : ($ts,*) -> $t)"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "carryWith", "content": "def carryWith (p : FSM arity) (carryState : p.α → Bool) (x : arity → BitStream) (n : Nat) : p.α → Bool := fun a =>\n  (p.changeInitCarry carryState).carry x n a"}, {"name": "changeInitCarry", "content": "def changeInitCarry (p : FSM arity) (c : p.α → Bool) : FSM arity :=\n  { p with initCarry := c }"}, {"name": "zero", "content": "def zero : FSM (Fin 0) :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := Empty.elim,\n    outputCirc := Circuit.fals\n  }"}, {"name": "carry", "content": "def carry (x : arity → BitStream) : ℕ → p.State\n  | 0 => p.initCarry\n  | n+1 => (p.nextBit (carry x n) (fun i => x i n)).1"}, {"name": "State", "content": "abbrev State : Type := p.α → Bool"}, {"name": "nextBit", "content": "def nextBit : p.State → (arity → Bool) → p.State × Bool :=\n  fun carry inputBits =>\n    let input := Sum.elim carry inputBits\n    let newState : p.State  := fun (a : p.α) => (p.nextStateCirc a).eval input\n    let outBit : Bool       := (p.outputCirc).eval input\n    (newState, outBit)"}, {"name": "rhs", "content": "def rhs := [poly q, n, hq | {\n^bb0(%a : !R):\n  return %a : !R\n\n}]"}, {"name": "AttrValue", "content": "inductive AttrValue where\n  | symbol: String -> AttrValue \n  | str : String -> AttrValue\n  | int : Int -> MLIRType φ -> AttrValue\n  | nat: Nat -> AttrValue\n  | bool : Bool -> AttrValue\n  | float : Float -> MLIRType φ -> AttrValue\n  | type : MLIRType φ -> AttrValue\n  | affine: AffineMap -> AttrValue\n  | permutation: List Nat -> AttrValue \n  | list: List AttrValue -> AttrValue\n  | nestedsymbol: AttrValue -> AttrValue -> AttrValue\n  | alias: String -> AttrValue\n  | dict: AttrDict -> AttrValue\n  | opaque_: (dialect: String) -> (value: String) -> AttrValue\n  | opaqueElements: (dialect: String) -> (value: String) -> (type: MLIRType φ) -> AttrValue\n  | unit: AttrValue"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}, {"name": "bb0", "content": "def bb0 : Region 0 := [mlir_region|\n{\n  ^bb0(%arg0: i32):\n    %0 = llvm.mlir.constant(8) : i32\n    %1 = llvm.mlir.constant(31) : i32\n    %2 = llvm.ashr %arg0, %1 : i32\n    %3 = llvm.and %2, %0 : i32\n    %4 = llvm.add %3, %2 : i32\n    llvm.return %4 : i32\n  }]"}, {"name": "Region", "content": "structure Region where\n  (name: String)\n  (args: List <| TypedSSAVal φ)\n  (ops: List Op)"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "carryWith_congrEnv", "content": "theorem carryWith_congrEnv {p : FSM arity}\n    {carryState : p.α → Bool} {x y : arity → BitStream} {n : Nat}\n    (h : ∀ a i, i < n → x a i = y a i) :\n    p.carryWith carryState x n = p.carryWith carryState y n"}, {"name": "carry_congrEnv", "content": "theorem carry_congrEnv {p : FSM arity}\n    {x y : arity → BitStream} {n : Nat} (h : ∀ a i, i < n → x a i = y a i) :\n    p.carry x n = p.carry y n"}], "used_local_defs": [{"name": "ReflectVerif.BvDecide.KInductionCircuits", "content": "structure KInductionCircuits {arity : Type _}\n  [DecidableEq arity] [Fintype arity] [Hashable arity] (fsm : FSM arity) (n : Nat) where\n  \n  cInitCarryAssignCirc : Circuit (Vars fsm.α arity 0)\n  \n  cSuccCarryAssignCirc : Circuit (Vars fsm.α arity (n+2))\n  \n  cOutAssignCirc : Circuit (Vars fsm.α arity (n + 2))\n  \n  cStatesUniqueCirc : Circuit (Vars fsm.α arity n)"}, {"name": "ReflectVerif.BvDecide.KInductionCircuits.envBitstream_set", "content": "def envBitstream_set (x : arity → BitStream) (n : Nat) (v : arity → Bool) :\n    arity → BitStream :=\n  fun a j => if j = n then v a else x a j"}], "used_local_lemmas": [], "local_ctx": "import Mathlib.Data.Bool.Basic\n\nimport Mathlib.Data.Fin.Basic\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Finset.Defs\n\nimport Mathlib.Data.Multiset.FinsetOps\n\nimport Blase.Fast.BitStream\n\nimport Blase.Fast.Defs\n\nimport Blase.Fast.FiniteStateMachine\n\nimport Blase.Fast.Decide\n\nimport Blase.SingleWidth.Syntax\n\nimport Lean.Meta.ForEachExpr\n\nimport Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec\n\nimport Blase.Fast.ForLean\n\nimport Blase.Vars\n\nimport Blase.EnvBitstream\n\nimport Lean\n\nopen Fin.NatCast\n\nnamespace ReflectVerif\n\nopen Lean Meta Elab Tactic\n\nnamespace BvDecide\n\nopen Std Sat AIG\n\nstructure KInductionCircuits {arity : Type _}\n  [DecidableEq arity] [Fintype arity] [Hashable arity] (fsm : FSM arity) (n : Nat) where\n  \n  cInitCarryAssignCirc : Circuit (Vars fsm.α arity 0)\n  \n  cSuccCarryAssignCirc : Circuit (Vars fsm.α arity (n+2))\n  \n  cOutAssignCirc : Circuit (Vars fsm.α arity (n + 2))\n  \n  cStatesUniqueCirc : Circuit (Vars fsm.α arity n)\n\nnamespace KInductionCircuits\n\nvariable {arity : Type _}\n  {fsm : FSM arity}\n\nvariable [DecidableEq arity] [Fintype arity] [Hashable arity]\n\ndef envBitstream_set (x : arity → BitStream) (n : Nat) (v : arity → Bool) :\n    arity → BitStream :=\n  fun a j => if j = n then v a else x a j", "target_theorem": "theorem FSM.carryWith_congrEnv_envBitstream_set_of_le (fsm : FSM arity)\n    (s0 : fsm.α → Bool) (env : arity → BitStream) (n : Nat) (v : arity → Bool)\n    (k : Nat) (hk : k ≤ n) :\n  fsm.carryWith s0 (envBitstream_set env n v) k =\n  fsm.carryWith s0 env k :=", "ground_truth_proof": ":= by\n  apply FSM.carryWith_congrEnv\n  intros a l hl\n  simp [envBitstream_set]\n  intros hcontra; omega", "nesting_depth": 5, "transitive_dep_count": 33, "subset_aristotle": false, "category": "Compiler"}
{"id": 301, "thm_name": "TermUnop.alt_lang", "thm_stmt": "lemma TermUnop.alt_lang {t : Term} (op : TermUnop) :\n  (op.subst_arity' ▸ (op.subst t).language) =\n    let lop : Set (BitVecs 2) := op.openTerm_arity' ▸ op.openTerm.language\n    let lop' : Set (BitVecs (t.arity + 2)) := lop.lift (λ i ↦ i.natAdd t.arity)\n    let lt : Set (BitVecs (t.arity + 2)) := t.language.lift (λ i ↦ i.castLE (by omega))\n    let l := lt ∩ lop'\n    l.proj (liftUnop t.arity)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Fin.natAdd", "module": "Init.Data.Fin.Basic"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "syntax \"xor\" : MLIR.Pretty.uniform_op", "content": "syntax \"xor\" : MLIR.Pretty.uniform_op\n\nsyntax \"max\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "sub", "content": "def sub (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).1"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "add", "content": "def add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1"}, {"name": "addAux", "content": "def addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "neg", "content": "def neg (x : BitStream) : BitStream :=\n  fun n => (negAux x n).1"}, {"name": "negAux", "content": "def negAux (x : BitStream) : Nat → Bool × Bool\n  | 0 => (x 0, !(x 0))\n  | n+1 =>\n    let borrow := (negAux x n).2\n    let a := x (n + 1)\n    (xor (!a) borrow, !a && borrow)"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "BitVecs.cast", "content": "def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=\n  { w := bvs.w, bvs := h ▸ bvs.bvs }"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "Term.language", "content": "def Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }"}, {"name": "Formula.arity", "content": "@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity"}, {"name": "Term.evalFinBV", "content": "@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n"}, {"name": "Term.arity", "content": "@[simp] def Term.arity : Term → Nat\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| ofNat _ => 0\n| Term.and t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.or t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.xor t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.not t => arity t\n| add t₁ t₂ => max (arity t₁) (arity t₂)\n| sub t₁ t₂ => max (arity t₁) (arity t₂)\n| neg t => arity t\n\n\n| shiftL t .. => arity t"}, {"name": "BitVec.transport", "content": "def BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=\n  BitVec.ofFn fun i => bv.getLsbD (f i)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "List.Vector.transport", "content": "def List.Vector.transport (v : Vector α m) (f : Fin n → Fin m) : Vector α n :=\n  Vector.ofFn fun i => v.get (f i)"}, {"name": "BitVecs'.transport", "content": "def BitVecs'.transport (f : Fin n → Fin m) (bvs' : BitVecs' m): BitVecs' n :=\n  bvs'.map fun bv => bv.transport f"}, {"name": "BitVecs.transport", "content": "def BitVecs.transport (f : Fin n → Fin m) (bvs : BitVecs m) : BitVecs n :=\n  { w := bvs.w, bvs := bvs.bvs.transport f }"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.ext_iff", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "ite_cond_eq_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}, {"name": "List.Vector.append_get_ge", "content": "@[simp]\nlemma List.Vector.append_get_ge {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: n ≤ i) :\n    (x ++ y).get i = y.get ((i.cast (Nat.add_comm n m) |>.subNat n hlt))"}, {"name": "List.Vector.append_get_lt", "content": "@[simp]\nlemma List.Vector.append_get_lt {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: i < n) :\n    (x ++ y).get i = x.get (i.castLT hlt)"}, {"name": "BitVecs.transport_getElem", "content": "@[simp]\nlemma BitVecs.transport_getElem {bvs : BitVecs m} (f : Fin n → Fin m) (i : Fin n) :\n    (bvs.transport f).bvs.get i = bvs.bvs.get (f i)"}], "used_local_defs": [{"name": "NFA.msbState", "content": "inductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype"}, {"name": "liftUnop", "content": "def liftUnop n : Fin (n + 1) → Fin (n + 2) :=\n  fun k =>\n    if k = n then Fin.last (n+1) else k.castLE (by admit /- proof elided -/\n    )"}, {"name": "TermBinop", "content": "inductive TermBinop where\n| and | or | xor | add | sub"}, {"name": "TermBinop.subst", "content": "def TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂"}, {"name": "TermBinop.openTerm", "content": "def TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)"}, {"name": "TermUnop", "content": "inductive TermUnop where\n| neg | not | shiftL (k : Nat)"}, {"name": "TermUnop.openTerm", "content": "def TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k"}, {"name": "TermUnop.openTerm_arity'", "content": "@[simp]\ndef TermUnop.openTerm_arity' (op : TermUnop) : op.openTerm.arity + 1 = 2 :="}, {"name": "TermUnop.subst", "content": "def TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k"}, {"name": "swapLastTwo", "content": "def swapLastTwo (x : Fin (n + 2)) : Fin (n + 2) :=\n  if x = Fin.last (n + 1) then n else if x = n then Fin.last (n + 1) else x"}], "used_local_lemmas": [{"name": "TermBinop.subst_arity'", "content": "lemma TermBinop.subst_arity' {op : TermBinop} : (op.subst t₁ t₂).arity + 1= t₁.arity ⊔ t₂.arity + 1"}, {"name": "BitVecs.cast_eq", "content": "@[simp]\nlemma BitVecs.cast_eq  (x : BitVecs n) (h : n = n') : h ▸ x = x.cast h"}, {"name": "Fin.natAdd_zero'", "content": "lemma Fin.natAdd_zero' [h : NeZero m] : Fin.natAdd (m := m) n 0 = n"}, {"name": "TermUnop.subst_arity'", "content": "@[simp]\nlemma TermUnop.subst_arity' {op : TermUnop} : (op.subst t).arity + 1 = t.arity + 1"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\nsection fsm\n\nvariable {arity : Type} [FinEnum arity]\n\nopen BitStream in\n\nend fsm\n\nsection nfas_relations\n\ninductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype\n\nend nfas_relations\n\ndef liftUnop n : Fin (n + 1) → Fin (n + 2) :=\n  fun k =>\n    if k = n then Fin.last (n+1) else k.castLE (by admit /- proof elided -/\n    )\n\ninductive TermBinop where\n| and | or | xor | add | sub\n\ndef TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂\n\ndef TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)\n\ninductive TermUnop where\n| neg | not | shiftL (k : Nat)\n\ndef TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k\n\n@[simp]\ndef TermUnop.openTerm_arity' (op : TermUnop) : op.openTerm.arity + 1 = 2 :=\n\ndef TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k\n\ndef swapLastTwo (x : Fin (n + 2)) : Fin (n + 2) :=\n  if x = Fin.last (n + 1) then n else if x = n then Fin.last (n + 1) else x", "target_theorem": "lemma TermUnop.alt_lang {t : Term} (op : TermUnop) :\n  (op.subst_arity' ▸ (op.subst t).language) =\n    let lop : Set (BitVecs 2) :=", "ground_truth_proof": ":= op.openTerm_arity' ▸ op.openTerm.language\n    let lop' : Set (BitVecs (t.arity + 2)) := lop.lift (λ i ↦ i.natAdd t.arity)\n    let lt : Set (BitVecs (t.arity + 2)) := t.language.lift (λ i ↦ i.castLE (by omega))\n    let l := lt ∩ lop'\n    l.proj (liftUnop t.arity)\n    := by\n  simp [Term.language]\n  ext bvs\n  simp\n\n  generalize_proofs h₁ h₂ h₃ h₄ h₅\n  constructor\n  · rintro heq\n    let bvs' := bvs.bvs ++ (t.evalFinBV (λ n ↦ bvs.bvs.get n) ::ᵥ List.Vector.nil) |>.transport swapLastTwo\n    use ⟨_, bvs'⟩\n    split_ands\n    · simp [bvs']\n      have heq : (swapLastTwo (Fin.castLE h₃ (Fin.last t.arity))) = Fin.last (t.arity + 1) := by\n        simp [swapLastTwo]\n        split_ifs with h₁ h₂\n        · exfalso; rw [Fin.ext_iff] at h₁\n          simp at h₁\n        · rfl\n        · exfalso; apply h₂; ext; simp; exact Eq.symm (Nat.mod_eq_of_lt h₃)\n      rw [heq]\n      rw [List.Vector.append_get_ge]\n      on_goal 2 => simp\n      simp\n      congr\n      ext1 x\n      rw [List.Vector.append_get_lt]\n      on_goal 2 => simp +arith\n      congr!\n    · rw [BitVecs.cast_eq] at heq ⊢\n      unfold BitVecs.cast\n      simp\n      simp [BitVecs.cast] at heq\n      have hget : bvs'.get (Fin.natAdd t.arity 1) = bvs.bvs.get t.arity := by\n        simp [bvs']\n        rw [List.Vector.append_get_lt]\n        on_goal 2 => simp +arith; exact Nat.mod_le t.arity (t.arity + 2)\n        · congr\n          ext\n          simp +arith\n      rw [hget]\n      convert heq using 1\n      · have hbvs' : bvs'.get t.arity = t.evalFinBV fun n => bvs.bvs.get n := by\n          simp [bvs']\n        simp [openTerm]\n        rcases op <;> simp [hbvs', Fin.natAdd_zero'] <;> rfl\n      · congr!;  ext; simp\n    · ext1\n      · simp\n      next i =>\n        simp [bvs', liftUnop]\n        split_ifs with h\n        · subst h\n          simp\n          rw [List.Vector.append_get_lt]\n          on_goal 2 => simp +arith; exact Nat.mod_le t.arity (t.arity + 2)\n          congr\n          ext; simp +arith\n        · simp [h]\n          congr!\n  · rintro ⟨bvs', ⟨⟨heq₁, heq₂⟩, heq₃⟩⟩\n    rw [BitVecs.cast_eq] at *\n    simp [BitVecs.cast] at *\n    rw [←heq₃]\n    conv_rhs =>\n      simp only [BitVecs.transport_getElem]\n      simp [liftUnop]\n      rw [ite_cond_eq_true]\n      rfl\n      tactic => simp only [eq_iff_iff, iff_true]; ext1; simp\n    convert heq₂ using 1\n    rcases op with _ | _ | _ <;>\n    . simp [subst, openTerm] at *; congr", "nesting_depth": 4, "transitive_dep_count": 63, "subset_aristotle": false, "category": "Compiler"}
{"id": 302, "thm_name": "formula_language", "thm_stmt": "theorem formula_language (φ : Formula) :\n    φ.language = { (bvs : BitVecs φ.arity) | φ.sat (fun k => bvs.bvs.get k) }", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/Defs.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.ForMathlib"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.card", "module": "Mathlib.Data.FinEnum"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "atom", "module": "Leanwuzla.Sexp.Basic"}, {"name": "Fin.isValue", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Fin"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Set.instMembership", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Mem", "module": "Mathlib.Data.Set.Defs"}, {"name": "List.Vector.get", "module": "Mathlib.Data.Vector.Defs"}], "used_repo_defs": [{"name": "syntax \"max\" : MLIR.Pretty.uniform_op", "content": "syntax \"max\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "WidthPredicate", "content": "inductive WidthPredicate\n| eq\n| neq\n| lt\n| le\n| gt\n| ge\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "Term.arity", "content": "@[simp] def Term.arity : Term → Nat\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| ofNat _ => 0\n| Term.and t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.or t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.xor t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.not t => arity t\n| add t₁ t₂ => max (arity t₁) (arity t₂)\n| sub t₁ t₂ => max (arity t₁) (arity t₂)\n| neg t => arity t\n\n\n| shiftL t .. => arity t"}, {"name": "BitVecs.transport", "content": "def BitVecs.transport (f : Fin n → Fin m) (bvs : BitVecs m) : BitVecs n :=\n  { w := bvs.w, bvs := bvs.bvs.transport f }"}, {"name": "BitVec.transport", "content": "def BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=\n  BitVec.ofFn fun i => bv.getLsbD (f i)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "List.Vector.transport", "content": "def List.Vector.transport (v : Vector α m) (f : Fin n → Fin m) : Vector α n :=\n  Vector.ofFn fun i => v.get (f i)"}, {"name": "BitVecs'.transport", "content": "def BitVecs'.transport (f : Fin n → Fin m) (bvs' : BitVecs' m): BitVecs' n :=\n  bvs'.map fun bv => bv.transport f"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "BitVecs.cast", "content": "def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=\n  { w := bvs.w, bvs := h ▸ bvs.bvs }"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.val_last", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "le_add_iff_nonneg_right", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "zero_le", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "Set.compl_def", "module": "Mathlib.Order.BooleanAlgebra.Set"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_inter_iff", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.preimage_setOf_eq", "module": "Mathlib.Data.Set.Image"}], "repo_lemmas": [{"name": "List.Vector.append_get_ge", "content": "@[simp]\nlemma List.Vector.append_get_ge {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: n ≤ i) :\n    (x ++ y).get i = y.get ((i.cast (Nat.add_comm n m) |>.subNat n hlt))"}, {"name": "List.Vector.append_get_lt", "content": "@[simp]\nlemma List.Vector.append_get_lt {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: i < n) :\n    (x ++ y).get i = x.get (i.castLT hlt)"}, {"name": "BitVecs.transport_getElem", "content": "@[simp]\nlemma BitVecs.transport_getElem {bvs : BitVecs m} (f : Fin n → Fin m) (i : Fin n) :\n    (bvs.transport f).bvs.get i = bvs.bvs.get (f i)"}], "used_local_defs": [{"name": "liftMaxSucc1", "content": "def liftMaxSucc1 (n m : Nat) : Fin (n + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = n then Fin.last (max n m) else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMaxSucc2", "content": "def liftMaxSucc2 (n m : Nat) : Fin (m + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = m then Fin.last (max n m + 1) else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftLast2", "content": "def liftLast2 n : Fin 2 → Fin (n + 2)\n| 0 => n\n| 1 => Fin.last (n + 1)"}, {"name": "liftExcept2", "content": "def liftExcept2 n : Fin n → Fin (n + 2) :=\n  fun k => Fin.castLE (by admit /- proof elided -/\n  ) k"}, {"name": "liftMax1", "content": "def liftMax1 (n m : Nat) : Fin n → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMax2", "content": "def liftMax2 (n m : Nat) : Fin m → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )"}, {"name": "Term.evalFinBV", "content": "@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n"}, {"name": "Term.language", "content": "def Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }"}, {"name": "RelationOrdering", "content": "inductive RelationOrdering\n| lt | le | gt | ge\nderiving Repr, Fintype"}, {"name": "Relation", "content": "inductive Relation\n| eq\n| signed (ord : RelationOrdering)\n| unsigned (ord : RelationOrdering)\nderiving Repr"}, {"name": "evalRelation", "content": "def evalRelation (rel : Relation) {w} (bv1 bv2 : BitVec w) : Prop :=\n  match rel with\n  | .eq => bv1 = bv2\n  | .signed .lt => bv1.slt bv2\n  | .signed .le => bv1.sle  bv2\n  | .signed .gt => bv2.slt bv1\n  | .signed .ge => bv2.sle bv1\n  | .unsigned .lt => bv1.ult bv2\n  | .unsigned .le => bv1.ule bv2\n  | .unsigned .gt => bv2.ult bv1\n  | .unsigned .ge => bv2.ule bv1"}, {"name": "Relation.language", "content": "@[simp]\ndef Relation.language (rel : Relation) : Set (BitVecs 2) :=\n  { bvs | evalRelation rel (bvs.bvs.get 0) (bvs.bvs.get 1) }"}, {"name": "Binop", "content": "inductive Binop\n| and | or | impl | equiv\nderiving Repr"}, {"name": "evalBinop", "content": "def evalBinop (op : Binop) (b1 b2 : Prop) : Prop :=\n  match op with\n  | .and => b1 ∧ b2\n  | .or => b1 ∨ b2\n  | .impl => b1 → b2\n  | .equiv => b1 ↔ b2"}, {"name": "langBinop", "content": "def langBinop (op : Binop) (l1 l2 : Set (BitVecs n)) : Set (BitVecs n) :=\n  match op with\n  | .and => l1 ∩ l2\n  | .or => l1 ∪ l2\n  | .impl => l1ᶜ ∪ l2\n  | .equiv => (l1ᶜ ∪ l2) ∩ (l2ᶜ ∪ l1)"}, {"name": "Unop", "content": "inductive Unop\n| neg\nderiving Repr"}, {"name": "Formula", "content": "inductive Formula : Type\n| width : WidthPredicate → Nat → Formula\n| atom : Relation → Term → Term → Formula\n| msbSet : Term → Formula\n| unop : Unop → Formula → Formula\n| binop : Binop → Formula → Formula → Formula\nderiving Repr"}, {"name": "Formula.arity", "content": "@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity"}, {"name": "WidthPredicate.sat", "content": "@[simp]\ndef WidthPredicate.sat (wp : WidthPredicate) (w n : Nat) : Bool :=\n  match wp with\n  | .eq => w = n\n  | .neq => w ≠ n\n  | .lt => w < n\n  | .le => w ≤ n\n  | .gt => w > n\n  | .ge => w ≥ n"}, {"name": "Formula.sat", "content": "@[simp]\ndef Formula.sat {w : Nat} (φ : Formula) (ρ : Fin φ.arity → BitVec w) : Prop :=\n  match φ with\n  | .width wp n => wp.sat w n\n  | .atom rel t1 t2 =>\n    let bv1 := t1.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let bv2 := t2.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalRelation rel bv1 bv2\n  | .unop .neg φ => ¬ φ.sat ρ\n  | .binop op φ1 φ2 =>\n    let b1 := φ1.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let b2 := φ2.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalBinop op b1 b2\n  | .msbSet t => (t.evalFinBV ρ).msb"}, {"name": "_root_.Set.lift", "content": "@[simp]\ndef _root_.Set.lift (f : Fin n → Fin m) (bvs : Set (BitVecs n)) : Set (BitVecs m) :=\n  BitVecs.transport f ⁻¹' bvs"}, {"name": "_root_.Set.proj", "content": "@[simp]\ndef _root_.Set.proj (f : Fin n → Fin m) (bvs : Set (BitVecs m)) : Set (BitVecs n) :=\n  BitVecs.transport f '' bvs"}, {"name": "langMsb", "content": "@[simp]\ndef langMsb : Set (BitVecs 1) := { bvs | bvs.bvs.get 0 |>.msb }"}, {"name": "Formula.language", "content": "@[simp]\ndef Formula.language (φ : Formula) : Set (BitVecs φ.arity) :=\n  match φ with\n  | .width wp n => { bvs | wp.sat bvs.w n }\n  | .atom rel t1 t2 =>\n    let l1 := t1.language.lift (liftMaxSucc1 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let l2 := t2.language.lift (liftMaxSucc2 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let lrel := rel.language.lift $ liftLast2 (max (FinEnum.card (Fin t1.arity)) (FinEnum.card (Fin t2.arity)))\n    let l := lrel ∩ l1 ∩ l2\n    l.proj (liftExcept2 _)\n  | .unop .neg φ => φ.languageᶜ\n  | .binop op φ1 φ2 =>\n    let l1 := φ1.language.lift $ liftMax1 φ1.arity φ2.arity\n    let l2 := φ2.language.lift $ liftMax2 φ1.arity φ2.arity\n    langBinop op l1 l2\n  | .msbSet t =>\n    let lmsb := langMsb.lift $ fun _ => Fin.last t.arity\n    let l' := t.language ∩ lmsb\n    l'.proj fun n => n.castLE (by admit /- proof elided -/\n    )"}], "used_local_lemmas": [{"name": "evalFin_eq", "content": "lemma evalFin_eq {t : Term} {vars1 : Fin t.arity → BitVec w1} {vars2 : Fin t.arity → BitVec w2} :\n    ∀ (heq : w1 = w2),\n    (∀ n, vars1 n = heq ▸ vars2 n) →\n    t.evalFinBV vars1 = heq ▸ t.evalFinBV vars2"}, {"name": "evalRelation_coe", "content": "@[simp]\nlemma evalRelation_coe (rel : Relation) (bv1 bv2 : BitVec w1) (heq : w1 = w2) :\n    evalRelation rel (heq ▸ bv1) (heq ▸ bv2) = evalRelation rel bv1 bv2"}, {"name": "helper1", "content": "lemma helper1 : (k = 0) → (x ::ᵥ vs).get k = x"}, {"name": "helper2", "content": "lemma helper2 : (k = 1) → (x ::ᵥ y ::ᵥ vs).get k = y"}, {"name": "formula_language_case_atom", "content": "lemma formula_language_case_atom :\n    let φ := Formula.atom rel t1 t2\n    φ.language = λ (bvs : BitVecs φ.arity) => φ.sat (fun k => bvs.bvs.get k)"}], "local_ctx": "import Blase.AutoStructs.ForMathlib\n\nimport Blase.SingleWidth.Defs\n\nopen Fin.NatCast\n\ndef liftMaxSucc1 (n m : Nat) : Fin (n + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = n then Fin.last (max n m) else k.castLE (by admit /- proof elided -/\n  )\n\ndef liftMaxSucc2 (n m : Nat) : Fin (m + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = m then Fin.last (max n m + 1) else k.castLE (by admit /- proof elided -/\n  )\n\ndef liftLast2 n : Fin 2 → Fin (n + 2)\n| 0 => n\n| 1 => Fin.last (n + 1)\n\ndef liftExcept2 n : Fin n → Fin (n + 2) :=\n  fun k => Fin.castLE (by admit /- proof elided -/\n  ) k\n\ndef liftMax1 (n m : Nat) : Fin n → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )\n\ndef liftMax2 (n m : Nat) : Fin m → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )\n\n@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n\n\ndef Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }\n\ninductive RelationOrdering\n| lt | le | gt | ge\nderiving Repr, Fintype\n\ninductive Relation\n| eq\n| signed (ord : RelationOrdering)\n| unsigned (ord : RelationOrdering)\nderiving Repr\n\ndef evalRelation (rel : Relation) {w} (bv1 bv2 : BitVec w) : Prop :=\n  match rel with\n  | .eq => bv1 = bv2\n  | .signed .lt => bv1.slt bv2\n  | .signed .le => bv1.sle  bv2\n  | .signed .gt => bv2.slt bv1\n  | .signed .ge => bv2.sle bv1\n  | .unsigned .lt => bv1.ult bv2\n  | .unsigned .le => bv1.ule bv2\n  | .unsigned .gt => bv2.ult bv1\n  | .unsigned .ge => bv2.ule bv1\n\n@[simp]\ndef Relation.language (rel : Relation) : Set (BitVecs 2) :=\n  { bvs | evalRelation rel (bvs.bvs.get 0) (bvs.bvs.get 1) }\n\ninductive Binop\n| and | or | impl | equiv\nderiving Repr\n\ndef evalBinop (op : Binop) (b1 b2 : Prop) : Prop :=\n  match op with\n  | .and => b1 ∧ b2\n  | .or => b1 ∨ b2\n  | .impl => b1 → b2\n  | .equiv => b1 ↔ b2\n\ndef langBinop (op : Binop) (l1 l2 : Set (BitVecs n)) : Set (BitVecs n) :=\n  match op with\n  | .and => l1 ∩ l2\n  | .or => l1 ∪ l2\n  | .impl => l1ᶜ ∪ l2\n  | .equiv => (l1ᶜ ∪ l2) ∩ (l2ᶜ ∪ l1)\n\ninductive Unop\n| neg\nderiving Repr\n\ninductive Formula : Type\n| width : WidthPredicate → Nat → Formula\n| atom : Relation → Term → Term → Formula\n| msbSet : Term → Formula\n| unop : Unop → Formula → Formula\n| binop : Binop → Formula → Formula → Formula\nderiving Repr\n\n@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity\n\n@[simp]\ndef WidthPredicate.sat (wp : WidthPredicate) (w n : Nat) : Bool :=\n  match wp with\n  | .eq => w = n\n  | .neq => w ≠ n\n  | .lt => w < n\n  | .le => w ≤ n\n  | .gt => w > n\n  | .ge => w ≥ n\n\n@[simp]\ndef Formula.sat {w : Nat} (φ : Formula) (ρ : Fin φ.arity → BitVec w) : Prop :=\n  match φ with\n  | .width wp n => wp.sat w n\n  | .atom rel t1 t2 =>\n    let bv1 := t1.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let bv2 := t2.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalRelation rel bv1 bv2\n  | .unop .neg φ => ¬ φ.sat ρ\n  | .binop op φ1 φ2 =>\n    let b1 := φ1.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let b2 := φ2.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalBinop op b1 b2\n  | .msbSet t => (t.evalFinBV ρ).msb\n\n@[simp]\ndef _root_.Set.lift (f : Fin n → Fin m) (bvs : Set (BitVecs n)) : Set (BitVecs m) :=\n  BitVecs.transport f ⁻¹' bvs\n\n@[simp]\ndef _root_.Set.proj (f : Fin n → Fin m) (bvs : Set (BitVecs m)) : Set (BitVecs n) :=\n  BitVecs.transport f '' bvs\n\n@[simp]\ndef langMsb : Set (BitVecs 1) := { bvs | bvs.bvs.get 0 |>.msb }\n\n@[simp]\ndef Formula.language (φ : Formula) : Set (BitVecs φ.arity) :=\n  match φ with\n  | .width wp n => { bvs | wp.sat bvs.w n }\n  | .atom rel t1 t2 =>\n    let l1 := t1.language.lift (liftMaxSucc1 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let l2 := t2.language.lift (liftMaxSucc2 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let lrel := rel.language.lift $ liftLast2 (max (FinEnum.card (Fin t1.arity)) (FinEnum.card (Fin t2.arity)))\n    let l := lrel ∩ l1 ∩ l2\n    l.proj (liftExcept2 _)\n  | .unop .neg φ => φ.languageᶜ\n  | .binop op φ1 φ2 =>\n    let l1 := φ1.language.lift $ liftMax1 φ1.arity φ2.arity\n    let l2 := φ2.language.lift $ liftMax2 φ1.arity φ2.arity\n    langBinop op l1 l2\n  | .msbSet t =>\n    let lmsb := langMsb.lift $ fun _ => Fin.last t.arity\n    let l' := t.language ∩ lmsb\n    l'.proj fun n => n.castLE (by admit /- proof elided -/\n    )", "target_theorem": "theorem formula_language (φ : Formula) :\n    φ.language = { (bvs : BitVecs φ.arity) | φ.sat (fun k => bvs.bvs.get k) } :=", "ground_truth_proof": ":= by\n  let n : Nat := φ.arity\n  induction φ\n  case width wp n =>\n    simp\n  case atom rel t1 t2 =>\n    apply formula_language_case_atom\n  case unop op φ ih =>\n    rcases op; simp [ih, Set.compl_def]\n  case binop op φ1 φ2 ih1 ih2 =>\n    unfold Formula.language\n    ext1 bvs\n    simp [ih1, ih2]\n    have heq1 : (φ1.sat fun k => bvs.bvs.get (liftMax1 φ1.arity φ2.arity k)) ↔\n           (φ1.sat fun n => bvs.bvs.get (Fin.castLE (by simp) n)) := by\n      congr!\n    have heq2 : (φ2.sat fun k => bvs.bvs.get (liftMax2 φ1.arity φ2.arity k)) = true ↔\n           (φ2.sat fun n => bvs.bvs.get (Fin.castLE (by simp) n)) = true := by\n      congr!\n    rcases op <;>\n      simp [evalBinop, langBinop, Set.instMembership] <;> simp_all <;> tauto\n  case msbSet t =>\n    ext1 bvs; simp only [Formula.arity, Formula.language, Set.proj, Set.lift, langMsb, Fin.isValue,\n      Set.preimage_setOf_eq, Set.mem_image, Set.mem_inter_iff,\n      Set.mem_setOf_eq, Formula.sat]\n    rcases bvs with ⟨w, bvs⟩\n    constructor\n    · rintro ⟨bvsb, ⟨ht, hmsb⟩, heq⟩\n      simp only [Fin.isValue, Formula.arity] at ht hmsb ⊢\n      unfold Term.language at ht\n      simp only [BitVecs.transport, List.Vector.transport] at hmsb\n      simp at ht; rw [←ht] at hmsb; rw [←hmsb]\n      simp [BitVecs.transport] at heq\n      obtain ⟨hw, hbvs⟩ := heq\n      simp; congr 1; simp [hw]\n      rcases hw; simp\n      congr 1; ext1 k\n      simp at hbvs; simp [←hbvs, List.Vector.transport]; congr\n    · intros heq\n      use ⟨w,\n        bvs ++ ((t.evalFinBV fun k => bvs.get $ k.castLE (by simp)) ::ᵥ List.Vector.nil)⟩\n      unfold Term.language\n      simp [BitVecs.transport, List.Vector.transport] at heq ⊢\n      constructor; assumption\n      ext1 k; simp; congr 1", "nesting_depth": 5, "transitive_dep_count": 80, "subset_aristotle": false, "category": "Compiler"}
{"id": 303, "thm_name": "createSink_trans", "thm_stmt": "@[grind =, simp]\nlemma createSink_trans [LawfulBEq A] {m : RawCNFA A} (hwf : m.WF) :\n    s₂ ∈ m.createSink.2.tr s₁ a ↔\n      (s₁ = m.stateMax ∧ s₂ = m.stateMax) ∨\n      (s₁ ∈ m.states ∧ s₂ ∈ m.states ∧ s₂ ∈ m.tr s₁ a)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/Basic.lean", "imports": ["import Blase.Blase.AutoStructs.ForLean", "import Blase.AutoStructs.BundledNfa", "import Mathlib.Algebra.Group.Nat.Range", "import Blase.FinEnum"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.toList", "module": "Mathlib.Data.FinEnum"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "LawfulBEq", "module": "Init.Core"}, {"name": "List", "module": "Init.Prelude"}, {"name": "LawfulHashable", "module": "Init.Data.LawfulHashable"}, {"name": "Inhabited", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Std.HashSet.mem_insert", "module": "Std.Data.HashSet.Lemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "beq_iff_eq", "module": "Init.Core"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "Std.HashMap.getD_eq_getD_getElem?", "module": "Std.Data.HashMap.Lemmas"}, {"name": "Finset.range_add", "module": "Mathlib.Algebra.Group.Nat.Range"}, {"name": "FinEnum.mem_toList", "module": "Mathlib.Data.FinEnum"}], "repo_lemmas": [{"name": "Std.HashMap.mem_of_getElem?", "content": "@[aesop 50% unsafe]\ntheorem Std.HashMap.mem_of_getElem? [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] {m : Std.HashMap K V} {k : K} :\n    m[k]? = some v → k ∈ m"}, {"name": "Std.HashMap.mem_iff_getElem?", "content": "theorem Std.HashMap.mem_iff_getElem? [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] [Inhabited V] {m : Std.HashMap K V} {k : K} :\n     k ∈ m ↔ ∃ v, m[k]? = some v"}], "used_local_defs": [{"name": "State", "content": "abbrev State := Nat"}, {"name": "RawCNFA", "content": "structure RawCNFA (A : Type 0) [BEq A] [Hashable A] [DecidableEq A] [FinEnum A] where\n  stateMax : State\n  initials : Std.HashSet State\n  finals : Std.HashSet State\n  trans : Std.HashMap (State × A) (Std.HashSet State)\nderiving Repr"}, {"name": "RawCNFA.tr", "content": "@[inline]\ndef RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅"}, {"name": "RawCNFA.states", "content": "def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax"}, {"name": "RawCNFA", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "RawCNFA.newState", "content": "def RawCNFA.newState (m : RawCNFA A) : State × RawCNFA A :=\n  let old := m.stateMax\n  let m := { m with stateMax := old + 1 }\n  (old, m)"}, {"name": "RawCNFA.addTrans", "content": "def RawCNFA.addTrans (m : RawCNFA A) (a : A) (s s' : State) : RawCNFA A :=\n  let ns := m.trans.getD (s, a) ∅\n  let ns := ns.insert s'\n  { m with trans :=  m.trans.insert (s, a) ns }"}, {"name": "RawCNFA.addInitial", "content": "def RawCNFA.addInitial (m : RawCNFA A) (s : State) : RawCNFA A :=\n  { m with initials := m.initials.insert s }"}, {"name": "RawCNFA.createSink", "content": "def RawCNFA.createSink (m : RawCNFA A) : State × RawCNFA A :=\n  let (s, m) := m.newState\n  let m := m.addInitial s\n  let m := FinEnum.toList (α := A).foldl (init := m) fun m a =>\n    m.addTrans a s s\n  (s, m)"}, {"name": "RawCNFA.WF", "content": "structure RawCNFA.WF (m : RawCNFA A) where\n  initials_lt : ∀ {s}, s ∈ m.initials → s ∈ m.states\n  finals_lt : ∀ {s}, s ∈ m.finals → s ∈ m.states\n  trans_src_lt : ∀ s_a ∈ m.trans, s_a.1 ∈ m.states\n  trans_tgt_lt : s' ∈ m.tr s a → s' ∈ m.states"}], "used_local_lemmas": [{"name": "RawCNFA.addInitial_tr", "content": "@[grind =, simp]\nlemma RawCNFA.addInitial_tr {m : RawCNFA A} : (m.addInitial s'').tr s a = m.tr s a"}, {"name": "states_addInitial", "content": "@[grind =, simp, aesop 50% unsafe]\nlemma states_addInitial (m : RawCNFA A) (s' : State) :\n    (m.addInitial s').states = m.states"}, {"name": "addTrans_tr", "content": "@[grind =] -- TODO: should I?\nlemma addTrans_tr (m : RawCNFA A) [LawfulBEq A] {a b : A} {s₁ s₁' s₂ : State} :\n    (m.addTrans a s₁ s₂).tr s₁' b = if s₁ = s₁' ∧ a = b then (m.tr s₁ a).insert s₂ else m.tr s₁' b"}, {"name": "mem_addTrans_tr", "content": "@[grind =, simp]\nlemma mem_addTrans_tr (m : RawCNFA A) [LawfulBEq A] (a : A) (s1 s2 : State) :\n    s' ∈ (m.addTrans a s1 s2).tr s b ↔\n      (s = s1 ∧ s' = s2 ∧ b = a) ∨ s' ∈ m.tr s b"}, {"name": "newState_tr", "content": "@[grind =, simp]\nlemma newState_tr {m : RawCNFA A} : m.newState.2.tr s a = m.tr s a"}, {"name": "states_newState", "content": "@[grind =, simp, aesop 50% unsafe]\nlemma states_newState (m : RawCNFA A) :\n    m.newState.2.states = m.states ∪ { m.stateMax }"}, {"name": "RawCNFA.WF.trans_src_lt'", "content": "@[grind ., simp, aesop 50% unsafe]\nlemma RawCNFA.WF.trans_src_lt' {m : RawCNFA A} (hwf : m.WF) :\n    ∀ {s a}, (s, a) ∈ m.trans → s ∈ m.states"}, {"name": "RawCNFA.WF.trans_src_lt''", "content": "@[grind ., simp, aesop 50% unsafe]\nlemma RawCNFA.WF.trans_src_lt'' [LawfulBEq A] {m : RawCNFA A} (hwf : m.WF) :\n    ∀ {s a s'}, s' ∈ m.tr s a → s ∈ m.states"}], "local_ctx": "import Mathlib.Algebra.Group.Nat.Range\n\nimport Blase.AutoStructs.BundledNfa\n\nimport Blase.FinEnum\n\nopen SetRel\n\nabbrev State := Nat\n\nstructure RawCNFA (A : Type 0) [BEq A] [Hashable A] [DecidableEq A] [FinEnum A] where\n  stateMax : State\n  initials : Std.HashSet State\n  finals : Std.HashSet State\n  trans : Std.HashMap (State × A) (Std.HashSet State)\nderiving Repr\n\nsection sim\n\nvariable {A : Type} [BEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\n@[inline]\ndef RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅\n\ndef RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax\n\ninstance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort\n\nend sim\n\nsection basics\n\nvariable {A : Type} [BEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\ndef RawCNFA.newState (m : RawCNFA A) : State × RawCNFA A :=\n  let old := m.stateMax\n  let m := { m with stateMax := old + 1 }\n  (old, m)\n\ndef RawCNFA.addTrans (m : RawCNFA A) (a : A) (s s' : State) : RawCNFA A :=\n  let ns := m.trans.getD (s, a) ∅\n  let ns := ns.insert s'\n  { m with trans :=  m.trans.insert (s, a) ns }\n\ndef RawCNFA.addInitial (m : RawCNFA A) (s : State) : RawCNFA A :=\n  { m with initials := m.initials.insert s }\n\ndef RawCNFA.createSink (m : RawCNFA A) : State × RawCNFA A :=\n  let (s, m) := m.newState\n  let m := m.addInitial s\n  let m := FinEnum.toList (α := A).foldl (init := m) fun m a =>\n    m.addTrans a s s\n  (s, m)\n\nstructure RawCNFA.WF (m : RawCNFA A) where\n  initials_lt : ∀ {s}, s ∈ m.initials → s ∈ m.states\n  finals_lt : ∀ {s}, s ∈ m.finals → s ∈ m.states\n  trans_src_lt : ∀ s_a ∈ m.trans, s_a.1 ∈ m.states\n  trans_tgt_lt : s' ∈ m.tr s a → s' ∈ m.states", "target_theorem": "@[grind =, simp]\nlemma createSink_trans [LawfulBEq A] {m : RawCNFA A} (hwf : m.WF) :\n    s₂ ∈ m.createSink.2.tr s₁ a ↔\n      (s₁ = m.stateMax ∧ s₂ = m.stateMax) ∨\n      (s₁ ∈ m.states ∧ s₂ ∈ m.states ∧ s₂ ∈ m.tr s₁ a) :=", "ground_truth_proof": ":= by\n  unfold RawCNFA.createSink\n  simp\n  let motive (mᵢ : RawCNFA A) (hwf : mᵢ.WF) (as : List A) :=\n    mᵢ.states = m.states ∪ {m.stateMax} →\n    ∀ a,\n      s₂ ∈ (as.foldl (λ (m' : RawCNFA A) a => m'.addTrans a m.stateMax m.stateMax)\n              (init := mᵢ)).tr s₁ a ↔\n        (s₁ = m.stateMax ∧ s₂ = m.stateMax ∧ a ∈ as) ∨\n        (s₂ ∈ mᵢ.tr s₁ a)\n  suffices h : motive (m.newState.2.addInitial m.stateMax) (by simp_all) (FinEnum.toList A) by\n    simp only [FinEnum.mem_toList, and_true, motive] at h\n    rw [h] <;> simp only [RawCNFA.addInitial_tr, newState_tr,\n                          states_addInitial, states_newState]\n    constructor; on_goal 2 => tauto\n    rintro (h | h); (exact .inl h); right\n    use (RawCNFA.WF.trans_src_lt'' hwf h),\n        (RawCNFA.WF.trans_tgt_lt hwf h),\n        h\n  generalize_proofs hwf'; revert hwf'\n  generalize (m.newState.2.addInitial m.stateMax) = mi\n  induction FinEnum.toList A generalizing mi\n  case nil =>\n    rintro hwf'; simp [motive]\n  case cons a as ih =>\n    simp [motive]\n    rintro hwf' hstates b\n    simp [motive] at ih\n    rw [ih]\n    · simp only [mem_addTrans_tr]; tauto\n    · simp [hwf', hstates]\n    · simp [hstates]", "nesting_depth": 4, "transitive_dep_count": 42, "subset_aristotle": false, "category": "Compiler"}
{"id": 304, "thm_name": "ReflectVerif.BvDecide.KInductionCircuits.IsLawful_mkSucc_of_IsLawful", "thm_stmt": "theorem IsLawful_mkSucc_of_IsLawful {arity : Type _}\n    [DecidableEq arity] [Fintype arity] [Hashable arity]\n    {fsm : FSM arity} {n : Nat}\n    (prev : KInductionCircuits fsm n)\n    (hPrev : prev.IsLawful) :\n    (mkSucc prev).IsLawful", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/KInduction/KInduction.lean", "imports": ["import Blase.Fast.Defs", "import Blase.Vars", "import Blase.Fast.ForLean", "import Blase.Blase.Vars", "import Lean.Meta.ForEachExpr", "import Mathlib.Data.Bool.Basic", "import Mathlib.Data.Finset.Defs", "import Blase.Fast.FiniteStateMachine", "import Blase.SingleWidth.Syntax", "import Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec", "import Blase.Blase.Fast.Circuit", "import Blase.Fast.BitStream", "import Blase.EnvBitstream", "import Mathlib.Data.Finset.Basic", "import Mathlib.Data.Fin.Basic", "import Blase.Fast.Decide", "import Lean", "import Mathlib.Data.Multiset.FinsetOps"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "input", "module": "Leanwuzla.Basic"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "FinEnum.toList", "module": "Mathlib.Data.FinEnum"}, {"name": "List.range", "module": "Init.Data.List.Basic"}, {"name": "Bool.xor", "module": "Init.Data.Bool"}, {"name": "Vector.mapM", "module": "Init.Data.Vector.Basic"}, {"name": "Subtype", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Vars.inputN", "content": "def Vars.inputN (inp : ι) (k : Nat) {n : Nat} (hkn : k < n := by admit /- proof elided -/\n) : Vars σ ι n :=\n  .inputs (Inputs.mk ⟨k, by admit /- proof elided -/\n  ⟩ inp)"}, {"name": "Vars", "content": "inductive Vars (σ : Type) (ι : Type) (n : Nat)\n| state (s : Inputs σ (n + 1))\n| inputs (is : Inputs ι n)\n| outputs (os : Fin n) \nderiving DecidableEq, Hashable"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Vars.stateN", "content": "def Vars.stateN (s : σ) (i : Nat) {n : Nat} (hin : i ≤ n := by admit /- proof elided -/\n) : Vars σ ι n :=\n  .state (Inputs.mk ⟨i, by admit /- proof elided -/\n  ⟩ s)"}, {"name": "Inputs", "content": "structure Inputs (ι : Type) (n : Nat) : Type  where\n  ix : Fin n\n  input : ι\nderiving DecidableEq, Hashable"}, {"name": "Vars.state0", "content": "def Vars.state0 (s : σ) {n : Nat} : Vars σ ι n :=\n  .state (Inputs.mk ⟨0, by admit /- proof elided -/\n  ⟩ s)"}, {"name": "bigOr", "content": "def bigOr {α : Type _}\n    (cs : List (Circuit α)) : Circuit α :=\n  match cs with\n  | [] => Circuit.fals\n  | c :: cs =>\n    c ||| (Circuit.bigOr cs)"}, {"name": "bigAnd", "content": "def bigAnd {α : Type _}\n    (cs : List (Circuit α)) : Circuit α :=\n  match cs with\n  | [] => Circuit.tru\n  | c :: cs =>\n    c &&& (Circuit.bigAnd cs)"}, {"name": "eval", "content": "@[simp]\ndef eval : Circuit α → (α → Bool) → Bool\n  | tru, _ => true\n  | fals, _ => false\n  | var b x, f => if b then f x else !(f x)\n  | and c₁ c₂, f => (eval c₁ f) && (eval c₂ f)\n  | or c₁ c₂, f => (eval c₁ f) || (eval c₂ f)\n  | xor c₁ c₂, f => Bool.xor (eval c₁ f) (eval c₂ f)"}, {"name": "map", "content": "def map : ∀ (_c : Circuit α) (_f : α → β), Circuit β\n  | tru, _ => tru\n  | fals, _ => fals\n  | var b x, f => var b (f x)\n  | and c₁ c₂, f => (map c₁ f) &&& (map c₂ f)\n  | or c₁ c₂, f => (map c₁ f) ||| (map c₂ f)\n  | xor c₁ c₂, f => (map c₁ f) ^^^ (map c₂ f)"}, {"name": "Vars.castLe", "content": "def Vars.castLe {n m : Nat} (v : Vars σ ι n) (hnm : n ≤ m) : Vars σ ι m :=\n  match v with\n  | .state ss => .state (ss.castLe (by admit /- proof elided -/\n  ))\n  | .inputs is => .inputs (is.castLe hnm)\n  | .outputs os =>\n    .outputs (os.castLE (by admit /- proof elided -/\n    ))"}, {"name": "castLe", "content": "def castLe (i : Inputs ι n) (hn : n ≤ m) : Inputs ι m where\n  ix := ⟨i.ix, by admit /- proof elided -/\n  ⟩\n  input := i.input"}], "lib_lemmas": [{"name": "List.mem_attach", "module": "Init.Data.List.Attach"}, {"name": "List.mem_map", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_range", "module": "Init.Data.List.Nat.Range"}, {"name": "forall_exists_index", "module": "Init.PropLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "eval_map", "content": "lemma eval_map {c : Circuit α} {f : α → β} {g : β → Bool} :\n    eval (map c f) g = eval c (λ x => g (f x))"}, {"name": "eval_bigOr_eq_false_iff", "content": "@[simp]\ntheorem eval_bigOr_eq_false_iff\n    (cs : List (Circuit α)) (env : α → Bool):\n    (Circuit.bigOr cs).eval env = false ↔\n    (∀ (c : Circuit α), c ∈ cs → c.eval env = false)"}, {"name": "Vars.castLe_inputs_eq_inputs", "content": "@[simp]\ntheorem Vars.castLe_inputs_eq_inputs {n i m : Nat} (hi : i < n) (hnm : n ≤ m) :\n  (Vars.inputN inp i hi : Vars σ ι n).castLe hnm =\n  Vars.inputN inp i (by omega)"}, {"name": "Vars.castLe_stateN_eq_stateN", "content": "@[simp]\ntheorem Vars.castLe_stateN_eq_stateN  {n i m : Nat} (hi : i ≤ n) (hnm : n ≤ m) :\n  (Vars.stateN s i : Vars σ ι n).castLe hnm =\n  Vars.stateN s i (hin := by omega)"}, {"name": "Vars.castLe_castLe_eq_castLe_self", "content": "@[simp]\ntheorem Vars.castLe_castLe_eq_castLe_self {α : Type _} {p q r : Nat}\n  (v : Vars α σ p) (h : p ≤ q) (h' : q ≤ r) :\n    (v.castLe h).castLe h' = v.castLe (by omega)"}], "used_local_defs": [{"name": "ReflectVerif.BvDecide.mkCarryAssignCircuitNAux", "content": "def mkCarryAssignCircuitNAux {arity : Type _}\n  [DecidableEq arity]\n  [Fintype arity]\n  [Hashable arity]\n  (p : FSM arity) (s : p.α) (n : Nat) : Circuit (Vars p.α arity (n + 1)) :=\n    (p.nextStateCirc s).map fun v =>\n      match v with\n        | .inl t => Vars.stateN t n\n        | .inr i => Vars.inputN i n"}, {"name": "ReflectVerif.BvDecide.mkCarryAssignCircuitN", "content": "def mkCarryAssignCircuitN {arity : Type _}\n  [DecidableEq arity]\n  [Fintype arity]\n  [Hashable arity]\n  (p : FSM arity) (n : Nat) :\n  Circuit (Vars p.α arity (n + 1)) :=\n  let carrys := FinEnum.toList p.α |>.map fun s =>\n    \n    Circuit.xor\n      (mkCarryAssignCircuitNAux p s n)\n      (Circuit.var true <| Vars.stateN s (n + 1))\n  Circuit.bigOr carrys"}, {"name": "ReflectVerif.BvDecide.mkOutputAssignCircuitNAux", "content": "def mkOutputAssignCircuitNAux {arity : Type _}\n  [DecidableEq arity]\n  [Fintype arity]\n  [Hashable arity]\n  (p : FSM arity) (n : Nat) : Circuit (Vars p.α arity (n + 1)) :=\n    (p.outputCirc).map fun v =>\n      match v with\n        | .inl s' => Vars.stateN s' n\n        | .inr i => Vars.inputN i n"}, {"name": "ReflectVerif.BvDecide.mkOutputAssignCircuitN", "content": "def mkOutputAssignCircuitN {arity : Type _}\n  [DecidableEq arity]\n  [Fintype arity]\n  [Hashable arity]\n  (p : FSM arity) (n : Nat) :\n  Circuit (Vars p.α arity (n + 1)) :=\n    Circuit.xor\n      (mkOutputAssignCircuitNAux p n)\n      (Circuit.var true <| Vars.outputs ⟨n, by admit /- proof elided -/\n      ⟩)"}, {"name": "ReflectVerif.BvDecide.mkStateNeqCircuit", "content": "def mkStateNeqCircuit\n  {arity : Type _} {i : Nat}\n  [DecidableEq arity] [Fintype arity] [Hashable arity]\n  (p : FSM arity) (s t : p.α → Circuit (Vars p.α arity i)) : Circuit (Vars p.α arity i) :=\n  Circuit.bigAnd <| FinEnum.toList p.α |>.map fun a => ~~~ (s a) ^^^ (t a)"}, {"name": "ReflectVerif.BvDecide.mkStateUniqueCircuitN", "content": "def mkStateUniqueCircuitN {arity : Type _}\n  [DecidableEq arity] [Fintype arity] [Hashable arity]\n  (p : FSM arity) (n : Nat) : Circuit (Vars p.α arity n) :=\n  let sn : p.α → Circuit (Vars p.α arity n) := fun s =>\n    Circuit.var true (Vars.stateN s n)\n  let circs := (List.range n).attach |>.map fun ⟨i, hi⟩ =>\n    let si : p.α → Circuit (Vars p.α arity n) := fun s =>\n      Circuit.var true (Vars.stateN s i (by admit /- proof elided -/\n      ))\n    (mkStateNeqCircuit p si sn)\n  Circuit.bigOr circs"}, {"name": "ReflectVerif.BvDecide.KInductionCircuits", "content": "structure KInductionCircuits {arity : Type _}\n  [DecidableEq arity] [Fintype arity] [Hashable arity] (fsm : FSM arity) (n : Nat) where\n  \n  cInitCarryAssignCirc : Circuit (Vars fsm.α arity 0)\n  \n  cSuccCarryAssignCirc : Circuit (Vars fsm.α arity (n+2))\n  \n  cOutAssignCirc : Circuit (Vars fsm.α arity (n + 2))\n  \n  cStatesUniqueCirc : Circuit (Vars fsm.α arity n)"}, {"name": "ReflectVerif.BvDecide.KInductionCircuits.IsLawful", "content": "structure KInductionCircuits.IsLawful {arity : Type _}\n    [DecidableEq arity] [Fintype arity] [Hashable arity] {fsm : FSM arity} {n : Nat}\n  (circs : KInductionCircuits fsm n) where\n  hCInitCarryAssignCirc :\n    ∀ {env : Vars fsm.α arity 0 → Bool},\n      (circs.cInitCarryAssignCirc.eval env = false)\n      ↔ (∀ (s : fsm.α), fsm.initCarry s = env (Vars.state0 s))\n\n  hCSuccCarryAssignCirc :\n    ∀ {env : Vars fsm.α arity (n + 2) → Bool},\n      (circs.cSuccCarryAssignCirc.eval env = false)\n      ↔ (∀ (s : fsm.α) (i : Nat) (hi : i < n + 2),\n        env (Vars.stateN s (i + 1)) =\n          ((mkCarryAssignCircuitNAux fsm s i).map\n            (fun v => v.castLe (by admit /- proof elided -/\n            ))).eval env)\n  hCOutAssignCirc :\n    ∀ {env : Vars fsm.α arity (n + 2) → Bool},\n      (circs.cOutAssignCirc.eval env = false)\n      ↔ (∀ (i : Nat) (hi : i < n + 2),\n        (fsm.outputCirc).eval\n          (fun x => match x with\n            | .inl s => env (Vars.stateN s i)\n            | .inr j => env (Vars.inputN j i)) =\n        env (Vars.outputs ⟨i, by admit /- proof elided -/\n        ⟩))\n  hCStatesUniqueCirc :\n    ∀ {env : Vars fsm.α arity (n) → Bool},\n      (circs.cStatesUniqueCirc.eval env = false)\n      ↔ (∀ (i j : Nat) (hij : i < j ∧ j ≤ n),\n        ∃ (s : fsm.α), env (Vars.stateN s i) ≠ env (Vars.stateN s j))"}, {"name": "ReflectVerif.BvDecide.KInductionCircuits.castCircLe", "content": "def castCircLe {n m : Nat} (c : Circuit (Vars fsm.α arity n)) (hnm : n ≤ m := by admit /- proof elided -/\n) :\n    Circuit (Vars fsm.α arity m) :=\n  c.map (fun v => v.castLe hnm)"}, {"name": "ReflectVerif.BvDecide.KInductionCircuits.mkSucc", "content": "def mkSucc\n    (prev : KInductionCircuits fsm n) :\n    KInductionCircuits fsm (n + 1) :=\n  let cInitCarryAssignCirc := prev.cInitCarryAssignCirc\n  { cInitCarryAssignCirc := cInitCarryAssignCirc\n  , cSuccCarryAssignCirc :=\n      (mkCarryAssignCircuitN fsm (n + 2)) |||\n      (castCircLe prev.cSuccCarryAssignCirc)\n  , cOutAssignCirc :=\n      (mkOutputAssignCircuitN fsm (n + 2)) |||\n      (castCircLe prev.cOutAssignCirc)\n  , cStatesUniqueCirc :=\n      mkStateUniqueCircuitN fsm (n + 1) |||\n      (castCircLe prev.cStatesUniqueCirc)\n  }"}], "used_local_lemmas": [{"name": "ReflectVerif.BvDecide.mkCarryAssignCircuitNAux_eval_eq", "content": "@[simp]\ntheorem mkCarryAssignCircuitNAux_eval_eq {arity : Type _}\n    [DecidableEq arity]\n    [Fintype arity]\n    [Hashable arity]\n    (p : FSM arity) (s : p.α) (n : Nat)\n    {env : Vars p.α arity (n + 1) → Bool} :\n    ((mkCarryAssignCircuitNAux p s n).eval env) = ((p.nextStateCirc s).eval\n      (fun x => match x with | .inl x => env (Vars.stateN x n) | .inr x => env (Vars.inputN x n)))"}, {"name": "ReflectVerif.BvDecide.mkStateUniqueCircuitN_eq_false_iff", "content": "theorem mkStateUniqueCircuitN_eq_false_iff {arity : Type _}\n  [DecidableEq arity] [Fintype arity] [Hashable arity]\n  (p : FSM arity) (n : Nat)\n  {env : Vars p.α arity n → Bool} :\n  ((mkStateUniqueCircuitN p n).eval env = false) ↔\n  (∀ (i : Nat) (hi : i < n), ∃ (s : p.α), env (Vars.stateN s i) ≠ env (Vars.stateN s n))"}], "local_ctx": "import Mathlib.Data.Bool.Basic\n\nimport Mathlib.Data.Fin.Basic\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Finset.Defs\n\nimport Mathlib.Data.Multiset.FinsetOps\n\nimport Blase.Fast.BitStream\n\nimport Blase.Fast.Defs\n\nimport Blase.Fast.FiniteStateMachine\n\nimport Blase.Fast.Decide\n\nimport Blase.SingleWidth.Syntax\n\nimport Lean.Meta.ForEachExpr\n\nimport Lean.Meta.Tactic.Simp.BuiltinSimprocs.BitVec\n\nimport Blase.Fast.ForLean\n\nimport Blase.Vars\n\nimport Blase.EnvBitstream\n\nimport Lean\n\nopen Fin.NatCast\n\nnamespace ReflectVerif\n\nopen Lean Meta Elab Tactic\n\nnamespace BvDecide\n\nopen Std Sat AIG\n\ndef mkCarryAssignCircuitNAux {arity : Type _}\n  [DecidableEq arity]\n  [Fintype arity]\n  [Hashable arity]\n  (p : FSM arity) (s : p.α) (n : Nat) : Circuit (Vars p.α arity (n + 1)) :=\n    (p.nextStateCirc s).map fun v =>\n      match v with\n        | .inl t => Vars.stateN t n\n        | .inr i => Vars.inputN i n\n\ndef mkCarryAssignCircuitN {arity : Type _}\n  [DecidableEq arity]\n  [Fintype arity]\n  [Hashable arity]\n  (p : FSM arity) (n : Nat) :\n  Circuit (Vars p.α arity (n + 1)) :=\n  let carrys := FinEnum.toList p.α |>.map fun s =>\n    \n    Circuit.xor\n      (mkCarryAssignCircuitNAux p s n)\n      (Circuit.var true <| Vars.stateN s (n + 1))\n  Circuit.bigOr carrys\n\ndef mkOutputAssignCircuitNAux {arity : Type _}\n  [DecidableEq arity]\n  [Fintype arity]\n  [Hashable arity]\n  (p : FSM arity) (n : Nat) : Circuit (Vars p.α arity (n + 1)) :=\n    (p.outputCirc).map fun v =>\n      match v with\n        | .inl s' => Vars.stateN s' n\n        | .inr i => Vars.inputN i n\n\ndef mkOutputAssignCircuitN {arity : Type _}\n  [DecidableEq arity]\n  [Fintype arity]\n  [Hashable arity]\n  (p : FSM arity) (n : Nat) :\n  Circuit (Vars p.α arity (n + 1)) :=\n    Circuit.xor\n      (mkOutputAssignCircuitNAux p n)\n      (Circuit.var true <| Vars.outputs ⟨n, by admit /- proof elided -/\n      ⟩)\n\ndef mkStateNeqCircuit\n  {arity : Type _} {i : Nat}\n  [DecidableEq arity] [Fintype arity] [Hashable arity]\n  (p : FSM arity) (s t : p.α → Circuit (Vars p.α arity i)) : Circuit (Vars p.α arity i) :=\n  Circuit.bigAnd <| FinEnum.toList p.α |>.map fun a => ~~~ (s a) ^^^ (t a)\n\ndef mkStateUniqueCircuitN {arity : Type _}\n  [DecidableEq arity] [Fintype arity] [Hashable arity]\n  (p : FSM arity) (n : Nat) : Circuit (Vars p.α arity n) :=\n  let sn : p.α → Circuit (Vars p.α arity n) := fun s =>\n    Circuit.var true (Vars.stateN s n)\n  let circs := (List.range n).attach |>.map fun ⟨i, hi⟩ =>\n    let si : p.α → Circuit (Vars p.α arity n) := fun s =>\n      Circuit.var true (Vars.stateN s i (by admit /- proof elided -/\n      ))\n    (mkStateNeqCircuit p si sn)\n  Circuit.bigOr circs\n\nstructure KInductionCircuits {arity : Type _}\n  [DecidableEq arity] [Fintype arity] [Hashable arity] (fsm : FSM arity) (n : Nat) where\n  \n  cInitCarryAssignCirc : Circuit (Vars fsm.α arity 0)\n  \n  cSuccCarryAssignCirc : Circuit (Vars fsm.α arity (n+2))\n  \n  cOutAssignCirc : Circuit (Vars fsm.α arity (n + 2))\n  \n  cStatesUniqueCirc : Circuit (Vars fsm.α arity n)\n\nstructure KInductionCircuits.IsLawful {arity : Type _}\n    [DecidableEq arity] [Fintype arity] [Hashable arity] {fsm : FSM arity} {n : Nat}\n  (circs : KInductionCircuits fsm n) where\n  hCInitCarryAssignCirc :\n    ∀ {env : Vars fsm.α arity 0 → Bool},\n      (circs.cInitCarryAssignCirc.eval env = false)\n      ↔ (∀ (s : fsm.α), fsm.initCarry s = env (Vars.state0 s))\n\n  hCSuccCarryAssignCirc :\n    ∀ {env : Vars fsm.α arity (n + 2) → Bool},\n      (circs.cSuccCarryAssignCirc.eval env = false)\n      ↔ (∀ (s : fsm.α) (i : Nat) (hi : i < n + 2),\n        env (Vars.stateN s (i + 1)) =\n          ((mkCarryAssignCircuitNAux fsm s i).map\n            (fun v => v.castLe (by admit /- proof elided -/\n            ))).eval env)\n  hCOutAssignCirc :\n    ∀ {env : Vars fsm.α arity (n + 2) → Bool},\n      (circs.cOutAssignCirc.eval env = false)\n      ↔ (∀ (i : Nat) (hi : i < n + 2),\n        (fsm.outputCirc).eval\n          (fun x => match x with\n            | .inl s => env (Vars.stateN s i)\n            | .inr j => env (Vars.inputN j i)) =\n        env (Vars.outputs ⟨i, by admit /- proof elided -/\n        ⟩))\n  hCStatesUniqueCirc :\n    ∀ {env : Vars fsm.α arity (n) → Bool},\n      (circs.cStatesUniqueCirc.eval env = false)\n      ↔ (∀ (i j : Nat) (hij : i < j ∧ j ≤ n),\n        ∃ (s : fsm.α), env (Vars.stateN s i) ≠ env (Vars.stateN s j))\n\nnamespace KInductionCircuits\n\nvariable {arity : Type _}\n  {fsm : FSM arity}\n\ndef castCircLe {n m : Nat} (c : Circuit (Vars fsm.α arity n)) (hnm : n ≤ m := by admit /- proof elided -/\n) :\n    Circuit (Vars fsm.α arity m) :=\n  c.map (fun v => v.castLe hnm)\n\nvariable [DecidableEq arity] [Fintype arity] [Hashable arity]\n\ndef mkSucc\n    (prev : KInductionCircuits fsm n) :\n    KInductionCircuits fsm (n + 1) :=\n  let cInitCarryAssignCirc := prev.cInitCarryAssignCirc\n  { cInitCarryAssignCirc := cInitCarryAssignCirc\n  , cSuccCarryAssignCirc :=\n      (mkCarryAssignCircuitN fsm (n + 2)) |||\n      (castCircLe prev.cSuccCarryAssignCirc)\n  , cOutAssignCirc :=\n      (mkOutputAssignCircuitN fsm (n + 2)) |||\n      (castCircLe prev.cOutAssignCirc)\n  , cStatesUniqueCirc :=\n      mkStateUniqueCircuitN fsm (n + 1) |||\n      (castCircLe prev.cStatesUniqueCirc)\n  }", "target_theorem": "theorem IsLawful_mkSucc_of_IsLawful {arity : Type _}\n    [DecidableEq arity] [Fintype arity] [Hashable arity]\n    {fsm : FSM arity} {n : Nat}\n    (prev : KInductionCircuits fsm n)\n    (hPrev : prev.IsLawful) :\n    (mkSucc prev).IsLawful :=", "ground_truth_proof": "where\n  hCInitCarryAssignCirc := by\n    simp only [mkSucc, castCircLe]\n    exact hPrev.hCInitCarryAssignCirc\n  hCSuccCarryAssignCirc := by\n    simp only [mkSucc, castCircLe, Circuit.eval_map]\n    simp [Circuit.eval_map]\n    intros env\n    constructor\n    · intros h s i hi\n      obtain ⟨h₁, h₂⟩ := h\n      rw [hPrev.hCSuccCarryAssignCirc] at h₂\n      by_cases hi : i < n + 2\n      · simp only [Vars.castLe_stateN_eq_stateN, Circuit.eval_map,\n        Vars.castLe_castLe_eq_castLe_self, mkCarryAssignCircuitNAux_eval_eq,\n        Vars.castLe_inputs_eq_inputs] at h₂\n        rw [h₂ s i hi]\n      · have hi : i = n + 2 := by omega\n        subst hi\n        apply h₁\n    · intros h\n      constructor\n      · intros s\n        simp only at h ⊢\n        rw [h s _ (by omega)]\n      · rw [hPrev.hCSuccCarryAssignCirc]\n        intros s i hi\n        simp only at h\n        simp only [Vars.castLe_stateN_eq_stateN, Circuit.eval_map,\n          Vars.castLe_castLe_eq_castLe_self, mkCarryAssignCircuitNAux_eval_eq,\n          Vars.castLe_inputs_eq_inputs]\n        rw [h s i (by omega)]\n\n  hCOutAssignCirc := by\n    simp only [mkSucc, castCircLe]\n    simp [Circuit.eval_map]\n    intros env\n    constructor\n    · intros h i hi\n      obtain ⟨h₁, h₂⟩ := h\n      rw [hPrev.hCOutAssignCirc] at h₂\n      by_cases hi : i < n + 2\n      · simp at h₁ h₂ ⊢\n        rw [h₂ i hi]\n      · have : i = n + 2 := by omega\n        subst this\n        apply h₁\n    · intros h\n      constructor\n      · simp only at h ⊢\n        rw [h]\n      · rw [hPrev.hCOutAssignCirc]\n        intros i hi\n        simp\n        apply h\n  hCStatesUniqueCirc := by\n    simp only [mkSucc, castCircLe]\n    simp [Circuit.eval_map]\n    intros env\n    constructor\n    · intros h i j hij hjn\n      simp [mkStateUniqueCircuitN_eq_false_iff] at h\n      simp [hPrev.hCStatesUniqueCirc] at h\n      obtain ⟨h₁, h₂⟩ := h\n      by_cases hj : j ≤ n\n      · grind\n      · have : j = n + 1 := by\n          omega\n        subst this\n        grind\n    · intros h\n      constructor\n      · rw [mkStateUniqueCircuitN_eq_false_iff]\n        intros i hi\n        apply h (j := n + 1)\n        omega\n        omega\n      · simp [hPrev.hCStatesUniqueCirc]\n        intros i j hij jn\n        simp at h ⊢\n        apply h i j\n        omega\n        omega", "nesting_depth": 4, "transitive_dep_count": 54, "subset_aristotle": false, "category": "Compiler"}
{"id": 305, "thm_name": "NFA'.autSignedCmp_correct", "thm_stmt": "lemma NFA'.autSignedCmp_correct cmp : autSignedCmp cmp |>.correct2 autSignedCmpSA cmp.srel", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.AutoStructs.Constructions", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "BitVec", "module": "Init.Prelude"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "BitVec.ofFin", "module": "Init.Prelude"}, {"name": "cmp", "module": "Mathlib.Data.Ordering.Basic"}, {"name": "NFA.stepSet", "module": "Mathlib.Computability.NFA"}, {"name": "BitVec.ult", "module": "Init.Data.BitVec.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.sle", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.slt", "module": "Init.Data.BitVec.Basic"}], "used_repo_defs": [{"name": "RelationOrdering", "content": "inductive RelationOrdering\n| lt | le | gt | ge\nderiving Repr, Fintype"}, {"name": "bv2", "content": "def bv2 : BitVec 4 := BitVec.ofNat 4 1 "}, {"name": "bv1", "content": "def bv1 : BitVec 4 := BitVec.ofNat 4 5 "}, {"name": "instFinEnumBV", "content": "instance instFinEnumBV : FinEnum (BitVec w) where\n  card := 2^w\n  equiv := {\n    toFun := fun x => x.toFin\n    invFun := fun x => BitVec.ofFin x\n    left_inv := by admit /- proof elided -/"}], "lib_lemmas": [{"name": "BitVec.toNat_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Nat.le_antisymm", "module": "Init.Prelude"}, {"name": "BitVec.toInt_inj", "module": "Init.Data.BitVec.Lemmas"}, {"name": "le_iff_lt_or_eq", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "NFA.correct", "content": "structure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q"}, {"name": "BVRel", "content": "abbrev BVRel := ∀ ⦃w⦄, BitVec w → BitVec w → Prop"}, {"name": "NFA'.sa2", "content": "def NFA'.sa2 (M : NFA' 2) := M.σ → BVRel"}, {"name": "NFA'.correct2", "content": "structure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)"}, {"name": "NFA.signedCmpState", "content": "inductive NFA.signedCmpState : Type where\n| eq | gt | lt | ltfin | gtfin\nderiving DecidableEq, Fintype"}, {"name": "NFA.signedCmpStep", "content": "def NFA.signedCmpStep (q : NFA.signedCmpState) (a : BitVec 2) : List NFA.signedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [.eq] | .eq, 1 => [.gt, .ltfin] | .eq, 2 => [ .lt, .gtfin ]\n  | .gt, 0 => [ .gt, .gtfin ] | .gt, 1 => [ .gt, .ltfin ] | .gt, 3 => [ .gt, .gtfin ] | .gt, 2 => [ .lt, .gtfin ]\n  | .lt, 0 => [ .lt, .ltfin ] | .lt, 1 => [ .gt, .ltfin ] | .lt, 2 => [ .lt, .gtfin ] | .lt, 3 => [ .lt, .ltfin ]\n  | .gtfin, _ => ∅\n  | .ltfin, _ => ∅"}, {"name": "NFA.autSignedCmp", "content": "def NFA.autSignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) signedCmpState where\n  step s a := { s' | s' ∈ signedCmpStep s a }\n  start := { s | s = signedCmpState.eq }\n  accept := { s | s ∈ match cmp with | .lt => [NFA.signedCmpState.ltfin] | .le => [.ltfin, .eq] | .gt => [.gtfin] | .ge => [.gtfin, .eq] }"}, {"name": "NFA'.autSignedCmp", "content": "def NFA'.autSignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autSignedCmp cmp⟩"}, {"name": "RelationOrdering.srel", "content": "def RelationOrdering.srel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.slt bv2\n  | .le => fun _ bv1 bv2 => bv1.sle bv2\n  | .gt => fun _ bv1 bv2 => bv2.slt bv1\n  | .ge => fun _ bv1 bv2 => bv2.sle bv1"}, {"name": "NFA'.autSignedCmpSA", "content": "def NFA'.autSignedCmpSA (q : NFA.signedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ltfin => fun _ bv1 bv2 => bv1.slt bv2\n  | .gtfin => fun _ bv1 bv2 => bv2.slt bv1"}], "used_local_lemmas": [{"name": "ucmp_tricho", "content": "@[simp]\nlemma ucmp_tricho {bv1 bv2 : BitVec w} : (bv2.ult bv1) = false → (bv1.ult bv2) = false → bv1 = bv2"}, {"name": "BitVec.sle_iff_slt_or_eq", "content": "private lemma BitVec.sle_iff_slt_or_eq {w : ℕ} (bv1 bv2 : BitVec w):\n    (bv1.sle bv2) = true ↔ (bv1.slt bv2) = true ∨ bv1 = bv2"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\nstructure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q\n\nabbrev BVRel := ∀ ⦃w⦄, BitVec w → BitVec w → Prop\n\ndef NFA'.sa2 (M : NFA' 2) := M.σ → BVRel\n\nstructure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)\n\nsection fsm\n\nvariable {arity : Type} [FinEnum arity]\n\nopen BitStream in\n\nend fsm\n\nsection nfas_relations\n\ninductive NFA.signedCmpState : Type where\n| eq | gt | lt | ltfin | gtfin\nderiving DecidableEq, Fintype\n\ndef NFA.signedCmpStep (q : NFA.signedCmpState) (a : BitVec 2) : List NFA.signedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [.eq] | .eq, 1 => [.gt, .ltfin] | .eq, 2 => [ .lt, .gtfin ]\n  | .gt, 0 => [ .gt, .gtfin ] | .gt, 1 => [ .gt, .ltfin ] | .gt, 3 => [ .gt, .gtfin ] | .gt, 2 => [ .lt, .gtfin ]\n  | .lt, 0 => [ .lt, .ltfin ] | .lt, 1 => [ .gt, .ltfin ] | .lt, 2 => [ .lt, .gtfin ] | .lt, 3 => [ .lt, .ltfin ]\n  | .gtfin, _ => ∅\n  | .ltfin, _ => ∅\n\ndef NFA.autSignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) signedCmpState where\n  step s a := { s' | s' ∈ signedCmpStep s a }\n  start := { s | s = signedCmpState.eq }\n  accept := { s | s ∈ match cmp with | .lt => [NFA.signedCmpState.ltfin] | .le => [.ltfin, .eq] | .gt => [.gtfin] | .ge => [.gtfin, .eq] }\n\ndef NFA'.autSignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autSignedCmp cmp⟩\n\ndef RelationOrdering.srel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.slt bv2\n  | .le => fun _ bv1 bv2 => bv1.sle bv2\n  | .gt => fun _ bv1 bv2 => bv2.slt bv1\n  | .ge => fun _ bv1 bv2 => bv2.sle bv1\n\ndef NFA'.autSignedCmpSA (q : NFA.signedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ltfin => fun _ bv1 bv2 => bv1.slt bv2\n  | .gtfin => fun _ bv1 bv2 => bv2.slt bv1", "target_theorem": "lemma NFA'.autSignedCmp_correct cmp : autSignedCmp cmp |>.correct2 autSignedCmpSA cmp.srel :=", "ground_truth_proof": ":= by\n  let getState {w} (bv1 bv2 : BitVec w) : NFA.signedCmpState :=\n    if bv2.ult bv1 then .gt else if bv1.ult bv2 then .lt else .eq\n  constructor <;> simp [NFA.autSignedCmp, autSignedCmp, autSignedCmpSA, RelationOrdering.srel]\n  · cases cmp <;> simp [BitVec.sle_iff_slt_or_eq]; tauto\n  · rintro (_ | _ | _) <;> simp\n  · rintro (_ | _ | _) a w bv1 bv2 <;> simp [NFA.stepSet]\n    · constructor\n      · rintro ⟨i, hi⟩; cases i <;> fin_cases a <;> simp_all [NFA.signedCmpStep, instFinEnumBV]\n      · rintro ⟨_, _⟩; use .eq; simp; fin_cases a <;> simp [instFinEnumBV] at * <;> tauto\n    · constructor\n      · rintro ⟨i, hi⟩; cases i <;> fin_cases a <;> simp_all [NFA.signedCmpStep, instFinEnumBV]\n      · rintro _; fin_cases a <;> simp [NFA.signedCmpStep, instFinEnumBV] at *\n        · use .gt; simp_all\n        · use (getState bv1 bv2); simp [getState]; split_ifs <;> simp_all; apply ucmp_tricho <;> assumption\n        · use .gt; simp_all\n    · constructor\n      · rintro ⟨i, hi⟩; cases i <;> fin_cases a <;> simp_all [NFA.signedCmpStep, instFinEnumBV]\n      · rintro _; fin_cases a <;> simp [NFA.signedCmpStep, instFinEnumBV] at *\n        · use .lt; simp_all\n        · use (getState bv1 bv2); simp [getState]; split_ifs <;> simp_all; apply ucmp_tricho <;> assumption\n        · use .lt; simp_all\n    · constructor\n      · rintro ⟨i, hi⟩; cases i <;> fin_cases a <;> simp_all [NFA.signedCmpStep, instFinEnumBV]\n      · rintro _; fin_cases a <;> simp [NFA.signedCmpStep, instFinEnumBV] at *\n        · use .lt; simp_all\n        · use (getState bv1 bv2); simp [getState]; split_ifs <;> simp_all; apply ucmp_tricho <;> assumption\n        · use .lt; simp_all\n    · constructor\n      · rintro ⟨i, hi⟩; cases i <;> fin_cases a <;> simp_all [NFA.signedCmpStep, instFinEnumBV]\n      · rintro _; fin_cases a <;> simp [NFA.signedCmpStep, instFinEnumBV] at *\n        · use .gt; simp_all\n        · use (getState bv1 bv2); simp [getState]; split_ifs <;> simp_all; apply ucmp_tricho <;> assumption\n        · use .gt; simp_all", "nesting_depth": 4, "transitive_dep_count": 39, "subset_aristotle": false, "category": "Compiler"}
{"id": 306, "thm_name": "product.f_spec", "thm_stmt": "lemma product.f_spec {m₁ m₂ : CNFA n} {s₁ : m₁.m.states} {s₂ : m₂.m.states} :\n    ∀ a s₁' s₂',\n      (a, (s₁', s₂')) ∈ f m₁ m₂ (s₁, s₂) ↔ s₁'.val ∈ m₁.m.tr s₁ a ∧ s₂'.val ∈ m₂.m.tr s₂ a", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/Constructions.lean", "imports": ["import Blase.AutoStructs.Worklist", "import Mathlib.Tactic.ApplyFun", "import Mathlib.Data.Fintype.Prod", "import Blase.Blase.AutoStructs.ForLean", "import Blase.Blase.AutoStructs.ForMathlib"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "FinEnum.toList", "module": "Mathlib.Data.FinEnum"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Nodup", "module": "Init.Data.List.Basic"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "LawfulBEq", "module": "Init.Core"}, {"name": "Function.Injective2", "module": "Mathlib.Logic.Function.Basic"}], "used_repo_defs": [{"name": "CNFA", "content": "structure CNFA (n : Nat) where\n  m : RawCNFA (BitVec n)\n  wf : m.WF"}, {"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "State", "content": "abbrev State := Nat"}, {"name": "Std.HashSet.toSet", "content": "def Std.HashSet.toSet [BEq α] [Hashable α] (m : HashSet α) : Set α := { x | x ∈ m }\n\naxiom hashMap_missing : ∀ {P : Prop}, P"}], "lib_lemmas": [{"name": "Array.mem_push", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.mem_toList_iff", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.toList_push", "module": "Init.Data.Array.Bootstrap"}, {"name": "Function.Injective2.eq_iff", "module": "Mathlib.Logic.Function.Basic"}, {"name": "List.Nodup.append", "module": "Mathlib.Data.List.Nodup"}, {"name": "List.append_assoc", "module": "Init.Data.List.Basic"}, {"name": "List.disjoint_singleton", "module": "Batteries.Data.List.Lemmas"}, {"name": "List.mem_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_singleton", "module": "Init.Data.List.Lemmas"}, {"name": "Set.mem_empty_iff_false", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.mem_insert_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "Set.union_singleton", "module": "Mathlib.Data.Set.Insert"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "iff_and_self", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "ne_or_eq", "module": "Mathlib.Logic.Basic"}, {"name": "Array.mem_def", "module": "Init.Data.Array.Basic"}, {"name": "Array.not_mem_empty", "module": "Init.Data.Array.Lemmas"}, {"name": "FinEnum.nodup_toList", "module": "Mathlib.Data.FinEnum"}, {"name": "List.Nodup.notMem", "module": "Mathlib.Data.List.Nodup"}, {"name": "List.dedup_eq_self", "module": "Mathlib.Data.List.Dedup"}, {"name": "List.foldl_nil", "module": "Init.Data.List.Basic"}, {"name": "List.mem_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.nodup_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.not_mem_nil", "module": "Init.Data.List.Lemmas"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "Std.HashSet.fold_induction", "content": "theorem Std.HashSet.fold_induction [BEq α] [LawfulBEq α] [Hashable α]\n  {f : β → α → β} {m : HashSet α} {motive : β → Set α → Prop} :\n    motive b ∅ →\n    (∀ b x s, x ∉ s → motive b s → motive (f b x) (s ∪ {x})) →\n    motive (m.fold f b) m.toSet"}, {"name": "Std.HashSet.toSet_toList[BEq", "content": "theorem Std.HashSet.toSet_toList[BEq α] [LawfulBEq α] [Hashable α] (m : HashSet α) : m.toSet = { x | x ∈ m.toList }"}, {"name": "Std.HashSet.mem_toSet", "content": "@[simp]\nlemma Std.HashSet.mem_toSet [BEq α] [Hashable α] (m : HashSet α) : x ∈ m.toSet ↔ x ∈ m"}, {"name": "Std.HashSet.mem_attachWith_mem", "content": "@[simp]\ntheorem Std.HashSet.mem_attachWith_mem [BEq α] [Hashable α] [LawfulBEq α] (m : HashSet α) {P H} (x : α) h :\n    ⟨x, h⟩ ∈ m.attachWith P H ↔ x ∈ m"}], "used_local_defs": [{"name": "product.prodArray'", "content": "@[inline]\ndef product.prodArray' (a : Array γ) :=\n  m₁.attachWith _ hm₁ |>.fold (init := a) fun is s1 =>\n    m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s1 s2)"}, {"name": "product", "content": "def product (final? : Bool → Bool → Bool) (m₁ m₂ : CNFA n) : CNFA n :=\n  worklistRun (m₁.m.states × m₂.m.states) final (product.inits m₁ m₂)\n    (by admit /- proof elided -/\n    ) f\nwhere final (ss : m₁.m.states × m₂.m.states) := final? (ss.1 ∈ m₁.m.finals) (ss.2 ∈ m₂.m.finals)\n      f (ss : m₁.m.states × m₂.m.states) :=\n        let (s1, s2) := ss\n        (FinEnum.toList (α := BitVec n)).foldl (init := Array.empty) fun as a =>\n          product.prodArray' (λ s₁ s₂ ↦ (a, (s₁, s₂)))\n            (fun s' => m₁.wf.trans_tgt_lt (s := s1) (a := a)) (fun s' => m₂.wf.trans_tgt_lt (s := s2) (a := a)) as"}], "used_local_lemmas": [{"name": "product.prodArray_spec_helper", "content": "include hinj in\nomit [BEq α] [Hashable α] [LawfulBEq α] in\nlemma product.prodArray_spec_helper\n    (is : Array γ) (hnd : is.toList.Nodup)\n    (s : S₁) (hnew : ∀ s₂, f s s₂ ∉ is):\n  let motive (a : Array γ) (S : Set S₂)  :=\n    a.toList.Nodup ∧\n    (∃ r, a.toList = is.toList ++ r ∧ (∀ z ∈ r, ∃ s₁ s₂, z = f s₁ s₂)) ∧\n    ∀ s1 s2, f s1 s2 ∈ a ↔ s1 ≠ s ∧ f s1 s2 ∈ is ∨ s1 = s ∧ s2 ∈ S\n  let body := m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s s2)\n  motive body (m₂.attachWith _ hm₂).toSet"}, {"name": "product.prodArray'_spec_full", "content": "include hinj in\nlemma product.prodArray'_spec_full {aᵢ : Array γ} (hnd: aᵢ.toList.Nodup) (hnin : ∀ s₁ s₂, f s₁ s₂ ∉ aᵢ) :\n    (product.prodArray' f hm₁ hm₂ aᵢ).toList.Nodup ∧\n    (∃ r, (product.prodArray' f hm₁ hm₂ aᵢ).toList = aᵢ.toList ++ r ∧ (∀ z ∈ r, ∃ s₁ s₂, z = f s₁ s₂)) ∧\n    ∀ s₁ s₂, f s₁ s₂ ∈ product.prodArray' f hm₁ hm₂ aᵢ ↔ (s₁.val ∈ m₁ ∧ s₂.val ∈ m₂)"}], "local_ctx": "import Mathlib.Data.Fintype.Prod\n\nimport Blase.AutoStructs.Worklist\n\nimport Mathlib.Tactic.ApplyFun\n\nopen SetRel\n\nsection sink\n\nvariable {A : Type} [BEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nend sink\n\nsection generic_prod\n\nvariable {α} [BEq α] [Hashable α] [LawfulBEq α]\n\nvariable {β} [BEq β] [Hashable β] [LawfulBEq β]\n\nvariable {S₁ : Finset α} {S₂ : Finset β}\n\nvariable {γ} (f : S₁ → S₂ → γ) (hinj : Function.Injective2 f)\n\nvariable {m₁ : Std.HashSet α} (hm₁ : ∀ s₁ ∈ m₁, s₁ ∈ S₁)\n\nvariable {m₂ : Std.HashSet β} (hm₂ : ∀ s₂ ∈ m₂, s₂ ∈ S₂)\n\n@[inline]\ndef product.prodArray' (a : Array γ) :=\n  m₁.attachWith _ hm₁ |>.fold (init := a) fun is s1 =>\n    m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s1 s2)\n\nend generic_prod\n\nsection product\n\nvariable {A : Type} [BEq A] [LawfulBEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\ndef product (final? : Bool → Bool → Bool) (m₁ m₂ : CNFA n) : CNFA n :=\n  worklistRun (m₁.m.states × m₂.m.states) final (product.inits m₁ m₂)\n    (by admit /- proof elided -/\n    ) f\nwhere final (ss : m₁.m.states × m₂.m.states) := final? (ss.1 ∈ m₁.m.finals) (ss.2 ∈ m₂.m.finals)\n      f (ss : m₁.m.states × m₂.m.states) :=\n        let (s1, s2) := ss\n        (FinEnum.toList (α := BitVec n)).foldl (init := Array.empty) fun as a =>\n          product.prodArray' (λ s₁ s₂ ↦ (a, (s₁, s₂)))\n            (fun s' => m₁.wf.trans_tgt_lt (s := s1) (a := a)) (fun s' => m₂.wf.trans_tgt_lt (s := s2) (a := a)) as", "target_theorem": "lemma product.f_spec {m₁ m₂ : CNFA n} {s₁ : m₁.m.states} {s₂ : m₂.m.states} :\n    ∀ a s₁' s₂',\n      (a, (s₁', s₂')) ∈ f m₁ m₂ (s₁, s₂) ↔ s₁'.val ∈ m₁.m.tr s₁ a ∧ s₂'.val ∈ m₂.m.tr s₂ a :=", "ground_truth_proof": ":= by\n  suffices heq :\n      ∀ as a (hnd : as.toList.Nodup) (hnd' : (FinEnum.toList (α := BitVec n)).Nodup)\n       (hnew : ∀ b s₁ s₂, (b, s₁, s₂) ∈ as → b ∉ (FinEnum.toList (α := BitVec n))) s₁' s₂',\n        (a, (s₁', s₂')) ∈\n          ((FinEnum.toList (α := BitVec n)).foldl (init := as) fun as a =>\n          product.prodArray' (λ s₁' s₂' ↦ (a, (s₁', s₂')))\n            (fun _ => m₁.wf.trans_tgt_lt (s := s₁) (a := a)) (fun _ => m₂.wf.trans_tgt_lt (s := s₂) (a := a)) as) ↔\n        (a, (s₁', s₂')) ∈ as ∨ a ∈ (FinEnum.toList (α := BitVec n)) ∧ s₁'.val ∈ m₁.m.tr s₁ a ∧ s₂'.val ∈ m₂.m.tr s₂ a by\n    rintro a s₁' s₂'; rw [f, heq]\n    · simp; rintro h; apply Array.not_mem_empty at h; trivial\n    · exact List.dedup_eq_self.mp rfl\n    · exact FinEnum.nodup_toList\n    · rintro _ _ _ h; apply Array.not_mem_empty at h; trivial\n  induction (FinEnum.toList (α := BitVec n))\n  case nil =>\n    simp only [List.foldl_nil, List.not_mem_nil, false_and, Subtype.forall]\n    intros _ _ _ _ _ _; tauto\n  case cons a as ih =>\n    rintro bs b hnd hnd' hnew s₁' s₂'\n    obtain ⟨hnd'', ⟨r, hr, hf⟩, hin⟩ := prodArray'_spec_full (fun s₁' s₂' => (a, s₁', s₂'))\n      (by rintro _ _ _ _ ⟨rfl, rfl, rfl⟩; tauto)\n      (fun _ => m₁.wf.trans_tgt_lt (s := s₁)) (fun _ => m₂.wf.trans_tgt_lt (s := s₂) (a := a))\n      hnd\n      (by dsimp; rintro s₁ s₂ hin; apply hnew at hin; simp at hin)\n    dsimp\n    have hmem : ∀ b s₁' s₂', b ≠ a → (b, s₁', s₂') ∈ prodArray' (fun s₁' s₂' => (a, s₁', s₂'))\n        (fun _ => m₁.wf.trans_tgt_lt (s := s₁) (a := a))\n        (fun _ => m₂.wf.trans_tgt_lt (s := s₂) (a := a)) bs →\n          (b, s₁', s₂') ∈ bs := by\n      rintro b s₁' s₂' hneq hin\n      rw [Array.mem_def] at hin ⊢; rw [hr] at hin; simp only [List.mem_append] at hin\n      rcases hin with hin | hin; assumption\n      obtain ⟨_, _, ⟨rfl, -, -⟩⟩ := hf _ hin; simp at hneq\n    rw [ih] <;> clear ih; rotate_left\n    · apply hnd''\n    · exact (List.nodup_cons.mp hnd').2\n    · rintro b s₁' s₂' hin; by_cases heq : b = a\n      · subst heq; apply List.Nodup.notMem hnd'\n      · specialize hnew b s₁' s₂' (hmem b s₁' s₂' heq hin); simp at hnew; simp [hnew]\n    by_cases heq :  b = a\n    · subst heq\n      simp [hin]; clear hin; constructor\n      · rintro (⟨h1, h2⟩ | ⟨_, h1, h2⟩) <;> right <;> simp [h1, h2]\n      · rintro (hc | ⟨h1, h2⟩)\n        · apply hnew at hc; simp at hc\n        · left; simp [h1, h2]\n    · simp only [List.mem_cons, heq, false_or]; constructor\n      · rintro (hin | hin)\n        · left; apply hmem _ _ _ heq hin\n        · right; exact hin\n      · rintro (hin | hin)\n        · left; rw [Array.mem_def] at hin ⊢; rw [hr]; simp [hin]\n        · right; exact hin", "nesting_depth": 4, "transitive_dep_count": 58, "subset_aristotle": false, "category": "Compiler"}
{"id": 307, "thm_name": "Zipper.denote_insertPureCom", "thm_stmt": "theorem denote_insertPureCom {zip : Zipper d Γ_in eff t₁} [LawfulMonad d.m]\n    {newCom : Com d zip.Γ_mid .pure newTys} {vs : HVector zip.Γ_mid.Var newTys} :\n    (zip.insertPureCom vs newCom).denote = (fun (V_in : Valuation Γ_in) => do\n      let V_mid ← zip.top.denote V_in\n      zip.bot.denote\n        ((Com.denoteLets newCom V_mid).comap <| newCom.outContextHom.with vs newCom.returnVars)\n      )", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Framework/Zipper.lean", "imports": ["import LeanMLIR.Transforms.Rewrite.Match", "import LeanMLIR.LeanMLIR.Framework.Basic", "import LeanMLIR.Framework.Basic", "import LeanMLIR.LeanMLIR.HVector"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "id", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"llvm.and\"     : MLIR.Pretty.uniform_op", "content": "syntax \"llvm.and\"     : MLIR.Pretty.uniform_op\n\nsyntax \"llvm.ashr\"    : MLIR.Pretty.exact_op\n\nsyntax \"llvm.add\"     : MLIR.Pretty.overflow_op\n\nsyntax \"llvm.return\"  : MLIR.Pretty.uniform_op"}, {"name": "notation:50 x \" ≤ₛ \" y => BitVec.sle x y", "content": "notation:50 x \" ≤ₛ \" y => BitVec.sle x y"}, {"name": "notation:50 x \" >ᵤ \" y => BitVec.ult y x", "content": "notation:50 x \" >ᵤ \" y => BitVec.ult y x"}, {"name": "notation:50 x \" ≥ᵤ \" y => BitVec.ule y x", "content": "notation:50 x \" ≥ᵤ \" y => BitVec.ule y x"}, {"name": "notation:50 x \" <ᵤ \" y => BitVec.ult x y", "content": "notation:50 x \" <ᵤ \" y => BitVec.ult x y"}, {"name": "notation:50 x \" ≥ₛ \" y => BitVec.sle y x", "content": "notation:50 x \" ≥ₛ \" y => BitVec.sle y x"}, {"name": "notation:50 x \" <ₛ \" y => BitVec.slt x y", "content": "notation:50 x \" <ₛ \" y => BitVec.slt x y"}, {"name": "notation:50 x \" >ₛ \" y => BitVec.slt y x", "content": "notation:50 x \" >ₛ \" y => BitVec.slt y x"}, {"name": "notation:50 x \" ≤ᵤ \" y => BitVec.ule x y", "content": "notation:50 x \" ≤ᵤ \" y => BitVec.ule x y"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $resName:mlir_op_operand = $name:InstCombine.cmp_op_name $x, $y $[: $t]?) => do\n    let some opName := extractOpName name.raw\n      | Macro.throwUnsupported\n    let t ← t.getDM `(mlir_type| _)\n    `(mlir_op| $resName:mlir_op_operand = $opName ($x, $y) : ($t, $t) -> (i1) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $resName:mlir_op_operand = $name:InstCombine.int_cast_op $x : $t to $t') => do\n    let some opName := extractOpName name.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $resName:mlir_op_operand = $opName ($x) : ($t) -> $t')"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant( $x $[: $inner_type]?)\n      $[: $outer_type]? ) => do\n       \n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      let inner_type := inner_type.getD outer_type\n      `(mlir_op| $res:mlir_op_operand = \"llvm.mlir.constant\"()\n          {value = $x:neg_num : $inner_type} : () -> ($outer_type) )\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant( ${ $x:term }) $[: $t]?) => do\n      let t ← t.getDM `(mlir_type| _)\n      let x ← `(MLIR.AST.AttrValue.int $x [mlir_type| $t])\n      `(mlir_op| $res:mlir_op_operand = \"llvm.mlir.constant\"() {value = $$($x) } : () -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (true) $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (1 : i1) : i1)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (false) $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (0 : i1) : i1)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant $x $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant($x $[: $t]?) $[: $t]?)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant ${ $x:term } $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant($$($x) $[: $t]?) $[: $t]?)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.icmp $p $x, $y $[: $t]?) => do\n    let t ← t.getDM `(mlir_type| _)\n    match p.getString with\n      | \"eq\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.eq\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ne\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ne\" ($x, $y) : ($t, $t) -> (i1))\n      | \"slt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.slt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sle\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sle\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sgt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sgt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sge\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sge\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ult\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ult\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ule\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ule\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ugt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ugt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"uge\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.uge\" ($x, $y) : ($t, $t) -> (i1))\n      | _ => Macro.throwErrorAt p s!\"unexpected predicate {p.getString}\""}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.select $c, $x, $y $[: $t]?) => do\n    let t ← t.getDM `(mlir_type| _)\n    `(mlir_op| $res:mlir_op_operand = \"llvm.select\" ($c, $x, $y) : (i1, $t, $t) -> ($t))"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "FlatCom", "content": "structure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts"}, {"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "Hom.with", "content": "def Hom.with [DecidableEq Ty] {Γ₁ Γ₂ : Ctxt Ty} (f : Γ₁.Hom Γ₂) {ts}\n    (v₁ : HVector Γ₁.Var ts) (v₂ : HVector Γ₂.Var ts) : Γ₁.Hom Γ₂ :=\n  fun _ w =>\n    match v₁.idxOf? w with\n    | none => f w\n    | some ⟨i, h⟩ => (v₂.get i).cast h"}, {"name": "Hom", "content": "abbrev Hom (Γ Γ' : Ctxt Ty) := ⦃t : Ty⦄ → Γ.Var t → Γ'.Var t"}, {"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "Valuation.instAppendHVector", "content": "@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V"}, {"name": "Com.outContextHom", "content": "def Com.outContextHom (com : Com d Γ eff t) : Γ.Hom com.outContext :=\n  com.outContextDiff.toHom"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) "}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "Com.outContextDiff", "content": "def Com.outContextDiff (com : Com d Γ eff ts) : Γ.Diff com.outContext :=\n  ⟨com.bvars, by admit /- proof elided -/\n      ⟩"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "Expr.bvars", "content": "@[simp, grind=] def Expr.bvars (e : Expr d Γ eff Δ) : Nat :=\n  (DialectSignature.returnTypes e.op).length"}, {"name": "returnTypes", "content": "def returnTypes  := Signature.returnTypes ∘ s.signature"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "Com.bvars", "content": "def Com.bvars : Com d Γ eff t → Nat :=\n  Com.rec'\n           (fun _ => 0)\n      (fun e _body bodySize => e.bvars + bodySize)"}, {"name": "Diff", "content": "def Diff (Γ₁ Γ₂ : Ctxt Ty) : Type :=\n  {d : Nat // Diff.Valid Γ₁ Γ₂ d}"}, {"name": "Diff.Valid", "content": "@[simp]\nabbrev Diff.Valid (Γ₁ Γ₂ : Ctxt Ty) (d : Nat) : Prop :=\n  ∀ {i t}, Γ₁[i]? = some t → Γ₂[i+d]? = some t"}, {"name": "toHom", "content": "def toHom (d : Diff Γ₁ Γ₂) : Hom Γ₁ Γ₂ :=\n  fun _ v => ⟨v.val + d.val, d.property v.property⟩"}, {"name": "castCtxt", "content": "def castCtxt (h_eq : Γ = Δ) : Γ.Var ty → Δ.Var ty\n  | ⟨i, h⟩ => ⟨i, h_eq ▸ h⟩"}, {"name": "Com.denoteLets", "content": "def Com.denoteLets : (com : Com d Γ eff ty) → (Γv : Valuation Γ) →\n    eff.toMonad d.m (com.outContext.Valuation)\n  | .rets _, V => pure V\n  | .var e body, V =>\n      e.denote V >>= body.denoteLets >>= fun V =>\n        return V.cast (by admit /- proof elided -/\n        )"}, {"name": "DialectDenote", "content": "class DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))"}, {"name": "Lets.denote", "content": "def Lets.denote [DialectSignature d] [DialectDenote d] {Γ₂}\n    (lets : Lets d Γ₁ eff Γ₂) (V : Valuation Γ₁) : (eff.toMonad d.m <| Valuation Γ₂) :=\n  match lets with\n  | .nil          => return V\n  | .var lets' e  => lets'.denote V >>= e.denote"}, {"name": "sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "regSig", "content": "def regSig       := Signature.regSig ∘ s.signature"}, {"name": "RegionSignature", "content": "abbrev RegionSignature Ty := List (Ctxt Ty × List Ty)"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "effectKind", "content": "def effectKind   := Signature.effectKind ∘ s.signature"}, {"name": "Dialect", "content": "structure Dialect where\n  (Op : Type)\n  (Ty : Type)\n  (m : Type → Type := Id)"}, {"name": "Op", "content": "inductive Op (q : Nat) (n : Nat)\n  | add : Op q n\n  | sub : Op q n\n  | mul : Op q n\n  | mul_constant : Op q n\n    \n    \n  | leading_term : Op q n\n  | monomial : Op q n\n  | monomial_mul : Op q n\n  | from_tensor : Op q n\n  | to_tensor : Op q n\n  | const (c : R q n) : Op q n\n  | const_int (c : Int) : Op q n\n  | const_idx (i : Nat) : Op q n"}, {"name": "HVector.denote", "content": "def HVector.denote :\n    {l : RegionSignature d.Ty} → (T : HVector (fun t => Com d t.1 .impure t.2) l) →\n    HVector (fun t => t.1.Valuation → EffectKind.impure.toMonad d.m (HVector toType t.2)) l\n  | _, .nil => HVector.nil\n  | _, .cons v vs => HVector.cons (v.denote) (HVector.denote vs)"}, {"name": "FlatCom.denote", "content": "@[simp] abbrev FlatCom.denote [DialectDenote d]\n    (flatCom : FlatCom d Γ eff Γ_out ts)\n    (V : Γ.Valuation) : eff.toMonad d.m (HVector toType ts) :=\n  flatCom.lets.denote V >>= (return flatCom.rets.map ·)"}, {"name": "RegionSignature.map", "content": "def RegionSignature.map (f : Ty → Ty') : RegionSignature Ty → RegionSignature Ty' :=\n  List.map fun ⟨Γ, ty⟩ => (Γ.map f, ty.map f)"}, {"name": "Signature.map", "content": "def Signature.map (f : Ty → Ty') : Signature Ty → Signature Ty' :=\n  fun sig => {\n    sig         := sig.sig.map f\n    regSig      := sig.regSig.map f\n    returnTypes := sig.returnTypes.map f\n  }"}, {"name": "map", "content": "def map (f : Ty₁ → Ty₂) : Ctxt Ty₁ → Ctxt Ty₂ :=\n  ofList ∘ (List.map f) ∘ toList"}, {"name": "Expr.denote", "content": "def Expr.denote {ty} (e : Expr d Γ eff ty) (V : Valuation Γ) :\n    eff.toMonad d.m (e.outContext.Valuation) :=\n  match e with\n  | ⟨op, ty_eq, heff, args, regArgs⟩ => do\n      let argsDenote := args.map V\n      let val ← EffectKind.liftEffect heff <| DialectDenote.denote op argsDenote regArgs.denote\n      return (val ++ V).cast (by admit /- proof elided -/\n      )"}, {"name": "Expr.op", "content": "def Expr.op {Γ : Ctxt d.Ty} {eff : EffectKind} {ty} (e : Expr d Γ eff ty) : d.Op :=\n  Expr.casesOn e (fun op _ _ _ _ => op)"}, {"name": "liftEffect", "content": "def liftEffect [Pure m] {e1 e2 : EffectKind} {α : Type}\n    (hle : e1 ≤ e2) (v1 : e1.toMonad m α) : e2.toMonad m α :=\n  match e1, e2, hle with\n    | .pure, .pure, _ | .impure, .impure, _ => v1\n    | .pure, .impure, _ => Pure.pure v1"}, {"name": "toMonad", "content": "def toMonad (e : EffectKind) (m : Type → Type) : Type → Type :=\n  match e with\n  | pure => Id\n  | impure => m"}, {"name": "Com.denote", "content": "def Com.denote : Com d Γ eff ty → (Γv : Valuation Γ) →\n    eff.toMonad d.m (HVector toType ty)\n  | .rets vs, Γv     => pure (vs.map Γv)\n  | .var e body, V => e.denote V >>= body.denote"}, {"name": "Com.ty", "content": "def Com.ty : Com d Γ eff [t] → d.Ty := fun _ => t"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "cast", "content": "def cast (h_eq : ty₁ = ty₂) : Γ.Var ty₁ → Γ.Var ty₂\n  | ⟨i, h⟩ => ⟨i, h_eq ▸ h⟩"}, {"name": "cast", "content": "def cast (h₁ : Γ = Γ') (h₂ : Δ = Δ') : Diff Γ Δ → Diff Γ' Δ'\n  | ⟨n, h⟩ => ⟨n, by admit /- proof elided -/\n  ⟩"}, {"name": "Com.castPureToEff", "content": "def Com.castPureToEff (eff : EffectKind) : Com d Γ .pure t → Com d Γ eff t :=\n  changeEffect (EffectKind.pure_le eff)"}, {"name": "Com.changeEffect", "content": "def Com.changeEffect {eff₁ eff₂ : EffectKind} (h : eff₁ ≤ eff₂) :\n    Com d Γ eff₁ t → Com d Γ eff₂ t := fun com =>\n  Com.rec' (motive := @fun Γ _ => eff₁ ≤ eff₂ → Com d Γ eff₂ t)\n            (fun v _h               => rets v)\n      (fun e _body castBody h => var (e.changeEffect h) (castBody h))\n    com h"}, {"name": "Expr.changeEffect", "content": "def Expr.changeEffect {eff₁ eff₂ : EffectKind} (h : eff₁ ≤ eff₂) :\n    Expr d Γ eff₁ t → Expr d Γ eff₂ t\n  | Expr.mk op ty_eq eff_le args regArgs =>\n    have heff : DialectSignature.effectKind op ≤ eff₂ := by admit /- proof elided -/"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "Expr.regArgs", "content": "def Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)"}, {"name": "Regions", "content": "abbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig"}, {"name": "com", "content": "def com := mkCom (d := InstCombine.MetaLLVM 0) bb0 |>.toOption |>.get (by admit /- proof elided -/\n)"}, {"name": "bb0", "content": "def bb0 : Region 0 := [mlir_region|\n{\n  ^bb0(%arg0: i32):\n    %0 = llvm.mlir.constant(8) : i32\n    %1 = llvm.mlir.constant(31) : i32\n    %2 = llvm.ashr %arg0, %1 : i32\n    %3 = llvm.and %2, %0 : i32\n    %4 = llvm.add %3, %2 : i32\n    llvm.return %4 : i32\n  }]"}, {"name": "Region", "content": "structure Region where\n  (name: String)\n  (args: List <| TypedSSAVal φ)\n  (ops: List Op)"}, {"name": "MetaLLVM", "content": "abbrev MetaLLVM (φ : Nat) : Dialect where\n  Op := MOp φ\n  Ty := MTy φ"}, {"name": "Ty", "content": "@[deprecated \"Use `LLVM.Ty` instead\" (since:=\"2025-04-30\")] abbrev Ty := LLVM.Ty"}, {"name": "Op", "content": "@[deprecated \"Use `LLVM.Op` instead\" (since:=\"2025-04-30\")] abbrev Op := LLVM.Op"}, {"name": "MOp", "content": "inductive MOp (φ : Nat) : Type\n  | unary   (w : Width φ) (op : MOp.UnaryOp φ) :  MOp φ\n  | binary  (w : Width φ) (op : MOp.BinaryOp) :  MOp φ\n  | select  (w : Width φ) : MOp φ\n  | icmp    (c : IntPred) (w : Width φ) : MOp φ\n   \n  | const (w : Width φ) (val : ℤ) : MOp φ\nderiving Repr, DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "binary", "content": "@[match_pattern] abbrev binary (w : Nat) (op : MOp.BinaryOp) : LLVM.Op :=\n  MOp.binary (.concrete w) op"}, {"name": "MOp.BinaryOp", "content": "inductive MOp.BinaryOp : Type\n  | and\n  | or   (disjoint : DisjointFlag := {disjoint := false} )\n  | xor\n  | shl  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | lshr (exact : ExactFlag := {exact := false} )\n  | ashr (exact : ExactFlag := {exact := false} )\n  | urem\n  | srem\n  | add  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | mul  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | sub  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | sdiv (exact : ExactFlag := {exact := false} )\n  | udiv (exact : ExactFlag := {exact := false} )\nderiving DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "LLVM", "content": "def LLVM : Dialect where\n  Op := MOp 0\n  Ty := MTy 0"}, {"name": "MTy", "content": "inductive MTy (φ : Nat)\n  | bitvec (w : Width φ) : MTy φ\n  deriving DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "Width", "content": "abbrev Width φ := ConcreteOrMVar Nat φ"}, {"name": "ConcreteOrMVar", "content": "inductive ConcreteOrMVar (α : Type u) (φ : Nat)\n  | concrete (a : α)\n  | mvar (i : Fin φ)\n  deriving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "const", "content": "@[match_pattern] abbrev const (w : Nat) (val : ℤ) : LLVM.Op        := MOp.const (.concrete w) val"}, {"name": "MOp.UnaryOp", "content": "inductive MOp.UnaryOp (φ : Nat) : Type\n  | neg\n  | not\n  | copy\n  | freeze\n  | trunc (w' : Width φ) (noWrapFlags : NoWrapFlags := {nsw := false, nuw := false} )\n  | zext  (w' : Width φ) (nneg : NonNegFlag := {nneg := false} )\n  | sext  (w' : Width φ)\nderiving Repr, DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "select", "content": "@[simp_llvm_option]\ndef select {w : Nat} (c? : IntW 1) (x? y? : IntW w ) : IntW w := do\n  let c ← c?\n  if c = 1#1 then x? else y?"}, {"name": "IntW", "content": "def IntW w := PoisonOr <| BitVec w"}, {"name": "PoisonOr", "content": "structure PoisonOr (α : Type) where\n  val : α\n  poisonous : Bool\nderiving Inhabited, DecidableEq"}, {"name": "icmp", "content": "@[simp_llvm_option]\ndef icmp {w : Nat} (c : IntPred) (x y : IntW w) : IntW 1 := do\n  let x' ← x\n  let y' ← y\n  icmp? c x' y'"}, {"name": "icmp?", "content": "@[simp_llvm]\ndef icmp? {w : Nat} (c : IntPred) (x y : BitVec w) : IntW 1 :=\n  .value ↑(icmp' c x y)"}, {"name": "IntPred", "content": "inductive IntPred where\n  | eq\n  | ne\n  | ugt\n  | uge\n  | ult\n  | ule\n  | sgt\n  | sge\n  | slt\n  | sle\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "icmp'", "content": "@[simp_llvm]\ndef icmp' {w : Nat} (c : IntPred) (x y : BitVec w) : Bool :=\n  match c with\n    | .eq => (x == y)\n    | .ne => (x != y)\n    | .sgt => (x >ₛ y)\n    | .sge => (x ≥ₛ y)\n    | .slt => (x <ₛ y)\n    | .sle => (x ≤ₛ y)\n    | .ugt => (x >ᵤ y)\n    | .uge => (x ≥ᵤ y)\n    | .ult => (x <ᵤ y)\n    | .ule => (x ≤ᵤ y)"}, {"name": "mkCom", "content": "def mkCom [TransformTy d φ] [TransformExpr d φ] [TransformReturn d φ]\n  (reg : MLIR.AST.Region φ) :\n  ExceptM d (Σ (Γ : Ctxt d.Ty) (eff : EffectKind) (ty : _), Com d Γ eff ty) :=\n  match reg.ops with\n  | [] => throw <| .generic \"Ill-formed region (empty)\"\n  | coms => BuilderM.runWithEmptyMapping <| do\n    let Γ ← declareBindings ∅ reg.args\n    let com ← mkComHelper Γ coms\n    return ⟨Γ, com⟩"}, {"name": "FlatCom.denoteLets", "content": "def FlatCom.denoteLets (flatCom : FlatCom d Γ eff Γ_out t) (Γv : Γ.Valuation) :\n    eff.toMonad d.m <| Γ_out.Valuation :=\n  flatCom.lets.denote Γv"}, {"name": "Com.toLets", "content": "def Com.toLets (com : Com d Γ eff t) : Lets d Γ eff com.outContext :=\n  Lets.nil.addComToEnd com"}, {"name": "Lets.castPureToEff", "content": "def Lets.castPureToEff (eff : EffectKind) : Lets d Γ_in .pure Γ_out → Lets d Γ_in eff Γ_out\n  | .nil => .nil\n  | .var body e => .var (body.castPureToEff eff) (e.castPureToEff eff)"}, {"name": "Expr.castPureToEff", "content": "def Expr.castPureToEff (eff : EffectKind) : Expr d Γ .pure t → Expr d Γ eff t :=\n  changeEffect (EffectKind.pure_le eff)"}, {"name": "Expr.returnVars", "content": "def Expr.returnVars (e : Expr d Γ eff tys) : HVector e.outContext.Var tys :=\n  .ofFn _ _ <| fun i => (Var.ofFin i).appendInl"}, {"name": "ofFin", "content": "def ofFin (i : Fin Γ.length) : Γ.Var (Γ[i]) :=\n  ⟨i.val, by admit /- proof elided -/\n  ⟩"}, {"name": "Com.returnVars", "content": "def Com.returnVars : (com : Com d Γ eff ts) → HVector (Var com.outContext) ts\n  | .rets vs => vs\n  | .var _ body => body.returnVars"}, {"name": "Valuation.comap", "content": "def Valuation.comap {Γi Γo : Ctxt Ty} (Γiv: Γi.Valuation) (hom : Ctxt.Hom Γo Γi) : Γo.Valuation :=\n  fun _to vo => Γiv (hom vo)"}, {"name": "map", "content": "def map (f : ∀ (a : α), A a → B a) :\n    ∀ {l : List α}, HVector A l → HVector B l\n  | [],   .nil        => .nil\n  | t::_, .cons a as  => .cons (f t a) (map f as)"}, {"name": "HVectorLiteral", "content": "structure HVectorLiteral where\n  u : Level\n  v : Level\n  α : Q(Type $u)\n  A : Q($α → Type $v)\n  elems : Array ((a : Q($α)) × Q($A $a))"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "TyDenote.toType", "content": "notation \"⟦\" x \"⟧\" => TyDenote.toType x"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Com.denoteLets_eq", "content": "theorem Com.denoteLets_eq {com : Com d Γ eff t} : com.denoteLets = com.toLets.denote"}, {"name": "Lets.denote_var", "content": "@[simp] theorem Lets.denote_var {lets : Lets d Γ_in eff Γ_out} {e : Expr d Γ_out eff t} :\n    (lets.var e).denote = fun V_in => lets.denote V_in >>= e.denote"}, {"name": "castCtxt_rfl", "content": "@[simp, grind=] theorem castCtxt_rfl (h : Γ = Γ) : v.castCtxt h = v"}, {"name": "Com.returnVars_castPureToEff", "content": "@[simp] theorem Com.returnVars_castPureToEff (eff : _) (com : Com d Γ .pure tys) :\n    (com.castPureToEff eff).returnVars = com.returnVars.map (fun _ v => v.castCtxt (by simp))"}, {"name": "Valuation.comap_apply", "content": "@[simp] theorem Valuation.comap_apply {Γi Γo : Ctxt Ty}\n    (V : Γi.Valuation) (f : Ctxt.Hom Γo Γi) (v : Γo.Var t) :\n    V.comap f v = V (f v)"}, {"name": "Com.denoteLets_castPureToEff", "content": "@[simp] theorem Com.denoteLets_castPureToEff {com : Com d Γ .pure ty} :\n    denoteLets (com.castPureToEff eff)\n    = fun V => pure (com.denoteLets V |>.comap fun _ v => v.castCtxt (by simp))"}, {"name": "Com.denoteLets_returnVars", "content": "@[simp] theorem Com.denoteLets_returnVars (c : Com d Γ .pure tys) (V : Valuation Γ) :\n    c.returnVars.map (c.denoteLets V) = c.denote V"}, {"name": "Id.bind_eq'", "content": "theorem Id.bind_eq' (x : Id α) (f : α → id β) : x >>= f = f x"}, {"name": "Id.pure_eq'", "content": "theorem Id.pure_eq' (a : α) : (pure a : Id α) = a"}, {"name": "Ctxt.Valuation.comap_outContextHom_denoteLets", "content": "@[simp] theorem Ctxt.Valuation.comap_outContextHom_denoteLets {com : Com d Γ .pure ty} {V} :\n    Valuation.comap (com.denoteLets V) com.outContextHom = V"}, {"name": "Com.bvars_castPureToEff", "content": "@[simp] theorem Com.bvars_castPureToEff {com : Com d Γ .pure ty} :\n    (com.castPureToEff eff).bvars = com.bvars"}, {"name": "Valuation.comap_with", "content": "@[simp] theorem Valuation.comap_with [DecidableEq Ty] {Γ Δ : Ctxt Ty}\n    {V : Valuation Γ} {map : Δ.Hom Γ} {vs : HVector Δ.Var ty} {ws : HVector Γ.Var ty} :\n    V.comap (map.with vs ws) = (V.comap map).reassignVars vs (ws.map V)"}, {"name": "map_map", "content": "theorem map_map {A B C : α → Type*} {l : List α} (t : HVector A l)\n    (f : ∀ a, A a → B a) (g : ∀ a, B a → C a) :\n    (t.map f).map g = t.map (fun a v => g a (f a v))"}, {"name": "castCtxt_castCtxt", "content": "@[simp, grind=] theorem castCtxt_castCtxt (h₁ : Γ = Δ) (h₂ : Δ = Ξ) :\n    (v.castCtxt h₁).castCtxt h₂ = v.castCtxt (by simp [*])"}], "used_local_defs": [{"name": "Zipper", "content": "structure Zipper (Γ_in : Ctxt d.Ty) (eff : EffectKind) (tys : List d.Ty) where\n   \n  {Γ_mid : Ctxt d.Ty}\n   \n  top : Lets d Γ_in eff Γ_mid\n   \n  bot : Com d Γ_mid eff tys"}, {"name": "Zipper.denote", "content": "def denote (zip : Zipper d Γ_in eff tys) (V_in : Valuation Γ_in) :\n    eff.toMonad d.m (HVector toType tys) :=\n  (zip.top.denote V_in) >>= zip.bot.denote"}, {"name": "Zipper.insertCom", "content": "def insertCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy) (newCom : Com d zip.Γ_mid eff newTy) :\n    Zipper d Γ_in eff ty :=\n  let top := zip.top.addComToEnd newCom\n  \n  let bot := zip.bot.changeVars <| newCom.outContextHom.with vs newCom.returnVars\n  \n  \n  { top, bot }"}, {"name": "Zipper.insertPureCom", "content": "def insertPureCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy)\n    (newCom : Com d zip.Γ_mid .pure newTy) : Zipper d Γ_in eff ty :=\n  zip.insertCom vs (newCom.castPureToEff eff)"}], "used_local_lemmas": [{"name": "Zipper.denote_insertCom", "content": "theorem denote_insertCom {zip : Zipper d Γ_in eff t₁} [LawfulMonad d.m]\n    {newCom : Com d zip.Γ_mid eff newTys} {vs : HVector zip.Γ_mid.Var newTys} :\n    (zip.insertCom vs newCom).denote = (fun (V_in : Valuation Γ_in) => do\n      let V_mid ← zip.top.denote V_in\n      let V_newMid ← newCom.denoteLets V_mid\n      zip.bot.denote\n        (V_newMid.comap <| newCom.outContextHom.with vs newCom.returnVars)\n      )"}], "local_ctx": "import LeanMLIR.Framework.Basic\n\nimport LeanMLIR.Transforms.Rewrite.Match\n\nopen Ctxt (Valuation Var Hom)\n\nvariable (d : Dialect) [DialectSignature d]\n\nstructure Zipper (Γ_in : Ctxt d.Ty) (eff : EffectKind) (tys : List d.Ty) where\n   \n  {Γ_mid : Ctxt d.Ty}\n   \n  top : Lets d Γ_in eff Γ_mid\n   \n  bot : Com d Γ_mid eff tys\n\nnamespace Zipper\n\nvariable {d}\n\nsection Denote\n\nvariable [TyDenote d.Ty] [DialectDenote d] [Monad d.m]\n\ndef denote (zip : Zipper d Γ_in eff tys) (V_in : Valuation Γ_in) :\n    eff.toMonad d.m (HVector toType tys) :=\n  (zip.top.denote V_in) >>= zip.bot.denote\n\nend Denote\n\nsection ToCom\n\nvariable {Γ_mid}\n\nvariable [TyDenote d.Ty] [DialectDenote d] [Monad d.m]\n\nend ToCom\n\nsection InsertCom\n\nvariable [DecidableEq d.Ty]\n\ndef insertCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy) (newCom : Com d zip.Γ_mid eff newTy) :\n    Zipper d Γ_in eff ty :=\n  let top := zip.top.addComToEnd newCom\n  \n  let bot := zip.bot.changeVars <| newCom.outContextHom.with vs newCom.returnVars\n  \n  \n  { top, bot }\n\ndef insertPureCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy)\n    (newCom : Com d zip.Γ_mid .pure newTy) : Zipper d Γ_in eff ty :=\n  zip.insertCom vs (newCom.castPureToEff eff)\n\nsection Lemmas\n\nvariable [TyDenote d.Ty] [DialectDenote d] [Monad d.m]", "target_theorem": "theorem denote_insertPureCom {zip : Zipper d Γ_in eff t₁} [LawfulMonad d.m]\n    {newCom : Com d zip.Γ_mid .pure newTys} {vs : HVector zip.Γ_mid.Var newTys} :\n    (zip.insertPureCom vs newCom).denote = (fun (V_in : Valuation Γ_in) => do\n      let V_mid ← zip.top.denote V_in\n      zip.bot.denote\n        ((Com.denoteLets newCom V_mid).comap <| newCom.outContextHom.with vs newCom.returnVars)\n      ) :=", "ground_truth_proof": ":= by\n  have (V_mid) (h : Com.outContext (Com.castPureToEff eff newCom) = Com.outContext newCom) :\n      ((Com.denoteLets newCom V_mid).comap fun x v => v.castCtxt h).comap\n        (newCom.castPureToEff eff).outContextHom\n      = (Com.denoteLets newCom V_mid).comap newCom.outContextHom := by\n    funext t' ⟨v', hv'⟩\n    simp only [Com.outContextHom, Com.outContextDiff, Com.bvars_castPureToEff]\n    rfl\n  funext V\n  simp only [insertPureCom, denote_insertCom, Com.denoteLets_castPureToEff,\n    Com.returnVars_castPureToEff, Valuation.comap_with, HVector.map_map, pure_bind, this,\n    Valuation.comap_outContextHom_denoteLets, Valuation.comap_apply, Var.castCtxt_castCtxt,\n    Var.castCtxt_rfl, Com.denoteLets_returnVars]", "nesting_depth": 11, "transitive_dep_count": 133, "subset_aristotle": false, "category": "Compiler"}
{"id": 308, "thm_name": "autMsbSet_accepts", "thm_stmt": "@[simp]\nlemma autMsbSet_accepts : NFA'.autMsbSet.accepts = langMsb", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.Blase.AutoStructs.ForLean", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "Language", "module": "Mathlib.Computability.Language"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "BitVec.ofFin", "module": "Init.Prelude"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}, {"name": "NFA.stepSet", "module": "Mathlib.Computability.NFA"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "List.Vector.ofFn", "module": "Mathlib.Data.Vector.Defs"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "List.Vector.replicate", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Function.RightInverse", "module": "Init.Data.Function"}], "used_repo_defs": [{"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "instFinEnumBV", "content": "instance instFinEnumBV : FinEnum (BitVec w) where\n  card := 2^w\n  equiv := {\n    toFun := fun x => x.toFin\n    invFun := fun x => BitVec.ofFin x\n    left_inv := by admit /- proof elided -/"}, {"name": "langMsb", "content": "@[simp]\ndef langMsb : Set (BitVecs 1) := { bvs | bvs.bvs.get 0 |>.msb }"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "head", "content": "def head (x : BitStream) : Bool      := x 0"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "enc", "content": "def enc (bvs : BitVecs n) : BitVecs' n :=\n  (List.finRange bvs.w).map (fun i =>\n    BitVec.ofFn (fun (k : Fin n) => (bvs.bvs.get k)[i]))"}, {"name": "BitVecs'", "content": "abbrev BitVecs' (n : Nat) := List (BitVec n)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "dec", "content": "@[simps]\ndef dec (bvs' : BitVecs' n) : BitVecs n where\n  w := bvs'.length\n  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k"}, {"name": "accepts", "content": "def accepts (M : NFA' n) : Set (BitVecs n) := dec '' M.accepts'"}, {"name": "NFA'", "content": "structure NFA' (n : Nat) where\n  σ : Type\n  M : NFA (BitVec n) σ"}, {"name": "accepts'", "content": "def accepts' (M : NFA' n) : Set (BitVecs' n) := M.M.accepts"}], "lib_lemmas": [{"name": "BitVec.eq_nil", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.getElem_one", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.getLsbD_eq_getElem", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.msb_eq_getLsbD_last", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.ofNat_eq_ofNat", "module": "Init.Data.BitVec.Basic"}, {"name": "List.getElem?_eq_getElem", "module": "Init.GetElem"}, {"name": "List.getLast?_eq_getElem?", "module": "Init.Data.List.Lemmas"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "decide_true", "module": "Init.Core"}], "repo_lemmas": [{"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}, {"name": "dec_enc'", "content": "@[simp]\nlemma dec_enc' : dec (enc bvs) = bvs"}, {"name": "dec_enc", "content": "@[simp]\nlemma dec_enc : Function.RightInverse (α := BitVecs' n) enc dec"}, {"name": "dec_enc_w", "content": "lemma dec_enc_w (bvs : BitVecs n) : (dec (enc bvs)).w = bvs.w"}, {"name": "BitVec.ofFn_getElem", "content": "@[simp]\ntheorem BitVec.ofFn_getElem {w : Nat} (f : Fin w → Bool) {i : Nat} (hi : i < w) :\n    (BitVec.ofFn f)[i] = f ⟨i, hi⟩"}, {"name": "BitVec.ofFn_getLsbD", "content": "@[simp]\ntheorem BitVec.ofFn_getLsbD {w : Nat} {f : Fin w → Bool} {i : Nat} (hi : i < w) :\n    (BitVec.ofFn f).getLsbD i = f ⟨i, hi⟩"}, {"name": "BitVec.ofFn_getLsbD_fin", "content": "theorem BitVec.ofFn_getLsbD_fin {w : Nat} {f : Fin w → Bool} {i : Fin w} :\n    (BitVec.ofFn f).getLsbD i = f i"}], "used_local_defs": [{"name": "NFA.sa", "content": "def NFA.sa (_ : NFA α σ) := σ → Language α"}, {"name": "NFA.correct", "content": "structure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q"}, {"name": "BVNRel", "content": "abbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop"}, {"name": "NFA'.sa", "content": "def NFA'.sa (M : NFA' n) := M.σ → BVNRel n"}, {"name": "NFA'.correct", "content": "structure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))"}, {"name": "NFA'.correct2", "content": "structure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)"}, {"name": "NFA.msbState", "content": "inductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype"}, {"name": "NFA.msbStep", "content": "def NFA.msbStep (q : NFA.msbState) (a : BitVec 1) : List NFA.msbState :=\n  match q, a with\n  | .i, 0 => [.i]\n  | .i, 1 => [.i, .f]\n  | _, _ => []"}, {"name": "NFA.autMsbSet", "content": "def NFA.autMsbSet : NFA (BitVec 1) msbState where\n  step s a := { s' | s' ∈ msbStep s a }\n  start := {.i}\n  accept := {.f}"}, {"name": "NFA'.autMsbSet", "content": "def NFA'.autMsbSet : NFA' 1 := ⟨_, NFA.autMsbSet⟩"}, {"name": "NFA.msbLang", "content": "def NFA.msbLang : Language (BitVec 1) := { bvs  | bvs.getLast? = some 1 }"}, {"name": "NFA.msbSA", "content": "def NFA.msbSA (q : msbState) : Language (BitVec 1) :=\n  match q with\n  | .i => ⊤\n  | .f => msbLang"}, {"name": "NFA.msbCorrect", "content": "def NFA.msbCorrect : NFA.autMsbSet.correct msbSA msbLang :="}], "used_local_lemmas": [{"name": "NFA.correct_spec", "content": "lemma NFA.correct_spec {M : NFA α σ} {ζ : M.sa} {L : Language α} :\n    M.correct ζ L → M.accepts = L"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\ndef NFA.sa (_ : NFA α σ) := σ → Language α\n\nstructure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q\n\nabbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop\n\ndef NFA'.sa (M : NFA' n) := M.σ → BVNRel n\n\nstructure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))\n\nstructure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)\n\nsection fsm\n\nvariable {arity : Type} [FinEnum arity]\n\nopen BitStream in\n\nend fsm\n\nsection nfas_relations\n\ninductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype\n\ndef NFA.msbStep (q : NFA.msbState) (a : BitVec 1) : List NFA.msbState :=\n  match q, a with\n  | .i, 0 => [.i]\n  | .i, 1 => [.i, .f]\n  | _, _ => []\n\ndef NFA.autMsbSet : NFA (BitVec 1) msbState where\n  step s a := { s' | s' ∈ msbStep s a }\n  start := {.i}\n  accept := {.f}\n\ndef NFA'.autMsbSet : NFA' 1 := ⟨_, NFA.autMsbSet⟩\n\ndef NFA.msbLang : Language (BitVec 1) := { bvs  | bvs.getLast? = some 1 }\n\ndef NFA.msbSA (q : msbState) : Language (BitVec 1) :=\n  match q with\n  | .i => ⊤\n  | .f => msbLang\n\ndef NFA.msbCorrect : NFA.autMsbSet.correct msbSA msbLang :=", "target_theorem": "@[simp]\nlemma autMsbSet_accepts : NFA'.autMsbSet.accepts = langMsb :=", "ground_truth_proof": ":= by\n  simp [NFA'.accepts, NFA'.accepts', NFA'.autMsbSet]\n  rw [NFA.correct_spec NFA.msbCorrect, NFA.msbLang]\n  ext bvs; simp only [BitVec.ofNat_eq_ofNat, Set.mem_image, Set.mem_setOf_eq]\n  constructor\n  · rintro ⟨bvs', hl, heq⟩\n    have _ : bvs'.length ≠ 0 := by cases bvs'; tauto; simp\n    rw [←heq]\n    simp [dec]\n    rw [BitVec.msb_eq_getLsbD_last]\n    rw [BitVec.ofFn_getLsbD (by omega)]\n    simp\n    rw [List.getLast?_eq_getElem?] at hl\n    rw [List.getElem?_eq_getElem (by omega)] at hl\n    injection hl\n    simp_all only [BitVec.getElem_one, decide_true]\n  · intros h; use enc bvs\n    simp only [dec_enc', and_true]\n    simp [enc]\n    have hw : bvs.w ≠ 0 := by\n      rcases bvs with ⟨w, bvs⟩; rintro rfl\n      simp_all [BitVec.eq_nil (bvs.head)]\n    use ⟨bvs.w - 1, by omega⟩\n    simp; rw [List.getLast?_eq_getElem?]\n    simp; constructor\n    · rw [List.getElem?_eq_getElem (by simp; omega)]; simp\n    · ext i hi; rw [BitVec.ofFn_getElem _ (by omega)]\n      rw [BitVec.msb_eq_getLsbD_last] at h\n      simp [←BitVec.getLsbD_eq_getElem]\n      obtain rfl : i = 0 := by omega\n      simp_all", "nesting_depth": 5, "transitive_dep_count": 71, "subset_aristotle": false, "category": "Compiler"}
{"id": 309, "thm_name": "decideIfZerosAux_correct", "thm_stmt": "theorem decideIfZerosAux_correct {arity : Type _} [DecidableEq arity]\n    (p : FSM arity) (c : Circuit p.α)\n    (hc : ∀ s, c.eval s = true →\n      ∃ m y, (p.changeInitCarry s).eval y m = true)\n    (hc₂ : ∀ (x : arity → Bool) (s : p.α → Bool),\n      (FSM.nextBit p s x).snd = true → Circuit.eval c s = true) :\n    decideIfZerosAux p c = true ↔ ∀ n x, p.eval x n = false", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/Fast/FiniteStateMachine.lean", "imports": ["import Blase.Fast.Defs", "import Mathlib.Data.Fintype.BigOperators", "import Blase.Fast.Circuit", "import Blase.Vars", "import Blase.Blase.Fast.Circuit", "import Mathlib.Data.Fintype.Pi", "import Mathlib.Data.Fintype.Sum", "import Mathlib.Data.Fintype.Card", "import Mathlib.Data.Fintype.Sigma", "import Mathlib.Tactic.Ring", "import Mathlib.Data.FinEnum", "import Blase.FinEnum", "import Mathlib.Tactic.Zify"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.card", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Bool.xor", "module": "Init.Data.Bool"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "id", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"slt\" : MLIR.Pretty.uniform_op", "content": "syntax \"slt\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "Term.eval", "content": "def Term.eval (t : Term) (vars : List BitStream) : BitStream :=\n  match t with\n  | var n       => vars.getD n default\n  | zero        => BitStream.zero\n  | one         => BitStream.one\n  | negOne      => BitStream.negOne\n  | ofNat n     => BitStream.ofNat n\n  | and t₁ t₂   => (t₁.eval vars) &&& (t₂.eval vars)\n  | or t₁ t₂    => (t₁.eval vars) ||| (t₂.eval vars)\n  | xor t₁ t₂   => (t₁.eval vars) ^^^ (t₂.eval vars)\n  | not t       => ~~~(t.eval vars)\n  | add t₁ t₂   => (Term.eval t₁ vars) + (Term.eval t₂ vars)\n  | sub t₁ t₂   => (Term.eval t₁ vars) - (Term.eval t₂ vars)\n  | neg t       => -(Term.eval t vars)\n\n\n  | shiftL t n  => BitStream.shiftLeft (Term.eval t vars) n"}, {"name": "Predicate.eval", "content": "def Predicate.eval (p : Predicate) (vars : List BitStream) : BitStream :=\n  match p with\n  | .width .eq n => BitStream.falseIffEq n\n  | .width .neq n => BitStream.falseIffNeq n\n  | .width .lt n => BitStream.falseIffLt n\n  | .width .le n => BitStream.falseIffLe n\n  | .width .gt n => BitStream.falseIffGt n\n  | .width .ge n => BitStream.falseIffGe n\n  | lor p q => Predicate.evalLor (p.eval vars) (q.eval vars)\n  | land p q => Predicate.evalLand (p.eval vars) (q.eval vars)\n  | binary .eq t₁ t₂ => Predicate.evalEq (t₁.eval vars) (t₂.eval vars)\n   \n  | binary .neq t1 t2 => Predicate.evalNeq (t1.eval vars) (t2.eval vars)\n  | binary .ult t₁ t₂ => Predicate.evalUlt (t₁.eval vars) (t₂.eval vars)\n  | binary .ule t₁ t₂ =>\n     Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalUlt (t₁.eval vars) (t₂.eval vars))\n  | binary .slt t₁ t₂ => Predicate.evalSlt (t₁.eval vars) (t₂.eval vars)\n  | binary .sle t₁ t₂ => Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalSlt (t₁.eval vars) (t₂.eval vars))"}, {"name": "Predicate.evalUlt", "content": "def Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true"}, {"name": "borrow", "content": "def borrow (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).2"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "Predicate.evalSlt", "content": "def Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))"}, {"name": "Predicate.evalMsbEq", "content": "def Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false"}, {"name": "Predicate.evalLand", "content": "def Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)"}, {"name": "Predicate.evalNeq", "content": "def Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd"}, {"name": "nxor", "content": "def nxor (a b : BitStream) : BitStream := fun i => a i == b i"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "Predicate.evalLor", "content": "def Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)"}, {"name": "Predicate.evalEq", "content": "def Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "falseIffNeq", "content": "abbrev falseIffNeq (n : Nat) : BitStream := fun i => decide (i == n)"}, {"name": "falseIffLt", "content": "abbrev falseIffLt (n : Nat) : BitStream := fun i => decide (i ≥ n)"}, {"name": "falseIffGe", "content": "abbrev falseIffGe (n : Nat) : BitStream := fun i => decide (i < n)"}, {"name": "falseIffEq", "content": "abbrev falseIffEq (n : Nat) : BitStream := fun i => decide (i != n)"}, {"name": "falseIffGt", "content": "abbrev falseIffGt (n : Nat) : BitStream := fun i => decide (i ≤ n)"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "falseIffLe", "content": "abbrev falseIffLe (n : Nat) : BitStream := fun i => decide (i > n)"}, {"name": "negOne", "content": "abbrev negOne : BitStream := fun _ => true"}, {"name": "shiftLeft", "content": "def shiftLeft (x : BitStream) (k : Nat) : BitStream :=\n  fun i => if i < k then false else x (i - k) "}, {"name": "ofNat", "content": "def ofNat (x : Nat) : BitStream :=\n  Nat.testBit x"}, {"name": "one", "content": "abbrev one    : BitStream := (· == 0)"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "eval", "content": "@[simp]\ndef eval : Circuit α → (α → Bool) → Bool\n  | tru, _ => true\n  | fals, _ => false\n  | var b x, f => if b then f x else !(f x)\n  | and c₁ c₂, f => (eval c₁ f) && (eval c₂ f)\n  | or c₁ c₂, f => (eval c₁ f) || (eval c₂ f)\n  | xor c₁ c₂, f => Bool.xor (eval c₁ f) (eval c₂ f)"}, {"name": "sumVarsRight", "content": "def sumVarsRight [DecidableEq α] [DecidableEq β] : Circuit (α ⊕ β) → List β\n  | tru => []\n  | fals => []\n  | var _ (Sum.inl _) => []\n  | var _ (Sum.inr x) => [x]\n  | and c₁ c₂ => (sumVarsRight c₁ ++ sumVarsRight c₂).dedup\n  | or c₁ c₂ => (sumVarsRight c₁ ++ sumVarsRight c₂).dedup\n  | xor c₁ c₂ => (sumVarsRight c₁ ++ sumVarsRight c₂).dedup"}, {"name": "fst", "content": "def fst {α β : Type _} [DecidableEq α] [DecidableEq β]\n    (c : Circuit (α ⊕ β)) : Circuit α :=\n  Circuit.bOr (c.sumVarsRight.pi (λ _ => [true, false]))\n  (λ x => Circuit.assignVars c\n    (λ i => Sum.rec (λ i _ => Sum.inl i) (λ i hi => Sum.inr (x i (by admit /- proof elided -/\n    ))) i))"}, {"name": "bOr", "content": "def bOr : ∀ (_s : List α) (_f : α → Circuit β), Circuit β\n| [], _ => fals\n| a::l, f => l.foldl (λ c x => c ||| (f x)) (f a)"}, {"name": "assignVars", "content": "def assignVars [DecidableEq α] :\n    ∀ (c : Circuit α) (_f : ∀ (a : α) (_ha : a ∈ c.vars), β ⊕ Bool), Circuit β\n  | tru, _ => tru\n  | fals, _ => fals\n  | var b x, f =>\n    Sum.elim\n      (var b)\n      (λ c : Bool => if Bool.xor b c then fals else tru)\n      (f x (by admit /- proof elided -/\n      ))\n  | and c₁ c₂, f => (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/\n  ))) &&&\n                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/\n                    )))\n  | or c₁ c₂, f =>  (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/\n  ))) |||\n                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/\n                    )))\n  | xor c₁ c₂, f => (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/\n  ))) ^^^\n                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/\n                    )))"}, {"name": "vars", "content": "def vars [DecidableEq α] : Circuit α → List α\n  | tru => []\n  | fals => []\n  | var _ x => [x]\n  | and c₁ c₂ => (vars c₁ ++ vars c₂).dedup\n  | or c₁ c₂ => (vars c₁ ++ vars c₂).dedup\n  | xor c₁ c₂ => (vars c₁ ++ vars c₂).dedup"}, {"name": "evalv", "content": "@[simp] def evalv [DecidableEq α] : ∀ (c : Circuit α), (∀ a ∈ vars c, Bool) → Bool\n  | tru, _ => true\n  | fals, _ => false\n  | var b x, f => if b then f x (by admit /- proof elided -/\n  ) else !(f x (by admit /- proof elided -/\n  ))\n  | and c₁ c₂, f => (evalv c₁ (fun i hi => f i (by admit /- proof elided -/\n  ))) &&\n    (evalv c₂ (fun i hi => f i (by admit /- proof elided -/\n    )))\n  | or c₁ c₂, f => (evalv c₁ (fun i hi => f i (by admit /- proof elided -/\n  ))) ||\n    (evalv c₂ (fun i hi => f i (by admit /- proof elided -/\n    )))\n  | xor c₁ c₂, f => Bool.xor (evalv c₁ (fun i hi => f i (by admit /- proof elided -/\n  )))\n    (evalv c₂ (fun i hi => f i (by admit /- proof elided -/\n    )))"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "simplifyOr", "content": "def simplifyOr : Circuit α → Circuit α → Circuit α\n  | tru, _ => tru\n  | _, tru => tru\n  | fals, c => c\n  | c, fals => c\n  | c₁, c₂ => or c₁ c₂"}, {"name": "bind", "content": "def bind : ∀ (_c : Circuit α) (_f : α → Circuit β), Circuit β\n  | tru, _ => tru\n  | fals, _ => fals\n  | var b x, f => if b then f x else ~~~ (f x)\n  | and c₁ c₂, f => (bind c₁ f) &&& (bind c₂ f)\n  | or c₁ c₂, f => (bind c₁ f) ||| (bind c₂ f)\n  | xor c₁ c₂, f => (bind c₁ f) ^^^ (bind c₂ f)"}], "lib_lemmas": [{"name": "Bool.not_eq_true", "module": "Init.SimpLemmas"}, {"name": "Finset.card_lt_card", "module": "Mathlib.Data.Finset.Card"}, {"name": "Finset.ssubset_iff", "module": "Mathlib.Data.Finset.Insert"}, {"name": "Finset.subset_iff", "module": "Mathlib.Data.Finset.Defs"}, {"name": "not_imp_not", "module": "Mathlib.Logic.Basic"}, {"name": "Bool.or_eq_true", "module": "Init.SimpLemmas"}, {"name": "Nat.rec_zero", "module": "Mathlib.Data.Nat.Init"}, {"name": "or_true", "module": "Init.SimpLemmas"}, {"name": "true_iff", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "le_def", "content": "lemma le_def : ∀ (c₁ c₂ : Circuit α), c₁ ≤ c₂ ↔ ∀ f, eval c₁ f → eval c₂ f"}, {"name": "eval_fst", "content": "theorem eval_fst {α β : Type _} [DecidableEq α] [DecidableEq β]\n    (c : Circuit (α ⊕ β)) (g : α → Bool) :\n    c.fst.eval g ↔ ∃ g' : β → Bool, c.eval (Sum.elim g g')"}, {"name": "eval_assignVars", "content": "lemma eval_assignVars [DecidableEq α] : ∀ {c : Circuit α}\n    {f : ∀ (a : α) (_ha : a ∈ c.vars), β ⊕ Bool} {g : β → Bool},\n    eval (assignVars c f) g = evalv c (λ a ha => Sum.elim g id (f a ha))"}, {"name": "eval_bOr", "content": "@[simp] lemma eval_bOr :\n  ∀ {s : List α} {f : α → Circuit β} {g : β → Bool},\n    eval (bOr s f) g = ∃ a ∈ s, eval (f a) g"}, {"name": "eval_foldl_or", "content": "@[simp] lemma eval_foldl_or :\n  ∀ (s : List α) (f : α → Circuit β) (c : Circuit β) (g : β → Bool),\n    (eval (s.foldl (λ c x => c ||| (f x)) c) g : Prop) ↔\n      eval c g ∨ (∃ a ∈ s, eval (f a) g)"}, {"name": "eval_or", "content": "@[simp] lemma eval_or : ∀ (c₁ c₂ : Circuit α) (f : α → Bool),\n    (eval (c₁ ||| c₂) f) = ((eval c₁ f) || (eval c₂ f))"}, {"name": "eval_eq_evalv", "content": "lemma eval_eq_evalv [DecidableEq α] : ∀ (c : Circuit α) (f : α → Bool),\n    eval c f = evalv c (λ x _ => f x)"}, {"name": "eval_bind", "content": "lemma eval_bind : ∀ (c : Circuit α) (f : α → Circuit β) (g : β → Bool),\n    eval (bind c f) g = eval c (λ a => eval (f a) g)"}], "used_local_defs": [{"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "FSM.State", "content": "abbrev State : Type := p.α → Bool"}, {"name": "FSM.nextBit", "content": "def nextBit : p.State → (arity → Bool) → p.State × Bool :=\n  fun carry inputBits =>\n    let input := Sum.elim carry inputBits\n    let newState : p.State  := fun (a : p.α) => (p.nextStateCirc a).eval input\n    let outBit : Bool       := (p.outputCirc).eval input\n    (newState, outBit)"}, {"name": "FSM.carry", "content": "def carry (x : arity → BitStream) : ℕ → p.State\n  | 0 => p.initCarry\n  | n+1 => (p.nextBit (carry x n) (fun i => x i n)).1"}, {"name": "FSM.eval", "content": "def eval (x : arity → BitStream) : BitStream :=\n  fun n => (p.nextBit (p.carry x n) (fun i => x i n)).2"}, {"name": "FSM.changeInitCarry", "content": "def changeInitCarry (p : FSM arity) (c : p.α → Bool) : FSM arity :=\n  { p with initCarry := c }"}, {"name": "card_compl", "content": "def card_compl [Fintype α] [DecidableEq α] (c : Circuit α) : ℕ :=\n  Finset.card $ (@Finset.univ (α → Bool) _).filter (fun a => c.eval a = false)"}, {"name": "decideIfZerosAux", "content": "def decideIfZerosAux {arity : Type _} [DecidableEq arity]\n    (p : FSM arity) (c : Circuit p.α) : Bool :=\n  \n  if c.eval p.initCarry \n  then false \n  else\n    \n    \n    have c' := (c.bind (p.nextStateCirc)).fst\n    if h : c' ≤ c then true\n    else\n      have _wf : card_compl (c' ||| c) < card_compl c :=\n        decideIfZeroAux_wf h\n      decideIfZerosAux p (c' ||| c)\n  termination_by card_compl c"}], "used_local_lemmas": [{"name": "FSM.carry_changeInitCarry_succ", "content": "theorem carry_changeInitCarry_succ\n    (p : FSM arity) (c : p.α → Bool) (x : arity → BitStream) : ∀ n,\n    (p.changeInitCarry c).carry x (n+1) =\n      (p.changeInitCarry (p.nextBit c (fun a => x a 0)).1).carry\n        (fun a i => x a (i+1)) n\n  | 0 => by simp [carry, changeInitCarry, nextBit]\n  | n+1 => by\n    rw [carry, carry_changeInitCarry_succ p _ _ n]\n    simp [nextBit, carry, changeInitCarry]"}, {"name": "FSM.eval_changeInitCarry_succ", "content": "theorem eval_changeInitCarry_succ\n    (p : FSM arity) (c : p.α → Bool) (x : arity → BitStream) (n : ℕ) :\n    (p.changeInitCarry c).eval x (n+1) =\n      (p.changeInitCarry (p.nextBit c (fun a => x a 0)).1).eval\n        (fun a i => x a (i+1)) n"}, {"name": "FSM.evalAux_eq_zero_of_set", "content": "theorem evalAux_eq_zero_of_set {arity : Type _} (p : FSM arity)\n    (R : Set (p.α → Bool)) (hR : ∀ x s, (p.nextBit s x).1 ∈ R → s ∈ R)\n    (hi : p.initCarry ∉ R) (hr1 : ∀ x s, (p.nextBit s x).2 = true → s ∈ R)\n    (x : arity → BitStream) (n : ℕ) : p.eval x n = false ∧ p.carry x n ∉ R"}, {"name": "FSM.eval_eq_zero_of_set", "content": "theorem eval_eq_zero_of_set {arity : Type _} (p : FSM arity)\n    (R : Set (p.α → Bool)) (hR : ∀ x s, (p.nextBit s x).1 ∈ R → s ∈ R)\n    (hi : p.initCarry ∉ R) (hr1 : ∀ x s, (p.nextBit s x).2 = true → s ∈ R) :\n    p.eval = fun _ _ => false"}, {"name": "decideIfZeroAux_wf", "content": "theorem decideIfZeroAux_wf {α : Type _} [Fintype α] [DecidableEq α]\n    {c c' : Circuit α} (h : ¬c' ≤ c) : card_compl (c' ||| c) < card_compl c"}], "local_ctx": "import Mathlib.Data.FinEnum\n\nimport Mathlib.Data.Fintype.Card\n\nimport Mathlib.Data.Fintype.Sum\n\nimport Mathlib.Data.Fintype.Sigma\n\nimport Mathlib.Data.Fintype.Pi\n\nimport Mathlib.Data.Fintype.BigOperators\n\nimport Mathlib.Tactic.Zify\n\nimport Mathlib.Tactic.Ring\n\nimport Blase.FinEnum\n\nimport Blase.Fast.Defs\n\nimport Blase.Fast.Circuit\n\nimport Blase.Vars\n\nopen Sum\n\nsection FSM\n\nvariable {α β α' β' : Type} {γ : β → Type}\n\nstructure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)\n\nopen Lean in\n\nopen Lean in\n\nopen Lean in\n\nopen Lean in\n\nnamespace FSM\n\nvariable {arity : Type} (p : FSM arity)\n\nabbrev State : Type := p.α → Bool\n\ndef nextBit : p.State → (arity → Bool) → p.State × Bool :=\n  fun carry inputBits =>\n    let input := Sum.elim carry inputBits\n    let newState : p.State  := fun (a : p.α) => (p.nextStateCirc a).eval input\n    let outBit : Bool       := (p.outputCirc).eval input\n    (newState, outBit)\n\ndef carry (x : arity → BitStream) : ℕ → p.State\n  | 0 => p.initCarry\n  | n+1 => (p.nextBit (carry x n) (fun i => x i n)).1\n\ndef eval (x : arity → BitStream) : BitStream :=\n  fun n => (p.nextBit (p.carry x n) (fun i => x i n)).2\n\ndef changeInitCarry (p : FSM arity) (c : p.α → Bool) : FSM arity :=\n  { p with initCarry := c }\n\nsection EvalInduction\n\nend EvalInduction\n\nend FSM\n\nnamespace FSM\n\nend FSM\n\nopen Term\n\ndef card_compl [Fintype α] [DecidableEq α] (c : Circuit α) : ℕ :=\n  Finset.card $ (@Finset.univ (α → Bool) _).filter (fun a => c.eval a = false)\n\ndef decideIfZerosAux {arity : Type _} [DecidableEq arity]\n    (p : FSM arity) (c : Circuit p.α) : Bool :=\n  \n  if c.eval p.initCarry \n  then false \n  else\n    \n    \n    have c' := (c.bind (p.nextStateCirc)).fst\n    if h : c' ≤ c then true\n    else\n      have _wf : card_compl (c' ||| c) < card_compl c :=\n        decideIfZeroAux_wf h\n      decideIfZerosAux p (c' ||| c)\n  termination_by card_compl c", "target_theorem": "theorem decideIfZerosAux_correct {arity : Type _} [DecidableEq arity]\n    (p : FSM arity) (c : Circuit p.α)\n    (hc : ∀ s, c.eval s = true →\n      ∃ m y, (p.changeInitCarry s).eval y m = true)\n    (hc₂ : ∀ (x : arity → Bool) (s : p.α → Bool),\n      (FSM.nextBit p s x).snd = true → Circuit.eval c s = true) :\n    decideIfZerosAux p c = true ↔ ∀ n x, p.eval x n = false :=", "ground_truth_proof": ":= by\n  rw [decideIfZerosAux]\n  split_ifs with h\n  · simp\n    exact hc p.initCarry h\n  · dsimp\n    split_ifs with h'\n    · simp only [true_iff]\n      intro n x\n      rw [p.eval_eq_zero_of_set {x | c.eval x = true}]\n      · intro y s\n        simp [Circuit.le_def, Circuit.eval_fst, Circuit.eval_bind] at h'\n        simp [FSM.nextBit]\n        apply h'\n      · assumption\n      · exact hc₂\n    · let c' := (c.bind (p.nextStateCirc)).fst\n      have _wf : card_compl (c' ||| c) < card_compl c :=\n        decideIfZeroAux_wf h'\n      apply decideIfZerosAux_correct p (c' ||| c)\n      simp [c', Circuit.eval_fst, Circuit.eval_bind]\n      intro s hs\n      rcases hs with ⟨x, hx⟩ | h\n      · rcases hc _ hx with ⟨m, y, hmy⟩\n        use (m+1)\n        use fun a i => Nat.casesOn i x (fun i a => y a i) a\n        rw [FSM.eval_changeInitCarry_succ]\n        rw [← hmy]\n        simp only [FSM.nextBit, Nat.rec_zero]\n      · exact hc _ h\n      · intro x s h\n        have := hc₂ _ _ h\n        simp only [Bool.or_eq_true, Circuit.eval_or, this, or_true]\ntermination_by card_compl c", "nesting_depth": 7, "transitive_dep_count": 97, "subset_aristotle": false, "category": "Compiler"}
{"id": 310, "thm_name": "MultiWidth.eval_fsmTermSlt_eq_decide_slt", "thm_stmt": "theorem eval_fsmTermSlt_eq_decide_slt {wcard tcard : Nat}\n    (tctx : Term.Ctx wcard tcard)\n    {wenv : WidthExpr.Env wcard}\n    (tenv : tctx.Env wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (a : Term bcard ncard icard pcard tctx (.bv w))\n    (b : Term bcard ncard icard pcard tctx (.bv w))\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (afsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm a))\n    (hafsm : HTermFSMToBitStream afsm)\n    (bfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm b))\n    (hbfsm : HTermFSMToBitStream bfsm)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (henv : HTermEnv fsmEnv tenv benv)\n    :\n    ((fsmTermSlt\n      wfsm\n      afsm\n      bfsm)).eval fsmEnv i =\n       decide (((a.toBV benv nenv ienv penv tenv).signExtend i).slt\n          ((b.toBV benv nenv ienv penv tenv).signExtend i))", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/MultiWidth/GoodFSM.lean", "imports": ["import Blase.MultiWidth.Defs", "import Blase.Vars", "import Blase.KInduction.KInduction", "import Lean", "import Blase.Blase.Fast.BitStream", "import Blase.Fast.FiniteStateMachine"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "Empty", "module": "Init.Prelude"}, {"name": "Empty.elim", "module": "Init.Core"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat.max", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.min", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.ofBool", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.signExtend", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "BitVec.carry", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Bool.atLeastTwo", "module": "Init.Data.BitVec.Bitblast"}], "used_repo_defs": [{"name": "syntax \"min\" : MLIR.Pretty.uniform_op", "content": "syntax \"min\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "composeBinaryAux'", "content": "def composeBinaryAux'\n    (p : FSM Bool)\n    (qtrue : FSM α)\n    (qfalse : FSM α) :\n    FSM α :=\n  p.compose (α)\n    (λ _ => α)\n    (λ _ i => i)\n    (λ b => match b with\n      | true => qtrue\n      | false => qfalse)"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "xor", "content": "def xor : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ^^^ Circuit.var true (inr false),\n    nextStateCirc := Empty.elim\n  }"}, {"name": "composeUnaryAux", "content": "def composeUnaryAux\n    (p : FSM Unit)\n    (q : FSM arity) :\n    FSM arity :=\n  p.compose\n    arity\n    _\n    (λ _ => id)\n    (λ _ => q)"}, {"name": "ls", "content": "def ls (b : Bool) : FSM Unit :=\n  { α := Unit,\n    initCarry := fun _ => b,\n    nextStateCirc := fun () => Circuit.var true (inr ()),\n    outputCirc := Circuit.var true (inl ())\n  }"}, {"name": "latchImmediate", "content": "def latchImmediate (initVal : Bool) : FSM Bool where\n  α := Unit\n  initCarry := fun _ => initVal\n  outputCirc :=\n    let xval := Circuit.var true (inr false)\n    let control := Circuit.var true (inr true)\n    let state := Circuit.var true (inl ())\n    Circuit.ite control xval state\n  nextStateCirc := fun () =>\n    let xval := Circuit.var true (inr false)\n    let control := Circuit.var true (inr true)\n    let state := Circuit.var true (inl ())\n    Circuit.ite control xval state"}, {"name": "ite", "content": "def ite (cond t f : Circuit α) : Circuit α :=\n  (cond &&& t) ||| (~~~ cond &&& f)"}, {"name": "TermFSM", "content": "structure TermFSM (wcard tcard bcard ncard icard pcard : Nat) (t : Nondep.Term) where\n  toFsmZext : FSM (StateSpace wcard tcard bcard ncard icard pcard)\n  width : NatFSM wcard tcard bcard ncard icard pcard t.width"}, {"name": "NatFSM", "content": "structure NatFSM (wcard tcard bcard ncard icard pcard : Nat) (v : Nondep.WidthExpr) where\n  toFsm : FSM (StateSpace wcard tcard bcard ncard icard pcard)"}, {"name": "StateSpace", "content": "inductive StateSpace (wcard tcard bcard ncard icard pcard : Nat)\n| widthVar (v : Fin wcard)\n| termVar (v : Fin tcard)\n| predVar (v : Fin pcard)\n| boolVar (v : Fin bcard)\nderiving DecidableEq, Repr, Hashable"}, {"name": "Term", "content": "inductive Term\n| ofNat (w : WidthExpr) (n : Nat) : Term\n| var (v : Nat) (w : WidthExpr) : Term\n| add (w : WidthExpr) (a b : Term) : Term\n| zext (a : Term) (wnew : WidthExpr) : Term\n| setWidth (a : Term) (wnew : WidthExpr) : Term\n| sext (a : Term) (wnew : WidthExpr) : Term\n| bor (w : WidthExpr) (a b : Term) : Term\n| band (w : WidthExpr) (a b : Term) : Term\n| bxor (w : WidthExpr) (a b : Term) : Term\n| bnot (w : WidthExpr)  (a : Term) : Term\n| boolVar (v : Nat) : Term\n| boolConst (b : Bool) : Term\n| shiftl (w : WidthExpr) (a : Term) (k : Nat) : Term\n| bvOfBool (b : Term) : Term\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr) : Term\n| binRel (k : BinaryRelationKind) (w : WidthExpr)\n    (a : Term) (b : Term) : Term\n| or (p1 p2 : Term) : Term\n| and (p1 p2 : Term) : Term\n| pvar (v : Nat) : Term\n| boolBinRel (k : BoolBinaryRelationKind)\n    (a b : Term) : Term\nderiving DecidableEq, Inhabited, Repr, Lean.ToExpr"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : Nat → WidthExpr → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : (k : Nat) → (v : WidthExpr) → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "HNatFSMToBitstream", "content": "structure HNatFSMToBitstream {wcard : Nat} {v : WidthExpr wcard} {tcard : Nat} {bcard : Nat} {pcard : Nat}\n   (fsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v)) : Prop where\n  heq :\n    ∀ (wenv : Fin wcard → Nat)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n    (henv : HWidthEnv fsmEnv wenv) →\n      fsm.toFsm.eval fsmEnv =\n      BitStream.ofNatUnary (v.toNat wenv)"}, {"name": "WidthExpr.toNat", "content": "def WidthExpr.toNat (e : WidthExpr wcard) (env : WidthExpr.Env wcard) : Nat :=\n  match e with\n  | .const n => n\n  | .var v => env v\n  | .min v w => Nat.min (v.toNat env) (w.toNat env)\n  | .max v w => Nat.max (v.toNat env) (w.toNat env)\n  | .addK v k => v.toNat env + k\n  | .kadd k v => k + v.toNat env"}, {"name": "WidthExpr", "content": "inductive WidthExpr (wcard : Nat) : Type\n| const (n : Nat) :  WidthExpr wcard\n| var : (v : Fin wcard) → WidthExpr wcard\n| min : (v w : WidthExpr wcard) → WidthExpr wcard\n| max : (v w : WidthExpr wcard) → WidthExpr wcard\n| addK : (v : WidthExpr wcard) → (k : Nat) → WidthExpr wcard\n| kadd : (k : Nat) → (v : WidthExpr wcard) → WidthExpr wcard"}, {"name": "WidthExpr.Env", "content": "abbrev WidthExpr.Env (wcard : Nat) : Type :=\n  Fin wcard → Nat"}, {"name": "HWidthEnv", "content": "structure HWidthEnv {wcard tcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (wenv : Fin wcard → Nat) : Prop where\n    heq_width : ∀ (v : Fin wcard),\n      fsmEnv (StateSpace.widthVar v) = BitStream.ofNatUnary (wenv v)"}, {"name": "HPredicateEnv", "content": "structure HPredicateEnv {wcard tcard bcard ncard icard pcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (penv : Fin pcard → Prop) : Prop where\n    heq_width : ∀ (v : Fin pcard),\n      fsmEnv (StateSpace.predVar v) = BitStream.ofProp (penv v)"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "ofNatUnary", "content": "abbrev ofNatUnary (n : Nat) : BitStream :=\n  fun i => decide (i < n)"}, {"name": "HPredFSMToBitStream", "content": "structure HPredFSMToBitStream {pcard : Nat}\n  {tctx : Term.Ctx wcard tcard}\n  {p : Term bcard ncard icard pcard tctx .prop}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard\n    (.ofDepTerm p)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (htenv : HTermEnv fsmEnv tenv benv) →\n      (hpenv : HPredicateEnv fsmEnv penv) →\n        p.toBV benv nenv ienv penv tenv  ↔ (fsm.toFsmZext.eval fsmEnv = .negOne)"}, {"name": "Term.Ctx", "content": "abbrev Term.Ctx (wcard : Nat) (tcard : Nat) : Type :=\n  Fin tcard → WidthExpr wcard"}, {"name": "Term.BoolEnv", "content": "def Term.BoolEnv (bcard : Nat) : Type := Fin bcard → Bool"}, {"name": "Term.IntEnv", "content": "def Term.IntEnv (icard : Nat) : Type := Fin icard → Nat"}, {"name": "HTermFSMToBitStream", "content": "structure HTermFSMToBitStream {w : WidthExpr wcard}\n  {tctx : Term.Ctx wcard tcard}\n  {t : Term bcard ncard icard pcard tctx (.bv w)}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm t)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (henv : HTermEnv fsmEnv tenv benv) →\n        fsm.toFsmZext.eval fsmEnv =\n        BitStream.ofBitVecZext (t.toBV benv nenv ienv penv tenv)"}, {"name": "Predicate.Env", "content": "def Predicate.Env (pcard : Nat) : Type :=\n  Fin pcard → Prop"}, {"name": "TermKind", "content": "inductive TermKind (wcard : Nat) : Type\n| bool\n| bv (w : WidthExpr wcard)  : TermKind wcard\n| prop\n| nat\n| int"}, {"name": "HTermEnv", "content": "structure HTermEnv {wcard tcard bcard : Nat}\n    {wenv : Fin wcard → Nat} {tctx : Term.Ctx wcard tcard}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (tenv : tctx.Env wenv)\n    (benv : Term.BoolEnv bcard) : Prop\n  extends HWidthEnv fsmEnv wenv where\n    heq_term : ∀ (v : Fin tcard),\n      fsmEnv (StateSpace.termVar v) = BitStream.ofBitVecZext (tenv v)\n    heq_bool : ∀ (v : Fin bcard),\n      fsmEnv (StateSpace.boolVar v) = BitStream.ofBool (benv v)"}, {"name": "BitStream.ofBool", "content": "noncomputable def BitStream.ofBool (b : Bool) : BitStream := fun _i => b"}, {"name": "Term.Ctx.Env", "content": "abbrev Term.Ctx.Env\n  (tctx : Term.Ctx wcard tcard)\n  (wenv : WidthExpr.Env wcard) :=\n  (v : Fin tcard) → BitVec ((tctx v).toNat wenv)"}, {"name": "ofBitVecZext", "content": "abbrev ofBitVecZext {w} (x : BitVec w) : BitStream :=\n  fun i => x.getLsbD i"}, {"name": "Term.NatEnv", "content": "def Term.NatEnv (ncard : Nat) : Type := Fin ncard → Nat"}, {"name": "Term.toBV", "content": "def Term.toBV {wenv : WidthExpr.Env wcard}\n    {tctx : Term.Ctx wcard tcard}\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (penv : Predicate.Env pcard)\n    (tenv : tctx.Env wenv)\n    (t : Term bcard ncard icard pcard tctx k) : k.denote wenv :=\nmatch t with\n| .ofNat w n => BitVec.ofNat (w.toNat wenv) n\n| .boolConst b => b\n| .var v => tenv.get v.1 v.2\n| .add (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a + b\n| .zext a v => (a.toBV benv nenv ienv penv tenv).zeroExtend (v.toNat wenv)\n| .setWidth a v => (a.toBV benv nenv ienv penv tenv).zeroExtend (v.toNat wenv)\n| .sext a v => (a.toBV benv nenv ienv penv tenv).signExtend (v.toNat wenv)\n| .bor a b (w := w) =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a ||| b\n| .band (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a &&& b\n| .bxor (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a ^^^ b\n| .bnot (w := w) a =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    ~~~ a\n| .boolVar v => benv v\n| .shiftl (w := w) a k =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    a <<< k\n| .bvOfBool b => BitVec.ofBool (b.toBV benv nenv ienv penv tenv)\n\n| .binWidthRel rel wa wb =>\n  match rel with\n  | .eq => wa.toNat wenv = wb.toNat wenv\n  | .le => wa.toNat wenv ≤ wb.toNat wenv\n| .binRel rel _w a b =>\n  match rel with\n  | .eq => a.toBV benv nenv ienv penv tenv = b.toBV benv nenv ienv penv tenv\n  | .ne => a.toBV benv nenv ienv penv tenv ≠ b.toBV benv nenv ienv penv tenv\n  | .ult => (a.toBV benv nenv ienv penv tenv).ult (b.toBV benv nenv ienv penv tenv) = true\n  | .ule => (a.toBV benv nenv ienv penv tenv).ule (b.toBV benv nenv ienv penv tenv) = true\n  | .slt => (a.toBV benv nenv ienv penv tenv).slt (b.toBV benv nenv ienv penv tenv) = true\n  | .sle => (a.toBV benv nenv ienv penv tenv).sle (b.toBV benv nenv ienv penv tenv) = true\n| .and p1 p2 => p1.toBV benv nenv ienv penv tenv  ∧ p2.toBV benv nenv ienv penv tenv\n| .or p1 p2 => p1.toBV benv nenv ienv penv tenv ∨ p2.toBV benv nenv ienv penv tenv\n| .boolBinRel rel a b =>\n  match rel with\n  \n  | .eq => (a.toBV benv nenv ienv penv tenv) = (b.toBV benv nenv ienv penv tenv)\n| .pvar v => penv v"}, {"name": "Term", "content": "inductive Term {wcard tcard : Nat} (bcard : Nat) (ncard : Nat) (icard : Nat) (pcard : Nat)\n  (tctx : Term.Ctx wcard tcard) : TermKind wcard → Type\n\n \n| ofNat (w : WidthExpr wcard) (n : Nat) : Term bcard ncard icard pcard tctx (.bv w)\n \n| var (v : Fin tcard) : Term bcard ncard icard pcard tctx (.bv (tctx v))\n \n| add (a : Term bcard ncard icard pcard tctx (.bv w))\n  (b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| shiftl (a : Term bcard ncard icard pcard tctx (.bv w)) (k : Nat) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bor (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| band (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bxor (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bnot (a : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| zext (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| setWidth (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| sext (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| bvOfBool (b : Term bcard ncard icard pcard tctx .bool) : Term bcard ncard icard pcard tctx (.bv (.const 1))\n\n| boolConst (b : Bool) : Term bcard ncard icard pcard tctx .bool\n| boolVar (v : Fin bcard) : Term bcard ncard icard pcard tctx .bool\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr wcard) :\n    Term bcard ncard icard pcard tctx .prop\n| binRel\n    (k : BinaryRelationKind)\n    (w : WidthExpr wcard)\n    (a : Term bcard ncard icard pcard tctx (.bv w))\n    (b : Term bcard ncard icard pcard tctx (.bv w)) :\n    Term bcard ncard icard pcard tctx .prop\n| and (p1 p2 : Term bcard ncard icard pcard tctx (.prop)) : Term bcard ncard icard pcard tctx (.prop)\n| or (p1 p2 : Term bcard ncard icard pcard tctx (.prop)) : Term bcard ncard icard pcard tctx (.prop)\n| pvar (v : Fin pcard) : Term bcard ncard icard pcard tctx (.prop) \n\n\n| boolBinRel\n  (k : BoolBinaryRelationKind)\n  (a b : Term bcard ncard icard pcard tctx .bool) :\n  Term bcard ncard icard pcard tctx (.prop)"}, {"name": "Term.Ctx.Env.get", "content": "def Term.Ctx.Env.get {tcard : Nat}\n  {wcard : Nat} {wenv : Fin wcard → Nat}\n  {tctx : Term.Ctx wcard tcard}\n  (tenv : tctx.Env wenv) (i : Nat) (hi : i < tcard) :\n  BitVec ((tctx ⟨i, hi⟩).toNat wenv) :=\n  tenv ⟨i, hi⟩"}, {"name": "BinaryRelationKind", "content": "inductive BinaryRelationKind\n| eq\n| ne\n| ule\n| slt\n| sle\n| ult \nderiving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "WidthBinaryRelationKind", "content": "inductive WidthBinaryRelationKind\n| eq\n| le\n\n\nderiving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "carry'", "content": "def carry' (initCarry : Bool) (x y : BitStream) : BitStream :=\n  fun n =>\n  match n with\n  | 0 => initCarry\n  | n + 1 => (addAux' initCarry x y n).2"}, {"name": "addAux'", "content": "def addAux' (carryIn : Bool) (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => carryIn\n    | i + 1 => (addAux' carryIn x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}], "lib_lemmas": [{"name": "BitVec.msb_eq_getLsbD_last", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.getElem_signExtend", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.of_length_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.slt_eq_not_carry", "module": "Init.Data.BitVec.Bitblast"}], "repo_lemmas": [{"name": "carry'_eq_carry", "content": "protected theorem carry'_eq_carry (x y : BitStream) (c : Bool)\n    (x' y' : BitVec w)\n    (hx : ∀ i, i < n → x'.getLsbD i = x i)\n    (hy : ∀ i,  i < n → y'.getLsbD i = y i) :\n    carry' c x y n = (BitVec.carry n x' y' c)"}, {"name": "carry'_succ", "content": "@[simp] theorem carry'_succ (initCarry : Bool) (x y : BitStream) :\n    (carry' initCarry x y (i + 1)) =\n    let out"}], "used_local_defs": [{"name": "MultiWidth.fsmMsb", "content": "def fsmMsb (x w : FSM α) : FSM α :=\n  composeBinaryAux'\n    (FSM.latchImmediate false)\n    (qfalse := x)\n    (qtrue := w)"}, {"name": "MultiWidth.fsmCarry''", "content": "def fsmCarry'' (initialCarryVal : Bool): FSM Bool :=\n  let outputCirc :=\n    let carry := Circuit.var true (Sum.inl ())\n    let a := Circuit.var true (Sum.inr true)\n    let b := Circuit.var true (Sum.inr false)\n    \n    ((a &&& b) ||| (a &&& carry) ||| (b &&& carry))\n  { α := Unit,\n    \n    \n    initCarry := fun () => initialCarryVal, \n    outputCirc := Circuit.var true (Sum.inl ()) ,\n    nextStateCirc := fun () => outputCirc\n  }"}, {"name": "MultiWidth.fsmMsbEq", "content": "def fsmMsbEq (a : FSM α) (b : FSM α) : FSM α :=\n  composeUnaryAux (FSM.ls false) <|\n    composeBinaryAux' FSM.xor a b"}, {"name": "MultiWidth.fsmTermSlt", "content": "def fsmTermSlt\n  {wcard tcard : Nat}\n  {w : Nondep.WidthExpr}\n  {a b : Nondep.Term}\n  (wfsm : NatFSM wcard tcard bcard ncard icard pcard w)\n  (afsm : TermFSM wcard tcard bcard ncard icard pcard a)\n  (bfsm : TermFSM wcard tcard bcard ncard icard pcard b)\n  : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n    let afsm := afsm.toFsmZext\n    let bfsm := bfsm.toFsmZext\n    let afsm := fsmMsb afsm wfsm.toFsm\n    let bfsm := fsmMsb bfsm wfsm.toFsm\n    let carryFsm :=\n      (~~~ (composeBinaryAux' (fsmCarry'' true)  afsm (~~~ bfsm)))\n    let xorFsm := fsmMsbEq afsm bfsm\n    let val := xorFsm ^^^ carryFsm\n    val"}], "used_local_lemmas": [{"name": "MultiWidth.eval_fsmMsb_eq", "content": "@[simp]\ntheorem eval_fsmMsb_eq {wcard bcard tcard : Nat}\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (x : Term bcard ncard icard pcard tctx (.bv w))\n    (xfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm x))\n    (hxfsm : HTermFSMToBitStream xfsm)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmMsb xfsm.toFsmZext wfsm.toFsm).eval fsmEnv = (fun i =>\n      BitStream.ofBitVecZext (x.toBV benv nenv ienv penv tenv) (min i (w.toNat wenv - 1)))"}, {"name": "MultiWidth.eval_fsmMsb_eq_BitStream_ofBitVecSext", "content": "theorem eval_fsmMsb_eq_BitStream_ofBitVecSext {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (x : Term bcard ncard icard pcard tctx (.bv w))\n    (xfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm x))\n    (hxfsm : HTermFSMToBitStream xfsm)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmMsb xfsm.toFsmZext wfsm.toFsm).eval fsmEnv =\n      BitStream.ofBitVecSext (x.toBV benv nenv ienv penv tenv)"}], "local_ctx": "import Blase.Fast.FiniteStateMachine\n\nimport Blase.Vars\n\nimport Blase.MultiWidth.Defs\n\nimport Blase.KInduction.KInduction\n\nimport Lean\n\nnamespace MultiWidth\n\ndef fsmMsb (x w : FSM α) : FSM α :=\n  composeBinaryAux'\n    (FSM.latchImmediate false)\n    (qfalse := x)\n    (qtrue := w)\n\ndef fsmCarry'' (initialCarryVal : Bool): FSM Bool :=\n  let outputCirc :=\n    let carry := Circuit.var true (Sum.inl ())\n    let a := Circuit.var true (Sum.inr true)\n    let b := Circuit.var true (Sum.inr false)\n    \n    ((a &&& b) ||| (a &&& carry) ||| (b &&& carry))\n  { α := Unit,\n    \n    \n    initCarry := fun () => initialCarryVal, \n    outputCirc := Circuit.var true (Sum.inl ()) ,\n    nextStateCirc := fun () => outputCirc\n  }\n\ndef fsmMsbEq (a : FSM α) (b : FSM α) : FSM α :=\n  composeUnaryAux (FSM.ls false) <|\n    composeBinaryAux' FSM.xor a b\n\ndef fsmTermSlt\n  {wcard tcard : Nat}\n  {w : Nondep.WidthExpr}\n  {a b : Nondep.Term}\n  (wfsm : NatFSM wcard tcard bcard ncard icard pcard w)\n  (afsm : TermFSM wcard tcard bcard ncard icard pcard a)\n  (bfsm : TermFSM wcard tcard bcard ncard icard pcard b)\n  : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n    let afsm := afsm.toFsmZext\n    let bfsm := bfsm.toFsmZext\n    let afsm := fsmMsb afsm wfsm.toFsm\n    let bfsm := fsmMsb bfsm wfsm.toFsm\n    let carryFsm :=\n      (~~~ (composeBinaryAux' (fsmCarry'' true)  afsm (~~~ bfsm)))\n    let xorFsm := fsmMsbEq afsm bfsm\n    let val := xorFsm ^^^ carryFsm\n    val", "target_theorem": "theorem eval_fsmTermSlt_eq_decide_slt {wcard tcard : Nat}\n    (tctx : Term.Ctx wcard tcard)\n    {wenv : WidthExpr.Env wcard}\n    (tenv : tctx.Env wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (a : Term bcard ncard icard pcard tctx (.bv w))\n    (b : Term bcard ncard icard pcard tctx (.bv w))\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (afsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm a))\n    (hafsm : HTermFSMToBitStream afsm)\n    (bfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm b))\n    (hbfsm : HTermFSMToBitStream bfsm)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (henv : HTermEnv fsmEnv tenv benv)\n    :\n    ((fsmTermSlt\n      wfsm\n      afsm\n      bfsm)).eval fsmEnv i =\n       decide (((a.toBV benv nenv ienv penv tenv).signExtend i).slt\n          ((b.toBV benv nenv ienv penv tenv).signExtend i)) :=", "ground_truth_proof": ":= by\n  simp [fsmTermSlt]\n  have := BitVec.slt_eq_not_carry\n    (x := (a.toBV benv nenv ienv penv tenv).signExtend i)\n    (y := (b.toBV benv nenv ienv penv tenv).signExtend i)\n  rw [this]\n  clear this\n  simp [eval_fsmMsb_eq_BitStream_ofBitVecSext\n        (hxfsm := hafsm) (hwfsm := hwfsm)\n        (benv := benv) (nenv := nenv) (ienv := ienv) (penv := penv)\n        (tenv := tenv) (htenv := henv)]\n  simp [eval_fsmMsb_eq_BitStream_ofBitVecSext (hxfsm := hbfsm) (hwfsm := hwfsm)\n    (benv := benv) (nenv := nenv) (ienv := ienv) (penv := penv)\n    (tenv := tenv) (htenv := henv)]\n  rw [BitStream.carry'_eq_carry\n      (x' := BitVec.signExtend i (Term.toBV benv nenv ienv penv tenv a))\n      (y' := ~~~ BitVec.signExtend i (Term.toBV benv nenv ienv penv tenv b))]\n  -- simp [fsmMsbEq]\n  · rcases i with rfl | i\n    · simp\n      simp [BitVec.of_length_zero]\n      simp [fsmMsbEq]\n    · simp [fsmMsbEq]\n      simp [eval_fsmMsb_eq_BitStream_ofBitVecSext\n            (hxfsm := hafsm) (hwfsm := hwfsm)\n            (benv := benv) (nenv := nenv) (ienv := ienv) (penv := penv)\n            (tenv := tenv) (htenv := henv)]\n      simp [eval_fsmMsb_eq_BitStream_ofBitVecSext (hxfsm := hbfsm) (hwfsm := hwfsm)\n        (benv := benv) (nenv := nenv) (ienv := ienv) (penv := penv)\n        (tenv := tenv) (htenv := henv)]\n      simp [BitStream.ofBitVecSext]\n      by_cases hi : i < w.toNat wenv\n      · simp [BitVec.msb_eq_getLsbD_last, BitVec.getElem_signExtend]\n        simp [hi]\n        grind [Bool]\n      · simp [BitVec.msb_eq_getLsbD_last, BitVec.getElem_signExtend]\n        simp [hi]\n        grind [Bool]\n  · intros j\n    intros hj\n    simp [hj]\n    simp [BitVec.getElem_signExtend]\n    simp [BitStream.ofBitVecSext]\n    by_cases hw : j < w.toNat wenv\n    · simp [hw]\n    · simp [hw]\n  · intros j\n    simp\n    intros hj\n    simp [hj]\n    simp [BitVec.getElem_signExtend]\n    simp [BitStream.ofBitVecSext]\n    by_cases hw : j < w.toNat wenv\n    · simp [hw]\n    · simp [hw]", "nesting_depth": 5, "transitive_dep_count": 89, "subset_aristotle": false, "category": "Compiler"}
{"id": 311, "thm_name": "MultiWidth.eval_fsmTermSle_eq_decide_sle", "thm_stmt": "theorem eval_fsmTermSle_eq_decide_sle {wcard tcard bcard : Nat}\n    (tctx : Term.Ctx wcard tcard)\n    {wenv : WidthExpr.Env wcard}\n    (tenv : tctx.Env wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (a : Term bcard ncard icard pcard tctx (.bv w))\n    (b : Term bcard ncard icard pcard tctx (.bv w))\n    (afsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm a))\n    (hafsm : HTermFSMToBitStream afsm)\n    (bfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm b))\n    (hbfsm : HTermFSMToBitStream bfsm)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (henv : HTermEnv fsmEnv tenv benv)\n    :\n    ((fsmTermSle\n      wfsm\n      afsm\n      bfsm)).eval fsmEnv i =\n       decide (((a.toBV benv nenv ienv penv tenv).signExtend i).sle\n       ((b.toBV benv nenv ienv penv tenv).signExtend i))", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/MultiWidth/GoodFSM.lean", "imports": ["import Blase.MultiWidth.Defs", "import Blase.Vars", "import Blase.KInduction.KInduction", "import Lean", "import Blase.Blase.Fast.BitStream", "import Blase.Fast.FiniteStateMachine"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "Empty", "module": "Init.Prelude"}, {"name": "Empty.elim", "module": "Init.Core"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat.max", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.min", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.ofBool", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.signExtend", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "BitVec.carry", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Bool.atLeastTwo", "module": "Init.Data.BitVec.Bitblast"}], "used_repo_defs": [{"name": "syntax \"min\" : MLIR.Pretty.uniform_op", "content": "syntax \"min\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "composeBinaryAux'", "content": "def composeBinaryAux'\n    (p : FSM Bool)\n    (qtrue : FSM α)\n    (qfalse : FSM α) :\n    FSM α :=\n  p.compose (α)\n    (λ _ => α)\n    (λ _ i => i)\n    (λ b => match b with\n      | true => qtrue\n      | false => qfalse)"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "xor", "content": "def xor : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ^^^ Circuit.var true (inr false),\n    nextStateCirc := Empty.elim\n  }"}, {"name": "composeUnaryAux", "content": "def composeUnaryAux\n    (p : FSM Unit)\n    (q : FSM arity) :\n    FSM arity :=\n  p.compose\n    arity\n    _\n    (λ _ => id)\n    (λ _ => q)"}, {"name": "ls", "content": "def ls (b : Bool) : FSM Unit :=\n  { α := Unit,\n    initCarry := fun _ => b,\n    nextStateCirc := fun () => Circuit.var true (inr ()),\n    outputCirc := Circuit.var true (inl ())\n  }"}, {"name": "latchImmediate", "content": "def latchImmediate (initVal : Bool) : FSM Bool where\n  α := Unit\n  initCarry := fun _ => initVal\n  outputCirc :=\n    let xval := Circuit.var true (inr false)\n    let control := Circuit.var true (inr true)\n    let state := Circuit.var true (inl ())\n    Circuit.ite control xval state\n  nextStateCirc := fun () =>\n    let xval := Circuit.var true (inr false)\n    let control := Circuit.var true (inr true)\n    let state := Circuit.var true (inl ())\n    Circuit.ite control xval state"}, {"name": "ite", "content": "def ite (cond t f : Circuit α) : Circuit α :=\n  (cond &&& t) ||| (~~~ cond &&& f)"}, {"name": "TermFSM", "content": "structure TermFSM (wcard tcard bcard ncard icard pcard : Nat) (t : Nondep.Term) where\n  toFsmZext : FSM (StateSpace wcard tcard bcard ncard icard pcard)\n  width : NatFSM wcard tcard bcard ncard icard pcard t.width"}, {"name": "NatFSM", "content": "structure NatFSM (wcard tcard bcard ncard icard pcard : Nat) (v : Nondep.WidthExpr) where\n  toFsm : FSM (StateSpace wcard tcard bcard ncard icard pcard)"}, {"name": "StateSpace", "content": "inductive StateSpace (wcard tcard bcard ncard icard pcard : Nat)\n| widthVar (v : Fin wcard)\n| termVar (v : Fin tcard)\n| predVar (v : Fin pcard)\n| boolVar (v : Fin bcard)\nderiving DecidableEq, Repr, Hashable"}, {"name": "Term", "content": "inductive Term\n| ofNat (w : WidthExpr) (n : Nat) : Term\n| var (v : Nat) (w : WidthExpr) : Term\n| add (w : WidthExpr) (a b : Term) : Term\n| zext (a : Term) (wnew : WidthExpr) : Term\n| setWidth (a : Term) (wnew : WidthExpr) : Term\n| sext (a : Term) (wnew : WidthExpr) : Term\n| bor (w : WidthExpr) (a b : Term) : Term\n| band (w : WidthExpr) (a b : Term) : Term\n| bxor (w : WidthExpr) (a b : Term) : Term\n| bnot (w : WidthExpr)  (a : Term) : Term\n| boolVar (v : Nat) : Term\n| boolConst (b : Bool) : Term\n| shiftl (w : WidthExpr) (a : Term) (k : Nat) : Term\n| bvOfBool (b : Term) : Term\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr) : Term\n| binRel (k : BinaryRelationKind) (w : WidthExpr)\n    (a : Term) (b : Term) : Term\n| or (p1 p2 : Term) : Term\n| and (p1 p2 : Term) : Term\n| pvar (v : Nat) : Term\n| boolBinRel (k : BoolBinaryRelationKind)\n    (a b : Term) : Term\nderiving DecidableEq, Inhabited, Repr, Lean.ToExpr"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : Nat → WidthExpr → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : (k : Nat) → (v : WidthExpr) → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "HNatFSMToBitstream", "content": "structure HNatFSMToBitstream {wcard : Nat} {v : WidthExpr wcard} {tcard : Nat} {bcard : Nat} {pcard : Nat}\n   (fsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v)) : Prop where\n  heq :\n    ∀ (wenv : Fin wcard → Nat)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n    (henv : HWidthEnv fsmEnv wenv) →\n      fsm.toFsm.eval fsmEnv =\n      BitStream.ofNatUnary (v.toNat wenv)"}, {"name": "WidthExpr.toNat", "content": "def WidthExpr.toNat (e : WidthExpr wcard) (env : WidthExpr.Env wcard) : Nat :=\n  match e with\n  | .const n => n\n  | .var v => env v\n  | .min v w => Nat.min (v.toNat env) (w.toNat env)\n  | .max v w => Nat.max (v.toNat env) (w.toNat env)\n  | .addK v k => v.toNat env + k\n  | .kadd k v => k + v.toNat env"}, {"name": "WidthExpr", "content": "inductive WidthExpr (wcard : Nat) : Type\n| const (n : Nat) :  WidthExpr wcard\n| var : (v : Fin wcard) → WidthExpr wcard\n| min : (v w : WidthExpr wcard) → WidthExpr wcard\n| max : (v w : WidthExpr wcard) → WidthExpr wcard\n| addK : (v : WidthExpr wcard) → (k : Nat) → WidthExpr wcard\n| kadd : (k : Nat) → (v : WidthExpr wcard) → WidthExpr wcard"}, {"name": "WidthExpr.Env", "content": "abbrev WidthExpr.Env (wcard : Nat) : Type :=\n  Fin wcard → Nat"}, {"name": "HWidthEnv", "content": "structure HWidthEnv {wcard tcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (wenv : Fin wcard → Nat) : Prop where\n    heq_width : ∀ (v : Fin wcard),\n      fsmEnv (StateSpace.widthVar v) = BitStream.ofNatUnary (wenv v)"}, {"name": "HPredicateEnv", "content": "structure HPredicateEnv {wcard tcard bcard ncard icard pcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (penv : Fin pcard → Prop) : Prop where\n    heq_width : ∀ (v : Fin pcard),\n      fsmEnv (StateSpace.predVar v) = BitStream.ofProp (penv v)"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "ofNatUnary", "content": "abbrev ofNatUnary (n : Nat) : BitStream :=\n  fun i => decide (i < n)"}, {"name": "HPredFSMToBitStream", "content": "structure HPredFSMToBitStream {pcard : Nat}\n  {tctx : Term.Ctx wcard tcard}\n  {p : Term bcard ncard icard pcard tctx .prop}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard\n    (.ofDepTerm p)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (htenv : HTermEnv fsmEnv tenv benv) →\n      (hpenv : HPredicateEnv fsmEnv penv) →\n        p.toBV benv nenv ienv penv tenv  ↔ (fsm.toFsmZext.eval fsmEnv = .negOne)"}, {"name": "Term.Ctx", "content": "abbrev Term.Ctx (wcard : Nat) (tcard : Nat) : Type :=\n  Fin tcard → WidthExpr wcard"}, {"name": "Term.BoolEnv", "content": "def Term.BoolEnv (bcard : Nat) : Type := Fin bcard → Bool"}, {"name": "Term.IntEnv", "content": "def Term.IntEnv (icard : Nat) : Type := Fin icard → Nat"}, {"name": "HTermFSMToBitStream", "content": "structure HTermFSMToBitStream {w : WidthExpr wcard}\n  {tctx : Term.Ctx wcard tcard}\n  {t : Term bcard ncard icard pcard tctx (.bv w)}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm t)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (henv : HTermEnv fsmEnv tenv benv) →\n        fsm.toFsmZext.eval fsmEnv =\n        BitStream.ofBitVecZext (t.toBV benv nenv ienv penv tenv)"}, {"name": "Predicate.Env", "content": "def Predicate.Env (pcard : Nat) : Type :=\n  Fin pcard → Prop"}, {"name": "TermKind", "content": "inductive TermKind (wcard : Nat) : Type\n| bool\n| bv (w : WidthExpr wcard)  : TermKind wcard\n| prop\n| nat\n| int"}, {"name": "HTermEnv", "content": "structure HTermEnv {wcard tcard bcard : Nat}\n    {wenv : Fin wcard → Nat} {tctx : Term.Ctx wcard tcard}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (tenv : tctx.Env wenv)\n    (benv : Term.BoolEnv bcard) : Prop\n  extends HWidthEnv fsmEnv wenv where\n    heq_term : ∀ (v : Fin tcard),\n      fsmEnv (StateSpace.termVar v) = BitStream.ofBitVecZext (tenv v)\n    heq_bool : ∀ (v : Fin bcard),\n      fsmEnv (StateSpace.boolVar v) = BitStream.ofBool (benv v)"}, {"name": "BitStream.ofBool", "content": "noncomputable def BitStream.ofBool (b : Bool) : BitStream := fun _i => b"}, {"name": "Term.Ctx.Env", "content": "abbrev Term.Ctx.Env\n  (tctx : Term.Ctx wcard tcard)\n  (wenv : WidthExpr.Env wcard) :=\n  (v : Fin tcard) → BitVec ((tctx v).toNat wenv)"}, {"name": "ofBitVecZext", "content": "abbrev ofBitVecZext {w} (x : BitVec w) : BitStream :=\n  fun i => x.getLsbD i"}, {"name": "Term.NatEnv", "content": "def Term.NatEnv (ncard : Nat) : Type := Fin ncard → Nat"}, {"name": "Term.toBV", "content": "def Term.toBV {wenv : WidthExpr.Env wcard}\n    {tctx : Term.Ctx wcard tcard}\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (penv : Predicate.Env pcard)\n    (tenv : tctx.Env wenv)\n    (t : Term bcard ncard icard pcard tctx k) : k.denote wenv :=\nmatch t with\n| .ofNat w n => BitVec.ofNat (w.toNat wenv) n\n| .boolConst b => b\n| .var v => tenv.get v.1 v.2\n| .add (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a + b\n| .zext a v => (a.toBV benv nenv ienv penv tenv).zeroExtend (v.toNat wenv)\n| .setWidth a v => (a.toBV benv nenv ienv penv tenv).zeroExtend (v.toNat wenv)\n| .sext a v => (a.toBV benv nenv ienv penv tenv).signExtend (v.toNat wenv)\n| .bor a b (w := w) =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a ||| b\n| .band (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a &&& b\n| .bxor (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a ^^^ b\n| .bnot (w := w) a =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    ~~~ a\n| .boolVar v => benv v\n| .shiftl (w := w) a k =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    a <<< k\n| .bvOfBool b => BitVec.ofBool (b.toBV benv nenv ienv penv tenv)\n\n| .binWidthRel rel wa wb =>\n  match rel with\n  | .eq => wa.toNat wenv = wb.toNat wenv\n  | .le => wa.toNat wenv ≤ wb.toNat wenv\n| .binRel rel _w a b =>\n  match rel with\n  | .eq => a.toBV benv nenv ienv penv tenv = b.toBV benv nenv ienv penv tenv\n  | .ne => a.toBV benv nenv ienv penv tenv ≠ b.toBV benv nenv ienv penv tenv\n  | .ult => (a.toBV benv nenv ienv penv tenv).ult (b.toBV benv nenv ienv penv tenv) = true\n  | .ule => (a.toBV benv nenv ienv penv tenv).ule (b.toBV benv nenv ienv penv tenv) = true\n  | .slt => (a.toBV benv nenv ienv penv tenv).slt (b.toBV benv nenv ienv penv tenv) = true\n  | .sle => (a.toBV benv nenv ienv penv tenv).sle (b.toBV benv nenv ienv penv tenv) = true\n| .and p1 p2 => p1.toBV benv nenv ienv penv tenv  ∧ p2.toBV benv nenv ienv penv tenv\n| .or p1 p2 => p1.toBV benv nenv ienv penv tenv ∨ p2.toBV benv nenv ienv penv tenv\n| .boolBinRel rel a b =>\n  match rel with\n  \n  | .eq => (a.toBV benv nenv ienv penv tenv) = (b.toBV benv nenv ienv penv tenv)\n| .pvar v => penv v"}, {"name": "Term", "content": "inductive Term {wcard tcard : Nat} (bcard : Nat) (ncard : Nat) (icard : Nat) (pcard : Nat)\n  (tctx : Term.Ctx wcard tcard) : TermKind wcard → Type\n\n \n| ofNat (w : WidthExpr wcard) (n : Nat) : Term bcard ncard icard pcard tctx (.bv w)\n \n| var (v : Fin tcard) : Term bcard ncard icard pcard tctx (.bv (tctx v))\n \n| add (a : Term bcard ncard icard pcard tctx (.bv w))\n  (b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| shiftl (a : Term bcard ncard icard pcard tctx (.bv w)) (k : Nat) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bor (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| band (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bxor (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bnot (a : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| zext (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| setWidth (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| sext (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| bvOfBool (b : Term bcard ncard icard pcard tctx .bool) : Term bcard ncard icard pcard tctx (.bv (.const 1))\n\n| boolConst (b : Bool) : Term bcard ncard icard pcard tctx .bool\n| boolVar (v : Fin bcard) : Term bcard ncard icard pcard tctx .bool\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr wcard) :\n    Term bcard ncard icard pcard tctx .prop\n| binRel\n    (k : BinaryRelationKind)\n    (w : WidthExpr wcard)\n    (a : Term bcard ncard icard pcard tctx (.bv w))\n    (b : Term bcard ncard icard pcard tctx (.bv w)) :\n    Term bcard ncard icard pcard tctx .prop\n| and (p1 p2 : Term bcard ncard icard pcard tctx (.prop)) : Term bcard ncard icard pcard tctx (.prop)\n| or (p1 p2 : Term bcard ncard icard pcard tctx (.prop)) : Term bcard ncard icard pcard tctx (.prop)\n| pvar (v : Fin pcard) : Term bcard ncard icard pcard tctx (.prop) \n\n\n| boolBinRel\n  (k : BoolBinaryRelationKind)\n  (a b : Term bcard ncard icard pcard tctx .bool) :\n  Term bcard ncard icard pcard tctx (.prop)"}, {"name": "Term.Ctx.Env.get", "content": "def Term.Ctx.Env.get {tcard : Nat}\n  {wcard : Nat} {wenv : Fin wcard → Nat}\n  {tctx : Term.Ctx wcard tcard}\n  (tenv : tctx.Env wenv) (i : Nat) (hi : i < tcard) :\n  BitVec ((tctx ⟨i, hi⟩).toNat wenv) :=\n  tenv ⟨i, hi⟩"}, {"name": "BinaryRelationKind", "content": "inductive BinaryRelationKind\n| eq\n| ne\n| ule\n| slt\n| sle\n| ult \nderiving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "WidthBinaryRelationKind", "content": "inductive WidthBinaryRelationKind\n| eq\n| le\n\n\nderiving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "carry'", "content": "def carry' (initCarry : Bool) (x y : BitStream) : BitStream :=\n  fun n =>\n  match n with\n  | 0 => initCarry\n  | n + 1 => (addAux' initCarry x y n).2"}, {"name": "addAux'", "content": "def addAux' (carryIn : Bool) (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => carryIn\n    | i + 1 => (addAux' carryIn x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}], "lib_lemmas": [{"name": "BitVec.msb_eq_getLsbD_last", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.getElem_signExtend", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.of_length_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.sle_eq_carry", "module": "Init.Data.BitVec.Bitblast"}], "repo_lemmas": [{"name": "carry'_eq_carry", "content": "protected theorem carry'_eq_carry (x y : BitStream) (c : Bool)\n    (x' y' : BitVec w)\n    (hx : ∀ i, i < n → x'.getLsbD i = x i)\n    (hy : ∀ i,  i < n → y'.getLsbD i = y i) :\n    carry' c x y n = (BitVec.carry n x' y' c)"}, {"name": "carry'_succ", "content": "@[simp] theorem carry'_succ (initCarry : Bool) (x y : BitStream) :\n    (carry' initCarry x y (i + 1)) =\n    let out"}], "used_local_defs": [{"name": "MultiWidth.fsmMsb", "content": "def fsmMsb (x w : FSM α) : FSM α :=\n  composeBinaryAux'\n    (FSM.latchImmediate false)\n    (qfalse := x)\n    (qtrue := w)"}, {"name": "MultiWidth.fsmCarry''", "content": "def fsmCarry'' (initialCarryVal : Bool): FSM Bool :=\n  let outputCirc :=\n    let carry := Circuit.var true (Sum.inl ())\n    let a := Circuit.var true (Sum.inr true)\n    let b := Circuit.var true (Sum.inr false)\n    \n    ((a &&& b) ||| (a &&& carry) ||| (b &&& carry))\n  { α := Unit,\n    \n    \n    initCarry := fun () => initialCarryVal, \n    outputCirc := Circuit.var true (Sum.inl ()) ,\n    nextStateCirc := fun () => outputCirc\n  }"}, {"name": "MultiWidth.fsmMsbEq", "content": "def fsmMsbEq (a : FSM α) (b : FSM α) : FSM α :=\n  composeUnaryAux (FSM.ls false) <|\n    composeBinaryAux' FSM.xor a b"}, {"name": "MultiWidth.fsmTermSle", "content": "def fsmTermSle\n  {wcard tcard : Nat}\n  {w : Nondep.WidthExpr}\n  {a b : Nondep.Term}\n  (wfsm : NatFSM wcard tcard bcard ncard icard pcard w)\n  (afsm : TermFSM wcard tcard bcard ncard icard pcard a)\n  (bfsm : TermFSM wcard tcard bcard ncard icard pcard b)\n  : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n    let afsm := afsm.toFsmZext\n    let bfsm := bfsm.toFsmZext\n    let afsm := fsmMsb afsm wfsm.toFsm\n    let bfsm := fsmMsb bfsm wfsm.toFsm\n    let carryFsm :=\n      ((composeBinaryAux' (fsmCarry'' true)  bfsm (~~~ afsm)))\n    let xorFsm := fsmMsbEq afsm bfsm\n    ~~~ ((~~~ xorFsm) ^^^ carryFsm)"}], "used_local_lemmas": [{"name": "MultiWidth.eval_fsmMsb_eq", "content": "@[simp]\ntheorem eval_fsmMsb_eq {wcard bcard tcard : Nat}\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (x : Term bcard ncard icard pcard tctx (.bv w))\n    (xfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm x))\n    (hxfsm : HTermFSMToBitStream xfsm)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmMsb xfsm.toFsmZext wfsm.toFsm).eval fsmEnv = (fun i =>\n      BitStream.ofBitVecZext (x.toBV benv nenv ienv penv tenv) (min i (w.toNat wenv - 1)))"}, {"name": "MultiWidth.eval_fsmMsb_eq_BitStream_ofBitVecSext", "content": "theorem eval_fsmMsb_eq_BitStream_ofBitVecSext {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (x : Term bcard ncard icard pcard tctx (.bv w))\n    (xfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm x))\n    (hxfsm : HTermFSMToBitStream xfsm)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmMsb xfsm.toFsmZext wfsm.toFsm).eval fsmEnv =\n      BitStream.ofBitVecSext (x.toBV benv nenv ienv penv tenv)"}], "local_ctx": "import Blase.Fast.FiniteStateMachine\n\nimport Blase.Vars\n\nimport Blase.MultiWidth.Defs\n\nimport Blase.KInduction.KInduction\n\nimport Lean\n\nnamespace MultiWidth\n\ndef fsmMsb (x w : FSM α) : FSM α :=\n  composeBinaryAux'\n    (FSM.latchImmediate false)\n    (qfalse := x)\n    (qtrue := w)\n\ndef fsmCarry'' (initialCarryVal : Bool): FSM Bool :=\n  let outputCirc :=\n    let carry := Circuit.var true (Sum.inl ())\n    let a := Circuit.var true (Sum.inr true)\n    let b := Circuit.var true (Sum.inr false)\n    \n    ((a &&& b) ||| (a &&& carry) ||| (b &&& carry))\n  { α := Unit,\n    \n    \n    initCarry := fun () => initialCarryVal, \n    outputCirc := Circuit.var true (Sum.inl ()) ,\n    nextStateCirc := fun () => outputCirc\n  }\n\ndef fsmMsbEq (a : FSM α) (b : FSM α) : FSM α :=\n  composeUnaryAux (FSM.ls false) <|\n    composeBinaryAux' FSM.xor a b\n\ndef fsmTermSle\n  {wcard tcard : Nat}\n  {w : Nondep.WidthExpr}\n  {a b : Nondep.Term}\n  (wfsm : NatFSM wcard tcard bcard ncard icard pcard w)\n  (afsm : TermFSM wcard tcard bcard ncard icard pcard a)\n  (bfsm : TermFSM wcard tcard bcard ncard icard pcard b)\n  : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n    let afsm := afsm.toFsmZext\n    let bfsm := bfsm.toFsmZext\n    let afsm := fsmMsb afsm wfsm.toFsm\n    let bfsm := fsmMsb bfsm wfsm.toFsm\n    let carryFsm :=\n      ((composeBinaryAux' (fsmCarry'' true)  bfsm (~~~ afsm)))\n    let xorFsm := fsmMsbEq afsm bfsm\n    ~~~ ((~~~ xorFsm) ^^^ carryFsm)", "target_theorem": "theorem eval_fsmTermSle_eq_decide_sle {wcard tcard bcard : Nat}\n    (tctx : Term.Ctx wcard tcard)\n    {wenv : WidthExpr.Env wcard}\n    (tenv : tctx.Env wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (a : Term bcard ncard icard pcard tctx (.bv w))\n    (b : Term bcard ncard icard pcard tctx (.bv w))\n    (afsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm a))\n    (hafsm : HTermFSMToBitStream afsm)\n    (bfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm b))\n    (hbfsm : HTermFSMToBitStream bfsm)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (henv : HTermEnv fsmEnv tenv benv)\n    :\n    ((fsmTermSle\n      wfsm\n      afsm\n      bfsm)).eval fsmEnv i =\n       decide (((a.toBV benv nenv ienv penv tenv).signExtend i).sle\n       ((b.toBV benv nenv ienv penv tenv).signExtend i)) :=", "ground_truth_proof": ":= by\n  have := BitVec.sle_eq_carry\n    (x := (a.toBV benv nenv ienv penv tenv).signExtend i)\n    (y := (b.toBV benv nenv ienv penv tenv).signExtend i)\n  rw [this]\n  clear this\n  simp [fsmTermSle]\n  simp [eval_fsmMsb_eq_BitStream_ofBitVecSext (hxfsm := hafsm) (hwfsm := hwfsm)\n        (benv := benv) (nenv := nenv) (ienv := ienv) (penv := penv) (tenv := tenv) (htenv := henv)]\n  simp [eval_fsmMsb_eq_BitStream_ofBitVecSext (hxfsm := hbfsm) (hwfsm := hwfsm)\n    (benv := benv) (nenv := nenv) (ienv := ienv) (penv := penv) (tenv := tenv) (htenv := henv)]\n  rw [BitStream.carry'_eq_carry\n      (x' := BitVec.signExtend i (Term.toBV benv nenv ienv penv tenv b))\n      (y' := ~~~ BitVec.signExtend i (Term.toBV benv nenv ienv penv tenv a))]\n  simp [fsmMsbEq]\n  · rcases i with rfl | i\n    · simp\n      simp [BitVec.of_length_zero]\n    · simp [eval_fsmMsb_eq_BitStream_ofBitVecSext (hxfsm := hafsm) (hwfsm := hwfsm)\n            (benv := benv) (nenv := nenv) (ienv := ienv) (penv := penv) (tenv := tenv) (htenv := henv)]\n      simp [eval_fsmMsb_eq_BitStream_ofBitVecSext (hxfsm := hbfsm) (hwfsm := hwfsm)\n        (benv := benv) (nenv := nenv) (ienv := ienv) (penv := penv) (tenv := tenv) (htenv := henv)]\n      simp [BitStream.ofBitVecSext]\n      by_cases hi : i < w.toNat wenv\n      · simp [BitVec.msb_eq_getLsbD_last, BitVec.getElem_signExtend]\n        simp [hi]\n        grind [Bool]\n      · simp [BitVec.msb_eq_getLsbD_last, BitVec.getElem_signExtend]\n        simp [hi]\n        grind [Bool]\n  · intros j\n    intros hj\n    simp [hj]\n    simp [BitVec.getElem_signExtend]\n    simp [BitStream.ofBitVecSext]\n    by_cases hw : j < w.toNat wenv\n    · simp [hw]\n    · simp [hw]\n  · intros j\n    simp\n    intros hj\n    simp [hj]\n    simp [BitVec.getElem_signExtend]\n    simp [BitStream.ofBitVecSext]\n    by_cases hw : j < w.toNat wenv\n    · simp [hw]\n    · simp [hw]", "nesting_depth": 5, "transitive_dep_count": 89, "subset_aristotle": false, "category": "Compiler"}
{"id": 312, "thm_name": "MultiWidth.fsmSext_eval_eq", "thm_stmt": "theorem fsmSext_eval_eq\n    (woldFsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep wold))\n    (wnewFsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep wnew))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (hwnew : HNatFSMToBitstream wnewFsm)\n    (hwold : HNatFSMToBitstream woldFsm)\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (t : Term bcard ncard icard pcard tctx (.bv wold))\n    (tFsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm t))\n    (htfsm : HTermFSMToBitStream tFsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmSext tFsm.toFsmZext woldFsm.toFsm  wnewFsm.toFsm).eval fsmEnv = fun i =>\n      ((BitStream.ofBitVecZext ((Term.sext t wnew).toBV benv nenv ienv penv tenv))) i", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/MultiWidth/GoodFSM.lean", "imports": ["import Blase.MultiWidth.Defs", "import Blase.Vars", "import Blase.KInduction.KInduction", "import Lean", "import Blase.Blase.Fast.FiniteStateMachine", "import Blase.Blase.Fast.BitStream", "import Blase.Fast.FiniteStateMachine"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat.max", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.min", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.ofBool", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Empty", "module": "Init.Prelude"}, {"name": "Empty.elim", "module": "Init.Core"}], "used_repo_defs": [{"name": "syntax \"min\" : MLIR.Pretty.uniform_op", "content": "syntax \"min\" : MLIR.Pretty.uniform_op\n\nsyntax \"slt\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "composeBinaryAux'", "content": "def composeBinaryAux'\n    (p : FSM Bool)\n    (qtrue : FSM α)\n    (qfalse : FSM α) :\n    FSM α :=\n  p.compose (α)\n    (λ _ => α)\n    (λ _ i => i)\n    (λ b => match b with\n      | true => qtrue\n      | false => qfalse)"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "latchImmediate", "content": "def latchImmediate (initVal : Bool) : FSM Bool where\n  α := Unit\n  initCarry := fun _ => initVal\n  outputCirc :=\n    let xval := Circuit.var true (inr false)\n    let control := Circuit.var true (inr true)\n    let state := Circuit.var true (inl ())\n    Circuit.ite control xval state\n  nextStateCirc := fun () =>\n    let xval := Circuit.var true (inr false)\n    let control := Circuit.var true (inr true)\n    let state := Circuit.var true (inl ())\n    Circuit.ite control xval state"}, {"name": "ite", "content": "def ite (cond t f : Circuit α) : Circuit α :=\n  (cond &&& t) ||| (~~~ cond &&& f)"}, {"name": "Term", "content": "inductive Term\n| ofNat (w : WidthExpr) (n : Nat) : Term\n| var (v : Nat) (w : WidthExpr) : Term\n| add (w : WidthExpr) (a b : Term) : Term\n| zext (a : Term) (wnew : WidthExpr) : Term\n| setWidth (a : Term) (wnew : WidthExpr) : Term\n| sext (a : Term) (wnew : WidthExpr) : Term\n| bor (w : WidthExpr) (a b : Term) : Term\n| band (w : WidthExpr) (a b : Term) : Term\n| bxor (w : WidthExpr) (a b : Term) : Term\n| bnot (w : WidthExpr)  (a : Term) : Term\n| mul (w : WidthExpr) (a b : Term) : Term\n| udiv (w : WidthExpr) (a b : Term) : Term\n| umod (w : WidthExpr) (a b : Term) : Term\n| boolVar (v : Nat) : Term\n| boolConst (b : Bool) : Term\n| shiftl (w : WidthExpr) (a : Term) (k : Nat) : Term\n| junk (s : String)  : Term \nderiving DecidableEq, Inhabited, Repr, Lean.ToExpr"}, {"name": "TermFSM", "content": "structure TermFSM (wcard tcard bcard ncard icard pcard : Nat) (t : Nondep.Term) where\n  toFsmZext : FSM (StateSpace wcard tcard bcard ncard icard pcard)\n  width : NatFSM wcard tcard bcard ncard icard pcard t.width"}, {"name": "NatFSM", "content": "structure NatFSM (wcard tcard bcard ncard icard pcard : Nat) (v : Nondep.WidthExpr) where\n  toFsm : FSM (StateSpace wcard tcard bcard ncard icard pcard)"}, {"name": "StateSpace", "content": "inductive StateSpace (wcard tcard bcard ncard icard pcard : Nat)\n| widthVar (v : Fin wcard)\n| termVar (v : Fin tcard)\n| predVar (v : Fin pcard)\n| boolVar (v : Fin bcard)\nderiving DecidableEq, Repr, Hashable"}, {"name": "Term", "content": "inductive Term\n| ofNat (w : WidthExpr) (n : Nat) : Term\n| var (v : Nat) (w : WidthExpr) : Term\n| add (w : WidthExpr) (a b : Term) : Term\n| zext (a : Term) (wnew : WidthExpr) : Term\n| setWidth (a : Term) (wnew : WidthExpr) : Term\n| sext (a : Term) (wnew : WidthExpr) : Term\n| bor (w : WidthExpr) (a b : Term) : Term\n| band (w : WidthExpr) (a b : Term) : Term\n| bxor (w : WidthExpr) (a b : Term) : Term\n| bnot (w : WidthExpr)  (a : Term) : Term\n| boolVar (v : Nat) : Term\n| boolConst (b : Bool) : Term\n| shiftl (w : WidthExpr) (a : Term) (k : Nat) : Term\n| bvOfBool (b : Term) : Term\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr) : Term\n| binRel (k : BinaryRelationKind) (w : WidthExpr)\n    (a : Term) (b : Term) : Term\n| or (p1 p2 : Term) : Term\n| and (p1 p2 : Term) : Term\n| pvar (v : Nat) : Term\n| boolBinRel (k : BoolBinaryRelationKind)\n    (a b : Term) : Term\nderiving DecidableEq, Inhabited, Repr, Lean.ToExpr"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : Nat → WidthExpr → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : (k : Nat) → (v : WidthExpr) → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "HNatFSMToBitstream", "content": "structure HNatFSMToBitstream {wcard : Nat} {v : WidthExpr wcard} {tcard : Nat} {bcard : Nat} {pcard : Nat}\n   (fsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v)) : Prop where\n  heq :\n    ∀ (wenv : Fin wcard → Nat)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n    (henv : HWidthEnv fsmEnv wenv) →\n      fsm.toFsm.eval fsmEnv =\n      BitStream.ofNatUnary (v.toNat wenv)"}, {"name": "WidthExpr.toNat", "content": "def WidthExpr.toNat (e : WidthExpr wcard) (env : WidthExpr.Env wcard) : Nat :=\n  match e with\n  | .const n => n\n  | .var v => env v\n  | .min v w => Nat.min (v.toNat env) (w.toNat env)\n  | .max v w => Nat.max (v.toNat env) (w.toNat env)\n  | .addK v k => v.toNat env + k\n  | .kadd k v => k + v.toNat env"}, {"name": "WidthExpr", "content": "inductive WidthExpr (wcard : Nat) : Type\n| const (n : Nat) :  WidthExpr wcard\n| var : (v : Fin wcard) → WidthExpr wcard\n| min : (v w : WidthExpr wcard) → WidthExpr wcard\n| max : (v w : WidthExpr wcard) → WidthExpr wcard\n| addK : (v : WidthExpr wcard) → (k : Nat) → WidthExpr wcard\n| kadd : (k : Nat) → (v : WidthExpr wcard) → WidthExpr wcard"}, {"name": "WidthExpr.Env", "content": "abbrev WidthExpr.Env (wcard : Nat) : Type :=\n  Fin wcard → Nat"}, {"name": "HWidthEnv", "content": "structure HWidthEnv {wcard tcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (wenv : Fin wcard → Nat) : Prop where\n    heq_width : ∀ (v : Fin wcard),\n      fsmEnv (StateSpace.widthVar v) = BitStream.ofNatUnary (wenv v)"}, {"name": "HPredicateEnv", "content": "structure HPredicateEnv {wcard tcard bcard ncard icard pcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (penv : Fin pcard → Prop) : Prop where\n    heq_width : ∀ (v : Fin pcard),\n      fsmEnv (StateSpace.predVar v) = BitStream.ofProp (penv v)"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "ofNatUnary", "content": "abbrev ofNatUnary (n : Nat) : BitStream :=\n  fun i => decide (i < n)"}, {"name": "HPredFSMToBitStream", "content": "structure HPredFSMToBitStream {pcard : Nat}\n  {tctx : Term.Ctx wcard tcard}\n  {p : Term bcard ncard icard pcard tctx .prop}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard\n    (.ofDepTerm p)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (htenv : HTermEnv fsmEnv tenv benv) →\n      (hpenv : HPredicateEnv fsmEnv penv) →\n        p.toBV benv nenv ienv penv tenv  ↔ (fsm.toFsmZext.eval fsmEnv = .negOne)"}, {"name": "Term.Ctx", "content": "abbrev Term.Ctx (wcard : Nat) (tcard : Nat) : Type :=\n  Fin tcard → WidthExpr wcard"}, {"name": "Term.BoolEnv", "content": "def Term.BoolEnv (bcard : Nat) : Type := Fin bcard → Bool"}, {"name": "Term.IntEnv", "content": "def Term.IntEnv (icard : Nat) : Type := Fin icard → Nat"}, {"name": "ofBitVecZext", "content": "abbrev ofBitVecZext {w} (x : BitVec w) : BitStream :=\n  fun i => x.getLsbD i"}, {"name": "HTermFSMToBitStream", "content": "structure HTermFSMToBitStream {w : WidthExpr wcard}\n  {tctx : Term.Ctx wcard tcard}\n  {t : Term bcard ncard icard pcard tctx (.bv w)}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm t)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (henv : HTermEnv fsmEnv tenv benv) →\n        fsm.toFsmZext.eval fsmEnv =\n        BitStream.ofBitVecZext (t.toBV benv nenv ienv penv tenv)"}, {"name": "Predicate.Env", "content": "def Predicate.Env (pcard : Nat) : Type :=\n  Fin pcard → Prop"}, {"name": "TermKind", "content": "inductive TermKind (wcard : Nat) : Type\n| bool\n| bv (w : WidthExpr wcard)  : TermKind wcard\n| prop\n| nat\n| int"}, {"name": "HTermEnv", "content": "structure HTermEnv {wcard tcard bcard : Nat}\n    {wenv : Fin wcard → Nat} {tctx : Term.Ctx wcard tcard}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (tenv : tctx.Env wenv)\n    (benv : Term.BoolEnv bcard) : Prop\n  extends HWidthEnv fsmEnv wenv where\n    heq_term : ∀ (v : Fin tcard),\n      fsmEnv (StateSpace.termVar v) = BitStream.ofBitVecZext (tenv v)\n    heq_bool : ∀ (v : Fin bcard),\n      fsmEnv (StateSpace.boolVar v) = BitStream.ofBool (benv v)"}, {"name": "BitStream.ofBool", "content": "noncomputable def BitStream.ofBool (b : Bool) : BitStream := fun _i => b"}, {"name": "Term.Ctx.Env", "content": "abbrev Term.Ctx.Env\n  (tctx : Term.Ctx wcard tcard)\n  (wenv : WidthExpr.Env wcard) :=\n  (v : Fin tcard) → BitVec ((tctx v).toNat wenv)"}, {"name": "Term.NatEnv", "content": "def Term.NatEnv (ncard : Nat) : Type := Fin ncard → Nat"}, {"name": "Term.toBV", "content": "def Term.toBV {wenv : WidthExpr.Env wcard}\n    {tctx : Term.Ctx wcard tcard}\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (penv : Predicate.Env pcard)\n    (tenv : tctx.Env wenv)\n    (t : Term bcard ncard icard pcard tctx k) : k.denote wenv :=\nmatch t with\n| .ofNat w n => BitVec.ofNat (w.toNat wenv) n\n| .boolConst b => b\n| .var v => tenv.get v.1 v.2\n| .add (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a + b\n| .zext a v => (a.toBV benv nenv ienv penv tenv).zeroExtend (v.toNat wenv)\n| .setWidth a v => (a.toBV benv nenv ienv penv tenv).zeroExtend (v.toNat wenv)\n| .sext a v => (a.toBV benv nenv ienv penv tenv).signExtend (v.toNat wenv)\n| .bor a b (w := w) =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a ||| b\n| .band (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a &&& b\n| .bxor (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a ^^^ b\n| .bnot (w := w) a =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    ~~~ a\n| .boolVar v => benv v\n| .shiftl (w := w) a k =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    a <<< k\n| .bvOfBool b => BitVec.ofBool (b.toBV benv nenv ienv penv tenv)\n\n| .binWidthRel rel wa wb =>\n  match rel with\n  | .eq => wa.toNat wenv = wb.toNat wenv\n  | .le => wa.toNat wenv ≤ wb.toNat wenv\n| .binRel rel _w a b =>\n  match rel with\n  | .eq => a.toBV benv nenv ienv penv tenv = b.toBV benv nenv ienv penv tenv\n  | .ne => a.toBV benv nenv ienv penv tenv ≠ b.toBV benv nenv ienv penv tenv\n  | .ult => (a.toBV benv nenv ienv penv tenv).ult (b.toBV benv nenv ienv penv tenv) = true\n  | .ule => (a.toBV benv nenv ienv penv tenv).ule (b.toBV benv nenv ienv penv tenv) = true\n  | .slt => (a.toBV benv nenv ienv penv tenv).slt (b.toBV benv nenv ienv penv tenv) = true\n  | .sle => (a.toBV benv nenv ienv penv tenv).sle (b.toBV benv nenv ienv penv tenv) = true\n| .and p1 p2 => p1.toBV benv nenv ienv penv tenv  ∧ p2.toBV benv nenv ienv penv tenv\n| .or p1 p2 => p1.toBV benv nenv ienv penv tenv ∨ p2.toBV benv nenv ienv penv tenv\n| .boolBinRel rel a b =>\n  match rel with\n  \n  | .eq => (a.toBV benv nenv ienv penv tenv) = (b.toBV benv nenv ienv penv tenv)\n| .pvar v => penv v"}, {"name": "Term", "content": "inductive Term {wcard tcard : Nat} (bcard : Nat) (ncard : Nat) (icard : Nat) (pcard : Nat)\n  (tctx : Term.Ctx wcard tcard) : TermKind wcard → Type\n\n \n| ofNat (w : WidthExpr wcard) (n : Nat) : Term bcard ncard icard pcard tctx (.bv w)\n \n| var (v : Fin tcard) : Term bcard ncard icard pcard tctx (.bv (tctx v))\n \n| add (a : Term bcard ncard icard pcard tctx (.bv w))\n  (b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| shiftl (a : Term bcard ncard icard pcard tctx (.bv w)) (k : Nat) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bor (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| band (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bxor (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bnot (a : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| zext (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| setWidth (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| sext (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| bvOfBool (b : Term bcard ncard icard pcard tctx .bool) : Term bcard ncard icard pcard tctx (.bv (.const 1))\n\n| boolConst (b : Bool) : Term bcard ncard icard pcard tctx .bool\n| boolVar (v : Fin bcard) : Term bcard ncard icard pcard tctx .bool\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr wcard) :\n    Term bcard ncard icard pcard tctx .prop\n| binRel\n    (k : BinaryRelationKind)\n    (w : WidthExpr wcard)\n    (a : Term bcard ncard icard pcard tctx (.bv w))\n    (b : Term bcard ncard icard pcard tctx (.bv w)) :\n    Term bcard ncard icard pcard tctx .prop\n| and (p1 p2 : Term bcard ncard icard pcard tctx (.prop)) : Term bcard ncard icard pcard tctx (.prop)\n| or (p1 p2 : Term bcard ncard icard pcard tctx (.prop)) : Term bcard ncard icard pcard tctx (.prop)\n| pvar (v : Fin pcard) : Term bcard ncard icard pcard tctx (.prop) \n\n\n| boolBinRel\n  (k : BoolBinaryRelationKind)\n  (a b : Term bcard ncard icard pcard tctx .bool) :\n  Term bcard ncard icard pcard tctx (.prop)"}, {"name": "Term.Ctx.Env.get", "content": "def Term.Ctx.Env.get {tcard : Nat}\n  {wcard : Nat} {wenv : Fin wcard → Nat}\n  {tctx : Term.Ctx wcard tcard}\n  (tenv : tctx.Env wenv) (i : Nat) (hi : i < tcard) :\n  BitVec ((tctx ⟨i, hi⟩).toNat wenv) :=\n  tenv ⟨i, hi⟩"}, {"name": "BinaryRelationKind", "content": "inductive BinaryRelationKind\n| eq\n| ne\n| ule\n| slt\n| sle\n| ult \nderiving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "WidthBinaryRelationKind", "content": "inductive WidthBinaryRelationKind\n| eq\n| le\n\n\nderiving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "eval", "content": "def eval (x : arity → BitStream) : BitStream :=\n  fun n => (p.nextBit (p.carry x n) (fun i => x i n)).2"}, {"name": "nextBit", "content": "def nextBit : p.State → (arity → Bool) → p.State × Bool :=\n  fun carry inputBits =>\n    let input := Sum.elim carry inputBits\n    let newState : p.State  := fun (a : p.α) => (p.nextStateCirc a).eval input\n    let outBit : Bool       := (p.outputCirc).eval input\n    (newState, outBit)"}, {"name": "State", "content": "abbrev State : Type := p.α → Bool"}, {"name": "carry", "content": "def carry (x : arity → BitStream) : ℕ → p.State\n  | 0 => p.initCarry\n  | n+1 => (p.nextBit (carry x n) (fun i => x i n)).1"}, {"name": "Term.eval", "content": "def Term.eval (t : Term) (vars : List BitStream) : BitStream :=\n  match t with\n  | var n       => vars.getD n default\n  | zero        => BitStream.zero\n  | one         => BitStream.one\n  | negOne      => BitStream.negOne\n  | ofNat n     => BitStream.ofNat n\n  | and t₁ t₂   => (t₁.eval vars) &&& (t₂.eval vars)\n  | or t₁ t₂    => (t₁.eval vars) ||| (t₂.eval vars)\n  | xor t₁ t₂   => (t₁.eval vars) ^^^ (t₂.eval vars)\n  | not t       => ~~~(t.eval vars)\n  | add t₁ t₂   => (Term.eval t₁ vars) + (Term.eval t₂ vars)\n  | sub t₁ t₂   => (Term.eval t₁ vars) - (Term.eval t₂ vars)\n  | neg t       => -(Term.eval t vars)\n\n\n  | shiftL t n  => BitStream.shiftLeft (Term.eval t vars) n"}, {"name": "Predicate.eval", "content": "def Predicate.eval (p : Predicate) (vars : List BitStream) : BitStream :=\n  match p with\n  | .width .eq n => BitStream.falseIffEq n\n  | .width .neq n => BitStream.falseIffNeq n\n  | .width .lt n => BitStream.falseIffLt n\n  | .width .le n => BitStream.falseIffLe n\n  | .width .gt n => BitStream.falseIffGt n\n  | .width .ge n => BitStream.falseIffGe n\n  | lor p q => Predicate.evalLor (p.eval vars) (q.eval vars)\n  | land p q => Predicate.evalLand (p.eval vars) (q.eval vars)\n  | binary .eq t₁ t₂ => Predicate.evalEq (t₁.eval vars) (t₂.eval vars)\n   \n  | binary .neq t1 t2 => Predicate.evalNeq (t1.eval vars) (t2.eval vars)\n  | binary .ult t₁ t₂ => Predicate.evalUlt (t₁.eval vars) (t₂.eval vars)\n  | binary .ule t₁ t₂ =>\n     Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalUlt (t₁.eval vars) (t₂.eval vars))\n  | binary .slt t₁ t₂ => Predicate.evalSlt (t₁.eval vars) (t₂.eval vars)\n  | binary .sle t₁ t₂ => Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalSlt (t₁.eval vars) (t₂.eval vars))"}, {"name": "Predicate.evalUlt", "content": "def Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true"}, {"name": "borrow", "content": "def borrow (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).2"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "Predicate.evalSlt", "content": "def Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))"}, {"name": "Predicate.evalMsbEq", "content": "def Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false"}, {"name": "Predicate.evalLand", "content": "def Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)"}, {"name": "Predicate.evalNeq", "content": "def Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd"}, {"name": "nxor", "content": "def nxor (a b : BitStream) : BitStream := fun i => a i == b i"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "Predicate.evalLor", "content": "def Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)"}, {"name": "Predicate.evalEq", "content": "def Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "falseIffNeq", "content": "abbrev falseIffNeq (n : Nat) : BitStream := fun i => decide (i == n)"}, {"name": "falseIffLt", "content": "abbrev falseIffLt (n : Nat) : BitStream := fun i => decide (i ≥ n)"}, {"name": "falseIffGe", "content": "abbrev falseIffGe (n : Nat) : BitStream := fun i => decide (i < n)"}, {"name": "falseIffEq", "content": "abbrev falseIffEq (n : Nat) : BitStream := fun i => decide (i != n)"}, {"name": "falseIffGt", "content": "abbrev falseIffGt (n : Nat) : BitStream := fun i => decide (i ≤ n)"}, {"name": "falseIffLe", "content": "abbrev falseIffLe (n : Nat) : BitStream := fun i => decide (i > n)"}, {"name": "negOne", "content": "abbrev negOne : BitStream := fun _ => true"}, {"name": "shiftLeft", "content": "def shiftLeft (x : BitStream) (k : Nat) : BitStream :=\n  fun i => if i < k then false else x (i - k) "}, {"name": "ofNat", "content": "def ofNat (x : Nat) : BitStream :=\n  Nat.testBit x"}, {"name": "one", "content": "abbrev one    : BitStream := (· == 0)"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "and", "content": "def and : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := fun a => a.elim,\n    outputCirc := Circuit.var true (inr true) &&& Circuit.var true (inr false),\n  }"}], "lib_lemmas": [{"name": "BitVec.getElem_signExtend", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.getLsbD_signExtend", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.msb_eq_getLsbD_last", "module": "Init.Data.BitVec.Lemmas"}], "repo_lemmas": [{"name": "and_eq", "content": "@[simp] theorem and_eq : (x &&& y) i = (x i && y i)"}, {"name": "FSM.eval_and'", "content": "@[simp]\ntheorem FSM.eval_and' (a b : FSM arity) : (a &&& b).eval env = a.eval env &&& b.eval env"}, {"name": "FSM.and_eq", "content": "theorem FSM.and_eq (a b : FSM arity) : (a &&& b) = composeBinaryAux' FSM.and a b"}], "used_local_defs": [{"name": "MultiWidth.fsmMsb", "content": "def fsmMsb (x w : FSM α) : FSM α :=\n  composeBinaryAux'\n    (FSM.latchImmediate false)\n    (qfalse := x)\n    (qtrue := w)"}, {"name": "MultiWidth.fsmSext", "content": "def fsmSext (x wold wnew : FSM α) : FSM α :=\n  (fsmMsb x wold) &&& wnew"}], "used_local_lemmas": [{"name": "MultiWidth.eval_fsmMsb_eq", "content": "@[simp]\ntheorem eval_fsmMsb_eq {wcard bcard tcard : Nat}\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (x : Term bcard ncard icard pcard tctx (.bv w))\n    (xfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm x))\n    (hxfsm : HTermFSMToBitStream xfsm)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmMsb xfsm.toFsmZext wfsm.toFsm).eval fsmEnv = (fun i =>\n      BitStream.ofBitVecZext (x.toBV benv nenv ienv penv tenv) (min i (w.toNat wenv - 1)))"}], "local_ctx": "import Blase.Fast.FiniteStateMachine\n\nimport Blase.Vars\n\nimport Blase.MultiWidth.Defs\n\nimport Blase.KInduction.KInduction\n\nimport Lean\n\nnamespace MultiWidth\n\ndef fsmMsb (x w : FSM α) : FSM α :=\n  composeBinaryAux'\n    (FSM.latchImmediate false)\n    (qfalse := x)\n    (qtrue := w)\n\ndef fsmSext (x wold wnew : FSM α) : FSM α :=\n  (fsmMsb x wold) &&& wnew", "target_theorem": "theorem fsmSext_eval_eq\n    (woldFsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep wold))\n    (wnewFsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep wnew))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (hwnew : HNatFSMToBitstream wnewFsm)\n    (hwold : HNatFSMToBitstream woldFsm)\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (t : Term bcard ncard icard pcard tctx (.bv wold))\n    (tFsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm t))\n    (htfsm : HTermFSMToBitStream tFsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmSext tFsm.toFsmZext woldFsm.toFsm  wnewFsm.toFsm).eval fsmEnv = fun i =>\n      ((BitStream.ofBitVecZext ((Term.sext t wnew).toBV benv nenv ienv penv tenv))) i :=", "ground_truth_proof": ":= by\n  ext i\n  rw [fsmSext]\n  simp [FSM.eval_and', BitStream.and_eq]\n  rw [hwnew.heq (henv := htenv.toHWidthEnv)]\n  rw [eval_fsmMsb_eq\n        (xfsm := tFsm) (wfsm := woldFsm) (htenv := htenv)\n        (hxfsm := htfsm) (hwfsm := hwold)]\n  simp\n  by_cases hwold : i < wold.toNat wenv\n  · simp [hwold]\n    by_cases hwnew : i < wnew.toNat wenv\n    · simp [hwnew]\n      simp [BitVec.getElem_signExtend]\n      simp [hwold]\n      congr; omega\n    · simp [hwnew]\n      simp [BitVec.getLsbD_signExtend]\n      omega\n  · by_cases hwnew : i < wnew.toNat wenv\n    · simp [hwnew]\n      simp at hwold\n      rw [BitVec.getElem_signExtend]\n      simp [show min i (wold.toNat wenv - 1) = wold.toNat wenv - 1 by omega]\n      simp [show ¬ i < wold.toNat wenv by omega]\n      rw [BitVec.msb_eq_getLsbD_last]\n    · simp [hwnew]\n      rw [BitVec.getLsbD_signExtend]\n      simp; omega", "nesting_depth": 7, "transitive_dep_count": 107, "subset_aristotle": false, "category": "Compiler"}
{"id": 313, "thm_name": "R.repLength_lt_n_plus_1", "thm_stmt": "theorem R.repLength_lt_n_plus_1 [Fact (q > 1)]: forall a : R q n, a.repLength < 2^n + 1", "lean_root": "lean-mlir", "rel_path": "SSA/Projects/FullyHomomorphicEncryption/Basic.lean", "imports": ["import Mathlib.Data.List.Basic", "import Mathlib.Data.List.ToFinsupp", "import Mathlib.RingTheory.Polynomial.Quotient", "import Mathlib.Data.ZMod.Defs", "import Mathlib.RingTheory.Ideal.Defs", "import Mathlib.RingTheory.Ideal.Basic", "import Mathlib.Data.ZMod.Basic", "import Mathlib.Algebra.MonoidAlgebra.Basic", "import LeanMLIR.Framework", "import Mathlib.Tactic.Cases", "import Mathlib.Algebra.Polynomial.RingDivision", "import Mathlib.Data.Finset.Sort"], "used_lib_defs": [{"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Function.surjInv", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.degree", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "WithBot", "module": "Mathlib.Order.TypeTags"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "OfNat.ofNat", "module": "Init.Prelude"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Fin.pos'", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.cast_ofNat", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_pow", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Polynomial.degree_X_pow", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.degree_add_eq_left_of_degree_lt", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.degree_one", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Nat.one_le_two_pow", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.monic_X_pow_add", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Option.some_inj", "module": "Init.Data.Option.Instances"}, {"name": "Polynomial.degree_modByMonic_lt", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "WithBot.lt_def", "module": "Mathlib.Order.WithBot"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}, {"name": "R.representative'", "content": "private noncomputable def R.representative' :\n    R q n → (ZMod q)[X] := Function.surjInv (R.surjective_fromPoly q n)"}, {"name": "R.representative", "content": "noncomputable def R.representative :\n    R q n → (ZMod q)[X] := fun x => R.representative' q n x %ₘ (f q n)"}, {"name": "R.repLength", "content": "noncomputable def R.repLength {q n} (a : R q n) : Nat := match\n  Polynomial.degree a.representative with\n    | none => 0\n    | some d => d + 1"}], "used_local_lemmas": [{"name": "f_deg_eq", "content": "theorem f_deg_eq : (f q n).degree = 2^n"}, {"name": "f_monic", "content": "theorem f_monic : Monic (f q n)"}], "local_ctx": "import Mathlib.RingTheory.Polynomial.Quotient\n\nimport Mathlib.RingTheory.Ideal.Defs\n\nimport Mathlib.RingTheory.Ideal.Basic\n\nimport Mathlib.Data.ZMod.Defs\n\nimport Mathlib.Data.ZMod.Basic\n\nimport Mathlib.Algebra.MonoidAlgebra.Basic\n\nimport Mathlib.Algebra.Polynomial.RingDivision\n\nimport Mathlib.Data.Finset.Sort\n\nimport Mathlib.Data.List.ToFinsupp\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Tactic.Cases\n\nimport LeanMLIR.Framework\n\nopen Polynomial -- for R[X] notation\n\nsection CommRing\n\nvariable (q t : Nat) [Fact (q > 1)] (n : Nat)\n\nnoncomputable def f : (ZMod q)[X] := X^(2^n) + 1\n\nabbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})\n\nend CommRing\n\nsection Representative\n\nvariable (q t n : Nat)\n\nprivate noncomputable def R.representative' :\n    R q n → (ZMod q)[X] := Function.surjInv (R.surjective_fromPoly q n)\n\nnoncomputable def R.representative :\n    R q n → (ZMod q)[X] := fun x => R.representative' q n x %ₘ (f q n)\n\nvariable [Fact (q > 1)]\n\nend Representative\n\nnoncomputable def R.repLength {q n} (a : R q n) : Nat := match\n  Polynomial.degree a.representative with\n    | none => 0\n    | some d => d + 1", "target_theorem": "theorem R.repLength_lt_n_plus_1 [Fact (q > 1)]: forall a : R q n, a.repLength < 2^n + 1 :=", "ground_truth_proof": ":= by\n  intro a\n  simp only [R.repLength, representative]\n  have : Polynomial.degree ( R.representative' q n a %ₘ f q n) < 2^n := by\n    rw [← f_deg_eq q n]\n    apply (Polynomial.degree_modByMonic_lt)\n    apply f_monic\n  /- simp only [LT.lt] at this -/\n  /- let ⟨val, VAL, VAL_EQN⟩ := this -/\n  rcases H : degree (R.representative' q n a %ₘ f q n) <;> simp\n  case some val' =>\n    rw [H] at this\n    rw [WithBot.lt_def] at this\n    simp at this\n    obtain ⟨val'', twon, hval', heq1, heq2⟩ := this\n    rw [Option.some_inj.mp heq1]\n    have := Option.some_inj.mp heq2\n    subst this\n    simp_all", "nesting_depth": 4, "transitive_dep_count": 31, "subset_aristotle": false, "category": "Compiler"}
{"id": 314, "thm_name": "Circuit.varsFinset_assignVars", "thm_stmt": "lemma varsFinset_assignVars [DecidableEq α] [DecidableEq β] :\n    ∀ (c : Circuit α) (f : ∀ (a : α) (_ha : a ∈ c.vars), β ⊕ Bool),\n      (c.assignVars f).varsFinset ⊆ c.varsFinset.biUnion\n        (fun a => if ha : a ∈ c.vars\n                  then\n                    match f a ha with\n                    | Sum.inl b => {b}\n                    | Sum.inr _ => ∅\n                  else ∅)\n  | tru, _ => by simp [assignVars, varsFinset, vars]\n  | fals, _ => by simp [vars, assignVars, varsFinset]\n  | var c v, f => by\n    intro x\n    simp [assignVars, varsFinset, vars]\n    split <;>\n    simp [*, vars]\n    split_ifs <;> simp [vars]\n  | and c₁ c₂, f => by\n    intro x\n    simp only [assignVars, Finset.mem_biUnion]\n    intro hx\n    replace hx := varsFinset_and _ _ hx\n    simp only [Finset.mem_union] at hx\n    cases hx with\n    | inl hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n    | inr hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n  | or c₁ c₂, f => by\n    intro x\n    simp only [assignVars, Finset.mem_biUnion]\n    intro hx\n    replace hx := varsFinset_or _ _ hx\n    simp only [Finset.mem_union] at hx\n    cases hx with\n    | inl hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n    | inr hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n  | xor c₁ c₂, f => by\n    intro x\n    simp only [assignVars, Finset.mem_biUnion]\n    intro hx\n    replace hx := varsFinset_xor _ _ hx\n    simp only [Finset.mem_union] at hx\n    cases hx with\n    | inl hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n    | inr hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/Fast/Circuit.lean", "imports": ["import Mathlib.Data.Fin.Basic", "import Mathlib.Data.Fintype.Basic", "import Mathlib.Data.Finset.Union", "import Mathlib.Data.Finset.Defs", "import Mathlib.Data.Finset.Card", "import Mathlib.Data.List.Pi", "import Mathlib.Data.Finset.Basic"], "used_lib_defs": [{"name": "BitVec.xor", "module": "Init.Data.BitVec.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "HOr", "module": "Init.Prelude"}, {"name": "HOr.hOr", "module": "Init.Prelude"}, {"name": "OrOp", "module": "Init.Prelude"}, {"name": "AndOp", "module": "Init.Prelude"}, {"name": "HAnd", "module": "Init.Prelude"}, {"name": "HAnd.hAnd", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}], "lib_lemmas": [{"name": "Finset.subset_iff", "module": "Mathlib.Data.Finset.Defs"}, {"name": "List.mem_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_dedup", "module": "Mathlib.Data.List.Dedup"}, {"name": "Finset.mem_union", "module": "Mathlib.Data.Finset.Lattice.Basic"}, {"name": "List.subset_def", "module": "Init.Data.List.Sublist"}, {"name": "Finset.mem_biUnion", "module": "Mathlib.Data.Finset.Union"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Circuit.vars", "content": "def vars [DecidableEq α] : Circuit α → List α\n  | tru => []\n  | fals => []\n  | var _ x => [x]\n  | and c₁ c₂ => (vars c₁ ++ vars c₂).dedup\n  | or c₁ c₂ => (vars c₁ ++ vars c₂).dedup\n  | xor c₁ c₂ => (vars c₁ ++ vars c₂).dedup"}, {"name": "Circuit.varsFinset", "content": "def varsFinset [DecidableEq α] (c : Circuit α) : Finset α :=\n  ⟨c.vars, nodup_vars c⟩"}, {"name": "Circuit.simplifyAnd", "content": "def simplifyAnd : Circuit α → Circuit α → Circuit α\n  | tru, c => c\n  | c, tru => c\n  | fals, _ => fals\n  | _, fals => fals\n  | c₁, c₂ => and c₁ c₂"}, {"name": "Circuit.simplifyOr", "content": "def simplifyOr : Circuit α → Circuit α → Circuit α\n  | tru, _ => tru\n  | _, tru => tru\n  | fals, c => c\n  | c, fals => c\n  | c₁, c₂ => or c₁ c₂"}, {"name": "Circuit.simplifyNot", "content": "def simplifyNot : Circuit α → Circuit α\n  | tru => fals\n  | fals => tru\n  | xor a b => xor (simplifyNot a) b\n  | and a b => or (simplifyNot a) (simplifyNot b)\n  | or a b => and (simplifyNot a) (simplifyNot b)\n  | var b a => var (!b) a"}, {"name": "Circuit.simplifyXor", "content": "@[simp]\ndef simplifyXor : Circuit α → Circuit α → Circuit α\n  | fals, c => c\n  | c, fals => c\n  | tru, c => ~~~ c\n  | c, tru => ~~~ c\n  | c₁, c₂ => xor c₁ c₂"}, {"name": "Circuit.assignVars", "content": "def assignVars [DecidableEq α] :\n    ∀ (c : Circuit α) (_f : ∀ (a : α) (_ha : a ∈ c.vars), β ⊕ Bool), Circuit β\n  | tru, _ => tru\n  | fals, _ => fals\n  | var b x, f =>\n    Sum.elim\n      (var b)\n      (λ c : Bool => if Bool.xor b c then fals else tru)\n      (f x (by admit /- proof elided -/\n      ))\n  | and c₁ c₂, f => (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/\n  ))) &&&\n                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/\n                    )))\n  | or c₁ c₂, f =>  (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/\n  ))) |||\n                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/\n                    )))\n  | xor c₁ c₂, f => (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/\n  ))) ^^^\n                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/\n                    )))"}], "used_local_lemmas": [{"name": "Circuit.mem_varsFinset", "content": "lemma mem_varsFinset [DecidableEq α] {c : Circuit α} :\n    ∀ {x : α}, x ∈ c.varsFinset ↔ x ∈ c.vars"}, {"name": "Circuit.varsFinset_and", "content": "theorem varsFinset_and [DecidableEq α] (c₁ c₂ : Circuit α) :\n    (varsFinset (c₁ &&& c₂)) ⊆ (varsFinset c₁ ∪ varsFinset c₂)"}, {"name": "Circuit.varsFinset_or", "content": "theorem varsFinset_or [DecidableEq α] (c₁ c₂ : Circuit α) :\n    (varsFinset (c₁ ||| c₂)) ⊆ (varsFinset c₁ ∪ varsFinset c₂)"}, {"name": "Circuit.simplifyNot_eq_complement", "content": "@[simp]\ntheorem simplifyNot_eq_complement (c : Circuit α) :\n    simplifyNot c = ~~~ c"}, {"name": "Circuit.vars_simplifyXor", "content": "theorem vars_simplifyXor [DecidableEq α] (c₁ c₂ : Circuit α) :\n    (vars (simplifyXor c₁ c₂)) ⊆ (vars c₁ ++ vars c₂).dedup"}, {"name": "Circuit.varsFinset_simplifyXor", "content": "theorem varsFinset_simplifyXor [DecidableEq α] (c₁ c₂ : Circuit α) :\n    (varsFinset (simplifyXor c₁ c₂)) ⊆ (varsFinset c₁ ∪ varsFinset c₂)"}, {"name": "Circuit.varsFinset_xor", "content": "theorem varsFinset_xor [DecidableEq α] (c₁ c₂ : Circuit α) :\n    (varsFinset (c₁ ^^^ c₂)) ⊆ (varsFinset c₁ ∪ varsFinset c₂)"}], "local_ctx": "import Mathlib.Data.Finset.Card\n\nimport Mathlib.Data.List.Pi\n\nimport Mathlib.Data.Finset.Union\n\nimport Mathlib.Data.Fin.Basic\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Finset.Defs\n\nimport Mathlib.Data.Fintype.Basic\n\nopen Std Sat AIG\n\ninductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq\n\nopen Lean in\n\nnamespace Circuit\n\nvariable {α : Type u} {β : Type v}\n\ndef vars [DecidableEq α] : Circuit α → List α\n  | tru => []\n  | fals => []\n  | var _ x => [x]\n  | and c₁ c₂ => (vars c₁ ++ vars c₂).dedup\n  | or c₁ c₂ => (vars c₁ ++ vars c₂).dedup\n  | xor c₁ c₂ => (vars c₁ ++ vars c₂).dedup\n\ndef varsFinset [DecidableEq α] (c : Circuit α) : Finset α :=\n  ⟨c.vars, nodup_vars c⟩\n\ndef simplifyAnd : Circuit α → Circuit α → Circuit α\n  | tru, c => c\n  | c, tru => c\n  | fals, _ => fals\n  | _, fals => fals\n  | c₁, c₂ => and c₁ c₂\n\ndef simplifyOr : Circuit α → Circuit α → Circuit α\n  | tru, _ => tru\n  | _, tru => tru\n  | fals, c => c\n  | c, fals => c\n  | c₁, c₂ => or c₁ c₂\n\ndef simplifyNot : Circuit α → Circuit α\n  | tru => fals\n  | fals => tru\n  | xor a b => xor (simplifyNot a) b\n  | and a b => or (simplifyNot a) (simplifyNot b)\n  | or a b => and (simplifyNot a) (simplifyNot b)\n  | var b a => var (!b) a\n\n@[simp]\ndef simplifyXor : Circuit α → Circuit α → Circuit α\n  | fals, c => c\n  | c, fals => c\n  | tru, c => ~~~ c\n  | c, tru => ~~~ c\n  | c₁, c₂ => xor c₁ c₂\n\ndef assignVars [DecidableEq α] :\n    ∀ (c : Circuit α) (_f : ∀ (a : α) (_ha : a ∈ c.vars), β ⊕ Bool), Circuit β\n  | tru, _ => tru\n  | fals, _ => fals\n  | var b x, f =>\n    Sum.elim\n      (var b)\n      (λ c : Bool => if Bool.xor b c then fals else tru)\n      (f x (by admit /- proof elided -/\n      ))\n  | and c₁ c₂, f => (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/\n  ))) &&&\n                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/\n                    )))\n  | or c₁ c₂, f =>  (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/\n  ))) |||\n                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/\n                    )))\n  | xor c₁ c₂, f => (assignVars c₁ (λ x hx => f x (by admit /- proof elided -/\n  ))) ^^^\n                    (assignVars c₂ (λ x hx => f x (by admit /- proof elided -/\n                    )))", "target_theorem": "lemma varsFinset_assignVars [DecidableEq α] [DecidableEq β] :\n    ∀ (c : Circuit α) (f : ∀ (a : α) (_ha : a ∈ c.vars), β ⊕ Bool),\n      (c.assignVars f).varsFinset ⊆ c.varsFinset.biUnion\n        (fun a => if ha : a ∈ c.vars\n                  then\n                    match f a ha with\n                    | Sum.inl b => {b}\n                    | Sum.inr _ => ∅\n                  else ∅) :=", "ground_truth_proof": ":= varsFinset_and _ _ hx\n    simp only [Finset.mem_union] at hx\n    cases hx with\n    | inl hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n    | inr hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n  | or c₁ c₂, f => by\n    intro x\n    simp only [assignVars, Finset.mem_biUnion]\n    intro hx\n    replace hx := varsFinset_or _ _ hx\n    simp only [Finset.mem_union] at hx\n    cases hx with\n    | inl hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n    | inr hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n  | xor c₁ c₂, f => by\n    intro x\n    simp only [assignVars, Finset.mem_biUnion]\n    intro hx\n    replace hx := varsFinset_xor _ _ hx\n    simp only [Finset.mem_union] at hx\n    cases hx with\n    | inl hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2\n    | inr hx =>\n      have := varsFinset_assignVars _ _ hx\n      simp only [Finset.mem_biUnion] at this\n      rcases this with ⟨a, ha⟩\n      use a\n      simp only [mem_varsFinset] at ha\n      simpa [ha.1, mem_varsFinset, vars] using ha.2", "nesting_depth": 4, "transitive_dep_count": 40, "subset_aristotle": false, "category": "Compiler"}
{"id": 315, "thm_name": "NFA'.autUnsignedCmp_correct", "thm_stmt": "lemma NFA'.autUnsignedCmp_correct cmp : autUnsignedCmp cmp |>.correct2 autUnsignedCmpSA cmp.urel", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.AutoStructs.Constructions", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "BitVec.ofFin", "module": "Init.Prelude"}, {"name": "cmp", "module": "Mathlib.Data.Ordering.Basic"}, {"name": "NFA.stepSet", "module": "Mathlib.Computability.NFA"}, {"name": "BitVec.ule", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ult", "module": "Init.Data.BitVec.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "bv2", "content": "def bv2 : BitVec 4 := BitVec.ofNat 4 1 "}, {"name": "bv1", "content": "def bv1 : BitVec 4 := BitVec.ofNat 4 5 "}, {"name": "RelationOrdering", "content": "inductive RelationOrdering\n| lt | le | gt | ge\nderiving Repr, Fintype"}, {"name": "instFinEnumBV", "content": "instance instFinEnumBV : FinEnum (BitVec w) where\n  card := 2^w\n  equiv := {\n    toFun := fun x => x.toFin\n    invFun := fun x => BitVec.ofFin x\n    left_inv := by admit /- proof elided -/"}], "lib_lemmas": [{"name": "BitVec.toNat_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "le_iff_lt_or_eq", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Nat.le_antisymm", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "NFA.correct", "content": "structure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q"}, {"name": "BVRel", "content": "abbrev BVRel := ∀ ⦃w⦄, BitVec w → BitVec w → Prop"}, {"name": "NFA'.sa2", "content": "def NFA'.sa2 (M : NFA' 2) := M.σ → BVRel"}, {"name": "NFA'.correct2", "content": "structure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)"}, {"name": "NFA.unsignedCmpState", "content": "inductive NFA.unsignedCmpState : Type where\n| eq | gt | lt\nderiving Fintype, DecidableEq"}, {"name": "NFA.unsignedCmpStep", "content": "def NFA.unsignedCmpStep (q : NFA.unsignedCmpState) (a : BitVec 2) : List NFA.unsignedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [ .eq ] | .eq, 1 => [ .gt ] | .eq, 2 => [ .lt ]\n  | .gt, 0 => [ .gt ] | .gt, 1 => [ .gt ] | .gt, 3 => [ .gt ] | .gt, 2 => [ .lt ]\n  | .lt, 0 => [ .lt ] | .lt, 1 => [ .gt ] | .lt, 2 => [ .lt ] | .lt, 3 => [ .lt ]"}, {"name": "NFA.autUnsignedCmp", "content": "def NFA.autUnsignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) unsignedCmpState where\n  step s a := { s' | s' ∈ unsignedCmpStep s a }\n  start := {s | s = .eq }\n  accept := { s | s ∈ match cmp with | .lt => [unsignedCmpState.lt] | .le => [.lt, .eq] | .gt => [.gt] | .ge => [.gt, .eq] }"}, {"name": "NFA'.autUnsignedCmp", "content": "def NFA'.autUnsignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autUnsignedCmp cmp⟩"}, {"name": "RelationOrdering.urel", "content": "def RelationOrdering.urel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .le => fun _ bv1 bv2 => bv1.ule bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ge => fun _ bv1 bv2 => bv2.ule bv1"}, {"name": "NFA'.autUnsignedCmpSA", "content": "def NFA'.autUnsignedCmpSA (q : NFA.unsignedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1"}], "used_local_lemmas": [{"name": "BitVec.ule_iff_ult_or_eq", "content": "lemma BitVec.ule_iff_ult_or_eq {w : ℕ} (bv1 bv2 : BitVec w):\n    (bv1.ule bv2) = true ↔ (bv1.ult bv2) = true ∨ bv1 = bv2"}, {"name": "ucmp_tricho", "content": "@[simp]\nlemma ucmp_tricho {bv1 bv2 : BitVec w} : (bv2.ult bv1) = false → (bv1.ult bv2) = false → bv1 = bv2"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\nstructure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q\n\nabbrev BVRel := ∀ ⦃w⦄, BitVec w → BitVec w → Prop\n\ndef NFA'.sa2 (M : NFA' 2) := M.σ → BVRel\n\nstructure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)\n\nsection fsm\n\nvariable {arity : Type} [FinEnum arity]\n\nopen BitStream in\n\nend fsm\n\nsection nfas_relations\n\ninductive NFA.unsignedCmpState : Type where\n| eq | gt | lt\nderiving Fintype, DecidableEq\n\ndef NFA.unsignedCmpStep (q : NFA.unsignedCmpState) (a : BitVec 2) : List NFA.unsignedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [ .eq ] | .eq, 1 => [ .gt ] | .eq, 2 => [ .lt ]\n  | .gt, 0 => [ .gt ] | .gt, 1 => [ .gt ] | .gt, 3 => [ .gt ] | .gt, 2 => [ .lt ]\n  | .lt, 0 => [ .lt ] | .lt, 1 => [ .gt ] | .lt, 2 => [ .lt ] | .lt, 3 => [ .lt ]\n\ndef NFA.autUnsignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) unsignedCmpState where\n  step s a := { s' | s' ∈ unsignedCmpStep s a }\n  start := {s | s = .eq }\n  accept := { s | s ∈ match cmp with | .lt => [unsignedCmpState.lt] | .le => [.lt, .eq] | .gt => [.gt] | .ge => [.gt, .eq] }\n\ndef NFA'.autUnsignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autUnsignedCmp cmp⟩\n\ndef RelationOrdering.urel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .le => fun _ bv1 bv2 => bv1.ule bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ge => fun _ bv1 bv2 => bv2.ule bv1\n\ndef NFA'.autUnsignedCmpSA (q : NFA.unsignedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1", "target_theorem": "lemma NFA'.autUnsignedCmp_correct cmp : autUnsignedCmp cmp |>.correct2 autUnsignedCmpSA cmp.urel :=", "ground_truth_proof": ":= by\n  let getState {w} (bv1 bv2 : BitVec w) : NFA.unsignedCmpState :=\n    if bv2.ult bv1 then .gt else if bv1.ult bv2 then .lt else .eq\n  constructor <;> simp [NFA.autUnsignedCmp, autUnsignedCmp, autUnsignedCmpSA, RelationOrdering.urel]\n  · rintro _ _ _; cases cmp <;> simp [BitVec.ule_iff_ult_or_eq]; tauto\n  · rintro (_ | _ | _) <;> simp\n  · rintro (_ | _ | _) a w bv1 bv2 <;> simp [NFA.stepSet, NFA.unsignedCmpStep]\n    · constructor\n      · rintro ⟨i, hi⟩; cases i <;> fin_cases a <;> simp_all [instFinEnumBV]\n      · rintro ⟨_, _⟩; use .eq; simp; fin_cases a <;> simp [instFinEnumBV] at * <;> tauto\n    · constructor\n      · rintro ⟨i, hi⟩; cases i <;> fin_cases a <;> simp_all [instFinEnumBV]\n      · rintro _; fin_cases a <;> simp [instFinEnumBV] at *\n        · use .gt; simp_all\n        · use (getState bv1 bv2); simp [getState]; split_ifs <;> simp_all; apply ucmp_tricho <;> assumption\n        · use .gt; simp_all\n    · constructor\n      · rintro ⟨i, hi⟩; cases i <;> fin_cases a <;> simp_all [instFinEnumBV]\n      · rintro _; fin_cases a <;> simp [instFinEnumBV] at *\n        · use .lt; simp_all\n        · use (getState bv1 bv2); simp [getState]; split_ifs <;> simp_all; apply ucmp_tricho <;> assumption\n        · use .lt; simp_all", "nesting_depth": 3, "transitive_dep_count": 37, "subset_aristotle": false, "category": "Compiler"}
{"id": 316, "thm_name": "AngelicChoice.ExtractNonDet.extract_refines_wp_weak", "thm_stmt": "omit [MAlgDet m l] in\nlemma ExtractNonDet.extract_refines_wp_weak (s : NonDetT m α) (inst : ExtractNonDet WeakFindable s) :\n  wp s.extractWeak post <= wp s post", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/NonDetT'/Extract.lean", "imports": ["import Loom.MonadAlgebras.WP.Gen", "import Loom.MonadAlgebras.WP.Liberal", "import Mathlib.Order.CompleteBooleanAlgebra", "import Mathlib.Logic.Function.Basic", "import Mathlib.Data.W.Basic", "import Loom.MonadAlgebras.NonDetT'.Basic", "import Loom.MonadAlgebras.WP.Basic", "import Mathlib.Order.Lattice", "import Mathlib.Data.FinEnum", "import Mathlib.Order.Basic"], "used_lib_defs": [{"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "inline", "module": "Init.Core"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Encodable", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Encodable.decode", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.MonadEnv", "module": "Lean.Environment"}, {"name": "Lean.SimpleScopedEnvExtension", "module": "Lean.ScopedEnvExtension"}, {"name": "Lean.SimplePersistentEnvExtension", "module": "Lean.EnvExtension"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "Lean.Order.bot", "module": "Init.Internal.Order.Basic"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "MonadNonDet", "content": "class MonadNonDet (m : Type u → Type v) where\n  pick : (τ : Type u) → [Inhabited τ] → m τ\n  \n  pickSuchThat : (τ : Type u) → (p : τ → Prop) → [Findable p] → m τ\n  assume : (as : Prop) → [Decidable as] → m PUnit.{u+1}\n  \n  rep {α : Type u} : α → (α → m (ForInStep α)) → m α"}, {"name": "NonDetT", "content": "inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α"}, {"name": "CCPOBot", "content": "class CCPOBot (m : Type u -> Type v)  where\n  compBot {α} : m α"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "MAlgDet", "content": "class MAlgDet (l : outParam (Type v)) [Monad m] [CompleteLattice l] [MAlgOrdered m l] where\n   \n  demonic {α ι : Type v} (c : m α) (p : ι -> α -> l) [Nonempty ι] :\n    ⨅ i, MAlg.lift c (p i) ≤ MAlg.lift c (fun x => ⨅ i, p i x)\n   \n  angelic {α ι : Type v} (c : m α) (p : ι -> α -> l) [Nonempty ι] :\n    ⨆ i, MAlg.lift c (p i) ≥ MAlg.lift c (fun x => ⨆ i, p i x)"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "CCPOBotLawful", "content": "class CCPOBotLawful (m : Type u -> Type v) [∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] where\n  prop {α} : CCPOBot.compBot (m := m) (α := α) = Lean.Order.bot"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} {α : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m α -> Cont l α\n  | .pure ret => pure ret\n  | .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | .pickCont τ p f => fun post => let p : Set τ := p; ⨅ a ∈ (p : Set τ), wp (f a) post"}, {"name": "NonDetT.μ", "content": "def NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}, {"name": "NonDetT.bind", "content": "def NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f"}, {"name": "WPGen.bind", "content": "def WPGen.bind {x : m α} {f : α -> m β} (wpg : WPGen x) (wpgf : ∀ a, WPGen (f a)) :\n  WPGen (x >>= f) where\n  get := fun post => wpg.get (fun a => (wpgf a).get post)\n  prop := by admit /- proof elided -/"}, {"name": "_root_.Lean.SimpleScopedEnvExtension.get", "content": "private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "Context", "content": "structure Context where\n  ref         : Syntax\n   \n  m           : Syntax\n   \n  returnType  : Syntax\n  mutableVars : VarSet := {}\n  insideFor   : Bool := false"}, {"name": "_root_.Lean.SimplePersistentEnvExtension.get", "content": "private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "WPGen", "content": "structure WPGen (x : m α) where\n  get : Cont l α\n  \n  prop : ∀ post, get post <= wp x post"}, {"name": "_root_.Lean.EnvExtension.get", "content": "private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "MAlgTotal", "content": "class MAlgTotal (m : Type u -> Type v) [Monad m] [∀ α, Lean.Order.CCPO (m α)]\n  [CompleteLattice l] [MAlgOrdered m l] where\n  bot_lift {α : Type u} (post : α -> l) :\n    MAlg.lift (Lean.Order.bot : m α) post <= ⊥"}], "lib_lemmas": [{"name": "ge_iff_le", "module": "Init.Core"}, {"name": "iSup_const", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "inf_comm", "module": "Mathlib.Order.Lattice"}, {"name": "le_iSup_of_le", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "monadLift_self", "module": "Init.Control.Lawful.Basic"}], "repo_lemmas": [{"name": "wp_pure", "content": "lemma wp_pure (x : α) (post : α -> l) : wp (m := m) (pure x) post = post x"}, {"name": "NonDetT.wp_vis", "content": "@[simp]\nlemma NonDetT.wp_vis {β : Type u} (x : m β) (f : β → NonDetT m α) post :\n  _root_.wp (NonDetT.vis x f) post = _root_.wp x fun a => _root_.wp (f a) post"}, {"name": "NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}, {"name": "NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α β : Type u} (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}, {"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}, {"name": "NonDetT.wp_pickCont", "content": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post"}, {"name": "NonDetT.wp_pickCont", "content": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⨆ a, ⌜p a⌝ ⊓ _root_.wp (f a) post"}, {"name": "wp_bot", "content": "@[simp]\nlemma wp_bot :\n  wp (bot : m α) = fun _ => (⊥ : l)"}, {"name": "wp_bind", "content": "lemma wp_bind {β} (x : m α) (f : α -> m β) (post : β -> l) :\n    wp (x >>= f) post = wp x (fun x => wp (f x) post)"}], "used_local_defs": [{"name": "findNat", "content": "def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0"}, {"name": "find", "content": "def find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode"}, {"name": "WeakFindable", "content": "class WeakFindable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_some_p : find () = some x -> p x"}, {"name": "WeakFindable", "content": "instance WeakFindable.of_Findable {α : Type u} (p : α -> Prop) [Findable p] : WeakFindable p where\n  find := Findable.find p\n  find_some_p := Findable.find_some_p"}, {"name": "ExtractNonDet", "content": "inductive ExtractNonDet (findable : {τ : Type u} -> (τ -> Prop) -> Type u) {m} : {α : Type u} -> NonDetT m α -> Type _ where\n  | pure {α} : ∀ (x : α), ExtractNonDet findable (NonDetT.pure x)\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) :\n      (∀ y, ExtractNonDet findable (f y)) → ExtractNonDet findable (.vis x f)\n  | pickSuchThat {α} (τ : Type u) (p : τ -> Prop) (f : τ → NonDetT m α)\n    {_ : findable p}\n     : (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont τ p f)\n  | assume {α} (p : PUnit -> Prop) (f : PUnit → NonDetT m α) {_ : Decidable (p .unit)} :\n    (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont PUnit p f)"}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pure' : ExtractNonDet findable (Pure.pure (f := NonDetT m) x) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.liftM (x : m α) :\n  ExtractNonDet findable (liftM (n := NonDetT m) x) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.assume' {p : Prop} [Decidable p] : ExtractNonDet findable (MonadNonDet.assume (m :=  NonDetT m) p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pickSuchThat' {τ : Type u} (p : τ -> Prop) [Findable p] :\n  ExtractNonDet Findable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pickSuchThat_weak {τ : Type u} (p : τ -> Prop) [WeakFindable p] :\n  ExtractNonDet WeakFindable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.if {p : Prop} {dec : Decidable p} {x y : NonDetT m α}\n  (_ : ExtractNonDet findable x) (_ : ExtractNonDet findable y) :\n  ExtractNonDet findable (if p then x else y) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.ForIn_list {xs : List α} {init : β} {f : α → β → NonDetT m (ForInStep β)}\n  (_ : ∀ a b, ExtractNonDet findable (f a b)) :\n  ExtractNonDet findable (forIn xs init f) :="}, {"name": "NonDetT.extractGen", "content": "@[simp, inline]\ndef NonDetT.extractGen {findable : {τ : Type u} -> (τ -> Prop) -> Type u} {α : Type u}\n  (findOf : ∀ {τ : Type u} (p : τ -> Prop), findable p -> Unit -> Option τ)\n  : (s : NonDetT m α) -> (ex : ExtractNonDet findable s := by admit /- proof elided -/\n  ) -> m α\n  | .pure x, _ => Pure.pure x\n  | .vis x f, .vis _ _ _ => liftM x >>= (fun x => extractGen findOf (f x))\n  | .pickCont _ p f, .pickSuchThat _ _ _ _ =>\n    match findOf p ‹_› () with\n    | none => CCPOBot.compBot\n    | some x =>  extractGen findOf (f x)\n  | .pickCont _ p f, .assume _ _ _ =>\n    if p .unit then\n      extractGen findOf (f .unit)\n    else CCPOBot.compBot"}, {"name": "NonDetT.extractWeak", "content": "def NonDetT.extractWeak {α : Type u} (s : NonDetT m α) (ex : ExtractNonDet WeakFindable s := by admit /- proof elided -/\n) : m α :=\n  NonDetT.extractGen WeakFindable.find s"}], "used_local_lemmas": [], "local_ctx": "import Mathlib.Logic.Function.Basic\n\nimport Mathlib.Order.CompleteBooleanAlgebra\n\nimport Mathlib.Order.Lattice\n\nimport Mathlib.Order.Basic\n\nimport Mathlib.Data.W.Basic\n\nimport Mathlib.Data.FinEnum\n\nimport Loom.MonadAlgebras.WP.Gen\n\nimport Loom.MonadAlgebras.WP.Liberal\n\nimport Loom.MonadAlgebras.NonDetT'.Basic\n\nopen Lean.Order\n\ndef findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0\n\ndef find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode\n\nclass WeakFindable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_some_p : find () = some x -> p x\n\ninstance WeakFindable.of_Findable {α : Type u} (p : α -> Prop) [Findable p] : WeakFindable p where\n  find := Findable.find p\n  find_some_p := Findable.find_some_p\n\ninductive ExtractNonDet (findable : {τ : Type u} -> (τ -> Prop) -> Type u) {m} : {α : Type u} -> NonDetT m α -> Type _ where\n  | pure {α} : ∀ (x : α), ExtractNonDet findable (NonDetT.pure x)\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) :\n      (∀ y, ExtractNonDet findable (f y)) → ExtractNonDet findable (.vis x f)\n  | pickSuchThat {α} (τ : Type u) (p : τ -> Prop) (f : τ → NonDetT m α)\n    {_ : findable p}\n     : (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont τ p f)\n  | assume {α} (p : PUnit -> Prop) (f : PUnit → NonDetT m α) {_ : Decidable (p .unit)} :\n    (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont PUnit p f)\n\ninstance ExtractNonDet.pure' : ExtractNonDet findable (Pure.pure (f := NonDetT m) x) :=\n\ninstance ExtractNonDet.liftM (x : m α) :\n  ExtractNonDet findable (liftM (n := NonDetT m) x) :=\n\ninstance ExtractNonDet.assume' {p : Prop} [Decidable p] : ExtractNonDet findable (MonadNonDet.assume (m :=  NonDetT m) p) :=\n\ninstance ExtractNonDet.pickSuchThat' {τ : Type u} (p : τ -> Prop) [Findable p] :\n  ExtractNonDet Findable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=\n\ninstance ExtractNonDet.pickSuchThat_weak {τ : Type u} (p : τ -> Prop) [WeakFindable p] :\n  ExtractNonDet WeakFindable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=\n\ninstance ExtractNonDet.if {p : Prop} {dec : Decidable p} {x y : NonDetT m α}\n  (_ : ExtractNonDet findable x) (_ : ExtractNonDet findable y) :\n  ExtractNonDet findable (if p then x else y) :=\n\ninstance ExtractNonDet.ForIn_list {xs : List α} {init : β} {f : α → β → NonDetT m (ForInStep β)}\n  (_ : ∀ a b, ExtractNonDet findable (f a b)) :\n  ExtractNonDet findable (forIn xs init f) :=\n\nvariable [Monad m] [CCPOBot m] [CompleteBooleanAlgebra l] [MAlgOrdered m l] [MAlgDet m l] [LawfulMonad m]\n\n@[simp, inline]\ndef NonDetT.extractGen {findable : {τ : Type u} -> (τ -> Prop) -> Type u} {α : Type u}\n  (findOf : ∀ {τ : Type u} (p : τ -> Prop), findable p -> Unit -> Option τ)\n  : (s : NonDetT m α) -> (ex : ExtractNonDet findable s := by admit /- proof elided -/\n  ) -> m α\n  | .pure x, _ => Pure.pure x\n  | .vis x f, .vis _ _ _ => liftM x >>= (fun x => extractGen findOf (f x))\n  | .pickCont _ p f, .pickSuchThat _ _ _ _ =>\n    match findOf p ‹_› () with\n    | none => CCPOBot.compBot\n    | some x =>  extractGen findOf (f x)\n  | .pickCont _ p f, .assume _ _ _ =>\n    if p .unit then\n      extractGen findOf (f .unit)\n    else CCPOBot.compBot\n\ndef NonDetT.extractWeak {α : Type u} (s : NonDetT m α) (ex : ExtractNonDet WeakFindable s := by admit /- proof elided -/\n) : m α :=\n  NonDetT.extractGen WeakFindable.find s\n\nnamespace DemonicChoice\n\nend DemonicChoice\n\nnamespace AngelicChoice\n\nvariable [∀ α, CCPO (m α)] [CCPOBotLawful m] [MAlgTotal m]", "target_theorem": "omit [MAlgDet m l] in\nlemma ExtractNonDet.extract_refines_wp_weak (s : NonDetT m α) (inst : ExtractNonDet WeakFindable s) :\n  wp s.extractWeak post <= wp s post :=", "ground_truth_proof": ":= by\n  unhygienic induction inst\n  { simp [wp_pure, NonDetT.extractWeak] }\n  { simp only [NonDetT.extractWeak, NonDetT.extractGen, monadLift_self, wp_bind, NonDetT.wp_vis];\n    apply wp_cons; aesop (add norm inf_comm) }\n  { simp only [NonDetT.extractWeak, NonDetT.extractGen, NonDetT.wp_pickCont]; split\n    { simp [*, CCPOBotLawful.prop, TotalCorrectness.wp_bot] }\n    apply le_iSup_of_le; simp; constructor; rotate_left\n    apply a_ih; rename_i h; simp [x.find_some_p h] }\n  simp only [NonDetT.extractWeak, NonDetT.extractGen, NonDetT.wp_pickCont]\n  have : ∀ a : PUnit.{u_1 + 1}, a = .unit := by simp\n  simp only [this, ge_iff_le]; split_ifs <;> simp [*, iSup_const, CCPOBotLawful.prop, TotalCorrectness.wp_bot]\n  apply a_ih", "nesting_depth": 5, "transitive_dep_count": 67, "subset_aristotle": false, "category": "Framework"}
{"id": 317, "thm_name": "denote_matchVar", "thm_stmt": "theorem denote_matchVar\n    {v w : Var _ t}\n    (mapOut : MatchVarResult lets v matchLets w mapIn)\n    (V : lets.ValidDenotation) :\n    (matchLets.denote (mapOut.val.mapValuation V.val) w)\n    = V.val v", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Transforms/Rewrite/Match.lean", "imports": ["import LeanMLIR.Framework", "import LeanMLIR.LeanMLIR.Framework.Basic", "import LeanMLIR.Transforms.Rewrite.Mapping", "import LeanMLIR.LeanMLIR.ErasedContext"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "AList.insert", "module": "Mathlib.Data.List.AList"}, {"name": "Valuation.map", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "Exists", "module": "Init.Core"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "Sigma.mk", "module": "Init.Core"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}, {"name": "id", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "HEq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"neg\" : MLIR.Pretty.uniform_op", "content": "syntax \"neg\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Mapping", "content": "abbrev Mapping (Γ Δ : Ctxt Ty) : Type :=\n  @AList (Σ t, Var Γ t) (fun x => Var Δ x.1)"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "last", "content": "@[match_pattern]\ndef last (Γ : Ctxt Ty) (t : Ty) : Ctxt.Var (Ctxt.cons t Γ) t :=\n  ⟨0, by admit /- proof elided -/\n  ⟩"}, {"name": "appendInl", "content": "def appendInl (v : Γ.Var t) : (Γ ++ Δ).Var t :=\n  ⟨v.val, by admit /- proof elided -/\n  ⟩"}, {"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "map", "content": "def map (f : ∀ (a : α), A a → B a) :\n    ∀ {l : List α}, HVector A l → HVector B l\n  | [],   .nil        => .nil\n  | t::_, .cons a as  => .cons (f t a) (map f as)"}, {"name": "map", "content": "def map (f : Ty₁ → Ty₂) : Ctxt Ty₁ → Ctxt Ty₂ :=\n  ofList ∘ (List.map f) ∘ toList"}, {"name": "cons", "content": "@[match_pattern]\ndef cons (hd : Ty) : Ctxt Ty → Ctxt Ty\n| ⟨tl⟩ => ⟨hd :: tl⟩"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "effectKind", "content": "def effectKind   := Signature.effectKind ∘ s.signature"}, {"name": "returnTypes", "content": "def returnTypes  := Signature.returnTypes ∘ s.signature"}, {"name": "Expr.ty", "content": "def Expr.ty : Expr d Γ eff [t] → d.Ty := fun _ => t"}, {"name": "Expr.op", "content": "def Expr.op {Γ : Ctxt d.Ty} {eff : EffectKind} {ty} (e : Expr d Γ eff ty) : d.Op :=\n  Expr.casesOn e (fun op _ _ _ _ => op)"}, {"name": "DialectDenote", "content": "class DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))"}, {"name": "Lets.denote", "content": "def Lets.denote [DialectSignature d] [DialectDenote d] {Γ₂}\n    (lets : Lets d Γ₁ eff Γ₂) (V : Valuation Γ₁) : (eff.toMonad d.m <| Valuation Γ₂) :=\n  match lets with\n  | .nil          => return V\n  | .var lets' e  => lets'.denote V >>= e.denote"}, {"name": "regSig", "content": "def regSig       := Signature.regSig ∘ s.signature"}, {"name": "RegionSignature", "content": "abbrev RegionSignature Ty := List (Ctxt Ty × List Ty)"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "Dialect", "content": "structure Dialect where\n  (Op : Type)\n  (Ty : Type)\n  (m : Type → Type := Id)"}, {"name": "Op", "content": "inductive Op (q : Nat) (n : Nat)\n  | add : Op q n\n  | sub : Op q n\n  | mul : Op q n\n  | mul_constant : Op q n\n    \n    \n  | leading_term : Op q n\n  | monomial : Op q n\n  | monomial_mul : Op q n\n  | from_tensor : Op q n\n  | to_tensor : Op q n\n  | const (c : R q n) : Op q n\n  | const_int (c : Int) : Op q n\n  | const_idx (i : Nat) : Op q n"}, {"name": "Valuation.instAppendHVector", "content": "@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V"}, {"name": "HVector.denote", "content": "def HVector.denote :\n    {l : RegionSignature d.Ty} → (T : HVector (fun t => Com d t.1 .impure t.2) l) →\n    HVector (fun t => t.1.Valuation → EffectKind.impure.toMonad d.m (HVector toType t.2)) l\n  | _, .nil => HVector.nil\n  | _, .cons v vs => HVector.cons (v.denote) (HVector.denote vs)"}, {"name": "FlatCom.denote", "content": "@[simp] abbrev FlatCom.denote [DialectDenote d]\n    (flatCom : FlatCom d Γ eff Γ_out ts)\n    (V : Γ.Valuation) : eff.toMonad d.m (HVector toType ts) :=\n  flatCom.lets.denote V >>= (return flatCom.rets.map ·)"}, {"name": "FlatCom", "content": "structure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts"}, {"name": "RegionSignature.map", "content": "def RegionSignature.map (f : Ty → Ty') : RegionSignature Ty → RegionSignature Ty' :=\n  List.map fun ⟨Γ, ty⟩ => (Γ.map f, ty.map f)"}, {"name": "Signature.map", "content": "def Signature.map (f : Ty → Ty') : Signature Ty → Signature Ty' :=\n  fun sig => {\n    sig         := sig.sig.map f\n    regSig      := sig.regSig.map f\n    returnTypes := sig.returnTypes.map f\n  }"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "Expr.denote", "content": "def Expr.denote {ty} (e : Expr d Γ eff ty) (V : Valuation Γ) :\n    eff.toMonad d.m (e.outContext.Valuation) :=\n  match e with\n  | ⟨op, ty_eq, heff, args, regArgs⟩ => do\n      let argsDenote := args.map V\n      let val ← EffectKind.liftEffect heff <| DialectDenote.denote op argsDenote regArgs.denote\n      return (val ++ V).cast (by admit /- proof elided -/\n      )"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) "}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "liftEffect", "content": "def liftEffect [Pure m] {e1 e2 : EffectKind} {α : Type}\n    (hle : e1 ≤ e2) (v1 : e1.toMonad m α) : e2.toMonad m α :=\n  match e1, e2, hle with\n    | .pure, .pure, _ | .impure, .impure, _ => v1\n    | .pure, .impure, _ => Pure.pure v1"}, {"name": "toMonad", "content": "def toMonad (e : EffectKind) (m : Type → Type) : Type → Type :=\n  match e with\n  | pure => Id\n  | impure => m"}, {"name": "Com.denote", "content": "def Com.denote : Com d Γ eff ty → (Γv : Valuation Γ) →\n    eff.toMonad d.m (HVector toType ty)\n  | .rets vs, Γv     => pure (vs.map Γv)\n  | .var e body, V => e.denote V >>= body.denote"}, {"name": "Com.ty", "content": "def Com.ty : Com d Γ eff [t] → d.Ty := fun _ => t"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "Expr.regArgs", "content": "def Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)"}, {"name": "Regions", "content": "abbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig"}, {"name": "Expr.denoteOp", "content": "def Expr.denoteOp (e : Expr d Γ eff ty) (V : Γ.Valuation) :\n    eff.toMonad d.m (HVector toType ty) :=\n  EffectKind.liftEffect e.eff_le <| cast (by admit /- proof elided -/\n  ) <|\n    DialectDenote.denote e.op (e.args.map V) e.regArgs.denote"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "TyDenote.toType", "content": "notation \"⟦\" x \"⟧\" => TyDenote.toType x"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "IsEmpty.exists_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "Option.isSome_none", "module": "Init.Data.Option.Basic"}, {"name": "iff_false", "module": "Init.SimpLemmas"}, {"name": "Option.get_some", "module": "Init.Data.Option.Basic"}, {"name": "Option.isSome_iff_exists", "module": "Init.Data.Option.Lemmas"}, {"name": "forall_exists_index", "module": "Init.PropLemmas"}, {"name": "Exists.choose_spec", "module": "Init.Classical"}], "repo_lemmas": [{"name": "map_cons", "content": "@[simp] theorem map_cons : (Γ.cons a).map f = (Γ.map f).cons (f a)"}, {"name": "eq.ty_eq", "content": "theorem eq.ty_eq {v : Γ.Var t} {w : Γ.Var u} (h : v.eq w) : t = u"}, {"name": "Expr.op_mk", "content": "@[simp]\ntheorem Expr.op_mk {Γ : Ctxt d.Ty} {ty} {eff : EffectKind} (op : d.Op)\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) (DialectSignature.sig op))\n    (regArgs) :\n    (Expr.mk op ty_eq eff_le args regArgs).op = op"}, {"name": "Expr.regArgs_mk", "content": "@[simp]\ntheorem Expr.regArgs_mk {Γ : Ctxt d.Ty} {ty eff op}\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) (DialectSignature.sig op)) (regArgs) :\n    (Expr.mk op ty_eq eff_le args regArgs).regArgs = regArgs"}, {"name": "appendCases_appendInl", "content": "@[simp] theorem appendCases_appendInl (v : Γ.Var t) :\n    appendCases (motive := motive) left right v.appendInl = (left v)"}, {"name": "Expr.args_mk", "content": "@[simp]\ntheorem Expr.args_mk {Γ : Ctxt d.Ty} {ty eff op}\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) (DialectSignature.sig op)) (regArgs) :\n    (Expr.mk op ty_eq eff_le args regArgs).args = args"}, {"name": "Id.bind_eq'", "content": "theorem Id.bind_eq' (x : Id α) (f : α → id β) : x >>= f = f x"}, {"name": "Expr.denoteOp_eq_denoteOp_of", "content": "@[simp] theorem Expr.denoteOp_eq_denoteOp_of {e₁ : Expr d Γ eff ty} {e₂ : Expr d Δ eff ty}\n    {Γv : Valuation Γ} {Δv : Valuation Δ}\n    (op_eq : e₁.op = e₂.op)\n    (h_args : HVector.map Γv (op_eq ▸ e₁.args)\n              = HVector.map Δv e₂.args)\n    (h_regArgs : HEq e₁.regArgs.denote e₂.regArgs.denote) :\n    e₁.denoteOp Γv = e₂.denoteOp Δv"}, {"name": "Id.pure_eq'", "content": "theorem Id.pure_eq' (a : α) : (pure a : Id α) = a"}], "used_local_defs": [{"name": "MatchVarM", "content": "abbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)"}, {"name": "MatchVar", "content": "abbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit"}, {"name": "MatchVarM.unifyVars", "content": "def MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)"}, {"name": "matchArg", "content": "def matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)"}, {"name": "matchVar", "content": "def matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/"}, {"name": "MatchVarResult", "content": "def MatchVarResult := { mapOut : Mapping _ _ //\n  ∃ (mapIn' mapOut' : Mapping _ _),\n    mapIn.entries ⊆ mapIn'.entries\n    ∧ mapOut'.entries ⊆ mapOut.entries\n    ∧ matchVar lets v matchLets w mapIn' = some ((), mapOut') }"}, {"name": "MatchArgResult", "content": "def MatchArgResult := { mapOut : Mapping _ _ //\n  ∃ (mapIn' mapOut' : Mapping _ _),\n    mapIn.entries ⊆ mapIn'.entries\n    ∧ mapOut'.entries ⊆ mapOut.entries\n    ∧ matchArg lets matchLets vs ws mapIn' = some ((), mapOut') }"}, {"name": "MatchVarResult.eqvVarLeft", "content": "def eqvVarLeft  :\n    MatchVarResult lets v (.var matchLets matchExpr) w.appendInr ma\n    ≃ MatchVarResult lets v matchLets w ma where\n  toFun := fun ⟨x, h⟩ => ⟨x, by admit /- proof elided -/\n  ⟩\n  invFun := fun ⟨x, h⟩ => ⟨x, by admit /- proof elided -/\n  ⟩"}, {"name": "MatchVarResult.toArgResult", "content": "noncomputable def toArgResult\n    (mapOut : MatchVarResult lets v (.var matchLets matchExpr) w.appendInl mapIn) :\n    let args := mapOut.getPureExpr_eq_some.choose\n    MatchArgResult lets matchLets args matchExpr.args mapIn :=\n  ⟨mapOut.1, by admit /- proof elided -/\n  ⟩"}], "used_local_lemmas": [{"name": "MatchVar.liftM_bind_eq_some_iff", "content": "@[simp]\ntheorem MatchVar.liftM_bind_eq_some_iff (x? : Option α)\n    (f : α → MatchVarM Δ Γ β) :\n    ((liftM x? >>= f) mapIn = some mapOut)\n    ↔ ( ∃ h : x?.isSome,\n        f (x?.get h) mapIn = some mapOut )"}, {"name": "matchVar_appendInl", "content": "theorem matchVar_appendInl {w : Var ⟨te⟩ t} :\n    matchVar lets v (.var matchLets matchExpr) w.appendInl ma = some ma' →\n    ∃ args,\n      lets.getPureExpr v\n        = some ⟨_, w, matchExpr.op, matchExpr.ty_eq, matchExpr.eff_le, args, matchExpr.regArgs⟩\n      ∧ matchArg lets matchLets args matchExpr.args ma = some ma'"}, {"name": "MatchVarResult.getPureExpr_eq_some", "content": "theorem getPureExpr_eq_some\n    (mapOut : MatchVarResult lets v (.var matchLets matchExpr) w.appendInl mapIn) :\n    ∃ args, lets.getPureExpr v = some ⟨te, w, ⟨\n        matchExpr.op,\n        matchExpr.ty_eq,\n        matchExpr.eff_le,\n        args,\n        matchExpr.regArgs\n      ⟩⟩"}, {"name": "HVector.map_eq_map_of_matchArg", "content": "theorem HVector.map_eq_map_of_matchArg\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}\n    {ma : Mapping Δ_in Γ_out}\n    {l : List d.Ty} {args₁ : HVector _ l} {args₂ : HVector _ l}\n    (mapOut : MatchArgResult lets matchLets args₁ args₂ ma)\n    (f₁ f₂ : (t : d.Ty) → Var _ t → ⟦t⟧)\n    (hf : ∀ {t v₁ v₂},\n      (mapOut' : MatchVarResult lets v₁ matchLets v₂ ma)\n      → mapOut'.val = mapOut.val\n      → f₂ t v₂ = f₁ t v₁) :\n    HVector.map f₂ args₂ = HVector.map f₁ args₁"}], "local_ctx": "import LeanMLIR.Framework\n\nimport LeanMLIR.Transforms.Rewrite.Mapping\n\nopen Ctxt (Var VarSet Valuation Hom)\n\nvariable {d} [DialectSignature d] [DecidableEq d.Ty]\n\nvariable {Γ : Ctxt d.Ty} {ty : d.Ty}\n\nabbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)\n\nabbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit\n\ndef MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)\n\nopen MatchVarM\n\nvariable [DecidableEq d.Op]\n\ndef matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)\n\ndef matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/\n\nend\n\nsection MatchVar\n\nvariable [DecidableEq d.Op] {Γ_in Γ_out Δ_in Δ_out t te}\n          {lets : Lets d Γ_in eff Γ_out} {v : Var Γ_out t}\n          {matchLets : Lets d Δ_in .pure Δ_out}\n          {matchExpr : Expr d Δ_out .pure te}\n\nvariable (lets v matchLets w) (mapIn : Mapping _ _) in\n\ndef MatchVarResult := { mapOut : Mapping _ _ //\n  ∃ (mapIn' mapOut' : Mapping _ _),\n    mapIn.entries ⊆ mapIn'.entries\n    ∧ mapOut'.entries ⊆ mapOut.entries\n    ∧ matchVar lets v matchLets w mapIn' = some ((), mapOut') }\n\nvariable (lets matchLets) {tys} (vs ws : HVector _ tys) (mapIn : Mapping _ _) in\n\ndef MatchArgResult := { mapOut : Mapping _ _ //\n  ∃ (mapIn' mapOut' : Mapping _ _),\n    mapIn.entries ⊆ mapIn'.entries\n    ∧ mapOut'.entries ⊆ mapOut.entries\n    ∧ matchArg lets matchLets vs ws mapIn' = some ((), mapOut') }\n\nnamespace MatchVarResult\n\nvariable [TyDenote d.Ty] [∀ (t : d.Ty), Inhabited ⟦t⟧] in\n\nsection Left\n\nvariable {w : Δ_out.Var t}\n\ndef eqvVarLeft  :\n    MatchVarResult lets v (.var matchLets matchExpr) w.appendInr ma\n    ≃ MatchVarResult lets v matchLets w ma where\n  toFun := fun ⟨x, h⟩ => ⟨x, by admit /- proof elided -/\n  ⟩\n  invFun := fun ⟨x, h⟩ => ⟨x, by admit /- proof elided -/\n  ⟩\n\nvariable {mapIn} (mapOut : MatchVarResult lets v (.var matchLets matchExpr) w.appendInr mapIn)\n\nend Left\n\nvariable {w : Var ⟨te⟩ _} {mapIn}\n\nnoncomputable def toArgResult\n    (mapOut : MatchVarResult lets v (.var matchLets matchExpr) w.appendInl mapIn) :\n    let args := mapOut.getPureExpr_eq_some.choose\n    MatchArgResult lets matchLets args matchExpr.args mapIn :=\n  ⟨mapOut.1, by admit /- proof elided -/\n  ⟩\n\nend MatchVarResult\n\nend MatchVar\n\nsection SubsetEntries\n\nopen MatchVar\n\nsection UnifyVars\n\nvariable {Δ Γ : Ctxt d.Ty} {t} (w : Δ.Var t) (v : Γ.Var t)\n\nend UnifyVars\n\nvariable [DecidableEq d.Op]\n\nend SubsetEntries\n\nnamespace MatchArgResult\n\nvariable [DecidableEq d.Op] {Γ_in Γ_out Δ_in Δ_out te}\n          {lets : Lets d Γ_in eff Γ_out}\n          {matchLets : Lets d Δ_in .pure Δ_out}\n          {matchExpr : Expr d Δ_out .pure te}\n          {u us}\n          {v : Γ_out.Var u} {vs : HVector Γ_out.Var us}\n          {w : Δ_out.Var u} {ws : HVector Δ_out.Var us}\n          {mapIn : Mapping _ _}\n          (mapOut : MatchArgResult lets matchLets (v ::ₕ vs) (w ::ₕ ws) mapIn)\n\nend MatchArgResult\n\nsection DenoteLemmas\n\nvariable [TyDenote d.Ty] [DecidableEq d.Op]\n\nvariable [∀ (t : d.Ty), Inhabited ⟦t⟧]\n\nvariable [Monad d.m] [LawfulMonad d.m] [DialectDenote d]\n\nsection DenoteIntoSubtype\n\nend DenoteIntoSubtype\n\nvariable {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty}\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}", "target_theorem": "theorem denote_matchVar\n    {v w : Var _ t}\n    (mapOut : MatchVarResult lets v matchLets w mapIn)\n    (V : lets.ValidDenotation) :\n    (matchLets.denote (mapOut.val.mapValuation V.val) w)\n    = V.val v :=", "ground_truth_proof": ":= by\n  induction matchLets generalizing v mapIn t\n  case nil => simp [Id.pure_eq']\n  case var t' matchLets matchExpr ih =>\n    cases w using Var.appendCases with\n    | right w =>\n      specialize ih mapOut.eqvVarLeft\n      simpa [Id.bind_eq'] using ih\n    | left w =>\n      let mapOut' := mapOut.toArgResult\n      have h := Exists.choose_spec mapOut.getPureExpr_eq_some\n\n      rw [← V.prop h]\n      simp\n      congr 1\n\n      apply Expr.denoteOp_eq_denoteOp_of (by rfl)\n      · simp only [Expr.op_mk, Expr.args_mk]\n        rw [HVector.map_eq_map_of_matchArg (f₁ := V.val) (mapOut := mapOut')]\n        · intro t v₁ v₂ mapOut'' mapOut_eq\n          simp [← ih mapOut'', mapOut_eq, mapOut']\n      · rfl", "nesting_depth": 8, "transitive_dep_count": 114, "subset_aristotle": false, "category": "Compiler"}
{"id": 318, "thm_name": "NFA.proj_eval", "thm_stmt": "@[simp]\nlemma proj_eval (M : NFA (BitVec m) σ) (f : Fin n → Fin m) :\n    (M.proj f).eval w =\n      ⋃ w' ∈ BitVecs'.transport f ⁻¹' {w}, M.eval w'", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/ForMathlib.lean", "imports": ["import Mathlib.Data.Rel", "import Mathlib.Data.FinEnum", "import Mathlib.Data.Vector.Basic", "import Blase.AutoStructs.ForLean", "import Mathlib.Computability.NFA"], "used_lib_defs": [{"name": "BitVec", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}, {"name": "List.getLast", "module": "Init.Data.List.Basic"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}, {"name": "NFA.stepSet", "module": "Mathlib.Computability.NFA"}], "used_repo_defs": [{"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "_root_.Set.proj", "content": "@[simp]\ndef _root_.Set.proj (f : Fin n → Fin m) (bvs : Set (BitVecs m)) : Set (BitVecs n) :=\n  BitVecs.transport f '' bvs"}], "lib_lemmas": [{"name": "List.append_inj_left'", "module": "Init.Data.List.Lemmas"}, {"name": "List.append_inj_right'", "module": "Init.Data.List.Lemmas"}, {"name": "List.dropLast_concat_getLast", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_cons", "module": "Init.Data.List.Basic"}, {"name": "List.map_dropLast", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_nil", "module": "Init.Data.List.Basic"}, {"name": "Set.ext", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.mem_iUnion", "module": "Mathlib.Order.SetNotation"}, {"name": "exists_prop", "module": "Init.PropLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "List.Vector.transport", "content": "def List.Vector.transport (v : Vector α m) (f : Fin n → Fin m) : Vector α n :=\n  Vector.ofFn fun i => v.get (f i)"}, {"name": "BitVec.transport", "content": "def BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=\n  BitVec.ofFn fun i => bv.getLsbD (f i)"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "BitVecs'", "content": "abbrev BitVecs' (n : Nat) := List (BitVec n)"}, {"name": "BitVecs.transport", "content": "def BitVecs.transport (f : Fin n → Fin m) (bvs : BitVecs m) : BitVecs n :=\n  { w := bvs.w, bvs := bvs.bvs.transport f }"}, {"name": "BitVecs'.transport", "content": "def BitVecs'.transport (f : Fin n → Fin m) (bvs' : BitVecs' m): BitVecs' n :=\n  bvs'.map fun bv => bv.transport f"}, {"name": "NFA.Deterministic", "content": "structure Deterministic (M : NFA α σ) : Prop where\n  start : M.start.Subsingleton\n  step : ∀ q a, M.step q a |>.Subsingleton"}, {"name": "NFA.proj", "content": "@[simps]\ndef proj (M: NFA (BitVec n1) σ) (f : Fin n2 → Fin n1) : NFA (BitVec n2) σ where\n  start := M.start\n  accept := M.accept\n  step q a := { q' | ∃ a', a'.transport f = a ∧ q' ∈ M.step q a' }"}], "used_local_lemmas": [], "local_ctx": "import Mathlib.Computability.NFA\n\nimport Mathlib.Data.FinEnum\n\nimport Mathlib.Data.Rel\n\nimport Mathlib.Data.Vector.Basic\n\nimport Blase.AutoStructs.ForLean\n\nopen Set\n\nopen Mathlib\n\nopen SetRel\n\ndef List.Vector.transport (v : Vector α m) (f : Fin n → Fin m) : Vector α n :=\n  Vector.ofFn fun i => v.get (f i)\n\ndef BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=\n  BitVec.ofFn fun i => bv.getLsbD (f i)\n\nstructure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n\n\nabbrev BitVecs' (n : Nat) := List (BitVec n)\n\ndef BitVecs.transport (f : Fin n → Fin m) (bvs : BitVecs m) : BitVecs n :=\n  { w := bvs.w, bvs := bvs.bvs.transport f }\n\ndef BitVecs'.transport (f : Fin n → Fin m) (bvs' : BitVecs' m): BitVecs' n :=\n  bvs'.map fun bv => bv.transport f\n\nnamespace NFA\n\nstructure Deterministic (M : NFA α σ) : Prop where\n  start : M.start.Subsingleton\n  step : ∀ q a, M.step q a |>.Subsingleton\n\n@[simps]\ndef proj (M: NFA (BitVec n1) σ) (f : Fin n2 → Fin n1) : NFA (BitVec n2) σ where\n  start := M.start\n  accept := M.accept\n  step q a := { q' | ∃ a', a'.transport f = a ∧ q' ∈ M.step q a' }", "target_theorem": "@[simp]\nlemma proj_eval (M : NFA (BitVec m) σ) (f : Fin n → Fin m) :\n    (M.proj f).eval w =\n      ⋃ w' ∈ BitVecs'.transport f ⁻¹' {w}, M.eval w' :=", "ground_truth_proof": ":= by\n  induction w using List.reverseRecOn\n  case nil => simp [proj, BitVecs'.transport]\n  case append_singleton w a ih =>\n    ext q; simp [BitVecs'.transport]; constructor\n    · rintro ⟨a', htr, S, hrS, hqS⟩\n      rcases hrS with ⟨q', rfl⟩\n      simp [ih] at hqS\n      rcases hqS with ⟨⟨w', htr', he⟩, hs⟩\n      use w' ++ [a']; constructor\n      · simp_all; exact htr'\n      · simp; use M.step q' a'; constructor\n        on_goal 2 => assumption\n        use q'; simp_all\n    · rintro ⟨wa', heq, he⟩\n      by_cases hemp : wa' = []\n      · simp_all\n      have hdl := List.dropLast_concat_getLast hemp\n      rw [←hdl] at heq he\n      simp only [List.map_append, List.map_dropLast, List.map_cons, List.map_nil] at heq\n      use List.getLast wa' hemp; constructor\n      · apply List.append_inj_right' at heq; simp_all\n      · obtain rfl := List.append_inj_left' heq (by simp)\n        simp at he; rcases he with ⟨S, ⟨q', rfl⟩, hqS⟩\n        simp at hqS; rcases hqS with ⟨he, hs⟩\n        simp only [stepSet, mem_iUnion, exists_prop]\n        use q'\n        simp [ih]; constructor\n        on_goal 2 => assumption\n        use wa'.dropLast\n        simp_all [BitVecs'.transport]", "nesting_depth": 4, "transitive_dep_count": 36, "subset_aristotle": false, "category": "Compiler"}
{"id": 319, "thm_name": "nfaOfFormula_bv_language", "thm_stmt": "theorem nfaOfFormula_bv_language φ :\n    (nfaOfFormula φ).bv_recognizes φ.language", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.Blase.AutoStructs.Basic", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.Blase.AutoStructs.Constructions", "import Blase.Blase.AutoStructs.ForLean", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.card", "module": "Mathlib.Data.FinEnum"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Array.foldl", "module": "Init.Data.Array.Basic"}, {"name": "Std.HashMap.emptyWithCapacity", "module": "Std.Data.HashMap.Basic"}, {"name": "Array.size", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Empty", "module": "Init.Prelude"}, {"name": "Empty.elim", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "cond", "module": "Init.Prelude"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "Array.emptyWithCapacity", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Array.empty", "module": "Init.Prelude"}, {"name": "FinEnum.toList", "module": "Mathlib.Data.FinEnum"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "atom", "module": "Leanwuzla.Sexp.Basic"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "FinEnum.equiv", "module": "Mathlib.Data.FinEnum"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.stepSet", "module": "Mathlib.Computability.NFA"}, {"name": "Subsingleton", "module": "Init.Core"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "Language", "module": "Mathlib.Computability.Language"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "List.Vector.ofFn", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.replicate", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}, {"name": "SetRel", "module": "Mathlib.Data.Rel"}, {"name": "Array.back?", "module": "Init.Data.Array.Basic"}, {"name": "Array.isEmpty", "module": "Init.Data.Array.Basic"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}, {"name": "L", "module": "Archive.Hairer"}, {"name": "Fin.mk", "module": "Init.Prelude"}, {"name": "Fin.cast", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.castLT", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.subNat", "module": "Init.Data.Fin.Basic"}, {"name": "List.Vector.get", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "DFA", "module": "Mathlib.Computability.DFA"}, {"name": "NFA.toDFA", "module": "Mathlib.Computability.NFA"}, {"name": "List.range", "module": "Init.Data.List.Basic"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Fin.natAdd", "module": "Init.Data.Fin.Basic"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "cmp", "module": "Mathlib.Data.Ordering.Basic"}, {"name": "BitVec.ofFin", "module": "Init.Prelude"}, {"name": "BitVec.ule", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ult", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.sle", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.slt", "module": "Init.Data.BitVec.Basic"}, {"name": "LawfulBEq", "module": "Init.Core"}, {"name": "Classical.propDecidable", "module": "Init.Classical"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Function.RightInverse", "module": "Init.Data.Function"}], "used_repo_defs": [{"name": "syntax \"max\" : MLIR.Pretty.uniform_op", "content": "syntax \"max\" : MLIR.Pretty.uniform_op\n\nsyntax \"slt\" : MLIR.Pretty.uniform_op\n\nsyntax \"xor\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "carry", "content": "def carry (initCarry : Bool) (x y : BitStream) : BitStream :=\n  fun n => (addAux' initCarry x y n).2"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "addAux'", "content": "def addAux' (carryIn : Bool) (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => carryIn\n    | i + 1 => (addAux' carryIn x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "worklistRun", "content": "def worklistRun (final : S → Bool) (inits : Array S)\n    (hinits : inits.toList.Nodup) (f : S → Array (BitVec n × S)) : CNFA n :=\n  ⟨worklistRun' _ S final inits hinits f, worklistRun'_wf (BitVec n) S⟩"}, {"name": "worklistRun'", "content": "def worklistRun' (final : S → Bool) (inits : Array S) (hinits : inits.toList.Nodup) (f : S → Array (A × S)) : RawCNFA A :=\n  let st0 := worklist.initState _ _ inits hinits final\n  go st0\nwhere go (st0 : worklist.St A S) : RawCNFA A :=\n  if hemp : st0.worklist.isEmpty then st0.m else\n  let sa? := st0.worklist.back?\n  match heq : sa? with\n  | some sa =>\n    let wl := st0.worklist.pop\n    let st1 := { st0 with worklist := wl,\n                          worklist_nodup := by admit /- proof elided -/"}, {"name": "worklist.St", "content": "structure worklist.St where\n  m : RawCNFA A\n  map : Std.HashMap S State := ∅\n  worklist : Array S := ∅\n  worklist_nodup : worklist.toList.Nodup\n  worklist_incl : ∀ sa ∈ worklist, sa ∈ map"}, {"name": "worklist.initState", "content": "def worklist.initState (inits : Array S) (hinits : inits.toList.Nodup) (final? : S → Bool) : worklist.St A S :=\n  let m := RawCNFA.empty (A := A)\n  let mapm := inits.foldl (init := (Std.HashMap.emptyWithCapacity, m)) fun (map, m) sa =>\n    let (s, m) := m.newState\n    let m := m.addInitial s\n    let m := if final? sa then m.addFinal s else m\n    (map.insert sa s, m)\n  let map := mapm.1\n  let m := mapm.2\n  let worklist_incl : ∀ sa ∈ inits, sa ∈ map :="}, {"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "State", "content": "abbrev State := Nat"}, {"name": "RawCNFA.empty", "content": "def RawCNFA.empty : RawCNFA A := {\n  stateMax := 0\n  initials := ∅\n  finals := ∅\n  trans := ∅\n}"}, {"name": "processOneElem", "content": "def processOneElem (final : S → Bool) (s : State) (st : worklist.St A S) : A × S → worklist.St A S :=\n  fun (a', sa') =>\n    let (s', st') := st.addOrCreateState _ _ (final sa') sa'\n    let m := st'.m.addTrans a' s s'\n    { st' with m }"}, {"name": "worklist.St.addOrCreateState", "content": "def worklist.St.addOrCreateState (st : worklist.St A S) (final? : Bool) (sa : S) : State × worklist.St A S :=\n  match heq : st.map[sa]? with\n  | some s => (s, st)\n  | none =>\n    let (s, m) := st.m.newState\n    let m := if final? then m.addFinal s else m\n    let map := st.map.insert sa s\n    let worklist := st.worklist.push sa\n    have worklist_nodup : worklist.toList.Nodup := by admit /- proof elided -/"}, {"name": "CNFA", "content": "structure CNFA (n : Nat) where\n  m : RawCNFA (BitVec n)\n  wf : m.WF"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "sub", "content": "def sub (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).1"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "ofTerm", "content": "abbrev ofTerm (t : Term) : FSM (Fin t.arity) := termEvalEqFSM t |>.toFSM"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "termEvalEqFSM", "content": "def termEvalEqFSM : ∀ (t : Term), FSMTermSolution t\n  | ofNat n =>\n    { toFSM := FSM.ofNat n,\n      good := by admit /- proof elided -/"}, {"name": "or", "content": "def or : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ||| Circuit.var true (inr false),\n    nextStateCirc := fun a => a.elim\n  }"}, {"name": "shiftLeft", "content": "def shiftLeft (n : Nat) : FSM Unit :=\n  match n with\n  | 0 => FSM.id\n  | n + 1 => composeUnaryAux (FSM.ls false) (shiftLeft n)"}, {"name": "id", "content": "def id : FSM Unit := {\n α := Empty,\n initCarry := Empty.elim,\n outputCirc := Circuit.var true (inr ()),\n nextStateCirc := Empty.elim\n}"}, {"name": "ls", "content": "def ls (b : Bool) : FSM Unit :=\n  { α := Unit,\n    initCarry := fun _ => b,\n    nextStateCirc := fun () => Circuit.var true (inr ()),\n    outputCirc := Circuit.var true (inl ())\n  }"}, {"name": "composeUnaryAux", "content": "def composeUnaryAux\n    (p : FSM Unit)\n    (q : FSM arity) :\n    FSM arity :=\n  p.compose\n    arity\n    _\n    (λ _ => id)\n    (λ _ => q)"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "and", "content": "def and : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := fun a => a.elim,\n    outputCirc := Circuit.var true (inr true) &&& Circuit.var true (inr false),\n  }"}, {"name": "xor", "content": "def xor : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ^^^ Circuit.var true (inr false),\n    nextStateCirc := Empty.elim\n  }"}, {"name": "neg", "content": "def neg : FSM Unit :=\n  { α := Unit,\n    i := by admit /- proof elided -/"}, {"name": "composeBinary", "content": "def composeBinary\n    (p : FSM Bool)\n    {t₁ t₂ : Term}\n    (q₁ : FSMTermSolution t₁)\n    (q₂ : FSMTermSolution t₂) :\n    FSM (Fin (max t₁.arity t₂.arity)) := composeBinaryAux p q₁.toFSM q₂.toFSM"}, {"name": "composeBinaryAux", "content": "def composeBinaryAux\n    (p : FSM Bool)\n    (q₁ : FSM (Fin a₁))\n    (q₂ : FSM (Fin a₂)) :\n    FSM (Fin (max a₁ a₂)) :=\n  p.compose (Fin (max a₁ a₂))\n    (λ b => Fin (cond b a₁ a₂))\n    (λ b i => Fin.castLE (by admit /- proof elided -/\n    ) i)\n    (λ b => match b with\n      | true => q₁\n      | false => q₂)"}, {"name": "FSMTermSolution", "content": "structure FSMTermSolution (t : Term) extends FSM (Fin t.arity) where\n  ( good : t.evalFin = toFSM.eval )"}, {"name": "Term.evalFin", "content": "@[simp] def Term.evalFin (t : Term) (vars : Fin (arity t) → BitStream) : BitStream :=\n  match t with\n  | var n => vars (Fin.last n)\n  | zero    => BitStream.zero\n  | one     => BitStream.one\n  | negOne  => BitStream.negOne\n  | ofNat n => BitStream.ofNat n\n  | and t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | or t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | xor t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | not t     => ~~~(t.evalFin vars)\n  | add t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | sub t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | neg t       => -(Term.evalFin t vars)\n \n \n  | shiftL t n  => BitStream.shiftLeft (Term.evalFin t vars) n"}, {"name": "Predicate.evalFin", "content": "@[simp] def Predicate.evalFin (p : Predicate) (vars : Fin (arity p) → BitStream) : BitStream :=\nmatch p with\n| .width .eq n => BitStream.falseIffEq n\n| .width .neq n => BitStream.falseIffNeq n\n| .width .lt n => BitStream.falseIffLt n\n| .width .le n => BitStream.falseIffLe n\n| .width .gt n => BitStream.falseIffGt n\n| .width .ge n => BitStream.falseIffGe n\n| .binary .eq t₁ t₂ =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalEq x₁ x₂\n| .binary .neq t₁ t₂  =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalNeq x₁ x₂\n| .land p q =>\n  \n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLand x₁ x₂\n| .lor p q =>\n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor x₁ x₂\n| .binary .slt p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalSlt x₁ x₂\n| .binary .sle p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalSlt x₁ x₂) (Predicate.evalEq x₁ x₂)\n| .binary .ult p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  (Predicate.evalUlt x₁ x₂)\n| .binary .ule p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalUlt x₁ x₂) (Predicate.evalEq x₁ x₂)"}, {"name": "Predicate.evalUlt", "content": "def Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true"}, {"name": "borrow", "content": "def borrow (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).2"}, {"name": "Predicate.evalLor", "content": "def Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)"}, {"name": "Predicate.evalSlt", "content": "def Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))"}, {"name": "Predicate.evalMsbEq", "content": "def Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false"}, {"name": "Predicate.evalLand", "content": "def Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)"}, {"name": "Predicate.evalNeq", "content": "def Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd"}, {"name": "nxor", "content": "def nxor (a b : BitStream) : BitStream := fun i => a i == b i"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "Predicate.evalEq", "content": "def Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "Predicate", "content": "inductive Predicate : Type where\n \n| width (wp : WidthPredicate) (n : Nat) : Predicate\n| binary (p : BinaryPredicate) (t₁ t₂ : Term)\n| land  (p q : Predicate) : Predicate\n| lor (p q : Predicate) : Predicate\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "falseIffNeq", "content": "abbrev falseIffNeq (n : Nat) : BitStream := fun i => decide (i == n)"}, {"name": "falseIffLt", "content": "abbrev falseIffLt (n : Nat) : BitStream := fun i => decide (i ≥ n)"}, {"name": "falseIffLe", "content": "abbrev falseIffLe (n : Nat) : BitStream := fun i => decide (i > n)"}, {"name": "falseIffGe", "content": "abbrev falseIffGe (n : Nat) : BitStream := fun i => decide (i < n)"}, {"name": "falseIffEq", "content": "abbrev falseIffEq (n : Nat) : BitStream := fun i => decide (i != n)"}, {"name": "falseIffGt", "content": "abbrev falseIffGt (n : Nat) : BitStream := fun i => decide (i ≤ n)"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "Term.arity", "content": "@[simp] def Term.arity : Term → Nat\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| ofNat _ => 0\n| Term.and t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.or t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.xor t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.not t => arity t\n| add t₁ t₂ => max (arity t₁) (arity t₂)\n| sub t₁ t₂ => max (arity t₁) (arity t₂)\n| neg t => arity t\n\n\n| shiftL t .. => arity t"}, {"name": "negOne", "content": "abbrev negOne : BitStream := fun _ => true"}, {"name": "shiftLeft", "content": "def shiftLeft (x : BitStream) (k : Nat) : BitStream :=\n  fun i => if i < k then false else x (i - k) "}, {"name": "ofNat", "content": "def ofNat (x : Nat) : BitStream :=\n  Nat.testBit x"}, {"name": "one", "content": "abbrev one    : BitStream := (· == 0)"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "ofNat", "content": "def ofNat (n : Nat)  : FSM (Fin 0) :=\n  match hn : n with\n  | 0 => FSM.zero\n\n  | n' + 1 =>\n    let bit := n.testBit 0\n    let m := n / 2\n    have h : m < n := by admit /- proof elided -/"}, {"name": "zero", "content": "def zero : FSM (Fin 0) :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := Empty.elim,\n    outputCirc := Circuit.fals\n  }"}, {"name": "composeUnary", "content": "def composeUnary\n    (p : FSM Unit)\n    {t : Term}\n    (q : FSMTermSolution t) :\n    FSM (Fin t.arity) := composeUnaryAux p q.toFSM"}, {"name": "one", "content": "def one : FSM (Fin 0) :=\n  { α := Unit,\n    i := by admit /- proof elided -/"}, {"name": "var", "content": "def var (n : ℕ) : FSM (Fin (n+1)) :=\n  { α := Empty,\n    i := by admit /- proof elided -/"}, {"name": "add", "content": "def add : FSM Bool :=\n  { α := Unit,\n    initCarry := λ _ => false,\n    nextStateCirc := fun () =>\n      Circuit.var true (inr true) &&& Circuit.var true (inr false) |||\n      Circuit.var true (inr true) &&& Circuit.var true (inl ()) |||\n      Circuit.var true (inr false) &&& Circuit.var true (inl ()),\n    outputCirc := Circuit.var true (inr true) ^^^\n                  Circuit.var true (inr false) ^^^\n                  Circuit.var true (inl ()),\n  }"}, {"name": "negOne", "content": "def negOne : FSM (Fin 0) :=\n  { α := Empty,\n    i := by admit /- proof elided -/"}, {"name": "sub", "content": "def sub : FSM Bool :=\n  { α := Unit,\n    initCarry := fun _ => false,\n    outputCirc := Circuit.var true (inr true) ^^^\n                  Circuit.var true (inr false) ^^^\n                  Circuit.var true (inl ()),\n    nextStateCirc := fun _ =>\n      (Circuit.var false (inr true) &&& Circuit.var true (inr false)) |||\n      (Circuit.var false (inr true) ^^^ Circuit.var true (inr false)) &&&\n      (Circuit.var true (inl ()))\n  }"}, {"name": "not", "content": "def not : FSM Unit :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := Empty.elim,\n    outputCirc := Circuit.var false (inr ())\n  }"}, {"name": "add", "content": "def add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1"}, {"name": "addAux", "content": "def addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "neg", "content": "def neg (x : BitStream) : BitStream :=\n  fun n => (negAux x n).1"}, {"name": "negAux", "content": "def negAux (x : BitStream) : Nat → Bool × Bool\n  | 0 => (x 0, !(x 0))\n  | n+1 =>\n    let borrow := (negAux x n).2\n    let a := x (n + 1)\n    (xor (!a) borrow, !a && borrow)"}, {"name": "CNFA.inter", "content": "def CNFA.inter (m1 m2 : CNFA n) : CNFA n := product (fun b1 b2 => b1 && b2) m1 m2"}, {"name": "product", "content": "def product (final? : Bool → Bool → Bool) (m₁ m₂ : CNFA n) : CNFA n :=\n  worklistRun (m₁.m.states × m₂.m.states) final (product.inits m₁ m₂)\n    (by admit /- proof elided -/\n    ) f\nwhere final (ss : m₁.m.states × m₂.m.states) := final? (ss.1 ∈ m₁.m.finals) (ss.2 ∈ m₂.m.finals)\n      f (ss : m₁.m.states × m₂.m.states) :=\n        let (s1, s2) := ss\n        (FinEnum.toList (α := BitVec n)).foldl (init := Array.empty) fun as a =>\n          product.prodArray' (λ s₁ s₂ ↦ (a, (s₁, s₂)))\n            (fun s' => m₁.wf.trans_tgt_lt (s := s1) (a := a)) (fun s' => m₂.wf.trans_tgt_lt (s := s2) (a := a)) as"}, {"name": "product.prodArray'", "content": "@[inline]\ndef product.prodArray' (a : Array γ) :=\n  m₁.attachWith _ hm₁ |>.fold (init := a) fun is s1 =>\n    m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s1 s2)"}, {"name": "product.inits_nodup", "content": "def product.inits_nodup : inits m₁ m₂ |>.toList.Nodup :="}, {"name": "product.inits", "content": "def product.inits (m₁ m₂ : CNFA n) :=\n  product.prodArray Prod.mk @m₁.wf.initials_lt @m₂.wf.initials_lt"}, {"name": "product.prodArray", "content": "@[inline]\ndef product.prodArray := prodArray' f hm₁ hm₂ (Array.emptyWithCapacity <| m₁.size * m₂.size)"}, {"name": "liftMaxSuccSucc2", "content": "def liftMaxSuccSucc2 (n m : Nat) : Fin (m + 1) → Fin (max n m + 3) :=\n  fun k => if _ : k = Fin.last m then max n m + 1 else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMaxSuccSucc1", "content": "def liftMaxSuccSucc1 (n m : Nat) : Fin (n + 1) → Fin (max n m + 3) :=\n  fun k => if _ : k = Fin.last n then (max n m).cast else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftLast3", "content": "def liftLast3 n : Fin 3 → Fin (n + 3)\n| 0 => n\n| 1 => n + 1\n| 2 => Fin.last (n + 2)"}, {"name": "Unop", "content": "inductive Unop\n| neg\nderiving Repr"}, {"name": "RelationOrdering", "content": "inductive RelationOrdering\n| lt | le | gt | ge\nderiving Repr, Fintype"}, {"name": "Relation", "content": "inductive Relation\n| eq\n| signed (ord : RelationOrdering)\n| unsigned (ord : RelationOrdering)\nderiving Repr"}, {"name": "WidthPredicate", "content": "inductive WidthPredicate\n| eq\n| neq\n| lt\n| le\n| gt\n| ge\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "Binop", "content": "inductive Binop\n| and | or | impl | equiv\nderiving Repr"}, {"name": "liftLast2", "content": "def liftLast2 n : Fin 2 → Fin (n + 2)\n| 0 => n\n| 1 => Fin.last (n + 1)"}, {"name": "liftMaxSucc2", "content": "def liftMaxSucc2 (n m : Nat) : Fin (m + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = m then Fin.last (max n m + 1) else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMaxSucc1", "content": "def liftMaxSucc1 (n m : Nat) : Fin (n + 1) → Fin (max n m + 2) :=\n  fun k => if _ : k = n then Fin.last (max n m) else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMax2", "content": "def liftMax2 (n m : Nat) : Fin m → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftExcept2", "content": "def liftExcept2 n : Fin n → Fin (n + 2) :=\n  fun k => Fin.castLE (by admit /- proof elided -/\n  ) k"}, {"name": "Formula.arity", "content": "@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity"}, {"name": "liftMax1", "content": "def liftMax1 (n m : Nat) : Fin n → Fin (max n m) :=\n  fun k => k.castLE (by admit /- proof elided -/\n  )"}, {"name": "Formula", "content": "inductive Formula : Type\n| width : WidthPredicate → Nat → Formula\n| atom : Relation → Term → Term → Formula\n| msbSet : Term → Formula\n| unop : Unop → Formula → Formula\n| binop : Binop → Formula → Formula → Formula\nderiving Repr"}, {"name": "CNFA.inter_bv_language", "content": "def CNFA.inter_bv_language (m₁ m₂ : CNFA n) :\n    m₁.bv_recognizes L₁ →\n    m₂.bv_recognizes L₂ →\n    (m₁.inter m₂).bv_recognizes (L₁ ∩ L₂) :="}, {"name": "HashSet.inter", "content": "def HashSet.inter [BEq A] [Hashable A] (m1 m2 : Std.HashSet A) : Std.HashSet A :=\n  m1.fold (init := ∅) fun mi x => if m2.contains x then mi.insert x else mi"}, {"name": "Formula.language", "content": "@[simp]\ndef Formula.language (φ : Formula) : Set (BitVecs φ.arity) :=\n  match φ with\n  | .width wp n => { bvs | wp.sat bvs.w n }\n  | .atom rel t1 t2 =>\n    let l1 := t1.language.lift (liftMaxSucc1 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let l2 := t2.language.lift (liftMaxSucc2 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity))\n    let lrel := rel.language.lift $ liftLast2 (max (FinEnum.card (Fin t1.arity)) (FinEnum.card (Fin t2.arity)))\n    let l := lrel ∩ l1 ∩ l2\n    l.proj (liftExcept2 _)\n  | .unop .neg φ => φ.languageᶜ\n  | .binop op φ1 φ2 =>\n    let l1 := φ1.language.lift $ liftMax1 φ1.arity φ2.arity\n    let l2 := φ2.language.lift $ liftMax2 φ1.arity φ2.arity\n    langBinop op l1 l2\n  | .msbSet t =>\n    let lmsb := langMsb.lift $ fun _ => Fin.last t.arity\n    let l' := t.language ∩ lmsb\n    l'.proj fun n => n.castLE (by admit /- proof elided -/\n    )"}, {"name": "WidthPredicate.sat", "content": "@[simp]\ndef WidthPredicate.sat (wp : WidthPredicate) (w n : Nat) : Bool :=\n  match wp with\n  | .eq => w = n\n  | .neq => w ≠ n\n  | .lt => w < n\n  | .le => w ≤ n\n  | .gt => w > n\n  | .ge => w ≥ n"}, {"name": "_root_.Set.proj", "content": "@[simp]\ndef _root_.Set.proj (f : Fin n → Fin m) (bvs : Set (BitVecs m)) : Set (BitVecs n) :=\n  BitVecs.transport f '' bvs"}, {"name": "BitVecs.transport", "content": "def BitVecs.transport (f : Fin n → Fin m) (bvs : BitVecs m) : BitVecs n :=\n  { w := bvs.w, bvs := bvs.bvs.transport f }"}, {"name": "BitVec.transport", "content": "def BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=\n  BitVec.ofFn fun i => bv.getLsbD (f i)"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "List.Vector.transport", "content": "def List.Vector.transport (v : Vector α m) (f : Fin n → Fin m) : Vector α n :=\n  Vector.ofFn fun i => v.get (f i)"}, {"name": "BitVecs'.transport", "content": "def BitVecs'.transport (f : Fin n → Fin m) (bvs' : BitVecs' m): BitVecs' n :=\n  bvs'.map fun bv => bv.transport f"}, {"name": "Term.language", "content": "def Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }"}, {"name": "Term.evalFinBV", "content": "@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n"}, {"name": "_root_.Set.lift", "content": "@[simp]\ndef _root_.Set.lift (f : Fin n → Fin m) (bvs : Set (BitVecs n)) : Set (BitVecs m) :=\n  BitVecs.transport f ⁻¹' bvs"}, {"name": "Formula.sat", "content": "@[simp]\ndef Formula.sat {w : Nat} (φ : Formula) (ρ : Fin φ.arity → BitVec w) : Prop :=\n  match φ with\n  | .width wp n => wp.sat w n\n  | .atom rel t1 t2 =>\n    let bv1 := t1.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let bv2 := t2.evalFinBV (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalRelation rel bv1 bv2\n  | .unop .neg φ => ¬ φ.sat ρ\n  | .binop op φ1 φ2 =>\n    let b1 := φ1.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    let b2 := φ2.sat (fun n => ρ $ Fin.castLE (by admit /- proof elided -/\n    ) n)\n    evalBinop op b1 b2\n  | .msbSet t => (t.evalFinBV ρ).msb"}, {"name": "evalBinop", "content": "def evalBinop (op : Binop) (b1 b2 : Prop) : Prop :=\n  match op with\n  | .and => b1 ∧ b2\n  | .or => b1 ∨ b2\n  | .impl => b1 → b2\n  | .equiv => b1 ↔ b2"}, {"name": "evalRelation", "content": "def evalRelation (rel : Relation) {w} (bv1 bv2 : BitVec w) : Prop :=\n  match rel with\n  | .eq => bv1 = bv2\n  | .signed .lt => bv1.slt bv2\n  | .signed .le => bv1.sle  bv2\n  | .signed .gt => bv2.slt bv1\n  | .signed .ge => bv2.sle bv1\n  | .unsigned .lt => bv1.ult bv2\n  | .unsigned .le => bv1.ule bv2\n  | .unsigned .gt => bv2.ult bv1\n  | .unsigned .ge => bv2.ule bv1"}, {"name": "langBinop", "content": "def langBinop (op : Binop) (l1 l2 : Set (BitVecs n)) : Set (BitVecs n) :=\n  match op with\n  | .and => l1 ∩ l2\n  | .or => l1 ∪ l2\n  | .impl => l1ᶜ ∪ l2\n  | .equiv => (l1ᶜ ∪ l2) ∩ (l2ᶜ ∪ l1)"}, {"name": "Relation.language", "content": "@[simp]\ndef Relation.language (rel : Relation) : Set (BitVecs 2) :=\n  { bvs | evalRelation rel (bvs.bvs.get 0) (bvs.bvs.get 1) }"}, {"name": "NFA'", "content": "structure NFA' (n : Nat) where\n  σ : Type\n  M : NFA (BitVec n) σ"}, {"name": "eval", "content": "def eval (x : arity → BitStream) : BitStream :=\n  fun n => (p.nextBit (p.carry x n) (fun i => x i n)).2"}, {"name": "nextBit", "content": "def nextBit : p.State → (arity → Bool) → p.State × Bool :=\n  fun carry inputBits =>\n    let input := Sum.elim carry inputBits\n    let newState : p.State  := fun (a : p.α) => (p.nextStateCirc a).eval input\n    let outBit : Bool       := (p.outputCirc).eval input\n    (newState, outBit)"}, {"name": "State", "content": "abbrev State : Type := p.α → Bool"}, {"name": "carry", "content": "def carry (x : arity → BitStream) : ℕ → p.State\n  | 0 => p.initCarry\n  | n+1 => (p.nextBit (carry x n) (fun i => x i n)).1"}, {"name": "carryBV", "content": "def carryBV (x : ar → BitVec w) : p.State :=\n  p.carry (fun ar => .ofBitVecSext (x ar)) w"}, {"name": "evalBV", "content": "def evalBV {w} (x : ar → BitVec w) : BitVec w :=\n  BitVec.ofFn fun k => p.eval (fun ar => .ofBitVecSext (x ar)) k"}, {"name": "ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}, {"name": "enc", "content": "def enc (bvs : BitVecs n) : BitVecs' n :=\n  (List.finRange bvs.w).map (fun i =>\n    BitVec.ofFn (fun (k : Fin n) => (bvs.bvs.get k)[i]))"}, {"name": "BitVecs'", "content": "abbrev BitVecs' (n : Nat) := List (BitVec n)"}, {"name": "dec", "content": "@[simps]\ndef dec (bvs' : BitVecs' n) : BitVecs n where\n  w := bvs'.length\n  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k"}, {"name": "accepts", "content": "def accepts (M : NFA' n) : Set (BitVecs n) := dec '' M.accepts'"}, {"name": "accepts'", "content": "def accepts' (M : NFA' n) : Set (BitVecs' n) := M.M.accepts"}, {"name": "worklistRun_spec", "content": "def worklistRun_spec : (worklistRun S final inits hinits f |>.Sim $ nfa' inits final f) :=\n  worklistRun'_spec inits final f"}, {"name": "nfa'", "content": "def nfa' : NFA' n :=\n  { σ := _, M := nfa inits final f }"}, {"name": "nfa", "content": "def nfa : NFA A S where\n  start := { sa | sa ∈ inits }\n  accept := { sa | final sa }\n  step sa a := { sa' | (a, sa') ∈ f sa }"}, {"name": "worklistRun'_spec", "content": "def worklistRun'_spec :\n    (worklistRun' A S final inits hinits f |>.Sim $ nfa inits final f) :="}, {"name": "StInv", "content": "structure StInv (m : RawCNFA A) (map : Std.HashMap S State) where\n  wf : m.WF\n  map_states : ∀ (sa : S) s, map[sa]? = some s → s ∈ m.states\n  map_surj : ∀ s : m.states, ∃ (sa : S), map[sa]? = some s.val\n  map_inj : ∀ {s} {sa sa' : S}, map[sa]? = some s → map[sa']? = some s → sa = sa'"}, {"name": "worklist.St.D", "content": "def worklist.St.D (st : worklist.St A S) : Set S := st.visited"}, {"name": "worklist.St.visited", "content": "def worklist.St.visited (st : worklist.St A S) : Set S := { s : S | s ∈ st.map ∧ s ∉ st.worklist }"}, {"name": "worklistGo_spec", "content": "def worklistGo_spec {st : worklist.St A S} (inv : StInv A S st.m st.map) :\n    st.sim inits final f ∅ →\n    (worklistRun'.go A S final f st |>.Sim $ nfa inits final f) :="}, {"name": "worklist.St.rel", "content": "def worklist.St.rel (st : worklist.St A S) : SetRel State S := {(s, sa) | st.map[sa]? = some s }"}, {"name": "processOneElem_mot", "content": "def processOneElem_mot (s : State) (sa : S) (n : ℕ) (st : worklist.St A S) : Prop :=\n  st.map[sa]? = some s ∧\n  sa ∈ st.visited ∧\n  StInv A S st.m st.map ∧\n  st.sim inits final f  {(sa1, a, sa') | sa1 = sa ∧ ∃ k ≥ n, (f sa)[k]? = some (a, sa') }"}, {"name": "worklist.St.sim", "content": "abbrev worklist.St.sim {st : worklist.St A S} (T : Set (S × A × S)) :=\n  st.m.Simul (nfa inits final f) st.rel st.D T"}, {"name": "RawCNFA.Sim", "content": "def RawCNFA.Sim (m : RawCNFA A) (A : NFA A S) := ∃ R, RawCNFA.Simul m A R ⊤ ∅"}, {"name": "RawCNFA.Simul", "content": "structure RawCNFA.Simul (m : RawCNFA A) (M : NFA A Q) (R : SetRel State Q) (D : Set Q) (T : Set (Q × A × Q)) where\n  accept {s q} : s ~[R] q → (s ∈ m.finals ↔ q ∈ M.accept)\n  initial₁ {s} : s ∈ m.initials → ∃ q ∈ M.start, s ~[R] q\n  initial₂ {q} : q ∈ M.start → ∃ s ∈ m.initials, s ~[R] q\n  trans_match₁ {s s' a q} : s ~[R] q → s' ∈ m.tr s a → ∃ q', q' ∈ M.step q a ∧ s' ~[R] q'\n  trans_match₂ {s a q q'} : s ~[R] q → q' ∈ M.step q a → q ∈ D → (q, a, q') ∉ T → ∃ s', s' ∈ m.tr s a ∧ s' ~[R] q'"}, {"name": "RawCNFA.SimulFun", "content": "structure RawCNFA.SimulFun (m : RawCNFA A) (M : NFA A Q) (f : m.states ≃ Q)  where\n  accept {q} : ((f.invFun q).val ∈ m.finals ↔ q ∈ M.accept)\n  initial {q} : q ∈ M.start ↔ (f.invFun q).val ∈ m.initials\n  trans_match {a q q'} : q' ∈ M.step q a ↔ (f.invFun q').val ∈ m.tr (f.invFun q) a"}, {"name": "RawCNFA.tr", "content": "@[inline]\ndef RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}, {"name": "CNFA.Sim", "content": "def CNFA.Sim (m : CNFA n) (M : NFA' n) :=\n  m.m.Sim M.M"}, {"name": "CNFA.bv_recognizes", "content": "def CNFA.bv_recognizes (m : CNFA n) (L : Set (BitVecs n)) :=\n  ∃ L', m.recognizes L' ∧ L = dec '' L'"}, {"name": "RawCNFA.recognizes", "content": "def RawCNFA.recognizes (m : RawCNFA A) (L : Language A) :=\n  ∃ (σ : Type) (M : NFA A σ), m.Sim M ∧ M.accepts = L"}, {"name": "CNFA.recognizes", "content": "def CNFA.recognizes (m : CNFA n) (L : Language (BitVec n)) :=\n  ∃ (M : NFA' n), m.Sim M ∧ M.M.accepts = L"}, {"name": "BitVecs.cast", "content": "def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=\n  { w := bvs.w, bvs := h ▸ bvs.bvs }"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "CNFA.minimize", "content": "def CNFA.minimize (m : CNFA n) : CNFA n :=\n  let mᵣ := m.reverse.determinize\n  mᵣ.reverse.determinize"}, {"name": "CNFA.determinize", "content": "def CNFA.determinize (m : CNFA n) : CNFA n :=\n  worklistRun (BitVec m.m.stateMax)\n    (fun ss => ss.any fun n b => b == true && n ∈ m.m.finals)\n    (determinize.inits m)\n    (by admit /- proof elided -/\n    )\n    f\nwhere\n  f := fun (ss : BitVec m.m.stateMax) =>\n        (FinEnum.toList (BitVec n)).foldl (init := Array.empty) fun ts a =>\n          let ss' := m.m.transSetBV ss a\n          ts.push (a, ss')"}, {"name": "CNFA.determinize.inits", "content": "def CNFA.determinize.inits (m : CNFA n) : Array (BitVec m.m.stateMax) :=\n  #[BitVec.ofFn (fun n => n ∈ m.m.initials)]"}, {"name": "CNFA.reverse", "content": "def CNFA.reverse (m : CNFA n) : CNFA n :=\n  ⟨m.m.reverse, RawCNFA.reverse_spec m.wf |>.1⟩"}, {"name": "RawCNFA.reverse", "content": "def RawCNFA.reverse (m : RawCNFA A) : RawCNFA A :=\n  let m' := { stateMax := m.stateMax, trans := Std.HashMap.emptyWithCapacity m.trans.size, initials := m.finals, finals := m.initials}\n  m.trans.fold (init := m') processState\nwhere\n  processState := fun m' (s, a) ss' =>\n    ss'.fold (init := m') fun m' s' => m'.addTrans a s' s"}, {"name": "CNFA.toNFA'", "content": "def CNFA.toNFA' (m : CNFA n) : NFA' n := ⟨_, m.toNFA⟩"}, {"name": "CNFA.toNFA", "content": "def CNFA.toNFA (m : CNFA n) : NFA (BitVec n) m.m.states where\n  start := { s | s.val ∈ m.m.initials }\n  accept := { s | s.val ∈ m.m.finals }\n  step s₁ a := { s₂ | s₂.val ∈ m.m.tr s₁.val a }"}, {"name": "RawCNFA.states", "content": "def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax"}, {"name": "reverse", "content": "def reverse (M : NFA' n) : NFA' n where\n  σ := _\n  M := M.M.reverse"}, {"name": "CNFA.determinize_spec", "content": "def CNFA.determinize_spec (m : CNFA n)\n  {M : NFA' n} (hsim : m.Sim M) :\n    m.determinize.Sim M.determinize :="}, {"name": "bv_to_set", "content": "private def bv_to_set (bv : BitVec w) : Set State :=\n  { s | bv.getLsbD s }"}, {"name": "_root_.SetRel.set_eq", "content": "structure _root_.SetRel.set_eq (R : SetRel α β) (A : Set α) (B : Set β) where\n  fwd : a ∈ A → ∃ b ∈ B, a ~[R] b\n  bwd : b ∈ B → ∃ a ∈ A, a ~[R] b"}, {"name": "RawCNFA.lift", "content": "@[inline]\ndef RawCNFA.lift (m₁: RawCNFA (BitVec n1)) (f : Fin n1 → Fin n2) : RawCNFA (BitVec n2) :=\n  let trans := (List.range m₁.stateMax).foldl (init := ∅) fun m2 s => processState m2 s\n  { m₁ with trans }\nwhere"}, {"name": "CNFA.lift", "content": "@[inline]\ndef CNFA.lift (m: CNFA n1) (f : Fin n1 → Fin n2) : CNFA n2 :=\n  ⟨m.m.lift f, m.m.lift_wf m.wf⟩"}, {"name": "RawCNFA.proj", "content": "@[inline]\ndef RawCNFA.proj (m1: RawCNFA (BitVec n1)) (f : Fin n2 → Fin n1) : RawCNFA (BitVec n2) :=\n  let trans := m1.trans.keysArray.foldl (init := Std.HashMap.emptyWithCapacity) process\n  { m1 with trans }\nwhere"}, {"name": "CNFA.proj_spec", "content": "def CNFA.proj_spec (m : CNFA n2) (f : Fin n1 → Fin n2) {M : NFA' n2} :\n    m.Sim M → (m.proj f |>.Sim (M.proj f)) :="}, {"name": "CNFA.proj", "content": "@[inline]\ndef CNFA.proj (m: CNFA n2) (f : Fin n1 → Fin n2) : CNFA n1 :=\n  ⟨m.m.proj f, m.m.proj_wf m.wf⟩"}, {"name": "CNFA.neg_spec", "content": "def CNFA.neg_spec (m : CNFA n)  {M : NFA' n} (hsim : m.Sim M) :\n    m.neg.Sim M.neg :="}, {"name": "CNFA.neg", "content": "def CNFA.neg (m : CNFA n) : CNFA n := m.determinize.flipFinals"}, {"name": "CNFA.flipFinals", "content": "def CNFA.flipFinals (m : CNFA n) : CNFA n := ⟨m.m.flipFinals, m.m.flipFinals_wf m.wf⟩"}, {"name": "RawCNFA.flipFinals", "content": "def RawCNFA.flipFinals (m : RawCNFA A) : RawCNFA A :=\n  let oldFinals := m.finals\n  let newFinals := (List.range m.stateMax).foldl (init := ∅) fun fins s =>\n    if oldFinals.contains s then fins else fins.insert s\n  { m with finals := newFinals }"}, {"name": "bv2", "content": "def bv2 : BitVec 4 := BitVec.ofNat 4 1 "}, {"name": "bv1", "content": "def bv1 : BitVec 4 := BitVec.ofNat 4 5 "}, {"name": "instFinEnumBV", "content": "instance instFinEnumBV : FinEnum (BitVec w) where\n  card := 2^w\n  equiv := {\n    toFun := fun x => x.toFin\n    invFun := fun x => BitVec.ofFin x\n    left_inv := by admit /- proof elided -/"}, {"name": "RawCNFA.WF", "content": "structure RawCNFA.WF (m : RawCNFA A) where\n  initials_lt : ∀ {s}, s ∈ m.initials → s ∈ m.states\n  finals_lt : ∀ {s}, s ∈ m.finals → s ∈ m.states\n  trans_src_lt : ∀ s_a ∈ m.trans, s_a.1 ∈ m.states\n  trans_tgt_lt : s' ∈ m.tr s a → s' ∈ m.states"}, {"name": "CNFA.product_spec", "content": "def CNFA.product_spec (final? : Bool → Bool → Bool) (m1 m2 : CNFA n)\n  {M1 : NFA' n} {M2 : NFA' n} :\n    m1.Sim M1 →\n    m2.Sim M2 →\n    (product final? m1 m2).Sim (NFA'.product (to_prop final?) M1 M2) :="}, {"name": "to_prop", "content": "noncomputable def to_prop (f : Bool → Bool → Bool) (p1 p2 : Prop) : Prop :=\n  f (@Decidable.decide p1 (Classical.propDecidable _)) (@Decidable.decide p2 (Classical.propDecidable _))"}, {"name": "CNFA.union", "content": "def CNFA.union (m1 m2 : CNFA n) : CNFA n :=\n  product (fun b1 b2 => b1 || b2) m1.addSink m2.addSink"}, {"name": "CNFA.addSink", "content": "@[inline]\ndef CNFA.addSink (m : CNFA n) : CNFA n := ⟨m.m.addSink, wf_createSink m.wf⟩"}, {"name": "RawCNFA.addSink", "content": "@[inline]\ndef RawCNFA.addSink (m : RawCNFA A) : RawCNFA A := m.createSink.2"}, {"name": "complete", "content": "noncomputable def complete (M : NFA' n) : NFA' n where\n  σ := _\n  M := M.M.complete"}, {"name": "RawCNFA.addTrans", "content": "def RawCNFA.addTrans (m : RawCNFA A) (a : A) (s s' : State) : RawCNFA A :=\n  let ns := m.trans.getD (s, a) ∅\n  let ns := ns.insert s'\n  { m with trans :=  m.trans.insert (s, a) ns }"}, {"name": "RawCNFA.newState", "content": "def RawCNFA.newState (m : RawCNFA A) : State × RawCNFA A :=\n  let old := m.stateMax\n  let m := { m with stateMax := old + 1 }\n  (old, m)"}, {"name": "RawCNFA.addInitial", "content": "def RawCNFA.addInitial (m : RawCNFA A) (s : State) : RawCNFA A :=\n  { m with initials := m.initials.insert s }"}, {"name": "langMsb", "content": "@[simp]\ndef langMsb : Set (BitVecs 1) := { bvs | bvs.bvs.get 0 |>.msb }"}, {"name": "head", "content": "def head (x : BitStream) : Bool      := x 0"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "List.nodup_singleton", "module": "Mathlib.Data.List.Nodup"}, {"name": "NFA.eval_append_singleton", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.eval_nil", "module": "Mathlib.Computability.NFA"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.add_def", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.castLE_castLE", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.le_of_eq", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Fin.ext_iff", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "ite_cond_eq_true", "module": "Init.SimpLemmas"}, {"name": "BitVec.toNat_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "le_iff_lt_or_eq", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Nat.le_antisymm", "module": "Init.Prelude"}, {"name": "BitVec.toInt_inj", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.zero_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "Finset.mem_range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Function.iterate_add", "module": "Mathlib.Logic.Function.Iterate"}, {"name": "eq_of_forall_lt_iff", "module": "Mathlib.Order.Basic"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "Fin.val_one", "module": "Init.Data.Fin.Lemmas"}, {"name": "eq_iff_eq_of_cmp_eq_cmp", "module": "Mathlib.Order.Compare"}, {"name": "BitVec.eq_nil", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.getElem_one", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.getLsbD_eq_getElem", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.msb_eq_getLsbD_last", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.ofNat_eq_ofNat", "module": "Init.Data.BitVec.Basic"}, {"name": "List.getElem?_eq_getElem", "module": "Init.GetElem"}, {"name": "List.getLast?_eq_getElem?", "module": "Init.Data.List.Lemmas"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "decide_true", "module": "Init.Core"}], "repo_lemmas": [{"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}, {"name": "bisim_comp", "content": "lemma bisim_comp (m : RawCNFA A) :\n    m.Sim M₁ → M₁.Bisim M₂ → m.Sim M₂"}, {"name": "bisimul_comp", "content": "lemma bisimul_comp {m : RawCNFA A} :\n    m.Simul M₁ R₁ ⊤ ∅ → M₁.Bisimul R₂ M₂ →\n    m.Simul M₂ (R₁.comp R₂) ⊤ ∅"}, {"name": "CNFA.bv_recognizes_equiv", "content": "lemma CNFA.bv_recognizes_equiv {m : CNFA n} :\n    m.bv_recognizes L ↔ ∃ (M : NFA' n), m.Sim M ∧ M.accepts = L"}, {"name": "List.Vector.append_get_ge", "content": "@[simp]\nlemma List.Vector.append_get_ge {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: n ≤ i) :\n    (x ++ y).get i = y.get ((i.cast (Nat.add_comm n m) |>.subNat n hlt))"}, {"name": "List.Vector.append_get_lt", "content": "@[simp]\nlemma List.Vector.append_get_lt {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: i < n) :\n    (x ++ y).get i = x.get (i.castLT hlt)"}, {"name": "CNFA.minimize_bv_language", "content": "lemma CNFA.minimize_bv_language {m : CNFA n} :\n    m.bv_recognizes L → m.minimize.bv_recognizes L"}, {"name": "CNFA.minimize_language", "content": "lemma CNFA.minimize_language {m : CNFA n} :\n    m.recognizes L → m.minimize.recognizes L"}, {"name": "CNFA.reverse_language", "content": "lemma CNFA.reverse_language {m : CNFA n} (hl : m.recognizes L) : m.reverse.recognizes L.reverse"}, {"name": "CNFA.reverse_spec", "content": "lemma CNFA.reverse_spec {m : CNFA n} : m.reverse.Sim m.toNFA'.reverse"}, {"name": "RawCNFA.reverse_spec", "content": "lemma RawCNFA.reverse_spec {m : RawCNFA A} (hwf : m.WF) :\n    let m'"}, {"name": "RawCNFA.reverse_spec_procesState", "content": "lemma RawCNFA.reverse_spec_procesState {m : RawCNFA A} (hwf : m.WF) s₀ a₀ ss' (hs₀ : s₀ ∈ m.states) :\n    let motive m' ss'"}, {"name": "CNFA.determinize_language", "content": "lemma CNFA.determinize_language {m : CNFA n} :\n    m.recognizes L → m.determinize.recognizes L"}, {"name": "CNFA.lift_bv_language", "content": "@[simp]\nlemma CNFA.lift_bv_language {m : CNFA n1} {f : Fin n1 → Fin n2} :\n    m.bv_recognizes L → (m.lift f |>.bv_recognizes (BitVecs.transport f ⁻¹' L))"}, {"name": "CNFA.lift_spec", "content": "lemma CNFA.lift_spec (m : CNFA n1) (f : Fin n1 → Fin n2) {M : NFA' n1} :\n    m.Sim M → (m.lift f |>.Sim (M.lift f))"}, {"name": "CNFA.proj_bv_language", "content": "lemma CNFA.proj_bv_language {m : CNFA n2} {f : Fin n1 → Fin n2} :\n    m.bv_recognizes L → (m.proj f |>.bv_recognizes (BitVecs.transport f '' L))"}, {"name": "BitVecs.transport_getElem", "content": "@[simp]\nlemma BitVecs.transport_getElem {bvs : BitVecs m} (f : Fin n → Fin m) (i : Fin n) :\n    (bvs.transport f).bvs.get i = bvs.bvs.get (f i)"}, {"name": "simulFun_sim", "content": "lemma simulFun_sim {m : CNFA n} f :\n    m.m.SimulFun M.M f → m.Sim M"}, {"name": "simulFun_sim_raw", "content": "lemma simulFun_sim_raw [LawfulBEq A] {m : RawCNFA A} (hwf : m.WF) f :\n    m.SimulFun M f → m.Sim M"}, {"name": "RawCNFA.Simul.initial", "content": "@[simp]\nlemma RawCNFA.Simul.initial {m : RawCNFA A} {M : NFA A Q} (hsim : m.Simul M R ⊤ ∅) :\n    R.set_eq m.initials.toSet M.start"}, {"name": "CNFA.inter_spec", "content": "lemma CNFA.inter_spec (m1 m2 : CNFA n)\n  {M1 : NFA' n} {M2 : NFA' n} :\n    m1.Sim M1 →\n    m2.Sim M2 →\n    (m1.inter m2).Sim (M1.inter M2)"}, {"name": "CNFA.union_spec", "content": "lemma CNFA.union_spec (m1 m2 : CNFA n)\n  {M1 : NFA' n} {M2 : NFA' n} :\n    m1.Sim M1 →\n    m2.Sim M2 →\n    (m1.union m2).Sim (M1.union M2)"}, {"name": "CNFA.addSink_spec", "content": "lemma CNFA.addSink_spec (m : CNFA n) (M : NFA' n) :\n    m.Sim M →\n    m.addSink.Sim M.complete"}, {"name": "wf_addTrans", "content": "@[grind ., simp, aesop 50% unsafe]\nlemma wf_addTrans [LawfulBEq A] (m : RawCNFA A) (hwf : m.WF) s a s' (hin : s ∈ m.states) (hin' : s' ∈ m.states) :\n    (m.addTrans a s s').WF"}, {"name": "RawCNFA.same_stateMax", "content": "@[grind =, simp]\nlemma RawCNFA.same_stateMax (m : RawCNFA A) x y (z : Std.HashMap (State × A) (Std.HashSet State)) :\n    (RawCNFA.mk m.stateMax x y z).states = m.states"}, {"name": "newState_eq", "content": "@[grind =, simp, aesop 50% unsafe]\nlemma newState_eq (m : RawCNFA A) :\n    m.newState.1 = m.stateMax"}, {"name": "addInitial_stateMax", "content": "@[grind =, simp]\nlemma addInitial_stateMax {m : RawCNFA A} : (m.addInitial s).stateMax = m.stateMax"}, {"name": "addTrans_stateMax", "content": "@[grind =, simp]\nlemma addTrans_stateMax {m : RawCNFA A} : (m.addTrans a s s').stateMax = m.stateMax"}, {"name": "dec_enc'", "content": "@[simp]\nlemma dec_enc' : dec (enc bvs) = bvs"}, {"name": "dec_enc", "content": "@[simp]\nlemma dec_enc : Function.RightInverse (α := BitVecs' n) enc dec"}, {"name": "dec_enc_w", "content": "lemma dec_enc_w (bvs : BitVecs n) : (dec (enc bvs)).w = bvs.w"}, {"name": "BitVec.ofFn_getElem", "content": "@[simp]\ntheorem BitVec.ofFn_getElem {w : Nat} (f : Fin w → Bool) {i : Nat} (hi : i < w) :\n    (BitVec.ofFn f)[i] = f ⟨i, hi⟩"}, {"name": "BitVec.ofFn_getLsbD", "content": "@[simp]\ntheorem BitVec.ofFn_getLsbD {w : Nat} {f : Fin w → Bool} {i : Nat} (hi : i < w) :\n    (BitVec.ofFn f).getLsbD i = f ⟨i, hi⟩"}, {"name": "BitVec.ofFn_getLsbD_fin", "content": "theorem BitVec.ofFn_getLsbD_fin {w : Nat} {f : Fin w → Bool} {i : Fin w} :\n    (BitVec.ofFn f).getLsbD i = f i"}], "used_local_defs": [{"name": "NFA.sa", "content": "def NFA.sa (_ : NFA α σ) := σ → Language α"}, {"name": "NFA.correct", "content": "structure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q"}, {"name": "BVRel", "content": "abbrev BVRel := ∀ ⦃w⦄, BitVec w → BitVec w → Prop"}, {"name": "BVNRel", "content": "abbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop"}, {"name": "NFA'.sa", "content": "def NFA'.sa (M : NFA' n) := M.σ → BVNRel n"}, {"name": "NFA'.sa2", "content": "def NFA'.sa2 (M : NFA' 2) := M.σ → BVRel"}, {"name": "langRel", "content": "def langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }"}, {"name": "langRel2", "content": "def langRel2 (R : BVRel) : Set (BitVecs 2) :=\n  { bvs | R (bvs.bvs.get 0) (bvs.bvs.get 1) }"}, {"name": "NFA'.correct", "content": "structure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))"}, {"name": "NFA'.correct2", "content": "structure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)"}, {"name": "Alphabet", "content": "abbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)"}, {"name": "finFunToBitVec", "content": "def finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)"}, {"name": "bitVecToFinFun", "content": "def bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]"}, {"name": "NFA.ofFSM", "content": "def NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }"}, {"name": "inFSMRel", "content": "@[simp]\nabbrev inFSMRel (p : FSM arity) {w} (bvn : List.Vector (BitVec w) _) :=\n  bvn.get (Fin.last (FinEnum.card arity)) = p.evalBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))"}, {"name": "NFA'.ofFSM_sa", "content": "def NFA'.ofFSM_sa (p : FSM arity) : (NFA'.ofFSM' p).sa := fun q _ bvn =>\n    inFSMRel p bvn ∧ q = p.carryBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))"}, {"name": "NFA'.ofFSM_correct", "content": "def NFA'.ofFSM_correct (p : FSM arity) :\n    (NFA'.ofFSM' p).correct (ofFSM_sa p) (fun _ bvn => inFSMRel p bvn) :="}, {"name": "CNFA.ofFSM", "content": "def CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where"}, {"name": "RawCNFA.autEq", "content": "def RawCNFA.autEq : RawCNFA (BitVec 2) :=\n  let m := RawCNFA.empty\n  let (s, m) := m.newState\n  let m := m.addInitial s\n  let m := m.addFinal s\n  let m := m.addTrans 0 s s\n  let m := m.addTrans 3 s s\n  m"}, {"name": "CNFA.autEq", "content": "def CNFA.autEq : CNFA 2 :=\n  ⟨RawCNFA.autEq, by admit /- proof elided -/\n  ⟩"}, {"name": "NFA.autEq", "content": "def NFA.autEq : NFA (BitVec 2) Unit :=\n  { start := ⊤, accept := ⊤, step _ a := { _s' | if a = 0 ∨ a = 3 then true else false }}"}, {"name": "NFA'.autEq", "content": "def NFA'.autEq : NFA' 2 :=\n  ⟨Unit, NFA.autEq⟩"}, {"name": "NFA'.eqRel", "content": "def NFA'.eqRel : BVRel := fun _ x y => x = y"}, {"name": "autEq_equiv", "content": "def autEq_equiv : CNFA.autEq.m.states ≃ NFA'.autEq.σ where\n  toFun := fun ⟨s, hs⟩ =>\n    match s with\n    | _ => ()\n  invFun q :=\n    match q with\n    | () => ⟨0, by admit /- proof elided -/\n    ⟩\n  left_inv := by admit /- proof elided -/"}, {"name": "RawCNFA.autUnsignedCmp", "content": "def RawCNFA.autUnsignedCmp (cmp: RelationOrdering) : RawCNFA (BitVec 2) :=\n  let m := RawCNFA.empty\n  let (seq, m) := m.newState\n  let (sgt, m) := m.newState\n  let (slt, m) := m.newState\n  let m := m.addInitial seq\n  let m := m.addManyTrans [0#2, 3#2] seq seq\n  let m := m.addTrans 1#2 seq sgt\n  let m := m.addTrans 2#2 seq slt\n  let m := m.addManyTrans [0#2, 1#2, 3#2] sgt sgt\n  let m := m.addTrans 2#2 sgt slt\n  let m := m.addManyTrans [0#2, 2#2, 3#2] slt slt\n  let mf := m.addTrans 1#2 slt sgt\n  match cmp with\n  | .lt => mf.addFinal slt\n  | .le => (mf.addFinal slt).addFinal seq\n  | .gt => mf.addFinal sgt\n  | .ge => (mf.addFinal sgt).addFinal seq"}, {"name": "CNFA.autUnsignedCmp", "content": "def CNFA.autUnsignedCmp (cmp: RelationOrdering) : CNFA 2 :=\n  ⟨RawCNFA.autUnsignedCmp cmp, RawCNFA.autoUnsignedCmp_wf⟩"}, {"name": "NFA.unsignedCmpState", "content": "inductive NFA.unsignedCmpState : Type where\n| eq | gt | lt\nderiving Fintype, DecidableEq"}, {"name": "NFA.unsignedCmpStep", "content": "def NFA.unsignedCmpStep (q : NFA.unsignedCmpState) (a : BitVec 2) : List NFA.unsignedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [ .eq ] | .eq, 1 => [ .gt ] | .eq, 2 => [ .lt ]\n  | .gt, 0 => [ .gt ] | .gt, 1 => [ .gt ] | .gt, 3 => [ .gt ] | .gt, 2 => [ .lt ]\n  | .lt, 0 => [ .lt ] | .lt, 1 => [ .gt ] | .lt, 2 => [ .lt ] | .lt, 3 => [ .lt ]"}, {"name": "NFA.autUnsignedCmp", "content": "def NFA.autUnsignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) unsignedCmpState where\n  step s a := { s' | s' ∈ unsignedCmpStep s a }\n  start := {s | s = .eq }\n  accept := { s | s ∈ match cmp with | .lt => [unsignedCmpState.lt] | .le => [.lt, .eq] | .gt => [.gt] | .ge => [.gt, .eq] }"}, {"name": "NFA'.autUnsignedCmp", "content": "def NFA'.autUnsignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autUnsignedCmp cmp⟩"}, {"name": "RelationOrdering.urel", "content": "def RelationOrdering.urel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .le => fun _ bv1 bv2 => bv1.ule bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ge => fun _ bv1 bv2 => bv2.ule bv1"}, {"name": "NFA'.autUnsignedCmpSA", "content": "def NFA'.autUnsignedCmpSA (q : NFA.unsignedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1"}, {"name": "RawCNFA.autSignedCmp", "content": "def RawCNFA.autSignedCmp (cmp: RelationOrdering) : RawCNFA (BitVec 2) :=\n  let (m, sltfin, sgtfin, seq) := m\n  match cmp with\n  | .lt => m.addFinal sltfin\n  | .le => (m.addFinal sltfin).addFinal seq\n  | .gt => m.addFinal sgtfin\n  | .ge => (m.addFinal sgtfin).addFinal seq\nwhere"}, {"name": "CNFA.autSignedCmp", "content": "def CNFA.autSignedCmp (cmp: RelationOrdering) : CNFA 2 :=\n  ⟨RawCNFA.autSignedCmp cmp, RawCNFA.autSignedCmp_wf⟩"}, {"name": "NFA.signedCmpState", "content": "inductive NFA.signedCmpState : Type where\n| eq | gt | lt | ltfin | gtfin\nderiving DecidableEq, Fintype"}, {"name": "NFA.signedCmpStep", "content": "def NFA.signedCmpStep (q : NFA.signedCmpState) (a : BitVec 2) : List NFA.signedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [.eq] | .eq, 1 => [.gt, .ltfin] | .eq, 2 => [ .lt, .gtfin ]\n  | .gt, 0 => [ .gt, .gtfin ] | .gt, 1 => [ .gt, .ltfin ] | .gt, 3 => [ .gt, .gtfin ] | .gt, 2 => [ .lt, .gtfin ]\n  | .lt, 0 => [ .lt, .ltfin ] | .lt, 1 => [ .gt, .ltfin ] | .lt, 2 => [ .lt, .gtfin ] | .lt, 3 => [ .lt, .ltfin ]\n  | .gtfin, _ => ∅\n  | .ltfin, _ => ∅"}, {"name": "NFA.autSignedCmp", "content": "def NFA.autSignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) signedCmpState where\n  step s a := { s' | s' ∈ signedCmpStep s a }\n  start := { s | s = signedCmpState.eq }\n  accept := { s | s ∈ match cmp with | .lt => [NFA.signedCmpState.ltfin] | .le => [.ltfin, .eq] | .gt => [.gtfin] | .ge => [.gtfin, .eq] }"}, {"name": "NFA'.autSignedCmp", "content": "def NFA'.autSignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autSignedCmp cmp⟩"}, {"name": "RelationOrdering.srel", "content": "def RelationOrdering.srel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.slt bv2\n  | .le => fun _ bv1 bv2 => bv1.sle bv2\n  | .gt => fun _ bv1 bv2 => bv2.slt bv1\n  | .ge => fun _ bv1 bv2 => bv2.sle bv1"}, {"name": "NFA'.autSignedCmpSA", "content": "def NFA'.autSignedCmpSA (q : NFA.signedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ltfin => fun _ bv1 bv2 => bv1.slt bv2\n  | .gtfin => fun _ bv1 bv2 => bv2.slt bv1"}, {"name": "unsigned_equiv", "content": "def unsigned_equiv cmp : (CNFA.autUnsignedCmp cmp).m.states ≃ (NFA'.autUnsignedCmp cmp).σ where\n  toFun := fun ⟨s, hs⟩ =>\n    match s with\n    | 0 => .eq\n    | 1 => .gt\n    | _ => .lt\n  invFun q :=\n    match q with\n    | .eq => ⟨0, by admit /- proof elided -/\n    ⟩\n    | .gt => ⟨1, by admit /- proof elided -/\n    ⟩\n    | .lt => ⟨2, by admit /- proof elided -/\n    ⟩\n  left_inv := by admit /- proof elided -/"}, {"name": "signed_equiv", "content": "def signed_equiv cmp : (CNFA.autSignedCmp cmp).m.states ≃ (NFA'.autSignedCmp cmp).σ where\n  toFun := fun ⟨s, hs⟩ =>\n    match s with\n    | 0 => .eq\n    | 1 => .gt\n    | 2 => .lt\n    | 3 => .gtfin\n    | _ => .ltfin\n  invFun q :=\n    match q with\n    | .eq => ⟨0, by admit /- proof elided -/\n    ⟩\n    | .gt => ⟨1, by admit /- proof elided -/\n    ⟩\n    | .lt => ⟨2, by admit /- proof elided -/\n    ⟩\n    | .gtfin => ⟨3, by admit /- proof elided -/\n    ⟩\n    | .ltfin => ⟨4, by admit /- proof elided -/\n    ⟩\n  left_inv := by admit /- proof elided -/"}, {"name": "RawCNFA.autMsbSet", "content": "def RawCNFA.autMsbSet : RawCNFA (BitVec 1) :=\n  let m := RawCNFA.empty\n  let (si, m) := m.newState\n  let (sf, m) := m.newState\n  let m := m.addInitial si\n  let m := m.addFinal sf\n  let m := m.addTrans 1 si sf\n  let m := m.addManyTrans [0, 1] si si\n  m"}, {"name": "CNFA.autMsbSet", "content": "@[inline]\ndef CNFA.autMsbSet : CNFA 1 :=\n  ⟨RawCNFA.autMsbSet, RawCNFA.autMsbSet_wf⟩"}, {"name": "NFA.msbState", "content": "inductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype"}, {"name": "NFA.msbStep", "content": "def NFA.msbStep (q : NFA.msbState) (a : BitVec 1) : List NFA.msbState :=\n  match q, a with\n  | .i, 0 => [.i]\n  | .i, 1 => [.i, .f]\n  | _, _ => []"}, {"name": "NFA.autMsbSet", "content": "def NFA.autMsbSet : NFA (BitVec 1) msbState where\n  step s a := { s' | s' ∈ msbStep s a }\n  start := {.i}\n  accept := {.f}"}, {"name": "NFA'.autMsbSet", "content": "def NFA'.autMsbSet : NFA' 1 := ⟨_, NFA.autMsbSet⟩"}, {"name": "NFA.msbLang", "content": "def NFA.msbLang : Language (BitVec 1) := { bvs  | bvs.getLast? = some 1 }"}, {"name": "NFA.msbSA", "content": "def NFA.msbSA (q : msbState) : Language (BitVec 1) :=\n  match q with\n  | .i => ⊤\n  | .f => msbLang"}, {"name": "NFA.msbCorrect", "content": "def NFA.msbCorrect : NFA.autMsbSet.correct msbSA msbLang :="}, {"name": "autMsb_equiv", "content": "def autMsb_equiv : CNFA.autMsbSet.m.states ≃ NFA'.autMsbSet.σ where\n  toFun := fun ⟨s, hs⟩ =>\n    match s with\n    | 0 => .i\n    | 1 => .f\n    | _ => .i\n  invFun q :=\n    match q with\n    | .i => ⟨0, by admit /- proof elided -/\n    ⟩\n    | .f => ⟨1, by admit /- proof elided -/\n    ⟩\n  left_inv := by admit /- proof elided -/"}, {"name": "WidthPredicate.final?", "content": "def WidthPredicate.final? (wp : WidthPredicate) (n : Nat) (s : State) : Bool :=\n  decide (wp.sat s n)"}, {"name": "RawCNFA.autWidth", "content": "def RawCNFA.autWidth (wp : WidthPredicate) (n : Nat) : RawCNFA (BitVec 0) :=\n  let m := (n+2).iterate f empty\n  let m := m.addInitial 0\n  m.addTrans (BitVec.zero 0) (n + 1) (n + 1)\nwhere\n  f m :=\n    let (s, m) := m.newState\n    let m := if wp.final? n s then m.addFinal s else m\n    if s > 0 then m.addTrans (BitVec.zero 0) (s-1) s else m"}, {"name": "CNFA.autWidth", "content": "def CNFA.autWidth (wp : WidthPredicate) (n : Nat) : CNFA 0 :=\n  ⟨RawCNFA.autWidth wp n, RawCNFA.autWidth_wf⟩"}, {"name": "NFA.autWidth", "content": "def NFA.autWidth (wp : WidthPredicate) (n : Nat) : NFA (BitVec 0) (Fin (n+2)) where\n  start := { 0 }\n  accept := { s | wp.final? n s }\n  step s₁ _ := { s₂ | if s₁ = Fin.last (n+1) then s₁ = s₂ else s₂ = s₁ + 1 }"}, {"name": "NFA'.autWidth", "content": "def NFA'.autWidth (wp : WidthPredicate) (n : Nat) : NFA' 0 := ⟨_, NFA.autWidth wp n⟩"}, {"name": "NFA'.autWidth_spec", "content": "@[simp]\ndef NFA'.autWidth_spec : (autWidth wp n).accepts = { bv | wp.sat bv.w n } :="}, {"name": "autWidth_equiv", "content": "def autWidth_equiv : (CNFA.autWidth wp n).m.states ≃ (NFA'.autWidth wp n).σ where\n  toFun := fun ⟨s, hs⟩ =>\n    Fin.mk s (by admit /- proof elided -/\n    )\n  invFun q := ⟨q.val, by admit /- proof elided -/\n  ⟩\n  left_inv := by admit /- proof elided -/"}, {"name": "Relation.autOfRelation", "content": "def Relation.autOfRelation : Relation → CNFA 2\n| .eq => CNFA.autEq\n| .signed ord => CNFA.autSignedCmp ord\n| .unsigned ord => CNFA.autUnsignedCmp ord"}, {"name": "Relation.absAutOfRelation", "content": "def Relation.absAutOfRelation (rel : Relation) : NFA' 2 :=\n  match rel with\n  | .eq => NFA'.autEq\n  | .unsigned cmp => NFA'.autUnsignedCmp cmp\n  | .signed cmp => NFA'.autSignedCmp cmp"}, {"name": "unopNfa", "content": "def unopNfa (op : Unop) (m : CNFA n) : CNFA n :=\n  match op with\n  | .neg => m.neg"}, {"name": "unopAbsNfa", "content": "def unopAbsNfa (op : Unop) (M : NFA' n) : NFA' n :=\n  match op with\n  | .neg => M.neg"}, {"name": "binopNfa", "content": "def binopNfa (op : Binop) (m1 m2 : CNFA n) : CNFA n :=\n  match op with\n  | .and => m1.inter m2\n  | .or => m1.union m2\n  | .impl => m1.neg.union m2\n  | .equiv => (m1.neg.union m2).inter (m2.neg.union m1)"}, {"name": "binopAbsNfa", "content": "def binopAbsNfa (op : Binop) (M1 M2: NFA' n) : NFA' n :=\n  match op with\n  | .and => M1.inter M2\n  | .or => M1.union M2\n  | .impl => M1.neg.union M2\n  | .equiv => (M1.neg.union M2).inter (M2.neg.union M1)"}, {"name": "liftOp", "content": "def liftOp n : Fin (n + 1) → Fin (n + 3) :=\n  fun k =>\n    if k = n then Fin.last (n+2) else k.castLE (by admit /- proof elided -/\n    )"}, {"name": "liftOp_unchanged", "content": "@[simp]\ndef liftOp_unchanged (k : Fin n) : liftOp n k.castSucc = k.castLE (by simp) :="}, {"name": "liftUnop", "content": "def liftUnop n : Fin (n + 1) → Fin (n + 2) :=\n  fun k =>\n    if k = n then Fin.last (n+1) else k.castLE (by admit /- proof elided -/\n    )"}, {"name": "TermBinop", "content": "inductive TermBinop where\n| and | or | xor | add | sub"}, {"name": "TermBinop.subst", "content": "def TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂"}, {"name": "TermBinop.openTerm", "content": "def TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)"}, {"name": "TermBinop.openTerm_arity", "content": "@[simp]\ndef TermBinop.openTerm_arity (op : TermBinop) : op.openTerm.arity + 1 = 3 :="}, {"name": "TermBinop.termGadget", "content": "def TermBinop.termGadget (t : TermBinop) : CNFA 3 :=\n  match t with\n  | .and => FSM.ofTerm (.and (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .or => FSM.ofTerm (.or (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .xor => FSM.ofTerm (.xor (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .add => FSM.ofTerm (.add (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .sub => FSM.ofTerm (.sub (.var 0) (.var 1)) |> CNFA.ofFSM"}, {"name": "autOfTermBinop", "content": "def autOfTermBinop (op : TermBinop) (m₁ : CNFA (n + 1)) (m₂ : CNFA (m + 1)) : CNFA ((n ⊔ m) + 1 ) :=\n  let mop : CNFA 3 := op.termGadget\n  let f₁ := liftMaxSuccSucc1 n m\n  let m1' := m₁.lift f₁\n  let f₂ := liftMaxSuccSucc2 n m\n  let m2' := m₂.lift f₂\n  let mop := mop.lift $ liftLast3 (max (FinEnum.card (Fin n)) (FinEnum.card (Fin m)))\n  let m := CNFA.inter m1' m2' |> CNFA.inter mop\n  let mfinal := m.proj (liftOp _)\n  mfinal.minimize"}, {"name": "swapLastTwoBlock", "content": "def swapLastTwoBlock (x : Fin (n + 3)) : Fin (n + 3) :=\n  if x = Fin.last (n+2) then n\n  else if x = n+1 then Fin.last (n + 2)\n  else if x = n then n + 1\n  else x"}, {"name": "TermUnop", "content": "inductive TermUnop where\n| neg | not | shiftL (k : Nat)"}, {"name": "TermUnop.openTerm", "content": "def TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k"}, {"name": "TermUnop.openTerm_arity", "content": "def TermUnop.openTerm_arity (op : TermUnop) : op.openTerm.arity = 1 :="}, {"name": "TermUnop.openTerm_arity'", "content": "@[simp]\ndef TermUnop.openTerm_arity' (op : TermUnop) : op.openTerm.arity + 1 = 2 :="}, {"name": "TermUnop.subst", "content": "def TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k"}, {"name": "TermUnop.termGadget", "content": "def TermUnop.termGadget (t : TermUnop) : CNFA 2 :=\n  match t with\n  | .neg => FSM.ofTerm (.neg (.var 0)) |> CNFA.ofFSM\n  | .not => FSM.ofTerm (.not (.var 0)) |> CNFA.ofFSM\n  | .shiftL k => FSM.ofTerm (.shiftL (.var 0) k) |> CNFA.ofFSM"}, {"name": "autOfTermUnop", "content": "def autOfTermUnop (op : TermUnop) (m : CNFA (n + 1)) : CNFA (n + 1) :=\n  let mop : CNFA 2 := op.termGadget\n  let mop : CNFA (n + 2) := mop.lift (λ i ↦ i.natAdd n)\n  let m : CNFA (n + 2) := m.lift (λ i ↦ i.castLE (by admit /- proof elided -/\n  ))\n  let m := CNFA.inter m mop\n  let mfinal := m.proj (liftUnop n)\n  mfinal.minimize"}, {"name": "nfaOfTerm", "content": "def nfaOfTerm (t : Term) : CNFA (t.arity + 1) :=\n  match t with\n  | .var n => FSM.ofTerm (.var n) |> CNFA.ofFSM\n  | .zero => FSM.ofTerm .zero |> CNFA.ofFSM\n  | .negOne => FSM.ofTerm .negOne |> CNFA.ofFSM\n  | .one => FSM.ofTerm .one |> CNFA.ofFSM\n  | .ofNat n => FSM.ofTerm (.ofNat n) |> CNFA.ofFSM\n  | .and t₁ t₂ => autOfTermBinop .and (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .or t₁ t₂ => autOfTermBinop .or (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .xor t₁ t₂ => autOfTermBinop .xor (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .add t₁ t₂ => autOfTermBinop .add (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .sub t₁ t₂ => autOfTermBinop .sub (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .neg t => autOfTermUnop .neg (nfaOfTerm t)\n  | .not t => autOfTermUnop .not (nfaOfTerm t)\n  | .shiftL t k => autOfTermUnop (.shiftL k) (nfaOfTerm t)"}, {"name": "swapLastTwo", "content": "def swapLastTwo (x : Fin (n + 2)) : Fin (n + 2) :=\n  if x = Fin.last (n + 1) then n else if x = n then Fin.last (n + 1) else x"}, {"name": "nfaOfFormula", "content": "def nfaOfFormula (φ : Formula) : CNFA φ.arity :=\n  match φ with\n  | .width wp n => CNFA.autWidth wp n\n  | .atom rel t1 t2 =>\n    let m1 := nfaOfTerm t1\n    let m2 := nfaOfTerm t2\n    let f1 := liftMaxSucc1 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity)\n    let m1' := m1.lift f1\n    let f2 := liftMaxSucc2 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity)\n    let m2' := m2.lift f2\n    let meq := rel.autOfRelation.lift $ liftLast2 (max (FinEnum.card (Fin t1.arity)) (FinEnum.card (Fin t2.arity)))\n    let m := CNFA.inter m1' m2' |> CNFA.inter meq\n    let mfinal := m.proj (liftExcept2 _)\n    mfinal\n  | .msbSet t =>\n    let m := (termEvalEqFSM t).toFSM |> CNFA.ofFSM\n    let mMsb := CNFA.autMsbSet.lift $ fun _ => Fin.last t.arity\n    let res := m.inter mMsb\n    res.proj $ fun n => n.castLE (by admit /- proof elided -/\n    )\n  | .unop op φ => unopNfa op (nfaOfFormula φ)\n  | .binop op φ1 φ2 =>\n    let m1 := (nfaOfFormula φ1).lift $ liftMax1 φ1.arity φ2.arity\n    let m2 := (nfaOfFormula φ2).lift $ liftMax2 φ1.arity φ2.arity\n    binopNfa op m1 m2"}], "used_local_lemmas": [{"name": "NFA.correct_spec", "content": "lemma NFA.correct_spec {M : NFA α σ} {ζ : M.sa} {L : Language α} :\n    M.correct ζ L → M.accepts = L"}, {"name": "in_enc", "content": "@[simp]\nlemma in_enc : x ∈ enc '' S ↔ dec x ∈ S"}, {"name": "dec_snoc_in_langRel", "content": "@[simp]\nlemma dec_snoc_in_langRel {n} {R : BVNRel n} {w : BitVecs' n} {a : BitVec n} :\n    dec (w ++ [a]) ∈ langRel R ↔\n      R (List.Vector.ofFn fun k => .cons (a.getLsbD k) ((dec w).bvs.get k))"}, {"name": "NFA'.correct_spec", "content": "lemma NFA'.correct_spec {M : NFA' n} {ζ : M.sa} {L : BVNRel n} :\n    M.correct ζ L → M.accepts = langRel L"}, {"name": "NFA'.correct2_spec", "content": "lemma NFA'.correct2_spec {M : NFA' 2} {ζ : M.sa2} {L : BVRel} :\n    M.correct2 ζ L → M.accepts = langRel2 L"}, {"name": "NFA'.ofFSM_spec", "content": "@[simp]\nlemma NFA'.ofFSM_spec (t : Term) :\n    (ofFSM (FSM.ofTerm t)).accepts = t.language"}, {"name": "CNFA.ofFSM_spec", "content": "lemma CNFA.ofFSM_spec (p : FSM arity) :\n    (CNFA.ofFSM p).Sim (NFA'.ofFSM p)"}, {"name": "CNFA.ofFSM_bv_language", "content": "lemma CNFA.ofFSM_bv_language :\n    (CNFA.ofFSM (FSM.ofTerm t)).bv_recognizes t.language"}, {"name": "NFA'.autEq_correct", "content": "lemma NFA'.autEq_correct : autEq.correct2 (fun _ => eqRel) eqRel"}, {"name": "CNFA.autEq_spec", "content": "lemma CNFA.autEq_spec : autEq.Sim NFA'.autEq"}, {"name": "BitVec.ule_iff_ult_or_eq", "content": "lemma BitVec.ule_iff_ult_or_eq {w : ℕ} (bv1 bv2 : BitVec w):\n    (bv1.ule bv2) = true ↔ (bv1.ult bv2) = true ∨ bv1 = bv2"}, {"name": "ucmp_tricho", "content": "@[simp]\nlemma ucmp_tricho {bv1 bv2 : BitVec w} : (bv2.ult bv1) = false → (bv1.ult bv2) = false → bv1 = bv2"}, {"name": "NFA'.autUnsignedCmp_correct", "content": "lemma NFA'.autUnsignedCmp_correct cmp : autUnsignedCmp cmp |>.correct2 autUnsignedCmpSA cmp.urel"}, {"name": "BitVec.sle_iff_slt_or_eq", "content": "private lemma BitVec.sle_iff_slt_or_eq {w : ℕ} (bv1 bv2 : BitVec w):\n    (bv1.sle bv2) = true ↔ (bv1.slt bv2) = true ∨ bv1 = bv2"}, {"name": "NFA'.autSignedCmp_correct", "content": "lemma NFA'.autSignedCmp_correct cmp : autSignedCmp cmp |>.correct2 autSignedCmpSA cmp.srel"}, {"name": "CNFA.autUnsignedCmp_spec", "content": "lemma CNFA.autUnsignedCmp_spec {cmp} : (CNFA.autUnsignedCmp cmp).Sim (NFA'.autUnsignedCmp cmp)"}, {"name": "CNFA.autSignedCmp_spec", "content": "lemma CNFA.autSignedCmp_spec {cmp} : (CNFA.autSignedCmp cmp).Sim (NFA'.autSignedCmp cmp)"}, {"name": "CNFA.autMsbSet_spec", "content": "lemma CNFA.autMsbSet_spec : CNFA.autMsbSet.Sim NFA'.autMsbSet"}, {"name": "autMsbSet_accepts", "content": "@[simp]\nlemma autMsbSet_accepts : NFA'.autMsbSet.accepts = langMsb"}, {"name": "CNFA.autMsbSet_bv_language", "content": "lemma CNFA.autMsbSet_bv_language : autMsbSet.bv_recognizes langMsb"}, {"name": "RawCNFA.autWidth_spec", "content": "lemma RawCNFA.autWidth_spec {wp : WidthPredicate} :\n  let m := RawCNFA.autWidth wp n\n  m.WF ∧ m.stateMax = n+2 ∧\n  (∀ s, s ∈ m.states → (s ∈ m.initials ↔ s = 0) ∧ (s ∈ m.finals ↔ wp.final? n s)) ∧\n  (∀ s s', s ∈ m.states → s' ∈ m.states → (s' ∈ m.tr s 0 ↔ if s = n+1 then s = s' else s' = s + 1))"}, {"name": "CNFA.autWidth_states", "content": "@[simp]\nlemma CNFA.autWidth_states: s ∈ (autWidth wp n).m.states ↔ s < n+2"}, {"name": "CNFA.autWidth_initials", "content": "lemma CNFA.autWidth_initials : s ∈ (autWidth wp n).m.initials ↔ s = 0"}, {"name": "CNFA.autWidth_finals", "content": "lemma CNFA.autWidth_finals (hn : s < n + 2) : s ∈ (autWidth wp n).m.finals ↔ wp.final? n s"}, {"name": "CNFA.autWidth_tr", "content": "lemma CNFA.autWidth_tr (hs : s < n + 2) (hs' : s' < n + 2) : s' ∈ (autWidth wp n).m.tr s 0 ↔ if s = n+1 then s = s' else s' = s + 1"}, {"name": "CNFA.autWidth_spec", "content": "lemma CNFA.autWidth_spec : autWidth wp n |>.Sim (NFA'.autWidth wp n)"}, {"name": "CNFA.autWidth_bv_language", "content": "lemma CNFA.autWidth_bv_language :\n    (autWidth wp n).bv_recognizes { bv | wp.sat bv.w n }"}, {"name": "autOfRelation_spec", "content": "lemma autOfRelation_spec (r : Relation) :\n  r.autOfRelation.Sim r.absAutOfRelation"}, {"name": "autOfRelation_accepts", "content": "@[simp]\nlemma autOfRelation_accepts (r : Relation) :\n    r.absAutOfRelation.accepts = r.language"}, {"name": "CNFA.autOfRelation_bv_language", "content": "lemma CNFA.autOfRelation_bv_language (r : Relation) :\n    (r.autOfRelation).bv_recognizes r.language"}, {"name": "unopNfa_spec", "content": "lemma unopNfa_spec (op : Unop) (m : CNFA n) (M : NFA' n) :\n    m.Sim M → (unopNfa op m).Sim (unopAbsNfa op M)"}, {"name": "unopNfa_accepts", "content": "lemma unopNfa_accepts (op : Unop) (M : NFA' n) :\n    (unopAbsNfa op M).accepts = M.acceptsᶜ"}, {"name": "unopNfa_bv_language", "content": "lemma unopNfa_bv_language (op : Unop) :\n    m.bv_recognizes L → (unopNfa op m).bv_recognizes Lᶜ"}, {"name": "binopNfa_bv_language", "content": "lemma binopNfa_bv_language (op : Binop) {m₁ m₂ : CNFA n}  :\n    m₁.bv_recognizes L₁ → m₂.bv_recognizes L₂ →\n    (binopNfa op m₁ m₂).bv_recognizes (langBinop op L₁ L₂)"}, {"name": "TermBinop.subst_arity'", "content": "lemma TermBinop.subst_arity' {op : TermBinop} : (op.subst t₁ t₂).arity + 1= t₁.arity ⊔ t₂.arity + 1"}, {"name": "BitVecs.cast_eq", "content": "@[simp]\nlemma BitVecs.cast_eq  (x : BitVecs n) (h : n = n') : h ▸ x = x.cast h"}, {"name": "Fin.natAdd_zero'", "content": "lemma Fin.natAdd_zero' [h : NeZero m] : Fin.natAdd (m := m) n 0 = n"}, {"name": "TermBinop.alt_lang", "content": "lemma TermBinop.alt_lang {t₁ t₂ : Term} (op : TermBinop) :\n  (op.subst_arity' ▸ (op.subst t₁ t₂).language) =\n    let lop : Set (BitVecs 3) := op.openTerm_arity ▸ op.openTerm.language\n    let lop' : Set (BitVecs ((t₁.arity ⊔ t₂.arity) + 3)) := lop.lift (liftLast3 (max t₁.arity t₂.arity))\n    let l₁ := t₁.language.lift (liftMaxSuccSucc1 t₁.arity t₂.arity)\n    let l₂ := t₂.language.lift (liftMaxSuccSucc2 t₁.arity t₂.arity)\n    let l := l₁ ∩ l₂ ∩ lop'\n    l.proj (liftOp _)"}, {"name": "TermUnop.subst_arity'", "content": "@[simp]\nlemma TermUnop.subst_arity' {op : TermUnop} : (op.subst t).arity + 1 = t.arity + 1"}, {"name": "autOfTermBinop_bv_language", "content": "lemma autOfTermBinop_bv_language op {t₁ t₂ : Term} (m₁ : CNFA (t₁.arity + 1)) (m₂ : CNFA (t₂.arity + 1)) :\n    m₁.bv_recognizes t₁.language →\n    m₂.bv_recognizes t₂.language →\n    (autOfTermBinop op m₁ m₂ |>.bv_recognizes (op.subst_arity' ▸ (op.subst t₁ t₂).language))"}, {"name": "TermUnop.alt_lang", "content": "lemma TermUnop.alt_lang {t : Term} (op : TermUnop) :\n  (op.subst_arity' ▸ (op.subst t).language) =\n    let lop : Set (BitVecs 2) := op.openTerm_arity' ▸ op.openTerm.language\n    let lop' : Set (BitVecs (t.arity + 2)) := lop.lift (λ i ↦ i.natAdd t.arity)\n    let lt : Set (BitVecs (t.arity + 2)) := t.language.lift (λ i ↦ i.castLE (by omega))\n    let l := lt ∩ lop'\n    l.proj (liftUnop t.arity)"}, {"name": "autOfTermUnop_bv_language", "content": "lemma autOfTermUnop_bv_language op {t : Term} (m : CNFA (t.arity + 1)) :\n    m.bv_recognizes t.language →\n    (autOfTermUnop op m |>.bv_recognizes (op.subst_arity' ▸ (op.subst t).language))"}, {"name": "nfaOfTerm_bv_language", "content": "lemma nfaOfTerm_bv_language (t : Term) :\n    nfaOfTerm t |>.bv_recognizes t.language"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\ndef NFA.sa (_ : NFA α σ) := σ → Language α\n\nstructure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q\n\nabbrev BVRel := ∀ ⦃w⦄, BitVec w → BitVec w → Prop\n\nabbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop\n\ndef NFA'.sa (M : NFA' n) := M.σ → BVNRel n\n\ndef NFA'.sa2 (M : NFA' 2) := M.σ → BVRel\n\ndef langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }\n\ndef langRel2 (R : BVRel) : Set (BitVecs 2) :=\n  { bvs | R (bvs.bvs.get 0) (bvs.bvs.get 1) }\n\nstructure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))\n\nstructure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)\n\nsection fsm\n\nabbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)\n\nvariable {arity : Type} [FinEnum arity]\n\ndef finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)\n\ndef bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]\n\ndef NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }\n\n@[simp]\nabbrev inFSMRel (p : FSM arity) {w} (bvn : List.Vector (BitVec w) _) :=\n  bvn.get (Fin.last (FinEnum.card arity)) = p.evalBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))\n\ndef NFA'.ofFSM_sa (p : FSM arity) : (NFA'.ofFSM' p).sa := fun q _ bvn =>\n    inFSMRel p bvn ∧ q = p.carryBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))\n\ndef NFA'.ofFSM_correct (p : FSM arity) :\n    (NFA'.ofFSM' p).correct (ofFSM_sa p) (fun _ bvn => inFSMRel p bvn) :=\n\nopen BitStream in\n\ndef CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where\n\nend fsm\n\nsection nfas_relations\n\ndef RawCNFA.autEq : RawCNFA (BitVec 2) :=\n  let m := RawCNFA.empty\n  let (s, m) := m.newState\n  let m := m.addInitial s\n  let m := m.addFinal s\n  let m := m.addTrans 0 s s\n  let m := m.addTrans 3 s s\n  m\n\ndef CNFA.autEq : CNFA 2 :=\n  ⟨RawCNFA.autEq, by admit /- proof elided -/\n  ⟩\n\ndef NFA.autEq : NFA (BitVec 2) Unit :=\n  { start := ⊤, accept := ⊤, step _ a := { _s' | if a = 0 ∨ a = 3 then true else false }}\n\ndef NFA'.autEq : NFA' 2 :=\n  ⟨Unit, NFA.autEq⟩\n\ndef NFA'.eqRel : BVRel := fun _ x y => x = y\n\ndef autEq_equiv : CNFA.autEq.m.states ≃ NFA'.autEq.σ where\n  toFun := fun ⟨s, hs⟩ =>\n    match s with\n    | _ => ()\n  invFun q :=\n    match q with\n    | () => ⟨0, by admit /- proof elided -/\n    ⟩\n  left_inv := by admit /- proof elided -/\n\ndef RawCNFA.autUnsignedCmp (cmp: RelationOrdering) : RawCNFA (BitVec 2) :=\n  let m := RawCNFA.empty\n  let (seq, m) := m.newState\n  let (sgt, m) := m.newState\n  let (slt, m) := m.newState\n  let m := m.addInitial seq\n  let m := m.addManyTrans [0#2, 3#2] seq seq\n  let m := m.addTrans 1#2 seq sgt\n  let m := m.addTrans 2#2 seq slt\n  let m := m.addManyTrans [0#2, 1#2, 3#2] sgt sgt\n  let m := m.addTrans 2#2 sgt slt\n  let m := m.addManyTrans [0#2, 2#2, 3#2] slt slt\n  let mf := m.addTrans 1#2 slt sgt\n  match cmp with\n  | .lt => mf.addFinal slt\n  | .le => (mf.addFinal slt).addFinal seq\n  | .gt => mf.addFinal sgt\n  | .ge => (mf.addFinal sgt).addFinal seq\n\ndef CNFA.autUnsignedCmp (cmp: RelationOrdering) : CNFA 2 :=\n  ⟨RawCNFA.autUnsignedCmp cmp, RawCNFA.autoUnsignedCmp_wf⟩\n\ninductive NFA.unsignedCmpState : Type where\n| eq | gt | lt\nderiving Fintype, DecidableEq\n\ndef NFA.unsignedCmpStep (q : NFA.unsignedCmpState) (a : BitVec 2) : List NFA.unsignedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [ .eq ] | .eq, 1 => [ .gt ] | .eq, 2 => [ .lt ]\n  | .gt, 0 => [ .gt ] | .gt, 1 => [ .gt ] | .gt, 3 => [ .gt ] | .gt, 2 => [ .lt ]\n  | .lt, 0 => [ .lt ] | .lt, 1 => [ .gt ] | .lt, 2 => [ .lt ] | .lt, 3 => [ .lt ]\n\ndef NFA.autUnsignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) unsignedCmpState where\n  step s a := { s' | s' ∈ unsignedCmpStep s a }\n  start := {s | s = .eq }\n  accept := { s | s ∈ match cmp with | .lt => [unsignedCmpState.lt] | .le => [.lt, .eq] | .gt => [.gt] | .ge => [.gt, .eq] }\n\ndef NFA'.autUnsignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autUnsignedCmp cmp⟩\n\ndef RelationOrdering.urel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .le => fun _ bv1 bv2 => bv1.ule bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ge => fun _ bv1 bv2 => bv2.ule bv1\n\ndef NFA'.autUnsignedCmpSA (q : NFA.unsignedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n\ndef RawCNFA.autSignedCmp (cmp: RelationOrdering) : RawCNFA (BitVec 2) :=\n  let (m, sltfin, sgtfin, seq) := m\n  match cmp with\n  | .lt => m.addFinal sltfin\n  | .le => (m.addFinal sltfin).addFinal seq\n  | .gt => m.addFinal sgtfin\n  | .ge => (m.addFinal sgtfin).addFinal seq\nwhere\n\ndef CNFA.autSignedCmp (cmp: RelationOrdering) : CNFA 2 :=\n  ⟨RawCNFA.autSignedCmp cmp, RawCNFA.autSignedCmp_wf⟩\n\ninductive NFA.signedCmpState : Type where\n| eq | gt | lt | ltfin | gtfin\nderiving DecidableEq, Fintype\n\ndef NFA.signedCmpStep (q : NFA.signedCmpState) (a : BitVec 2) : List NFA.signedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [.eq] | .eq, 1 => [.gt, .ltfin] | .eq, 2 => [ .lt, .gtfin ]\n  | .gt, 0 => [ .gt, .gtfin ] | .gt, 1 => [ .gt, .ltfin ] | .gt, 3 => [ .gt, .gtfin ] | .gt, 2 => [ .lt, .gtfin ]\n  | .lt, 0 => [ .lt, .ltfin ] | .lt, 1 => [ .gt, .ltfin ] | .lt, 2 => [ .lt, .gtfin ] | .lt, 3 => [ .lt, .ltfin ]\n  | .gtfin, _ => ∅\n  | .ltfin, _ => ∅\n\ndef NFA.autSignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) signedCmpState where\n  step s a := { s' | s' ∈ signedCmpStep s a }\n  start := { s | s = signedCmpState.eq }\n  accept := { s | s ∈ match cmp with | .lt => [NFA.signedCmpState.ltfin] | .le => [.ltfin, .eq] | .gt => [.gtfin] | .ge => [.gtfin, .eq] }\n\ndef NFA'.autSignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autSignedCmp cmp⟩\n\ndef RelationOrdering.srel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.slt bv2\n  | .le => fun _ bv1 bv2 => bv1.sle bv2\n  | .gt => fun _ bv1 bv2 => bv2.slt bv1\n  | .ge => fun _ bv1 bv2 => bv2.sle bv1\n\ndef NFA'.autSignedCmpSA (q : NFA.signedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ltfin => fun _ bv1 bv2 => bv1.slt bv2\n  | .gtfin => fun _ bv1 bv2 => bv2.slt bv1\n\ndef unsigned_equiv cmp : (CNFA.autUnsignedCmp cmp).m.states ≃ (NFA'.autUnsignedCmp cmp).σ where\n  toFun := fun ⟨s, hs⟩ =>\n    match s with\n    | 0 => .eq\n    | 1 => .gt\n    | _ => .lt\n  invFun q :=\n    match q with\n    | .eq => ⟨0, by admit /- proof elided -/\n    ⟩\n    | .gt => ⟨1, by admit /- proof elided -/\n    ⟩\n    | .lt => ⟨2, by admit /- proof elided -/\n    ⟩\n  left_inv := by admit /- proof elided -/\n\ndef signed_equiv cmp : (CNFA.autSignedCmp cmp).m.states ≃ (NFA'.autSignedCmp cmp).σ where\n  toFun := fun ⟨s, hs⟩ =>\n    match s with\n    | 0 => .eq\n    | 1 => .gt\n    | 2 => .lt\n    | 3 => .gtfin\n    | _ => .ltfin\n  invFun q :=\n    match q with\n    | .eq => ⟨0, by admit /- proof elided -/\n    ⟩\n    | .gt => ⟨1, by admit /- proof elided -/\n    ⟩\n    | .lt => ⟨2, by admit /- proof elided -/\n    ⟩\n    | .gtfin => ⟨3, by admit /- proof elided -/\n    ⟩\n    | .ltfin => ⟨4, by admit /- proof elided -/\n    ⟩\n  left_inv := by admit /- proof elided -/\n\ndef RawCNFA.autMsbSet : RawCNFA (BitVec 1) :=\n  let m := RawCNFA.empty\n  let (si, m) := m.newState\n  let (sf, m) := m.newState\n  let m := m.addInitial si\n  let m := m.addFinal sf\n  let m := m.addTrans 1 si sf\n  let m := m.addManyTrans [0, 1] si si\n  m\n\n@[inline]\ndef CNFA.autMsbSet : CNFA 1 :=\n  ⟨RawCNFA.autMsbSet, RawCNFA.autMsbSet_wf⟩\n\ninductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype\n\ndef NFA.msbStep (q : NFA.msbState) (a : BitVec 1) : List NFA.msbState :=\n  match q, a with\n  | .i, 0 => [.i]\n  | .i, 1 => [.i, .f]\n  | _, _ => []\n\ndef NFA.autMsbSet : NFA (BitVec 1) msbState where\n  step s a := { s' | s' ∈ msbStep s a }\n  start := {.i}\n  accept := {.f}\n\ndef NFA'.autMsbSet : NFA' 1 := ⟨_, NFA.autMsbSet⟩\n\ndef NFA.msbLang : Language (BitVec 1) := { bvs  | bvs.getLast? = some 1 }\n\ndef NFA.msbSA (q : msbState) : Language (BitVec 1) :=\n  match q with\n  | .i => ⊤\n  | .f => msbLang\n\ndef NFA.msbCorrect : NFA.autMsbSet.correct msbSA msbLang :=\n\ndef autMsb_equiv : CNFA.autMsbSet.m.states ≃ NFA'.autMsbSet.σ where\n  toFun := fun ⟨s, hs⟩ =>\n    match s with\n    | 0 => .i\n    | 1 => .f\n    | _ => .i\n  invFun q :=\n    match q with\n    | .i => ⟨0, by admit /- proof elided -/\n    ⟩\n    | .f => ⟨1, by admit /- proof elided -/\n    ⟩\n  left_inv := by admit /- proof elided -/\n\ndef WidthPredicate.final? (wp : WidthPredicate) (n : Nat) (s : State) : Bool :=\n  decide (wp.sat s n)\n\ndef RawCNFA.autWidth (wp : WidthPredicate) (n : Nat) : RawCNFA (BitVec 0) :=\n  let m := (n+2).iterate f empty\n  let m := m.addInitial 0\n  m.addTrans (BitVec.zero 0) (n + 1) (n + 1)\nwhere\n  f m :=\n    let (s, m) := m.newState\n    let m := if wp.final? n s then m.addFinal s else m\n    if s > 0 then m.addTrans (BitVec.zero 0) (s-1) s else m\n\ndef CNFA.autWidth (wp : WidthPredicate) (n : Nat) : CNFA 0 :=\n  ⟨RawCNFA.autWidth wp n, RawCNFA.autWidth_wf⟩\n\ndef NFA.autWidth (wp : WidthPredicate) (n : Nat) : NFA (BitVec 0) (Fin (n+2)) where\n  start := { 0 }\n  accept := { s | wp.final? n s }\n  step s₁ _ := { s₂ | if s₁ = Fin.last (n+1) then s₁ = s₂ else s₂ = s₁ + 1 }\n\ndef NFA'.autWidth (wp : WidthPredicate) (n : Nat) : NFA' 0 := ⟨_, NFA.autWidth wp n⟩\n\n@[simp]\ndef NFA'.autWidth_spec : (autWidth wp n).accepts = { bv | wp.sat bv.w n } :=\n\ndef autWidth_equiv : (CNFA.autWidth wp n).m.states ≃ (NFA'.autWidth wp n).σ where\n  toFun := fun ⟨s, hs⟩ =>\n    Fin.mk s (by admit /- proof elided -/\n    )\n  invFun q := ⟨q.val, by admit /- proof elided -/\n  ⟩\n  left_inv := by admit /- proof elided -/\n\nend nfas_relations\n\ndef Relation.autOfRelation : Relation → CNFA 2\n| .eq => CNFA.autEq\n| .signed ord => CNFA.autSignedCmp ord\n| .unsigned ord => CNFA.autUnsignedCmp ord\n\ndef Relation.absAutOfRelation (rel : Relation) : NFA' 2 :=\n  match rel with\n  | .eq => NFA'.autEq\n  | .unsigned cmp => NFA'.autUnsignedCmp cmp\n  | .signed cmp => NFA'.autSignedCmp cmp\n\ndef unopNfa (op : Unop) (m : CNFA n) : CNFA n :=\n  match op with\n  | .neg => m.neg\n\ndef unopAbsNfa (op : Unop) (M : NFA' n) : NFA' n :=\n  match op with\n  | .neg => M.neg\n\ndef binopNfa (op : Binop) (m1 m2 : CNFA n) : CNFA n :=\n  match op with\n  | .and => m1.inter m2\n  | .or => m1.union m2\n  | .impl => m1.neg.union m2\n  | .equiv => (m1.neg.union m2).inter (m2.neg.union m1)\n\ndef binopAbsNfa (op : Binop) (M1 M2: NFA' n) : NFA' n :=\n  match op with\n  | .and => M1.inter M2\n  | .or => M1.union M2\n  | .impl => M1.neg.union M2\n  | .equiv => (M1.neg.union M2).inter (M2.neg.union M1)\n\ndef liftOp n : Fin (n + 1) → Fin (n + 3) :=\n  fun k =>\n    if k = n then Fin.last (n+2) else k.castLE (by admit /- proof elided -/\n    )\n\n@[simp]\ndef liftOp_unchanged (k : Fin n) : liftOp n k.castSucc = k.castLE (by simp) :=\n\ndef liftUnop n : Fin (n + 1) → Fin (n + 2) :=\n  fun k =>\n    if k = n then Fin.last (n+1) else k.castLE (by admit /- proof elided -/\n    )\n\ninductive TermBinop where\n| and | or | xor | add | sub\n\ndef TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂\n\ndef TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)\n\n@[simp]\ndef TermBinop.openTerm_arity (op : TermBinop) : op.openTerm.arity + 1 = 3 :=\n\ndef TermBinop.termGadget (t : TermBinop) : CNFA 3 :=\n  match t with\n  | .and => FSM.ofTerm (.and (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .or => FSM.ofTerm (.or (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .xor => FSM.ofTerm (.xor (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .add => FSM.ofTerm (.add (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .sub => FSM.ofTerm (.sub (.var 0) (.var 1)) |> CNFA.ofFSM\n\ndef autOfTermBinop (op : TermBinop) (m₁ : CNFA (n + 1)) (m₂ : CNFA (m + 1)) : CNFA ((n ⊔ m) + 1 ) :=\n  let mop : CNFA 3 := op.termGadget\n  let f₁ := liftMaxSuccSucc1 n m\n  let m1' := m₁.lift f₁\n  let f₂ := liftMaxSuccSucc2 n m\n  let m2' := m₂.lift f₂\n  let mop := mop.lift $ liftLast3 (max (FinEnum.card (Fin n)) (FinEnum.card (Fin m)))\n  let m := CNFA.inter m1' m2' |> CNFA.inter mop\n  let mfinal := m.proj (liftOp _)\n  mfinal.minimize\n\ndef swapLastTwoBlock (x : Fin (n + 3)) : Fin (n + 3) :=\n  if x = Fin.last (n+2) then n\n  else if x = n+1 then Fin.last (n + 2)\n  else if x = n then n + 1\n  else x\n\ninductive TermUnop where\n| neg | not | shiftL (k : Nat)\n\ndef TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k\n\ndef TermUnop.openTerm_arity (op : TermUnop) : op.openTerm.arity = 1 :=\n\n@[simp]\ndef TermUnop.openTerm_arity' (op : TermUnop) : op.openTerm.arity + 1 = 2 :=\n\ndef TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k\n\ndef TermUnop.termGadget (t : TermUnop) : CNFA 2 :=\n  match t with\n  | .neg => FSM.ofTerm (.neg (.var 0)) |> CNFA.ofFSM\n  | .not => FSM.ofTerm (.not (.var 0)) |> CNFA.ofFSM\n  | .shiftL k => FSM.ofTerm (.shiftL (.var 0) k) |> CNFA.ofFSM\n\ndef autOfTermUnop (op : TermUnop) (m : CNFA (n + 1)) : CNFA (n + 1) :=\n  let mop : CNFA 2 := op.termGadget\n  let mop : CNFA (n + 2) := mop.lift (λ i ↦ i.natAdd n)\n  let m : CNFA (n + 2) := m.lift (λ i ↦ i.castLE (by admit /- proof elided -/\n  ))\n  let m := CNFA.inter m mop\n  let mfinal := m.proj (liftUnop n)\n  mfinal.minimize\n\ndef nfaOfTerm (t : Term) : CNFA (t.arity + 1) :=\n  match t with\n  | .var n => FSM.ofTerm (.var n) |> CNFA.ofFSM\n  | .zero => FSM.ofTerm .zero |> CNFA.ofFSM\n  | .negOne => FSM.ofTerm .negOne |> CNFA.ofFSM\n  | .one => FSM.ofTerm .one |> CNFA.ofFSM\n  | .ofNat n => FSM.ofTerm (.ofNat n) |> CNFA.ofFSM\n  | .and t₁ t₂ => autOfTermBinop .and (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .or t₁ t₂ => autOfTermBinop .or (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .xor t₁ t₂ => autOfTermBinop .xor (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .add t₁ t₂ => autOfTermBinop .add (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .sub t₁ t₂ => autOfTermBinop .sub (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .neg t => autOfTermUnop .neg (nfaOfTerm t)\n  | .not t => autOfTermUnop .not (nfaOfTerm t)\n  | .shiftL t k => autOfTermUnop (.shiftL k) (nfaOfTerm t)\n\ndef swapLastTwo (x : Fin (n + 2)) : Fin (n + 2) :=\n  if x = Fin.last (n + 1) then n else if x = n then Fin.last (n + 1) else x\n\ndef nfaOfFormula (φ : Formula) : CNFA φ.arity :=\n  match φ with\n  | .width wp n => CNFA.autWidth wp n\n  | .atom rel t1 t2 =>\n    let m1 := nfaOfTerm t1\n    let m2 := nfaOfTerm t2\n    let f1 := liftMaxSucc1 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity)\n    let m1' := m1.lift f1\n    let f2 := liftMaxSucc2 (FinEnum.card $ Fin t1.arity) (FinEnum.card $ Fin t2.arity)\n    let m2' := m2.lift f2\n    let meq := rel.autOfRelation.lift $ liftLast2 (max (FinEnum.card (Fin t1.arity)) (FinEnum.card (Fin t2.arity)))\n    let m := CNFA.inter m1' m2' |> CNFA.inter meq\n    let mfinal := m.proj (liftExcept2 _)\n    mfinal\n  | .msbSet t =>\n    let m := (termEvalEqFSM t).toFSM |> CNFA.ofFSM\n    let mMsb := CNFA.autMsbSet.lift $ fun _ => Fin.last t.arity\n    let res := m.inter mMsb\n    res.proj $ fun n => n.castLE (by admit /- proof elided -/\n    )\n  | .unop op φ => unopNfa op (nfaOfFormula φ)\n  | .binop op φ1 φ2 =>\n    let m1 := (nfaOfFormula φ1).lift $ liftMax1 φ1.arity φ2.arity\n    let m2 := (nfaOfFormula φ2).lift $ liftMax2 φ1.arity φ2.arity\n    binopNfa op m1 m2", "target_theorem": "theorem nfaOfFormula_bv_language φ :\n    (nfaOfFormula φ).bv_recognizes φ.language :=", "ground_truth_proof": ":= by\n  induction φ\n  case width rel n =>\n    simp only [nfaOfFormula]\n    apply CNFA.autWidth_bv_language\n  case atom rel t1 t2 =>\n    simp only [nfaOfFormula, Formula.language]\n    apply CNFA.proj_bv_language\n    ac_nf\n    apply CNFA.inter_bv_language\n    · apply CNFA.lift_bv_language\n      exact CNFA.autOfRelation_bv_language rel\n    · apply CNFA.inter_bv_language\n      · apply CNFA.lift_bv_language\n        exact nfaOfTerm_bv_language t1\n      · apply CNFA.lift_bv_language\n        exact nfaOfTerm_bv_language t2\n  case msbSet t =>\n    simp only [nfaOfFormula, Formula.language]\n    apply CNFA.proj_bv_language\n    apply CNFA.inter_bv_language\n    · exact CNFA.ofFSM_bv_language\n    · apply CNFA.lift_bv_language\n      apply CNFA.autMsbSet_bv_language\n  case unop op φ ih =>\n    simp only [nfaOfFormula, Formula.language]\n    exact unopNfa_bv_language op ih\n  case binop op φ₁ φ2 ih₁ ih₂ =>\n    simp only [nfaOfFormula, Formula.language]\n    apply binopNfa_bv_language op\n    · apply CNFA.lift_bv_language; assumption\n    · apply CNFA.lift_bv_language; assumption", "nesting_depth": 13, "transitive_dep_count": 480, "subset_aristotle": false, "category": "Compiler"}
{"id": 320, "thm_name": "CNFA.ofFSM.f_spec", "thm_stmt": "@[simp]\nlemma CNFA.ofFSM.f_spec {p : FSM arity} {s s' : BitVec (FinEnum.card p.α)} :\n    (a, s') ∈ f p s ↔ bitVecToFinFun s' ∈ (NFA.ofFSM p).step (bitVecToFinFun s) a", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.card", "module": "Mathlib.Data.FinEnum"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Array.foldl", "module": "Init.Data.Array.Basic"}, {"name": "Std.HashMap.emptyWithCapacity", "module": "Std.Data.HashMap.Basic"}, {"name": "Array.size", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "FinEnum.equiv", "module": "Mathlib.Data.FinEnum"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "FinEnum.toList", "module": "Mathlib.Data.FinEnum"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.cast", "module": "Init.Data.BitVec.Basic"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "Function.LeftInverse", "module": "Init.Data.Function"}], "used_repo_defs": [{"name": "carry", "content": "def carry (initCarry : Bool) (x y : BitStream) : BitStream :=\n  fun n => (addAux' initCarry x y n).2"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "addAux'", "content": "def addAux' (carryIn : Bool) (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => carryIn\n    | i + 1 => (addAux' carryIn x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "worklistRun", "content": "def worklistRun (final : S → Bool) (inits : Array S)\n    (hinits : inits.toList.Nodup) (f : S → Array (BitVec n × S)) : CNFA n :=\n  ⟨worklistRun' _ S final inits hinits f, worklistRun'_wf (BitVec n) S⟩"}, {"name": "worklistRun'", "content": "def worklistRun' (final : S → Bool) (inits : Array S) (hinits : inits.toList.Nodup) (f : S → Array (A × S)) : RawCNFA A :=\n  let st0 := worklist.initState _ _ inits hinits final\n  go st0\nwhere go (st0 : worklist.St A S) : RawCNFA A :=\n  if hemp : st0.worklist.isEmpty then st0.m else\n  let sa? := st0.worklist.back?\n  match heq : sa? with\n  | some sa =>\n    let wl := st0.worklist.pop\n    let st1 := { st0 with worklist := wl,\n                          worklist_nodup := by admit /- proof elided -/"}, {"name": "worklist.St", "content": "structure worklist.St where\n  m : RawCNFA A\n  map : Std.HashMap S State := ∅\n  worklist : Array S := ∅\n  worklist_nodup : worklist.toList.Nodup\n  worklist_incl : ∀ sa ∈ worklist, sa ∈ map"}, {"name": "worklist.initState", "content": "def worklist.initState (inits : Array S) (hinits : inits.toList.Nodup) (final? : S → Bool) : worklist.St A S :=\n  let m := RawCNFA.empty (A := A)\n  let mapm := inits.foldl (init := (Std.HashMap.emptyWithCapacity, m)) fun (map, m) sa =>\n    let (s, m) := m.newState\n    let m := m.addInitial s\n    let m := if final? sa then m.addFinal s else m\n    (map.insert sa s, m)\n  let map := mapm.1\n  let m := mapm.2\n  let worklist_incl : ∀ sa ∈ inits, sa ∈ map :="}, {"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "State", "content": "abbrev State := Nat"}, {"name": "RawCNFA.empty", "content": "def RawCNFA.empty : RawCNFA A := {\n  stateMax := 0\n  initials := ∅\n  finals := ∅\n  trans := ∅\n}"}, {"name": "processOneElem", "content": "def processOneElem (final : S → Bool) (s : State) (st : worklist.St A S) : A × S → worklist.St A S :=\n  fun (a', sa') =>\n    let (s', st') := st.addOrCreateState _ _ (final sa') sa'\n    let m := st'.m.addTrans a' s s'\n    { st' with m }"}, {"name": "worklist.St.addOrCreateState", "content": "def worklist.St.addOrCreateState (st : worklist.St A S) (final? : Bool) (sa : S) : State × worklist.St A S :=\n  match heq : st.map[sa]? with\n  | some s => (s, st)\n  | none =>\n    let (s, m) := st.m.newState\n    let m := if final? then m.addFinal s else m\n    let map := st.map.insert sa s\n    let worklist := st.worklist.push sa\n    have worklist_nodup : worklist.toList.Nodup := by admit /- proof elided -/"}, {"name": "CNFA", "content": "structure CNFA (n : Nat) where\n  m : RawCNFA (BitVec n)\n  wf : m.WF"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "bv2", "content": "def bv2 : BitVec 4 := BitVec.ofNat 4 1 "}, {"name": "bv1", "content": "def bv1 : BitVec 4 := BitVec.ofNat 4 5"}], "lib_lemmas": [{"name": "List.nodup_singleton", "module": "Mathlib.Data.List.Nodup"}, {"name": "BitVec.eq_of_getLsbD_eq", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.getLsbD_append", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.ofBool_eq_iff_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Function.LeftInverse.injective", "module": "Init.Data.Function"}, {"name": "BitVec.cons_msb_setWidth", "module": "Init.Data.BitVec.Bootstrap"}, {"name": "BitVec.setWidth_cons", "module": "Init.Data.BitVec.Lemmas"}, {"name": "List.foldl_cons", "module": "Init.Data.List.Basic"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "or_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}], "used_local_defs": [{"name": "Alphabet", "content": "abbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)"}, {"name": "finFunToBitVec", "content": "def finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)"}, {"name": "bitVecToFinFun", "content": "def bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]"}, {"name": "NFA.ofFSM", "content": "def NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }"}, {"name": "CNFA.ofFSM", "content": "def CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where"}], "used_local_lemmas": [{"name": "BitVec.cast_inj", "content": "@[simp]\ntheorem BitVec.cast_inj (h : w = w') {x y : BitVec w} : BitVec.cast h x = BitVec.cast h y ↔ x = y"}, {"name": "BitVec.append_inj", "content": "@[simp]\ntheorem BitVec.append_inj {x1 x2 : BitVec w} {y1 y2 : BitVec w'} :\n    x1 ++ y1 = x2 ++ y2 ↔ x1 = x2 ∧ y1 = y2"}, {"name": "BitVec.cons_inj", "content": "@[simp]\nlemma BitVec.cons_inj : cons b1 bv1 = cons b2 bv2 ↔ (b1 = b2) ∧ bv1 = bv2"}, {"name": "bitVecToFinFun_rinv", "content": "@[simp]\nlemma bitVecToFinFun_rinv (c : carry → Bool) [FinEnum carry]:\n    bitVecToFinFun (finFunToBitVec c) = c"}, {"name": "bitVecToFinFun_linv", "content": "@[simp]\nlemma bitVecToFinFun_linv [FinEnum ar] (bv : BitVec $ FinEnum.card ar) :\n    finFunToBitVec (bitVecToFinFun bv) = bv"}, {"name": "bitVecToFinFun_inj", "content": "@[simp]\nlemma bitVecToFinFun_inj [FinEnum ar] : Function.Injective (bitVecToFinFun (ar := ar))"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\nsection fsm\n\nabbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)\n\nvariable {arity : Type} [FinEnum arity]\n\ndef finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)\n\ndef bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]\n\ndef NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }\n\nopen BitStream in\n\ndef CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where", "target_theorem": "@[simp]\nlemma CNFA.ofFSM.f_spec {p : FSM arity} {s s' : BitVec (FinEnum.card p.α)} :\n    (a, s') ∈ f p s ↔ bitVecToFinFun s' ∈ (NFA.ofFSM p).step (bitVecToFinFun s) a :=", "ground_truth_proof": ":= by\n  let motive (as : List (BitVec (FinEnum.card arity))) := ∀ (acc : Array _) a s',\n    ((a, s') ∈ as.foldl (init := acc) (process p s))\n      ↔ (a, s') ∈ acc ∨ (a.setWidth (FinEnum.card arity) ∈ as) ∧\n          bitVecToFinFun s' ∈ (NFA.ofFSM p).step (bitVecToFinFun s) a\n  suffices h : motive (FinEnum.toList (BitVec (FinEnum.card arity))) by\n    specialize h #[]\n    simp at h\n    rw [←h]\n    rfl\n  generalize FinEnum.toList (BitVec (FinEnum.card arity)) = qs\n  induction qs\n  case nil => simp [motive]\n  case cons a as ih =>\n    rintro acc b s'\n    simp only [List.foldl_cons]; rw [ih]\n    simp [process]\n    constructor\n    · rintro ((hacc | ⟨rfl, rfl⟩) | ⟨hin₁, hin₂⟩)\n      · exact .inl hacc\n      · right; simp [NFA.ofFSM]; constructor <;> rfl\n      · right; simp_all only [or_true, and_self, motive]\n    · rintro (hacc | ⟨(rfl | hold), hst⟩)\n      · tauto\n      · simp [NFA.ofFSM] at hst; left; right\n        rcases hst with ⟨hs', hb⟩\n        constructor\n        · rw [←BitVec.cons_msb_setWidth b]\n          simp_all only [BitVec.setWidth_cons, BitVec.cons_inj, and_true]; rfl\n        · apply_fun bitVecToFinFun <;> simp only [hs', bitVecToFinFun_rinv, bitVecToFinFun_inj]\n          rfl\n      · tauto", "nesting_depth": 5, "transitive_dep_count": 82, "subset_aristotle": false, "category": "Compiler"}
{"id": 321, "thm_name": "Predicate.evalFin_eq_eval", "thm_stmt": "lemma Predicate.evalFin_eq_eval (p : Predicate)\n   (varsList : List BitStream) (varsFin : Fin p.arity → BitStream)\n   (hvars : ∀ (i : Fin p.arity), varsList.getD i default = (varsFin i)) :\n    Predicate.evalFin p varsFin  = Predicate.eval p varsList", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/Fast/Lemmas.lean", "imports": ["import Blase.Fast.Defs", "import Mathlib.Data.Fintype.BigOperators", "import Blase.Fast.BitStream", "import Mathlib.Data.Fintype.Sum", "import Mathlib.Data.Fintype.Card", "import Mathlib.Data.Fintype.Sigma", "import Mathlib.Tactic.Ring", "import Blase.Blase.Fast.BitStream", "import Mathlib.Tactic.Zify"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "List", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"max\" : MLIR.Pretty.uniform_op", "content": "syntax \"max\" : MLIR.Pretty.uniform_op\n\nsyntax \"slt\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Predicate.evalUlt", "content": "def Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "borrow", "content": "def borrow (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).2"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "Predicate", "content": "inductive Predicate : Type where\n \n| width (wp : WidthPredicate) (n : Nat) : Predicate\n| binary (p : BinaryPredicate) (t₁ t₂ : Term)\n| land  (p q : Predicate) : Predicate\n| lor (p q : Predicate) : Predicate\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "Predicate.evalFin", "content": "@[simp] def Predicate.evalFin (p : Predicate) (vars : Fin (arity p) → BitStream) : BitStream :=\nmatch p with\n| .width .eq n => BitStream.falseIffEq n\n| .width .neq n => BitStream.falseIffNeq n\n| .width .lt n => BitStream.falseIffLt n\n| .width .le n => BitStream.falseIffLe n\n| .width .gt n => BitStream.falseIffGt n\n| .width .ge n => BitStream.falseIffGe n\n| .binary .eq t₁ t₂ =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalEq x₁ x₂\n| .binary .neq t₁ t₂  =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalNeq x₁ x₂\n| .land p q =>\n  \n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLand x₁ x₂\n| .lor p q =>\n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor x₁ x₂\n| .binary .slt p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalSlt x₁ x₂\n| .binary .sle p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalSlt x₁ x₂) (Predicate.evalEq x₁ x₂)\n| .binary .ult p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  (Predicate.evalUlt x₁ x₂)\n| .binary .ule p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalUlt x₁ x₂) (Predicate.evalEq x₁ x₂)"}, {"name": "Predicate.evalLor", "content": "def Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)"}, {"name": "Term.evalFin", "content": "@[simp] def Term.evalFin (t : Term) (vars : Fin (arity t) → BitStream) : BitStream :=\n  match t with\n  | var n => vars (Fin.last n)\n  | zero    => BitStream.zero\n  | one     => BitStream.one\n  | negOne  => BitStream.negOne\n  | ofNat n => BitStream.ofNat n\n  | and t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | or t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | xor t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | not t     => ~~~(t.evalFin vars)\n  | add t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | sub t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | neg t       => -(Term.evalFin t vars)\n \n \n  | shiftL t n  => BitStream.shiftLeft (Term.evalFin t vars) n"}, {"name": "negOne", "content": "abbrev negOne : BitStream := fun _ => true"}, {"name": "shiftLeft", "content": "def shiftLeft (x : BitStream) (k : Nat) : BitStream :=\n  fun i => if i < k then false else x (i - k) "}, {"name": "ofNat", "content": "def ofNat (x : Nat) : BitStream :=\n  Nat.testBit x"}, {"name": "one", "content": "abbrev one    : BitStream := (· == 0)"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "Term.arity", "content": "@[simp] def Term.arity : Term → Nat\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| ofNat _ => 0\n| Term.and t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.or t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.xor t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.not t => arity t\n| add t₁ t₂ => max (arity t₁) (arity t₂)\n| sub t₁ t₂ => max (arity t₁) (arity t₂)\n| neg t => arity t\n\n\n| shiftL t .. => arity t"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "Predicate.evalSlt", "content": "def Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))"}, {"name": "Predicate.evalMsbEq", "content": "def Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false"}, {"name": "Predicate.evalLand", "content": "def Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)"}, {"name": "Predicate.evalNeq", "content": "def Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd"}, {"name": "nxor", "content": "def nxor (a b : BitStream) : BitStream := fun i => a i == b i"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "Predicate.evalEq", "content": "def Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "falseIffNeq", "content": "abbrev falseIffNeq (n : Nat) : BitStream := fun i => decide (i == n)"}, {"name": "falseIffLt", "content": "abbrev falseIffLt (n : Nat) : BitStream := fun i => decide (i ≥ n)"}, {"name": "falseIffLe", "content": "abbrev falseIffLe (n : Nat) : BitStream := fun i => decide (i > n)"}, {"name": "falseIffGe", "content": "abbrev falseIffGe (n : Nat) : BitStream := fun i => decide (i < n)"}, {"name": "falseIffEq", "content": "abbrev falseIffEq (n : Nat) : BitStream := fun i => decide (i != n)"}, {"name": "falseIffGt", "content": "abbrev falseIffGt (n : Nat) : BitStream := fun i => decide (i ≤ n)"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "Predicate.arity", "content": "@[simp] def Predicate.arity : Predicate → Nat\n| .width _ _ => 0\n| .binary .eq t1 t2 => max t1.arity t2.arity\n| .binary .neq t₁ t₂ => max t₁.arity t₂.arity\n| .binary .ult t₁ t₂ => max t₁.arity t₂.arity\n| .binary .ule t₁ t₂ => t₁.arity ⊔ t₂.arity ⊔ (t₁.arity ⊔ t₂.arity)\n| .binary .slt t₁ t₂ => (t₁.arity ⊔ t₂.arity ⊔ (t₁.arity ⊔ t₂.arity))\n| .binary .sle t₁ t₂ => (t₁.arity ⊔ t₂.arity ⊔ (t₁.arity ⊔ t₂.arity) ⊔ (t₁.arity ⊔ t₂.arity))\n| .lor p q => max p.arity q.arity\n| .land p q => max p.arity q.arity"}, {"name": "BinaryPredicate", "content": "inductive BinaryPredicate\n| eq\n| neq\n| ult\n| ule\n| slt\n| sle\nderiving Repr, Lean.ToExpr"}, {"name": "WidthPredicate", "content": "inductive WidthPredicate\n| eq\n| neq\n| lt\n| le\n| gt\n| ge\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "Predicate.eval", "content": "def Predicate.eval (p : Predicate) (vars : List BitStream) : BitStream :=\n  match p with\n  | .width .eq n => BitStream.falseIffEq n\n  | .width .neq n => BitStream.falseIffNeq n\n  | .width .lt n => BitStream.falseIffLt n\n  | .width .le n => BitStream.falseIffLe n\n  | .width .gt n => BitStream.falseIffGt n\n  | .width .ge n => BitStream.falseIffGe n\n  | lor p q => Predicate.evalLor (p.eval vars) (q.eval vars)\n  | land p q => Predicate.evalLand (p.eval vars) (q.eval vars)\n  | binary .eq t₁ t₂ => Predicate.evalEq (t₁.eval vars) (t₂.eval vars)\n   \n  | binary .neq t1 t2 => Predicate.evalNeq (t1.eval vars) (t2.eval vars)\n  | binary .ult t₁ t₂ => Predicate.evalUlt (t₁.eval vars) (t₂.eval vars)\n  | binary .ule t₁ t₂ =>\n     Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalUlt (t₁.eval vars) (t₂.eval vars))\n  | binary .slt t₁ t₂ => Predicate.evalSlt (t₁.eval vars) (t₂.eval vars)\n  | binary .sle t₁ t₂ => Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalSlt (t₁.eval vars) (t₂.eval vars))"}, {"name": "Term.eval", "content": "def Term.eval (t : Term) (vars : List BitStream) : BitStream :=\n  match t with\n  | var n       => vars.getD n default\n  | zero        => BitStream.zero\n  | one         => BitStream.one\n  | negOne      => BitStream.negOne\n  | ofNat n     => BitStream.ofNat n\n  | and t₁ t₂   => (t₁.eval vars) &&& (t₂.eval vars)\n  | or t₁ t₂    => (t₁.eval vars) ||| (t₂.eval vars)\n  | xor t₁ t₂   => (t₁.eval vars) ^^^ (t₂.eval vars)\n  | not t       => ~~~(t.eval vars)\n  | add t₁ t₂   => (Term.eval t₁ vars) + (Term.eval t₂ vars)\n  | sub t₁ t₂   => (Term.eval t₁ vars) - (Term.eval t₂ vars)\n  | neg t       => -(Term.eval t vars)\n\n\n  | shiftL t n  => BitStream.shiftLeft (Term.eval t vars) n"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "and_comm", "content": "theorem and_comm (x y : BitStream) : x &&& y = y &&& x"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Term.evalFin_eq_eval", "content": "lemma Term.evalFin_eq_eval (t : Term)\n   (varsList : List BitStream) (varsFin : Fin t.arity → BitStream)\n   (hvars : ∀ (i : Fin t.arity), varsList.getD i default = (varsFin i)) :\n    Term.evalFin t varsFin = Term.eval t varsList"}], "local_ctx": "import Mathlib.Data.Fintype.Card\n\nimport Mathlib.Data.Fintype.Sum\n\nimport Mathlib.Data.Fintype.Sigma\n\nimport Mathlib.Data.Fintype.BigOperators\n\nimport Mathlib.Tactic.Zify\n\nimport Mathlib.Tactic.Ring\n\nimport Blase.Fast.Defs\n\nimport Blase.Fast.BitStream\n\nopen Term", "target_theorem": "lemma Predicate.evalFin_eq_eval (p : Predicate)\n   (varsList : List BitStream) (varsFin : Fin p.arity → BitStream)\n   (hvars : ∀ (i : Fin p.arity), varsList.getD i default = (varsFin i)) :\n    Predicate.evalFin p varsFin  = Predicate.eval p varsList :=", "ground_truth_proof": ":= by\n  induction p generalizing varsList <;>\n    dsimp -failIfUnchanged [Predicate.evalFin, Predicate.eval, Predicate.arity] at *\n  case width rel n =>\n    rcases rel <;> dsimp -failIfUnchanged [Predicate.evalFin, Predicate.eval, Predicate.arity] at *\n  case binary ap t₁ t₂ =>\n    rcases ap <;>\n    · dsimp [Predicate.evalFin, Predicate.eval, Predicate.arity] at *\n      simp [evalEq, evalNeq, evalUlt, evalSlt, evalLor]\n      rw [Term.evalFin_eq_eval _ varsList]\n      · rw [Term.evalFin_eq_eval _ varsList]\n        try rw [BitStream.and_comm]\n        · intros i\n          rw [hvars ⟨i, by omega⟩]\n          rfl\n      · intros i\n        rw [hvars ⟨i, by omega⟩]\n        rfl\n  case land p q hp hq =>\n    simp [evalLand]\n    rw [hp varsList]\n    · rw [hq varsList]\n      · intros i\n        rw [hvars ⟨i, by omega⟩]\n        rfl\n    · intros i\n      rw [hvars ⟨i, by omega⟩]\n      rfl\n  case lor p q hp hq =>\n    simp [evalLor]\n    rw [hp varsList]\n    · rw [hq varsList]\n      · intros i\n        rw [hvars ⟨i, by omega⟩]\n        rfl\n    · intros i\n      rw [hvars ⟨i, by omega⟩]\n      rfl", "nesting_depth": 4, "transitive_dep_count": 46, "subset_aristotle": false, "category": "Compiler"}
{"id": 322, "thm_name": "BitStream.ofBitVecZext_add_EqualUpTo", "thm_stmt": "theorem ofBitVecZext_add_EqualUpTo :\n    ofBitVecZext (x + y) ≈ʷ (ofBitVecZext x) + (ofBitVecZext y)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/Fast/BitStream.lean", "imports": ["import Mathlib.Logic.Function.Iterate", "import Mathlib.Tactic.NormNum"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Add", "module": "Init.Prelude"}, {"name": "Add.add", "module": "Init.Prelude"}, {"name": "BitVec.carry", "module": "Init.Data.BitVec.Bitblast"}, {"name": "BitVec.getLsbD", "module": "Init.Data.BitVec.Basic"}, {"name": "HAdd", "module": "Init.Prelude"}, {"name": "HAdd.hAdd", "module": "Init.Prelude"}, {"name": "Int.succ", "module": "Mathlib.Data.Int.Init"}, {"name": "Prod.mk", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.add_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_two_eq_zero_or_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.add_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.carry_succ", "module": "Init.Data.BitVec.Bitblast"}, {"name": "BitVec.getElem_add", "module": "Init.Data.BitVec.Bitblast"}, {"name": "BitVec.toNat_add", "module": "Init.Data.BitVec.Lemmas"}, {"name": "Bool.decide_and", "module": "Init.Data.Bool"}, {"name": "Bool.decide_iff_dist", "module": "Init.Data.Bool"}, {"name": "Bool.not_eq_eq_eq_not", "module": "Init.SimpLemmas"}, {"name": "Bool.toNat_false", "module": "Init.Data.Bool"}, {"name": "Nat.add_eq", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mod_two_pos_mod_two_eq_one", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "Nat.testBit_zero", "module": "Init.Data.Nat.Bitwise.Lemmas"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "decide_not", "module": "Init.SimpLemmas"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "pow_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "BitStream.ofBitVecZext", "content": "abbrev ofBitVecZext {w} (x : BitVec w) : BitStream :=\n  fun i => x.getLsbD i"}, {"name": "BitStream.addAux", "content": "def addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "BitStream.add", "content": "def add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1"}, {"name": "BitStream.zero", "content": "abbrev zero   : BitStream := fun _ => false"}], "used_local_lemmas": [{"name": "BitStream.ofBitVecZext_eq_getLsbD", "content": "@[simp]\ntheorem ofBitVecZext_eq_getLsbD (x : BitVec w) (i : Nat) :\n  ofBitVecZext x i = x.getLsbD i"}, {"name": "BitStream.addAux_zero", "content": "@[simp] theorem addAux_zero (x y : BitStream) : (x.addAux y 0) =\n  ((x 0) ^^ (y 0), (x 0) && (y 0))"}, {"name": "BitStream.two_le_add_iff_odd_and_odd", "content": "private theorem two_le_add_iff_odd_and_odd (n m : Nat) :\n    2 ≤ n % 2 + m % 2 ↔ n % 2 = 1 ∧ m % 2 = 1"}, {"name": "BitStream.add_odd_iff_neq", "content": "private theorem add_odd_iff_neq (n m : Nat) :\n    (n + m) % 2 = 1 ↔ (n % 2 = 1) ≠ (m % 2 = 1)"}], "local_ctx": "import Mathlib.Tactic.NormNum\n\nimport Mathlib.Logic.Function.Iterate\n\nsection UpStream\n\nnamespace Int\n\nend Int\n\nend UpStream\n\ndef BitStream : Type := Nat → Bool\n\nnamespace BitStream\n\nsection Basic\n\nsection Lemmas\n\nend Lemmas\n\nend Basic\n\nsection OfNat\n\nend OfNat\n\nsection ToBitVec\n\nabbrev ofBitVecZext {w} (x : BitVec w) : BitStream :=\n  fun i => x.getLsbD i\n\nsection Lemmas\n\nend Lemmas\n\nend ToBitVec\n\nsection BitwiseOps\n\nsection Lemmas\n\nvariable {w : Nat}\n\nvariable (x y : BitStream) (i : Nat)\n\nvariable (x y : BitVec (w+1))\n\nend Lemmas\n\nend BitwiseOps\n\nsection Scan\n\nend Scan\n\nsection FindIndex\n\nsection Arith\n\ndef addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)\n\ndef add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1\n\nabbrev zero   : BitStream := fun _ => false\n\nsection Lemmas\n\nvariable {w : Nat} {x y : BitVec w} {a b a' b' : BitStream}\n\nlocal infix:20 \" ≈ʷ \" => EqualUpTo w", "target_theorem": "theorem ofBitVecZext_add_EqualUpTo :\n    ofBitVecZext (x + y) ≈ʷ (ofBitVecZext x) + (ofBitVecZext y) :=", "ground_truth_proof": ":= by\n  intros n a\n  have add_lemma : ⟨(x + y).getLsbD n, BitVec.carry (n + 1) x y false ⟩ = (ofBitVecZext x).addAux (ofBitVecZext y) n := by\n    induction n\n    case zero =>\n      simp only [zero_add, addAux_zero, ofBitVecZext_eq_getLsbD, Prod.mk.injEq]\n      simp only [BitVec.getLsbD, BitVec.toNat_add, Nat.testBit_zero, Nat.mod_two_pos_mod_two_eq_one,\n        a, true_and]\n      simp only [add_odd_iff_neq, ne_eq, eq_iff_iff, decide_not, Bool.decide_iff_dist,\n        Bool.not_eq_eq_eq_not, BitVec.carry, pow_one, Bool.toNat_false, add_zero, ge_iff_le,\n        two_le_add_iff_odd_and_odd, Bool.decide_and, and_true]\n      bv_decide\n    case succ i ih =>\n      simp [addAux, ← ih (by omega), BitVec.adcb, a, BitVec.carry_succ, BitVec.getElem_add];\n  simp [HAdd.hAdd, Add.add, BitStream.add, ← add_lemma, a, -BitVec.add_eq, -Nat.add_eq]", "nesting_depth": 3, "transitive_dep_count": 43, "subset_aristotle": false, "category": "Compiler"}
{"id": 323, "thm_name": "product.sim", "thm_stmt": "lemma product.sim {m1 m2 : CNFA n}:\n    m1.Sim M1 → m2.Sim M2 →\n    (nfa (product.inits m1 m2) (final final? m1 m2) (f m1 m2)).Bisim (M1.M.product (to_prop final?) M2.M)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/Constructions.lean", "imports": ["import Blase.AutoStructs.Worklist", "import Mathlib.Tactic.ApplyFun", "import Mathlib.Data.Fintype.Prod", "import Blase.Blase.AutoStructs.ForLean", "import Blase.Blase.AutoStructs.ForMathlib"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Array.emptyWithCapacity", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Classical.propDecidable", "module": "Init.Classical"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "SetRel", "module": "Mathlib.Data.Rel"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "LawfulBEq", "module": "Init.Core"}, {"name": "Function.Injective2", "module": "Mathlib.Logic.Function.Basic"}, {"name": "List.Nodup", "module": "Init.Data.List.Basic"}, {"name": "FinEnum.toList", "module": "Mathlib.Data.FinEnum"}], "used_repo_defs": [{"name": "CNFA", "content": "structure CNFA (n : Nat) where\n  m : RawCNFA (BitVec n)\n  wf : m.WF"}, {"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "State", "content": "abbrev State := Nat"}, {"name": "product", "content": "def product (final? : Prop → Prop → Prop) (M N : NFA' n) : NFA' n where\n  σ := _\n  M := M.M.product final? N.M"}, {"name": "NFA'", "content": "structure NFA' (n : Nat) where\n  σ : Type\n  M : NFA (BitVec n) σ"}, {"name": "nfa", "content": "def nfa : NFA A S where\n  start := { sa | sa ∈ inits }\n  accept := { sa | final sa }\n  step sa a := { sa' | (a, sa') ∈ f sa }"}, {"name": "Std.HashSet.toSet", "content": "def Std.HashSet.toSet [BEq α] [Hashable α] (m : HashSet α) : Set α := { x | x ∈ m }\n\naxiom hashMap_missing : ∀ {P : Prop}, P"}], "lib_lemmas": [{"name": "Array.mem_push", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.mem_toList_iff", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.toList_push", "module": "Init.Data.Array.Bootstrap"}, {"name": "Function.Injective2.eq_iff", "module": "Mathlib.Logic.Function.Basic"}, {"name": "List.Nodup.append", "module": "Mathlib.Data.List.Nodup"}, {"name": "List.append_assoc", "module": "Init.Data.List.Basic"}, {"name": "List.disjoint_singleton", "module": "Batteries.Data.List.Lemmas"}, {"name": "List.mem_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.mem_singleton", "module": "Init.Data.List.Lemmas"}, {"name": "Set.mem_empty_iff_false", "module": "Mathlib.Data.Set.Basic"}, {"name": "Set.mem_insert_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "Set.union_singleton", "module": "Mathlib.Data.Set.Insert"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "iff_and_self", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "ne_or_eq", "module": "Mathlib.Logic.Basic"}, {"name": "Array.emptyWithCapacity_eq", "module": "Init.Data.Array.Basic"}, {"name": "and_self", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "Array.mem_def", "module": "Init.Data.Array.Basic"}, {"name": "Array.not_mem_empty", "module": "Init.Data.Array.Lemmas"}, {"name": "FinEnum.nodup_toList", "module": "Mathlib.Data.FinEnum"}, {"name": "List.Nodup.notMem", "module": "Mathlib.Data.List.Nodup"}, {"name": "List.dedup_eq_self", "module": "Mathlib.Data.List.Dedup"}, {"name": "List.foldl_nil", "module": "Init.Data.List.Basic"}, {"name": "List.mem_cons", "module": "Init.Data.List.Lemmas"}, {"name": "List.nodup_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.not_mem_nil", "module": "Init.Data.List.Lemmas"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}], "repo_lemmas": [{"name": "Std.HashSet.fold_induction", "content": "theorem Std.HashSet.fold_induction [BEq α] [LawfulBEq α] [Hashable α]\n  {f : β → α → β} {m : HashSet α} {motive : β → Set α → Prop} :\n    motive b ∅ →\n    (∀ b x s, x ∉ s → motive b s → motive (f b x) (s ∪ {x})) →\n    motive (m.fold f b) m.toSet"}, {"name": "Std.HashSet.toSet_toList[BEq", "content": "theorem Std.HashSet.toSet_toList[BEq α] [LawfulBEq α] [Hashable α] (m : HashSet α) : m.toSet = { x | x ∈ m.toList }"}, {"name": "Std.HashSet.mem_toSet", "content": "@[simp]\nlemma Std.HashSet.mem_toSet [BEq α] [Hashable α] (m : HashSet α) : x ∈ m.toSet ↔ x ∈ m"}, {"name": "Std.HashSet.mem_attachWith_mem", "content": "@[simp]\ntheorem Std.HashSet.mem_attachWith_mem [BEq α] [Hashable α] [LawfulBEq α] (m : HashSet α) {P H} (x : α) h :\n    ⟨x, h⟩ ∈ m.attachWith P H ↔ x ∈ m"}], "used_local_defs": [{"name": "product.prodArray'", "content": "@[inline]\ndef product.prodArray' (a : Array γ) :=\n  m₁.attachWith _ hm₁ |>.fold (init := a) fun is s1 =>\n    m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s1 s2)"}, {"name": "product.prodArray", "content": "@[inline]\ndef product.prodArray := prodArray' f hm₁ hm₂ (Array.emptyWithCapacity <| m₁.size * m₂.size)"}, {"name": "product.inits", "content": "def product.inits (m₁ m₂ : CNFA n) :=\n  product.prodArray Prod.mk @m₁.wf.initials_lt @m₂.wf.initials_lt"}, {"name": "product", "content": "def product (final? : Bool → Bool → Bool) (m₁ m₂ : CNFA n) : CNFA n :=\n  worklistRun (m₁.m.states × m₂.m.states) final (product.inits m₁ m₂)\n    (by admit /- proof elided -/\n    ) f\nwhere final (ss : m₁.m.states × m₂.m.states) := final? (ss.1 ∈ m₁.m.finals) (ss.2 ∈ m₂.m.finals)\n      f (ss : m₁.m.states × m₂.m.states) :=\n        let (s1, s2) := ss\n        (FinEnum.toList (α := BitVec n)).foldl (init := Array.empty) fun as a =>\n          product.prodArray' (λ s₁ s₂ ↦ (a, (s₁, s₂)))\n            (fun s' => m₁.wf.trans_tgt_lt (s := s1) (a := a)) (fun s' => m₂.wf.trans_tgt_lt (s := s2) (a := a)) as"}, {"name": "to_prop", "content": "noncomputable def to_prop (f : Bool → Bool → Bool) (p1 p2 : Prop) : Prop :=\n  f (@Decidable.decide p1 (Classical.propDecidable _)) (@Decidable.decide p2 (Classical.propDecidable _))"}], "used_local_lemmas": [{"name": "product.prodArray_spec_helper", "content": "include hinj in\nomit [BEq α] [Hashable α] [LawfulBEq α] in\nlemma product.prodArray_spec_helper\n    (is : Array γ) (hnd : is.toList.Nodup)\n    (s : S₁) (hnew : ∀ s₂, f s s₂ ∉ is):\n  let motive (a : Array γ) (S : Set S₂)  :=\n    a.toList.Nodup ∧\n    (∃ r, a.toList = is.toList ++ r ∧ (∀ z ∈ r, ∃ s₁ s₂, z = f s₁ s₂)) ∧\n    ∀ s1 s2, f s1 s2 ∈ a ↔ s1 ≠ s ∧ f s1 s2 ∈ is ∨ s1 = s ∧ s2 ∈ S\n  let body := m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s s2)\n  motive body (m₂.attachWith _ hm₂).toSet"}, {"name": "product.prodArray'_spec_full", "content": "include hinj in\nlemma product.prodArray'_spec_full {aᵢ : Array γ} (hnd: aᵢ.toList.Nodup) (hnin : ∀ s₁ s₂, f s₁ s₂ ∉ aᵢ) :\n    (product.prodArray' f hm₁ hm₂ aᵢ).toList.Nodup ∧\n    (∃ r, (product.prodArray' f hm₁ hm₂ aᵢ).toList = aᵢ.toList ++ r ∧ (∀ z ∈ r, ∃ s₁ s₂, z = f s₁ s₂)) ∧\n    ∀ s₁ s₂, f s₁ s₂ ∈ product.prodArray' f hm₁ hm₂ aᵢ ↔ (s₁.val ∈ m₁ ∧ s₂.val ∈ m₂)"}, {"name": "product.prodArray_spec_full", "content": "include hinj in\nlemma product.prodArray_spec_full :\n    (product.prodArray f hm₁ hm₂).toList.Nodup ∧\n    ∀ s₁ s₂, f s₁ s₂ ∈ product.prodArray f hm₁ hm₂ ↔ (s₁.val ∈ m₁ ∧ s₂.val ∈ m₂)"}, {"name": "product.prodArray_spec", "content": "include hinj in\n@[simp]\nlemma product.prodArray_spec :\n    ∀ s₁ s₂, f s₁ s₂ ∈ product.prodArray f hm₁ hm₂ ↔ (s₁.val ∈ m₁ ∧ s₂.val ∈ m₂)"}, {"name": "product.inits_spec", "content": "@[simp]\nlemma product.inits_spec :\n    ∀ s₁ s₂, (s₁, s₂) ∈ inits m₁ m₂ ↔ (s₁.val ∈ m₁.m.initials ∧ s₂.val ∈ m₂.m.initials)"}, {"name": "product.f_spec", "content": "lemma product.f_spec {m₁ m₂ : CNFA n} {s₁ : m₁.m.states} {s₂ : m₂.m.states} :\n    ∀ a s₁' s₂',\n      (a, (s₁', s₂')) ∈ f m₁ m₂ (s₁, s₂) ↔ s₁'.val ∈ m₁.m.tr s₁ a ∧ s₂'.val ∈ m₂.m.tr s₂ a"}], "local_ctx": "import Mathlib.Data.Fintype.Prod\n\nimport Blase.AutoStructs.Worklist\n\nimport Mathlib.Tactic.ApplyFun\n\nopen SetRel\n\nsection sink\n\nvariable {A : Type} [BEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nend sink\n\nsection generic_prod\n\nvariable {α} [BEq α] [Hashable α] [LawfulBEq α]\n\nvariable {β} [BEq β] [Hashable β] [LawfulBEq β]\n\nvariable {S₁ : Finset α} {S₂ : Finset β}\n\nvariable {γ} (f : S₁ → S₂ → γ) (hinj : Function.Injective2 f)\n\nvariable {m₁ : Std.HashSet α} (hm₁ : ∀ s₁ ∈ m₁, s₁ ∈ S₁)\n\nvariable {m₂ : Std.HashSet β} (hm₂ : ∀ s₂ ∈ m₂, s₂ ∈ S₂)\n\n@[inline]\ndef product.prodArray' (a : Array γ) :=\n  m₁.attachWith _ hm₁ |>.fold (init := a) fun is s1 =>\n    m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s1 s2)\n\n@[inline]\ndef product.prodArray := prodArray' f hm₁ hm₂ (Array.emptyWithCapacity <| m₁.size * m₂.size)\n\nend generic_prod\n\nsection product\n\nvariable {A : Type} [BEq A] [LawfulBEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\ndef product.inits (m₁ m₂ : CNFA n) :=\n  product.prodArray Prod.mk @m₁.wf.initials_lt @m₂.wf.initials_lt\n\ndef product (final? : Bool → Bool → Bool) (m₁ m₂ : CNFA n) : CNFA n :=\n  worklistRun (m₁.m.states × m₂.m.states) final (product.inits m₁ m₂)\n    (by admit /- proof elided -/\n    ) f\nwhere final (ss : m₁.m.states × m₂.m.states) := final? (ss.1 ∈ m₁.m.finals) (ss.2 ∈ m₂.m.finals)\n      f (ss : m₁.m.states × m₂.m.states) :=\n        let (s1, s2) := ss\n        (FinEnum.toList (α := BitVec n)).foldl (init := Array.empty) fun as a =>\n          product.prodArray' (λ s₁ s₂ ↦ (a, (s₁, s₂)))\n            (fun s' => m₁.wf.trans_tgt_lt (s := s1) (a := a)) (fun s' => m₂.wf.trans_tgt_lt (s := s2) (a := a)) as\n\nnoncomputable def to_prop (f : Bool → Bool → Bool) (p1 p2 : Prop) : Prop :=\n  f (@Decidable.decide p1 (Classical.propDecidable _)) (@Decidable.decide p2 (Classical.propDecidable _))", "target_theorem": "lemma product.sim {m1 m2 : CNFA n}:\n    m1.Sim M1 → m2.Sim M2 →\n    (nfa (product.inits m1 m2) (final final? m1 m2) (f m1 m2)).Bisim (M1.M.product (to_prop final?) M2.M) :=", "ground_truth_proof": ":= by\n  rintro ⟨R₁, hsim₁⟩ ⟨R₂, hsim₂⟩\n  let R : SetRel (m1.m.states × m2.m.states) (M1.σ × M2.σ) :=\n    {((s₁, s₂), (q₁, q₂)) | s₁.val ~[R₁] q₁ ∧ s₂.val ~[R₂] q₂ }\n  use R; constructor\n  · rintro ⟨s₁, s₂⟩ ⟨q₁, q₂⟩ ⟨hR₁, hR₂⟩\n    simp [nfa, to_prop, final]\n    rw [←hsim₁.accept hR₁, ←hsim₂.accept hR₂]; congr\n  · constructor\n    · rintro ⟨s₁, s₂⟩ hstart\n      simp [nfa, inits_spec] at hstart; rcases hstart with ⟨h₁, h₂⟩\n      obtain ⟨q₁, hq₁, hR₁⟩ := hsim₁.initial₁ h₁\n      obtain ⟨q₂, hq₂, hR₂⟩ := hsim₂.initial₁ h₂\n      use (q₁, q₂); simp [NFA.product, R, *]\n    · rintro ⟨q₁, q₂⟩ ⟨hst₁, hst₂⟩\n      apply hsim₁.initial₂ at hst₁; obtain ⟨s₁, hi₁, hR₁⟩ := hst₁\n      apply hsim₂.initial₂ at hst₂; obtain ⟨s₂, hi₂, hR₂⟩ := hst₂\n      simp only [nfa, Set.mem_setOf_eq, Prod.exists, inits_spec]\n      have hin₁ : s₁ ∈ m1.m.states := by apply m1.wf.initials_lt hi₁\n      have hin₂ : s₂ ∈ m2.m.states := by apply m2.wf.initials_lt hi₂\n      use ⟨s₁, hin₁⟩, ⟨s₂, hin₂⟩\n      simp_all [R]\n  · rintro ⟨s₁, s₂⟩ ⟨q₁, q₂⟩ a ⟨s₁', s₂'⟩ ⟨hR₁, hR₂⟩ hst\n    simp [nfa, f_spec] at hst; rcases hst with ⟨hst₁, hst₂⟩\n    obtain ⟨q₁', hst₁, hR₁'⟩ := hsim₁.trans_match₁ hR₁ hst₁\n    obtain ⟨q₂', hst₂, hR₂'⟩ := hsim₂.trans_match₁ hR₂ hst₂\n    simp [NFA.product]\n    use q₁', q₂'\n    simp_all [R]\n  · rintro ⟨s₁, s₂⟩ ⟨q₁, q₂⟩ a ⟨q₁', q₂'⟩ ⟨hR₁, hR₂⟩ hst\n    simp [NFA.product] at hst; rcases hst with ⟨hst₁, hst₂⟩\n    obtain ⟨s₁', hst₁, hR₁'⟩ := hsim₁.trans_match₂ hR₁ hst₁ (by simp) (by simp)\n    obtain ⟨s₂', hst₂, hR₂'⟩ := hsim₂.trans_match₂ hR₂ hst₂ (by simp) (by simp)\n    have hin₁ : s₁' ∈ m1.m.states := by apply m1.wf.trans_tgt_lt hst₁;\n    have hin₂ : s₂' ∈ m2.m.states := by apply m2.wf.trans_tgt_lt hst₂\n    simp only [nfa, Set.mem_setOf_eq, Prod.exists, f_spec]\n    use ⟨s₁', hin₁⟩, ⟨s₂', hin₂⟩\n    simp_all [R]", "nesting_depth": 7, "transitive_dep_count": 80, "subset_aristotle": false, "category": "Compiler"}
{"id": 324, "thm_name": "matchVar_appendInl", "thm_stmt": "theorem matchVar_appendInl {w : Var ⟨te⟩ t} :\n    matchVar lets v (.var matchLets matchExpr) w.appendInl ma = some ma' →\n    ∃ args,\n      lets.getPureExpr v\n        = some ⟨_, w, matchExpr.op, matchExpr.ty_eq, matchExpr.eff_le, args, matchExpr.regArgs⟩\n      ∧ matchArg lets matchLets args matchExpr.args ma = some ma'", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Transforms/Rewrite/Match.lean", "imports": ["import LeanMLIR.Framework", "import LeanMLIR.LeanMLIR.Framework.Basic", "import LeanMLIR.Transforms.Rewrite.Mapping", "import LeanMLIR.LeanMLIR.ErasedContext"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "AList.insert", "module": "Mathlib.Data.List.AList"}, {"name": "String", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Valuation.map", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "Sigma.mk", "module": "Init.Core"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}], "used_repo_defs": [{"name": "syntax \"neg\" : MLIR.Pretty.uniform_op", "content": "syntax \"neg\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Mapping", "content": "abbrev Mapping (Γ Δ : Ctxt Ty) : Type :=\n  @AList (Σ t, Var Γ t) (fun x => Var Δ x.1)"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "last", "content": "@[match_pattern]\ndef last (Γ : Ctxt Ty) (t : Ty) : Ctxt.Var (Ctxt.cons t Γ) t :=\n  ⟨0, by admit /- proof elided -/\n  ⟩"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "appendInl", "content": "def appendInl (v : Γ.Var t) : (Γ ++ Δ).Var t :=\n  ⟨v.val, by admit /- proof elided -/\n  ⟩"}, {"name": "sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "effectKind", "content": "def effectKind   := Signature.effectKind ∘ s.signature"}, {"name": "returnTypes", "content": "def returnTypes  := Signature.returnTypes ∘ s.signature"}, {"name": "Expr.ty", "content": "def Expr.ty : Expr d Γ eff [t] → d.Ty := fun _ => t"}, {"name": "Expr.op", "content": "def Expr.op {Γ : Ctxt d.Ty} {eff : EffectKind} {ty} (e : Expr d Γ eff ty) : d.Op :=\n  Expr.casesOn e (fun op _ _ _ _ => op)"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "IsEmpty.exists_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "Option.isSome_none", "module": "Init.Data.Option.Basic"}, {"name": "iff_false", "module": "Init.SimpLemmas"}, {"name": "Option.get_some", "module": "Init.Data.Option.Basic"}, {"name": "Option.isSome_iff_exists", "module": "Init.Data.Option.Lemmas"}, {"name": "forall_exists_index", "module": "Init.PropLemmas"}], "repo_lemmas": [{"name": "eq.ty_eq", "content": "theorem eq.ty_eq {v : Γ.Var t} {w : Γ.Var u} (h : v.eq w) : t = u"}, {"name": "Expr.op_mk", "content": "@[simp]\ntheorem Expr.op_mk {Γ : Ctxt d.Ty} {ty} {eff : EffectKind} (op : d.Op)\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) (DialectSignature.sig op))\n    (regArgs) :\n    (Expr.mk op ty_eq eff_le args regArgs).op = op"}, {"name": "Expr.regArgs_mk", "content": "@[simp]\ntheorem Expr.regArgs_mk {Γ : Ctxt d.Ty} {ty eff op}\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) (DialectSignature.sig op)) (regArgs) :\n    (Expr.mk op ty_eq eff_le args regArgs).regArgs = regArgs"}, {"name": "appendCases_appendInl", "content": "@[simp] theorem appendCases_appendInl (v : Γ.Var t) :\n    appendCases (motive := motive) left right v.appendInl = (left v)"}, {"name": "Expr.args_mk", "content": "@[simp]\ntheorem Expr.args_mk {Γ : Ctxt d.Ty} {ty eff op}\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) (DialectSignature.sig op)) (regArgs) :\n    (Expr.mk op ty_eq eff_le args regArgs).args = args"}], "used_local_defs": [{"name": "MatchVarM", "content": "abbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)"}, {"name": "MatchVar", "content": "abbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit"}, {"name": "MatchVarM.unifyVars", "content": "def MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)"}, {"name": "matchArg", "content": "def matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)"}, {"name": "matchVar", "content": "def matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "MatchVar.liftM_bind_eq_some_iff", "content": "@[simp]\ntheorem MatchVar.liftM_bind_eq_some_iff (x? : Option α)\n    (f : α → MatchVarM Δ Γ β) :\n    ((liftM x? >>= f) mapIn = some mapOut)\n    ↔ ( ∃ h : x?.isSome,\n        f (x?.get h) mapIn = some mapOut )"}], "local_ctx": "import LeanMLIR.Framework\n\nimport LeanMLIR.Transforms.Rewrite.Mapping\n\nopen Ctxt (Var VarSet Valuation Hom)\n\nvariable {d} [DialectSignature d] [DecidableEq d.Ty]\n\nvariable {Γ : Ctxt d.Ty} {ty : d.Ty}\n\nabbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)\n\nabbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit\n\ndef MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)\n\nopen MatchVarM\n\nvariable [DecidableEq d.Op]\n\ndef matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)\n\ndef matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/\n\nend\n\nsection MatchVar\n\nvariable [DecidableEq d.Op] {Γ_in Γ_out Δ_in Δ_out t te}\n          {lets : Lets d Γ_in eff Γ_out} {v : Var Γ_out t}\n          {matchLets : Lets d Δ_in .pure Δ_out}\n          {matchExpr : Expr d Δ_out .pure te}", "target_theorem": "theorem matchVar_appendInl {w : Var ⟨te⟩ t} :\n    matchVar lets v (.var matchLets matchExpr) w.appendInl ma = some ma' →\n    ∃ args,\n      lets.getPureExpr v\n        = some ⟨_, w, matchExpr.op, matchExpr.ty_eq, matchExpr.eff_le, args, matchExpr.regArgs⟩\n      ∧ matchArg lets matchLets args matchExpr.args ma = some ma' :=", "ground_truth_proof": ":= by\n  unfold matchVar\n  simp only [Var.appendCases_appendInl]\n  simp only [MatchVar.liftM_bind_eq_some_iff, forall_exists_index]\n  generalize h_e? : lets.getPureExpr v = e?\n  intro h_isSome h\n  split_ifs at h\n  case neg => contradiction\n  case neg => contradiction\n  case pos h_e? hw =>\n    rcases matchExpr with ⟨mOp, _, _, mArgs, mRegArgs⟩\n    rcases h_e? with ⟨(rfl : _ = mOp), (rfl : _ = mRegArgs)⟩\n    rcases Option.isSome_iff_exists.mp h_isSome with ⟨⟨tys', w', e'⟩, rfl⟩\n    simp only [Option.get_some, Expr.op_mk, Expr.regArgs_mk, Option.some.injEq, Sigma.mk.injEq,\n      Expr.args_mk]\n    obtain rfl : te = tys' := by\n      simpa [e'.ty_eq]\n    rcases e' with ⟨_op, _ty_eq, _eff_le, args, regArgs⟩\n    obtain rfl : w = w' := by\n      simp_all [Option.get_some]\n\n    exact ⟨args, ⟨rfl, by rfl⟩, h⟩", "nesting_depth": 7, "transitive_dep_count": 63, "subset_aristotle": false, "category": "Compiler"}
{"id": 325, "thm_name": "RawCNFA.reverse_spec", "thm_stmt": "lemma RawCNFA.reverse_spec {m : RawCNFA A} (hwf : m.WF) :\n    let m' := m.reverse\n    m'.WF ∧ m'.stateMax = m.stateMax ∧ m'.initials = m.finals ∧ m'.finals = m.initials ∧\n      ∀ s a s', s' ∈ m'.tr s a ↔ s ∈ m.tr s' a", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/Constructions.lean", "imports": ["import Blase.AutoStructs.Worklist", "import Mathlib.Tactic.ApplyFun", "import Mathlib.Data.Fintype.Prod", "import Blase.Blase.AutoStructs.ForLean", "import Blase.Blase.AutoStructs.ForMathlib"], "used_lib_defs": [{"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.emptyWithCapacity", "module": "Std.Data.HashMap.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "LawfulBEq", "module": "Init.Core"}, {"name": "Function.update", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.snd", "module": "Init.Prelude"}, {"name": "Std.HashMap.map", "module": "Std.Data.HashMap.AdditionalOperations"}, {"name": "LawfulHashable", "module": "Init.Data.LawfulHashable"}, {"name": "Inhabited", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "State", "content": "abbrev State := Nat"}, {"name": "RawCNFA.states", "content": "def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax"}, {"name": "RawCNFA.tr", "content": "@[inline]\ndef RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅"}, {"name": "Std.HashSet.toSet", "content": "def Std.HashSet.toSet [BEq α] [Hashable α] (m : HashSet α) : Set α := { x | x ∈ m }"}, {"name": "Std.HashMap.toPFun", "content": "def Std.HashMap.toPFun [BEq α] [Hashable α] (m : HashMap α β) (x : α) : Option β := m[x]?"}], "lib_lemmas": [{"name": "Function.update_apply", "module": "Mathlib.Logic.Function.Basic"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "Std.HashSet.fold_induction", "content": "theorem Std.HashSet.fold_induction [BEq α] [LawfulBEq α] [Hashable α]\n  {f : β → α → β} {m : HashSet α} {motive : β → Set α → Prop} :\n    motive b ∅ →\n    (∀ b x s, x ∉ s → motive b s → motive (f b x) (s ∪ {x})) →\n    motive (m.fold f b) m.toSet"}, {"name": "Std.HashSet.toSet_toList[BEq", "content": "theorem Std.HashSet.toSet_toList[BEq α] [LawfulBEq α] [Hashable α] (m : HashSet α) : m.toSet = { x | x ∈ m.toList }"}, {"name": "Std.HashMap.fold_induction", "content": "theorem Std.HashMap.fold_induction [BEq α] [LawfulBEq α] [DecidableEq α] [Hashable α]\n  {f : γ → α → β → γ} {m : HashMap α β} {motive : γ → (α → Option β) → Prop} :\n    motive b (λ _ ↦ none) →\n    (∀ b x y m, m x = none → motive b m → motive (f b x y) (Function.update m x y)) →\n    motive (m.fold f b) m.toPFun"}, {"name": "Std.HashMap.toPFun_toList[BEq", "content": "theorem Std.HashMap.toPFun_toList[BEq α] [LawfulBEq α] [Hashable α] (m : HashMap α β) :\n    m.toPFun = λ k ↦ m.toList.find? (λ x ↦ x.1 == k) |>.map Prod.snd"}, {"name": "Std.HashMap.mem_of_getElem?", "content": "@[aesop 50% unsafe]\ntheorem Std.HashMap.mem_of_getElem? [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] {m : Std.HashMap K V} {k : K} :\n    m[k]? = some v → k ∈ m"}, {"name": "Std.HashMap.mem_iff_getElem?", "content": "theorem Std.HashMap.mem_iff_getElem? [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] [Inhabited V] {m : Std.HashMap K V} {k : K} :\n     k ∈ m ↔ ∃ v, m[k]? = some v"}], "used_local_defs": [{"name": "RawCNFA.reverse", "content": "def RawCNFA.reverse (m : RawCNFA A) : RawCNFA A :=\n  let m' := { stateMax := m.stateMax, trans := Std.HashMap.emptyWithCapacity m.trans.size, initials := m.finals, finals := m.initials}\n  m.trans.fold (init := m') processState\nwhere\n  processState := fun m' (s, a) ss' =>\n    ss'.fold (init := m') fun m' s' => m'.addTrans a s' s"}], "used_local_lemmas": [{"name": "RawCNFA.reverse_spec_procesState", "content": "lemma RawCNFA.reverse_spec_procesState {m : RawCNFA A} (hwf : m.WF) s₀ a₀ ss' (hs₀ : s₀ ∈ m.states) :\n    let motive m' ss' :=\n      (∀ s ∈ ss', s ∈ m.states) →\n        m'.WF ∧ m'.stateMax = m.stateMax ∧ m'.initials = m.initials ∧ m'.finals = m.finals ∧\n          ∀ s a s', s' ∈ m'.tr s a ↔ (s' ∈ m.tr s a ∨ s' = s₀ ∧ a = a₀ ∧ s ∈ ss')\n    motive (RawCNFA.reverse.processState m (s₀, a₀) ss') ss'.toSet"}], "local_ctx": "import Mathlib.Data.Fintype.Prod\n\nimport Blase.AutoStructs.Worklist\n\nimport Mathlib.Tactic.ApplyFun\n\nopen SetRel\n\nsection sink\n\nvariable {A : Type} [BEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nend sink\n\nsection generic_prod\n\nvariable {α} [BEq α] [Hashable α] [LawfulBEq α]\n\nvariable {β} [BEq β] [Hashable β] [LawfulBEq β]\n\nvariable {S₁ : Finset α} {S₂ : Finset β}\n\nvariable {γ} (f : S₁ → S₂ → γ) (hinj : Function.Injective2 f)\n\nvariable {m₁ : Std.HashSet α} (hm₁ : ∀ s₁ ∈ m₁, s₁ ∈ S₁)\n\nvariable {m₂ : Std.HashSet β} (hm₂ : ∀ s₂ ∈ m₂, s₂ ∈ S₂)\n\nend generic_prod\n\nsection product\n\nvariable {A : Type} [BEq A] [LawfulBEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nend product\n\nsection determinization\n\nvariable {A : Type} [BEq A] [LawfulBEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\ndef RawCNFA.reverse (m : RawCNFA A) : RawCNFA A :=\n  let m' := { stateMax := m.stateMax, trans := Std.HashMap.emptyWithCapacity m.trans.size, initials := m.finals, finals := m.initials}\n  m.trans.fold (init := m') processState\nwhere\n  processState := fun m' (s, a) ss' =>\n    ss'.fold (init := m') fun m' s' => m'.addTrans a s' s", "target_theorem": "lemma RawCNFA.reverse_spec {m : RawCNFA A} (hwf : m.WF) :\n    let m' :=", "ground_truth_proof": ":= m.reverse\n    m'.WF ∧ m'.stateMax = m.stateMax ∧ m'.initials = m.finals ∧ m'.finals = m.initials ∧\n      ∀ s a s', s' ∈ m'.tr s a ↔ s ∈ m.tr s' a := by\n  let motive (m' : RawCNFA A) (trs : (State × A) → Option (Std.HashSet State)) :=\n    (∀ s a ss', trs (s, a) = some ss' → s ∈ m.states ∧ ∀ s' ∈ ss', s' ∈ m.states) →\n    m'.WF ∧ m'.stateMax = m.stateMax ∧ m'.initials = m.finals ∧ m'.finals = m.initials ∧\n      ∀ s a s', s' ∈ m'.tr s a ↔ ∃ ss', trs (s', a) = some ss' ∧ s ∈ ss' -- s' -a-> s is a transition\n  suffices h : motive (m.reverse) (m.trans.toPFun) by\n    specialize h (by rintro s a ss' heq; exact ⟨hwf.trans_src_lt _ (Std.HashMap.mem_of_getElem? heq), fun s' hin => hwf.trans_tgt_lt_internal heq hin⟩)\n    simp_all\n\n  apply Std.HashMap.fold_induction\n  · simp [motive, tr]\n    constructor <;> simp_all [tr]\n  · rintro m' ⟨s, a⟩ ss' trs hnew ih\n    rintro hsts\n    specialize ih (by rintro s₀ a₀ ss₀ heq; apply hsts _ a₀; simp [Function.update_apply]; split <;> simp_all)\n    rcases ih with ⟨hwf', hm', his', hfs', htrs'⟩\n    specialize hsts s a ss' (by simp)\n    have hs : s ∈ m'.states := by simp_all [states]\n    have hss' : ∀ s ∈ ss'.toSet, s ∈ m'.states := by simp_all [states]\n    obtain ⟨hwf'', hm'', his'', hfs'', htrs''⟩ := reverse_spec_procesState hwf' s a ss' hs hss'\n    simp_all only [true_and]\n    rintro s₁ b s₂\n    by_cases hcond : s₂ = s ∧ b = a\n    · rcases hcond with ⟨rfl, rfl, hin⟩\n      simp_all\n    · simp_all", "nesting_depth": 3, "transitive_dep_count": 37, "subset_aristotle": false, "category": "Compiler"}
{"id": 326, "thm_name": "denote_rewriteAt", "thm_stmt": "theorem denote_rewriteAt [LawfulMonad d.m]\n    {lhs rhs : Com d Γ₁ .pure t₁}\n    (hl : lhs.denote = rhs.denote)\n    {hlhs : ∀ t (v : Var Γ₁ t), ⟨t, v⟩ ∈ lhs.vars}\n    {pos : ℕ} {target : Com d Γ₂ eff t₂}\n    {rew : Com d Γ₂ eff t₂}\n    (hrew : rew ∈ rewriteAt lhs rhs hlhs pos target) :\n    rew.denote = target.denote", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Transforms/Rewrite/Rewrite.lean", "imports": ["import LeanMLIR.Transforms.Rewrite.Match", "import LeanMLIR.Framework", "import LeanMLIR.LeanMLIR.Transforms.Rewrite.Match", "import LeanMLIR.LeanMLIR.Framework.Zipper", "import LeanMLIR.LeanMLIR.Framework.Basic", "import LeanMLIR.Framework.Zipper"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "AList.insert", "module": "Mathlib.Data.List.AList"}, {"name": "Valuation.map", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "Valuation.mk", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"neg\" : MLIR.Pretty.uniform_op", "content": "syntax \"neg\" : MLIR.Pretty.uniform_op\n\nsyntax \"llvm.and\"     : MLIR.Pretty.uniform_op\n\nsyntax \"llvm.ashr\"    : MLIR.Pretty.exact_op\n\nsyntax \"llvm.add\"     : MLIR.Pretty.overflow_op\n\nsyntax \"llvm.return\"  : MLIR.Pretty.uniform_op"}, {"name": "notation:50 x \" ≤ₛ \" y => BitVec.sle x y", "content": "notation:50 x \" ≤ₛ \" y => BitVec.sle x y"}, {"name": "notation:50 x \" >ᵤ \" y => BitVec.ult y x", "content": "notation:50 x \" >ᵤ \" y => BitVec.ult y x"}, {"name": "notation:50 x \" ≥ᵤ \" y => BitVec.ule y x", "content": "notation:50 x \" ≥ᵤ \" y => BitVec.ule y x"}, {"name": "notation:50 x \" <ᵤ \" y => BitVec.ult x y", "content": "notation:50 x \" <ᵤ \" y => BitVec.ult x y"}, {"name": "notation:50 x \" ≥ₛ \" y => BitVec.sle y x", "content": "notation:50 x \" ≥ₛ \" y => BitVec.sle y x"}, {"name": "notation:50 x \" <ₛ \" y => BitVec.slt x y", "content": "notation:50 x \" <ₛ \" y => BitVec.slt x y"}, {"name": "notation:50 x \" >ₛ \" y => BitVec.slt y x", "content": "notation:50 x \" >ₛ \" y => BitVec.slt y x"}, {"name": "notation:50 x \" ≤ᵤ \" y => BitVec.ule x y", "content": "notation:50 x \" ≤ᵤ \" y => BitVec.ule x y"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $resName:mlir_op_operand = $name:InstCombine.cmp_op_name $x, $y $[: $t]?) => do\n    let some opName := extractOpName name.raw\n      | Macro.throwUnsupported\n    let t ← t.getDM `(mlir_type| _)\n    `(mlir_op| $resName:mlir_op_operand = $opName ($x, $y) : ($t, $t) -> (i1) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $resName:mlir_op_operand = $name:InstCombine.int_cast_op $x : $t to $t') => do\n    let some opName := extractOpName name.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $resName:mlir_op_operand = $opName ($x) : ($t) -> $t')"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant( $x $[: $inner_type]?)\n      $[: $outer_type]? ) => do\n       \n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      let inner_type := inner_type.getD outer_type\n      `(mlir_op| $res:mlir_op_operand = \"llvm.mlir.constant\"()\n          {value = $x:neg_num : $inner_type} : () -> ($outer_type) )\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant( ${ $x:term }) $[: $t]?) => do\n      let t ← t.getDM `(mlir_type| _)\n      let x ← `(MLIR.AST.AttrValue.int $x [mlir_type| $t])\n      `(mlir_op| $res:mlir_op_operand = \"llvm.mlir.constant\"() {value = $$($x) } : () -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (true) $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (1 : i1) : i1)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (false) $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (0 : i1) : i1)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant $x $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant($x $[: $t]?) $[: $t]?)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant ${ $x:term } $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant($$($x) $[: $t]?) $[: $t]?)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.icmp $p $x, $y $[: $t]?) => do\n    let t ← t.getDM `(mlir_type| _)\n    match p.getString with\n      | \"eq\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.eq\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ne\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ne\" ($x, $y) : ($t, $t) -> (i1))\n      | \"slt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.slt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sle\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sle\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sgt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sgt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sge\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sge\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ult\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ult\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ule\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ule\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ugt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ugt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"uge\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.uge\" ($x, $y) : ($t, $t) -> (i1))\n      | _ => Macro.throwErrorAt p s!\"unexpected predicate {p.getString}\""}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.select $c, $x, $y $[: $t]?) => do\n    let t ← t.getDM `(mlir_type| _)\n    `(mlir_op| $res:mlir_op_operand = \"llvm.select\" ($c, $x, $y) : (i1, $t, $t) -> ($t))"}, {"name": "Zipper", "content": "structure Zipper (Γ_in : Ctxt d.Ty) (eff : EffectKind) (tys : List d.Ty) where\n   \n  {Γ_mid : Ctxt d.Ty}\n   \n  top : Lets d Γ_in eff Γ_mid\n   \n  bot : Com d Γ_mid eff tys"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "FlatCom", "content": "structure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts"}, {"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "matchArgRes", "content": "def matchArgRes (lets : Lets d Γ_in eff Γ_out)\n    (matchLets : Lets d Δ_in .pure Δ_out)\n    (vs : HVector Γ_out.Var ts)\n    (ws : HVector Δ_out.Var ts) :\n    Option (MatchArgResult lets matchLets vs ws ∅) := do\n  (matchArg lets matchLets vs ws ∅).attach.map fun ⟨⟨_, _⟩, h⟩ => .mk h"}, {"name": "matchArg", "content": "def matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)"}, {"name": "matchVar", "content": "def matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/"}, {"name": "MatchVar", "content": "abbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit"}, {"name": "MatchVarM", "content": "abbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)"}, {"name": "Mapping", "content": "abbrev Mapping (Γ Δ : Ctxt Ty) : Type :=\n  @AList (Σ t, Var Γ t) (fun x => Var Δ x.1)"}, {"name": "MatchVarM.unifyVars", "content": "def MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "last", "content": "@[match_pattern]\ndef last (Γ : Ctxt Ty) (t : Ty) : Ctxt.Var (Ctxt.cons t Γ) t :=\n  ⟨0, by admit /- proof elided -/\n  ⟩"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "MatchArgResult", "content": "def MatchArgResult := { mapOut : Mapping _ _ //\n  ∃ (mapIn' mapOut' : Mapping _ _),\n    mapIn.entries ⊆ mapIn'.entries\n    ∧ mapOut'.entries ⊆ mapOut.entries\n    ∧ matchArg lets matchLets vs ws mapIn' = some ((), mapOut') }"}, {"name": "toHom", "content": "def toHom (d : Diff Γ₁ Γ₂) : Hom Γ₁ Γ₂ :=\n  fun _ v => ⟨v.val + d.val, d.property v.property⟩"}, {"name": "Hom", "content": "abbrev Hom (Γ Γ' : Ctxt Ty) := ⦃t : Ty⦄ → Γ.Var t → Γ'.Var t"}, {"name": "Diff", "content": "def Diff (Γ₁ Γ₂ : Ctxt Ty) : Type :=\n  {d : Nat // Diff.Valid Γ₁ Γ₂ d}"}, {"name": "Diff.Valid", "content": "@[simp]\nabbrev Diff.Valid (Γ₁ Γ₂ : Ctxt Ty) (d : Nat) : Prop :=\n  ∀ {i t}, Γ₁[i]? = some t → Γ₂[i+d]? = some t"}, {"name": "Expr.pdenoteOp", "content": "@[simp] abbrev Expr.pdenoteOp :\n    Expr d Γ .pure ty → Γ.Valuation → (HVector toType ty) :=\n  Expr.denoteOp"}, {"name": "Expr.ty", "content": "def Expr.ty : Expr d Γ eff [t] → d.Ty := fun _ => t"}, {"name": "Expr.denoteOp", "content": "def Expr.denoteOp (e : Expr d Γ eff ty) (V : Γ.Valuation) :\n    eff.toMonad d.m (HVector toType ty) :=\n  EffectKind.liftEffect e.eff_le <| cast (by admit /- proof elided -/\n  ) <|\n    DialectDenote.denote e.op (e.args.map V) e.regArgs.denote"}, {"name": "DialectDenote", "content": "class DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))"}, {"name": "Lets.denote", "content": "def Lets.denote [DialectSignature d] [DialectDenote d] {Γ₂}\n    (lets : Lets d Γ₁ eff Γ₂) (V : Valuation Γ₁) : (eff.toMonad d.m <| Valuation Γ₂) :=\n  match lets with\n  | .nil          => return V\n  | .var lets' e  => lets'.denote V >>= e.denote"}, {"name": "sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "regSig", "content": "def regSig       := Signature.regSig ∘ s.signature"}, {"name": "RegionSignature", "content": "abbrev RegionSignature Ty := List (Ctxt Ty × List Ty)"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "effectKind", "content": "def effectKind   := Signature.effectKind ∘ s.signature"}, {"name": "returnTypes", "content": "def returnTypes  := Signature.returnTypes ∘ s.signature"}, {"name": "Dialect", "content": "structure Dialect where\n  (Op : Type)\n  (Ty : Type)\n  (m : Type → Type := Id)"}, {"name": "Op", "content": "inductive Op (q : Nat) (n : Nat)\n  | add : Op q n\n  | sub : Op q n\n  | mul : Op q n\n  | mul_constant : Op q n\n    \n    \n  | leading_term : Op q n\n  | monomial : Op q n\n  | monomial_mul : Op q n\n  | from_tensor : Op q n\n  | to_tensor : Op q n\n  | const (c : R q n) : Op q n\n  | const_int (c : Int) : Op q n\n  | const_idx (i : Nat) : Op q n"}, {"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "Valuation.instAppendHVector", "content": "@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V"}, {"name": "HVector.denote", "content": "def HVector.denote :\n    {l : RegionSignature d.Ty} → (T : HVector (fun t => Com d t.1 .impure t.2) l) →\n    HVector (fun t => t.1.Valuation → EffectKind.impure.toMonad d.m (HVector toType t.2)) l\n  | _, .nil => HVector.nil\n  | _, .cons v vs => HVector.cons (v.denote) (HVector.denote vs)"}, {"name": "FlatCom.denote", "content": "@[simp] abbrev FlatCom.denote [DialectDenote d]\n    (flatCom : FlatCom d Γ eff Γ_out ts)\n    (V : Γ.Valuation) : eff.toMonad d.m (HVector toType ts) :=\n  flatCom.lets.denote V >>= (return flatCom.rets.map ·)"}, {"name": "RegionSignature.map", "content": "def RegionSignature.map (f : Ty → Ty') : RegionSignature Ty → RegionSignature Ty' :=\n  List.map fun ⟨Γ, ty⟩ => (Γ.map f, ty.map f)"}, {"name": "Signature.map", "content": "def Signature.map (f : Ty → Ty') : Signature Ty → Signature Ty' :=\n  fun sig => {\n    sig         := sig.sig.map f\n    regSig      := sig.regSig.map f\n    returnTypes := sig.returnTypes.map f\n  }"}, {"name": "map", "content": "def map (f : Ty₁ → Ty₂) : Ctxt Ty₁ → Ctxt Ty₂ :=\n  ofList ∘ (List.map f) ∘ toList"}, {"name": "Expr.denote", "content": "def Expr.denote {ty} (e : Expr d Γ eff ty) (V : Valuation Γ) :\n    eff.toMonad d.m (e.outContext.Valuation) :=\n  match e with\n  | ⟨op, ty_eq, heff, args, regArgs⟩ => do\n      let argsDenote := args.map V\n      let val ← EffectKind.liftEffect heff <| DialectDenote.denote op argsDenote regArgs.denote\n      return (val ++ V).cast (by admit /- proof elided -/\n      )"}, {"name": "Expr.op", "content": "def Expr.op {Γ : Ctxt d.Ty} {eff : EffectKind} {ty} (e : Expr d Γ eff ty) : d.Op :=\n  Expr.casesOn e (fun op _ _ _ _ => op)"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) "}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "liftEffect", "content": "def liftEffect [Pure m] {e1 e2 : EffectKind} {α : Type}\n    (hle : e1 ≤ e2) (v1 : e1.toMonad m α) : e2.toMonad m α :=\n  match e1, e2, hle with\n    | .pure, .pure, _ | .impure, .impure, _ => v1\n    | .pure, .impure, _ => Pure.pure v1"}, {"name": "toMonad", "content": "def toMonad (e : EffectKind) (m : Type → Type) : Type → Type :=\n  match e with\n  | pure => Id\n  | impure => m"}, {"name": "Com.denote", "content": "def Com.denote : Com d Γ eff ty → (Γv : Valuation Γ) →\n    eff.toMonad d.m (HVector toType ty)\n  | .rets vs, Γv     => pure (vs.map Γv)\n  | .var e body, V => e.denote V >>= body.denote"}, {"name": "Com.ty", "content": "def Com.ty : Com d Γ eff [t] → d.Ty := fun _ => t"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "Expr.regArgs", "content": "def Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)"}, {"name": "Regions", "content": "abbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig"}, {"name": "FlatCom.denoteLets", "content": "def FlatCom.denoteLets (flatCom : FlatCom d Γ eff Γ_out t) (Γv : Γ.Valuation) :\n    eff.toMonad d.m <| Γ_out.Valuation :=\n  flatCom.lets.denote Γv"}, {"name": "Com.toLets", "content": "def Com.toLets (com : Com d Γ eff t) : Lets d Γ eff com.outContext :=\n  Lets.nil.addComToEnd com"}, {"name": "Com.denoteLets", "content": "def Com.denoteLets : (com : Com d Γ eff ty) → (Γv : Valuation Γ) →\n    eff.toMonad d.m (com.outContext.Valuation)\n  | .rets _, V => pure V\n  | .var e body, V =>\n      e.denote V >>= body.denoteLets >>= fun V =>\n        return V.cast (by admit /- proof elided -/\n        )"}, {"name": "cast", "content": "def cast (h_eq : ty₁ = ty₂) : Γ.Var ty₁ → Γ.Var ty₂\n  | ⟨i, h⟩ => ⟨i, h_eq ▸ h⟩"}, {"name": "cast", "content": "def cast (h₁ : Γ = Γ') (h₂ : Δ = Δ') : Diff Γ Δ → Diff Γ' Δ'\n  | ⟨n, h⟩ => ⟨n, by admit /- proof elided -/\n  ⟩"}, {"name": "toCom", "content": "def toCom (zip : Zipper d Γ_in eff ty) : Com d Γ_in eff ty :=\n  go zip.top zip.bot\n  where\n    go : {Γ_mid : _} → Lets d Γ_in eff Γ_mid → Com d Γ_mid eff ty → Com d Γ_in eff ty\n      | _, .nil, com          => com\n      | _, .var body e, com  => go body (.var e com)"}, {"name": "denote", "content": "def denote (zip : Zipper d Γ_in eff tys) (V_in : Valuation Γ_in) :\n    eff.toMonad d.m (HVector toType tys) :=\n  (zip.top.denote V_in) >>= zip.bot.denote"}, {"name": "Expr.returnVars", "content": "def Expr.returnVars (e : Expr d Γ eff tys) : HVector e.outContext.Var tys :=\n  .ofFn _ _ <| fun i => (Var.ofFin i).appendInl"}, {"name": "ofFin", "content": "def ofFin (i : Fin Γ.length) : Γ.Var (Γ[i]) :=\n  ⟨i.val, by admit /- proof elided -/\n  ⟩"}, {"name": "Com.returnVars", "content": "def Com.returnVars : (com : Com d Γ eff ts) → HVector (Var com.outContext) ts\n  | .rets vs => vs\n  | .var _ body => body.returnVars"}, {"name": "MatchVarResult.toHom", "content": "def MatchVarResult.toHom\n    (map : MatchVarResult lets v matchLets w mapIn)\n    (hvars : ∀ t (v : Var Δ_in t), ⟨t, v⟩ ∈ matchLets.vars w) :\n    Δ_in.Hom Γ_out :=\n  map.val.toHom <| map.isTotal_of hvars"}, {"name": "MatchVarResult.isTotal_of", "content": "def MatchVarResult.isTotal_of\n    (map : MatchVarResult lets v matchLets w mapIn)\n    (hvars : ∀ t (v : Var Δ_in t), ⟨t, v⟩ ∈ matchLets.vars w) :\n    map.val.IsTotal :="}, {"name": "MatchArgResult.isTotal_of", "content": "def MatchArgResult.isTotal_of\n    (map : MatchArgResult lets matchLets vs ws mapIn)\n    (hvars : ∀ t (v : Var Δ_in t), ⟨t, v⟩ ∈ matchLets.varsOfVec ws) :\n    map.val.IsTotal :="}, {"name": "MatchArgResult.toHom", "content": "def MatchArgResult.toHom\n    (map : MatchArgResult lets matchLets vs ws mapIn)\n    (hvars : ∀ t (v : Var Δ_in t), ⟨t, v⟩ ∈ matchLets.varsOfVec ws) :\n    Δ_in.Hom Γ_out :=\n  map.val.toHom <| map.isTotal_of hvars\n\nvariable\n  {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty}\n  {lets : Lets d Γ_in eff Γ_out}\n  {matchTy}\n  {v : Var Γ_out matchTy}\n  {matchLets : Lets d Δ_in .pure Δ_out}\n  {w : Var Δ_out matchTy}\nin"}, {"name": "Valuation.comap", "content": "def Valuation.comap {Γi Γo : Ctxt Ty} (Γiv: Γi.Valuation) (hom : Ctxt.Hom Γo Γi) : Γo.Valuation :=\n  fun _to vo => Γiv (hom vo)"}, {"name": "map", "content": "def map (f : ∀ (a : α), A a → B a) :\n    ∀ {l : List α}, HVector A l → HVector B l\n  | [],   .nil        => .nil\n  | t::_, .cons a as  => .cons (f t a) (map f as)"}, {"name": "MatchVarResult", "content": "def MatchVarResult := { mapOut : Mapping _ _ //\n  ∃ (mapIn' mapOut' : Mapping _ _),\n    mapIn.entries ⊆ mapIn'.entries\n    ∧ mapOut'.entries ⊆ mapOut.entries\n    ∧ matchVar lets v matchLets w mapIn' = some ((), mapOut') }"}, {"name": "toArgResult", "content": "noncomputable def toArgResult\n    (mapOut : MatchVarResult lets v (.var matchLets matchExpr) w.appendInl mapIn) :\n    let args := mapOut.getPureExpr_eq_some.choose\n    MatchArgResult lets matchLets args matchExpr.args mapIn :=\n  ⟨mapOut.1, by admit /- proof elided -/\n  ⟩"}, {"name": "appendInl", "content": "def appendInl (v : Γ.Var t) : (Γ ++ Δ).Var t :=\n  ⟨v.val, by admit /- proof elided -/\n  ⟩"}, {"name": "eqvVarLeft", "content": "def eqvVarLeft  :\n    MatchVarResult lets v (.var matchLets matchExpr) w.appendInr ma\n    ≃ MatchVarResult lets v matchLets w ma where\n  toFun := fun ⟨x, h⟩ => ⟨x, by admit /- proof elided -/\n  ⟩\n  invFun := fun ⟨x, h⟩ => ⟨x, by admit /- proof elided -/\n  ⟩"}, {"name": "Expr.changeVars", "content": "def Expr.changeVars (varsMap : Γ.Hom Γ') {ty} (e : Expr d Γ eff ty) :\n    Expr d Γ' eff ty :=\n  ⟨e.op, e.ty_eq, e.eff_le, e.args.map varsMap, e.regArgs⟩"}, {"name": "Com.changeVars", "content": "def Com.changeVars : Com d Γ eff ty →\n    (varsMap : Γ.Hom Γ') →\n    Com d Γ' eff ty\n  |  .rets e => fun varsMap => .rets (e.map varsMap)\n  |  .var e body => fun varsMap => .var (e.changeVars varsMap)\n      (body.changeVars (fun _ v => varsMap.append v))"}, {"name": "Hom.append", "content": "def Hom.append {ζ : Ctxt Ty} (f : Γ.Hom Δ) : Hom (ζ ++ Γ) (ζ ++ Δ) :=\n  fun _ => Var.appendCases\n    (fun v => v.appendInl)\n    (fun v => (f v).appendInr)"}, {"name": "append", "content": "def append (d₁ : Diff Γ₁ Γ₂) (d₂ : Diff Γ₂ Γ₃) : Diff Γ₁ Γ₃ :=\n  {val := d₁.val + d₂.val,  property := append_valid d₁.property d₂.property}"}, {"name": "insertPureCom", "content": "def insertPureCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy)\n    (newCom : Com d zip.Γ_mid .pure newTy) : Zipper d Γ_in eff ty :=\n  zip.insertCom vs (newCom.castPureToEff eff)"}, {"name": "insertCom", "content": "def insertCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy) (newCom : Com d zip.Γ_mid eff newTy) :\n    Zipper d Γ_in eff ty :=\n  let top := zip.top.addComToEnd newCom\n  \n  let bot := zip.bot.changeVars <| newCom.outContextHom.with vs newCom.returnVars\n  \n  \n  { top, bot }"}, {"name": "Hom.with", "content": "def Hom.with [DecidableEq Ty] {Γ₁ Γ₂ : Ctxt Ty} (f : Γ₁.Hom Γ₂) {ts}\n    (v₁ : HVector Γ₁.Var ts) (v₂ : HVector Γ₂.Var ts) : Γ₁.Hom Γ₂ :=\n  fun _ w =>\n    match v₁.idxOf? w with\n    | none => f w\n    | some ⟨i, h⟩ => (v₂.get i).cast h"}, {"name": "Com.outContextHom", "content": "def Com.outContextHom (com : Com d Γ eff t) : Γ.Hom com.outContext :=\n  com.outContextDiff.toHom"}, {"name": "Com.outContextDiff", "content": "def Com.outContextDiff (com : Com d Γ eff ts) : Γ.Diff com.outContext :=\n  ⟨com.bvars, by admit /- proof elided -/\n      ⟩"}, {"name": "Expr.bvars", "content": "@[simp, grind=] def Expr.bvars (e : Expr d Γ eff Δ) : Nat :=\n  (DialectSignature.returnTypes e.op).length"}, {"name": "Com.bvars", "content": "def Com.bvars : Com d Γ eff t → Nat :=\n  Com.rec'\n           (fun _ => 0)\n      (fun e _body bodySize => e.bvars + bodySize)"}, {"name": "castCtxt", "content": "def castCtxt (h_eq : Γ = Δ) : Γ.Var ty → Δ.Var ty\n  | ⟨i, h⟩ => ⟨i, h_eq ▸ h⟩"}, {"name": "Com.castPureToEff", "content": "def Com.castPureToEff (eff : EffectKind) : Com d Γ .pure t → Com d Γ eff t :=\n  changeEffect (EffectKind.pure_le eff)"}, {"name": "Com.changeEffect", "content": "def Com.changeEffect {eff₁ eff₂ : EffectKind} (h : eff₁ ≤ eff₂) :\n    Com d Γ eff₁ t → Com d Γ eff₂ t := fun com =>\n  Com.rec' (motive := @fun Γ _ => eff₁ ≤ eff₂ → Com d Γ eff₂ t)\n            (fun v _h               => rets v)\n      (fun e _body castBody h => var (e.changeEffect h) (castBody h))\n    com h"}, {"name": "Expr.changeEffect", "content": "def Expr.changeEffect {eff₁ eff₂ : EffectKind} (h : eff₁ ≤ eff₂) :\n    Expr d Γ eff₁ t → Expr d Γ eff₂ t\n  | Expr.mk op ty_eq eff_le args regArgs =>\n    have heff : DialectSignature.effectKind op ≤ eff₂ := by admit /- proof elided -/"}, {"name": "com", "content": "def com := mkCom (d := InstCombine.MetaLLVM 0) bb0 |>.toOption |>.get (by admit /- proof elided -/\n)"}, {"name": "bb0", "content": "def bb0 : Region 0 := [mlir_region|\n{\n  ^bb0(%arg0: i32):\n    %0 = llvm.mlir.constant(8) : i32\n    %1 = llvm.mlir.constant(31) : i32\n    %2 = llvm.ashr %arg0, %1 : i32\n    %3 = llvm.and %2, %0 : i32\n    %4 = llvm.add %3, %2 : i32\n    llvm.return %4 : i32\n  }]"}, {"name": "Region", "content": "structure Region where\n  (name: String)\n  (args: List <| TypedSSAVal φ)\n  (ops: List Op)"}, {"name": "MetaLLVM", "content": "abbrev MetaLLVM (φ : Nat) : Dialect where\n  Op := MOp φ\n  Ty := MTy φ"}, {"name": "Ty", "content": "@[deprecated \"Use `LLVM.Ty` instead\" (since:=\"2025-04-30\")] abbrev Ty := LLVM.Ty"}, {"name": "Op", "content": "@[deprecated \"Use `LLVM.Op` instead\" (since:=\"2025-04-30\")] abbrev Op := LLVM.Op"}, {"name": "MOp", "content": "inductive MOp (φ : Nat) : Type\n  | unary   (w : Width φ) (op : MOp.UnaryOp φ) :  MOp φ\n  | binary  (w : Width φ) (op : MOp.BinaryOp) :  MOp φ\n  | select  (w : Width φ) : MOp φ\n  | icmp    (c : IntPred) (w : Width φ) : MOp φ\n   \n  | const (w : Width φ) (val : ℤ) : MOp φ\nderiving Repr, DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "binary", "content": "@[match_pattern] abbrev binary (w : Nat) (op : MOp.BinaryOp) : LLVM.Op :=\n  MOp.binary (.concrete w) op"}, {"name": "MOp.BinaryOp", "content": "inductive MOp.BinaryOp : Type\n  | and\n  | or   (disjoint : DisjointFlag := {disjoint := false} )\n  | xor\n  | shl  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | lshr (exact : ExactFlag := {exact := false} )\n  | ashr (exact : ExactFlag := {exact := false} )\n  | urem\n  | srem\n  | add  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | mul  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | sub  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | sdiv (exact : ExactFlag := {exact := false} )\n  | udiv (exact : ExactFlag := {exact := false} )\nderiving DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "LLVM", "content": "def LLVM : Dialect where\n  Op := MOp 0\n  Ty := MTy 0"}, {"name": "MTy", "content": "inductive MTy (φ : Nat)\n  | bitvec (w : Width φ) : MTy φ\n  deriving DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "Width", "content": "abbrev Width φ := ConcreteOrMVar Nat φ"}, {"name": "ConcreteOrMVar", "content": "inductive ConcreteOrMVar (α : Type u) (φ : Nat)\n  | concrete (a : α)\n  | mvar (i : Fin φ)\n  deriving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "const", "content": "@[match_pattern] abbrev const (w : Nat) (val : ℤ) : LLVM.Op        := MOp.const (.concrete w) val"}, {"name": "MOp.UnaryOp", "content": "inductive MOp.UnaryOp (φ : Nat) : Type\n  | neg\n  | not\n  | copy\n  | freeze\n  | trunc (w' : Width φ) (noWrapFlags : NoWrapFlags := {nsw := false, nuw := false} )\n  | zext  (w' : Width φ) (nneg : NonNegFlag := {nneg := false} )\n  | sext  (w' : Width φ)\nderiving Repr, DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "select", "content": "@[simp_llvm_option]\ndef select {w : Nat} (c? : IntW 1) (x? y? : IntW w ) : IntW w := do\n  let c ← c?\n  if c = 1#1 then x? else y?"}, {"name": "IntW", "content": "def IntW w := PoisonOr <| BitVec w"}, {"name": "PoisonOr", "content": "structure PoisonOr (α : Type) where\n  val : α\n  poisonous : Bool\nderiving Inhabited, DecidableEq"}, {"name": "icmp", "content": "@[simp_llvm_option]\ndef icmp {w : Nat} (c : IntPred) (x y : IntW w) : IntW 1 := do\n  let x' ← x\n  let y' ← y\n  icmp? c x' y'"}, {"name": "icmp?", "content": "@[simp_llvm]\ndef icmp? {w : Nat} (c : IntPred) (x y : BitVec w) : IntW 1 :=\n  .value ↑(icmp' c x y)"}, {"name": "IntPred", "content": "inductive IntPred where\n  | eq\n  | ne\n  | ugt\n  | uge\n  | ult\n  | ule\n  | sgt\n  | sge\n  | slt\n  | sle\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "icmp'", "content": "@[simp_llvm]\ndef icmp' {w : Nat} (c : IntPred) (x y : BitVec w) : Bool :=\n  match c with\n    | .eq => (x == y)\n    | .ne => (x != y)\n    | .sgt => (x >ₛ y)\n    | .sge => (x ≥ₛ y)\n    | .slt => (x <ₛ y)\n    | .sle => (x ≤ₛ y)\n    | .ugt => (x >ᵤ y)\n    | .uge => (x ≥ᵤ y)\n    | .ult => (x <ᵤ y)\n    | .ule => (x ≤ᵤ y)"}, {"name": "mkCom", "content": "def mkCom [TransformTy d φ] [TransformExpr d φ] [TransformReturn d φ]\n  (reg : MLIR.AST.Region φ) :\n  ExceptM d (Σ (Γ : Ctxt d.Ty) (eff : EffectKind) (ty : _), Com d Γ eff ty) :=\n  match reg.ops with\n  | [] => throw <| .generic \"Ill-formed region (empty)\"\n  | coms => BuilderM.runWithEmptyMapping <| do\n    let Γ ← declareBindings ∅ reg.args\n    let com ← mkComHelper Γ coms\n    return ⟨Γ, com⟩"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "TyDenote.toType", "content": "notation \"⟦\" x \"⟧\" => TyDenote.toType x"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Option.mem_def", "module": "Init.Data.Option.Instances"}, {"name": "Option.bind_eq_bind", "module": "Init.Data.Option.Lemmas"}, {"name": "Option.bind_eq_some_iff", "module": "Init.Data.Option.Lemmas"}, {"name": "Option.dite_none_right_eq_some", "module": "Init.Data.Option.Lemmas"}, {"name": "Option.pure_def", "module": "Init.Data.Option.Lemmas"}, {"name": "cast_eq", "module": "Init.Core"}], "repo_lemmas": [{"name": "Com.denoteLets_eq", "content": "theorem Com.denoteLets_eq {com : Com d Γ eff t} : com.denoteLets = com.toLets.denote"}, {"name": "Lets.denote_var", "content": "@[simp] theorem Lets.denote_var {lets : Lets d Γ_in eff Γ_out} {e : Expr d Γ_out eff t} :\n    (lets.var e).denote = fun V_in => lets.denote V_in >>= e.denote"}, {"name": "denote_toCom", "content": "@[simp] theorem denote_toCom [LawfulMonad d.m] (zip : Zipper d Γ_in eff ty) :\n    zip.toCom.denote = zip.denote"}, {"name": "Com.denoteLets_returnVars", "content": "@[simp] theorem Com.denoteLets_returnVars (c : Com d Γ .pure tys) (V : Valuation Γ) :\n    c.returnVars.map (c.denoteLets V) = c.denote V"}, {"name": "Id.bind_eq'", "content": "theorem Id.bind_eq' (x : Id α) (f : α → id β) : x >>= f = f x"}, {"name": "Id.pure_eq'", "content": "theorem Id.pure_eq' (a : α) : (pure a : Id α) = a"}, {"name": "denote_matchLets_of", "content": "theorem denote_matchLets_of\n    (map : MatchArgResult lets matchLets vs ws mapIn)\n    (hvars : ∀ t (v : Var Δ_in t), ⟨t, v⟩ ∈ matchLets.varsOfVec ws)\n    (V : lets.ValidDenotation) :\n    ws.map (matchLets.denote (V.val.comap <| map.toHom h)) = vs.map V.val"}, {"name": "denote_matchArg", "content": "theorem denote_matchArg\n    {vs ws : HVector (Var _) ts}\n    (mapOut : MatchArgResult lets matchLets vs ws mapIn)\n    (V : lets.ValidDenotation) :\n    HVector.map (matchLets.denote (mapOut.val.mapValuation V.val)) ws = HVector.map (V.val) vs"}, {"name": "HVector.map_eq_map_of_matchArg", "content": "theorem HVector.map_eq_map_of_matchArg\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}\n    {ma : Mapping Δ_in Γ_out}\n    {l : List d.Ty} {args₁ : HVector _ l} {args₂ : HVector _ l}\n    (mapOut : MatchArgResult lets matchLets args₁ args₂ ma)\n    (f₁ f₂ : (t : d.Ty) → Var _ t → ⟦t⟧)\n    (hf : ∀ {t v₁ v₂},\n      (mapOut' : MatchVarResult lets v₁ matchLets v₂ ma)\n      → mapOut'.val = mapOut.val\n      → f₂ t v₂ = f₁ t v₁) :\n    HVector.map f₂ args₂ = HVector.map f₁ args₁"}, {"name": "denote_matchVar", "content": "theorem denote_matchVar\n    {v w : Var _ t}\n    (mapOut : MatchVarResult lets v matchLets w mapIn)\n    (V : lets.ValidDenotation) :\n    (matchLets.denote (mapOut.val.mapValuation V.val) w)\n    = V.val v"}, {"name": "getPureExpr_eq_some", "content": "theorem getPureExpr_eq_some\n    (mapOut : MatchVarResult lets v (.var matchLets matchExpr) w.appendInl mapIn) :\n    ∃ args, lets.getPureExpr v = some ⟨te, w, ⟨\n        matchExpr.op,\n        matchExpr.ty_eq,\n        matchExpr.eff_le,\n        args,\n        matchExpr.regArgs\n      ⟩⟩"}, {"name": "matchVar_appendInl", "content": "theorem matchVar_appendInl {w : Var ⟨te⟩ t} :\n    matchVar lets v (.var matchLets matchExpr) w.appendInl ma = some ma' →\n    ∃ args,\n      lets.getPureExpr v\n        = some ⟨_, w, matchExpr.op, matchExpr.ty_eq, matchExpr.eff_le, args, matchExpr.regArgs⟩\n      ∧ matchArg lets matchLets args matchExpr.args ma = some ma'"}, {"name": "MatchVar.liftM_bind_eq_some_iff", "content": "@[simp]\ntheorem MatchVar.liftM_bind_eq_some_iff (x? : Option α)\n    (f : α → MatchVarM Δ Γ β) :\n    ((liftM x? >>= f) mapIn = some mapOut)\n    ↔ ( ∃ h : x?.isSome,\n        f (x?.get h) mapIn = some mapOut )"}, {"name": "Com.denote_changeVars", "content": "@[simp] theorem Com.denote_changeVars\n    (varsMap : Γ.Hom Γ') (c : Com d Γ eff ty) :\n    (c.changeVars varsMap).denote =\n    fun V => c.denote (V.comap varsMap)"}, {"name": "denote_insertPureCom_eq_of", "content": "theorem denote_insertPureCom_eq_of [LawfulMonad d.m]\n    {zip : Zipper d Γ_in eff tys} {vs}\n    {newCom : Com d zip.Γ_mid .pure newTys} {V_in : Valuation Γ_in}\n    (h : ∀ V : zip.top.ValidDenotation,\n        newCom.denote V.val = vs.map V.val) :\n    (zip.insertPureCom vs newCom).denote V_in = zip.denote V_in"}, {"name": "denote_insertPureCom", "content": "theorem denote_insertPureCom {zip : Zipper d Γ_in eff t₁} [LawfulMonad d.m]\n    {newCom : Com d zip.Γ_mid .pure newTys} {vs : HVector zip.Γ_mid.Var newTys} :\n    (zip.insertPureCom vs newCom).denote = (fun (V_in : Valuation Γ_in) => do\n      let V_mid ← zip.top.denote V_in\n      zip.bot.denote\n        ((Com.denoteLets newCom V_mid).comap <| newCom.outContextHom.with vs newCom.returnVars)\n      )"}, {"name": "denote_insertCom", "content": "theorem denote_insertCom {zip : Zipper d Γ_in eff t₁} [LawfulMonad d.m]\n    {newCom : Com d zip.Γ_mid eff newTys} {vs : HVector zip.Γ_mid.Var newTys} :\n    (zip.insertCom vs newCom).denote = (fun (V_in : Valuation Γ_in) => do\n      let V_mid ← zip.top.denote V_in\n      let V_newMid ← newCom.denoteLets V_mid\n      zip.bot.denote\n        (V_newMid.comap <| newCom.outContextHom.with vs newCom.returnVars)\n      )"}], "used_local_defs": [{"name": "SplitProgramResult", "content": "structure SplitProgramResult extends Zipper d Γ eff t where\n  {midTypes : List d.Ty}\n  midRet : HVector toZipper.Γ_mid.Var midTypes"}, {"name": "splitProgramAtAux", "content": "def splitProgramAtAux : (pos : ℕ) → (lets : Lets d Γ₁ eff Γ₂) →\n    (prog : Com d Γ₂ eff t) →\n    Option (SplitProgramResult d Γ₁ eff t)\n  | 0, lets, .var e body => some {\n      top := lets.var e\n      bot := body\n      midRet := e.returnVars\n    }\n  | _, _, .rets _ => none\n  | n+1, lets, .var e body =>\n    splitProgramAtAux n (lets.var e) body"}, {"name": "splitProgramAt", "content": "def splitProgramAt (pos : ℕ) (prog : Com d Γ eff t) :\n    Option (SplitProgramResult d Γ eff t) :=\n  splitProgramAtAux pos .nil prog"}, {"name": "rewriteAt", "content": "def rewriteAt\n    (lhs rhs : Com d Γ₁ .pure ts₁)\n    (hlhs : ∀ t (v : Var Γ₁ t), ⟨t, v⟩ ∈ lhs.vars)\n    (pos : ℕ) (target : Com d Γ₂ eff t₂) :\n    Option (Com d Γ₂ eff t₂) := do\n  let splitRes ← splitProgramAt pos target\n  if h : ts₁ = splitRes.midTypes then\n    let m ← matchArgRes splitRes.top lhs.toLets splitRes.midRet (h ▸ lhs.returnVars)\n    let m := m.toHom <| by\n      subst h; exact hlhs\n    let rhs := rhs.changeVars m\n    let zip := splitRes.insertPureCom splitRes.midRet (cast (by admit /- proof elided -/\n    ) rhs)\n    return zip.toCom\n  else none"}], "used_local_lemmas": [{"name": "denote_splitProgramAtAux", "content": "theorem denote_splitProgramAtAux [LawfulMonad d.m] :\n    {pos : ℕ} → {lets : Lets d Γ₁ eff Γ₂} →\n    {prog : Com d Γ₂ eff t} →\n    {res : _} → (hres : res ∈ splitProgramAtAux pos lets prog) →\n    (V : Valuation Γ₁) →\n    res.denote V = (lets.denote V) >>= prog.denote\n  | 0, lets, .var e body, res, hres, V => by\n    obtain rfl"}, {"name": "denote_splitProgramAt", "content": "@[simp]\ntheorem denote_splitProgramAt [LawfulMonad d.m] {pos : ℕ} {prog : Com d Γ eff t}\n    {res : _} (hres : res ∈ splitProgramAt pos prog) :\n    res.denote = prog.denote"}], "local_ctx": "import LeanMLIR.Framework\n\nimport LeanMLIR.Framework.Zipper\n\nimport LeanMLIR.Transforms.Rewrite.Match\n\nopen Ctxt (Var VarSet Valuation)\n\nvariable {d} [DialectSignature d] [TyDenote d.Ty] [DialectDenote d] [Monad d.m]\n\nvariable [DecidableEq d.Ty] [DecidableEq d.Op]\n\nvariable [∀ (t : d.Ty), Inhabited (toType t)]\n\nsection SplitProgram\n\nvariable (d Γ eff t) in\n\nstructure SplitProgramResult extends Zipper d Γ eff t where\n  {midTypes : List d.Ty}\n  midRet : HVector toZipper.Γ_mid.Var midTypes\n\ndef splitProgramAtAux : (pos : ℕ) → (lets : Lets d Γ₁ eff Γ₂) →\n    (prog : Com d Γ₂ eff t) →\n    Option (SplitProgramResult d Γ₁ eff t)\n  | 0, lets, .var e body => some {\n      top := lets.var e\n      bot := body\n      midRet := e.returnVars\n    }\n  | _, _, .rets _ => none\n  | n+1, lets, .var e body =>\n    splitProgramAtAux n (lets.var e) body\n\ndef splitProgramAt (pos : ℕ) (prog : Com d Γ eff t) :\n    Option (SplitProgramResult d Γ eff t) :=\n  splitProgramAtAux pos .nil prog\n\nend SplitProgram\n\ndef rewriteAt\n    (lhs rhs : Com d Γ₁ .pure ts₁)\n    (hlhs : ∀ t (v : Var Γ₁ t), ⟨t, v⟩ ∈ lhs.vars)\n    (pos : ℕ) (target : Com d Γ₂ eff t₂) :\n    Option (Com d Γ₂ eff t₂) := do\n  let splitRes ← splitProgramAt pos target\n  if h : ts₁ = splitRes.midTypes then\n    let m ← matchArgRes splitRes.top lhs.toLets splitRes.midRet (h ▸ lhs.returnVars)\n    let m := m.toHom <| by\n      subst h; exact hlhs\n    let rhs := rhs.changeVars m\n    let zip := splitRes.insertPureCom splitRes.midRet (cast (by admit /- proof elided -/\n    ) rhs)\n    return zip.toCom\n  else none", "target_theorem": "theorem denote_rewriteAt [LawfulMonad d.m]\n    {lhs rhs : Com d Γ₁ .pure t₁}\n    (hl : lhs.denote = rhs.denote)\n    {hlhs : ∀ t (v : Var Γ₁ t), ⟨t, v⟩ ∈ lhs.vars}\n    {pos : ℕ} {target : Com d Γ₂ eff t₂}\n    {rew : Com d Γ₂ eff t₂}\n    (hrew : rew ∈ rewriteAt lhs rhs hlhs pos target) :\n    rew.denote = target.denote :=", "ground_truth_proof": ":= by\n  funext V\n  simp only [rewriteAt, Option.pure_def, Option.bind_eq_bind, Option.mem_def,\n    Option.bind_eq_some_iff, Option.dite_none_right_eq_some, Option.some.injEq] at hrew\n  rcases hrew with ⟨res, h_split, rfl, varMap', -, rfl⟩\n  dsimp only at varMap'\n  simp only [cast_eq, Zipper.denote_toCom, ← denote_splitProgramAt h_split]\n  apply Zipper.denote_insertPureCom_eq_of\n  intro V\n  simp only [Expr.pdenoteOp, Com.denote_changeVars, ← hl]\n  rw [← Com.denoteLets_returnVars, Com.denoteLets_eq]\n  apply denote_matchLets_of varMap' hlhs", "nesting_depth": 13, "transitive_dep_count": 179, "subset_aristotle": false, "category": "Compiler"}
{"id": 327, "thm_name": "nfaOfTerm_bv_language", "thm_stmt": "lemma nfaOfTerm_bv_language (t : Term) :\n    nfaOfTerm t |>.bv_recognizes t.language", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.Blase.AutoStructs.Basic", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.Blase.AutoStructs.Constructions", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.card", "module": "Mathlib.Data.FinEnum"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Array.foldl", "module": "Init.Data.Array.Basic"}, {"name": "Std.HashMap.emptyWithCapacity", "module": "Std.Data.HashMap.Basic"}, {"name": "Array.size", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Empty", "module": "Init.Prelude"}, {"name": "Empty.elim", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "cond", "module": "Init.Prelude"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "Array.emptyWithCapacity", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Array.empty", "module": "Init.Prelude"}, {"name": "FinEnum.toList", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.equiv", "module": "Mathlib.Data.FinEnum"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "NFA.stepSet", "module": "Mathlib.Computability.NFA"}, {"name": "Subsingleton", "module": "Init.Core"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "Language", "module": "Mathlib.Computability.Language"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "List.Vector.ofFn", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.replicate", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}, {"name": "SetRel", "module": "Mathlib.Data.Rel"}, {"name": "Array.back?", "module": "Init.Data.Array.Basic"}, {"name": "Array.isEmpty", "module": "Init.Data.Array.Basic"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}, {"name": "L", "module": "Archive.Hairer"}, {"name": "Fin.mk", "module": "Init.Prelude"}, {"name": "Fin.cast", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.castLT", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.subNat", "module": "Init.Data.Fin.Basic"}, {"name": "List.Vector.get", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "DFA", "module": "Mathlib.Computability.DFA"}, {"name": "NFA.toDFA", "module": "Mathlib.Computability.NFA"}, {"name": "List.range", "module": "Init.Data.List.Basic"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Fin.natAdd", "module": "Init.Data.Fin.Basic"}, {"name": "NeZero", "module": "Init.Data.NeZero"}], "used_repo_defs": [{"name": "syntax \"max\" : MLIR.Pretty.uniform_op", "content": "syntax \"max\" : MLIR.Pretty.uniform_op\n\nsyntax \"slt\" : MLIR.Pretty.uniform_op\n\nsyntax \"xor\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "carry", "content": "def carry (initCarry : Bool) (x y : BitStream) : BitStream :=\n  fun n => (addAux' initCarry x y n).2"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "addAux'", "content": "def addAux' (carryIn : Bool) (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => carryIn\n    | i + 1 => (addAux' carryIn x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "worklistRun", "content": "def worklistRun (final : S → Bool) (inits : Array S)\n    (hinits : inits.toList.Nodup) (f : S → Array (BitVec n × S)) : CNFA n :=\n  ⟨worklistRun' _ S final inits hinits f, worklistRun'_wf (BitVec n) S⟩"}, {"name": "worklistRun'", "content": "def worklistRun' (final : S → Bool) (inits : Array S) (hinits : inits.toList.Nodup) (f : S → Array (A × S)) : RawCNFA A :=\n  let st0 := worklist.initState _ _ inits hinits final\n  go st0\nwhere go (st0 : worklist.St A S) : RawCNFA A :=\n  if hemp : st0.worklist.isEmpty then st0.m else\n  let sa? := st0.worklist.back?\n  match heq : sa? with\n  | some sa =>\n    let wl := st0.worklist.pop\n    let st1 := { st0 with worklist := wl,\n                          worklist_nodup := by admit /- proof elided -/"}, {"name": "worklist.St", "content": "structure worklist.St where\n  m : RawCNFA A\n  map : Std.HashMap S State := ∅\n  worklist : Array S := ∅\n  worklist_nodup : worklist.toList.Nodup\n  worklist_incl : ∀ sa ∈ worklist, sa ∈ map"}, {"name": "worklist.initState", "content": "def worklist.initState (inits : Array S) (hinits : inits.toList.Nodup) (final? : S → Bool) : worklist.St A S :=\n  let m := RawCNFA.empty (A := A)\n  let mapm := inits.foldl (init := (Std.HashMap.emptyWithCapacity, m)) fun (map, m) sa =>\n    let (s, m) := m.newState\n    let m := m.addInitial s\n    let m := if final? sa then m.addFinal s else m\n    (map.insert sa s, m)\n  let map := mapm.1\n  let m := mapm.2\n  let worklist_incl : ∀ sa ∈ inits, sa ∈ map :="}, {"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "State", "content": "abbrev State := Nat"}, {"name": "RawCNFA.empty", "content": "def RawCNFA.empty : RawCNFA A := {\n  stateMax := 0\n  initials := ∅\n  finals := ∅\n  trans := ∅\n}"}, {"name": "processOneElem", "content": "def processOneElem (final : S → Bool) (s : State) (st : worklist.St A S) : A × S → worklist.St A S :=\n  fun (a', sa') =>\n    let (s', st') := st.addOrCreateState _ _ (final sa') sa'\n    let m := st'.m.addTrans a' s s'\n    { st' with m }"}, {"name": "worklist.St.addOrCreateState", "content": "def worklist.St.addOrCreateState (st : worklist.St A S) (final? : Bool) (sa : S) : State × worklist.St A S :=\n  match heq : st.map[sa]? with\n  | some s => (s, st)\n  | none =>\n    let (s, m) := st.m.newState\n    let m := if final? then m.addFinal s else m\n    let map := st.map.insert sa s\n    let worklist := st.worklist.push sa\n    have worklist_nodup : worklist.toList.Nodup := by admit /- proof elided -/"}, {"name": "CNFA", "content": "structure CNFA (n : Nat) where\n  m : RawCNFA (BitVec n)\n  wf : m.WF"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "sub", "content": "def sub (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).1"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "ofTerm", "content": "abbrev ofTerm (t : Term) : FSM (Fin t.arity) := termEvalEqFSM t |>.toFSM"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "termEvalEqFSM", "content": "def termEvalEqFSM : ∀ (t : Term), FSMTermSolution t\n  | ofNat n =>\n    { toFSM := FSM.ofNat n,\n      good := by admit /- proof elided -/"}, {"name": "or", "content": "def or : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ||| Circuit.var true (inr false),\n    nextStateCirc := fun a => a.elim\n  }"}, {"name": "shiftLeft", "content": "def shiftLeft (n : Nat) : FSM Unit :=\n  match n with\n  | 0 => FSM.id\n  | n + 1 => composeUnaryAux (FSM.ls false) (shiftLeft n)"}, {"name": "id", "content": "def id : FSM Unit := {\n α := Empty,\n initCarry := Empty.elim,\n outputCirc := Circuit.var true (inr ()),\n nextStateCirc := Empty.elim\n}"}, {"name": "ls", "content": "def ls (b : Bool) : FSM Unit :=\n  { α := Unit,\n    initCarry := fun _ => b,\n    nextStateCirc := fun () => Circuit.var true (inr ()),\n    outputCirc := Circuit.var true (inl ())\n  }"}, {"name": "composeUnaryAux", "content": "def composeUnaryAux\n    (p : FSM Unit)\n    (q : FSM arity) :\n    FSM arity :=\n  p.compose\n    arity\n    _\n    (λ _ => id)\n    (λ _ => q)"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "and", "content": "def and : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := fun a => a.elim,\n    outputCirc := Circuit.var true (inr true) &&& Circuit.var true (inr false),\n  }"}, {"name": "xor", "content": "def xor : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ^^^ Circuit.var true (inr false),\n    nextStateCirc := Empty.elim\n  }"}, {"name": "neg", "content": "def neg : FSM Unit :=\n  { α := Unit,\n    i := by admit /- proof elided -/"}, {"name": "composeBinary", "content": "def composeBinary\n    (p : FSM Bool)\n    {t₁ t₂ : Term}\n    (q₁ : FSMTermSolution t₁)\n    (q₂ : FSMTermSolution t₂) :\n    FSM (Fin (max t₁.arity t₂.arity)) := composeBinaryAux p q₁.toFSM q₂.toFSM"}, {"name": "composeBinaryAux", "content": "def composeBinaryAux\n    (p : FSM Bool)\n    (q₁ : FSM (Fin a₁))\n    (q₂ : FSM (Fin a₂)) :\n    FSM (Fin (max a₁ a₂)) :=\n  p.compose (Fin (max a₁ a₂))\n    (λ b => Fin (cond b a₁ a₂))\n    (λ b i => Fin.castLE (by admit /- proof elided -/\n    ) i)\n    (λ b => match b with\n      | true => q₁\n      | false => q₂)"}, {"name": "FSMTermSolution", "content": "structure FSMTermSolution (t : Term) extends FSM (Fin t.arity) where\n  ( good : t.evalFin = toFSM.eval )"}, {"name": "Term.evalFin", "content": "@[simp] def Term.evalFin (t : Term) (vars : Fin (arity t) → BitStream) : BitStream :=\n  match t with\n  | var n => vars (Fin.last n)\n  | zero    => BitStream.zero\n  | one     => BitStream.one\n  | negOne  => BitStream.negOne\n  | ofNat n => BitStream.ofNat n\n  | and t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | or t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | xor t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | not t     => ~~~(t.evalFin vars)\n  | add t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | sub t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | neg t       => -(Term.evalFin t vars)\n \n \n  | shiftL t n  => BitStream.shiftLeft (Term.evalFin t vars) n"}, {"name": "Predicate.evalFin", "content": "@[simp] def Predicate.evalFin (p : Predicate) (vars : Fin (arity p) → BitStream) : BitStream :=\nmatch p with\n| .width .eq n => BitStream.falseIffEq n\n| .width .neq n => BitStream.falseIffNeq n\n| .width .lt n => BitStream.falseIffLt n\n| .width .le n => BitStream.falseIffLe n\n| .width .gt n => BitStream.falseIffGt n\n| .width .ge n => BitStream.falseIffGe n\n| .binary .eq t₁ t₂ =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalEq x₁ x₂\n| .binary .neq t₁ t₂  =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalNeq x₁ x₂\n| .land p q =>\n  \n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLand x₁ x₂\n| .lor p q =>\n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor x₁ x₂\n| .binary .slt p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalSlt x₁ x₂\n| .binary .sle p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalSlt x₁ x₂) (Predicate.evalEq x₁ x₂)\n| .binary .ult p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  (Predicate.evalUlt x₁ x₂)\n| .binary .ule p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalUlt x₁ x₂) (Predicate.evalEq x₁ x₂)"}, {"name": "Predicate.evalUlt", "content": "def Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true"}, {"name": "borrow", "content": "def borrow (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).2"}, {"name": "Predicate.evalLor", "content": "def Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)"}, {"name": "Predicate.evalSlt", "content": "def Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))"}, {"name": "Predicate.evalMsbEq", "content": "def Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false"}, {"name": "Predicate.evalLand", "content": "def Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)"}, {"name": "Predicate.evalNeq", "content": "def Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd"}, {"name": "nxor", "content": "def nxor (a b : BitStream) : BitStream := fun i => a i == b i"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "Predicate.evalEq", "content": "def Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "Predicate", "content": "inductive Predicate : Type where\n \n| width (wp : WidthPredicate) (n : Nat) : Predicate\n| binary (p : BinaryPredicate) (t₁ t₂ : Term)\n| land  (p q : Predicate) : Predicate\n| lor (p q : Predicate) : Predicate\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "falseIffNeq", "content": "abbrev falseIffNeq (n : Nat) : BitStream := fun i => decide (i == n)"}, {"name": "falseIffLt", "content": "abbrev falseIffLt (n : Nat) : BitStream := fun i => decide (i ≥ n)"}, {"name": "falseIffLe", "content": "abbrev falseIffLe (n : Nat) : BitStream := fun i => decide (i > n)"}, {"name": "falseIffGe", "content": "abbrev falseIffGe (n : Nat) : BitStream := fun i => decide (i < n)"}, {"name": "falseIffEq", "content": "abbrev falseIffEq (n : Nat) : BitStream := fun i => decide (i != n)"}, {"name": "falseIffGt", "content": "abbrev falseIffGt (n : Nat) : BitStream := fun i => decide (i ≤ n)"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "Term.arity", "content": "@[simp] def Term.arity : Term → Nat\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| ofNat _ => 0\n| Term.and t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.or t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.xor t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.not t => arity t\n| add t₁ t₂ => max (arity t₁) (arity t₂)\n| sub t₁ t₂ => max (arity t₁) (arity t₂)\n| neg t => arity t\n\n\n| shiftL t .. => arity t"}, {"name": "negOne", "content": "abbrev negOne : BitStream := fun _ => true"}, {"name": "shiftLeft", "content": "def shiftLeft (x : BitStream) (k : Nat) : BitStream :=\n  fun i => if i < k then false else x (i - k) "}, {"name": "ofNat", "content": "def ofNat (x : Nat) : BitStream :=\n  Nat.testBit x"}, {"name": "one", "content": "abbrev one    : BitStream := (· == 0)"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "ofNat", "content": "def ofNat (n : Nat)  : FSM (Fin 0) :=\n  match hn : n with\n  | 0 => FSM.zero\n\n  | n' + 1 =>\n    let bit := n.testBit 0\n    let m := n / 2\n    have h : m < n := by admit /- proof elided -/"}, {"name": "zero", "content": "def zero : FSM (Fin 0) :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := Empty.elim,\n    outputCirc := Circuit.fals\n  }"}, {"name": "composeUnary", "content": "def composeUnary\n    (p : FSM Unit)\n    {t : Term}\n    (q : FSMTermSolution t) :\n    FSM (Fin t.arity) := composeUnaryAux p q.toFSM"}, {"name": "one", "content": "def one : FSM (Fin 0) :=\n  { α := Unit,\n    i := by admit /- proof elided -/"}, {"name": "var", "content": "def var (n : ℕ) : FSM (Fin (n+1)) :=\n  { α := Empty,\n    i := by admit /- proof elided -/"}, {"name": "add", "content": "def add : FSM Bool :=\n  { α := Unit,\n    initCarry := λ _ => false,\n    nextStateCirc := fun () =>\n      Circuit.var true (inr true) &&& Circuit.var true (inr false) |||\n      Circuit.var true (inr true) &&& Circuit.var true (inl ()) |||\n      Circuit.var true (inr false) &&& Circuit.var true (inl ()),\n    outputCirc := Circuit.var true (inr true) ^^^\n                  Circuit.var true (inr false) ^^^\n                  Circuit.var true (inl ()),\n  }"}, {"name": "negOne", "content": "def negOne : FSM (Fin 0) :=\n  { α := Empty,\n    i := by admit /- proof elided -/"}, {"name": "sub", "content": "def sub : FSM Bool :=\n  { α := Unit,\n    initCarry := fun _ => false,\n    outputCirc := Circuit.var true (inr true) ^^^\n                  Circuit.var true (inr false) ^^^\n                  Circuit.var true (inl ()),\n    nextStateCirc := fun _ =>\n      (Circuit.var false (inr true) &&& Circuit.var true (inr false)) |||\n      (Circuit.var false (inr true) ^^^ Circuit.var true (inr false)) &&&\n      (Circuit.var true (inl ()))\n  }"}, {"name": "not", "content": "def not : FSM Unit :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := Empty.elim,\n    outputCirc := Circuit.var false (inr ())\n  }"}, {"name": "add", "content": "def add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1"}, {"name": "addAux", "content": "def addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "neg", "content": "def neg (x : BitStream) : BitStream :=\n  fun n => (negAux x n).1"}, {"name": "negAux", "content": "def negAux (x : BitStream) : Nat → Bool × Bool\n  | 0 => (x 0, !(x 0))\n  | n+1 =>\n    let borrow := (negAux x n).2\n    let a := x (n + 1)\n    (xor (!a) borrow, !a && borrow)"}, {"name": "CNFA.inter", "content": "def CNFA.inter (m1 m2 : CNFA n) : CNFA n := product (fun b1 b2 => b1 && b2) m1 m2"}, {"name": "product", "content": "def product (final? : Bool → Bool → Bool) (m₁ m₂ : CNFA n) : CNFA n :=\n  worklistRun (m₁.m.states × m₂.m.states) final (product.inits m₁ m₂)\n    (by admit /- proof elided -/\n    ) f\nwhere final (ss : m₁.m.states × m₂.m.states) := final? (ss.1 ∈ m₁.m.finals) (ss.2 ∈ m₂.m.finals)\n      f (ss : m₁.m.states × m₂.m.states) :=\n        let (s1, s2) := ss\n        (FinEnum.toList (α := BitVec n)).foldl (init := Array.empty) fun as a =>\n          product.prodArray' (λ s₁ s₂ ↦ (a, (s₁, s₂)))\n            (fun s' => m₁.wf.trans_tgt_lt (s := s1) (a := a)) (fun s' => m₂.wf.trans_tgt_lt (s := s2) (a := a)) as"}, {"name": "product.prodArray'", "content": "@[inline]\ndef product.prodArray' (a : Array γ) :=\n  m₁.attachWith _ hm₁ |>.fold (init := a) fun is s1 =>\n    m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s1 s2)"}, {"name": "product.inits_nodup", "content": "def product.inits_nodup : inits m₁ m₂ |>.toList.Nodup :="}, {"name": "product.inits", "content": "def product.inits (m₁ m₂ : CNFA n) :=\n  product.prodArray Prod.mk @m₁.wf.initials_lt @m₂.wf.initials_lt"}, {"name": "product.prodArray", "content": "@[inline]\ndef product.prodArray := prodArray' f hm₁ hm₂ (Array.emptyWithCapacity <| m₁.size * m₂.size)"}, {"name": "liftMaxSuccSucc2", "content": "def liftMaxSuccSucc2 (n m : Nat) : Fin (m + 1) → Fin (max n m + 3) :=\n  fun k => if _ : k = Fin.last m then max n m + 1 else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMaxSuccSucc1", "content": "def liftMaxSuccSucc1 (n m : Nat) : Fin (n + 1) → Fin (max n m + 3) :=\n  fun k => if _ : k = Fin.last n then (max n m).cast else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftLast3", "content": "def liftLast3 n : Fin 3 → Fin (n + 3)\n| 0 => n\n| 1 => n + 1\n| 2 => Fin.last (n + 2)"}, {"name": "CNFA.inter_bv_language", "content": "def CNFA.inter_bv_language (m₁ m₂ : CNFA n) :\n    m₁.bv_recognizes L₁ →\n    m₂.bv_recognizes L₂ →\n    (m₁.inter m₂).bv_recognizes (L₁ ∩ L₂) :="}, {"name": "HashSet.inter", "content": "def HashSet.inter [BEq A] [Hashable A] (m1 m2 : Std.HashSet A) : Std.HashSet A :=\n  m1.fold (init := ∅) fun mi x => if m2.contains x then mi.insert x else mi"}, {"name": "NFA'", "content": "structure NFA' (n : Nat) where\n  σ : Type\n  M : NFA (BitVec n) σ"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "eval", "content": "def eval (x : arity → BitStream) : BitStream :=\n  fun n => (p.nextBit (p.carry x n) (fun i => x i n)).2"}, {"name": "nextBit", "content": "def nextBit : p.State → (arity → Bool) → p.State × Bool :=\n  fun carry inputBits =>\n    let input := Sum.elim carry inputBits\n    let newState : p.State  := fun (a : p.α) => (p.nextStateCirc a).eval input\n    let outBit : Bool       := (p.outputCirc).eval input\n    (newState, outBit)"}, {"name": "State", "content": "abbrev State : Type := p.α → Bool"}, {"name": "carry", "content": "def carry (x : arity → BitStream) : ℕ → p.State\n  | 0 => p.initCarry\n  | n+1 => (p.nextBit (carry x n) (fun i => x i n)).1"}, {"name": "carryBV", "content": "def carryBV (x : ar → BitVec w) : p.State :=\n  p.carry (fun ar => .ofBitVecSext (x ar)) w"}, {"name": "evalBV", "content": "def evalBV {w} (x : ar → BitVec w) : BitVec w :=\n  BitVec.ofFn fun k => p.eval (fun ar => .ofBitVecSext (x ar)) k"}, {"name": "ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}, {"name": "Term.language", "content": "def Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }"}, {"name": "Formula.arity", "content": "@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity"}, {"name": "Term.evalFinBV", "content": "@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n"}, {"name": "enc", "content": "def enc (bvs : BitVecs n) : BitVecs' n :=\n  (List.finRange bvs.w).map (fun i =>\n    BitVec.ofFn (fun (k : Fin n) => (bvs.bvs.get k)[i]))"}, {"name": "BitVecs'", "content": "abbrev BitVecs' (n : Nat) := List (BitVec n)"}, {"name": "dec", "content": "@[simps]\ndef dec (bvs' : BitVecs' n) : BitVecs n where\n  w := bvs'.length\n  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k"}, {"name": "accepts", "content": "def accepts (M : NFA' n) : Set (BitVecs n) := dec '' M.accepts'"}, {"name": "accepts'", "content": "def accepts' (M : NFA' n) : Set (BitVecs' n) := M.M.accepts"}, {"name": "worklistRun_spec", "content": "def worklistRun_spec : (worklistRun S final inits hinits f |>.Sim $ nfa' inits final f) :=\n  worklistRun'_spec inits final f"}, {"name": "nfa'", "content": "def nfa' : NFA' n :=\n  { σ := _, M := nfa inits final f }"}, {"name": "nfa", "content": "def nfa : NFA A S where\n  start := { sa | sa ∈ inits }\n  accept := { sa | final sa }\n  step sa a := { sa' | (a, sa') ∈ f sa }"}, {"name": "worklistRun'_spec", "content": "def worklistRun'_spec :\n    (worklistRun' A S final inits hinits f |>.Sim $ nfa inits final f) :="}, {"name": "StInv", "content": "structure StInv (m : RawCNFA A) (map : Std.HashMap S State) where\n  wf : m.WF\n  map_states : ∀ (sa : S) s, map[sa]? = some s → s ∈ m.states\n  map_surj : ∀ s : m.states, ∃ (sa : S), map[sa]? = some s.val\n  map_inj : ∀ {s} {sa sa' : S}, map[sa]? = some s → map[sa']? = some s → sa = sa'"}, {"name": "worklist.St.D", "content": "def worklist.St.D (st : worklist.St A S) : Set S := st.visited"}, {"name": "worklist.St.visited", "content": "def worklist.St.visited (st : worklist.St A S) : Set S := { s : S | s ∈ st.map ∧ s ∉ st.worklist }"}, {"name": "worklistGo_spec", "content": "def worklistGo_spec {st : worklist.St A S} (inv : StInv A S st.m st.map) :\n    st.sim inits final f ∅ →\n    (worklistRun'.go A S final f st |>.Sim $ nfa inits final f) :="}, {"name": "worklist.St.rel", "content": "def worklist.St.rel (st : worklist.St A S) : SetRel State S := {(s, sa) | st.map[sa]? = some s }"}, {"name": "processOneElem_mot", "content": "def processOneElem_mot (s : State) (sa : S) (n : ℕ) (st : worklist.St A S) : Prop :=\n  st.map[sa]? = some s ∧\n  sa ∈ st.visited ∧\n  StInv A S st.m st.map ∧\n  st.sim inits final f  {(sa1, a, sa') | sa1 = sa ∧ ∃ k ≥ n, (f sa)[k]? = some (a, sa') }"}, {"name": "worklist.St.sim", "content": "abbrev worklist.St.sim {st : worklist.St A S} (T : Set (S × A × S)) :=\n  st.m.Simul (nfa inits final f) st.rel st.D T"}, {"name": "RawCNFA.Sim", "content": "def RawCNFA.Sim (m : RawCNFA A) (A : NFA A S) := ∃ R, RawCNFA.Simul m A R ⊤ ∅"}, {"name": "RawCNFA.Simul", "content": "structure RawCNFA.Simul (m : RawCNFA A) (M : NFA A Q) (R : SetRel State Q) (D : Set Q) (T : Set (Q × A × Q)) where\n  accept {s q} : s ~[R] q → (s ∈ m.finals ↔ q ∈ M.accept)\n  initial₁ {s} : s ∈ m.initials → ∃ q ∈ M.start, s ~[R] q\n  initial₂ {q} : q ∈ M.start → ∃ s ∈ m.initials, s ~[R] q\n  trans_match₁ {s s' a q} : s ~[R] q → s' ∈ m.tr s a → ∃ q', q' ∈ M.step q a ∧ s' ~[R] q'\n  trans_match₂ {s a q q'} : s ~[R] q → q' ∈ M.step q a → q ∈ D → (q, a, q') ∉ T → ∃ s', s' ∈ m.tr s a ∧ s' ~[R] q'"}, {"name": "RawCNFA.SimulFun", "content": "structure RawCNFA.SimulFun (m : RawCNFA A) (M : NFA A Q) (f : m.states ≃ Q)  where\n  accept {q} : ((f.invFun q).val ∈ m.finals ↔ q ∈ M.accept)\n  initial {q} : q ∈ M.start ↔ (f.invFun q).val ∈ m.initials\n  trans_match {a q q'} : q' ∈ M.step q a ↔ (f.invFun q').val ∈ m.tr (f.invFun q) a"}, {"name": "RawCNFA.tr", "content": "@[inline]\ndef RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}, {"name": "CNFA.Sim", "content": "def CNFA.Sim (m : CNFA n) (M : NFA' n) :=\n  m.m.Sim M.M"}, {"name": "CNFA.bv_recognizes", "content": "def CNFA.bv_recognizes (m : CNFA n) (L : Set (BitVecs n)) :=\n  ∃ L', m.recognizes L' ∧ L = dec '' L'"}, {"name": "RawCNFA.recognizes", "content": "def RawCNFA.recognizes (m : RawCNFA A) (L : Language A) :=\n  ∃ (σ : Type) (M : NFA A σ), m.Sim M ∧ M.accepts = L"}, {"name": "CNFA.recognizes", "content": "def CNFA.recognizes (m : CNFA n) (L : Language (BitVec n)) :=\n  ∃ (M : NFA' n), m.Sim M ∧ M.M.accepts = L"}, {"name": "BitVecs.cast", "content": "def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=\n  { w := bvs.w, bvs := h ▸ bvs.bvs }"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "CNFA.minimize", "content": "def CNFA.minimize (m : CNFA n) : CNFA n :=\n  let mᵣ := m.reverse.determinize\n  mᵣ.reverse.determinize"}, {"name": "CNFA.determinize", "content": "def CNFA.determinize (m : CNFA n) : CNFA n :=\n  worklistRun (BitVec m.m.stateMax)\n    (fun ss => ss.any fun n b => b == true && n ∈ m.m.finals)\n    (determinize.inits m)\n    (by admit /- proof elided -/\n    )\n    f\nwhere\n  f := fun (ss : BitVec m.m.stateMax) =>\n        (FinEnum.toList (BitVec n)).foldl (init := Array.empty) fun ts a =>\n          let ss' := m.m.transSetBV ss a\n          ts.push (a, ss')"}, {"name": "CNFA.determinize.inits", "content": "def CNFA.determinize.inits (m : CNFA n) : Array (BitVec m.m.stateMax) :=\n  #[BitVec.ofFn (fun n => n ∈ m.m.initials)]"}, {"name": "CNFA.reverse", "content": "def CNFA.reverse (m : CNFA n) : CNFA n :=\n  ⟨m.m.reverse, RawCNFA.reverse_spec m.wf |>.1⟩"}, {"name": "RawCNFA.reverse", "content": "def RawCNFA.reverse (m : RawCNFA A) : RawCNFA A :=\n  let m' := { stateMax := m.stateMax, trans := Std.HashMap.emptyWithCapacity m.trans.size, initials := m.finals, finals := m.initials}\n  m.trans.fold (init := m') processState\nwhere\n  processState := fun m' (s, a) ss' =>\n    ss'.fold (init := m') fun m' s' => m'.addTrans a s' s"}, {"name": "CNFA.toNFA'", "content": "def CNFA.toNFA' (m : CNFA n) : NFA' n := ⟨_, m.toNFA⟩"}, {"name": "CNFA.toNFA", "content": "def CNFA.toNFA (m : CNFA n) : NFA (BitVec n) m.m.states where\n  start := { s | s.val ∈ m.m.initials }\n  accept := { s | s.val ∈ m.m.finals }\n  step s₁ a := { s₂ | s₂.val ∈ m.m.tr s₁.val a }"}, {"name": "RawCNFA.states", "content": "def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax"}, {"name": "reverse", "content": "def reverse (M : NFA' n) : NFA' n where\n  σ := _\n  M := M.M.reverse"}, {"name": "CNFA.determinize_spec", "content": "def CNFA.determinize_spec (m : CNFA n)\n  {M : NFA' n} (hsim : m.Sim M) :\n    m.determinize.Sim M.determinize :="}, {"name": "bv_to_set", "content": "private def bv_to_set (bv : BitVec w) : Set State :=\n  { s | bv.getLsbD s }"}, {"name": "_root_.SetRel.set_eq", "content": "structure _root_.SetRel.set_eq (R : SetRel α β) (A : Set α) (B : Set β) where\n  fwd : a ∈ A → ∃ b ∈ B, a ~[R] b\n  bwd : b ∈ B → ∃ a ∈ A, a ~[R] b"}, {"name": "RawCNFA.lift", "content": "@[inline]\ndef RawCNFA.lift (m₁: RawCNFA (BitVec n1)) (f : Fin n1 → Fin n2) : RawCNFA (BitVec n2) :=\n  let trans := (List.range m₁.stateMax).foldl (init := ∅) fun m2 s => processState m2 s\n  { m₁ with trans }\nwhere"}, {"name": "CNFA.lift", "content": "@[inline]\ndef CNFA.lift (m: CNFA n1) (f : Fin n1 → Fin n2) : CNFA n2 :=\n  ⟨m.m.lift f, m.m.lift_wf m.wf⟩"}, {"name": "BitVecs.transport", "content": "def BitVecs.transport (f : Fin n → Fin m) (bvs : BitVecs m) : BitVecs n :=\n  { w := bvs.w, bvs := bvs.bvs.transport f }"}, {"name": "BitVec.transport", "content": "def BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=\n  BitVec.ofFn fun i => bv.getLsbD (f i)"}, {"name": "List.Vector.transport", "content": "def List.Vector.transport (v : Vector α m) (f : Fin n → Fin m) : Vector α n :=\n  Vector.ofFn fun i => v.get (f i)"}, {"name": "BitVecs'.transport", "content": "def BitVecs'.transport (f : Fin n → Fin m) (bvs' : BitVecs' m): BitVecs' n :=\n  bvs'.map fun bv => bv.transport f"}, {"name": "RawCNFA.proj", "content": "@[inline]\ndef RawCNFA.proj (m1: RawCNFA (BitVec n1)) (f : Fin n2 → Fin n1) : RawCNFA (BitVec n2) :=\n  let trans := m1.trans.keysArray.foldl (init := Std.HashMap.emptyWithCapacity) process\n  { m1 with trans }\nwhere"}, {"name": "CNFA.proj_spec", "content": "def CNFA.proj_spec (m : CNFA n2) (f : Fin n1 → Fin n2) {M : NFA' n2} :\n    m.Sim M → (m.proj f |>.Sim (M.proj f)) :="}, {"name": "CNFA.proj", "content": "@[inline]\ndef CNFA.proj (m: CNFA n2) (f : Fin n1 → Fin n2) : CNFA n1 :=\n  ⟨m.m.proj f, m.m.proj_wf m.wf⟩"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "List.nodup_singleton", "module": "Mathlib.Data.List.Nodup"}, {"name": "NFA.eval_append_singleton", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.eval_nil", "module": "Mathlib.Computability.NFA"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.add_def", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.castLE_castLE", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.le_of_eq", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Fin.ext_iff", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "ite_cond_eq_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}, {"name": "bisim_comp", "content": "lemma bisim_comp (m : RawCNFA A) :\n    m.Sim M₁ → M₁.Bisim M₂ → m.Sim M₂"}, {"name": "bisimul_comp", "content": "lemma bisimul_comp {m : RawCNFA A} :\n    m.Simul M₁ R₁ ⊤ ∅ → M₁.Bisimul R₂ M₂ →\n    m.Simul M₂ (R₁.comp R₂) ⊤ ∅"}, {"name": "CNFA.bv_recognizes_equiv", "content": "lemma CNFA.bv_recognizes_equiv {m : CNFA n} :\n    m.bv_recognizes L ↔ ∃ (M : NFA' n), m.Sim M ∧ M.accepts = L"}, {"name": "List.Vector.append_get_ge", "content": "@[simp]\nlemma List.Vector.append_get_ge {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: n ≤ i) :\n    (x ++ y).get i = y.get ((i.cast (Nat.add_comm n m) |>.subNat n hlt))"}, {"name": "List.Vector.append_get_lt", "content": "@[simp]\nlemma List.Vector.append_get_lt {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: i < n) :\n    (x ++ y).get i = x.get (i.castLT hlt)"}, {"name": "CNFA.minimize_bv_language", "content": "lemma CNFA.minimize_bv_language {m : CNFA n} :\n    m.bv_recognizes L → m.minimize.bv_recognizes L"}, {"name": "CNFA.minimize_language", "content": "lemma CNFA.minimize_language {m : CNFA n} :\n    m.recognizes L → m.minimize.recognizes L"}, {"name": "CNFA.reverse_language", "content": "lemma CNFA.reverse_language {m : CNFA n} (hl : m.recognizes L) : m.reverse.recognizes L.reverse"}, {"name": "CNFA.reverse_spec", "content": "lemma CNFA.reverse_spec {m : CNFA n} : m.reverse.Sim m.toNFA'.reverse"}, {"name": "RawCNFA.reverse_spec", "content": "lemma RawCNFA.reverse_spec {m : RawCNFA A} (hwf : m.WF) :\n    let m'"}, {"name": "RawCNFA.reverse_spec_procesState", "content": "lemma RawCNFA.reverse_spec_procesState {m : RawCNFA A} (hwf : m.WF) s₀ a₀ ss' (hs₀ : s₀ ∈ m.states) :\n    let motive m' ss'"}, {"name": "CNFA.determinize_language", "content": "lemma CNFA.determinize_language {m : CNFA n} :\n    m.recognizes L → m.determinize.recognizes L"}, {"name": "CNFA.lift_bv_language", "content": "@[simp]\nlemma CNFA.lift_bv_language {m : CNFA n1} {f : Fin n1 → Fin n2} :\n    m.bv_recognizes L → (m.lift f |>.bv_recognizes (BitVecs.transport f ⁻¹' L))"}, {"name": "CNFA.lift_spec", "content": "lemma CNFA.lift_spec (m : CNFA n1) (f : Fin n1 → Fin n2) {M : NFA' n1} :\n    m.Sim M → (m.lift f |>.Sim (M.lift f))"}, {"name": "CNFA.proj_bv_language", "content": "lemma CNFA.proj_bv_language {m : CNFA n2} {f : Fin n1 → Fin n2} :\n    m.bv_recognizes L → (m.proj f |>.bv_recognizes (BitVecs.transport f '' L))"}, {"name": "BitVecs.transport_getElem", "content": "@[simp]\nlemma BitVecs.transport_getElem {bvs : BitVecs m} (f : Fin n → Fin m) (i : Fin n) :\n    (bvs.transport f).bvs.get i = bvs.bvs.get (f i)"}], "used_local_defs": [{"name": "NFA.sa", "content": "def NFA.sa (_ : NFA α σ) := σ → Language α"}, {"name": "NFA.correct", "content": "structure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q"}, {"name": "BVNRel", "content": "abbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop"}, {"name": "NFA'.sa", "content": "def NFA'.sa (M : NFA' n) := M.σ → BVNRel n"}, {"name": "langRel", "content": "def langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }"}, {"name": "NFA'.correct", "content": "structure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))"}, {"name": "NFA'.correct2", "content": "structure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)"}, {"name": "Alphabet", "content": "abbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)"}, {"name": "finFunToBitVec", "content": "def finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)"}, {"name": "bitVecToFinFun", "content": "def bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]"}, {"name": "NFA.ofFSM", "content": "def NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }"}, {"name": "inFSMRel", "content": "@[simp]\nabbrev inFSMRel (p : FSM arity) {w} (bvn : List.Vector (BitVec w) _) :=\n  bvn.get (Fin.last (FinEnum.card arity)) = p.evalBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))"}, {"name": "NFA'.ofFSM_sa", "content": "def NFA'.ofFSM_sa (p : FSM arity) : (NFA'.ofFSM' p).sa := fun q _ bvn =>\n    inFSMRel p bvn ∧ q = p.carryBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))"}, {"name": "NFA'.ofFSM_correct", "content": "def NFA'.ofFSM_correct (p : FSM arity) :\n    (NFA'.ofFSM' p).correct (ofFSM_sa p) (fun _ bvn => inFSMRel p bvn) :="}, {"name": "CNFA.ofFSM", "content": "def CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where"}, {"name": "NFA.msbState", "content": "inductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype"}, {"name": "liftOp", "content": "def liftOp n : Fin (n + 1) → Fin (n + 3) :=\n  fun k =>\n    if k = n then Fin.last (n+2) else k.castLE (by admit /- proof elided -/\n    )"}, {"name": "liftOp_unchanged", "content": "@[simp]\ndef liftOp_unchanged (k : Fin n) : liftOp n k.castSucc = k.castLE (by simp) :="}, {"name": "liftUnop", "content": "def liftUnop n : Fin (n + 1) → Fin (n + 2) :=\n  fun k =>\n    if k = n then Fin.last (n+1) else k.castLE (by admit /- proof elided -/\n    )"}, {"name": "TermBinop", "content": "inductive TermBinop where\n| and | or | xor | add | sub"}, {"name": "TermBinop.subst", "content": "def TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂"}, {"name": "TermBinop.openTerm", "content": "def TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)"}, {"name": "TermBinop.openTerm_arity", "content": "@[simp]\ndef TermBinop.openTerm_arity (op : TermBinop) : op.openTerm.arity + 1 = 3 :="}, {"name": "TermBinop.termGadget", "content": "def TermBinop.termGadget (t : TermBinop) : CNFA 3 :=\n  match t with\n  | .and => FSM.ofTerm (.and (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .or => FSM.ofTerm (.or (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .xor => FSM.ofTerm (.xor (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .add => FSM.ofTerm (.add (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .sub => FSM.ofTerm (.sub (.var 0) (.var 1)) |> CNFA.ofFSM"}, {"name": "autOfTermBinop", "content": "def autOfTermBinop (op : TermBinop) (m₁ : CNFA (n + 1)) (m₂ : CNFA (m + 1)) : CNFA ((n ⊔ m) + 1 ) :=\n  let mop : CNFA 3 := op.termGadget\n  let f₁ := liftMaxSuccSucc1 n m\n  let m1' := m₁.lift f₁\n  let f₂ := liftMaxSuccSucc2 n m\n  let m2' := m₂.lift f₂\n  let mop := mop.lift $ liftLast3 (max (FinEnum.card (Fin n)) (FinEnum.card (Fin m)))\n  let m := CNFA.inter m1' m2' |> CNFA.inter mop\n  let mfinal := m.proj (liftOp _)\n  mfinal.minimize"}, {"name": "swapLastTwoBlock", "content": "def swapLastTwoBlock (x : Fin (n + 3)) : Fin (n + 3) :=\n  if x = Fin.last (n+2) then n\n  else if x = n+1 then Fin.last (n + 2)\n  else if x = n then n + 1\n  else x"}, {"name": "TermUnop", "content": "inductive TermUnop where\n| neg | not | shiftL (k : Nat)"}, {"name": "TermUnop.openTerm", "content": "def TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k"}, {"name": "TermUnop.openTerm_arity", "content": "def TermUnop.openTerm_arity (op : TermUnop) : op.openTerm.arity = 1 :="}, {"name": "TermUnop.openTerm_arity'", "content": "@[simp]\ndef TermUnop.openTerm_arity' (op : TermUnop) : op.openTerm.arity + 1 = 2 :="}, {"name": "TermUnop.subst", "content": "def TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k"}, {"name": "TermUnop.termGadget", "content": "def TermUnop.termGadget (t : TermUnop) : CNFA 2 :=\n  match t with\n  | .neg => FSM.ofTerm (.neg (.var 0)) |> CNFA.ofFSM\n  | .not => FSM.ofTerm (.not (.var 0)) |> CNFA.ofFSM\n  | .shiftL k => FSM.ofTerm (.shiftL (.var 0) k) |> CNFA.ofFSM"}, {"name": "autOfTermUnop", "content": "def autOfTermUnop (op : TermUnop) (m : CNFA (n + 1)) : CNFA (n + 1) :=\n  let mop : CNFA 2 := op.termGadget\n  let mop : CNFA (n + 2) := mop.lift (λ i ↦ i.natAdd n)\n  let m : CNFA (n + 2) := m.lift (λ i ↦ i.castLE (by admit /- proof elided -/\n  ))\n  let m := CNFA.inter m mop\n  let mfinal := m.proj (liftUnop n)\n  mfinal.minimize"}, {"name": "nfaOfTerm", "content": "def nfaOfTerm (t : Term) : CNFA (t.arity + 1) :=\n  match t with\n  | .var n => FSM.ofTerm (.var n) |> CNFA.ofFSM\n  | .zero => FSM.ofTerm .zero |> CNFA.ofFSM\n  | .negOne => FSM.ofTerm .negOne |> CNFA.ofFSM\n  | .one => FSM.ofTerm .one |> CNFA.ofFSM\n  | .ofNat n => FSM.ofTerm (.ofNat n) |> CNFA.ofFSM\n  | .and t₁ t₂ => autOfTermBinop .and (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .or t₁ t₂ => autOfTermBinop .or (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .xor t₁ t₂ => autOfTermBinop .xor (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .add t₁ t₂ => autOfTermBinop .add (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .sub t₁ t₂ => autOfTermBinop .sub (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .neg t => autOfTermUnop .neg (nfaOfTerm t)\n  | .not t => autOfTermUnop .not (nfaOfTerm t)\n  | .shiftL t k => autOfTermUnop (.shiftL k) (nfaOfTerm t)"}, {"name": "swapLastTwo", "content": "def swapLastTwo (x : Fin (n + 2)) : Fin (n + 2) :=\n  if x = Fin.last (n + 1) then n else if x = n then Fin.last (n + 1) else x"}], "used_local_lemmas": [{"name": "NFA.correct_spec", "content": "lemma NFA.correct_spec {M : NFA α σ} {ζ : M.sa} {L : Language α} :\n    M.correct ζ L → M.accepts = L"}, {"name": "in_enc", "content": "@[simp]\nlemma in_enc : x ∈ enc '' S ↔ dec x ∈ S"}, {"name": "dec_snoc_in_langRel", "content": "@[simp]\nlemma dec_snoc_in_langRel {n} {R : BVNRel n} {w : BitVecs' n} {a : BitVec n} :\n    dec (w ++ [a]) ∈ langRel R ↔\n      R (List.Vector.ofFn fun k => .cons (a.getLsbD k) ((dec w).bvs.get k))"}, {"name": "NFA'.correct_spec", "content": "lemma NFA'.correct_spec {M : NFA' n} {ζ : M.sa} {L : BVNRel n} :\n    M.correct ζ L → M.accepts = langRel L"}, {"name": "NFA'.ofFSM_spec", "content": "@[simp]\nlemma NFA'.ofFSM_spec (t : Term) :\n    (ofFSM (FSM.ofTerm t)).accepts = t.language"}, {"name": "CNFA.ofFSM_spec", "content": "lemma CNFA.ofFSM_spec (p : FSM arity) :\n    (CNFA.ofFSM p).Sim (NFA'.ofFSM p)"}, {"name": "CNFA.ofFSM_bv_language", "content": "lemma CNFA.ofFSM_bv_language :\n    (CNFA.ofFSM (FSM.ofTerm t)).bv_recognizes t.language"}, {"name": "TermBinop.subst_arity'", "content": "lemma TermBinop.subst_arity' {op : TermBinop} : (op.subst t₁ t₂).arity + 1= t₁.arity ⊔ t₂.arity + 1"}, {"name": "BitVecs.cast_eq", "content": "@[simp]\nlemma BitVecs.cast_eq  (x : BitVecs n) (h : n = n') : h ▸ x = x.cast h"}, {"name": "Fin.natAdd_zero'", "content": "lemma Fin.natAdd_zero' [h : NeZero m] : Fin.natAdd (m := m) n 0 = n"}, {"name": "TermBinop.alt_lang", "content": "lemma TermBinop.alt_lang {t₁ t₂ : Term} (op : TermBinop) :\n  (op.subst_arity' ▸ (op.subst t₁ t₂).language) =\n    let lop : Set (BitVecs 3) := op.openTerm_arity ▸ op.openTerm.language\n    let lop' : Set (BitVecs ((t₁.arity ⊔ t₂.arity) + 3)) := lop.lift (liftLast3 (max t₁.arity t₂.arity))\n    let l₁ := t₁.language.lift (liftMaxSuccSucc1 t₁.arity t₂.arity)\n    let l₂ := t₂.language.lift (liftMaxSuccSucc2 t₁.arity t₂.arity)\n    let l := l₁ ∩ l₂ ∩ lop'\n    l.proj (liftOp _)"}, {"name": "TermUnop.subst_arity'", "content": "@[simp]\nlemma TermUnop.subst_arity' {op : TermUnop} : (op.subst t).arity + 1 = t.arity + 1"}, {"name": "autOfTermBinop_bv_language", "content": "lemma autOfTermBinop_bv_language op {t₁ t₂ : Term} (m₁ : CNFA (t₁.arity + 1)) (m₂ : CNFA (t₂.arity + 1)) :\n    m₁.bv_recognizes t₁.language →\n    m₂.bv_recognizes t₂.language →\n    (autOfTermBinop op m₁ m₂ |>.bv_recognizes (op.subst_arity' ▸ (op.subst t₁ t₂).language))"}, {"name": "TermUnop.alt_lang", "content": "lemma TermUnop.alt_lang {t : Term} (op : TermUnop) :\n  (op.subst_arity' ▸ (op.subst t).language) =\n    let lop : Set (BitVecs 2) := op.openTerm_arity' ▸ op.openTerm.language\n    let lop' : Set (BitVecs (t.arity + 2)) := lop.lift (λ i ↦ i.natAdd t.arity)\n    let lt : Set (BitVecs (t.arity + 2)) := t.language.lift (λ i ↦ i.castLE (by omega))\n    let l := lt ∩ lop'\n    l.proj (liftUnop t.arity)"}, {"name": "autOfTermUnop_bv_language", "content": "lemma autOfTermUnop_bv_language op {t : Term} (m : CNFA (t.arity + 1)) :\n    m.bv_recognizes t.language →\n    (autOfTermUnop op m |>.bv_recognizes (op.subst_arity' ▸ (op.subst t).language))"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\ndef NFA.sa (_ : NFA α σ) := σ → Language α\n\nstructure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q\n\nabbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop\n\ndef NFA'.sa (M : NFA' n) := M.σ → BVNRel n\n\ndef langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }\n\nstructure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))\n\nstructure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)\n\nsection fsm\n\nabbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)\n\nvariable {arity : Type} [FinEnum arity]\n\ndef finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)\n\ndef bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]\n\ndef NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }\n\n@[simp]\nabbrev inFSMRel (p : FSM arity) {w} (bvn : List.Vector (BitVec w) _) :=\n  bvn.get (Fin.last (FinEnum.card arity)) = p.evalBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))\n\ndef NFA'.ofFSM_sa (p : FSM arity) : (NFA'.ofFSM' p).sa := fun q _ bvn =>\n    inFSMRel p bvn ∧ q = p.carryBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))\n\ndef NFA'.ofFSM_correct (p : FSM arity) :\n    (NFA'.ofFSM' p).correct (ofFSM_sa p) (fun _ bvn => inFSMRel p bvn) :=\n\nopen BitStream in\n\ndef CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where\n\nend fsm\n\nsection nfas_relations\n\ninductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype\n\nend nfas_relations\n\ndef liftOp n : Fin (n + 1) → Fin (n + 3) :=\n  fun k =>\n    if k = n then Fin.last (n+2) else k.castLE (by admit /- proof elided -/\n    )\n\n@[simp]\ndef liftOp_unchanged (k : Fin n) : liftOp n k.castSucc = k.castLE (by simp) :=\n\ndef liftUnop n : Fin (n + 1) → Fin (n + 2) :=\n  fun k =>\n    if k = n then Fin.last (n+1) else k.castLE (by admit /- proof elided -/\n    )\n\ninductive TermBinop where\n| and | or | xor | add | sub\n\ndef TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂\n\ndef TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)\n\n@[simp]\ndef TermBinop.openTerm_arity (op : TermBinop) : op.openTerm.arity + 1 = 3 :=\n\ndef TermBinop.termGadget (t : TermBinop) : CNFA 3 :=\n  match t with\n  | .and => FSM.ofTerm (.and (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .or => FSM.ofTerm (.or (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .xor => FSM.ofTerm (.xor (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .add => FSM.ofTerm (.add (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .sub => FSM.ofTerm (.sub (.var 0) (.var 1)) |> CNFA.ofFSM\n\ndef autOfTermBinop (op : TermBinop) (m₁ : CNFA (n + 1)) (m₂ : CNFA (m + 1)) : CNFA ((n ⊔ m) + 1 ) :=\n  let mop : CNFA 3 := op.termGadget\n  let f₁ := liftMaxSuccSucc1 n m\n  let m1' := m₁.lift f₁\n  let f₂ := liftMaxSuccSucc2 n m\n  let m2' := m₂.lift f₂\n  let mop := mop.lift $ liftLast3 (max (FinEnum.card (Fin n)) (FinEnum.card (Fin m)))\n  let m := CNFA.inter m1' m2' |> CNFA.inter mop\n  let mfinal := m.proj (liftOp _)\n  mfinal.minimize\n\ndef swapLastTwoBlock (x : Fin (n + 3)) : Fin (n + 3) :=\n  if x = Fin.last (n+2) then n\n  else if x = n+1 then Fin.last (n + 2)\n  else if x = n then n + 1\n  else x\n\ninductive TermUnop where\n| neg | not | shiftL (k : Nat)\n\ndef TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k\n\ndef TermUnop.openTerm_arity (op : TermUnop) : op.openTerm.arity = 1 :=\n\n@[simp]\ndef TermUnop.openTerm_arity' (op : TermUnop) : op.openTerm.arity + 1 = 2 :=\n\ndef TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k\n\ndef TermUnop.termGadget (t : TermUnop) : CNFA 2 :=\n  match t with\n  | .neg => FSM.ofTerm (.neg (.var 0)) |> CNFA.ofFSM\n  | .not => FSM.ofTerm (.not (.var 0)) |> CNFA.ofFSM\n  | .shiftL k => FSM.ofTerm (.shiftL (.var 0) k) |> CNFA.ofFSM\n\ndef autOfTermUnop (op : TermUnop) (m : CNFA (n + 1)) : CNFA (n + 1) :=\n  let mop : CNFA 2 := op.termGadget\n  let mop : CNFA (n + 2) := mop.lift (λ i ↦ i.natAdd n)\n  let m : CNFA (n + 2) := m.lift (λ i ↦ i.castLE (by admit /- proof elided -/\n  ))\n  let m := CNFA.inter m mop\n  let mfinal := m.proj (liftUnop n)\n  mfinal.minimize\n\ndef nfaOfTerm (t : Term) : CNFA (t.arity + 1) :=\n  match t with\n  | .var n => FSM.ofTerm (.var n) |> CNFA.ofFSM\n  | .zero => FSM.ofTerm .zero |> CNFA.ofFSM\n  | .negOne => FSM.ofTerm .negOne |> CNFA.ofFSM\n  | .one => FSM.ofTerm .one |> CNFA.ofFSM\n  | .ofNat n => FSM.ofTerm (.ofNat n) |> CNFA.ofFSM\n  | .and t₁ t₂ => autOfTermBinop .and (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .or t₁ t₂ => autOfTermBinop .or (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .xor t₁ t₂ => autOfTermBinop .xor (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .add t₁ t₂ => autOfTermBinop .add (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .sub t₁ t₂ => autOfTermBinop .sub (nfaOfTerm t₁) (nfaOfTerm t₂)\n  | .neg t => autOfTermUnop .neg (nfaOfTerm t)\n  | .not t => autOfTermUnop .not (nfaOfTerm t)\n  | .shiftL t k => autOfTermUnop (.shiftL k) (nfaOfTerm t)\n\ndef swapLastTwo (x : Fin (n + 2)) : Fin (n + 2) :=\n  if x = Fin.last (n + 1) then n else if x = n then Fin.last (n + 1) else x", "target_theorem": "lemma nfaOfTerm_bv_language (t : Term) :\n    nfaOfTerm t |>.bv_recognizes t.language :=", "ground_truth_proof": ":= by\n  induction t\n  case var x =>\n    simp only [nfaOfTerm]\n    exact CNFA.ofFSM_bv_language\n  case zero =>\n    simp only [nfaOfTerm]\n    exact CNFA.ofFSM_bv_language\n  case one =>\n    simp only [nfaOfTerm]\n    exact CNFA.ofFSM_bv_language\n  case negOne =>\n    simp only [nfaOfTerm]\n    exact CNFA.ofFSM_bv_language\n  case ofNat k =>\n    simp only [nfaOfTerm]\n    exact CNFA.ofFSM_bv_language\n  case neg t ih =>\n    simp only [nfaOfTerm]\n    apply autOfTermUnop_bv_language; assumption\n  case not t ih =>\n    simp only [nfaOfTerm]\n    apply autOfTermUnop_bv_language; assumption\n  case shiftL k t ih =>\n    simp only [nfaOfTerm]\n    apply autOfTermUnop_bv_language; assumption\n  case and t₁ t₂ ih₁ ih₂ =>\n    simp only [nfaOfTerm]\n    apply autOfTermBinop_bv_language <;> assumption\n  case or t₁ t₂ ih₁ ih₂ =>\n    simp only [nfaOfTerm]\n    apply autOfTermBinop_bv_language <;> assumption\n  case xor t₁ t₂ ih₁ ih₂ =>\n    simp only [nfaOfTerm]\n    apply autOfTermBinop_bv_language <;> assumption\n  case add t₁ t₂ ih₁ ih₂ =>\n    simp only [nfaOfTerm]\n    apply autOfTermBinop_bv_language <;> assumption\n  case sub t₁ t₂ ih₁ ih₂ =>\n    simp only [nfaOfTerm]\n    apply autOfTermBinop_bv_language <;> assumption", "nesting_depth": 12, "transitive_dep_count": 311, "subset_aristotle": false, "category": "Compiler"}
{"id": 328, "thm_name": "isMonotone_matchVarArg_aux", "thm_stmt": "theorem isMonotone_matchVarArg_aux (lets : Lets d Γ_in eff Γ_out) :\n    (\n     ∀  (Δ_out : Ctxt d.Ty)\n        (matchLets : Lets d Δ_in EffectKind.pure Δ_out) (l : List d.Ty)\n        (argsl : HVector Γ_out.Var l) (argsr : HVector Δ_out.Var l),\n        (matchArg lets matchLets argsl argsr).IsMonotone\n    )\n    ∧ (\n      ∀ (Δ_out : Ctxt d.Ty) (t : d.Ty) (v : Γ_out.Var t)\n        (matchLets : Lets d Δ_in EffectKind.pure Δ_out)\n        (w : Var Δ_out t),\n        (matchVar lets v matchLets w).IsMonotone\n    )", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Transforms/Rewrite/Match.lean", "imports": ["import LeanMLIR.Framework", "import LeanMLIR.Transforms.Rewrite.Mapping"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "AList.insert", "module": "Mathlib.Data.List.AList"}, {"name": "String", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Valuation.map", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Subset", "module": "Mathlib.Data.Set.Defs"}, {"name": "List.Subset", "module": "Init.Data.List.Basic"}, {"name": "StateT.bind", "module": "Init.Control.State"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}], "used_repo_defs": [{"name": "Mapping", "content": "abbrev Mapping (Γ Δ : Ctxt Ty) : Type :=\n  @AList (Σ t, Var Γ t) (fun x => Var Δ x.1)"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "last", "content": "@[match_pattern]\ndef last (Γ : Ctxt Ty) (t : Ty) : Ctxt.Var (Ctxt.cons t Γ) t :=\n  ⟨0, by admit /- proof elided -/\n  ⟩"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Option.bind_eq_some_iff", "module": "Init.Data.Option.Lemmas"}, {"name": "AList.entries_insert_of_notMem", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup_eq_none", "module": "Mathlib.Data.List.AList"}, {"name": "List.subset_cons_of_subset", "module": "Init.Data.List.Sublist"}, {"name": "Option.mem_def", "module": "Init.Data.Option.Instances"}, {"name": "IsEmpty.forall_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "forall_eq'", "module": "Init.PropLemmas"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "appendCases_appendInl", "content": "@[simp] theorem appendCases_appendInl (v : Γ.Var t) :\n    appendCases (motive := motive) left right v.appendInl = (left v)"}], "used_local_defs": [{"name": "MatchVarM", "content": "abbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)"}, {"name": "MatchVar", "content": "abbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit"}, {"name": "MatchVarM.unifyVars", "content": "def MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)"}, {"name": "matchArg", "content": "def matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)"}, {"name": "matchVar", "content": "def matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/"}, {"name": "MatchVar.IsMonotone", "content": "def MatchVar.IsMonotone (f : MatchVar Δ Γ) : Prop :=\n    ∀ mapIn, ∀ mapOut ∈ f mapIn,\n      mapIn.entries ⊆ mapOut.2.entries"}], "used_local_lemmas": [{"name": "unifyVars_eq_some_iff", "content": "@[simp]\ntheorem unifyVars_eq_some_iff :\n    unifyVars w v mapIn = some ((), mapOut)\n    ↔ ( mapIn.lookup ⟨t, w⟩ = none ∧ mapIn.insert ⟨t, w⟩ v = mapOut\n        ∨ mapIn.lookup ⟨t, w⟩ = v ∧ mapIn = mapOut\n      )"}, {"name": "MatchVar.isMonotone_bind", "content": "@[simp]\ntheorem MatchVar.isMonotone_bind {f : MatchVar Δ Γ} {g : Unit → MatchVar Δ Γ} :\n    f.IsMonotone → (g ()).IsMonotone → IsMonotone (f >>= g)"}, {"name": "MatchVar.isMonotone_bind_liftM", "content": "@[simp]\ntheorem MatchVar.isMonotone_bind_liftM {x? : Option α} {g : α → MatchVar Δ Γ} :\n    IsMonotone (liftM x? >>= g) ↔ (∀ x ∈ x?, (g x).IsMonotone)"}, {"name": "MatchVar.isMonotone_none", "content": "@[simp] theorem MatchVar.isMonotone_none : IsMonotone (none : MatchVar Δ Γ)"}, {"name": "MatchVar.isMonotone_unifyVars", "content": "theorem MatchVar.isMonotone_unifyVars  : IsMonotone (unifyVars w v)"}], "local_ctx": "import LeanMLIR.Framework\n\nimport LeanMLIR.Transforms.Rewrite.Mapping\n\nopen Ctxt (Var VarSet Valuation Hom)\n\nvariable {d} [DialectSignature d] [DecidableEq d.Ty]\n\nvariable {Γ : Ctxt d.Ty} {ty : d.Ty}\n\nabbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)\n\nabbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit\n\ndef MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)\n\nopen MatchVarM\n\nvariable [DecidableEq d.Op]\n\ndef matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)\n\ndef matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/\n\nend\n\nsection MatchVar\n\nvariable [DecidableEq d.Op] {Γ_in Γ_out Δ_in Δ_out t te}\n          {lets : Lets d Γ_in eff Γ_out} {v : Var Γ_out t}\n          {matchLets : Lets d Δ_in .pure Δ_out}\n          {matchExpr : Expr d Δ_out .pure te}\n\nvariable (lets v matchLets w) (mapIn : Mapping _ _) in\n\nvariable (lets matchLets) {tys} (vs ws : HVector _ tys) (mapIn : Mapping _ _) in\n\nnamespace MatchVarResult\n\nvariable [TyDenote d.Ty] [∀ (t : d.Ty), Inhabited ⟦t⟧] in\n\nsection Left\n\nvariable {w : Δ_out.Var t}\n\nvariable {mapIn} (mapOut : MatchVarResult lets v (.var matchLets matchExpr) w.appendInr mapIn)\n\nend Left\n\nvariable {w : Var ⟨te⟩ _} {mapIn}\n\nend MatchVarResult\n\nend MatchVar\n\nsection SubsetEntries\n\ndef MatchVar.IsMonotone (f : MatchVar Δ Γ) : Prop :=\n    ∀ mapIn, ∀ mapOut ∈ f mapIn,\n      mapIn.entries ⊆ mapOut.2.entries\n\nopen MatchVar\n\nsection UnifyVars\n\nvariable {Δ Γ : Ctxt d.Ty} {t} (w : Δ.Var t) (v : Γ.Var t)\n\nend UnifyVars\n\nvariable [DecidableEq d.Op]", "target_theorem": "theorem isMonotone_matchVarArg_aux (lets : Lets d Γ_in eff Γ_out) :\n    (\n     ∀  (Δ_out : Ctxt d.Ty)\n        (matchLets : Lets d Δ_in EffectKind.pure Δ_out) (l : List d.Ty)\n        (argsl : HVector Γ_out.Var l) (argsr : HVector Δ_out.Var l),\n        (matchArg lets matchLets argsl argsr).IsMonotone\n    )\n    ∧ (\n      ∀ (Δ_out : Ctxt d.Ty) (t : d.Ty) (v : Γ_out.Var t)\n        (matchLets : Lets d Δ_in EffectKind.pure Δ_out)\n        (w : Var Δ_out t),\n        (matchVar lets v matchLets w).IsMonotone\n    ) :=", "ground_truth_proof": ":= by\n  apply matchArg.mutual_induct (d:=d)\n  · intro _ _ mapIn ⟨(), mapOut⟩ hvarMap\n    obtain rfl : mapIn = mapOut := by\n      simp only [matchArg] at hvarMap\n      change some ((), _) = some ((), _) at hvarMap\n      simp_all\n    exact Set.Subset.refl _\n\n  · intros\n    simp only [matchArg]\n    apply isMonotone_bind <;> assumption\n\n  · intro _ _ Δ_out u matchLets matchExpr l h ih\n    cases l using Var.appendCases with\n    | right _ => simp [ih]\n    | left _ =>\n      simp only [matchVar, Var.appendCases_appendInl, isMonotone_bind_liftM,\n      Option.mem_def]\n      intro ⟨_, e⟩ h_getPureExpr\n      split_ifs\n      · apply h; assumption\n      · exact isMonotone_none\n      · exact isMonotone_none\n\n  · simp [isMonotone_unifyVars]", "nesting_depth": 7, "transitive_dep_count": 61, "subset_aristotle": false, "category": "Compiler"}
{"id": 329, "thm_name": "Deleted.toHom_append", "thm_stmt": "@[simp] lemma Deleted.toHom_append {Γ Γ' : Ctxt Ty} {vs : DeleteRange Γ}\n    (DEL : Deleted (⟨us⟩ ++ Γ) vs.appendInl (⟨us⟩ ++ Γ')) :\n    DEL.toHom\n    = have DEL' : Deleted Γ vs Γ'", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Transforms/DCE.lean", "imports": ["import Mathlib.Tactic.DepRewrite", "import LeanMLIR.Framework", "import Mathlib.Tactic.Linarith", "import LeanMLIR.LeanMLIR.ErasedContext"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "List.ofFn", "module": "Init.Data.List.OfFn"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List.length", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "length", "content": "@[grind=]\ndef length (Γ : Ctxt Ty) : Nat := Γ.toList.length"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "Hom.append", "content": "def Hom.append {ζ : Ctxt Ty} (f : Γ.Hom Δ) : Hom (ζ ++ Γ) (ζ ++ Δ) :=\n  fun _ => Var.appendCases\n    (fun v => v.appendInl)\n    (fun v => (f v).appendInr)"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "appendInl", "content": "def appendInl (v : Γ.Var t) : (Γ ++ Δ).Var t :=\n  ⟨v.val, by admit /- proof elided -/\n  ⟩"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "toHom", "content": "def toHom (d : Diff Γ₁ Γ₂) : Hom Γ₁ Γ₂ :=\n  fun _ v => ⟨v.val + d.val, d.property v.property⟩"}, {"name": "Hom", "content": "abbrev Hom (Γ Γ' : Ctxt Ty) := ⦃t : Ty⦄ → Γ.Var t → Γ'.Var t"}, {"name": "Diff", "content": "def Diff (Γ₁ Γ₂ : Ctxt Ty) : Type :=\n  {d : Nat // Diff.Valid Γ₁ Γ₂ d}"}, {"name": "Diff.Valid", "content": "@[simp]\nabbrev Diff.Valid (Γ₁ Γ₂ : Ctxt Ty) (d : Nat) : Prop :=\n  ∀ {i t}, Γ₁[i]? = some t → Γ₂[i+d]? = some t"}], "lib_lemmas": [{"name": "Fin.coe_cast", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.getElem?_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.getElem?_ofFn", "module": "Init.Data.List.OfFn"}, {"name": "List.getElem_append_right", "module": "Init.Data.List.BasicAux"}, {"name": "List.length_append", "module": "Init.Data.List.Basic"}, {"name": "Nat.ge_of_not_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Option.dite_none_right_eq_some", "module": "Init.Data.Option.Lemmas"}, {"name": "Valuation.ext", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "dite_eq_ite", "module": "Init.ByCases"}], "repo_lemmas": [{"name": "ofList_append", "content": "@[simp] theorem ofList_append {ts us : List Ty} :\n    Ctxt.ofList ts ++ Ctxt.ofList us = Ctxt.ofList (ts ++ us)"}, {"name": "length_ofList", "content": "@[simp, grind=] theorem length_ofList : (ofList ts).length = ts.length"}, {"name": "getElem?_ofList", "content": "@[simp, grind=] theorem getElem?_ofList (i : Nat) : (ofList ts)[i]? = ts[i]?"}, {"name": "getElem_ofList", "content": "@[simp, grind=] theorem getElem_ofList (i : Nat) (h : _) : (ofList ts)[i]'h = ts[i]'h"}, {"name": "val_lt", "content": "theorem val_lt (v : Γ.Var t) : v.val < Γ.length"}, {"name": "castCtxt_rfl", "content": "@[simp, grind=] theorem castCtxt_rfl (h : Γ = Γ) : v.castCtxt h = v"}, {"name": "val_castCtxt", "content": "@[simp, grind=] theorem val_castCtxt : (castCtxt h v).val = v.val"}], "used_local_defs": [{"name": "DCE.DeleteRange", "content": "structure DeleteRange (Γ : Ctxt Ty) where\n   \n  start : Fin (Γ.length + 1)\n   \n  num : Fin (Γ.length + 1 - start.val)"}, {"name": "DCE.DeleteRange.appendInl", "content": "def appendInl {Γ : Ctxt Ty} {ts : List Ty}\n    (r : DeleteRange Γ) : DeleteRange (⟨ts⟩ ++ Γ) where\n  start := ⟨r.start + ts.length, by admit /- proof elided -/\n  ⟩\n  num := ⟨r.num, by admit /- proof elided -/\n  ⟩"}, {"name": "Ctxt.delete", "content": "def Ctxt.delete (Γ : Ctxt Ty) (vs : DeleteRange Γ) : Ctxt Ty :=\n  Ctxt.ofList <| List.ofFn (n := Γ.length - vs.num.val) fun i =>\n    have := vs.start.prop\n    if hi : i.val < vs.start then\n      Γ[i.val]\n    else\n      Γ[i.val + vs.num]"}, {"name": "Hom.delete", "content": "def Hom.delete {Γ : Ctxt Ty} (delv : DeleteRange Γ) : Hom (Γ.delete delv) Γ :=\n  fun t' v =>\n    let idx :=\n      if v.val < delv.start then\n        v.val\n      else\n        v.val + delv.num\n    ⟨idx, by admit /- proof elided -/\n    ⟩"}, {"name": "Deleted", "content": "def Deleted (Γ: Ctxt Ty) (vs : DeleteRange Γ) (Γ' : Ctxt Ty) : Prop :=\n  Γ' = Γ.delete vs"}, {"name": "Deleted.toHom", "content": "def Deleted.toHom (h : Deleted Γ r Γ') : Γ'.Hom Γ :=\n  fun _ v => Hom.delete r (v.castCtxt h)"}], "used_local_lemmas": [{"name": "Ctxt.delete_append_appendInl", "content": "@[simp] theorem Ctxt.delete_append_appendInl {Γ : Ctxt Ty} {us : List Ty}\n    {r : DeleteRange Γ} :\n    (⟨us⟩ ++ Γ).delete r.appendInl = ⟨us⟩ ++ (Γ.delete r)"}], "local_ctx": "import LeanMLIR.Framework\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Tactic.DepRewrite\n\nopen Ctxt (Var Valuation Hom)\n\nnamespace DCE\n\nstructure DeleteRange (Γ : Ctxt Ty) where\n   \n  start : Fin (Γ.length + 1)\n   \n  num : Fin (Γ.length + 1 - start.val)\n\nopen DCE (DeleteRange)\n\nnamespace DeleteRange\n\ndef appendInl {Γ : Ctxt Ty} {ts : List Ty}\n    (r : DeleteRange Γ) : DeleteRange (⟨ts⟩ ++ Γ) where\n  start := ⟨r.start + ts.length, by admit /- proof elided -/\n  ⟩\n  num := ⟨r.num, by admit /- proof elided -/\n  ⟩\n\nsection Lemmas\n\nend Lemmas\n\nend DeleteRange\n\nend DCE\n\nopen DCE (DeleteRange)\n\ndef Ctxt.delete (Γ : Ctxt Ty) (vs : DeleteRange Γ) : Ctxt Ty :=\n  Ctxt.ofList <| List.ofFn (n := Γ.length - vs.num.val) fun i =>\n    have := vs.start.prop\n    if hi : i.val < vs.start then\n      Γ[i.val]\n    else\n      Γ[i.val + vs.num]\n\nsection Lemmas\n\nvariable {Γ : Ctxt Ty}\n\nend Lemmas\n\ndef Hom.delete {Γ : Ctxt Ty} (delv : DeleteRange Γ) : Hom (Γ.delete delv) Γ :=\n  fun t' v =>\n    let idx :=\n      if v.val < delv.start then\n        v.val\n      else\n        v.val + delv.num\n    ⟨idx, by admit /- proof elided -/\n    ⟩\n\ndef Deleted (Γ: Ctxt Ty) (vs : DeleteRange Γ) (Γ' : Ctxt Ty) : Prop :=\n  Γ' = Γ.delete vs\n\ndef Deleted.toHom (h : Deleted Γ r Γ') : Γ'.Hom Γ :=\n  fun _ v => Hom.delete r (v.castCtxt h)", "target_theorem": "@[simp] lemma Deleted.toHom_append {Γ Γ' : Ctxt Ty} {vs : DeleteRange Γ}\n    (DEL : Deleted (⟨us⟩ ++ Γ) vs.appendInl (⟨us⟩ ++ Γ')) :\n    DEL.toHom\n    = have DEL' : Deleted Γ vs Γ' :=", "ground_truth_proof": ":= by\n        rcases Γ'\n        simp only [Deleted, Ctxt.delete_append_appendInl] at DEL ⊢\n        injection DEL\n        simp_all [Ctxt.delete]\n      DEL'.toHom.append := by\n  have DEL' : Deleted Γ vs Γ' := by\n    rcases Γ'\n    simp only [Deleted, Ctxt.delete_append_appendInl] at DEL ⊢\n    injection DEL\n    simp_all [Ctxt.delete]\n  subst DEL'\n  funext t v\n  apply Subtype.ext\n  simp only [toHom, Hom.delete, Var.val_castCtxt, DeleteRange.val_start_appendInl,\n    DeleteRange.val_num_appendInl, Hom.append, Var.castCtxt_rfl]\n  cases v using Var.appendCases with\n  | right _  => simp; grind\n  | left v =>\n    have := v.val_lt\n    simp; grind", "nesting_depth": 3, "transitive_dep_count": 45, "subset_aristotle": false, "category": "Compiler"}
{"id": 330, "thm_name": "Poly.fromTensor_toTensor", "thm_stmt": "theorem fromTensor_toTensor [hqgt1 : Fact (q > 1)] (a : R q n)\n    (adeg : (R.representative q n a).natDegree + 1 < 2^n) :\n  R.fromTensor a.toTensor = a", "lean_root": "lean-mlir", "rel_path": "SSA/Projects/FullyHomomorphicEncryption/Statements.lean", "imports": ["import Mathlib.Data.List.Basic", "import Batteries.Data.List.Lemmas", "import SSA.Projects.FullyHomomorphicEncryption.Basic"], "used_lib_defs": [{"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Function.surjInv", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.degree", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Ideal", "module": "Mathlib.RingTheory.Ideal.Defs"}, {"name": "Ideal.Quotient.mk", "module": "Mathlib.RingTheory.Ideal.Quotient.Defs"}, {"name": "Ideal.span", "module": "Mathlib.RingTheory.Ideal.Span"}, {"name": "Polynomial.monomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Polynomial.coeff", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "List.range", "module": "Init.Data.List.Basic"}, {"name": "Polynomial.map", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "Fact", "module": "Mathlib.Logic.Basic"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "ZMod.cast", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "WithBot", "module": "Mathlib.Order.TypeTags"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "Int.cast", "module": "Init.Data.Int.Basic"}, {"name": "IntCast", "module": "Init.Data.Int.Basic"}, {"name": "IntCast.intCast", "module": "Init.Data.Int.Basic"}, {"name": "List.toFinsupp", "module": "Mathlib.Data.List.ToFinsupp"}, {"name": "Polynomial.ofFinsupp", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "List.getD", "module": "Init.Data.List.BasicAux"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Polynomial.toFinsupp", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}], "used_repo_defs": [{"name": "syntax \"neg\" : MLIR.Pretty.uniform_op", "content": "syntax \"neg\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "R.repLength", "content": "noncomputable def R.repLength {q n} (a : R q n) : Nat := match\n  Polynomial.degree a.representative with\n    | none => 0\n    | some d => d + 1"}, {"name": "R.representative", "content": "noncomputable def R.representative :\n    R q n → (ZMod q)[X] := fun x => R.representative' q n x %ₘ (f q n)"}, {"name": "R.representative'", "content": "private noncomputable def R.representative' :\n    R q n → (ZMod q)[X] := Function.surjInv (R.surjective_fromPoly q n)"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "R.fromTensor", "content": "noncomputable def R.fromTensor {q n} (coeffs : List Int) : R q n :=\n  coeffs.zipIdx.foldl (init := 0) fun res (c, i) =>\n    res + R.monomial ↑c i"}, {"name": "R.monomial", "content": "noncomputable def R.monomial {q n : Nat} (c : ZMod q) (i : Nat): R q n :=\n  R.fromPoly (Polynomial.monomial i c)"}, {"name": "R.fromPoly", "content": "abbrev R.fromPoly {q n : Nat} : (ZMod q)[X] →+* R q n := Ideal.Quotient.mk (Ideal.span {f q n})"}, {"name": "R.toTensor", "content": "noncomputable def R.toTensor {q n} (a : R q n) : List Int :=\n  List.range a.repLength |>.map fun i =>\n        a.coeff i |>.toInt"}, {"name": "R.coeff", "content": "noncomputable def R.coeff {q n} (a : R q n) (i : Nat) : ZMod q :=\n  Polynomial.coeff a.representative i"}, {"name": "ZMod.toInt", "content": "def ZMod.toInt (x : ZMod q) : Int := ZMod.cast x"}, {"name": "R.fromTensorFinsupp", "content": "noncomputable def R.fromTensorFinsupp (q : Nat) (coeffs : List Int) : (ZMod q)[X] :=\n  Polynomial.ofFinsupp (List.toFinsupp (coeffs.map Int.cast))"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}], "lib_lemmas": [{"name": "List.getD_eq_default", "module": "Mathlib.Data.List.GetD"}, {"name": "List.getD_eq_getElem", "module": "Mathlib.Data.List.GetD"}, {"name": "Nat.not_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.coeff_eq_zero_of_degree_lt", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "WithBot.bot_lt_coe", "module": "Mathlib.Order.WithBot"}, {"name": "WithBot.coe_lt_coe", "module": "Mathlib.Order.WithBot"}, {"name": "ZMod.cast_zero", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Fact.elim", "module": "Mathlib.Logic.Basic"}, {"name": "Nat.cast_add", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_one", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.succ_eq_add_one", "module": "Init.Data.Nat.Basic"}, {"name": "ZMod.cast_eq_val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.natCast_val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_intCast", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "not_lt_zero'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}, {"name": "Polynomial.coeff_inj", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.degree_eq_bot", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "ZMod.coe_intCast", "module": "Mathlib.Data.ZMod.Basic"}], "repo_lemmas": [{"name": "R.fromPoly_kernel_eq_zero", "content": "@[simp]\ntheorem R.fromPoly_kernel_eq_zero (x : (ZMod q)[X]) : R.fromPoly (n := n) (f q n * x) = 0"}, {"name": "R.representative_fromPoly", "content": "theorem R.representative_fromPoly :\n    forall a : (ZMod q)[X], (R.fromPoly (n:=n) a).representative = a %ₘ (f q n)"}, {"name": "R.fromPoly_rep'_eq_ideal", "content": "theorem R.fromPoly_rep'_eq_ideal :\n    forall a : (ZMod q)[X],\n      ∃ i ∈ Ideal.span {f q n}, (R.fromPoly (n:=n) a).representative' = a + i"}, {"name": "f_monic", "content": "theorem f_monic : Monic (f q n)"}, {"name": "R.coeff_fromTensor", "content": "theorem R.coeff_fromTensor (tensor : List Int)\n    (htensorlen : tensor.length < 2^n) :\n    (R.fromTensor (q := q) (n := n) tensor).coeff i = (tensor.getD i 0)"}, {"name": "R.fromTensorFinsupp_coeffs", "content": "theorem R.fromTensorFinsupp_coeffs (coeffs : List Int) :\n  Polynomial.coeff (fromTensorFinsupp q coeffs) i = ↑(List.getD coeffs i 0)"}, {"name": "R.fromTensorFinsupp_degree", "content": "theorem R.fromTensorFinsupp_degree (q : Nat) (coeffs : List Int):\n  (R.fromTensorFinsupp q coeffs).degree ≤ coeffs.length"}, {"name": "Polynomial.degree_toFinsupp", "content": "theorem Polynomial.degree_toFinsupp [Semiring M] [DecidableEq M]\n  (xs : List M) :\n  degree { toFinsupp := List.toFinsupp (l := xs) } ≤ List.length xs"}, {"name": "R.fromTensor_eq_fromTensorFinsupp_fromPoly", "content": "theorem R.fromTensor_eq_fromTensorFinsupp_fromPoly {coeffs : List Int} :\n    R.fromTensor (q := q) (n := n) coeffs =\n  R.fromPoly (q := q) (n := n) (R.fromTensorFinsupp q coeffs)"}, {"name": "R.fromTensorFinsupp_concat_monomial", "content": "theorem R.fromTensorFinsupp_concat_monomial (c : Int) (cs : List Int) :\n    (R.fromTensorFinsupp q (cs ++ [c])) =\n      (R.fromTensorFinsupp q cs) +\n        (Polynomial.monomial cs.length (Int.cast c : (ZMod q)))"}, {"name": "R.representative_fromPoly_eq", "content": "@[simp]\ntheorem R.representative_fromPoly_eq (x : (ZMod q)[X]) (DEGREE: x.degree < (f q n).degree) :\n   R.representative q n (R.fromPoly (n:=n) x) = x"}, {"name": "f_deg_eq", "content": "theorem f_deg_eq : (f q n).degree = 2^n"}, {"name": "R.fromPoly_representative", "content": "@[simp]\ntheorem R.fromPoly_representative [Fact (q > 1)]:\n    forall a : R q n, (R.fromPoly (n:=n) (R.representative q n a)) = a"}, {"name": "R.repLength_leq_representative_degree_plus_1", "content": "theorem R.repLength_leq_representative_degree_plus_1 (a : R q n) :\n  a.repLength ≤ (R.representative q n a).natDegree + 1"}, {"name": "R.toTensor_length", "content": "theorem R.toTensor_length {q n} (a : R q n) :\n    (R.toTensor a).length = a.repLength"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Poly.eq_iff_rep_eq", "content": "theorem eq_iff_rep_eq [Fact (q > 1)] (a b : R q n) :\n    a.representative = b.representative ↔ a = b"}, {"name": "Poly.from_poly_zero", "content": "theorem from_poly_zero : R.fromPoly (0 : (ZMod q)[X]) (n := n) = (0 : R q n)"}, {"name": "Poly.rep_zero", "content": "theorem rep_zero [Fact (q > 1)]: R.representative q n 0 = 0"}, {"name": "R.toTensor_getD", "content": "theorem R.toTensor_getD [Fact (q > 1)] (a : R q n) (i : Nat) :\n    a.toTensor.getD i 0 = (a.coeff i).toInt"}, {"name": "R.toTensor_getD'", "content": "theorem R.toTensor_getD' [hqgt1 : Fact (q > 1)] (a : R q n) (i : Nat) :\n    ↑(a.toTensor.getD i 0) = a.coeff i"}, {"name": "Poly.eq_iff_coeff_eq", "content": "theorem eq_iff_coeff_eq [hqgt1 : Fact (q > 1)] (a b : R q n) :\n    a = b ↔ Polynomial.coeff a.representative = Polynomial.coeff b.representative"}, {"name": "Poly.toTensor_fromTensor", "content": "theorem toTensor_fromTensor [hqgt1 : Fact (q > 1)] (tensor : List Int) (i : Nat)\n  (htensorlen : List.length tensor < 2 ^ n) :\n  (R.fromTensor tensor (q:=q) (n :=n)).toTensor.getD i 0 = (tensor.getD i 0) % q"}], "local_ctx": "import SSA.Projects.FullyHomomorphicEncryption.Basic\n\nimport Batteries.Data.List.Lemmas\n\nimport Mathlib.Data.List.Basic\n\nnamespace Poly\n\nopen Polynomial in\n\nopen Polynomial in\n\nend Poly\n\nnamespace Poly", "target_theorem": "theorem fromTensor_toTensor [hqgt1 : Fact (q > 1)] (a : R q n)\n    (adeg : (R.representative q n a).natDegree + 1 < 2^n) :\n  R.fromTensor a.toTensor = a :=", "ground_truth_proof": ":= by\n  cases h : Polynomial.degree (R.representative q n a) with\n    | bot =>\n        have h' :=  Polynomial.degree_eq_bot.1 h\n        rw [← rep_zero] at h'\n        have h'' := (eq_iff_rep_eq _ _).1 h'\n        simp only [R.fromTensor, R.toTensor, R.repLength]\n        rw [h, h'']\n        simp\n    | coe deg =>\n        apply (eq_iff_coeff_eq _ _).2\n        have hCoeff := R.toTensor_getD' (q := q) (n := n)\n        unfold R.coeff at hCoeff\n        apply funext\n        intro i\n        rw [← hCoeff, ← hCoeff]\n        rw [toTensor_fromTensor]\n        rw [← ZMod.coe_intCast]\n        norm_cast\n        · simp only [R.toTensor_length]\n          have hdeg := R.repLength_leq_representative_degree_plus_1 a\n          linarith", "nesting_depth": 5, "transitive_dep_count": 83, "subset_aristotle": false, "category": "Compiler"}
{"id": 331, "thm_name": "NFA'.correct_spec", "thm_stmt": "lemma NFA'.correct_spec {M : NFA' n} {ζ : M.sa} {L : BVNRel n} :\n    M.correct ζ L → M.accepts = langRel L", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "BitVec", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "Language", "module": "Mathlib.Computability.Language"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "List.Vector.ofFn", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.replicate", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}], "used_repo_defs": [{"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "enc", "content": "def enc (bvs : BitVecs n) : BitVecs' n :=\n  (List.finRange bvs.w).map (fun i =>\n    BitVec.ofFn (fun (k : Fin n) => (bvs.bvs.get k)[i]))"}, {"name": "BitVecs'", "content": "abbrev BitVecs' (n : Nat) := List (BitVec n)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "dec", "content": "@[simps]\ndef dec (bvs' : BitVecs' n) : BitVecs n where\n  w := bvs'.length\n  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k"}, {"name": "accepts", "content": "def accepts (M : NFA' n) : Set (BitVecs n) := dec '' M.accepts'"}, {"name": "NFA'", "content": "structure NFA' (n : Nat) where\n  σ : Type\n  M : NFA (BitVec n) σ"}, {"name": "accepts'", "content": "def accepts' (M : NFA' n) : Set (BitVecs' n) := M.M.accepts"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}], "lib_lemmas": [{"name": "NFA.eval_append_singleton", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.eval_nil", "module": "Mathlib.Computability.NFA"}], "repo_lemmas": [{"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}], "used_local_defs": [{"name": "NFA.sa", "content": "def NFA.sa (_ : NFA α σ) := σ → Language α"}, {"name": "NFA.correct", "content": "structure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q"}, {"name": "BVNRel", "content": "abbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop"}, {"name": "NFA'.sa", "content": "def NFA'.sa (M : NFA' n) := M.σ → BVNRel n"}, {"name": "langRel", "content": "def langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }"}, {"name": "NFA'.correct", "content": "structure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))"}, {"name": "NFA'.correct2", "content": "structure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)"}], "used_local_lemmas": [{"name": "NFA.correct_spec", "content": "lemma NFA.correct_spec {M : NFA α σ} {ζ : M.sa} {L : Language α} :\n    M.correct ζ L → M.accepts = L"}, {"name": "in_enc", "content": "@[simp]\nlemma in_enc : x ∈ enc '' S ↔ dec x ∈ S"}, {"name": "dec_snoc_in_langRel", "content": "@[simp]\nlemma dec_snoc_in_langRel {n} {R : BVNRel n} {w : BitVecs' n} {a : BitVec n} :\n    dec (w ++ [a]) ∈ langRel R ↔\n      R (List.Vector.ofFn fun k => .cons (a.getLsbD k) ((dec w).bvs.get k))"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\ndef NFA.sa (_ : NFA α σ) := σ → Language α\n\nstructure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q\n\nabbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop\n\ndef NFA'.sa (M : NFA' n) := M.σ → BVNRel n\n\ndef langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }\n\nstructure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))\n\nstructure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)", "target_theorem": "lemma NFA'.correct_spec {M : NFA' n} {ζ : M.sa} {L : BVNRel n} :\n    M.correct ζ L → M.accepts = langRel L :=", "ground_truth_proof": ":= by\n  rintro ⟨h1, h2, h3⟩\n  simp [accepts, accepts']\n  have heq : dec '' (enc '' langRel L) = langRel L := by simp\n  rw [←heq]\n  congr!\n  suffices h : M.M.correct (fun q => enc '' langRel (ζ q)) (enc '' langRel L) by\n    apply NFA.correct_spec h\n  constructor\n  · intros w; rw [in_enc]; simp [langRel, h1]; simp_rw [@in_enc _ _ w]; rfl\n  intros w; induction w using List.reverseRecOn\n  case nil =>\n    intros q; simp only [NFA.eval_nil]; rw [in_enc]; simp [h2, langRel]\n  case append_singleton w a ih =>\n    rintro q\n    simp only [NFA.eval_append_singleton]\n    rw [in_enc]\n    have h : M.M.eval w = { q | w ∈ enc '' langRel (ζ q) } := by\n      ext; rw [ih]; dsimp; rfl\n    rw [dec_snoc_in_langRel]\n    rw [h]; simp_rw [in_enc]\n    simp [langRel, h3]", "nesting_depth": 3, "transitive_dep_count": 37, "subset_aristotle": false, "category": "Compiler"}
{"id": 332, "thm_name": "mem_matchVar", "thm_stmt": "theorem mem_matchVar {Δ_out}\n    {varMap : Mapping Δ_in Γ_out} {ma : Mapping Δ_in Γ_out}\n    {lets : Lets d Γ_in eff Γ_out} {v : Var Γ_out t} /- : -/\n    {matchLets : Lets d Δ_in .pure Δ_out}  {w : Var Δ_out t}\n    (hvarMap : ((), varMap) ∈ matchVar lets v matchLets w ma)\n    {t': _ } {v' : _}\n    (hMatchLets : ⟨t', v'⟩ ∈ matchLets.vars w) :\n  ⟨t', v'⟩ ∈ varMap :=\n  match matchLets /- , hvarMap, t', v' -/ with\n  | .nil => by\n    revert hMatchLets\n    simp only [Lets.vars, VarSet.ofVar, Finset.mem_singleton, Sigma.mk.inj_iff, and_imp]\n    rintro ⟨⟩ ⟨⟩\n    simp only [matchVar, Option.mem_def, unifyVars_eq_some_iff] at hvarMap\n    rcases hvarMap with ⟨_, rfl⟩ | ⟨h_lookup, rfl⟩\n    · simp\n    · simp [← AList.lookup_isSome, h_lookup]\n\n  | .var matchLets matchE => by\n    simp only [matchVar, Option.mem_def] at hvarMap\n    cases w using Var.appendCases with\n    | right w =>\n        simp only [Var.appendCases_appendInr] at hvarMap\n        apply mem_matchVar hvarMap\n        simpa [Lets.vars] using hMatchLets\n    | left w =>\n        simp only [Var.appendCases_appendInl, MatchVar.liftM_bind_eq_some_iff] at hvarMap\n        rcases hvarMap with ⟨h_isSome, hvarMap⟩\n        split_ifs at hvarMap with h_pure h_var <;> (try contradiction)\n        subst h_var\n        apply mem_matchArg hvarMap\n        rcases matchE with ⟨matchOp, _⟩\n        obtain rfl : matchOp = _ := h_pure.1.symm\n        simpa [Lets.vars] using hMatchLets", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Transforms/Rewrite/Match.lean", "imports": ["import LeanMLIR.Framework", "import LeanMLIR.LeanMLIR.Framework.Basic", "import LeanMLIR.Transforms.Rewrite.Mapping"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "AList.insert", "module": "Mathlib.Data.List.AList"}, {"name": "Valuation.map", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "Sigma.mk", "module": "Init.Core"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}, {"name": "StateT.bind", "module": "Init.Control.State"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Subset", "module": "Mathlib.Data.Set.Defs"}, {"name": "List.Subset", "module": "Init.Data.List.Basic"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}], "used_repo_defs": [{"name": "Mapping", "content": "abbrev Mapping (Γ Δ : Ctxt Ty) : Type :=\n  @AList (Σ t, Var Γ t) (fun x => Var Δ x.1)"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "last", "content": "@[match_pattern]\ndef last (Γ : Ctxt Ty) (t : Ty) : Ctxt.Var (Ctxt.cons t Γ) t :=\n  ⟨0, by admit /- proof elided -/\n  ⟩"}, {"name": "Lets.vars", "content": "def Lets.vars : Lets d Γ_in eff Γ_out → Var Γ_out t → VarSet Γ_in\n  | .nil, v => VarSet.ofVar v\n  | .var lets e, v => by admit /- proof elided -/\n      | right v => exact lets.vars v\n      | left _ => exact lets.varsOfVec e.args"}, {"name": "Com.vars", "content": "def Com.vars (com : Com d Γ eff ts) : VarSet Γ :=\n  com.toLets.varsOfVec com.returnVars"}, {"name": "Expr.returnVars", "content": "def Expr.returnVars (e : Expr d Γ eff tys) : HVector e.outContext.Var tys :=\n  .ofFn _ _ <| fun i => (Var.ofFin i).appendInl"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) "}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "ofFin", "content": "def ofFin (i : Fin Γ.length) : Γ.Var (Γ[i]) :=\n  ⟨i.val, by admit /- proof elided -/\n  ⟩"}, {"name": "Lets.varsOfVec", "content": "def Lets.varsOfVec (lets : Lets d Γ_in eff Γ_out) (vs : HVector Γ_out.Var ts) :\n    VarSet Γ_in :=\n  (vs.vars).biUnion (fun v => lets.vars v.2)"}, {"name": "HVector.vars", "content": "def HVector.vars {l : List d.Ty} (T : HVector (Var Γ) l) : VarSet Γ :=\n  T.foldl (fun _ s a => insert ⟨_, a⟩ s) ∅"}, {"name": "VarSet", "content": "abbrev VarSet (Γ : Ctxt Ty) : Type :=\n  Finset (Σ t, Γ.Var t)"}, {"name": "Com.returnVars", "content": "def Com.returnVars : (com : Com d Γ eff ts) → HVector (Var com.outContext) ts\n  | .rets vs => vs\n  | .var _ body => body.returnVars"}, {"name": "Com.toLets", "content": "def Com.toLets (com : Com d Γ eff t) : Lets d Γ eff com.outContext :=\n  Lets.nil.addComToEnd com"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "ofVar", "content": "@[simp]\ndef ofVar {Γ : Ctxt Ty} (v : Γ.Var t) : VarSet Γ :=\n  {⟨_, v⟩}"}, {"name": "foldl", "content": "def foldl {B : Type*} (f : ∀ (a : α), B → A a → B) :\n    ∀ {l : List α}, B → HVector A l → B\n  | [],   b, .nil       => b\n  | t::_, b, .cons a as => foldl f (f t b a) as"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "IsEmpty.exists_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "Option.isSome_none", "module": "Init.Data.Option.Basic"}, {"name": "iff_false", "module": "Init.SimpLemmas"}, {"name": "AList.lookup_isSome", "module": "Mathlib.Data.List.AList"}, {"name": "Finset.mem_singleton", "module": "Mathlib.Data.Finset.Insert"}, {"name": "Option.mem_def", "module": "Init.Data.Option.Instances"}, {"name": "Sigma.mk.inj_iff", "module": "Mathlib.Data.Sigma.Basic"}, {"name": "and_imp", "module": "Init.SimpLemmas"}, {"name": "Option.bind_eq_some_iff", "module": "Init.Data.Option.Lemmas"}, {"name": "AList.entries_insert_of_notMem", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup_eq_none", "module": "Mathlib.Data.List.AList"}, {"name": "List.subset_cons_of_subset", "module": "Init.Data.List.Sublist"}, {"name": "IsEmpty.forall_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "forall_eq'", "module": "Init.PropLemmas"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "AList.keys_subset_keys_of_entries_subset_entries", "module": "Mathlib.Data.List.AList"}, {"name": "Finset.biUnion_insert", "module": "Mathlib.Data.Finset.Union"}, {"name": "Finset.mem_biUnion", "module": "Mathlib.Data.Finset.Union"}, {"name": "Finset.mem_union", "module": "Mathlib.Data.Finset.Lattice.Basic"}], "repo_lemmas": [{"name": "appendCases_appendInr", "content": "@[simp] theorem appendCases_appendInr (v : Γ.Var t) :\n    appendCases (motive := motive) left right v.appendInr = (right v)"}, {"name": "appendCases_appendInl", "content": "@[simp] theorem appendCases_appendInl (v : Γ.Var t) :\n    appendCases (motive := motive) left right v.appendInl = (left v)"}, {"name": "HVector.vars_cons", "content": "@[simp] theorem HVector.vars_cons {t  : d.Ty} {l : List d.Ty}\n    (v : Var Γ t) (T : HVector (Var Γ) l) :\n    (HVector.cons v T).vars = insert ⟨_, v⟩ T.vars"}], "used_local_defs": [{"name": "MatchVarM", "content": "abbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)"}, {"name": "MatchVar", "content": "abbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit"}, {"name": "MatchVarM.unifyVars", "content": "def MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)"}, {"name": "matchArg", "content": "def matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)"}, {"name": "matchVar", "content": "def matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/"}, {"name": "MatchVar.IsMonotone", "content": "def MatchVar.IsMonotone (f : MatchVar Δ Γ) : Prop :=\n    ∀ mapIn, ∀ mapOut ∈ f mapIn,\n      mapIn.entries ⊆ mapOut.2.entries"}], "used_local_lemmas": [{"name": "unifyVars_eq_some_iff", "content": "@[simp]\ntheorem unifyVars_eq_some_iff :\n    unifyVars w v mapIn = some ((), mapOut)\n    ↔ ( mapIn.lookup ⟨t, w⟩ = none ∧ mapIn.insert ⟨t, w⟩ v = mapOut\n        ∨ mapIn.lookup ⟨t, w⟩ = v ∧ mapIn = mapOut\n      )"}, {"name": "MatchVar.liftM_bind_eq_some_iff", "content": "@[simp]\ntheorem MatchVar.liftM_bind_eq_some_iff (x? : Option α)\n    (f : α → MatchVarM Δ Γ β) :\n    ((liftM x? >>= f) mapIn = some mapOut)\n    ↔ ( ∃ h : x?.isSome,\n        f (x?.get h) mapIn = some mapOut )"}, {"name": "MatchVar.isMonotone_bind", "content": "@[simp]\ntheorem MatchVar.isMonotone_bind {f : MatchVar Δ Γ} {g : Unit → MatchVar Δ Γ} :\n    f.IsMonotone → (g ()).IsMonotone → IsMonotone (f >>= g)"}, {"name": "MatchVar.isMonotone_bind_liftM", "content": "@[simp]\ntheorem MatchVar.isMonotone_bind_liftM {x? : Option α} {g : α → MatchVar Δ Γ} :\n    IsMonotone (liftM x? >>= g) ↔ (∀ x ∈ x?, (g x).IsMonotone)"}, {"name": "MatchVar.isMonotone_none", "content": "@[simp] theorem MatchVar.isMonotone_none : IsMonotone (none : MatchVar Δ Γ)"}, {"name": "MatchVar.isMonotone_unifyVars", "content": "theorem MatchVar.isMonotone_unifyVars  : IsMonotone (unifyVars w v)"}, {"name": "isMonotone_matchVarArg_aux", "content": "theorem isMonotone_matchVarArg_aux (lets : Lets d Γ_in eff Γ_out) :\n    (\n     ∀  (Δ_out : Ctxt d.Ty)\n        (matchLets : Lets d Δ_in EffectKind.pure Δ_out) (l : List d.Ty)\n        (argsl : HVector Γ_out.Var l) (argsr : HVector Δ_out.Var l),\n        (matchArg lets matchLets argsl argsr).IsMonotone\n    )\n    ∧ (\n      ∀ (Δ_out : Ctxt d.Ty) (t : d.Ty) (v : Γ_out.Var t)\n        (matchLets : Lets d Δ_in EffectKind.pure Δ_out)\n        (w : Var Δ_out t),\n        (matchVar lets v matchLets w).IsMonotone\n    )"}, {"name": "isMonotone_matchArg", "content": "theorem isMonotone_matchArg [DecidableEq d.Op]\n    {Γ_out Δ_in Δ_out : Ctxt d.Ty}\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}\n    {l : List d.Ty}\n    {argsl : HVector (Var Γ_out) l}\n    {argsr : HVector (Var Δ_out) l} :\n    (matchArg lets matchLets argsl argsr).IsMonotone"}, {"name": "mem_matchArg", "content": "theorem mem_matchArg {Δ_out}\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}\n    {l : List d.Ty} {argsₗ : HVector (Var Γ_out) l}\n    {argsᵣ : HVector (Var Δ_out) l} {ma : Mapping Δ_in Γ_out}\n    {varMap : Mapping Δ_in Γ_out}\n    (hvarMap : ((), varMap) ∈ matchArg lets matchLets argsₗ argsᵣ ma)\n    {t' v'} : ⟨t', v'⟩ ∈ matchLets.varsOfVec argsᵣ → ⟨t', v'⟩ ∈ varMap"}], "local_ctx": "import LeanMLIR.Framework\n\nimport LeanMLIR.Transforms.Rewrite.Mapping\n\nopen Ctxt (Var VarSet Valuation Hom)\n\nvariable {d} [DialectSignature d] [DecidableEq d.Ty]\n\nvariable {Γ : Ctxt d.Ty} {ty : d.Ty}\n\nabbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)\n\nabbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit\n\ndef MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)\n\nopen MatchVarM\n\nvariable [DecidableEq d.Op]\n\ndef matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)\n\ndef matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/\n\nend\n\nsection MatchVar\n\nvariable [DecidableEq d.Op] {Γ_in Γ_out Δ_in Δ_out t te}\n          {lets : Lets d Γ_in eff Γ_out} {v : Var Γ_out t}\n          {matchLets : Lets d Δ_in .pure Δ_out}\n          {matchExpr : Expr d Δ_out .pure te}\n\nvariable (lets v matchLets w) (mapIn : Mapping _ _) in\n\nvariable (lets matchLets) {tys} (vs ws : HVector _ tys) (mapIn : Mapping _ _) in\n\nnamespace MatchVarResult\n\nvariable [TyDenote d.Ty] [∀ (t : d.Ty), Inhabited ⟦t⟧] in\n\nsection Left\n\nvariable {w : Δ_out.Var t}\n\nvariable {mapIn} (mapOut : MatchVarResult lets v (.var matchLets matchExpr) w.appendInr mapIn)\n\nend Left\n\nvariable {w : Var ⟨te⟩ _} {mapIn}\n\nend MatchVarResult\n\nend MatchVar\n\nsection SubsetEntries\n\ndef MatchVar.IsMonotone (f : MatchVar Δ Γ) : Prop :=\n    ∀ mapIn, ∀ mapOut ∈ f mapIn,\n      mapIn.entries ⊆ mapOut.2.entries\n\nopen MatchVar\n\nsection UnifyVars\n\nvariable {Δ Γ : Ctxt d.Ty} {t} (w : Δ.Var t) (v : Γ.Var t)\n\nend UnifyVars\n\nvariable [DecidableEq d.Op]\n\nend SubsetEntries\n\nnamespace MatchArgResult\n\nvariable [DecidableEq d.Op] {Γ_in Γ_out Δ_in Δ_out te}\n          {lets : Lets d Γ_in eff Γ_out}\n          {matchLets : Lets d Δ_in .pure Δ_out}\n          {matchExpr : Expr d Δ_out .pure te}\n          {u us}\n          {v : Γ_out.Var u} {vs : HVector Γ_out.Var us}\n          {w : Δ_out.Var u} {ws : HVector Δ_out.Var us}\n          {mapIn : Mapping _ _}\n          (mapOut : MatchArgResult lets matchLets (v ::ₕ vs) (w ::ₕ ws) mapIn)\n\nend MatchArgResult\n\nsection DenoteLemmas\n\nvariable [TyDenote d.Ty] [DecidableEq d.Op]\n\nvariable [∀ (t : d.Ty), Inhabited ⟦t⟧]\n\nvariable [Monad d.m] [LawfulMonad d.m] [DialectDenote d]\n\nsection DenoteIntoSubtype\n\nend DenoteIntoSubtype\n\nvariable {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty}\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}", "target_theorem": "theorem mem_matchVar {Δ_out}\n    {varMap : Mapping Δ_in Γ_out} {ma : Mapping Δ_in Γ_out}\n    {lets : Lets d Γ_in eff Γ_out} {v : Var Γ_out t} /- : -/\n    {matchLets : Lets d Δ_in .pure Δ_out}  {w : Var Δ_out t}\n    (hvarMap : ((), varMap) ∈ matchVar lets v matchLets w ma)\n    {t': _ } {v' : _}\n    (hMatchLets : ⟨t', v'⟩ ∈ matchLets.vars w) :\n  ⟨t', v'⟩ ∈ varMap :=", "ground_truth_proof": ":=\n  match matchLets /- , hvarMap, t', v' -/ with\n  | .nil => by\n    revert hMatchLets\n    simp only [Lets.vars, VarSet.ofVar, Finset.mem_singleton, Sigma.mk.inj_iff, and_imp]\n    rintro ⟨⟩ ⟨⟩\n    simp only [matchVar, Option.mem_def, unifyVars_eq_some_iff] at hvarMap\n    rcases hvarMap with ⟨_, rfl⟩ | ⟨h_lookup, rfl⟩\n    · simp\n    · simp [← AList.lookup_isSome, h_lookup]\n\n  | .var matchLets matchE => by\n    simp only [matchVar, Option.mem_def] at hvarMap\n    cases w using Var.appendCases with\n    | right w =>\n        simp only [Var.appendCases_appendInr] at hvarMap\n        apply mem_matchVar hvarMap\n        simpa [Lets.vars] using hMatchLets\n    | left w =>\n        simp only [Var.appendCases_appendInl, MatchVar.liftM_bind_eq_some_iff] at hvarMap\n        rcases hvarMap with ⟨h_isSome, hvarMap⟩\n        split_ifs at hvarMap with h_pure h_var <;> (try contradiction)\n        subst h_var\n        apply mem_matchArg hvarMap\n        rcases matchE with ⟨matchOp, _⟩\n        obtain rfl : matchOp = _ := h_pure.1.symm\n        simpa [Lets.vars] using hMatchLets", "nesting_depth": 7, "transitive_dep_count": 104, "subset_aristotle": false, "category": "Compiler"}
{"id": 333, "thm_name": "MultiWidth.eval_fsmMsb_eq_BitStream_ofBitVecSext", "thm_stmt": "theorem eval_fsmMsb_eq_BitStream_ofBitVecSext {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (x : Term bcard ncard icard pcard tctx (.bv w))\n    (xfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm x))\n    (hxfsm : HTermFSMToBitStream xfsm)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmMsb xfsm.toFsmZext wfsm.toFsm).eval fsmEnv =\n      BitStream.ofBitVecSext (x.toBV benv nenv ienv penv tenv)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/MultiWidth/GoodFSM.lean", "imports": ["import Blase.MultiWidth.Defs", "import Blase.Vars", "import Blase.KInduction.KInduction", "import Lean", "import Blase.Fast.FiniteStateMachine"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat.max", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.min", "module": "Init.Data.Nat.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.ofBool", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"min\" : MLIR.Pretty.uniform_op", "content": "syntax \"min\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "composeBinaryAux'", "content": "def composeBinaryAux'\n    (p : FSM Bool)\n    (qtrue : FSM α)\n    (qfalse : FSM α) :\n    FSM α :=\n  p.compose (α)\n    (λ _ => α)\n    (λ _ i => i)\n    (λ b => match b with\n      | true => qtrue\n      | false => qfalse)"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "latchImmediate", "content": "def latchImmediate (initVal : Bool) : FSM Bool where\n  α := Unit\n  initCarry := fun _ => initVal\n  outputCirc :=\n    let xval := Circuit.var true (inr false)\n    let control := Circuit.var true (inr true)\n    let state := Circuit.var true (inl ())\n    Circuit.ite control xval state\n  nextStateCirc := fun () =>\n    let xval := Circuit.var true (inr false)\n    let control := Circuit.var true (inr true)\n    let state := Circuit.var true (inl ())\n    Circuit.ite control xval state"}, {"name": "ite", "content": "def ite (cond t f : Circuit α) : Circuit α :=\n  (cond &&& t) ||| (~~~ cond &&& f)"}, {"name": "TermFSM", "content": "structure TermFSM (wcard tcard bcard ncard icard pcard : Nat) (t : Nondep.Term) where\n  toFsmZext : FSM (StateSpace wcard tcard bcard ncard icard pcard)\n  width : NatFSM wcard tcard bcard ncard icard pcard t.width"}, {"name": "NatFSM", "content": "structure NatFSM (wcard tcard bcard ncard icard pcard : Nat) (v : Nondep.WidthExpr) where\n  toFsm : FSM (StateSpace wcard tcard bcard ncard icard pcard)"}, {"name": "StateSpace", "content": "inductive StateSpace (wcard tcard bcard ncard icard pcard : Nat)\n| widthVar (v : Fin wcard)\n| termVar (v : Fin tcard)\n| predVar (v : Fin pcard)\n| boolVar (v : Fin bcard)\nderiving DecidableEq, Repr, Hashable"}, {"name": "Term", "content": "inductive Term\n| ofNat (w : WidthExpr) (n : Nat) : Term\n| var (v : Nat) (w : WidthExpr) : Term\n| add (w : WidthExpr) (a b : Term) : Term\n| zext (a : Term) (wnew : WidthExpr) : Term\n| setWidth (a : Term) (wnew : WidthExpr) : Term\n| sext (a : Term) (wnew : WidthExpr) : Term\n| bor (w : WidthExpr) (a b : Term) : Term\n| band (w : WidthExpr) (a b : Term) : Term\n| bxor (w : WidthExpr) (a b : Term) : Term\n| bnot (w : WidthExpr)  (a : Term) : Term\n| boolVar (v : Nat) : Term\n| boolConst (b : Bool) : Term\n| shiftl (w : WidthExpr) (a : Term) (k : Nat) : Term\n| bvOfBool (b : Term) : Term\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr) : Term\n| binRel (k : BinaryRelationKind) (w : WidthExpr)\n    (a : Term) (b : Term) : Term\n| or (p1 p2 : Term) : Term\n| and (p1 p2 : Term) : Term\n| pvar (v : Nat) : Term\n| boolBinRel (k : BoolBinaryRelationKind)\n    (a b : Term) : Term\nderiving DecidableEq, Inhabited, Repr, Lean.ToExpr"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : Nat → WidthExpr → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : (k : Nat) → (v : WidthExpr) → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "HNatFSMToBitstream", "content": "structure HNatFSMToBitstream {wcard : Nat} {v : WidthExpr wcard} {tcard : Nat} {bcard : Nat} {pcard : Nat}\n   (fsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v)) : Prop where\n  heq :\n    ∀ (wenv : Fin wcard → Nat)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n    (henv : HWidthEnv fsmEnv wenv) →\n      fsm.toFsm.eval fsmEnv =\n      BitStream.ofNatUnary (v.toNat wenv)"}, {"name": "WidthExpr.toNat", "content": "def WidthExpr.toNat (e : WidthExpr wcard) (env : WidthExpr.Env wcard) : Nat :=\n  match e with\n  | .const n => n\n  | .var v => env v\n  | .min v w => Nat.min (v.toNat env) (w.toNat env)\n  | .max v w => Nat.max (v.toNat env) (w.toNat env)\n  | .addK v k => v.toNat env + k\n  | .kadd k v => k + v.toNat env"}, {"name": "WidthExpr", "content": "inductive WidthExpr (wcard : Nat) : Type\n| const (n : Nat) :  WidthExpr wcard\n| var : (v : Fin wcard) → WidthExpr wcard\n| min : (v w : WidthExpr wcard) → WidthExpr wcard\n| max : (v w : WidthExpr wcard) → WidthExpr wcard\n| addK : (v : WidthExpr wcard) → (k : Nat) → WidthExpr wcard\n| kadd : (k : Nat) → (v : WidthExpr wcard) → WidthExpr wcard"}, {"name": "WidthExpr.Env", "content": "abbrev WidthExpr.Env (wcard : Nat) : Type :=\n  Fin wcard → Nat"}, {"name": "HWidthEnv", "content": "structure HWidthEnv {wcard tcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (wenv : Fin wcard → Nat) : Prop where\n    heq_width : ∀ (v : Fin wcard),\n      fsmEnv (StateSpace.widthVar v) = BitStream.ofNatUnary (wenv v)"}, {"name": "HPredicateEnv", "content": "structure HPredicateEnv {wcard tcard bcard ncard icard pcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (penv : Fin pcard → Prop) : Prop where\n    heq_width : ∀ (v : Fin pcard),\n      fsmEnv (StateSpace.predVar v) = BitStream.ofProp (penv v)"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "ofNatUnary", "content": "abbrev ofNatUnary (n : Nat) : BitStream :=\n  fun i => decide (i < n)"}, {"name": "HPredFSMToBitStream", "content": "structure HPredFSMToBitStream {pcard : Nat}\n  {tctx : Term.Ctx wcard tcard}\n  {p : Term bcard ncard icard pcard tctx .prop}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard\n    (.ofDepTerm p)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (htenv : HTermEnv fsmEnv tenv benv) →\n      (hpenv : HPredicateEnv fsmEnv penv) →\n        p.toBV benv nenv ienv penv tenv  ↔ (fsm.toFsmZext.eval fsmEnv = .negOne)"}, {"name": "Term.Ctx", "content": "abbrev Term.Ctx (wcard : Nat) (tcard : Nat) : Type :=\n  Fin tcard → WidthExpr wcard"}, {"name": "Term.BoolEnv", "content": "def Term.BoolEnv (bcard : Nat) : Type := Fin bcard → Bool"}, {"name": "Term.IntEnv", "content": "def Term.IntEnv (icard : Nat) : Type := Fin icard → Nat"}, {"name": "HTermFSMToBitStream", "content": "structure HTermFSMToBitStream {w : WidthExpr wcard}\n  {tctx : Term.Ctx wcard tcard}\n  {t : Term bcard ncard icard pcard tctx (.bv w)}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm t)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (henv : HTermEnv fsmEnv tenv benv) →\n        fsm.toFsmZext.eval fsmEnv =\n        BitStream.ofBitVecZext (t.toBV benv nenv ienv penv tenv)"}, {"name": "Predicate.Env", "content": "def Predicate.Env (pcard : Nat) : Type :=\n  Fin pcard → Prop"}, {"name": "TermKind", "content": "inductive TermKind (wcard : Nat) : Type\n| bool\n| bv (w : WidthExpr wcard)  : TermKind wcard\n| prop\n| nat\n| int"}, {"name": "HTermEnv", "content": "structure HTermEnv {wcard tcard bcard : Nat}\n    {wenv : Fin wcard → Nat} {tctx : Term.Ctx wcard tcard}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (tenv : tctx.Env wenv)\n    (benv : Term.BoolEnv bcard) : Prop\n  extends HWidthEnv fsmEnv wenv where\n    heq_term : ∀ (v : Fin tcard),\n      fsmEnv (StateSpace.termVar v) = BitStream.ofBitVecZext (tenv v)\n    heq_bool : ∀ (v : Fin bcard),\n      fsmEnv (StateSpace.boolVar v) = BitStream.ofBool (benv v)"}, {"name": "BitStream.ofBool", "content": "noncomputable def BitStream.ofBool (b : Bool) : BitStream := fun _i => b"}, {"name": "Term.Ctx.Env", "content": "abbrev Term.Ctx.Env\n  (tctx : Term.Ctx wcard tcard)\n  (wenv : WidthExpr.Env wcard) :=\n  (v : Fin tcard) → BitVec ((tctx v).toNat wenv)"}, {"name": "ofBitVecZext", "content": "abbrev ofBitVecZext {w} (x : BitVec w) : BitStream :=\n  fun i => x.getLsbD i"}, {"name": "Term.NatEnv", "content": "def Term.NatEnv (ncard : Nat) : Type := Fin ncard → Nat"}, {"name": "Term.toBV", "content": "def Term.toBV {wenv : WidthExpr.Env wcard}\n    {tctx : Term.Ctx wcard tcard}\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (penv : Predicate.Env pcard)\n    (tenv : tctx.Env wenv)\n    (t : Term bcard ncard icard pcard tctx k) : k.denote wenv :=\nmatch t with\n| .ofNat w n => BitVec.ofNat (w.toNat wenv) n\n| .boolConst b => b\n| .var v => tenv.get v.1 v.2\n| .add (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a + b\n| .zext a v => (a.toBV benv nenv ienv penv tenv).zeroExtend (v.toNat wenv)\n| .setWidth a v => (a.toBV benv nenv ienv penv tenv).zeroExtend (v.toNat wenv)\n| .sext a v => (a.toBV benv nenv ienv penv tenv).signExtend (v.toNat wenv)\n| .bor a b (w := w) =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a ||| b\n| .band (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a &&& b\n| .bxor (w := w) a b =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    let b : BitVec (w.toNat wenv) := (b.toBV benv nenv ienv penv tenv)\n    a ^^^ b\n| .bnot (w := w) a =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    ~~~ a\n| .boolVar v => benv v\n| .shiftl (w := w) a k =>\n    let a : BitVec (w.toNat wenv) := (a.toBV benv nenv ienv penv tenv)\n    a <<< k\n| .bvOfBool b => BitVec.ofBool (b.toBV benv nenv ienv penv tenv)\n\n| .binWidthRel rel wa wb =>\n  match rel with\n  | .eq => wa.toNat wenv = wb.toNat wenv\n  | .le => wa.toNat wenv ≤ wb.toNat wenv\n| .binRel rel _w a b =>\n  match rel with\n  | .eq => a.toBV benv nenv ienv penv tenv = b.toBV benv nenv ienv penv tenv\n  | .ne => a.toBV benv nenv ienv penv tenv ≠ b.toBV benv nenv ienv penv tenv\n  | .ult => (a.toBV benv nenv ienv penv tenv).ult (b.toBV benv nenv ienv penv tenv) = true\n  | .ule => (a.toBV benv nenv ienv penv tenv).ule (b.toBV benv nenv ienv penv tenv) = true\n  | .slt => (a.toBV benv nenv ienv penv tenv).slt (b.toBV benv nenv ienv penv tenv) = true\n  | .sle => (a.toBV benv nenv ienv penv tenv).sle (b.toBV benv nenv ienv penv tenv) = true\n| .and p1 p2 => p1.toBV benv nenv ienv penv tenv  ∧ p2.toBV benv nenv ienv penv tenv\n| .or p1 p2 => p1.toBV benv nenv ienv penv tenv ∨ p2.toBV benv nenv ienv penv tenv\n| .boolBinRel rel a b =>\n  match rel with\n  \n  | .eq => (a.toBV benv nenv ienv penv tenv) = (b.toBV benv nenv ienv penv tenv)\n| .pvar v => penv v"}, {"name": "Term", "content": "inductive Term {wcard tcard : Nat} (bcard : Nat) (ncard : Nat) (icard : Nat) (pcard : Nat)\n  (tctx : Term.Ctx wcard tcard) : TermKind wcard → Type\n\n \n| ofNat (w : WidthExpr wcard) (n : Nat) : Term bcard ncard icard pcard tctx (.bv w)\n \n| var (v : Fin tcard) : Term bcard ncard icard pcard tctx (.bv (tctx v))\n \n| add (a : Term bcard ncard icard pcard tctx (.bv w))\n  (b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| shiftl (a : Term bcard ncard icard pcard tctx (.bv w)) (k : Nat) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bor (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| band (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bxor (a b : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| bnot (a : Term bcard ncard icard pcard tctx (.bv w)) : Term bcard ncard icard pcard tctx (.bv w)\n \n| zext (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| setWidth (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| sext (a : Term bcard ncard icard pcard tctx (.bv w)) (v : WidthExpr wcard) : Term bcard ncard icard pcard tctx (.bv v)\n \n| bvOfBool (b : Term bcard ncard icard pcard tctx .bool) : Term bcard ncard icard pcard tctx (.bv (.const 1))\n\n| boolConst (b : Bool) : Term bcard ncard icard pcard tctx .bool\n| boolVar (v : Fin bcard) : Term bcard ncard icard pcard tctx .bool\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr wcard) :\n    Term bcard ncard icard pcard tctx .prop\n| binRel\n    (k : BinaryRelationKind)\n    (w : WidthExpr wcard)\n    (a : Term bcard ncard icard pcard tctx (.bv w))\n    (b : Term bcard ncard icard pcard tctx (.bv w)) :\n    Term bcard ncard icard pcard tctx .prop\n| and (p1 p2 : Term bcard ncard icard pcard tctx (.prop)) : Term bcard ncard icard pcard tctx (.prop)\n| or (p1 p2 : Term bcard ncard icard pcard tctx (.prop)) : Term bcard ncard icard pcard tctx (.prop)\n| pvar (v : Fin pcard) : Term bcard ncard icard pcard tctx (.prop) \n\n\n| boolBinRel\n  (k : BoolBinaryRelationKind)\n  (a b : Term bcard ncard icard pcard tctx .bool) :\n  Term bcard ncard icard pcard tctx (.prop)"}, {"name": "Term.Ctx.Env.get", "content": "def Term.Ctx.Env.get {tcard : Nat}\n  {wcard : Nat} {wenv : Fin wcard → Nat}\n  {tctx : Term.Ctx wcard tcard}\n  (tenv : tctx.Env wenv) (i : Nat) (hi : i < tcard) :\n  BitVec ((tctx ⟨i, hi⟩).toNat wenv) :=\n  tenv ⟨i, hi⟩"}, {"name": "BinaryRelationKind", "content": "inductive BinaryRelationKind\n| eq\n| ne\n| ule\n| slt\n| sle\n| ult \nderiving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "WidthBinaryRelationKind", "content": "inductive WidthBinaryRelationKind\n| eq\n| le\n\n\nderiving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}], "lib_lemmas": [{"name": "BitVec.msb_eq_getLsbD_last", "module": "Init.Data.BitVec.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "MultiWidth.fsmMsb", "content": "def fsmMsb (x w : FSM α) : FSM α :=\n  composeBinaryAux'\n    (FSM.latchImmediate false)\n    (qfalse := x)\n    (qtrue := w)"}], "used_local_lemmas": [{"name": "MultiWidth.eval_fsmMsb_eq", "content": "@[simp]\ntheorem eval_fsmMsb_eq {wcard bcard tcard : Nat}\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (x : Term bcard ncard icard pcard tctx (.bv w))\n    (xfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm x))\n    (hxfsm : HTermFSMToBitStream xfsm)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmMsb xfsm.toFsmZext wfsm.toFsm).eval fsmEnv = (fun i =>\n      BitStream.ofBitVecZext (x.toBV benv nenv ienv penv tenv) (min i (w.toNat wenv - 1)))"}], "local_ctx": "import Blase.Fast.FiniteStateMachine\n\nimport Blase.Vars\n\nimport Blase.MultiWidth.Defs\n\nimport Blase.KInduction.KInduction\n\nimport Lean\n\nnamespace MultiWidth\n\ndef fsmMsb (x w : FSM α) : FSM α :=\n  composeBinaryAux'\n    (FSM.latchImmediate false)\n    (qfalse := x)\n    (qtrue := w)", "target_theorem": "theorem eval_fsmMsb_eq_BitStream_ofBitVecSext {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    {tctx : Term.Ctx wcard tcard}\n    (tenv : Term.Ctx.Env tctx wenv)\n    (benv : Term.BoolEnv bcard)\n    (nenv : Term.NatEnv ncard)\n    (ienv : Term.IntEnv icard)\n    (w : WidthExpr wcard)\n    (x : Term bcard ncard icard pcard tctx (.bv w))\n    (xfsm : TermFSM wcard tcard bcard ncard icard pcard (.ofDepTerm x))\n    (hxfsm : HTermFSMToBitStream xfsm)\n    (wfsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    (hwfsm : HNatFSMToBitstream wfsm)\n    (htenv : HTermEnv fsmEnv tenv benv) :\n    (fsmMsb xfsm.toFsmZext wfsm.toFsm).eval fsmEnv =\n      BitStream.ofBitVecSext (x.toBV benv nenv ienv penv tenv) :=", "ground_truth_proof": ":= by\n  rw [eval_fsmMsb_eq (wfsm := wfsm) (hwfsm := hwfsm) (hxfsm := hxfsm)\n    (tenv := tenv) (htenv := htenv)]\n  ext i\n  simp [BitStream.ofBitVecSext]\n  by_cases hi : i < w.toNat wenv\n  · simp [hi]\n    congr\n    omega\n  · simp [hi, BitVec.msb_eq_getLsbD_last]\n    congr\n    omega", "nesting_depth": 5, "transitive_dep_count": 66, "subset_aristotle": false, "category": "Compiler"}
{"id": 334, "thm_name": "MultiWidth.eval_fsmUltUnary_eq_decide", "thm_stmt": "theorem eval_fsmUltUnary_eq_decide\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (henv : HWidthEnv fsmEnv wenv)\n    (ha : HNatFSMToBitstream a) (hb : HNatFSMToBitstream b) :\n    ((fsmUltUnary a b).eval fsmEnv) i =\n   (decide (min i (v.toNat wenv) < min i (w.toNat wenv)))", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/MultiWidth/GoodFSM.lean", "imports": ["import Blase.MultiWidth.Defs", "import Blase.Vars", "import Blase.KInduction.KInduction", "import Lean", "import Blase.Blase.Fast.BitStream", "import Blase.Fast.FiniteStateMachine"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Empty", "module": "Init.Prelude"}, {"name": "Empty.elim", "module": "Init.Core"}, {"name": "Nat.max", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.min", "module": "Init.Data.Nat.Basic"}, {"name": "Decidable", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"min\" : MLIR.Pretty.uniform_op", "content": "syntax \"min\" : MLIR.Pretty.uniform_op\n\nsyntax \"neg\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "ls", "content": "def ls (b : Bool) : FSM Unit :=\n  { α := Unit,\n    initCarry := fun _ => b,\n    nextStateCirc := fun () => Circuit.var true (inr ()),\n    outputCirc := Circuit.var true (inl ())\n  }"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "StateSpace", "content": "inductive StateSpace (wcard tcard bcard ncard icard pcard : Nat)\n| widthVar (v : Fin wcard)\n| termVar (v : Fin tcard)\n| predVar (v : Fin pcard)\n| boolVar (v : Fin bcard)\nderiving DecidableEq, Repr, Hashable"}, {"name": "Term", "content": "inductive Term\n| ofNat (w : WidthExpr) (n : Nat) : Term\n| var (v : Nat) (w : WidthExpr) : Term\n| add (w : WidthExpr) (a b : Term) : Term\n| zext (a : Term) (wnew : WidthExpr) : Term\n| setWidth (a : Term) (wnew : WidthExpr) : Term\n| sext (a : Term) (wnew : WidthExpr) : Term\n| bor (w : WidthExpr) (a b : Term) : Term\n| band (w : WidthExpr) (a b : Term) : Term\n| bxor (w : WidthExpr) (a b : Term) : Term\n| bnot (w : WidthExpr)  (a : Term) : Term\n| boolVar (v : Nat) : Term\n| boolConst (b : Bool) : Term\n| shiftl (w : WidthExpr) (a : Term) (k : Nat) : Term\n| bvOfBool (b : Term) : Term\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr) : Term\n| binRel (k : BinaryRelationKind) (w : WidthExpr)\n    (a : Term) (b : Term) : Term\n| or (p1 p2 : Term) : Term\n| and (p1 p2 : Term) : Term\n| pvar (v : Nat) : Term\n| boolBinRel (k : BoolBinaryRelationKind)\n    (a b : Term) : Term\nderiving DecidableEq, Inhabited, Repr, Lean.ToExpr"}, {"name": "composeUnaryAux", "content": "def composeUnaryAux\n    (p : FSM Unit)\n    (q : FSM arity) :\n    FSM arity :=\n  p.compose\n    arity\n    _\n    (λ _ => id)\n    (λ _ => q)"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "NatFSM", "content": "structure NatFSM (wcard tcard bcard ncard icard pcard : Nat) (v : Nondep.WidthExpr) where\n  toFsm : FSM (StateSpace wcard tcard bcard ncard icard pcard)"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : Nat → WidthExpr → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "scanAnd", "content": "def scanAnd  : FSM Unit :=\n  {\n   α := Unit,\n   initCarry := fun () => true,\n   outputCirc := Circuit.var true (inl ()) &&& Circuit.var true (inr ()),\n   nextStateCirc := fun () => (Circuit.var true (inl ())) &&& (Circuit.var true (inr ()))\n  }"}, {"name": "scanOr", "content": "def scanOr  : FSM Unit :=\n  {\n   α := Unit,\n   initCarry := fun () => false,\n   outputCirc := Circuit.var true (inl ()) ||| Circuit.var true (inr ()),\n   nextStateCirc := fun () => Circuit.var true (inl ()) ||| Circuit.var true (inr ())\n  }"}, {"name": "composeBinaryAux'", "content": "def composeBinaryAux'\n    (p : FSM Bool)\n    (qtrue : FSM α)\n    (qfalse : FSM α) :\n    FSM α :=\n  p.compose (α)\n    (λ _ => α)\n    (λ _ i => i)\n    (λ b => match b with\n      | true => qtrue\n      | false => qfalse)"}, {"name": "and", "content": "def and : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := fun a => a.elim,\n    outputCirc := Circuit.var true (inr true) &&& Circuit.var true (inr false),\n  }"}, {"name": "WidthExpr.Env", "content": "abbrev WidthExpr.Env (wcard : Nat) : Type :=\n  Fin wcard → Nat"}, {"name": "HWidthEnv", "content": "structure HWidthEnv {wcard tcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (wenv : Fin wcard → Nat) : Prop where\n    heq_width : ∀ (v : Fin wcard),\n      fsmEnv (StateSpace.widthVar v) = BitStream.ofNatUnary (wenv v)"}, {"name": "HPredicateEnv", "content": "structure HPredicateEnv {wcard tcard bcard ncard icard pcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (penv : Fin pcard → Prop) : Prop where\n    heq_width : ∀ (v : Fin pcard),\n      fsmEnv (StateSpace.predVar v) = BitStream.ofProp (penv v)"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "ofNatUnary", "content": "abbrev ofNatUnary (n : Nat) : BitStream :=\n  fun i => decide (i < n)"}, {"name": "HNatFSMToBitstream", "content": "structure HNatFSMToBitstream {wcard : Nat} {v : WidthExpr wcard} {tcard : Nat} {bcard : Nat} {pcard : Nat}\n   (fsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v)) : Prop where\n  heq :\n    ∀ (wenv : Fin wcard → Nat)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n    (henv : HWidthEnv fsmEnv wenv) →\n      fsm.toFsm.eval fsmEnv =\n      BitStream.ofNatUnary (v.toNat wenv)"}, {"name": "WidthExpr.toNat", "content": "def WidthExpr.toNat (e : WidthExpr wcard) (env : WidthExpr.Env wcard) : Nat :=\n  match e with\n  | .const n => n\n  | .var v => env v\n  | .min v w => Nat.min (v.toNat env) (w.toNat env)\n  | .max v w => Nat.max (v.toNat env) (w.toNat env)\n  | .addK v k => v.toNat env + k\n  | .kadd k v => k + v.toNat env"}, {"name": "WidthExpr", "content": "inductive WidthExpr (wcard : Nat) : Type\n| const (n : Nat) :  WidthExpr wcard\n| var : (v : Fin wcard) → WidthExpr wcard\n| min : (v w : WidthExpr wcard) → WidthExpr wcard\n| max : (v w : WidthExpr wcard) → WidthExpr wcard\n| addK : (v : WidthExpr wcard) → (k : Nat) → WidthExpr wcard\n| kadd : (k : Nat) → (v : WidthExpr wcard) → WidthExpr wcard"}, {"name": "HPredFSMToBitStream", "content": "structure HPredFSMToBitStream {pcard : Nat}\n  {tctx : Term.Ctx wcard tcard}\n  {p : Term bcard ncard icard pcard tctx .prop}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard\n    (.ofDepTerm p)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (htenv : HTermEnv fsmEnv tenv benv) →\n      (hpenv : HPredicateEnv fsmEnv penv) →\n        p.toBV benv nenv ienv penv tenv  ↔ (fsm.toFsmZext.eval fsmEnv = .negOne)"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}], "lib_lemmas": [{"name": "decide_eq_true_iff", "module": "Init.PropLemmas"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "inf_of_le_left", "module": "Mathlib.Order.Lattice"}, {"name": "le_refl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "left_eq_inf", "module": "Mathlib.Order.Lattice"}, {"name": "not_iff", "module": "Mathlib.Logic.Basic"}, {"name": "not_le", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "decide_eq_decide", "module": "Init.PropLemmas"}], "repo_lemmas": [{"name": "scanOr_eq_decide", "content": "theorem scanOr_eq_decide (s : BitStream) (n : Nat) :\n    s.scanOr n = decide (∃ (i : Nat), i ≤ n ∧ s i = true)"}, {"name": "scanOr_true_iff", "content": "theorem scanOr_true_iff (s : BitStream) (n : Nat)\n    : s.scanOr n = true ↔ ∃ (i : Nat), (i ≤ n) ∧ s i = true"}, {"name": "scanOr_false_iff", "content": "theorem scanOr_false_iff (s : BitStream) (n : Nat) : s.scanOr n = false ↔ ∀ (i : Nat), (hi : i ≤ n) → s i = false"}, {"name": "scanOr_succ", "content": "@[simp]\ntheorem scanOr_succ (s : BitStream) : scanOr s (n+1) = ((s.scanOr n) || s (n + 1))"}, {"name": "scanAnd_eq_decide", "content": "theorem scanAnd_eq_decide (s : BitStream) (n : Nat) :\n    s.scanAnd n = decide (∀ (i : Nat), i ≤ n → s i = true)"}, {"name": "scanAnd_true_iff", "content": "theorem scanAnd_true_iff (s : BitStream) (n : Nat) :\n    s.scanAnd n = true ↔ ∀ (i : Nat), (hi : i ≤ n) → s i = true"}, {"name": "scanAnd_succ", "content": "@[simp] theorem scanAnd_succ (s : BitStream) : scanAnd s (n+1) = ((s.scanAnd n) && s (n + 1))"}], "used_local_defs": [{"name": "MultiWidth.NatFSM.fsmUnaryIndexUle", "content": "def NatFSM.fsmUnaryIndexUle (a : NatFSM wcard tcard bcard ncard icard pcard v) :\n    FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeUnaryAux (FSM.ls true) a.toFsm"}, {"name": "MultiWidth.fsmUnaryUle", "content": "def fsmUnaryUle (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w)) : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n composeUnaryAux FSM.scanAnd (b.fsmUnaryIndexUle ||| ~~~ a.fsmUnaryIndexUle)"}, {"name": "MultiWidth.fsmUnaryNeqUpto", "content": "def fsmUnaryNeqUpto (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w)) : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeUnaryAux FSM.scanOr (a.fsmUnaryIndexUle ^^^ b.fsmUnaryIndexUle)"}, {"name": "MultiWidth.fsmUltUnary", "content": "def fsmUltUnary\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w)) : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeBinaryAux' FSM.and (fsmUnaryUle a b) (fsmUnaryNeqUpto a b)"}], "used_local_lemmas": [{"name": "MultiWidth.decide_eq_eq_decide_iff_decide", "content": "private theorem decide_eq_eq_decide_iff_decide {P Q : Prop}\n  [Decidable P] [Decidable Q] :\n  (decide P = decide Q) = decide (P ↔ Q)"}, {"name": "MultiWidth.not_decide_eq_decide_lnot", "content": "private theorem not_decide_eq_decide_lnot {P : Prop}\n  [Decidable P] :\n    (!(decide P)) = (decide (¬ P))"}, {"name": "MultiWidth.decide_and_decide_eq_decide", "content": "private theorem decide_and_decide_eq_decide {P Q : Prop}\n  [Decidable P] [Decidable Q] :\n  (decide P && decide Q) = decide (P ∧ Q)"}, {"name": "MultiWidth.decide_or_decide_eq_decide", "content": "private theorem decide_or_decide_eq_decide {P Q : Prop}\n  [Decidable P] [Decidable Q] :\n  (decide P || decide Q) = decide (P ∨ Q)"}, {"name": "MultiWidth.min_eq_of_not_le", "content": "@[simp]\nprivate theorem min_eq_of_not_le {a b : Nat} (hab : ¬ a ≤ b) : min a b = b"}, {"name": "MultiWidth.min_eq_of_not_le'", "content": "@[simp]\nprivate theorem min_eq_of_not_le' {a b : Nat} (hab : ¬ a ≤ b) : min b a = b"}, {"name": "MultiWidth.HNatFSMToBitstream.fsmIndexUle_eval_eq", "content": "@[simp]\ntheorem HNatFSMToBitstream.fsmIndexUle_eval_eq\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (henv : HWidthEnv fsmEnv wenv)\n    (ha : HNatFSMToBitstream a) :\n    (NatFSM.fsmUnaryIndexUle a).eval fsmEnv = fun i =>\n    decide (i ≤ v.toNat wenv)"}, {"name": "MultiWidth.eval_fsmUnaryUle_eq_decide", "content": "theorem eval_fsmUnaryUle_eq_decide\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (henv : HWidthEnv fsmEnv wenv)\n    (ha : HNatFSMToBitstream a) (hb : HNatFSMToBitstream b) :\n    ((fsmUnaryUle a b).eval fsmEnv) i =\n    decide (min i (v.toNat wenv) ≤ min i (w.toNat wenv))"}, {"name": "MultiWidth.eval_fsmUnaryUle_eq_lt_or_decide", "content": "@[simp]\ntheorem eval_fsmUnaryUle_eq_lt_or_decide\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (henv : HWidthEnv fsmEnv wenv)\n    (ha : HNatFSMToBitstream a) (hb : HNatFSMToBitstream b) :\n    ((fsmUnaryUle a b).eval fsmEnv) i =\n    decide (i ≤ min (v.toNat wenv) (w.toNat wenv) ∨ (v.toNat wenv) ≤ (w.toNat wenv))"}, {"name": "MultiWidth.eval_fsmUnaryNeqUpto_eq_decide", "content": "@[simp]\ntheorem eval_fsmUnaryNeqUpto_eq_decide\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (henv : HWidthEnv fsmEnv wenv)\n    (ha : HNatFSMToBitstream a) (hb : HNatFSMToBitstream b) :\n    ((fsmUnaryNeqUpto a b).eval fsmEnv) i =\n    (decide (min i (v.toNat wenv) ≠ min i (w.toNat wenv)))"}], "local_ctx": "import Blase.Fast.FiniteStateMachine\n\nimport Blase.Vars\n\nimport Blase.MultiWidth.Defs\n\nimport Blase.KInduction.KInduction\n\nimport Lean\n\nnamespace MultiWidth\n\ndef NatFSM.fsmUnaryIndexUle (a : NatFSM wcard tcard bcard ncard icard pcard v) :\n    FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeUnaryAux (FSM.ls true) a.toFsm\n\ndef fsmUnaryUle (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w)) : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n composeUnaryAux FSM.scanAnd (b.fsmUnaryIndexUle ||| ~~~ a.fsmUnaryIndexUle)\n\ndef fsmUnaryNeqUpto (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w)) : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeUnaryAux FSM.scanOr (a.fsmUnaryIndexUle ^^^ b.fsmUnaryIndexUle)\n\ndef fsmUltUnary\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w)) : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeBinaryAux' FSM.and (fsmUnaryUle a b) (fsmUnaryNeqUpto a b)", "target_theorem": "theorem eval_fsmUltUnary_eq_decide\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (henv : HWidthEnv fsmEnv wenv)\n    (ha : HNatFSMToBitstream a) (hb : HNatFSMToBitstream b) :\n    ((fsmUltUnary a b).eval fsmEnv) i =\n   (decide (min i (v.toNat wenv) < min i (w.toNat wenv))) :=", "ground_truth_proof": ":= by\n  simp [fsmUltUnary]\n  rw [eval_fsmUnaryUle_eq_lt_or_decide (wenv := wenv) (henv := henv) (ha := ha) (hb := hb)]\n  rw [eval_fsmUnaryNeqUpto_eq_decide (wenv := wenv) (henv := henv) (ha := ha) (hb := hb)]\n  simp\n  generalize v.toNat wenv = v'\n  generalize w.toNat wenv = w'\n  simp only [not_decide_eq_decide_lnot,\n    decide_and_decide_eq_decide,\n    decide_or_decide_eq_decide, decide_eq_decide]\n  omega", "nesting_depth": 6, "transitive_dep_count": 78, "subset_aristotle": false, "category": "Compiler"}
{"id": 335, "thm_name": "MultiWidth.eval_fsmUnaryNeqUpto_eq_decide", "thm_stmt": "@[simp]\ntheorem eval_fsmUnaryNeqUpto_eq_decide\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (henv : HWidthEnv fsmEnv wenv)\n    (ha : HNatFSMToBitstream a) (hb : HNatFSMToBitstream b) :\n    ((fsmUnaryNeqUpto a b).eval fsmEnv) i =\n    (decide (min i (v.toNat wenv) ≠ min i (w.toNat wenv)))", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/MultiWidth/GoodFSM.lean", "imports": ["import Blase.MultiWidth.Defs", "import Blase.Vars", "import Blase.KInduction.KInduction", "import Lean", "import Blase.Blase.Fast.BitStream", "import Blase.Fast.FiniteStateMachine"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Nat.max", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.min", "module": "Init.Data.Nat.Basic"}, {"name": "Decidable", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"min\" : MLIR.Pretty.uniform_op", "content": "syntax \"min\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "ls", "content": "def ls (b : Bool) : FSM Unit :=\n  { α := Unit,\n    initCarry := fun _ => b,\n    nextStateCirc := fun () => Circuit.var true (inr ()),\n    outputCirc := Circuit.var true (inl ())\n  }"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "StateSpace", "content": "inductive StateSpace (wcard tcard bcard ncard icard pcard : Nat)\n| widthVar (v : Fin wcard)\n| termVar (v : Fin tcard)\n| predVar (v : Fin pcard)\n| boolVar (v : Fin bcard)\nderiving DecidableEq, Repr, Hashable"}, {"name": "Term", "content": "inductive Term\n| ofNat (w : WidthExpr) (n : Nat) : Term\n| var (v : Nat) (w : WidthExpr) : Term\n| add (w : WidthExpr) (a b : Term) : Term\n| zext (a : Term) (wnew : WidthExpr) : Term\n| setWidth (a : Term) (wnew : WidthExpr) : Term\n| sext (a : Term) (wnew : WidthExpr) : Term\n| bor (w : WidthExpr) (a b : Term) : Term\n| band (w : WidthExpr) (a b : Term) : Term\n| bxor (w : WidthExpr) (a b : Term) : Term\n| bnot (w : WidthExpr)  (a : Term) : Term\n| boolVar (v : Nat) : Term\n| boolConst (b : Bool) : Term\n| shiftl (w : WidthExpr) (a : Term) (k : Nat) : Term\n| bvOfBool (b : Term) : Term\n| binWidthRel (k : WidthBinaryRelationKind) (wa wb : WidthExpr) : Term\n| binRel (k : BinaryRelationKind) (w : WidthExpr)\n    (a : Term) (b : Term) : Term\n| or (p1 p2 : Term) : Term\n| and (p1 p2 : Term) : Term\n| pvar (v : Nat) : Term\n| boolBinRel (k : BoolBinaryRelationKind)\n    (a b : Term) : Term\nderiving DecidableEq, Inhabited, Repr, Lean.ToExpr"}, {"name": "composeUnaryAux", "content": "def composeUnaryAux\n    (p : FSM Unit)\n    (q : FSM arity) :\n    FSM arity :=\n  p.compose\n    arity\n    _\n    (λ _ => id)\n    (λ _ => q)"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "NatFSM", "content": "structure NatFSM (wcard tcard bcard ncard icard pcard : Nat) (v : Nondep.WidthExpr) where\n  toFsm : FSM (StateSpace wcard tcard bcard ncard icard pcard)"}, {"name": "WidthExpr", "content": "inductive WidthExpr where\n| const : Nat → WidthExpr\n| var : Nat → WidthExpr\n| max : WidthExpr → WidthExpr → WidthExpr\n| min : WidthExpr → WidthExpr → WidthExpr\n| addK : WidthExpr → Nat → WidthExpr\n| kadd : Nat → WidthExpr → WidthExpr\nderiving Inhabited, Repr, Hashable, DecidableEq, Lean.ToExpr"}, {"name": "scanOr", "content": "def scanOr  : FSM Unit :=\n  {\n   α := Unit,\n   initCarry := fun () => false,\n   outputCirc := Circuit.var true (inl ()) ||| Circuit.var true (inr ()),\n   nextStateCirc := fun () => Circuit.var true (inl ()) ||| Circuit.var true (inr ())\n  }"}, {"name": "WidthExpr.Env", "content": "abbrev WidthExpr.Env (wcard : Nat) : Type :=\n  Fin wcard → Nat"}, {"name": "HWidthEnv", "content": "structure HWidthEnv {wcard tcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (wenv : Fin wcard → Nat) : Prop where\n    heq_width : ∀ (v : Fin wcard),\n      fsmEnv (StateSpace.widthVar v) = BitStream.ofNatUnary (wenv v)"}, {"name": "HPredicateEnv", "content": "structure HPredicateEnv {wcard tcard bcard ncard icard pcard : Nat}\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream)\n    (penv : Fin pcard → Prop) : Prop where\n    heq_width : ∀ (v : Fin pcard),\n      fsmEnv (StateSpace.predVar v) = BitStream.ofProp (penv v)"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "ofNatUnary", "content": "abbrev ofNatUnary (n : Nat) : BitStream :=\n  fun i => decide (i < n)"}, {"name": "HNatFSMToBitstream", "content": "structure HNatFSMToBitstream {wcard : Nat} {v : WidthExpr wcard} {tcard : Nat} {bcard : Nat} {pcard : Nat}\n   (fsm : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v)) : Prop where\n  heq :\n    ∀ (wenv : Fin wcard → Nat)\n    (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n    (henv : HWidthEnv fsmEnv wenv) →\n      fsm.toFsm.eval fsmEnv =\n      BitStream.ofNatUnary (v.toNat wenv)"}, {"name": "WidthExpr.toNat", "content": "def WidthExpr.toNat (e : WidthExpr wcard) (env : WidthExpr.Env wcard) : Nat :=\n  match e with\n  | .const n => n\n  | .var v => env v\n  | .min v w => Nat.min (v.toNat env) (w.toNat env)\n  | .max v w => Nat.max (v.toNat env) (w.toNat env)\n  | .addK v k => v.toNat env + k\n  | .kadd k v => k + v.toNat env"}, {"name": "WidthExpr", "content": "inductive WidthExpr (wcard : Nat) : Type\n| const (n : Nat) :  WidthExpr wcard\n| var : (v : Fin wcard) → WidthExpr wcard\n| min : (v w : WidthExpr wcard) → WidthExpr wcard\n| max : (v w : WidthExpr wcard) → WidthExpr wcard\n| addK : (v : WidthExpr wcard) → (k : Nat) → WidthExpr wcard\n| kadd : (k : Nat) → (v : WidthExpr wcard) → WidthExpr wcard"}, {"name": "HPredFSMToBitStream", "content": "structure HPredFSMToBitStream {pcard : Nat}\n  {tctx : Term.Ctx wcard tcard}\n  {p : Term bcard ncard icard pcard tctx .prop}\n  (fsm : TermFSM wcard tcard bcard ncard icard pcard\n    (.ofDepTerm p)) : Prop where\n  heq :\n    ∀ {wenv : WidthExpr.Env wcard}\n      (benv : Term.BoolEnv bcard)\n      (nenv : Term.NatEnv ncard)\n      (ienv : Term.IntEnv icard)\n      (penv : Predicate.Env pcard) (tenv : tctx.Env wenv)\n      (fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream),\n      (htenv : HTermEnv fsmEnv tenv benv) →\n      (hpenv : HPredicateEnv fsmEnv penv) →\n        p.toBV benv nenv ienv penv tenv  ↔ (fsm.toFsmZext.eval fsmEnv = .negOne)"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}], "lib_lemmas": [{"name": "decide_eq_true_iff", "module": "Init.PropLemmas"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "inf_of_le_left", "module": "Mathlib.Order.Lattice"}, {"name": "le_refl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "left_eq_inf", "module": "Mathlib.Order.Lattice"}, {"name": "not_iff", "module": "Mathlib.Logic.Basic"}, {"name": "not_le", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}], "repo_lemmas": [{"name": "scanOr_eq_decide", "content": "theorem scanOr_eq_decide (s : BitStream) (n : Nat) :\n    s.scanOr n = decide (∃ (i : Nat), i ≤ n ∧ s i = true)"}, {"name": "scanOr_true_iff", "content": "theorem scanOr_true_iff (s : BitStream) (n : Nat)\n    : s.scanOr n = true ↔ ∃ (i : Nat), (i ≤ n) ∧ s i = true"}, {"name": "scanOr_false_iff", "content": "theorem scanOr_false_iff (s : BitStream) (n : Nat) : s.scanOr n = false ↔ ∀ (i : Nat), (hi : i ≤ n) → s i = false"}, {"name": "scanOr_succ", "content": "@[simp]\ntheorem scanOr_succ (s : BitStream) : scanOr s (n+1) = ((s.scanOr n) || s (n + 1))"}], "used_local_defs": [{"name": "MultiWidth.NatFSM.fsmUnaryIndexUle", "content": "def NatFSM.fsmUnaryIndexUle (a : NatFSM wcard tcard bcard ncard icard pcard v) :\n    FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeUnaryAux (FSM.ls true) a.toFsm"}, {"name": "MultiWidth.fsmUnaryNeqUpto", "content": "def fsmUnaryNeqUpto (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w)) : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeUnaryAux FSM.scanOr (a.fsmUnaryIndexUle ^^^ b.fsmUnaryIndexUle)"}], "used_local_lemmas": [{"name": "MultiWidth.decide_eq_eq_decide_iff_decide", "content": "private theorem decide_eq_eq_decide_iff_decide {P Q : Prop}\n  [Decidable P] [Decidable Q] :\n  (decide P = decide Q) = decide (P ↔ Q)"}, {"name": "MultiWidth.not_decide_eq_decide_lnot", "content": "private theorem not_decide_eq_decide_lnot {P : Prop}\n  [Decidable P] :\n    (!(decide P)) = (decide (¬ P))"}, {"name": "MultiWidth.min_eq_of_not_le", "content": "@[simp]\nprivate theorem min_eq_of_not_le {a b : Nat} (hab : ¬ a ≤ b) : min a b = b"}, {"name": "MultiWidth.min_eq_of_not_le'", "content": "@[simp]\nprivate theorem min_eq_of_not_le' {a b : Nat} (hab : ¬ a ≤ b) : min b a = b"}, {"name": "MultiWidth.HNatFSMToBitstream.fsmIndexUle_eval_eq", "content": "@[simp]\ntheorem HNatFSMToBitstream.fsmIndexUle_eval_eq\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (henv : HWidthEnv fsmEnv wenv)\n    (ha : HNatFSMToBitstream a) :\n    (NatFSM.fsmUnaryIndexUle a).eval fsmEnv = fun i =>\n    decide (i ≤ v.toNat wenv)"}], "local_ctx": "import Blase.Fast.FiniteStateMachine\n\nimport Blase.Vars\n\nimport Blase.MultiWidth.Defs\n\nimport Blase.KInduction.KInduction\n\nimport Lean\n\nnamespace MultiWidth\n\ndef NatFSM.fsmUnaryIndexUle (a : NatFSM wcard tcard bcard ncard icard pcard v) :\n    FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeUnaryAux (FSM.ls true) a.toFsm\n\ndef fsmUnaryNeqUpto (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w)) : FSM (StateSpace wcard tcard bcard ncard icard pcard) :=\n  composeUnaryAux FSM.scanOr (a.fsmUnaryIndexUle ^^^ b.fsmUnaryIndexUle)", "target_theorem": "@[simp]\ntheorem eval_fsmUnaryNeqUpto_eq_decide\n    (a : NatFSM wcard tcard bcard ncard icard pcard (.ofDep v))\n    (b : NatFSM wcard tcard bcard ncard icard pcard (.ofDep w))\n    {wenv : WidthExpr.Env wcard}\n    {fsmEnv : StateSpace wcard tcard bcard ncard icard pcard → BitStream}\n    (henv : HWidthEnv fsmEnv wenv)\n    (ha : HNatFSMToBitstream a) (hb : HNatFSMToBitstream b) :\n    ((fsmUnaryNeqUpto a b).eval fsmEnv) i =\n    (decide (min i (v.toNat wenv) ≠ min i (w.toNat wenv))) :=", "ground_truth_proof": ":= by\n  simp [fsmUnaryNeqUpto]\n  rw [ha.fsmIndexUle_eval_eq (henv := henv)]\n  rw [hb.fsmIndexUle_eval_eq (henv := henv)]\n  simp [BitStream.scanOr_eq_decide]\n  rw [not_decide_eq_decide_lnot]\n  rw [decide_eq_eq_decide_iff_decide]\n  rw [decide_eq_true_iff]\n  constructor\n  · intros hi\n    obtain ⟨j, hj₁, hj₂⟩ := hi\n    by_cases hiv : v.toNat wenv < i\n    · simp only [not_le, hiv, min_eq_of_not_le]\n      omega\n    · simp only [not_lt] at hiv\n      simp only [hiv, inf_of_le_left, left_eq_inf, not_le]\n      omega\n  · intros hivw\n    simp only [not_iff, not_le]\n    by_cases hiv : i < (v.toNat wenv)\n    · simp only [not_le, hiv, min_eq_of_not_le', left_eq_inf] at hivw ⊢\n      exists i\n      omega\n    · simp only [not_lt] at hiv; simp [hiv] at hivw\n      by_cases hiw : i < (w.toNat wenv)\n      · simp only [not_le, hiw, min_eq_of_not_le'] at hivw\n        have hiv' : (v.toNat wenv) ≤ i := by omega\n        exists i\n        omega\n      · simp only [not_lt] at hiw; simp [hiw] at hivw\n        by_cases hvw : v.toNat wenv < w.toNat wenv\n        · exists (w.toNat wenv)\n          simp only [le_refl, iff_true]\n          omega\n        · simp only [not_lt] at hvw\n          exists (v.toNat wenv)\n          omega", "nesting_depth": 5, "transitive_dep_count": 61, "subset_aristotle": false, "category": "Compiler"}
{"id": 336, "thm_name": "CNFA.autWidth_spec", "thm_stmt": "lemma CNFA.autWidth_spec : autWidth wp n |>.Sim (NFA'.autWidth wp n)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.Blase.AutoStructs.Basic", "import Blase.AutoStructs.Constructions", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "Fin.mk", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "BitVec.ofFin", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "SetRel", "module": "Mathlib.Data.Rel"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "LawfulBEq", "module": "Init.Core"}], "used_repo_defs": [{"name": "State", "content": "abbrev State := Nat"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "RawCNFA.empty", "content": "def RawCNFA.empty : RawCNFA A := {\n  stateMax := 0\n  initials := ∅\n  finals := ∅\n  trans := ∅\n}"}, {"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "WidthPredicate", "content": "inductive WidthPredicate\n| eq\n| neq\n| lt\n| le\n| gt\n| ge\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "instFinEnumBV", "content": "instance instFinEnumBV : FinEnum (BitVec w) where\n  card := 2^w\n  equiv := {\n    toFun := fun x => x.toFin\n    invFun := fun x => BitVec.ofFin x\n    left_inv := by admit /- proof elided -/"}, {"name": "CNFA.Sim", "content": "def CNFA.Sim (m : CNFA n) (M : NFA' n) :=\n  m.m.Sim M.M"}, {"name": "RawCNFA.Sim", "content": "def RawCNFA.Sim (m : RawCNFA A) (A : NFA A S) := ∃ R, RawCNFA.Simul m A R ⊤ ∅"}, {"name": "RawCNFA.Simul", "content": "structure RawCNFA.Simul (m : RawCNFA A) (M : NFA A Q) (R : SetRel State Q) (D : Set Q) (T : Set (Q × A × Q)) where\n  accept {s q} : s ~[R] q → (s ∈ m.finals ↔ q ∈ M.accept)\n  initial₁ {s} : s ∈ m.initials → ∃ q ∈ M.start, s ~[R] q\n  initial₂ {q} : q ∈ M.start → ∃ s ∈ m.initials, s ~[R] q\n  trans_match₁ {s s' a q} : s ~[R] q → s' ∈ m.tr s a → ∃ q', q' ∈ M.step q a ∧ s' ~[R] q'\n  trans_match₂ {s a q q'} : s ~[R] q → q' ∈ M.step q a → q ∈ D → (q, a, q') ∉ T → ∃ s', s' ∈ m.tr s a ∧ s' ~[R] q'"}, {"name": "RawCNFA.SimulFun", "content": "structure RawCNFA.SimulFun (m : RawCNFA A) (M : NFA A Q) (f : m.states ≃ Q)  where\n  accept {q} : ((f.invFun q).val ∈ m.finals ↔ q ∈ M.accept)\n  initial {q} : q ∈ M.start ↔ (f.invFun q).val ∈ m.initials\n  trans_match {a q q'} : q' ∈ M.step q a ↔ (f.invFun q').val ∈ m.tr (f.invFun q) a"}, {"name": "RawCNFA.tr", "content": "@[inline]\ndef RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}, {"name": "CNFA", "content": "structure CNFA (n : Nat) where\n  m : RawCNFA (BitVec n)\n  wf : m.WF"}, {"name": "NFA'", "content": "structure NFA' (n : Nat) where\n  σ : Type\n  M : NFA (BitVec n) σ"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "RawCNFA.states", "content": "def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax"}, {"name": "RawCNFA.WF", "content": "structure RawCNFA.WF (m : RawCNFA A) where\n  initials_lt : ∀ {s}, s ∈ m.initials → s ∈ m.states\n  finals_lt : ∀ {s}, s ∈ m.finals → s ∈ m.states\n  trans_src_lt : ∀ s_a ∈ m.trans, s_a.1 ∈ m.states\n  trans_tgt_lt : s' ∈ m.tr s a → s' ∈ m.states"}, {"name": "RawCNFA.addTrans", "content": "def RawCNFA.addTrans (m : RawCNFA A) (a : A) (s s' : State) : RawCNFA A :=\n  let ns := m.trans.getD (s, a) ∅\n  let ns := ns.insert s'\n  { m with trans :=  m.trans.insert (s, a) ns }"}, {"name": "RawCNFA.newState", "content": "def RawCNFA.newState (m : RawCNFA A) : State × RawCNFA A :=\n  let old := m.stateMax\n  let m := { m with stateMax := old + 1 }\n  (old, m)"}, {"name": "RawCNFA.addInitial", "content": "def RawCNFA.addInitial (m : RawCNFA A) (s : State) : RawCNFA A :=\n  { m with initials := m.initials.insert s }"}], "lib_lemmas": [{"name": "BitVec.zero_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "Finset.mem_range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Function.iterate_add", "module": "Mathlib.Logic.Function.Iterate"}, {"name": "eq_of_forall_lt_iff", "module": "Mathlib.Order.Basic"}, {"name": "gt_iff_lt", "module": "Init.Core"}, {"name": "Fin.add_def", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.val_one", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "eq_iff_eq_of_cmp_eq_cmp", "module": "Mathlib.Order.Compare"}], "repo_lemmas": [{"name": "wf_addTrans", "content": "@[grind ., simp, aesop 50% unsafe]\nlemma wf_addTrans [LawfulBEq A] (m : RawCNFA A) (hwf : m.WF) s a s' (hin : s ∈ m.states) (hin' : s' ∈ m.states) :\n    (m.addTrans a s s').WF"}, {"name": "RawCNFA.same_stateMax", "content": "@[grind =, simp]\nlemma RawCNFA.same_stateMax (m : RawCNFA A) x y (z : Std.HashMap (State × A) (Std.HashSet State)) :\n    (RawCNFA.mk m.stateMax x y z).states = m.states"}, {"name": "newState_eq", "content": "@[grind =, simp, aesop 50% unsafe]\nlemma newState_eq (m : RawCNFA A) :\n    m.newState.1 = m.stateMax"}, {"name": "addInitial_stateMax", "content": "@[grind =, simp]\nlemma addInitial_stateMax {m : RawCNFA A} : (m.addInitial s).stateMax = m.stateMax"}, {"name": "addTrans_stateMax", "content": "@[grind =, simp]\nlemma addTrans_stateMax {m : RawCNFA A} : (m.addTrans a s s').stateMax = m.stateMax"}, {"name": "simulFun_sim", "content": "lemma simulFun_sim {m : CNFA n} f :\n    m.m.SimulFun M.M f → m.Sim M"}, {"name": "simulFun_sim_raw", "content": "lemma simulFun_sim_raw [LawfulBEq A] {m : RawCNFA A} (hwf : m.WF) f :\n    m.SimulFun M f → m.Sim M"}, {"name": "RawCNFA.Simul.initial", "content": "@[simp]\nlemma RawCNFA.Simul.initial {m : RawCNFA A} {M : NFA A Q} (hsim : m.Simul M R ⊤ ∅) :\n    R.set_eq m.initials.toSet M.start"}], "used_local_defs": [{"name": "WidthPredicate.final?", "content": "def WidthPredicate.final? (wp : WidthPredicate) (n : Nat) (s : State) : Bool :=\n  decide (wp.sat s n)"}, {"name": "RawCNFA.autWidth", "content": "def RawCNFA.autWidth (wp : WidthPredicate) (n : Nat) : RawCNFA (BitVec 0) :=\n  let m := (n+2).iterate f empty\n  let m := m.addInitial 0\n  m.addTrans (BitVec.zero 0) (n + 1) (n + 1)\nwhere\n  f m :=\n    let (s, m) := m.newState\n    let m := if wp.final? n s then m.addFinal s else m\n    if s > 0 then m.addTrans (BitVec.zero 0) (s-1) s else m"}, {"name": "CNFA.autWidth", "content": "def CNFA.autWidth (wp : WidthPredicate) (n : Nat) : CNFA 0 :=\n  ⟨RawCNFA.autWidth wp n, RawCNFA.autWidth_wf⟩"}, {"name": "NFA.autWidth", "content": "def NFA.autWidth (wp : WidthPredicate) (n : Nat) : NFA (BitVec 0) (Fin (n+2)) where\n  start := { 0 }\n  accept := { s | wp.final? n s }\n  step s₁ _ := { s₂ | if s₁ = Fin.last (n+1) then s₁ = s₂ else s₂ = s₁ + 1 }"}, {"name": "NFA'.autWidth", "content": "def NFA'.autWidth (wp : WidthPredicate) (n : Nat) : NFA' 0 := ⟨_, NFA.autWidth wp n⟩"}, {"name": "autWidth_equiv", "content": "def autWidth_equiv : (CNFA.autWidth wp n).m.states ≃ (NFA'.autWidth wp n).σ where\n  toFun := fun ⟨s, hs⟩ =>\n    Fin.mk s (by admit /- proof elided -/\n    )\n  invFun q := ⟨q.val, by admit /- proof elided -/\n  ⟩\n  left_inv := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "RawCNFA.autWidth_spec", "content": "lemma RawCNFA.autWidth_spec {wp : WidthPredicate} :\n  let m := RawCNFA.autWidth wp n\n  m.WF ∧ m.stateMax = n+2 ∧\n  (∀ s, s ∈ m.states → (s ∈ m.initials ↔ s = 0) ∧ (s ∈ m.finals ↔ wp.final? n s)) ∧\n  (∀ s s', s ∈ m.states → s' ∈ m.states → (s' ∈ m.tr s 0 ↔ if s = n+1 then s = s' else s' = s + 1))"}, {"name": "CNFA.autWidth_states", "content": "@[simp]\nlemma CNFA.autWidth_states: s ∈ (autWidth wp n).m.states ↔ s < n+2"}, {"name": "CNFA.autWidth_initials", "content": "lemma CNFA.autWidth_initials : s ∈ (autWidth wp n).m.initials ↔ s = 0"}, {"name": "CNFA.autWidth_finals", "content": "lemma CNFA.autWidth_finals (hn : s < n + 2) : s ∈ (autWidth wp n).m.finals ↔ wp.final? n s"}, {"name": "CNFA.autWidth_tr", "content": "lemma CNFA.autWidth_tr (hs : s < n + 2) (hs' : s' < n + 2) : s' ∈ (autWidth wp n).m.tr s 0 ↔ if s = n+1 then s = s' else s' = s + 1"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\nsection fsm\n\nvariable {arity : Type} [FinEnum arity]\n\nopen BitStream in\n\nend fsm\n\nsection nfas_relations\n\ndef WidthPredicate.final? (wp : WidthPredicate) (n : Nat) (s : State) : Bool :=\n  decide (wp.sat s n)\n\ndef RawCNFA.autWidth (wp : WidthPredicate) (n : Nat) : RawCNFA (BitVec 0) :=\n  let m := (n+2).iterate f empty\n  let m := m.addInitial 0\n  m.addTrans (BitVec.zero 0) (n + 1) (n + 1)\nwhere\n  f m :=\n    let (s, m) := m.newState\n    let m := if wp.final? n s then m.addFinal s else m\n    if s > 0 then m.addTrans (BitVec.zero 0) (s-1) s else m\n\ndef CNFA.autWidth (wp : WidthPredicate) (n : Nat) : CNFA 0 :=\n  ⟨RawCNFA.autWidth wp n, RawCNFA.autWidth_wf⟩\n\ndef NFA.autWidth (wp : WidthPredicate) (n : Nat) : NFA (BitVec 0) (Fin (n+2)) where\n  start := { 0 }\n  accept := { s | wp.final? n s }\n  step s₁ _ := { s₂ | if s₁ = Fin.last (n+1) then s₁ = s₂ else s₂ = s₁ + 1 }\n\ndef NFA'.autWidth (wp : WidthPredicate) (n : Nat) : NFA' 0 := ⟨_, NFA.autWidth wp n⟩\n\ndef autWidth_equiv : (CNFA.autWidth wp n).m.states ≃ (NFA'.autWidth wp n).σ where\n  toFun := fun ⟨s, hs⟩ =>\n    Fin.mk s (by admit /- proof elided -/\n    )\n  invFun q := ⟨q.val, by admit /- proof elided -/\n  ⟩\n  left_inv := by admit /- proof elided -/", "target_theorem": "lemma CNFA.autWidth_spec : autWidth wp n |>.Sim (NFA'.autWidth wp n) :=", "ground_truth_proof": ":= by\n  apply simulFun_sim autWidth_equiv; simp [autWidth_equiv]; constructor\n  · rintro q; simp_all [NFA'.autWidth, NFA.autWidth]; apply autWidth_finals (q.isLt)\n  · rintro q; simp_all [NFA'.autWidth, NFA.autWidth]; rw [autWidth_initials]; exact\n    eq_iff_eq_of_cmp_eq_cmp rfl\n  · rintro a q q'; fin_cases a; simp_all [NFA'.autWidth, NFA.autWidth, instFinEnumBV];\n    have h := @autWidth_tr q.val n q'.val wp q.isLt q'.isLt\n    unfold State at *\n    simp_all\n    rcases q with ⟨q, hq⟩\n    rcases q' with ⟨q', hq'⟩\n    simp [Fin.last]\n    split\n    · rfl\n    · rw [Fin.add_def]\n      simp only [Fin.val_one, Fin.mk.injEq]\n      rw [Nat.mod_eq_of_lt (by omega)]", "nesting_depth": 5, "transitive_dep_count": 77, "subset_aristotle": false, "category": "Compiler"}
{"id": 337, "thm_name": "Ctxt.delete_append_appendInl", "thm_stmt": "@[simp] theorem Ctxt.delete_append_appendInl {Γ : Ctxt Ty} {us : List Ty}\n    {r : DeleteRange Γ} :\n    (⟨us⟩ ++ Γ).delete r.appendInl = ⟨us⟩ ++ (Γ.delete r)", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Transforms/DCE.lean", "imports": ["import Mathlib.Tactic.DepRewrite", "import LeanMLIR.Framework", "import Mathlib.Tactic.Linarith", "import LeanMLIR.LeanMLIR.ErasedContext"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "List.ofFn", "module": "Init.Data.List.OfFn"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List.length", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "length", "content": "@[grind=]\ndef length (Γ : Ctxt Ty) : Nat := Γ.toList.length"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "appendInl", "content": "def appendInl (v : Γ.Var t) : (Γ ++ Δ).Var t :=\n  ⟨v.val, by admit /- proof elided -/\n  ⟩"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}], "lib_lemmas": [{"name": "Fin.coe_cast", "module": "Init.Data.Fin.Lemmas"}, {"name": "List.getElem?_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.getElem?_ofFn", "module": "Init.Data.List.OfFn"}, {"name": "List.getElem_append_right", "module": "Init.Data.List.BasicAux"}, {"name": "List.length_append", "module": "Init.Data.List.Basic"}, {"name": "Nat.ge_of_not_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Option.dite_none_right_eq_some", "module": "Init.Data.Option.Lemmas"}, {"name": "Valuation.ext", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "dite_eq_ite", "module": "Init.ByCases"}], "repo_lemmas": [{"name": "ofList_append", "content": "@[simp] theorem ofList_append {ts us : List Ty} :\n    Ctxt.ofList ts ++ Ctxt.ofList us = Ctxt.ofList (ts ++ us)"}, {"name": "length_ofList", "content": "@[simp, grind=] theorem length_ofList : (ofList ts).length = ts.length"}, {"name": "getElem?_ofList", "content": "@[simp, grind=] theorem getElem?_ofList (i : Nat) : (ofList ts)[i]? = ts[i]?"}, {"name": "getElem_ofList", "content": "@[simp, grind=] theorem getElem_ofList (i : Nat) (h : _) : (ofList ts)[i]'h = ts[i]'h"}], "used_local_defs": [{"name": "DCE.DeleteRange", "content": "structure DeleteRange (Γ : Ctxt Ty) where\n   \n  start : Fin (Γ.length + 1)\n   \n  num : Fin (Γ.length + 1 - start.val)"}, {"name": "DCE.DeleteRange.appendInl", "content": "def appendInl {Γ : Ctxt Ty} {ts : List Ty}\n    (r : DeleteRange Γ) : DeleteRange (⟨ts⟩ ++ Γ) where\n  start := ⟨r.start + ts.length, by admit /- proof elided -/\n  ⟩\n  num := ⟨r.num, by admit /- proof elided -/\n  ⟩"}, {"name": "Ctxt.delete", "content": "def Ctxt.delete (Γ : Ctxt Ty) (vs : DeleteRange Γ) : Ctxt Ty :=\n  Ctxt.ofList <| List.ofFn (n := Γ.length - vs.num.val) fun i =>\n    have := vs.start.prop\n    if hi : i.val < vs.start then\n      Γ[i.val]\n    else\n      Γ[i.val + vs.num]"}], "used_local_lemmas": [], "local_ctx": "import LeanMLIR.Framework\n\nimport Mathlib.Tactic.Linarith\n\nimport Mathlib.Tactic.DepRewrite\n\nopen Ctxt (Var Valuation Hom)\n\nnamespace DCE\n\nstructure DeleteRange (Γ : Ctxt Ty) where\n   \n  start : Fin (Γ.length + 1)\n   \n  num : Fin (Γ.length + 1 - start.val)\n\nopen DCE (DeleteRange)\n\nnamespace DeleteRange\n\ndef appendInl {Γ : Ctxt Ty} {ts : List Ty}\n    (r : DeleteRange Γ) : DeleteRange (⟨ts⟩ ++ Γ) where\n  start := ⟨r.start + ts.length, by admit /- proof elided -/\n  ⟩\n  num := ⟨r.num, by admit /- proof elided -/\n  ⟩\n\nsection Lemmas\n\nend Lemmas\n\nend DeleteRange\n\nend DCE\n\nopen DCE (DeleteRange)\n\ndef Ctxt.delete (Γ : Ctxt Ty) (vs : DeleteRange Γ) : Ctxt Ty :=\n  Ctxt.ofList <| List.ofFn (n := Γ.length - vs.num.val) fun i =>\n    have := vs.start.prop\n    if hi : i.val < vs.start then\n      Γ[i.val]\n    else\n      Γ[i.val + vs.num]", "target_theorem": "@[simp] theorem Ctxt.delete_append_appendInl {Γ : Ctxt Ty} {us : List Ty}\n    {r : DeleteRange Γ} :\n    (⟨us⟩ ++ Γ).delete r.appendInl = ⟨us⟩ ++ (Γ.delete r) :=", "ground_truth_proof": ":= by\n  rcases Γ with ⟨Γ⟩\n  ext i t\n  simp only [delete, ofList_append, length_ofList, List.length_append,\n    DeleteRange.val_start_appendInl, DeleteRange.val_num_appendInl, Fin.coe_cast, getElem_ofList,\n    dite_eq_ite, getElem?_ofList, List.getElem?_ofFn, Option.dite_none_right_eq_some,\n    Option.some.injEq]\n  rw [List.getElem?_append]\n  by_cases hi : i < us.length <;> simp only [reduceIte, hi]\n  · have : i < us.length + Γ.length - r.num := by grind\n    have : i < r.start + us.length := by grind\n    simp [*]\n  · simp only [List.getElem?_ofFn, Option.dite_none_right_eq_some, Option.some.injEq]\n    simp only [\n      List.getElem_append_right (Nat.ge_of_not_lt hi),\n      List.getElem_append_right (by grind : us.length ≤ i + r.num),\n      length_ofList\n    ]\n    split <;> constructor <;> grind", "nesting_depth": 3, "transitive_dep_count": 33, "subset_aristotle": false, "category": "Compiler"}
{"id": 338, "thm_name": "Lets.getPureExpr_var_appendInl", "thm_stmt": "@[simp] theorem Lets.getPureExpr_var_appendInl (lets : Lets d Γ_in eff Γ_out)\n    (e : Expr d Γ_out eff ty) (v : Var ⟨ty⟩ u) :\n    getPureExpr (lets.var e) v.appendInl\n    = e.toPure?.map (fun e => ⟨_, v, e.changeVars <| e.contextHom⟩)", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Framework/Basic.lean", "imports": ["import LeanMLIR.HVector", "import LeanMLIR.ErasedContext", "import SSA/Projects/CIRCT/HSxComb/HSxCombFunctor.lean", "import SSA/Projects/CIRCT/DCxComb/DCxCombFunctor.lean", "import SSA/Projects/Tensor2D/Tensor2D.lean", "import SSA/Projects/RISCV64/Base.lean", "import SSA/Projects/ModArith/Basic.lean", "import SSA/Projects/Scf/ScfFunctor.lean", "import Mathlib.Data.Finset.Union", "import LeanMLIR.LeanMLIR.ErasedContext", "import LeanMLIR.Framework.Dialect", "import LeanMLIR/LeanMLIR/Transforms/CSE.lean", "import LeanMLIR/LeanMLIR/Examples.lean", "import LeanMLIR/LeanMLIR/Transforms/DCE.lean", "import LeanMLIR.LeanMLIR.HVector", "import SSA/Projects/FullyHomomorphicEncryption/Basic.lean", "import LeanMLIR/LeanMLIR/Dialects/LLVM/Basic.lean", "import SSA/Projects/Tensor1D/Tensor1D.lean", "import LeanMLIR/LeanMLIR/Framework/Macro.lean", "import SSA/Projects/LLVMRiscV/LLVMAndRiscv.lean", "import LeanMLIR.EffectKind"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "decidable_of_iff", "module": "Init.PropLemmas"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Valuation.mk", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Function.comp", "module": "Init.Prelude"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.matchAlts", "module": "Lean.Parser.Term"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Stream'", "module": "Mathlib.Data.Stream.Defs"}], "used_repo_defs": [{"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "map", "content": "def map (f : Ty₁ → Ty₂) : Ctxt Ty₁ → Ctxt Ty₂ :=\n  ofList ∘ (List.map f) ∘ toList"}, {"name": "Hom", "content": "abbrev Hom (Γ Γ' : Ctxt Ty) := ⦃t : Ty⦄ → Γ.Var t → Γ'.Var t"}, {"name": "dropUntil", "content": "def dropUntil : Ctxt Ty :=\n  ⟨Γ.toList.drop (v.val + 1)⟩"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "Hom.castCodomain", "content": "def Hom.castCodomain (h : Δ = Δ') (f : Γ.Hom Δ) : Γ.Hom Δ' :=\n  fun _t v => (f v).castCtxt h"}, {"name": "appendInl", "content": "def appendInl (v : Γ.Var t) : (Γ ++ Δ).Var t :=\n  ⟨v.val, by admit /- proof elided -/\n  ⟩"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "dropUntilHom", "content": "abbrev dropUntilHom : Hom (Γ.dropUntil v) Γ := dropUntilDiff.toHom"}, {"name": "dropUntilDiff", "content": "def dropUntilDiff : Diff (Γ.dropUntil v) Γ :=\n  ⟨v.val+1, by admit /- proof elided -/\n  ⟩"}, {"name": "emptyElim", "content": "def emptyElim {α : Sort _} {t : Ty} : Ctxt.Var ∅ t → α :=\n  fun ⟨_, h⟩ => by admit /- proof elided -/"}, {"name": "cons", "content": "@[match_pattern]\ndef cons (hd : Ty) : Ctxt Ty → Ctxt Ty\n| ⟨tl⟩ => ⟨hd :: tl⟩"}, {"name": "Diff", "content": "def Diff (Γ₁ Γ₂ : Ctxt Ty) : Type :=\n  {d : Nat // Diff.Valid Γ₁ Γ₂ d}"}, {"name": "Diff.Valid", "content": "@[simp]\nabbrev Diff.Valid (Γ₁ Γ₂ : Ctxt Ty) (d : Nat) : Prop :=\n  ∀ {i t}, Γ₁[i]? = some t → Γ₂[i+d]? = some t"}, {"name": "Hom.id", "content": "@[simp] abbrev Hom.id {Γ : Ctxt Ty} : Γ.Hom Γ :=\n  fun _ v => v"}, {"name": "map", "content": "def map (f : ∀ (a : α), A a → B a) :\n    ∀ {l : List α}, HVector A l → HVector B l\n  | [],   .nil        => .nil\n  | t::_, .cons a as  => .cons (f t a) (map f as)"}, {"name": "HVectorLiteral", "content": "structure HVectorLiteral where\n  u : Level\n  v : Level\n  α : Q(Type $u)\n  A : Q($α → Type $v)\n  elems : Array ((a : Q($α)) × Q($A $a))"}, {"name": "length", "content": "@[grind=]\ndef length (Γ : Ctxt Ty) : Nat := Γ.toList.length\n\n      instance : DialectSignature $dialect where\n        signature := fun op => match op with $matchAlts:matchAlts\n    )"}, {"name": "(q", "content": "noncomputable instance (q : ℕ) [Fact (q > 1)] : DialectDenote (ModArith q) where\ndenote\n  | .add, arg, _ =>\n      \n      (fun args : R q × R q => args.1 + args.2) arg.toPair\n  | .sub, arg, _ =>\n      \n      (fun args : R q × R q => args.1 - args.2) arg.toPair\n  | .mul, arg, _ =>\n      \n      (fun args : R q × R q => args.1 * args.2) arg.toPair\n  | .const _ c, _, _ =>\n      \n      c"}, {"name": "Op.signature", "content": "@[simp, reducible]\ndef Op.signature : Op q n → Signature (Ty q n) :=\n  fun o => {sig := Op.sig o, returnTypes := [Op.outTy o], regSig := []}"}, {"name": "Op.sig", "content": "@[simp, reducible]\ndef Op.sig : Op q n → List (Ty q n)\n| Op.add => [Ty.polynomialLike, Ty.polynomialLike]\n| Op.sub => [Ty.polynomialLike, Ty.polynomialLike]\n| Op.mul => [Ty.polynomialLike, Ty.polynomialLike]\n| Op.mul_constant => [Ty.polynomialLike, Ty.integer]\n| Op.leading_term => [Ty.polynomialLike]\n| Op.monomial => [Ty.integer, Ty.index]\n| Op.monomial_mul => [Ty.polynomialLike, Ty.index]\n| Op.from_tensor => [Ty.tensor]\n| Op.to_tensor => [Ty.polynomialLike]\n| Op.const _ => []\n| Op.const_int _ => []\n| Op.const_idx _ => []"}, {"name": "Op", "content": "inductive Op (q : Nat) (n : Nat)\n  | add : Op q n\n  | sub : Op q n\n  | mul : Op q n\n  | mul_constant : Op q n\n    \n    \n  | leading_term : Op q n\n  | monomial : Op q n\n  | monomial_mul : Op q n\n  | from_tensor : Op q n\n  | to_tensor : Op q n\n  | const (c : R q n) : Op q n\n  | const_int (c : Int) : Op q n\n  | const_idx (i : Nat) : Op q n"}, {"name": "Op.outTy", "content": "@[simp, reducible]\ndef Op.outTy : Op q n → Ty q n\n| Op.add | Op.sub | Op.mul | Op.mul_constant | Op.leading_term | Op.monomial\n| Op.monomial_mul | Op.from_tensor | Op.const _  => Ty.polynomialLike\n| Op.to_tensor => Ty.tensor\n| Op.const_int _ => Ty.integer\n| Op.const_idx _ => Ty.index"}, {"name": "Op.regSig", "content": "@[reducible, simp]\ndef Op.regSig : Op → RegionSignature Ty\n  | .map2d => [([Ty.int], [.int])]\n  | _ => []"}, {"name": "", "content": "instance : DialectSignature Ex where\n  signature\n    | .add    => ⟨[.nat, .nat], [], [.nat], .pure⟩\n    | .beq    => ⟨[.nat, .nat], [], [.bool], .pure⟩\n    | .cst _  => ⟨[], [], [.nat], .pure⟩"}, {"name": "ExOp", "content": "inductive ExOp :  Type\n  | add : ExOp\n  | beq : ExOp\n  | cst : ℕ → ExOp\n  deriving DecidableEq"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  | bool\n  deriving DecidableEq"}, {"name": "add", "content": "def add {Γ : Ctxt _} (e₁ e₂ : Ctxt.Var Γ .nat) : Expr Ex Γ .pure [.nat] :=\n  Expr.mk\n    (op := .add)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "Ex", "content": "abbrev Ex : Dialect where\n  Op := ExOp\n  Ty := ExTy"}, {"name": "cst", "content": "def cst {Γ : Ctxt _} (n : ℕ) : Expr Ex Γ .pure [.nat]  :=\n  Expr.mk\n    (op := .cst n)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "Dialect", "content": "structure Dialect where\n  (Op : Type)\n  (Ty : Type)\n  (m : Type → Type := Id)"}, {"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "", "content": "@[reducible]\ninstance : DialectSignature Tensor2D where\n  signature op := { sig := op.sig, regSig := op.regSig, returnTypes := [op.outTy] }"}, {"name": "", "content": "instance : DialectSignature Tensor1D where\n  signature op := { sig := op.sig, regSig := op.regSig, returnTypes := [op.outTy], effectKind := .pure }"}, {"name": "", "content": "instance : DialectSignature RV64 where\n  signature o := {sig := Op.sig o, returnTypes := [Op.outTy o], regSig := []}"}, {"name": "Op.sig", "content": "@[simp, reducible]\ndef Op.sig : Op → List Ty\n  | .li _ => []\n  | .mulh  => [Ty.bv, Ty.bv]\n  | .mulhu  => [Ty.bv, Ty.bv]\n  | .mulhsu  => [Ty.bv, Ty.bv]\n  | .divu =>  [Ty.bv, Ty.bv]\n  | .remuw  => [Ty.bv, Ty.bv]\n  | .remu  =>  [Ty.bv, Ty.bv]\n  | .addiw (_imm : BitVec 12) => [Ty.bv]\n  | .lui (_imm : BitVec 20) => [Ty.bv]\n  | .auipc (_imm : BitVec 20)  => [Ty.bv]\n  | .slliw (_shamt : BitVec 5)  => [Ty.bv]\n  | .srliw (_shamt : BitVec 5) => [Ty.bv]\n  | .sraiw (_shamt : BitVec 5) => [Ty.bv]\n  | .slli (_shamt : BitVec 6) => [Ty.bv]\n  | .srli (_shamt : BitVec 6) => [Ty.bv]\n  | .srai (_shamt : BitVec 6) => [Ty.bv]\n  | .addw => [Ty.bv, Ty.bv]\n  | .subw => [Ty.bv, Ty.bv]\n  | .sllw => [Ty.bv, Ty.bv]\n  | .srlw => [Ty.bv, Ty.bv]\n  | .sraw => [Ty.bv, Ty.bv]\n  | .add => [Ty.bv, Ty.bv]\n  | .slt => [Ty.bv, Ty.bv]\n  | .sltu => [Ty.bv, Ty.bv]\n  | .and => [Ty.bv, Ty.bv]\n  | .or => [Ty.bv, Ty.bv]\n  | .xor => [Ty.bv, Ty.bv]\n  | .sll => [Ty.bv, Ty.bv]\n  | .srl => [Ty.bv, Ty.bv]\n  | .sub => [Ty.bv, Ty.bv]\n  | .sra => [Ty.bv, Ty.bv]\n  | .remw  => [Ty.bv, Ty.bv]\n  | .rem  =>  [Ty.bv, Ty.bv]\n  | .mul => [Ty.bv, Ty.bv]\n  | .mulw => [Ty.bv, Ty.bv]\n  | .div  =>  [Ty.bv, Ty.bv]\n  | .divw  =>  [Ty.bv, Ty.bv]\n  | .divuw  =>  [Ty.bv, Ty.bv]\n  | .addi (_imm : BitVec 12) => [Ty.bv]\n  | .slti (_imm : BitVec 12) => [Ty.bv]\n  | .sltiu (_imm : BitVec 12) => [Ty.bv]\n  | .andi (_imm : BitVec 12) => [Ty.bv]\n  | .ori (_imm : BitVec 12) => [Ty.bv]\n  | .xori (_imm : BitVec 12) => [Ty.bv]\n  | .bclr => [Ty.bv, Ty.bv]\n  | .bext => [Ty.bv, Ty.bv]\n  | .binv => [Ty.bv, Ty.bv]\n  | .bset  => [Ty.bv, Ty.bv]\n  | .bclri (_shamt : BitVec 6) => [Ty.bv]\n  | .bexti (_shamt : BitVec 6) => [Ty.bv]\n  | .binvi (_shamt : BitVec 6) => [Ty.bv]\n  | .bseti (_shamt : BitVec 6) => [Ty.bv]\n  | .adduw => [Ty.bv, Ty.bv]\n  | .sh1adduw => [Ty.bv, Ty.bv]\n  | .sh2adduw => [Ty.bv, Ty.bv]\n  | .sh3adduw => [Ty.bv, Ty.bv]\n  | .sh1add => [Ty.bv, Ty.bv]\n  | .sh2add => [Ty.bv, Ty.bv]\n  | .sh3add => [Ty.bv, Ty.bv]\n  | .slliuw (_shamt : BitVec 6) => [Ty.bv]\n  | .andn => [Ty.bv, Ty.bv]\n  | .orn => [Ty.bv, Ty.bv]\n  | .xnor => [Ty.bv, Ty.bv]\n  | .clz\n  | .clzw\n  | .ctz\n  | .ctzw\n  | .max => [Ty.bv, Ty.bv]\n  | .maxu => [Ty.bv, Ty.bv]\n  | .min  => [Ty.bv, Ty.bv]\n  | .minu  => [Ty.bv, Ty.bv]\n  | .sextb => [Ty.bv]\n  | .sexth => [Ty.bv]\n  | .zexth => [Ty.bv]\n  | .rol => [Ty.bv, Ty.bv]\n  | .rolw => [Ty.bv, Ty.bv]\n  | .ror => [Ty.bv, Ty.bv]\n  | .rori (_shamt : BitVec 6) =>[Ty.bv]\n  | .roriw (_shamt : BitVec 5) =>[Ty.bv]\n  | .rorw => [Ty.bv, Ty.bv]\n  | .pack => [Ty.bv, Ty.bv]\n  | .packh => [Ty.bv, Ty.bv]\n  | .packw => [Ty.bv, Ty.bv]\n  | .mv => [Ty.bv]\n  | .not => [Ty.bv]\n  | .neg => [Ty.bv]\n  | .negw => [Ty.bv]\n  | .sextw => [Ty.bv]\n  | .zextb => [Ty.bv]\n  | .zextw => [Ty.bv]\n  | .seqz => [Ty.bv]\n  | .snez => [Ty.bv]\n  | .sltz => [Ty.bv]\n  | .sgtz => [Ty.bv]"}, {"name": "Op", "content": "inductive Op\n   \n  | li : (val : BitVec 64) → Op\n  | lui (imm : BitVec 20)\n  | auipc (imm : BitVec 20)\n  | addi (imm : BitVec 12)\n  | andi (imm : BitVec 12)\n  | ori (imm : BitVec 12)\n  | xori (imm : BitVec 12)\n  | addiw (imm : BitVec 12)\n  | add\n  | slli (shamt : BitVec 6)\n  | sub\n  | and\n  | or\n  | xor\n  | sll\n  | srl\n  | sra\n  | addw\n  | subw\n  | sllw\n  | srlw\n  | sraw\n  | slti (imm : BitVec 12)\n  | sltiu (imm : BitVec 12)\n  | srli (shamt : BitVec 6)\n  | srai (shamt : BitVec 6)\n  | slliw (shamt : BitVec 5)\n  | srliw (shamt : BitVec 5)\n  | sraiw (shamt : BitVec 5)\n  | slt\n  | sltu\n   \n  | mul\n  | mulw\n  | mulh\n  | mulhu\n  | mulhsu\n  | divw\n  | divuw\n  | div\n  | divu\n  | remw\n  | rem\n  | remuw\n  | remu\n   \n   \n  | adduw\n  | sh1adduw\n  | sh2adduw\n  | sh3adduw\n  | sh1add\n  | sh2add\n  | sh3add\n  | slliuw (shamt : BitVec 6)\n   \n  | andn\n  | orn\n  | xnor\n  | clz\n  | clzw\n  | ctz\n  | ctzw\n  | max\n  | maxu\n  | min\n  | minu\n  | sextb\n  | sexth\n  | zexth\n  | rol\n  | rolw\n  | ror\n  | rori (_shamt : BitVec 6)\n  | roriw (_shamt : BitVec 5)\n  | rorw\n   \n  | bclr\n  | bclri (shamt : BitVec 6)\n  | bext\n  | bexti (shamt : BitVec 6)\n  | binv\n  | binvi (shamt : BitVec 6)\n  | bset\n  | bseti (shamt : BitVec 6)\n   \n  | pack\n  | packh\n  | packw\n   \n  | mv\n  | not\n  | neg\n  | negw\n  | sextw\n  | zextb\n  | zextw\n  | seqz\n  | snez\n  | sltz\n  | sgtz\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "Ty", "content": "inductive Ty\n  | bv : Ty\n  deriving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "", "content": "instance : DialectSignature (FHE q n) := ⟨Op.signature⟩"}, {"name": "", "content": "instance : DialectSignature LLVM where\n  signature op := ⟨op.sig, [], [op.outTy], .pure⟩"}, {"name": "", "content": "instance : DialectSignature HSxComb where\n  signature := fun op =>\n    match op with\n    | .comb o => liftSig (signature o) \n    \n    \n    | .hs o => MLIR2Handshake.instDialectSignatureHandshake.signature o"}, {"name": "Op", "content": "inductive Op : Type _\n  | comb (o : MLIR2Comb.Comb.Op)\n  | hs (o : MLIR2Handshake.Handshake.Op)\n  deriving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "liftSig", "content": "def liftSig (sig : Signature MLIR2Comb.Ty) : Signature MLIR2Handshake.Ty :=\n  Signature.mk (sig.sig.map liftTy) [] (liftTy sig.outTy)"}, {"name": "liftTy", "content": "def liftTy : MLIR2Comb.Ty → MLIR2Handshake.Ty\n| .bitvec w => .stream (.bitvec w)"}, {"name": "Ty", "content": "inductive Ty\n| stream (ty2 : Ty2) : Ty \n| stream2 (ty2 : Ty2) : Ty \n| stream2token (ty2 : Ty2) : Ty \nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "Ty", "content": "inductive Ty\n| bitvec (w : Nat) : Ty \nderiving DecidableEq, Repr, ToExpr"}, {"name": "Ty2", "content": "inductive Ty2\n  | bitvec (w : Nat) : Ty2\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "map", "content": "def map {α β : Type} (s : Stream α) (f : α → β) : Stream β :=\n  fun i => (s i).map f"}, {"name": "Stream", "content": "def Stream (β : Type) := Stream' (Option β)"}, {"name": "", "content": "instance : DialectSignature DCxComb where\n  signature := fun op =>\n    match op with\n    | .comb o => liftSig (signature o) \n    \n    \n    | .dc o => MLIR2DC.instDialectSignatureDC.signature o"}, {"name": "Op", "content": "inductive Op : Type _\n  | comb (o : MLIR2Comb.Comb.Op)\n  | dc (o : MLIR2DC.DC.Op)\n  deriving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "liftSig", "content": "def liftSig (sig : Signature MLIR2Comb.Ty) : Signature MLIR2DC.Ty :=\n  Signature.mk (sig.sig.map liftTy) [] (liftTy sig.outTy)"}, {"name": "liftTy", "content": "def liftTy : MLIR2Comb.Ty → MLIR2DC.Ty\n| .bitvec w => .valuestream w"}, {"name": "Ty", "content": "inductive Ty\n| tokenstream : Ty\n| tokenstream2 : Ty\n| valuestream (w : Nat) : Ty \n| valuestream2 (w : Nat) : Ty \n| valuetokenstream (w : Nat) : Ty \n| variadicvaluetokenstream (w : Nat) : Ty \nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "[SIG", "content": "instance [SIG : DialectSignature d] [DENOTE : DialectDenote d] {Γ : Ctxt d.Ty} {t}\n    (com : Com d Γ .pure t) : Inhabited (DCEType com) where\n  default :=\n    ⟨Γ, Hom.id, com, by admit /- proof elided -/\n    ⟩"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  | bool\n  deriving DecidableEq, Repr"}, {"name": "cst", "content": "def cst {Γ : Ctxt _} (n : ℕ) : Expr Ex Γ .pure [.nat] :=\n  Expr.mk\n    (op := .cst n)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "LLVMPlusRiscVSignature", "content": "@[simp]\ninstance LLVMPlusRiscVSignature : DialectSignature LLVMPlusRiscV where\n  signature\n  | .llvm llvmOp => .llvm <$> DialectSignature.signature llvmOp\n  | .riscv riscvOp => .riscv <$> DialectSignature.signature riscvOp\n  | .castRiscv w =>\n      {sig := [Ty.riscv .bv], returnTypes := [Ty.llvm (.bitvec w)], regSig := []}\n  | .castLLVM w =>\n      {sig := [Ty.llvm (.bitvec w)], returnTypes := [Ty.riscv .bv], regSig := []}"}, {"name": "Op", "content": "inductive Op where\n  | llvm : LLVM.Op -> Op\n  | riscv : RISCV64.RV64.Op -> Op\n  | castRiscv : Nat → Op\n  | castLLVM : Nat → Op\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "Ty", "content": "inductive Ty where\n  | llvm : LLVM.Ty -> Ty\n  | riscv : RISCV64.RV64.Ty -> Ty\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "", "content": "instance : DialectSignature ExOp ExTy where\n  signature\n    | .add    => ⟨[.nat, .nat], [], .nat, .pure⟩\n    | .beq    => ⟨[.nat, .nat], [], .bool, .pure⟩\n    | .cst _  => ⟨[], [], .nat, .pure⟩"}, {"name": "ExOp", "content": "inductive ExOp :  Type\n  | add : ExOp\n  | beq : ExOp\n  | cst : ℕ → ExOp\n  deriving DecidableEq, Repr"}, {"name": "add", "content": "def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .nat) : Expr Γ .nat :=\n  Expr.mk\n    (op := .add)\n    (ty_eq := rfl)\n    (eff_le := EffectKind.le_refl _)\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  deriving DecidableEq, Repr"}, {"name": "Expr", "content": "abbrev Expr (Γ) (ty) := _root_.Expr ExOp Γ .pure ty"}, {"name": "cst", "content": "def cst {Γ : Ctxt _} (n : ℕ) : Expr Γ .nat  :=\n  Expr.mk\n    (op := .cst n)\n    (ty_eq := rfl)\n    (eff_le := EffectKind.le_refl _)\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "", "content": "instance : DialectSignature ExOp ExTy where\n  signature\n  | .add    => ⟨[.nat, .nat], [], .nat, .pure⟩\n  | .runK _ => ⟨[.nat], [([.nat], .nat)], .nat, .pure⟩"}, {"name": "ExOp", "content": "inductive ExOp :  Type\n  | add : ExOp\n  | runK : ℕ → ExOp\n  deriving DecidableEq, Repr"}, {"name": "add", "content": "def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .nat) : Expr Γ .nat :=\n  Expr.mk\n    (op := .add)\n    (ty_eq := rfl)\n    (eff_le := EffectKind.pure_le _)\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "[TyDenote", "content": "@[reducible]\ninstance [TyDenote d.Ty] [DialectSignature d] [DialectDenote d]\n    [B : HasBool d] [N : HasNat d] [I : HasInt d] : DialectSignature (Scf d) where\n   signature\n   | .coe o => signature (d:=d) o\n    | .if t t' => ⟨[B.ty, t], [(⟨[t]⟩, [t']), (⟨[t]⟩, [t'])], [t'], .impure⟩\n      \n      \n      \n      \n      \n    | .for t => ⟨[ I.ty,  I.ty,  N.ty, t], [(⟨[I.ty, t]⟩, [t])], [t], .impure⟩\n    | .run t => ⟨[t], [(⟨[t]⟩, [t])], [t], .impure⟩\n    | .iterate _k => ⟨[I.ty], [(⟨[I.ty]⟩, [I.ty])], [I.ty], .impure⟩"}, {"name": "HasTy", "content": "class HasTy (d : Dialect) (DenotedTy : Type) [TyDenote d.Ty] [DialectSignature d] where\n    ty : d.Ty\n    denote_eq : toType ty = DenotedTy := by admit /- proof elided -/"}, {"name": "Scf.Op", "content": "inductive Scf.Op (Op' Ty' : Type) (m') [TyDenote Ty'] [DialectSignature ⟨Op', Ty', m'⟩]\n    [DialectDenote ⟨Op', Ty', m'⟩] : Type _\n  | coe (o : Op')\n  | iterate (k : ℕ) \n  | run (inputty : Ty')  \n  | if (inputty retty' : Ty')  \n  | for (ty : Ty')\n  deriving DecidableEq, Repr"}, {"name": "iterate", "content": "@[simp_denote] def iterate {Γ : Ctxt _} (k : Nat) (input : Var Γ Arith.Ty.int)\n    (body : Com ScfArith ⟨[.int]⟩ .impure .int) : Expr ScfArith Γ .impure .int :=\n  Expr.mk\n    (op := .iterate k)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons input .nil)\n    (regArgs := HVector.cons body HVector.nil)"}, {"name": "ScfArith", "content": "abbrev ScfArith := Scf Arith"}, {"name": "Scf", "content": "def Scf (d : Dialect) [TyDenote d.Ty] [DialectSignature d] [DialectDenote d] : Dialect where\n  Op := Scf.Op d.Op d.Ty d.m\n  Ty := d.Ty\n  m  := d.m"}, {"name": "Op", "content": "inductive Op\n  | add : Op  \n  | add_nat : Op  \n  | axpy : Op  \n  | neg : Op  \n  | const : (val : ℤ) → Op\n  | const_nat : (val : ℕ) → Op"}, {"name": "run", "content": "@[simp_denote]\ndef run {Γ : Ctxt _} {t : Arith.Ty} (v : Var Γ t) (body : Com ScfArith ⟨[t]⟩ .impure t) :\n    Expr ScfArith Γ .impure t :=\n  Expr.mk\n    (op := .run t)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons v .nil)\n    (regArgs := HVector.cons body <| HVector.nil)"}, {"name": "Ty", "content": "inductive Ty\n| int\n| bool\n| nat\n deriving DecidableEq, Repr"}, {"name": "Valuation.instAppendHVector", "content": "@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V"}, {"name": "neg", "content": "@[simp_denote] def neg {Γ : Ctxt _} (a : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .neg)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "axpy", "content": "@[simp_denote] def axpy {Γ : Ctxt _} (a : Var Γ .int) (x : Var Γ .nat) (b: Var Γ .int) :\n    Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .axpy)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons x <| .cons b .nil)\n    (regArgs := .nil)"}, {"name": "add_nat", "content": "@[simp_denote] def add_nat (e₁ e₂ : Var Γ .nat) : Expr ScfArith Γ .pure .nat :=\n  Expr.mk\n    (op := .coe <| .add_nat)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "Ty", "content": "inductive Ty\n  | int\n   \n  | int2\n  deriving DecidableEq, Lean.ToExpr"}, {"name": "Op", "content": "inductive Op\n  | noop\n  | mkPair\n  | unPair\n  deriving Lean.ToExpr"}, {"name": "Arith", "content": "abbrev Arith : Dialect := {Op, Ty}"}, {"name": "", "content": "@[reducible]\ninstance : DialectSignature Arith where\n  signature\n    | .axpy => ⟨[.int, .nat, .int], [], [.int], .pure⟩\n    | .neg => ⟨[.int], [], [.int], .pure⟩\n    | .const _ => ⟨[], [], [.int], .pure⟩\n    | .const_nat _ => ⟨[], [], [.nat], .pure⟩\n    | .add   => ⟨[.int, .int], [], [.int], .pure⟩\n    | .add_nat   => ⟨[.nat, .nat], [], [.nat], .pure⟩"}, {"name": "add", "content": "@[simp_denote] def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .add)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "TyDenote.toType", "content": "notation \"⟦\" x \"⟧\" => TyDenote.toType x"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Option.map_map", "module": "Init.Data.Option.Lemmas"}], "repo_lemmas": [{"name": "map_map", "content": "theorem map_map {A B C : α → Type*} {l : List α} (t : HVector A l)\n    (f : ∀ a, A a → B a) (g : ∀ a, B a → C a) :\n    (t.map f).map g = t.map (fun a v => g a (f a v))"}, {"name": "val_lt", "content": "theorem val_lt (v : Γ.Var t) : v.val < Γ.length"}, {"name": "length_ofList", "content": "@[simp, grind=] theorem length_ofList : (ofList ts).length = ts.length"}, {"name": "getElem?_ofList", "content": "@[simp, grind=] theorem getElem?_ofList (i : Nat) : (ofList ts)[i]? = ts[i]?"}, {"name": "appendCases_appendInl", "content": "@[simp] theorem appendCases_appendInl (v : Γ.Var t) :\n    appendCases (motive := motive) left right v.appendInl = (left v)"}], "used_local_defs": [{"name": "RegionSignature", "content": "abbrev RegionSignature Ty := List (Ctxt Ty × List Ty)"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "Signature.mk", "content": "abbrev Signature.mk (sig : List Ty) (regSig : RegionSignature Ty) (returnTypes : List Ty) : Signature Ty :=\n { sig, regSig, returnTypes }"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "DialectSignature.sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "DialectSignature.regSig", "content": "def regSig       := Signature.regSig ∘ s.signature"}, {"name": "DialectSignature.returnTypes", "content": "def returnTypes  := Signature.returnTypes ∘ s.signature"}, {"name": "DialectSignature.effectKind", "content": "def effectKind   := Signature.effectKind ∘ s.signature"}, {"name": "DialectDenote", "content": "class DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))"}, {"name": "Expr", "content": "inductive Expr : (Γ : Ctxt d.Ty) → (eff : EffectKind) → (ty : List d.Ty) → Type where\n  | mk {Γ} {ty} (op : d.Op)\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) <| DialectSignature.sig op)\n     \n    (regArgs : HVector (fun t : Ctxt d.Ty × List d.Ty => Com t.1 .impure t.2)\n      (DialectSignature.regSig op)) : Expr Γ eff ty"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "Regions", "content": "abbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "HVector", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "Expr", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Com.decidableEq", "content": "protected instance Com.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty]\n    {Γ : Ctxt d.Ty} {eff : EffectKind} {tys : List d.Ty} : DecidableEq (Com d Γ eff tys)\n  | .rets v₁, .rets v₂ => decidable_of_iff (v₁ = v₂) (by admit /- proof elided -/\n  )\n  | .var (ty := ty₁) e₁ body₁, .var (ty := ty₂) e₂ body₂ =>\n    if hα : ty₁ = ty₂\n    then by\n      subst hα\n      letI := Expr.decidableEq e₁ e₂\n      letI := Com.decidableEq body₁ body₂\n      exact decidable_of_iff (e₁ = e₂ ∧ body₁ = body₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )\n  | .rets _, .var _ _ => isFalse (fun h => Com.noConfusion h)\n  | .var _ _, .rets _ => isFalse (fun h => Com.noConfusion h)"}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "Expr.regArgs", "content": "def Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r)"}, {"name": "Expr.contextHom", "content": "abbrev Expr.contextHom (e : Expr d Γ eff ts) : Γ.Hom e.outContext :=\n  Hom.id.appendCodomain"}, {"name": "Expr.changeVars", "content": "def Expr.changeVars (varsMap : Γ.Hom Γ') {ty} (e : Expr d Γ eff ty) :\n    Expr d Γ' eff ty :=\n  ⟨e.op, e.ty_eq, e.eff_le, e.args.map varsMap, e.regArgs⟩"}, {"name": "FlatCom", "content": "structure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts"}, {"name": "Lets.getPureExprAux", "content": "def Lets.getPureExprAux {Γ₁ Γ₂ : Ctxt d.Ty} {t} : Lets d Γ₁ eff Γ₂ → (v : Var Γ₂ t) →\n    Option (Σ ts, (Var ⟨ts⟩ t) × Expr d (Γ₂.dropUntil v) .pure ts)\n  | .nil, _ => none\n  | .var (Γ_out := Γ_out) (t := t) lets e, v => by admit /- proof elided -/\n    | right v =>\n        apply cast ?_ <| Lets.getPureExprAux lets v\n        simp\n    | left v =>\n        have h : (Ctxt.dropUntil t v) ++ Γ_out = e.outContext.dropUntil v.appendInl := by admit /- proof elided -/"}, {"name": "Lets.getPureExpr", "content": "def Lets.getPureExpr {Γ₁ Γ₂ : Ctxt d.Ty} (lets : Lets d Γ₁ eff Γ₂) {t : d.Ty} (v : Var Γ₂ t) :\n    Option (Σ ts, (Var ⟨ts⟩ t) × Expr d Γ₂ .pure ts) :=\n  (getPureExprAux lets v).map fun ⟨_, v, e⟩ =>\n    ⟨_, v, e.changeVars Ctxt.dropUntilHom⟩"}], "used_local_lemmas": [{"name": "Expr.changeVars_changeVars", "content": "@[simp] theorem Expr.changeVars_changeVars (e : Expr d Γ eff ty) (f : Γ.Hom Δ) (g : Δ.Hom Ξ) :\n    (e.changeVars f).changeVars g = e.changeVars (f.comp g)"}], "local_ctx": "import LeanMLIR.ErasedContext\n\nimport LeanMLIR.HVector\n\nimport LeanMLIR.EffectKind\n\nimport LeanMLIR.Framework.Dialect\n\nimport Mathlib.Data.Finset.Union\n\nopen Ctxt (Var VarSet Valuation Hom)\n\nopen TyDenote (toType)\n\nabbrev RegionSignature Ty := List (Ctxt Ty × List Ty)\n\nstructure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure\n\nabbrev Signature.mk (sig : List Ty) (regSig : RegionSignature Ty) (returnTypes : List Ty) : Signature Ty :=\n { sig, regSig, returnTypes }\n\nclass DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty\n\nnamespace DialectSignature\n\nvariable {d} [s : DialectSignature d]\n\ndef sig          := Signature.sig ∘ s.signature\n\ndef regSig       := Signature.regSig ∘ s.signature\n\ndef returnTypes  := Signature.returnTypes ∘ s.signature\n\ndef effectKind   := Signature.effectKind ∘ s.signature\n\nend DialectSignature\n\nclass DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))\n\nsection DataStructures\n\nvariable (d : Dialect) [DialectSignature d]\n\ninductive Expr : (Γ : Ctxt d.Ty) → (eff : EffectKind) → (ty : List d.Ty) → Type where\n  | mk {Γ} {ty} (op : d.Op)\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) <| DialectSignature.sig op)\n     \n    (regArgs : HVector (fun t : Ctxt d.Ty × List d.Ty => Com t.1 .impure t.2)\n      (DialectSignature.regSig op)) : Expr Γ eff ty\n\ninductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β\n\nend\n\nabbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ\n\nabbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig\n\ninductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext\n\nvariable {d} [DialectSignature d]\n\nprotected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )\n\nprotected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )\n\nprotected instance Com.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty]\n    {Γ : Ctxt d.Ty} {eff : EffectKind} {tys : List d.Ty} : DecidableEq (Com d Γ eff tys)\n  | .rets v₁, .rets v₂ => decidable_of_iff (v₁ = v₂) (by admit /- proof elided -/\n  )\n  | .var (ty := ty₁) e₁ body₁, .var (ty := ty₂) e₂ body₂ =>\n    if hα : ty₁ = ty₂\n    then by\n      subst hα\n      letI := Expr.decidableEq e₁ e₂\n      letI := Com.decidableEq body₁ body₂\n      exact decidable_of_iff (e₁ = e₂ ∧ body₁ = body₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )\n  | .rets _, .var _ _ => isFalse (fun h => Com.noConfusion h)\n  | .var _ _, .rets _ => isFalse (fun h => Com.noConfusion h)\n\nend -- decEq\n\nend DataStructures\n\nvariable {d : Dialect} [DialectSignature d]\n\nsection Rec\n\nvariable {eff t} {motive : ∀ {Γ}, Com d Γ eff t → Sort u}\n          (rets : ∀ {Γ : Ctxt _} , (v : HVector Γ.Var t) → motive (Com.rets v))\n          (var : ∀ {Γ} {u},\n            (e : Expr d Γ eff u) → (body : Com d e.outContext eff t) →\n              motive body → motive (Com.var e body))\n\ndef Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl\n\nvariable {rets} {var} {Γ : Ctxt _}\n\nend Rec\n\ndef Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)\n\ndef Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)\n\nsection Lemmas\n\nnamespace Com\n\nend Com\n\nend Lemmas\n\ndef Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) \n\nabbrev Expr.contextHom (e : Expr d Γ eff ts) : Γ.Hom e.outContext :=\n  Hom.id.appendCodomain\n\nsection Lemmas\n\nend Lemmas\n\nvariable [TyDenote d.Ty] [DialectDenote d] [DecidableEq d.Ty] [Monad d.m] [LawfulMonad d.m]\n\nend\n\nsection Unfoldings\n\nopen EffectKind (liftEffect)\n\nend Unfoldings\n\nsection Lemmas\n\nend Lemmas\n\ndef Expr.changeVars (varsMap : Γ.Hom Γ') {ty} (e : Expr d Γ eff ty) :\n    Expr d Γ' eff ty :=\n  ⟨e.op, e.ty_eq, e.eff_le, e.args.map varsMap, e.regArgs⟩\n\nsection Lemmas\n\nvariable {Γ Γ' : Ctxt d.Ty} {t} (f : Γ.Hom Γ') (e : Expr d Γ eff t) (V : Γ'.Valuation)\n\nend Lemmas\n\nstructure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts\n\nsection Lemmas\n\nend Lemmas\n\nsection toPureLemmas\n\nvariable {Γ eff ty} {e : Expr d Γ eff ty} (h : e.HasPureOp)\n\nend toPureLemmas\n\nsection DenoteInsert\n\nend DenoteInsert\n\ndef Lets.getPureExprAux {Γ₁ Γ₂ : Ctxt d.Ty} {t} : Lets d Γ₁ eff Γ₂ → (v : Var Γ₂ t) →\n    Option (Σ ts, (Var ⟨ts⟩ t) × Expr d (Γ₂.dropUntil v) .pure ts)\n  | .nil, _ => none\n  | .var (Γ_out := Γ_out) (t := t) lets e, v => by admit /- proof elided -/\n    | right v =>\n        apply cast ?_ <| Lets.getPureExprAux lets v\n        simp\n    | left v =>\n        have h : (Ctxt.dropUntil t v) ++ Γ_out = e.outContext.dropUntil v.appendInl := by admit /- proof elided -/\n\ndef Lets.getPureExpr {Γ₁ Γ₂ : Ctxt d.Ty} (lets : Lets d Γ₁ eff Γ₂) {t : d.Ty} (v : Var Γ₂ t) :\n    Option (Σ ts, (Var ⟨ts⟩ t) × Expr d Γ₂ .pure ts) :=\n  (getPureExprAux lets v).map fun ⟨_, v, e⟩ =>\n    ⟨_, v, e.changeVars Ctxt.dropUntilHom⟩", "target_theorem": "@[simp] theorem Lets.getPureExpr_var_appendInl (lets : Lets d Γ_in eff Γ_out)\n    (e : Expr d Γ_out eff ty) (v : Var ⟨ty⟩ u) :\n    getPureExpr (lets.var e) v.appendInl\n    = e.toPure?.map (fun e => ⟨_, v, e.changeVars <| e.contextHom⟩) :=", "ground_truth_proof": ":= by\n  simp only [getPureExpr, getPureExprAux, Ctxt.getElem?_ofList, Var.appendCases_appendInl,\n    Option.map_map]\n  congr 1\n  funext e\n  simp only [Expr.changeVars_changeVars, Function.comp]\n  congr 3\n  funext _ v'\n  apply Subtype.ext\n  have := v.val_lt\n  simp; grind", "nesting_depth": 6, "transitive_dep_count": 70, "subset_aristotle": false, "category": "Compiler"}
{"id": 339, "thm_name": "Expr.denote_castPureToEff", "thm_stmt": "@[simp] theorem Expr.denote_castPureToEff {e : Expr d Γ .pure t} :\n    denote (e.castPureToEff eff) = fun V => pure (e.denote V)", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Framework/Basic.lean", "imports": ["import LeanMLIR.HVector", "import LeanMLIR.ErasedContext", "import SSA/Projects/CIRCT/HSxComb/HSxCombFunctor.lean", "import SSA/Projects/CIRCT/DCxComb/DCxCombFunctor.lean", "import SSA/Projects/Tensor2D/Tensor2D.lean", "import SSA/Projects/RISCV64/Base.lean", "import SSA/Projects/ModArith/Basic.lean", "import LeanMLIR.LeanMLIR.EffectKind", "import SSA/Projects/PaperExamples/PaperExamples.lean", "import SSA/Projects/Scf/ScfFunctor.lean", "import Mathlib.Data.Finset.Union", "import LeanMLIR.Framework.Dialect", "import LeanMLIR/LeanMLIR/Transforms/CSE.lean", "import SSA/Projects/FullyHomomorphicEncryption/Basic.lean", "import LeanMLIR/LeanMLIR/Transforms/DCE.lean", "import LeanMLIR/LeanMLIR/Examples.lean", "import LeanMLIR/LeanMLIR/Dialects/LLVM/Basic.lean", "import SSA/Projects/Tensor1D/Tensor1D.lean", "import LeanMLIR/LeanMLIR/Framework/Macro.lean", "import SSA/Projects/LLVMRiscV/LLVMAndRiscv.lean", "import LeanMLIR.EffectKind"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "decidable_of_iff", "module": "Init.PropLemmas"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Valuation.mk", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.matchAlts", "module": "Lean.Parser.Term"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "RV64.sub", "module": "RISCV.Instructions"}, {"name": "RV64.sltiu", "module": "RISCV.Instructions"}, {"name": "RV64.slt", "module": "RISCV.Instructions"}, {"name": "RV64.andi", "module": "RISCV.Instructions"}, {"name": "RV64.sltu", "module": "RISCV.Instructions"}, {"name": "RV64.adduw", "module": "RISCV.Instructions"}, {"name": "RV64.addiw", "module": "RISCV.Instructions"}, {"name": "RV64.xori", "module": "RISCV.Instructions"}, {"name": "RV64.addi", "module": "RISCV.Instructions"}, {"name": "RV64.subw", "module": "RISCV.Instructions"}, {"name": "RV64.add", "module": "RISCV.Instructions"}, {"name": "RV64.addw", "module": "RISCV.Instructions"}, {"name": "RV64.andn", "module": "RISCV.Instructions"}, {"name": "RV64.auipc", "module": "RISCV.Instructions"}, {"name": "RV64.bclr", "module": "RISCV.Instructions"}, {"name": "RV64.bclri", "module": "RISCV.Instructions"}, {"name": "RV64.bext", "module": "RISCV.Instructions"}, {"name": "RV64.bexti", "module": "RISCV.Instructions"}, {"name": "RV64.binv", "module": "RISCV.Instructions"}, {"name": "RV64.binvi", "module": "RISCV.Instructions"}, {"name": "RV64.bset", "module": "RISCV.Instructions"}, {"name": "RV64.bseti", "module": "RISCV.Instructions"}, {"name": "RV64.clz", "module": "RISCV.Instructions"}, {"name": "RV64.clzw", "module": "RISCV.Instructions"}, {"name": "RV64.ctz", "module": "RISCV.Instructions"}, {"name": "RV64.ctzw", "module": "RISCV.Instructions"}, {"name": "RV64.div", "module": "RISCV.Instructions"}, {"name": "RV64.divu", "module": "RISCV.Instructions"}, {"name": "RV64.divuw", "module": "RISCV.Instructions"}, {"name": "RV64.divw", "module": "RISCV.Instructions"}, {"name": "RV64.lui", "module": "RISCV.Instructions"}, {"name": "RV64.max", "module": "RISCV.Instructions"}, {"name": "RV64.maxu", "module": "RISCV.Instructions"}, {"name": "RV64.min", "module": "RISCV.Instructions"}, {"name": "RV64.minu", "module": "RISCV.Instructions"}, {"name": "RV64.mul", "module": "RISCV.Instructions"}, {"name": "RV64.mulh", "module": "RISCV.Instructions"}, {"name": "RV64.mulhsu", "module": "RISCV.Instructions"}, {"name": "RV64.mulhu", "module": "RISCV.Instructions"}, {"name": "RV64.mulw", "module": "RISCV.Instructions"}, {"name": "RV64.ori", "module": "RISCV.Instructions"}, {"name": "RV64.orn", "module": "RISCV.Instructions"}, {"name": "RV64.pack", "module": "RISCV.Instructions"}, {"name": "RV64.packh", "module": "RISCV.Instructions"}, {"name": "RV64.packw", "module": "RISCV.Instructions"}, {"name": "RV64.rem", "module": "RISCV.Instructions"}, {"name": "RV64.remu", "module": "RISCV.Instructions"}, {"name": "RV64.remuw", "module": "RISCV.Instructions"}, {"name": "RV64.remw", "module": "RISCV.Instructions"}, {"name": "RV64.rol", "module": "RISCV.Instructions"}, {"name": "RV64.rolw", "module": "RISCV.Instructions"}, {"name": "RV64.ror", "module": "RISCV.Instructions"}, {"name": "RV64.rori", "module": "RISCV.Instructions"}, {"name": "RV64.roriw", "module": "RISCV.Instructions"}, {"name": "RV64.rorw", "module": "RISCV.Instructions"}, {"name": "RV64.sextb", "module": "RISCV.Instructions"}, {"name": "RV64.sexth", "module": "RISCV.Instructions"}, {"name": "RV64.sh1add", "module": "RISCV.Instructions"}, {"name": "RV64.sh1adduw", "module": "RISCV.Instructions"}, {"name": "RV64.sh2add", "module": "RISCV.Instructions"}, {"name": "RV64.sh2adduw", "module": "RISCV.Instructions"}, {"name": "RV64.sh3add", "module": "RISCV.Instructions"}, {"name": "RV64.sh3adduw", "module": "RISCV.Instructions"}, {"name": "RV64.sll", "module": "RISCV.Instructions"}, {"name": "RV64.slli", "module": "RISCV.Instructions"}, {"name": "RV64.slliuw", "module": "RISCV.Instructions"}, {"name": "RV64.slliw", "module": "RISCV.Instructions"}, {"name": "RV64.sllw", "module": "RISCV.Instructions"}, {"name": "RV64.slti", "module": "RISCV.Instructions"}, {"name": "RV64.sra", "module": "RISCV.Instructions"}, {"name": "RV64.srai", "module": "RISCV.Instructions"}, {"name": "RV64.sraiw", "module": "RISCV.Instructions"}, {"name": "RV64.sraw", "module": "RISCV.Instructions"}, {"name": "RV64.srl", "module": "RISCV.Instructions"}, {"name": "RV64.srli", "module": "RISCV.Instructions"}, {"name": "RV64.srliw", "module": "RISCV.Instructions"}, {"name": "RV64.srlw", "module": "RISCV.Instructions"}, {"name": "RV64.xnor", "module": "RISCV.Instructions"}, {"name": "RV64.xor", "module": "RISCV.Instructions"}, {"name": "RV64.zexth", "module": "RISCV.Instructions"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Ideal", "module": "Mathlib.RingTheory.Ideal.Defs"}, {"name": "Ideal.Quotient.mk", "module": "Mathlib.RingTheory.Ideal.Quotient.Defs"}, {"name": "Ideal.span", "module": "Mathlib.RingTheory.Ideal.Span"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.monomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Function.surjInv", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Polynomial.coeff", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.degree", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "List.range", "module": "Init.Data.List.Basic"}, {"name": "Polynomial.map", "module": "Mathlib.Algebra.Polynomial.Eval.Defs"}, {"name": "List.replicate", "module": "Init.Data.List.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "BitVec.signExtend", "module": "Init.Data.BitVec.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "cast", "module": "Init.Prelude"}, {"name": "cond", "module": "Init.Prelude"}, {"name": "Stream'", "module": "Mathlib.Data.Stream.Defs"}], "used_repo_defs": [{"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "Valuation.instAppendHVector", "content": "@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V"}, {"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "liftEffect", "content": "def liftEffect [Pure m] {e1 e2 : EffectKind} {α : Type}\n    (hle : e1 ≤ e2) (v1 : e1.toMonad m α) : e2.toMonad m α :=\n  match e1, e2, hle with\n    | .pure, .pure, _ | .impure, .impure, _ => v1\n    | .pure, .impure, _ => Pure.pure v1"}, {"name": "toMonad", "content": "def toMonad (e : EffectKind) (m : Type → Type) : Type → Type :=\n  match e with\n  | pure => Id\n  | impure => m"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "Dialect", "content": "structure Dialect where\n  (Op : Type)\n  (Ty : Type)\n  (m : Type → Type := Id)"}, {"name": "Op", "content": "inductive Op (q : Nat) (n : Nat)\n  | add : Op q n\n  | sub : Op q n\n  | mul : Op q n\n  | mul_constant : Op q n\n    \n    \n  | leading_term : Op q n\n  | monomial : Op q n\n  | monomial_mul : Op q n\n  | from_tensor : Op q n\n  | to_tensor : Op q n\n  | const (c : R q n) : Op q n\n  | const_int (c : Int) : Op q n\n  | const_idx (i : Nat) : Op q n"}, {"name": "map", "content": "def map (f : Ty₁ → Ty₂) : Ctxt Ty₁ → Ctxt Ty₂ :=\n  ofList ∘ (List.map f) ∘ toList\n\n    @[simp_denote]\n    instance : DialectDenote $dialect where\n      denote := fun op => match op with $matchAlts:matchAlts\n  )"}, {"name": "(q", "content": "noncomputable instance (q : ℕ) [Fact (q > 1)] : DialectDenote (ModArith q) where\ndenote\n  | .add, arg, _ =>\n      \n      (fun args : R q × R q => args.1 + args.2) arg.toPair\n  | .sub, arg, _ =>\n      \n      (fun args : R q × R q => args.1 - args.2) arg.toPair\n  | .mul, arg, _ =>\n      \n      (fun args : R q × R q => args.1 * args.2) arg.toPair\n  | .const _ c, _, _ =>\n      \n      c"}, {"name": "", "content": "@[reducible]\ninstance : DialectDenote Ex where\n  denote\n    | .cst n, _, _ => n ::ₕ .nil\n    | .add, .cons (a : Nat) (.cons b .nil), _ => a + b ::ₕ .nil\n    | .beq, .cons (a : Nat) (.cons b .nil), _ => (a == b) ::ₕ .nil"}, {"name": "ExOp", "content": "inductive ExOp :  Type\n  | add : ExOp\n  | beq : ExOp\n  | cst : ℕ → ExOp\n  deriving DecidableEq"}, {"name": "add", "content": "def add {Γ : Ctxt _} (e₁ e₂ : Ctxt.Var Γ .nat) : Expr Ex Γ .pure [.nat] :=\n  Expr.mk\n    (op := .add)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "Ex", "content": "abbrev Ex : Dialect where\n  Op := ExOp\n  Ty := ExTy"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  | bool\n  deriving DecidableEq"}, {"name": "cst", "content": "def cst {Γ : Ctxt _} (n : ℕ) : Expr Ex Γ .pure [.nat]  :=\n  Expr.mk\n    (op := .cst n)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "cons", "content": "@[match_pattern]\ndef cons (hd : Ty) : Ctxt Ty → Ctxt Ty\n| ⟨tl⟩ => ⟨hd :: tl⟩"}, {"name": "", "content": "@[simp, reducible]\ninstance : DialectDenote RV64 where\n  denote o args _ := [o.denote args]ₕ"}, {"name": "Op.denote", "content": "@[simp, simp_denote]\nabbrev Op.denote : (o : RV64.Op) → HVector toType o.sig → ⟦o.outTy⟧\n  | .li imm, _  => imm\n  | .addiw imm, regs => RV64.addiw imm (regs.getN 0)\n  | .lui imm, regs  => RV64.lui imm\n  | .auipc imm, regs => RV64.auipc imm (regs.getN 0)\n  | .slliw shamt, regs => RV64.slliw shamt (regs.getN 0)\n  | .srliw shamt, regs => RV64.srliw shamt (regs.getN 0)\n  | .sraiw shamt, regs => RV64.sraiw shamt (regs.getN 0)\n  | .slli shamt, regs => RV64.slli shamt (regs.getN 0)\n  | .srli shamt, regs => RV64.srli shamt (regs.getN 0)\n  | .srai shamt, regs => RV64.srai shamt (regs.getN 0)\n  | .addw, regs => RV64.addw (regs.getN 1) (regs.getN 0)\n  | .subw, regs => RV64.subw (regs.getN 1) (regs.getN 0)\n  | .sllw, regs => RV64.sllw (regs.getN 1) (regs.getN 0)\n  | .srlw, regs => RV64.srlw (regs.getN 1) (regs.getN 0)\n  | .sraw, regs => RV64.sraw (regs.getN 1) (regs.getN 0)\n  | .add, regs => RV64.add (regs.getN 1) (regs.getN 0)\n  | .slt, regs => RV64.slt (regs.getN 1) (regs.getN 0)\n  | .sltu, regs => RV64.sltu (regs.getN 1) (regs.getN 0)\n  | .and, regs => RV64.and (regs.getN 1) (regs.getN 0)\n  | .or, regs => RV64.or (regs.getN 1) (regs.getN 0)\n  | .xor, regs => RV64.xor (regs.getN 1) (regs.getN 0)\n  | .sll, regs => RV64.sll (regs.getN 1) (regs.getN 0)\n  | .srl, regs => RV64.srl (regs.getN 1) (regs.getN 0)\n  | .sub, regs => RV64.sub (regs.getN 1) (regs.getN 0)\n  | .sra, regs => RV64.sra (regs.getN 1) (regs.getN 0)\n  | .remw, regs => RV64.remw (regs.getN 1) (regs.getN 0)\n  | .remuw, regs => RV64.remuw (regs.getN 1) (regs.getN 0)\n  | .rem, regs => RV64.rem (regs.getN 1) (regs.getN 0)\n  | .remu, regs => RV64.remu (regs.getN 1) (regs.getN 0)\n  | .mulhu,regs => RV64.mulhu (regs.getN 1) (regs.getN 0)\n  | .mul ,regs => RV64.mul (regs.getN 1) (regs.getN 0)\n  | .mulhsu ,regs => RV64.mulhsu (regs.getN 1) (regs.getN 0)\n  | .mulh,regs => RV64.mulh (regs.getN 1) (regs.getN 0)\n  | .mulw,  regs => RV64.mulw (regs.getN 1) (regs.getN 0)\n  | .div, regs => RV64.div (regs.getN 1) (regs.getN 0)\n  | .divu, regs => RV64.divu (regs.getN 1) (regs.getN 0)\n  | .divw, regs => RV64.divw (regs.getN 1) (regs.getN 0)\n  | .divuw, regs => RV64.divuw (regs.getN 1) (regs.getN 0)\n  | .addi imm, reg => RV64.addi imm (reg.getN 0)\n  | .slti imm, reg => RV64.slti imm (reg.getN 0)\n  | .sltiu imm, reg => RV64.sltiu imm (reg.getN 0)\n  | .andi imm, reg => RV64.andi imm (reg.getN 0)\n  | .ori imm, reg => RV64.ori imm (reg.getN 0)\n  | .xori imm, reg => RV64.xori imm (reg.getN 0)\n  | .bclr, regs => RV64.bclr (regs.getN 1) (regs.getN 0)\n  | .bext, regs => RV64.bext (regs.getN 1) (regs.getN 0)\n  | .binv, regs => RV64.binv (regs.getN 1) (regs.getN 0)\n  | .bset, regs => RV64.bset (regs.getN 1) (regs.getN 0)\n  | .bclri shamt , reg => RV64.bclri shamt (reg.getN 0)\n  | .bexti shamt, reg => RV64.bexti shamt (reg.getN 0)\n  | .binvi shamt, reg => RV64.binvi shamt (reg.getN 0)\n  | .bseti shamt, reg => RV64.bseti shamt (reg.getN 0)\n  | .adduw, regs => RV64.adduw (regs.getN 1) (regs.getN 0)\n  | .sh1adduw , regs => RV64.sh1adduw (regs.getN 1) (regs.getN 0)\n  | .sh2adduw, regs => RV64.sh2adduw (regs.getN 1) (regs.getN 0)\n  | .sh3adduw, regs => RV64.sh3adduw (regs.getN 1) (regs.getN 0)\n  | .sh1add, regs => RV64.sh1add (regs.getN 1) (regs.getN 0)\n  | .sh2add, regs => RV64.sh2add (regs.getN 1) (regs.getN 0)\n  | .sh3add, regs => RV64.sh3add (regs.getN 1) (regs.getN 0)\n  | .slliuw shamt, regs => RV64.slliuw shamt (regs.getN 0)\n  | .andn, regs => RV64.andn (regs.getN 1) (regs.getN 0)\n  | .orn, regs => RV64.orn (regs.getN 1) (regs.getN 0)\n  | .xnor, regs => RV64.xnor (regs.getN 1) (regs.getN 0)\n  | .clz, regs => RV64.clz (regs.getN 1)\n  | .clzw, regs => RV64.clzw (regs.getN 1)\n  | .ctz, regs => RV64.ctz (regs.getN 1)\n  | .ctzw, regs => RV64.ctzw (regs.getN 1)\n  | .max, regs => RV64.max (regs.getN 1) (regs.getN 0)\n  | .maxu, regs => RV64.maxu (regs.getN 1) (regs.getN 0)\n  | .min, regs => RV64.min (regs.getN 1) (regs.getN 0)\n  | .minu, regs => RV64.minu (regs.getN 1) (regs.getN 0)\n  | .sextb, reg => RV64.sextb (reg.getN 0)\n  | .sexth, reg => RV64.sexth (reg.getN 0)\n  | .zexth, reg => RV64.zexth (reg.getN 0)\n  | .rol, regs => RV64.rol (regs.getN 1) (regs.getN 0)\n  | .rolw, regs => RV64.rolw (regs.getN 1) (regs.getN 0)\n  | .ror, regs => RV64.ror (regs.getN 1) (regs.getN 0)\n  | .rori shamt, regs => RV64.rori shamt (regs.getN 0)\n  | .roriw shamt, regs => RV64.roriw shamt (regs.getN 0)\n  | .rorw, regs => RV64.rorw (regs.getN 1) (regs.getN 0)\n  | .pack, regs => RV64.pack (regs.getN 1) (regs.getN 0)\n  | .packh, regs => RV64.packh (regs.getN 1) (regs.getN 0)\n  | .packw, regs => RV64.packw (regs.getN 1) (regs.getN 0)\n  \n  | .mv, regs  => RV64.mv_pseudo (regs.getN 0)\n  | .not, regs => RV64.not_pseudo (regs.getN 0)\n  | .neg, regs => RV64.neg_pseudo (regs.getN 0)\n  | .negw, regs => RV64.negw_pseudo (regs.getN 0)\n  | .sextw, regs => RV64.sextw_pseudo (regs.getN 0)\n  | .zextb, regs => RV64.zextb_pseudo (regs.getN 0)\n  | .zextw, regs => RV64.zextw_pseudo (regs.getN 0)\n  | .seqz, regs => RV64.seqz_pseudo (regs.getN 0)\n  | .snez, regs => RV64.snez_pseudo (regs.getN 0)\n  | .sltz, regs => RV64.sltz_pseudo (regs.getN 0)\n  | .sgtz, regs => RV64.sgtz_pseudo (regs.getN 0)"}, {"name": "Op", "content": "inductive Op\n   \n  | li : (val : BitVec 64) → Op\n  | lui (imm : BitVec 20)\n  | auipc (imm : BitVec 20)\n  | addi (imm : BitVec 12)\n  | andi (imm : BitVec 12)\n  | ori (imm : BitVec 12)\n  | xori (imm : BitVec 12)\n  | addiw (imm : BitVec 12)\n  | add\n  | slli (shamt : BitVec 6)\n  | sub\n  | and\n  | or\n  | xor\n  | sll\n  | srl\n  | sra\n  | addw\n  | subw\n  | sllw\n  | srlw\n  | sraw\n  | slti (imm : BitVec 12)\n  | sltiu (imm : BitVec 12)\n  | srli (shamt : BitVec 6)\n  | srai (shamt : BitVec 6)\n  | slliw (shamt : BitVec 5)\n  | srliw (shamt : BitVec 5)\n  | sraiw (shamt : BitVec 5)\n  | slt\n  | sltu\n   \n  | mul\n  | mulw\n  | mulh\n  | mulhu\n  | mulhsu\n  | divw\n  | divuw\n  | div\n  | divu\n  | remw\n  | rem\n  | remuw\n  | remu\n   \n   \n  | adduw\n  | sh1adduw\n  | sh2adduw\n  | sh3adduw\n  | sh1add\n  | sh2add\n  | sh3add\n  | slliuw (shamt : BitVec 6)\n   \n  | andn\n  | orn\n  | xnor\n  | clz\n  | clzw\n  | ctz\n  | ctzw\n  | max\n  | maxu\n  | min\n  | minu\n  | sextb\n  | sexth\n  | zexth\n  | rol\n  | rolw\n  | ror\n  | rori (_shamt : BitVec 6)\n  | roriw (_shamt : BitVec 5)\n  | rorw\n   \n  | bclr\n  | bclri (shamt : BitVec 6)\n  | bext\n  | bexti (shamt : BitVec 6)\n  | binv\n  | binvi (shamt : BitVec 6)\n  | bset\n  | bseti (shamt : BitVec 6)\n   \n  | pack\n  | packh\n  | packw\n   \n  | mv\n  | not\n  | neg\n  | negw\n  | sextw\n  | zextb\n  | zextw\n  | seqz\n  | snez\n  | sltz\n  | sgtz\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "RV64", "content": "@[simp]\nabbrev RV64 : Dialect where\n  Op := Op\n  Ty := Ty"}, {"name": "Ty", "content": "inductive Ty\n  | bv : Ty\n  deriving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "neg_pseudo", "content": "@[simp_riscv]\ndef neg_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.sub rs1_val 0"}, {"name": "seqz_pseudo", "content": "@[simp_riscv]\ndef seqz_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.sltiu 1 rs1_val"}, {"name": "sgtz_pseudo", "content": "@[simp_riscv]\ndef sgtz_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.slt rs1_val 0"}, {"name": "zextb_pseudo", "content": "@[simp_riscv]\ndef zextb_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.andi 255 rs1_val"}, {"name": "snez_pseudo", "content": "@[simp_riscv]\ndef snez_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.sltu rs1_val 0"}, {"name": "sltz_pseudo", "content": "@[simp_riscv]\ndef sltz_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.slt 0 rs1_val"}, {"name": "zextw_pseudo", "content": "@[simp_riscv]\ndef zextw_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.adduw 0 rs1_val"}, {"name": "sextw_pseudo", "content": "@[simp_riscv]\ndef sextw_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.addiw 0 rs1_val"}, {"name": "not_pseudo", "content": "@[simp_riscv]\ndef not_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.xori (-1) rs1_val"}, {"name": "mv_pseudo", "content": "@[simp_riscv]\ndef mv_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.addi 0 rs1_val"}, {"name": "negw_pseudo", "content": "@[simp_riscv]\ndef negw_pseudo (rs1_val : BitVec 64) : BitVec 64 :=\n  RV64.subw rs1_val 0"}, {"name": "", "content": "@[simp]\nnoncomputable instance : DialectDenote (FHE q n) where\n    denote\n    | Op.add, arg, _ => [(fun args : R q n × R q n => args.1 + args.2) arg.toPair]ₕ\n    | Op.sub, arg, _ => [(fun args : R q n × R q n => args.1 - args.2) arg.toPair]ₕ\n    | Op.mul, arg, _ => [(fun args : R q n × R q n => args.1 * args.2) arg.toPair]ₕ\n    | Op.mul_constant, arg, _ => [(fun args : R q n × Int => args.1 * ↑(args.2)) arg.toPair]ₕ\n    | Op.leading_term, arg, _ => [R.leadingTerm arg.toSingle]ₕ\n    | Op.monomial, arg, _ => [(fun args => R.monomial ↑(args.1) args.2) arg.toPair]ₕ\n    | Op.monomial_mul, arg, _ => [(fun args : R q n × Nat => args.1 * R.monomial 1 args.2) arg.toPair]ₕ\n    | Op.from_tensor, arg, _ => [R.fromTensor arg.toSingle]ₕ\n    | Op.to_tensor, arg, _ => [R.toTensor' arg.toSingle]ₕ\n    | Op.const c, _arg, _\n    | Op.const_int c, _, _\n    | Op.const_idx c, _, _ => [c]ₕ"}, {"name": "R.fromTensor", "content": "noncomputable def R.fromTensor {q n} (coeffs : List Int) : R q n :=\n  coeffs.zipIdx.foldl (init := 0) fun res (c, i) =>\n    res + R.monomial ↑c i"}, {"name": "R.monomial", "content": "noncomputable def R.monomial {q n : Nat} (c : ZMod q) (i : Nat): R q n :=\n  R.fromPoly (Polynomial.monomial i c)"}, {"name": "R.fromPoly", "content": "abbrev R.fromPoly {q n : Nat} : (ZMod q)[X] →+* R q n := Ideal.Quotient.mk (Ideal.span {f q n})"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "R.leadingTerm", "content": "noncomputable def R.leadingTerm {q n} (a : R q n) : R q n :=\n  let deg? := Polynomial.degree a.representative\n  match deg? with\n    | .none => 0\n    | .some deg =>  R.monomial (a.coeff deg) deg"}, {"name": "R.representative", "content": "noncomputable def R.representative :\n    R q n → (ZMod q)[X] := fun x => R.representative' q n x %ₘ (f q n)"}, {"name": "R.representative'", "content": "private noncomputable def R.representative' :\n    R q n → (ZMod q)[X] := Function.surjInv (R.surjective_fromPoly q n)"}, {"name": "R.coeff", "content": "noncomputable def R.coeff {q n} (a : R q n) (i : Nat) : ZMod q :=\n  Polynomial.coeff a.representative i"}, {"name": "R.toTensor'", "content": "noncomputable def R.toTensor' {q n} (a : R q n) : List Int :=\n  let t := a.toTensor\n  t ++ List.replicate (2^n - t.length + 1) 0"}, {"name": "R.toTensor", "content": "noncomputable def R.toTensor {q n} (a : R q n) : List Int :=\n  List.range a.repLength |>.map fun i =>\n        a.coeff i |>.toInt"}, {"name": "R.repLength", "content": "noncomputable def R.repLength {q n} (a : R q n) : Nat := match\n  Polynomial.degree a.representative with\n    | none => 0\n    | some d => d + 1"}, {"name": "", "content": "instance : DialectDenote LLVM := ⟨\n  fun o args _ => [Op.denote o args]ₕ\n⟩"}, {"name": "", "content": "@[reducible]\ninstance : DialectDenote SimpleReg where\n  denote\n    | .const n, _, _ => BitVec.ofInt 32 n ::ₕ .nil\n    | .add, [(a : BitVec 32), (b : BitVec 32)]ₕ , _ => a + b ::ₕ .nil\n    | .iterate k, [(x : BitVec 32)]ₕ, [(f : _ → _)]ₕ =>\n      let f := fun y => (f y).getN 0\n      let f' (v :  BitVec 32) : BitVec 32 := f (Ctxt.Valuation.nil.cons v)\n      let y := k.iterate f' x\n      [y]ₕ"}, {"name": "Op", "content": "inductive Op :  Type\n  | add : Op\n  | const : (val : ℤ) → Op\n  | iterate (k : ℕ) : Op\n  deriving DecidableEq"}, {"name": "add", "content": "@[simp_denote]\ndef add {Γ : Ctxt _} (e₁ e₂ : Var Γ int) : Expr SimpleReg Γ .pure [int] :=\n  Expr.mk\n    (op := .add)\n    (eff_le := by admit /- proof elided -/\n    )\n    (ty_eq := rfl)\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "SimpleReg.int", "content": "abbrev SimpleReg.int : SimpleReg.Ty := .int"}, {"name": "SimpleReg", "content": "abbrev SimpleReg : Dialect where\n  Op := Op\n  Ty := Ty"}, {"name": "iterate", "content": "@[simp_denote]\ndef iterate {Γ : Ctxt _} (k : Nat) (input : Var Γ int) (body : Com SimpleReg ⟨[int]⟩ .impure [int]) :\n    Expr SimpleReg Γ .pure [int] :=\n  Expr.mk\n    (op := Op.iterate k)\n    (eff_le := by admit /- proof elided -/\n    )\n    (ty_eq := rfl)\n    (args := .cons input .nil)\n    (regArgs := HVector.cons body HVector.nil)"}, {"name": "Ty", "content": "inductive Ty\n  | int\n  deriving DecidableEq"}, {"name": "[SIG", "content": "instance [SIG : DialectSignature d] [DENOTE : DialectDenote d] {Γ : Ctxt d.Ty} {t}\n    (com : Com d Γ .pure t) : Inhabited (DCEType com) where\n  default :=\n    ⟨Γ, Hom.id, com, by admit /- proof elided -/\n    ⟩"}, {"name": "Hom.id", "content": "@[simp] abbrev Hom.id {Γ : Ctxt Ty} : Γ.Hom Γ :=\n  fun _ v => v"}, {"name": "", "content": "@[reducible]\ninstance : DialectDenote Ex where\n  denote\n    | .cst n, _, _ => n ::ₕ .nil\n    | .add, (a : Nat) ::ₕ b ::ₕ .nil, _ => a + b    ::ₕ .nil\n    | .beq, (a : Nat) ::ₕ b ::ₕ .nil, _ => (a == b) ::ₕ .nil"}, {"name": "cst", "content": "def cst {Γ : Ctxt _} (n : ℕ) : Expr Ex Γ .pure [.nat] :=\n  Expr.mk\n    (op := .cst n)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  | bool\n  deriving DecidableEq, Repr"}, {"name": "", "content": "@[simp, reducible]\ninstance : DialectDenote (LLVMPlusRiscV) where\n  denote\n  | .llvm llvmOp, args, .nil => do\n      let xs ← DialectDenote.denote llvmOp (llvmArgsFromHybrid args) .nil\n      return xs.map' Ty.llvm (fun t x => x)\n  | .riscv (riscvOp), args, .nil => do\n      let xs ← DialectDenote.denote riscvOp (riscvArgsFromHybrid args) .nil\n      return xs.map' Ty.riscv (fun t x => x)\n  | .castRiscv _ , elemToCast, _ =>\n    let toCast : BitVec 64 :=\n      elemToCast.getN 0 (by admit /- proof elided -/\n      )\n    [castriscvToLLVM toCast]ₕ\n  | .castLLVM _,\n    (elemToCast : HVector TyDenote.toType [Ty.llvm (.bitvec _)]), _ =>\n    let toCast : PoisonOr (BitVec _) :=\n      elemToCast.getN 0 (by admit /- proof elided -/\n      )\n    [castLLVMToriscv toCast]ₕ"}, {"name": "Op", "content": "inductive Op where\n  | llvm : LLVM.Op -> Op\n  | riscv : RISCV64.RV64.Op -> Op\n  | castRiscv : Nat → Op\n  | castLLVM : Nat → Op\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "castLLVMToriscv", "content": "@[simp_riscv]\ndef castLLVMToriscv (toCast : PoisonOr (BitVec w)) : BitVec 64 :=\n  BitVec.signExtend 64 (toCast.toOption.getD 0#w)"}, {"name": "PoisonOr", "content": "structure PoisonOr (α : Type) where\n  ofOption :: toOption : Option α\n  deriving DecidableEq"}, {"name": "castriscvToLLVM", "content": "@[simp_riscv]\ndef castriscvToLLVM (toCast : BitVec 64) : PoisonOr (BitVec w) :=\n  .value (BitVec.signExtend w toCast)"}, {"name": "llvmArgsFromHybrid", "content": "@[simp_denote]\ndef llvmArgsFromHybrid : {tys : List LLVM.Ty} →\n  HVector TyDenote.toType (tys.map LLVMRiscV.Ty.llvm) → HVector TyDenote.toType tys\n  | [], .nil => .nil\n  | _ :: _, .cons x xs => .cons x (llvmArgsFromHybrid xs)"}, {"name": "Ty", "content": "inductive Ty where\n  | llvm : LLVM.Ty -> Ty\n  | riscv : RISCV64.RV64.Ty -> Ty\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "riscvArgsFromHybrid", "content": "@[simp_denote]\ndef riscvArgsFromHybrid : {tys : List RISCV64.RV64.Ty} →\n  HVector TyDenote.toType (tys.map LLVMRiscV.Ty.riscv) → HVector TyDenote.toType tys\n  | [], .nil => .nil\n  | _ :: _, .cons x xs => .cons x (riscvArgsFromHybrid xs)"}, {"name": "", "content": "@[reducible]\ninstance : DialectDenote ExOp ExTy where\n  denote\n    | .cst n, _, _ => n\n    | .add, .cons (a : Nat) (.cons b .nil), _ => a + b\n    | .beq, .cons (a : Nat) (.cons b .nil), _ => a == b"}, {"name": "ExOp", "content": "inductive ExOp :  Type\n  | add : ExOp\n  | beq : ExOp\n  | cst : ℕ → ExOp\n  deriving DecidableEq, Repr"}, {"name": "add", "content": "def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .nat) : Expr Γ .nat :=\n  Expr.mk\n    (op := .add)\n    (ty_eq := rfl)\n    (eff_le := EffectKind.le_refl _)\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  deriving DecidableEq, Repr"}, {"name": "Expr", "content": "abbrev Expr (Γ) (ty) := _root_.Expr ExOp Γ .pure ty"}, {"name": "cst", "content": "def cst {Γ : Ctxt _} (n : ℕ) : Expr Γ .nat  :=\n  Expr.mk\n    (op := .cst n)\n    (ty_eq := rfl)\n    (eff_le := EffectKind.le_refl _)\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "", "content": "@[reducible]\ninstance : DialectDenote ExOp ExTy where\n  denote\n    | .add, .cons (a : Nat) (.cons b .nil), _ => a + b\n    | .runK (k : Nat), (.cons (v : Nat) .nil), (.cons rgn _nil) =>\n      k.iterate (fun val => rgn (fun _ty _var => val)) v"}, {"name": "ExOp", "content": "inductive ExOp :  Type\n  | add : ExOp\n  | runK : ℕ → ExOp\n  deriving DecidableEq, Repr"}, {"name": "add", "content": "def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .nat) : Expr Γ .nat :=\n  Expr.mk\n    (op := .add)\n    (ty_eq := rfl)\n    (eff_le := EffectKind.pure_le _)\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "[Monad", "content": "@[reducible]\ninstance [Monad d.m] : DialectDenote (Scf d) where\n  denote\n    | .coe o', args', regArgs' =>\n        let denote' := DialectDenote.denote o'\n        by\n         exact denote' args' regArgs'\n    | .if t t', (.cons (cond ) (.cons v .nil)),\n         (.cons (f : Ctxt.Valuation ⟨[t]⟩ → d.m (HVector toType [t'])) (.cons (g : _ → _) .nil)) =>\n         let body := if B.denote_eq ▸ cond then f else g\n      body (Ctxt.Valuation.nil.cons v)\n    | .run _t, (.cons v .nil), (.cons (f : _ → _) .nil) =>\n        f (Ctxt.Valuation.nil.cons v)\n    | .for ty, (.cons istart (.cons istep (.cons niter (.cons vstart .nil)))),\n        (.cons (f : _  → _) .nil) => do\n        let istart : ℤ := Z.denote_eq ▸ istart\n        let istep : ℤ := Z.denote_eq ▸ istep\n        let niter : ℕ := N.denote_eq ▸ niter\n        let f' : LoopBody (d.m ⟦ty⟧) := fun i v => do\n          let v ← v\n          let i := Z.denote_eq.symm ▸ i\n          let xs ← f (Valuation.ofPair i v)\n          return xs.get (0 : Fin 1)\n        let to_iterate := f'.counterDecorator (α := d.m ⟦ty⟧) (δ := istep)\n        let loop_fn := niter.iterate (op := to_iterate)\n        let x ← (loop_fn (istart, pure vstart)).2\n        return [x]ₕ\n\n    | .iterate k, (.cons (x) .nil), (.cons (f : _ → _) .nil) => do\n      let x : ℤ := Z.denote_eq ▸ x\n      let coe : ℤ = toType Z.ty := Z.denote_eq.symm\n      let f' (v : d.m ℤ) : d.m ℤ := do\n        let v ← v\n        let xs ← f (Ctxt.Valuation.nil.cons (cast coe v))\n        let x := xs.getN 0\n        return coe ▸ x\n      let y ← (k.iterate f' (pure x))\n      return [cast Z.denote_eq.symm y]ₕ"}, {"name": "counterDecorator", "content": "def counterDecorator (δ : Int) (f : LoopBody α) : Int × α → Int × α :=\n  fun (i, v) => (i + δ, f i v)"}, {"name": "Scf.Op", "content": "inductive Scf.Op (Op' Ty' : Type) (m') [TyDenote Ty'] [DialectSignature ⟨Op', Ty', m'⟩]\n    [DialectDenote ⟨Op', Ty', m'⟩] : Type _\n  | coe (o : Op')\n  | iterate (k : ℕ) \n  | run (inputty : Ty')  \n  | if (inputty retty' : Ty')  \n  | for (ty : Ty')\n  deriving DecidableEq, Repr"}, {"name": "HasTy", "content": "class HasTy (d : Dialect) (DenotedTy : Type) [TyDenote d.Ty] [DialectSignature d] where\n    ty : d.Ty\n    denote_eq : toType ty = DenotedTy := by admit /- proof elided -/"}, {"name": "LoopBody", "content": "abbrev LoopBody (t : Type) : Type := Int → t → t"}, {"name": "iterate", "content": "@[simp_denote] def iterate {Γ : Ctxt _} (k : Nat) (input : Var Γ Arith.Ty.int)\n    (body : Com ScfArith ⟨[.int]⟩ .impure .int) : Expr ScfArith Γ .impure .int :=\n  Expr.mk\n    (op := .iterate k)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons input .nil)\n    (regArgs := HVector.cons body HVector.nil)"}, {"name": "ScfArith", "content": "abbrev ScfArith := Scf Arith"}, {"name": "Scf", "content": "def Scf (d : Dialect) [TyDenote d.Ty] [DialectSignature d] [DialectDenote d] : Dialect where\n  Op := Scf.Op d.Op d.Ty d.m\n  Ty := d.Ty\n  m  := d.m"}, {"name": "Op", "content": "inductive Op\n  | add : Op  \n  | add_nat : Op  \n  | axpy : Op  \n  | neg : Op  \n  | const : (val : ℤ) → Op\n  | const_nat : (val : ℕ) → Op"}, {"name": "run", "content": "@[simp_denote]\ndef run {Γ : Ctxt _} {t : Arith.Ty} (v : Var Γ t) (body : Com ScfArith ⟨[t]⟩ .impure t) :\n    Expr ScfArith Γ .impure t :=\n  Expr.mk\n    (op := .run t)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons v .nil)\n    (regArgs := HVector.cons body <| HVector.nil)"}, {"name": "Ty", "content": "inductive Ty\n| int\n| bool\n| nat\n deriving DecidableEq, Repr"}, {"name": "neg", "content": "@[simp_denote] def neg {Γ : Ctxt _} (a : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .neg)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "axpy", "content": "@[simp_denote] def axpy {Γ : Ctxt _} (a : Var Γ .int) (x : Var Γ .nat) (b: Var Γ .int) :\n    Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .axpy)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons x <| .cons b .nil)\n    (regArgs := .nil)"}, {"name": "add_nat", "content": "@[simp_denote] def add_nat (e₁ e₂ : Var Γ .nat) : Expr ScfArith Γ .pure .nat :=\n  Expr.mk\n    (op := .coe <| .add_nat)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "Ty", "content": "inductive Ty\n  | int\n   \n  | int2\n  deriving DecidableEq, Lean.ToExpr"}, {"name": "Op", "content": "inductive Op\n  | noop\n  | mkPair\n  | unPair\n  deriving Lean.ToExpr"}, {"name": "Arith", "content": "abbrev Arith : Dialect := {Op, Ty}"}, {"name": "Valuation.ofPair", "content": "def Valuation.ofPair  {t₁ t₂ : Ty} (v₁: ⟦t₁⟧) (v₂ : ⟦t₂⟧) :\n    Valuation (Ctxt.ofList [t₁, t₂]) :=\n  Valuation.ofHVector (.cons v₁ <| .cons v₂ <| .nil )"}, {"name": "", "content": "@[reducible]\ninstance : DialectDenote Arith where\n  denote\n    | .const n, _, _ => [n]ₕ\n    | .const_nat n, _, _ => [n]ₕ\n    | .neg, .cons (a : ℤ ) .nil, _ => [-a]ₕ\n    | .axpy, .cons (a : ℤ) (.cons (x : ℕ) (.cons (b : ℤ) .nil)), _ => [a * (x : ℤ) + b]ₕ\n    | .add, .cons (a : ℤ) (.cons (b : ℤ) .nil), _ => [a + b]ₕ\n    | .add_nat, .cons (a : ℕ) (.cons (b : ℕ) .nil), _ => [a + b]ₕ"}, {"name": "add", "content": "@[simp_denote] def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .add)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)\n\n      instance : DialectSignature $dialect where\n        signature := fun op => match op with $matchAlts:matchAlts\n    )"}, {"name": "Op.signature", "content": "@[simp, reducible]\ndef Op.signature : Op q n → Signature (Ty q n) :=\n  fun o => {sig := Op.sig o, returnTypes := [Op.outTy o], regSig := []}"}, {"name": "Op.sig", "content": "@[simp, reducible]\ndef Op.sig : Op q n → List (Ty q n)\n| Op.add => [Ty.polynomialLike, Ty.polynomialLike]\n| Op.sub => [Ty.polynomialLike, Ty.polynomialLike]\n| Op.mul => [Ty.polynomialLike, Ty.polynomialLike]\n| Op.mul_constant => [Ty.polynomialLike, Ty.integer]\n| Op.leading_term => [Ty.polynomialLike]\n| Op.monomial => [Ty.integer, Ty.index]\n| Op.monomial_mul => [Ty.polynomialLike, Ty.index]\n| Op.from_tensor => [Ty.tensor]\n| Op.to_tensor => [Ty.polynomialLike]\n| Op.const _ => []\n| Op.const_int _ => []\n| Op.const_idx _ => []"}, {"name": "Op.outTy", "content": "@[simp, reducible]\ndef Op.outTy : Op q n → Ty q n\n| Op.add | Op.sub | Op.mul | Op.mul_constant | Op.leading_term | Op.monomial\n| Op.monomial_mul | Op.from_tensor | Op.const _  => Ty.polynomialLike\n| Op.to_tensor => Ty.tensor\n| Op.const_int _ => Ty.integer\n| Op.const_idx _ => Ty.index"}, {"name": "Op.regSig", "content": "@[reducible, simp]\ndef Op.regSig : Op → RegionSignature Ty\n  | .map2d => [([Ty.int], [.int])]\n  | _ => []"}, {"name": "", "content": "instance : DialectSignature Ex where\n  signature\n    | .add    => ⟨[.nat, .nat], [], [.nat], .pure⟩\n    | .beq    => ⟨[.nat, .nat], [], [.bool], .pure⟩\n    | .cst _  => ⟨[], [], [.nat], .pure⟩"}, {"name": "", "content": "@[reducible]\ninstance : DialectSignature Tensor2D where\n  signature op := { sig := op.sig, regSig := op.regSig, returnTypes := [op.outTy] }"}, {"name": "", "content": "instance : DialectSignature Tensor1D where\n  signature op := { sig := op.sig, regSig := op.regSig, returnTypes := [op.outTy], effectKind := .pure }"}, {"name": "", "content": "instance : DialectSignature RV64 where\n  signature o := {sig := Op.sig o, returnTypes := [Op.outTy o], regSig := []}"}, {"name": "Op.sig", "content": "@[simp, reducible]\ndef Op.sig : Op → List Ty\n  | .li _ => []\n  | .mulh  => [Ty.bv, Ty.bv]\n  | .mulhu  => [Ty.bv, Ty.bv]\n  | .mulhsu  => [Ty.bv, Ty.bv]\n  | .divu =>  [Ty.bv, Ty.bv]\n  | .remuw  => [Ty.bv, Ty.bv]\n  | .remu  =>  [Ty.bv, Ty.bv]\n  | .addiw (_imm : BitVec 12) => [Ty.bv]\n  | .lui (_imm : BitVec 20) => [Ty.bv]\n  | .auipc (_imm : BitVec 20)  => [Ty.bv]\n  | .slliw (_shamt : BitVec 5)  => [Ty.bv]\n  | .srliw (_shamt : BitVec 5) => [Ty.bv]\n  | .sraiw (_shamt : BitVec 5) => [Ty.bv]\n  | .slli (_shamt : BitVec 6) => [Ty.bv]\n  | .srli (_shamt : BitVec 6) => [Ty.bv]\n  | .srai (_shamt : BitVec 6) => [Ty.bv]\n  | .addw => [Ty.bv, Ty.bv]\n  | .subw => [Ty.bv, Ty.bv]\n  | .sllw => [Ty.bv, Ty.bv]\n  | .srlw => [Ty.bv, Ty.bv]\n  | .sraw => [Ty.bv, Ty.bv]\n  | .add => [Ty.bv, Ty.bv]\n  | .slt => [Ty.bv, Ty.bv]\n  | .sltu => [Ty.bv, Ty.bv]\n  | .and => [Ty.bv, Ty.bv]\n  | .or => [Ty.bv, Ty.bv]\n  | .xor => [Ty.bv, Ty.bv]\n  | .sll => [Ty.bv, Ty.bv]\n  | .srl => [Ty.bv, Ty.bv]\n  | .sub => [Ty.bv, Ty.bv]\n  | .sra => [Ty.bv, Ty.bv]\n  | .remw  => [Ty.bv, Ty.bv]\n  | .rem  =>  [Ty.bv, Ty.bv]\n  | .mul => [Ty.bv, Ty.bv]\n  | .mulw => [Ty.bv, Ty.bv]\n  | .div  =>  [Ty.bv, Ty.bv]\n  | .divw  =>  [Ty.bv, Ty.bv]\n  | .divuw  =>  [Ty.bv, Ty.bv]\n  | .addi (_imm : BitVec 12) => [Ty.bv]\n  | .slti (_imm : BitVec 12) => [Ty.bv]\n  | .sltiu (_imm : BitVec 12) => [Ty.bv]\n  | .andi (_imm : BitVec 12) => [Ty.bv]\n  | .ori (_imm : BitVec 12) => [Ty.bv]\n  | .xori (_imm : BitVec 12) => [Ty.bv]\n  | .bclr => [Ty.bv, Ty.bv]\n  | .bext => [Ty.bv, Ty.bv]\n  | .binv => [Ty.bv, Ty.bv]\n  | .bset  => [Ty.bv, Ty.bv]\n  | .bclri (_shamt : BitVec 6) => [Ty.bv]\n  | .bexti (_shamt : BitVec 6) => [Ty.bv]\n  | .binvi (_shamt : BitVec 6) => [Ty.bv]\n  | .bseti (_shamt : BitVec 6) => [Ty.bv]\n  | .adduw => [Ty.bv, Ty.bv]\n  | .sh1adduw => [Ty.bv, Ty.bv]\n  | .sh2adduw => [Ty.bv, Ty.bv]\n  | .sh3adduw => [Ty.bv, Ty.bv]\n  | .sh1add => [Ty.bv, Ty.bv]\n  | .sh2add => [Ty.bv, Ty.bv]\n  | .sh3add => [Ty.bv, Ty.bv]\n  | .slliuw (_shamt : BitVec 6) => [Ty.bv]\n  | .andn => [Ty.bv, Ty.bv]\n  | .orn => [Ty.bv, Ty.bv]\n  | .xnor => [Ty.bv, Ty.bv]\n  | .clz\n  | .clzw\n  | .ctz\n  | .ctzw\n  | .max => [Ty.bv, Ty.bv]\n  | .maxu => [Ty.bv, Ty.bv]\n  | .min  => [Ty.bv, Ty.bv]\n  | .minu  => [Ty.bv, Ty.bv]\n  | .sextb => [Ty.bv]\n  | .sexth => [Ty.bv]\n  | .zexth => [Ty.bv]\n  | .rol => [Ty.bv, Ty.bv]\n  | .rolw => [Ty.bv, Ty.bv]\n  | .ror => [Ty.bv, Ty.bv]\n  | .rori (_shamt : BitVec 6) =>[Ty.bv]\n  | .roriw (_shamt : BitVec 5) =>[Ty.bv]\n  | .rorw => [Ty.bv, Ty.bv]\n  | .pack => [Ty.bv, Ty.bv]\n  | .packh => [Ty.bv, Ty.bv]\n  | .packw => [Ty.bv, Ty.bv]\n  | .mv => [Ty.bv]\n  | .not => [Ty.bv]\n  | .neg => [Ty.bv]\n  | .negw => [Ty.bv]\n  | .sextw => [Ty.bv]\n  | .zextb => [Ty.bv]\n  | .zextw => [Ty.bv]\n  | .seqz => [Ty.bv]\n  | .snez => [Ty.bv]\n  | .sltz => [Ty.bv]\n  | .sgtz => [Ty.bv]"}, {"name": "", "content": "instance : DialectSignature (FHE q n) := ⟨Op.signature⟩"}, {"name": "", "content": "instance : DialectSignature LLVM where\n  signature op := ⟨op.sig, [], [op.outTy], .pure⟩"}, {"name": "", "content": "instance : DialectSignature HSxComb where\n  signature := fun op =>\n    match op with\n    | .comb o => liftSig (signature o) \n    \n    \n    | .hs o => MLIR2Handshake.instDialectSignatureHandshake.signature o"}, {"name": "Op", "content": "inductive Op : Type _\n  | comb (o : MLIR2Comb.Comb.Op)\n  | hs (o : MLIR2Handshake.Handshake.Op)\n  deriving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "liftSig", "content": "def liftSig (sig : Signature MLIR2Comb.Ty) : Signature MLIR2Handshake.Ty :=\n  Signature.mk (sig.sig.map liftTy) [] (liftTy sig.outTy)"}, {"name": "liftTy", "content": "def liftTy : MLIR2Comb.Ty → MLIR2Handshake.Ty\n| .bitvec w => .stream (.bitvec w)"}, {"name": "Ty", "content": "inductive Ty\n| stream (ty2 : Ty2) : Ty \n| stream2 (ty2 : Ty2) : Ty \n| stream2token (ty2 : Ty2) : Ty \nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "Ty", "content": "inductive Ty\n| bitvec (w : Nat) : Ty \nderiving DecidableEq, Repr, ToExpr"}, {"name": "Ty2", "content": "inductive Ty2\n  | bitvec (w : Nat) : Ty2\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "map", "content": "def map {α β : Type} (s : Stream α) (f : α → β) : Stream β :=\n  fun i => (s i).map f"}, {"name": "Stream", "content": "def Stream (β : Type) := Stream' (Option β)"}, {"name": "", "content": "instance : DialectSignature DCxComb where\n  signature := fun op =>\n    match op with\n    | .comb o => liftSig (signature o) \n    \n    \n    | .dc o => MLIR2DC.instDialectSignatureDC.signature o"}, {"name": "Op", "content": "inductive Op : Type _\n  | comb (o : MLIR2Comb.Comb.Op)\n  | dc (o : MLIR2DC.DC.Op)\n  deriving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "liftSig", "content": "def liftSig (sig : Signature MLIR2Comb.Ty) : Signature MLIR2DC.Ty :=\n  Signature.mk (sig.sig.map liftTy) [] (liftTy sig.outTy)"}, {"name": "liftTy", "content": "def liftTy : MLIR2Comb.Ty → MLIR2DC.Ty\n| .bitvec w => .valuestream w"}, {"name": "Ty", "content": "inductive Ty\n| tokenstream : Ty\n| tokenstream2 : Ty\n| valuestream (w : Nat) : Ty \n| valuestream2 (w : Nat) : Ty \n| valuetokenstream (w : Nat) : Ty \n| variadicvaluetokenstream (w : Nat) : Ty \nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "LLVMPlusRiscVSignature", "content": "@[simp]\ninstance LLVMPlusRiscVSignature : DialectSignature LLVMPlusRiscV where\n  signature\n  | .llvm llvmOp => .llvm <$> DialectSignature.signature llvmOp\n  | .riscv riscvOp => .riscv <$> DialectSignature.signature riscvOp\n  | .castRiscv w =>\n      {sig := [Ty.riscv .bv], returnTypes := [Ty.llvm (.bitvec w)], regSig := []}\n  | .castLLVM w =>\n      {sig := [Ty.llvm (.bitvec w)], returnTypes := [Ty.riscv .bv], regSig := []}"}, {"name": "", "content": "instance : DialectSignature ExOp ExTy where\n  signature\n    | .add    => ⟨[.nat, .nat], [], .nat, .pure⟩\n    | .beq    => ⟨[.nat, .nat], [], .bool, .pure⟩\n    | .cst _  => ⟨[], [], .nat, .pure⟩"}, {"name": "", "content": "instance : DialectSignature ExOp ExTy where\n  signature\n  | .add    => ⟨[.nat, .nat], [], .nat, .pure⟩\n  | .runK _ => ⟨[.nat], [([.nat], .nat)], .nat, .pure⟩"}, {"name": "[TyDenote", "content": "@[reducible]\ninstance [TyDenote d.Ty] [DialectSignature d] [DialectDenote d]\n    [B : HasBool d] [N : HasNat d] [I : HasInt d] : DialectSignature (Scf d) where\n   signature\n   | .coe o => signature (d:=d) o\n    | .if t t' => ⟨[B.ty, t], [(⟨[t]⟩, [t']), (⟨[t]⟩, [t'])], [t'], .impure⟩\n      \n      \n      \n      \n      \n    | .for t => ⟨[ I.ty,  I.ty,  N.ty, t], [(⟨[I.ty, t]⟩, [t])], [t], .impure⟩\n    | .run t => ⟨[t], [(⟨[t]⟩, [t])], [t], .impure⟩\n    | .iterate _k => ⟨[I.ty], [(⟨[I.ty]⟩, [I.ty])], [I.ty], .impure⟩"}, {"name": "", "content": "@[reducible]\ninstance : DialectSignature Arith where\n  signature\n    | .axpy => ⟨[.int, .nat, .int], [], [.int], .pure⟩\n    | .neg => ⟨[.int], [], [.int], .pure⟩\n    | .const _ => ⟨[], [], [.int], .pure⟩\n    | .const_nat _ => ⟨[], [], [.nat], .pure⟩\n    | .add   => ⟨[.int, .int], [], [.int], .pure⟩\n    | .add_nat   => ⟨[.nat, .nat], [], [.nat], .pure⟩"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "TyDenote.toType", "content": "notation \"⟦\" x \"⟧\" => TyDenote.toType x"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "pure_liftEffect", "content": "@[simp]\ntheorem pure_liftEffect {eff₁ eff₂ : EffectKind}\n    (hle : eff₁ ≤ .pure) [Monad m] (x : eff₁.toMonad m α) :\n    (Pure.pure (liftEffect hle x) : eff₂.toMonad m α)\n    = liftEffect (by cases hle; constructor) x"}, {"name": "eq_of_le_pure", "content": "@[simp]\ntheorem eq_of_le_pure {e : EffectKind}\n    (he : e ≤ pure) : e = pure"}, {"name": "pure_map", "content": "theorem pure_map (f : α → β) (x : pure.toMonad m α) (eff : EffectKind) :\n    (Pure.pure (f <$> x : pure.toMonad m _) : eff.toMonad m _) = f <$> (Pure.pure x)"}], "used_local_defs": [{"name": "RegionSignature", "content": "abbrev RegionSignature Ty := List (Ctxt Ty × List Ty)"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "Signature.mk", "content": "abbrev Signature.mk (sig : List Ty) (regSig : RegionSignature Ty) (returnTypes : List Ty) : Signature Ty :=\n { sig, regSig, returnTypes }"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "DialectSignature.sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "DialectSignature.regSig", "content": "def regSig       := Signature.regSig ∘ s.signature"}, {"name": "DialectSignature.returnTypes", "content": "def returnTypes  := Signature.returnTypes ∘ s.signature"}, {"name": "DialectSignature.effectKind", "content": "def effectKind   := Signature.effectKind ∘ s.signature"}, {"name": "DialectDenote", "content": "class DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))"}, {"name": "Expr", "content": "inductive Expr : (Γ : Ctxt d.Ty) → (eff : EffectKind) → (ty : List d.Ty) → Type where\n  | mk {Γ} {ty} (op : d.Op)\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) <| DialectSignature.sig op)\n     \n    (regArgs : HVector (fun t : Ctxt d.Ty × List d.Ty => Com t.1 .impure t.2)\n      (DialectSignature.regSig op)) : Expr Γ eff ty"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "Regions", "content": "abbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "HVector", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "Expr", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Com.decidableEq", "content": "protected instance Com.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty]\n    {Γ : Ctxt d.Ty} {eff : EffectKind} {tys : List d.Ty} : DecidableEq (Com d Γ eff tys)\n  | .rets v₁, .rets v₂ => decidable_of_iff (v₁ = v₂) (by admit /- proof elided -/\n  )\n  | .var (ty := ty₁) e₁ body₁, .var (ty := ty₂) e₂ body₂ =>\n    if hα : ty₁ = ty₂\n    then by\n      subst hα\n      letI := Expr.decidableEq e₁ e₂\n      letI := Com.decidableEq body₁ body₂\n      exact decidable_of_iff (e₁ = e₂ ∧ body₁ = body₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )\n  | .rets _, .var _ _ => isFalse (fun h => Com.noConfusion h)\n  | .var _ _, .rets _ => isFalse (fun h => Com.noConfusion h)"}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Expr.op", "content": "def Expr.op {Γ : Ctxt d.Ty} {eff : EffectKind} {ty} (e : Expr d Γ eff ty) : d.Op :=\n  Expr.casesOn e (fun op _ _ _ _ => op)"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "Expr.regArgs", "content": "def Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r)"}, {"name": "HVector.denote", "content": "def HVector.denote :\n    {l : RegionSignature d.Ty} → (T : HVector (fun t => Com d t.1 .impure t.2) l) →\n    HVector (fun t => t.1.Valuation → EffectKind.impure.toMonad d.m (HVector toType t.2)) l\n  | _, .nil => HVector.nil\n  | _, .cons v vs => HVector.cons (v.denote) (HVector.denote vs)"}, {"name": "Expr.denote", "content": "def Expr.denote {ty} (e : Expr d Γ eff ty) (V : Valuation Γ) :\n    eff.toMonad d.m (e.outContext.Valuation) :=\n  match e with\n  | ⟨op, ty_eq, heff, args, regArgs⟩ => do\n      let argsDenote := args.map V\n      let val ← EffectKind.liftEffect heff <| DialectDenote.denote op argsDenote regArgs.denote\n      return (val ++ V).cast (by admit /- proof elided -/\n      )"}, {"name": "Com.denote", "content": "def Com.denote : Com d Γ eff ty → (Γv : Valuation Γ) →\n    eff.toMonad d.m (HVector toType ty)\n  | .rets vs, Γv     => pure (vs.map Γv)\n  | .var e body, V => e.denote V >>= body.denote"}, {"name": "Lets.denote", "content": "def Lets.denote [DialectSignature d] [DialectDenote d] {Γ₂}\n    (lets : Lets d Γ₁ eff Γ₂) (V : Valuation Γ₁) : (eff.toMonad d.m <| Valuation Γ₂) :=\n  match lets with\n  | .nil          => return V\n  | .var lets' e  => lets'.denote V >>= e.denote"}, {"name": "Expr.denoteOp", "content": "def Expr.denoteOp (e : Expr d Γ eff ty) (V : Γ.Valuation) :\n    eff.toMonad d.m (HVector toType ty) :=\n  EffectKind.liftEffect e.eff_le <| cast (by admit /- proof elided -/\n  ) <|\n    DialectDenote.denote e.op (e.args.map V) e.regArgs.denote"}, {"name": "FlatCom", "content": "structure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts"}, {"name": "FlatCom.denote", "content": "@[simp] abbrev FlatCom.denote [DialectDenote d]\n    (flatCom : FlatCom d Γ eff Γ_out ts)\n    (V : Γ.Valuation) : eff.toMonad d.m (HVector toType ts) :=\n  flatCom.lets.denote V >>= (return flatCom.rets.map ·)"}, {"name": "Expr.changeEffect", "content": "def Expr.changeEffect {eff₁ eff₂ : EffectKind} (h : eff₁ ≤ eff₂) :\n    Expr d Γ eff₁ t → Expr d Γ eff₂ t\n  | Expr.mk op ty_eq eff_le args regArgs =>\n    have heff : DialectSignature.effectKind op ≤ eff₂ := by admit /- proof elided -/"}, {"name": "Expr.castPureToEff", "content": "def Expr.castPureToEff (eff : EffectKind) : Expr d Γ .pure t → Expr d Γ eff t :=\n  changeEffect (EffectKind.pure_le eff)"}], "used_local_lemmas": [{"name": "Expr.op_mk", "content": "@[simp]\ntheorem Expr.op_mk {Γ : Ctxt d.Ty} {ty} {eff : EffectKind} (op : d.Op)\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) (DialectSignature.sig op))\n    (regArgs) :\n    (Expr.mk op ty_eq eff_le args regArgs).op = op"}, {"name": "Expr.args_mk", "content": "@[simp]\ntheorem Expr.args_mk {Γ : Ctxt d.Ty} {ty eff op}\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) (DialectSignature.sig op)) (regArgs) :\n    (Expr.mk op ty_eq eff_le args regArgs).args = args"}, {"name": "Expr.regArgs_mk", "content": "@[simp]\ntheorem Expr.regArgs_mk {Γ : Ctxt d.Ty} {ty eff op}\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) (DialectSignature.sig op)) (regArgs) :\n    (Expr.mk op ty_eq eff_le args regArgs).regArgs = regArgs"}, {"name": "Expr.denote_unfold", "content": "theorem Expr.denote_unfold (e : Expr d Γ eff ty) :\n    e.denote = fun V => (· ++ V) <$> (e.denoteOp V)"}], "local_ctx": "import LeanMLIR.ErasedContext\n\nimport LeanMLIR.HVector\n\nimport LeanMLIR.EffectKind\n\nimport LeanMLIR.Framework.Dialect\n\nimport Mathlib.Data.Finset.Union\n\nopen Ctxt (Var VarSet Valuation Hom)\n\nopen TyDenote (toType)\n\nabbrev RegionSignature Ty := List (Ctxt Ty × List Ty)\n\nstructure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure\n\nabbrev Signature.mk (sig : List Ty) (regSig : RegionSignature Ty) (returnTypes : List Ty) : Signature Ty :=\n { sig, regSig, returnTypes }\n\nclass DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty\n\nnamespace DialectSignature\n\nvariable {d} [s : DialectSignature d]\n\ndef sig          := Signature.sig ∘ s.signature\n\ndef regSig       := Signature.regSig ∘ s.signature\n\ndef returnTypes  := Signature.returnTypes ∘ s.signature\n\ndef effectKind   := Signature.effectKind ∘ s.signature\n\nend DialectSignature\n\nclass DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))\n\nsection DataStructures\n\nvariable (d : Dialect) [DialectSignature d]\n\ninductive Expr : (Γ : Ctxt d.Ty) → (eff : EffectKind) → (ty : List d.Ty) → Type where\n  | mk {Γ} {ty} (op : d.Op)\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) <| DialectSignature.sig op)\n     \n    (regArgs : HVector (fun t : Ctxt d.Ty × List d.Ty => Com t.1 .impure t.2)\n      (DialectSignature.regSig op)) : Expr Γ eff ty\n\ninductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β\n\nend\n\nabbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ\n\nabbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig\n\ninductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext\n\nvariable {d} [DialectSignature d]\n\nprotected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )\n\nprotected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )\n\nprotected instance Com.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty]\n    {Γ : Ctxt d.Ty} {eff : EffectKind} {tys : List d.Ty} : DecidableEq (Com d Γ eff tys)\n  | .rets v₁, .rets v₂ => decidable_of_iff (v₁ = v₂) (by admit /- proof elided -/\n  )\n  | .var (ty := ty₁) e₁ body₁, .var (ty := ty₂) e₂ body₂ =>\n    if hα : ty₁ = ty₂\n    then by\n      subst hα\n      letI := Expr.decidableEq e₁ e₂\n      letI := Com.decidableEq body₁ body₂\n      exact decidable_of_iff (e₁ = e₂ ∧ body₁ = body₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )\n  | .rets _, .var _ _ => isFalse (fun h => Com.noConfusion h)\n  | .var _ _, .rets _ => isFalse (fun h => Com.noConfusion h)\n\nend -- decEq\n\nend DataStructures\n\nvariable {d : Dialect} [DialectSignature d]\n\nsection Rec\n\nvariable {eff t} {motive : ∀ {Γ}, Com d Γ eff t → Sort u}\n          (rets : ∀ {Γ : Ctxt _} , (v : HVector Γ.Var t) → motive (Com.rets v))\n          (var : ∀ {Γ} {u},\n            (e : Expr d Γ eff u) → (body : Com d e.outContext eff t) →\n              motive body → motive (Com.var e body))\n\ndef Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl\n\nvariable {rets} {var} {Γ : Ctxt _}\n\nend Rec\n\ndef Expr.op {Γ : Ctxt d.Ty} {eff : EffectKind} {ty} (e : Expr d Γ eff ty) : d.Op :=\n  Expr.casesOn e (fun op _ _ _ _ => op)\n\ndef Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)\n\ndef Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)\n\nsection Lemmas\n\nnamespace Com\n\nend Com\n\nend Lemmas\n\ndef Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) \n\nsection Lemmas\n\nend Lemmas\n\nvariable [TyDenote d.Ty] [DialectDenote d] [DecidableEq d.Ty] [Monad d.m] [LawfulMonad d.m]\n\ndef HVector.denote :\n    {l : RegionSignature d.Ty} → (T : HVector (fun t => Com d t.1 .impure t.2) l) →\n    HVector (fun t => t.1.Valuation → EffectKind.impure.toMonad d.m (HVector toType t.2)) l\n  | _, .nil => HVector.nil\n  | _, .cons v vs => HVector.cons (v.denote) (HVector.denote vs)\n\ndef Expr.denote {ty} (e : Expr d Γ eff ty) (V : Valuation Γ) :\n    eff.toMonad d.m (e.outContext.Valuation) :=\n  match e with\n  | ⟨op, ty_eq, heff, args, regArgs⟩ => do\n      let argsDenote := args.map V\n      let val ← EffectKind.liftEffect heff <| DialectDenote.denote op argsDenote regArgs.denote\n      return (val ++ V).cast (by admit /- proof elided -/\n      )\n\ndef Com.denote : Com d Γ eff ty → (Γv : Valuation Γ) →\n    eff.toMonad d.m (HVector toType ty)\n  | .rets vs, Γv     => pure (vs.map Γv)\n  | .var e body, V => e.denote V >>= body.denote\n\nend\n\ndef Lets.denote [DialectSignature d] [DialectDenote d] {Γ₂}\n    (lets : Lets d Γ₁ eff Γ₂) (V : Valuation Γ₁) : (eff.toMonad d.m <| Valuation Γ₂) :=\n  match lets with\n  | .nil          => return V\n  | .var lets' e  => lets'.denote V >>= e.denote\n\nsection Unfoldings\n\nopen EffectKind (liftEffect)\n\ndef Expr.denoteOp (e : Expr d Γ eff ty) (V : Γ.Valuation) :\n    eff.toMonad d.m (HVector toType ty) :=\n  EffectKind.liftEffect e.eff_le <| cast (by admit /- proof elided -/\n  ) <|\n    DialectDenote.denote e.op (e.args.map V) e.regArgs.denote\n\nend Unfoldings\n\nsection Lemmas\n\nend Lemmas\n\nsection Lemmas\n\nvariable {Γ Γ' : Ctxt d.Ty} {t} (f : Γ.Hom Γ') (e : Expr d Γ eff t) (V : Γ'.Valuation)\n\nend Lemmas\n\nstructure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts\n\n@[simp] abbrev FlatCom.denote [DialectDenote d]\n    (flatCom : FlatCom d Γ eff Γ_out ts)\n    (V : Γ.Valuation) : eff.toMonad d.m (HVector toType ts) :=\n  flatCom.lets.denote V >>= (return flatCom.rets.map ·)\n\ndef Expr.changeEffect {eff₁ eff₂ : EffectKind} (h : eff₁ ≤ eff₂) :\n    Expr d Γ eff₁ t → Expr d Γ eff₂ t\n  | Expr.mk op ty_eq eff_le args regArgs =>\n    have heff : DialectSignature.effectKind op ≤ eff₂ := by admit /- proof elided -/\n\ndef Expr.castPureToEff (eff : EffectKind) : Expr d Γ .pure t → Expr d Γ eff t :=\n  changeEffect (EffectKind.pure_le eff)\n\nsection Lemmas", "target_theorem": "@[simp] theorem Expr.denote_castPureToEff {e : Expr d Γ .pure t} :\n    denote (e.castPureToEff eff) = fun V => pure (e.denote V) :=", "ground_truth_proof": ":= by\n  rcases e with ⟨op, rfl, eff_le, _, _⟩\n  cases eff\n  case pure => rfl\n  case impure =>\n    funext V\n    simp only [castPureToEff, changeEffect, denote_unfold, denoteOp, op_mk, args_mk, regArgs_mk,\n      EffectKind.pure_map, EffectKind.pure_liftEffect]", "nesting_depth": 6, "transitive_dep_count": 69, "subset_aristotle": false, "category": "Compiler"}
{"id": 340, "thm_name": "CIRCTStream.Stream.removeNone_equiv", "thm_stmt": "theorem removeNone_equiv (x : Stream α) :\n    x.removeNone ~ x", "lean_root": "lean-mlir", "rel_path": "SSA/Projects/CIRCT/Stream/WeakBisim.lean", "imports": ["import SSA.Projects.CIRCT.Stream.Stream", "import Mathlib.Data.Stream.Init", "import Mathlib.Logic.Function.Iterate"], "used_lib_defs": [{"name": "Option", "module": "Init.Prelude"}, {"name": "Stream'", "module": "Mathlib.Data.Stream.Defs"}, {"name": "Stream'.const", "module": "Mathlib.Data.Stream.Defs"}, {"name": "Stream'.head", "module": "Mathlib.Data.Stream.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.strongRecOn", "module": "Init.WF"}, {"name": "Stream'.drop", "module": "Mathlib.Data.Stream.Defs"}, {"name": "Stream'.get", "module": "Mathlib.Data.Stream.Defs"}, {"name": "Stream'.tail", "module": "Mathlib.Data.Stream.Defs"}, {"name": "Stream'.corec", "module": "Mathlib.Data.Stream.Defs"}], "used_repo_defs": [{"name": "syntax \"neg\" : MLIR.Pretty.uniform_op", "content": "syntax \"neg\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "Stream", "content": "def Stream (β : Type) := Stream' (Option β)"}, {"name": "head", "content": "def head : Stream α → Option α := Stream'.head"}, {"name": "tail", "content": "def tail : Stream α → Stream α := Stream'.tail"}, {"name": "corec", "content": "def corec {α} {β} (s0 : β) (f : β → (Option α × β)) : Stream α :=\n  Stream'.corec (f · |>.fst) (f · |>.snd) s0"}], "lib_lemmas": [{"name": "Nat.not_lt_zero", "module": "Init.Prelude"}, {"name": "Nat.zero_add", "module": "Init.Data.Nat.Basic"}, {"name": "false_implies", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "CIRCTStream.Stream.stuck", "content": "def stuck (α : Type) : Stream α := Stream'.const none"}, {"name": "CIRCTStream.Stream.nonesUntilSome", "content": "noncomputable def nonesUntilSome (x : Stream α) (not_stuck : x ≠ stuck α) : Nat :=\n  prop.choose\nwhere\n  prop : ∃ i, (x.drop i).head.isSome ∧ ∀ j < i, x.get j = none := by admit /- proof elided -/"}, {"name": "CIRCTStream.Stream.dropLeadingNones", "content": "noncomputable def dropLeadingNones (x : Stream α) (not_stuck : x ≠ stuck α) : Stream α:=\n  x.drop (nonesUntilSome x not_stuck)"}, {"name": "CIRCTStream.Stream.removeNone", "content": "noncomputable def removeNone (x : Stream α) : Stream α :=\n  Stream.corec x fun x =>\n    if h : x ≠ stuck α then\n      let x := x.dropLeadingNones h\n      (x.head, x.tail)\n    else\n      (none, x)"}], "used_local_lemmas": [{"name": "CIRCTStream.Stream.nonesUntilSome_spec", "content": "theorem nonesUntilSome_spec (x : Stream α) (not_stuck : x ≠ stuck α) :\n    (dropLeadingNones x not_stuck).head.isSome\n    ∧ ∀ j < nonesUntilSome x not_stuck, x.get j = none"}, {"name": "CIRCTStream.Stream.head_removeNone", "content": "@[simp] theorem head_removeNone (x : Stream α) :\n    x.removeNone.head =\n      if h : x ≠ stuck α then\n        (x.dropLeadingNones h).head\n      else\n        none"}], "local_ctx": "import SSA.Projects.CIRCT.Stream.Stream\n\nimport Mathlib.Logic.Function.Iterate\n\nimport Mathlib.Data.Stream.Init\n\nnamespace CIRCTStream\n\nnamespace Stream\n\nnamespace Bisim\n\nscoped infix:50 \" ~ \" => Bisim\n\nend Bisim\n\nopen Bisim\n\ndef stuck (α : Type) : Stream α := Stream'.const none\n\nnoncomputable def nonesUntilSome (x : Stream α) (not_stuck : x ≠ stuck α) : Nat :=\n  prop.choose\nwhere\n  prop : ∃ i, (x.drop i).head.isSome ∧ ∀ j < i, x.get j = none := by admit /- proof elided -/\n\nnoncomputable def dropLeadingNones (x : Stream α) (not_stuck : x ≠ stuck α) : Stream α:=\n  x.drop (nonesUntilSome x not_stuck)\n\nopen Classical in\n\nnoncomputable def removeNone (x : Stream α) : Stream α :=\n  Stream.corec x fun x =>\n    if h : x ≠ stuck α then\n      let x := x.dropLeadingNones h\n      (x.head, x.tail)\n    else\n      (none, x)\n\nopen Classical in\n\nopen Classical in\n\nopen Classical in", "target_theorem": "theorem removeNone_equiv (x : Stream α) :\n    x.removeNone ~ x :=", "ground_truth_proof": ":= by\n  apply Bisim.coinduct (· = ·.removeNone)\n  · rintro _ x rfl\n    · use 0\n      simp only [Nat.zero_add, Nat.not_lt_zero, false_implies, implies_true, true_and]\n      by_cases x_eq_stuck : x = stuck α\n      case pos =>\n        subst x_eq_stuck\n        refine ⟨0, ?_, ?_, by intros; contradiction⟩\n        · show tail _ = removeNone (stuck α).tail; simp\n        · show head _ = none; simp\n      case neg =>\n        have ⟨_, h2⟩ := nonesUntilSome_spec x x_eq_stuck\n        refine ⟨nonesUntilSome x x_eq_stuck, ?_, ?_, h2⟩\n        · show x.removeNone.tail = removeNone (Stream'.drop _ x)\n          have (w : Nat) : x.drop (w + 1) = tail (x.drop w) := by simp [tail]\n          rw [this]\n          simp [x_eq_stuck]\n          rfl\n        · show x.removeNone.head = _\n          simp only [head_removeNone, ne_eq, x_eq_stuck, dropLeadingNones]\n          simp [head]\n  · rfl", "nesting_depth": 4, "transitive_dep_count": 26, "subset_aristotle": false, "category": "Compiler"}
{"id": 341, "thm_name": "autOfTermUnop_bv_language", "thm_stmt": "lemma autOfTermUnop_bv_language op {t : Term} (m : CNFA (t.arity + 1)) :\n    m.bv_recognizes t.language →\n    (autOfTermUnop op m |>.bv_recognizes (op.subst_arity' ▸ (op.subst t).language))", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.Blase.AutoStructs.Basic", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.Blase.AutoStructs.Constructions", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.card", "module": "Mathlib.Data.FinEnum"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Array.foldl", "module": "Init.Data.Array.Basic"}, {"name": "Std.HashMap.emptyWithCapacity", "module": "Std.Data.HashMap.Basic"}, {"name": "Array.size", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Empty", "module": "Init.Prelude"}, {"name": "Empty.elim", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "cond", "module": "Init.Prelude"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "Array.emptyWithCapacity", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Array.empty", "module": "Init.Prelude"}, {"name": "FinEnum.toList", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.equiv", "module": "Mathlib.Data.FinEnum"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "NFA.stepSet", "module": "Mathlib.Computability.NFA"}, {"name": "Subsingleton", "module": "Init.Core"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "Language", "module": "Mathlib.Computability.Language"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "List.Vector.ofFn", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.replicate", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}, {"name": "SetRel", "module": "Mathlib.Data.Rel"}, {"name": "Array.back?", "module": "Init.Data.Array.Basic"}, {"name": "Array.isEmpty", "module": "Init.Data.Array.Basic"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}, {"name": "L", "module": "Archive.Hairer"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Fin.natAdd", "module": "Init.Data.Fin.Basic"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "DFA", "module": "Mathlib.Computability.DFA"}, {"name": "NFA.toDFA", "module": "Mathlib.Computability.NFA"}, {"name": "List.range", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "syntax \"max\" : MLIR.Pretty.uniform_op", "content": "syntax \"max\" : MLIR.Pretty.uniform_op\n\nsyntax \"slt\" : MLIR.Pretty.uniform_op\n\nsyntax \"xor\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "carry", "content": "def carry (initCarry : Bool) (x y : BitStream) : BitStream :=\n  fun n => (addAux' initCarry x y n).2"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "addAux'", "content": "def addAux' (carryIn : Bool) (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => carryIn\n    | i + 1 => (addAux' carryIn x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "worklistRun", "content": "def worklistRun (final : S → Bool) (inits : Array S)\n    (hinits : inits.toList.Nodup) (f : S → Array (BitVec n × S)) : CNFA n :=\n  ⟨worklistRun' _ S final inits hinits f, worklistRun'_wf (BitVec n) S⟩"}, {"name": "worklistRun'", "content": "def worklistRun' (final : S → Bool) (inits : Array S) (hinits : inits.toList.Nodup) (f : S → Array (A × S)) : RawCNFA A :=\n  let st0 := worklist.initState _ _ inits hinits final\n  go st0\nwhere go (st0 : worklist.St A S) : RawCNFA A :=\n  if hemp : st0.worklist.isEmpty then st0.m else\n  let sa? := st0.worklist.back?\n  match heq : sa? with\n  | some sa =>\n    let wl := st0.worklist.pop\n    let st1 := { st0 with worklist := wl,\n                          worklist_nodup := by admit /- proof elided -/"}, {"name": "worklist.St", "content": "structure worklist.St where\n  m : RawCNFA A\n  map : Std.HashMap S State := ∅\n  worklist : Array S := ∅\n  worklist_nodup : worklist.toList.Nodup\n  worklist_incl : ∀ sa ∈ worklist, sa ∈ map"}, {"name": "worklist.initState", "content": "def worklist.initState (inits : Array S) (hinits : inits.toList.Nodup) (final? : S → Bool) : worklist.St A S :=\n  let m := RawCNFA.empty (A := A)\n  let mapm := inits.foldl (init := (Std.HashMap.emptyWithCapacity, m)) fun (map, m) sa =>\n    let (s, m) := m.newState\n    let m := m.addInitial s\n    let m := if final? sa then m.addFinal s else m\n    (map.insert sa s, m)\n  let map := mapm.1\n  let m := mapm.2\n  let worklist_incl : ∀ sa ∈ inits, sa ∈ map :="}, {"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "State", "content": "abbrev State := Nat"}, {"name": "RawCNFA.empty", "content": "def RawCNFA.empty : RawCNFA A := {\n  stateMax := 0\n  initials := ∅\n  finals := ∅\n  trans := ∅\n}"}, {"name": "processOneElem", "content": "def processOneElem (final : S → Bool) (s : State) (st : worklist.St A S) : A × S → worklist.St A S :=\n  fun (a', sa') =>\n    let (s', st') := st.addOrCreateState _ _ (final sa') sa'\n    let m := st'.m.addTrans a' s s'\n    { st' with m }"}, {"name": "worklist.St.addOrCreateState", "content": "def worklist.St.addOrCreateState (st : worklist.St A S) (final? : Bool) (sa : S) : State × worklist.St A S :=\n  match heq : st.map[sa]? with\n  | some s => (s, st)\n  | none =>\n    let (s, m) := st.m.newState\n    let m := if final? then m.addFinal s else m\n    let map := st.map.insert sa s\n    let worklist := st.worklist.push sa\n    have worklist_nodup : worklist.toList.Nodup := by admit /- proof elided -/"}, {"name": "CNFA", "content": "structure CNFA (n : Nat) where\n  m : RawCNFA (BitVec n)\n  wf : m.WF"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "sub", "content": "def sub (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).1"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "ofTerm", "content": "abbrev ofTerm (t : Term) : FSM (Fin t.arity) := termEvalEqFSM t |>.toFSM"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "termEvalEqFSM", "content": "def termEvalEqFSM : ∀ (t : Term), FSMTermSolution t\n  | ofNat n =>\n    { toFSM := FSM.ofNat n,\n      good := by admit /- proof elided -/"}, {"name": "or", "content": "def or : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ||| Circuit.var true (inr false),\n    nextStateCirc := fun a => a.elim\n  }"}, {"name": "shiftLeft", "content": "def shiftLeft (n : Nat) : FSM Unit :=\n  match n with\n  | 0 => FSM.id\n  | n + 1 => composeUnaryAux (FSM.ls false) (shiftLeft n)"}, {"name": "id", "content": "def id : FSM Unit := {\n α := Empty,\n initCarry := Empty.elim,\n outputCirc := Circuit.var true (inr ()),\n nextStateCirc := Empty.elim\n}"}, {"name": "ls", "content": "def ls (b : Bool) : FSM Unit :=\n  { α := Unit,\n    initCarry := fun _ => b,\n    nextStateCirc := fun () => Circuit.var true (inr ()),\n    outputCirc := Circuit.var true (inl ())\n  }"}, {"name": "composeUnaryAux", "content": "def composeUnaryAux\n    (p : FSM Unit)\n    (q : FSM arity) :\n    FSM arity :=\n  p.compose\n    arity\n    _\n    (λ _ => id)\n    (λ _ => q)"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "and", "content": "def and : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := fun a => a.elim,\n    outputCirc := Circuit.var true (inr true) &&& Circuit.var true (inr false),\n  }"}, {"name": "xor", "content": "def xor : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ^^^ Circuit.var true (inr false),\n    nextStateCirc := Empty.elim\n  }"}, {"name": "neg", "content": "def neg : FSM Unit :=\n  { α := Unit,\n    i := by admit /- proof elided -/"}, {"name": "composeBinary", "content": "def composeBinary\n    (p : FSM Bool)\n    {t₁ t₂ : Term}\n    (q₁ : FSMTermSolution t₁)\n    (q₂ : FSMTermSolution t₂) :\n    FSM (Fin (max t₁.arity t₂.arity)) := composeBinaryAux p q₁.toFSM q₂.toFSM"}, {"name": "composeBinaryAux", "content": "def composeBinaryAux\n    (p : FSM Bool)\n    (q₁ : FSM (Fin a₁))\n    (q₂ : FSM (Fin a₂)) :\n    FSM (Fin (max a₁ a₂)) :=\n  p.compose (Fin (max a₁ a₂))\n    (λ b => Fin (cond b a₁ a₂))\n    (λ b i => Fin.castLE (by admit /- proof elided -/\n    ) i)\n    (λ b => match b with\n      | true => q₁\n      | false => q₂)"}, {"name": "FSMTermSolution", "content": "structure FSMTermSolution (t : Term) extends FSM (Fin t.arity) where\n  ( good : t.evalFin = toFSM.eval )"}, {"name": "Term.evalFin", "content": "@[simp] def Term.evalFin (t : Term) (vars : Fin (arity t) → BitStream) : BitStream :=\n  match t with\n  | var n => vars (Fin.last n)\n  | zero    => BitStream.zero\n  | one     => BitStream.one\n  | negOne  => BitStream.negOne\n  | ofNat n => BitStream.ofNat n\n  | and t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | or t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | xor t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | not t     => ~~~(t.evalFin vars)\n  | add t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | sub t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | neg t       => -(Term.evalFin t vars)\n \n \n  | shiftL t n  => BitStream.shiftLeft (Term.evalFin t vars) n"}, {"name": "Predicate.evalFin", "content": "@[simp] def Predicate.evalFin (p : Predicate) (vars : Fin (arity p) → BitStream) : BitStream :=\nmatch p with\n| .width .eq n => BitStream.falseIffEq n\n| .width .neq n => BitStream.falseIffNeq n\n| .width .lt n => BitStream.falseIffLt n\n| .width .le n => BitStream.falseIffLe n\n| .width .gt n => BitStream.falseIffGt n\n| .width .ge n => BitStream.falseIffGe n\n| .binary .eq t₁ t₂ =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalEq x₁ x₂\n| .binary .neq t₁ t₂  =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalNeq x₁ x₂\n| .land p q =>\n  \n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLand x₁ x₂\n| .lor p q =>\n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor x₁ x₂\n| .binary .slt p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalSlt x₁ x₂\n| .binary .sle p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalSlt x₁ x₂) (Predicate.evalEq x₁ x₂)\n| .binary .ult p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  (Predicate.evalUlt x₁ x₂)\n| .binary .ule p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalUlt x₁ x₂) (Predicate.evalEq x₁ x₂)"}, {"name": "Predicate.evalUlt", "content": "def Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true"}, {"name": "borrow", "content": "def borrow (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).2"}, {"name": "Predicate.evalLor", "content": "def Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)"}, {"name": "Predicate.evalSlt", "content": "def Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))"}, {"name": "Predicate.evalMsbEq", "content": "def Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false"}, {"name": "Predicate.evalLand", "content": "def Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)"}, {"name": "Predicate.evalNeq", "content": "def Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd"}, {"name": "nxor", "content": "def nxor (a b : BitStream) : BitStream := fun i => a i == b i"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "Predicate.evalEq", "content": "def Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "Predicate", "content": "inductive Predicate : Type where\n \n| width (wp : WidthPredicate) (n : Nat) : Predicate\n| binary (p : BinaryPredicate) (t₁ t₂ : Term)\n| land  (p q : Predicate) : Predicate\n| lor (p q : Predicate) : Predicate\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "falseIffNeq", "content": "abbrev falseIffNeq (n : Nat) : BitStream := fun i => decide (i == n)"}, {"name": "falseIffLt", "content": "abbrev falseIffLt (n : Nat) : BitStream := fun i => decide (i ≥ n)"}, {"name": "falseIffLe", "content": "abbrev falseIffLe (n : Nat) : BitStream := fun i => decide (i > n)"}, {"name": "falseIffGe", "content": "abbrev falseIffGe (n : Nat) : BitStream := fun i => decide (i < n)"}, {"name": "falseIffEq", "content": "abbrev falseIffEq (n : Nat) : BitStream := fun i => decide (i != n)"}, {"name": "falseIffGt", "content": "abbrev falseIffGt (n : Nat) : BitStream := fun i => decide (i ≤ n)"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "Term.arity", "content": "@[simp] def Term.arity : Term → Nat\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| ofNat _ => 0\n| Term.and t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.or t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.xor t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.not t => arity t\n| add t₁ t₂ => max (arity t₁) (arity t₂)\n| sub t₁ t₂ => max (arity t₁) (arity t₂)\n| neg t => arity t\n\n\n| shiftL t .. => arity t"}, {"name": "negOne", "content": "abbrev negOne : BitStream := fun _ => true"}, {"name": "shiftLeft", "content": "def shiftLeft (x : BitStream) (k : Nat) : BitStream :=\n  fun i => if i < k then false else x (i - k) "}, {"name": "ofNat", "content": "def ofNat (x : Nat) : BitStream :=\n  Nat.testBit x"}, {"name": "one", "content": "abbrev one    : BitStream := (· == 0)"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "ofNat", "content": "def ofNat (n : Nat)  : FSM (Fin 0) :=\n  match hn : n with\n  | 0 => FSM.zero\n\n  | n' + 1 =>\n    let bit := n.testBit 0\n    let m := n / 2\n    have h : m < n := by admit /- proof elided -/"}, {"name": "zero", "content": "def zero : FSM (Fin 0) :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := Empty.elim,\n    outputCirc := Circuit.fals\n  }"}, {"name": "composeUnary", "content": "def composeUnary\n    (p : FSM Unit)\n    {t : Term}\n    (q : FSMTermSolution t) :\n    FSM (Fin t.arity) := composeUnaryAux p q.toFSM"}, {"name": "one", "content": "def one : FSM (Fin 0) :=\n  { α := Unit,\n    i := by admit /- proof elided -/"}, {"name": "var", "content": "def var (n : ℕ) : FSM (Fin (n+1)) :=\n  { α := Empty,\n    i := by admit /- proof elided -/"}, {"name": "add", "content": "def add : FSM Bool :=\n  { α := Unit,\n    initCarry := λ _ => false,\n    nextStateCirc := fun () =>\n      Circuit.var true (inr true) &&& Circuit.var true (inr false) |||\n      Circuit.var true (inr true) &&& Circuit.var true (inl ()) |||\n      Circuit.var true (inr false) &&& Circuit.var true (inl ()),\n    outputCirc := Circuit.var true (inr true) ^^^\n                  Circuit.var true (inr false) ^^^\n                  Circuit.var true (inl ()),\n  }"}, {"name": "negOne", "content": "def negOne : FSM (Fin 0) :=\n  { α := Empty,\n    i := by admit /- proof elided -/"}, {"name": "sub", "content": "def sub : FSM Bool :=\n  { α := Unit,\n    initCarry := fun _ => false,\n    outputCirc := Circuit.var true (inr true) ^^^\n                  Circuit.var true (inr false) ^^^\n                  Circuit.var true (inl ()),\n    nextStateCirc := fun _ =>\n      (Circuit.var false (inr true) &&& Circuit.var true (inr false)) |||\n      (Circuit.var false (inr true) ^^^ Circuit.var true (inr false)) &&&\n      (Circuit.var true (inl ()))\n  }"}, {"name": "not", "content": "def not : FSM Unit :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := Empty.elim,\n    outputCirc := Circuit.var false (inr ())\n  }"}, {"name": "add", "content": "def add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1"}, {"name": "addAux", "content": "def addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "neg", "content": "def neg (x : BitStream) : BitStream :=\n  fun n => (negAux x n).1"}, {"name": "negAux", "content": "def negAux (x : BitStream) : Nat → Bool × Bool\n  | 0 => (x 0, !(x 0))\n  | n+1 =>\n    let borrow := (negAux x n).2\n    let a := x (n + 1)\n    (xor (!a) borrow, !a && borrow)"}, {"name": "CNFA.inter", "content": "def CNFA.inter (m1 m2 : CNFA n) : CNFA n := product (fun b1 b2 => b1 && b2) m1 m2"}, {"name": "product", "content": "def product (final? : Bool → Bool → Bool) (m₁ m₂ : CNFA n) : CNFA n :=\n  worklistRun (m₁.m.states × m₂.m.states) final (product.inits m₁ m₂)\n    (by admit /- proof elided -/\n    ) f\nwhere final (ss : m₁.m.states × m₂.m.states) := final? (ss.1 ∈ m₁.m.finals) (ss.2 ∈ m₂.m.finals)\n      f (ss : m₁.m.states × m₂.m.states) :=\n        let (s1, s2) := ss\n        (FinEnum.toList (α := BitVec n)).foldl (init := Array.empty) fun as a =>\n          product.prodArray' (λ s₁ s₂ ↦ (a, (s₁, s₂)))\n            (fun s' => m₁.wf.trans_tgt_lt (s := s1) (a := a)) (fun s' => m₂.wf.trans_tgt_lt (s := s2) (a := a)) as"}, {"name": "product.prodArray'", "content": "@[inline]\ndef product.prodArray' (a : Array γ) :=\n  m₁.attachWith _ hm₁ |>.fold (init := a) fun is s1 =>\n    m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s1 s2)"}, {"name": "product.inits_nodup", "content": "def product.inits_nodup : inits m₁ m₂ |>.toList.Nodup :="}, {"name": "product.inits", "content": "def product.inits (m₁ m₂ : CNFA n) :=\n  product.prodArray Prod.mk @m₁.wf.initials_lt @m₂.wf.initials_lt"}, {"name": "product.prodArray", "content": "@[inline]\ndef product.prodArray := prodArray' f hm₁ hm₂ (Array.emptyWithCapacity <| m₁.size * m₂.size)"}, {"name": "CNFA.inter_bv_language", "content": "def CNFA.inter_bv_language (m₁ m₂ : CNFA n) :\n    m₁.bv_recognizes L₁ →\n    m₂.bv_recognizes L₂ →\n    (m₁.inter m₂).bv_recognizes (L₁ ∩ L₂) :="}, {"name": "HashSet.inter", "content": "def HashSet.inter [BEq A] [Hashable A] (m1 m2 : Std.HashSet A) : Std.HashSet A :=\n  m1.fold (init := ∅) fun mi x => if m2.contains x then mi.insert x else mi"}, {"name": "NFA'", "content": "structure NFA' (n : Nat) where\n  σ : Type\n  M : NFA (BitVec n) σ"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "eval", "content": "def eval (x : arity → BitStream) : BitStream :=\n  fun n => (p.nextBit (p.carry x n) (fun i => x i n)).2"}, {"name": "nextBit", "content": "def nextBit : p.State → (arity → Bool) → p.State × Bool :=\n  fun carry inputBits =>\n    let input := Sum.elim carry inputBits\n    let newState : p.State  := fun (a : p.α) => (p.nextStateCirc a).eval input\n    let outBit : Bool       := (p.outputCirc).eval input\n    (newState, outBit)"}, {"name": "State", "content": "abbrev State : Type := p.α → Bool"}, {"name": "carry", "content": "def carry (x : arity → BitStream) : ℕ → p.State\n  | 0 => p.initCarry\n  | n+1 => (p.nextBit (carry x n) (fun i => x i n)).1"}, {"name": "carryBV", "content": "def carryBV (x : ar → BitVec w) : p.State :=\n  p.carry (fun ar => .ofBitVecSext (x ar)) w"}, {"name": "evalBV", "content": "def evalBV {w} (x : ar → BitVec w) : BitVec w :=\n  BitVec.ofFn fun k => p.eval (fun ar => .ofBitVecSext (x ar)) k"}, {"name": "ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}, {"name": "Term.language", "content": "def Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }"}, {"name": "Formula.arity", "content": "@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity"}, {"name": "Term.evalFinBV", "content": "@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n"}, {"name": "enc", "content": "def enc (bvs : BitVecs n) : BitVecs' n :=\n  (List.finRange bvs.w).map (fun i =>\n    BitVec.ofFn (fun (k : Fin n) => (bvs.bvs.get k)[i]))"}, {"name": "BitVecs'", "content": "abbrev BitVecs' (n : Nat) := List (BitVec n)"}, {"name": "dec", "content": "@[simps]\ndef dec (bvs' : BitVecs' n) : BitVecs n where\n  w := bvs'.length\n  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k"}, {"name": "accepts", "content": "def accepts (M : NFA' n) : Set (BitVecs n) := dec '' M.accepts'"}, {"name": "accepts'", "content": "def accepts' (M : NFA' n) : Set (BitVecs' n) := M.M.accepts"}, {"name": "worklistRun_spec", "content": "def worklistRun_spec : (worklistRun S final inits hinits f |>.Sim $ nfa' inits final f) :=\n  worklistRun'_spec inits final f"}, {"name": "nfa'", "content": "def nfa' : NFA' n :=\n  { σ := _, M := nfa inits final f }"}, {"name": "nfa", "content": "def nfa : NFA A S where\n  start := { sa | sa ∈ inits }\n  accept := { sa | final sa }\n  step sa a := { sa' | (a, sa') ∈ f sa }"}, {"name": "worklistRun'_spec", "content": "def worklistRun'_spec :\n    (worklistRun' A S final inits hinits f |>.Sim $ nfa inits final f) :="}, {"name": "StInv", "content": "structure StInv (m : RawCNFA A) (map : Std.HashMap S State) where\n  wf : m.WF\n  map_states : ∀ (sa : S) s, map[sa]? = some s → s ∈ m.states\n  map_surj : ∀ s : m.states, ∃ (sa : S), map[sa]? = some s.val\n  map_inj : ∀ {s} {sa sa' : S}, map[sa]? = some s → map[sa']? = some s → sa = sa'"}, {"name": "worklist.St.D", "content": "def worklist.St.D (st : worklist.St A S) : Set S := st.visited"}, {"name": "worklist.St.visited", "content": "def worklist.St.visited (st : worklist.St A S) : Set S := { s : S | s ∈ st.map ∧ s ∉ st.worklist }"}, {"name": "worklistGo_spec", "content": "def worklistGo_spec {st : worklist.St A S} (inv : StInv A S st.m st.map) :\n    st.sim inits final f ∅ →\n    (worklistRun'.go A S final f st |>.Sim $ nfa inits final f) :="}, {"name": "worklist.St.rel", "content": "def worklist.St.rel (st : worklist.St A S) : SetRel State S := {(s, sa) | st.map[sa]? = some s }"}, {"name": "processOneElem_mot", "content": "def processOneElem_mot (s : State) (sa : S) (n : ℕ) (st : worklist.St A S) : Prop :=\n  st.map[sa]? = some s ∧\n  sa ∈ st.visited ∧\n  StInv A S st.m st.map ∧\n  st.sim inits final f  {(sa1, a, sa') | sa1 = sa ∧ ∃ k ≥ n, (f sa)[k]? = some (a, sa') }"}, {"name": "worklist.St.sim", "content": "abbrev worklist.St.sim {st : worklist.St A S} (T : Set (S × A × S)) :=\n  st.m.Simul (nfa inits final f) st.rel st.D T"}, {"name": "RawCNFA.Sim", "content": "def RawCNFA.Sim (m : RawCNFA A) (A : NFA A S) := ∃ R, RawCNFA.Simul m A R ⊤ ∅"}, {"name": "RawCNFA.Simul", "content": "structure RawCNFA.Simul (m : RawCNFA A) (M : NFA A Q) (R : SetRel State Q) (D : Set Q) (T : Set (Q × A × Q)) where\n  accept {s q} : s ~[R] q → (s ∈ m.finals ↔ q ∈ M.accept)\n  initial₁ {s} : s ∈ m.initials → ∃ q ∈ M.start, s ~[R] q\n  initial₂ {q} : q ∈ M.start → ∃ s ∈ m.initials, s ~[R] q\n  trans_match₁ {s s' a q} : s ~[R] q → s' ∈ m.tr s a → ∃ q', q' ∈ M.step q a ∧ s' ~[R] q'\n  trans_match₂ {s a q q'} : s ~[R] q → q' ∈ M.step q a → q ∈ D → (q, a, q') ∉ T → ∃ s', s' ∈ m.tr s a ∧ s' ~[R] q'"}, {"name": "RawCNFA.SimulFun", "content": "structure RawCNFA.SimulFun (m : RawCNFA A) (M : NFA A Q) (f : m.states ≃ Q)  where\n  accept {q} : ((f.invFun q).val ∈ m.finals ↔ q ∈ M.accept)\n  initial {q} : q ∈ M.start ↔ (f.invFun q).val ∈ m.initials\n  trans_match {a q q'} : q' ∈ M.step q a ↔ (f.invFun q').val ∈ m.tr (f.invFun q) a"}, {"name": "RawCNFA.tr", "content": "@[inline]\ndef RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}, {"name": "CNFA.Sim", "content": "def CNFA.Sim (m : CNFA n) (M : NFA' n) :=\n  m.m.Sim M.M"}, {"name": "CNFA.bv_recognizes", "content": "def CNFA.bv_recognizes (m : CNFA n) (L : Set (BitVecs n)) :=\n  ∃ L', m.recognizes L' ∧ L = dec '' L'"}, {"name": "RawCNFA.recognizes", "content": "def RawCNFA.recognizes (m : RawCNFA A) (L : Language A) :=\n  ∃ (σ : Type) (M : NFA A σ), m.Sim M ∧ M.accepts = L"}, {"name": "CNFA.recognizes", "content": "def CNFA.recognizes (m : CNFA n) (L : Language (BitVec n)) :=\n  ∃ (M : NFA' n), m.Sim M ∧ M.M.accepts = L"}, {"name": "BitVecs.cast", "content": "def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=\n  { w := bvs.w, bvs := h ▸ bvs.bvs }"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "BitVec.transport", "content": "def BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=\n  BitVec.ofFn fun i => bv.getLsbD (f i)"}, {"name": "List.Vector.transport", "content": "def List.Vector.transport (v : Vector α m) (f : Fin n → Fin m) : Vector α n :=\n  Vector.ofFn fun i => v.get (f i)"}, {"name": "BitVecs'.transport", "content": "def BitVecs'.transport (f : Fin n → Fin m) (bvs' : BitVecs' m): BitVecs' n :=\n  bvs'.map fun bv => bv.transport f"}, {"name": "BitVecs.transport", "content": "def BitVecs.transport (f : Fin n → Fin m) (bvs : BitVecs m) : BitVecs n :=\n  { w := bvs.w, bvs := bvs.bvs.transport f }"}, {"name": "CNFA.minimize", "content": "def CNFA.minimize (m : CNFA n) : CNFA n :=\n  let mᵣ := m.reverse.determinize\n  mᵣ.reverse.determinize"}, {"name": "CNFA.determinize", "content": "def CNFA.determinize (m : CNFA n) : CNFA n :=\n  worklistRun (BitVec m.m.stateMax)\n    (fun ss => ss.any fun n b => b == true && n ∈ m.m.finals)\n    (determinize.inits m)\n    (by admit /- proof elided -/\n    )\n    f\nwhere\n  f := fun (ss : BitVec m.m.stateMax) =>\n        (FinEnum.toList (BitVec n)).foldl (init := Array.empty) fun ts a =>\n          let ss' := m.m.transSetBV ss a\n          ts.push (a, ss')"}, {"name": "CNFA.determinize.inits", "content": "def CNFA.determinize.inits (m : CNFA n) : Array (BitVec m.m.stateMax) :=\n  #[BitVec.ofFn (fun n => n ∈ m.m.initials)]"}, {"name": "CNFA.reverse", "content": "def CNFA.reverse (m : CNFA n) : CNFA n :=\n  ⟨m.m.reverse, RawCNFA.reverse_spec m.wf |>.1⟩"}, {"name": "RawCNFA.reverse", "content": "def RawCNFA.reverse (m : RawCNFA A) : RawCNFA A :=\n  let m' := { stateMax := m.stateMax, trans := Std.HashMap.emptyWithCapacity m.trans.size, initials := m.finals, finals := m.initials}\n  m.trans.fold (init := m') processState\nwhere\n  processState := fun m' (s, a) ss' =>\n    ss'.fold (init := m') fun m' s' => m'.addTrans a s' s"}, {"name": "CNFA.toNFA'", "content": "def CNFA.toNFA' (m : CNFA n) : NFA' n := ⟨_, m.toNFA⟩"}, {"name": "CNFA.toNFA", "content": "def CNFA.toNFA (m : CNFA n) : NFA (BitVec n) m.m.states where\n  start := { s | s.val ∈ m.m.initials }\n  accept := { s | s.val ∈ m.m.finals }\n  step s₁ a := { s₂ | s₂.val ∈ m.m.tr s₁.val a }"}, {"name": "RawCNFA.states", "content": "def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax"}, {"name": "reverse", "content": "def reverse (M : NFA' n) : NFA' n where\n  σ := _\n  M := M.M.reverse"}, {"name": "CNFA.determinize_spec", "content": "def CNFA.determinize_spec (m : CNFA n)\n  {M : NFA' n} (hsim : m.Sim M) :\n    m.determinize.Sim M.determinize :="}, {"name": "bv_to_set", "content": "private def bv_to_set (bv : BitVec w) : Set State :=\n  { s | bv.getLsbD s }"}, {"name": "_root_.SetRel.set_eq", "content": "structure _root_.SetRel.set_eq (R : SetRel α β) (A : Set α) (B : Set β) where\n  fwd : a ∈ A → ∃ b ∈ B, a ~[R] b\n  bwd : b ∈ B → ∃ a ∈ A, a ~[R] b"}, {"name": "RawCNFA.lift", "content": "@[inline]\ndef RawCNFA.lift (m₁: RawCNFA (BitVec n1)) (f : Fin n1 → Fin n2) : RawCNFA (BitVec n2) :=\n  let trans := (List.range m₁.stateMax).foldl (init := ∅) fun m2 s => processState m2 s\n  { m₁ with trans }\nwhere"}, {"name": "CNFA.lift", "content": "@[inline]\ndef CNFA.lift (m: CNFA n1) (f : Fin n1 → Fin n2) : CNFA n2 :=\n  ⟨m.m.lift f, m.m.lift_wf m.wf⟩"}, {"name": "RawCNFA.proj", "content": "@[inline]\ndef RawCNFA.proj (m1: RawCNFA (BitVec n1)) (f : Fin n2 → Fin n1) : RawCNFA (BitVec n2) :=\n  let trans := m1.trans.keysArray.foldl (init := Std.HashMap.emptyWithCapacity) process\n  { m1 with trans }\nwhere"}, {"name": "CNFA.proj_spec", "content": "def CNFA.proj_spec (m : CNFA n2) (f : Fin n1 → Fin n2) {M : NFA' n2} :\n    m.Sim M → (m.proj f |>.Sim (M.proj f)) :="}, {"name": "CNFA.proj", "content": "@[inline]\ndef CNFA.proj (m: CNFA n2) (f : Fin n1 → Fin n2) : CNFA n1 :=\n  ⟨m.m.proj f, m.m.proj_wf m.wf⟩"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "List.nodup_singleton", "module": "Mathlib.Data.List.Nodup"}, {"name": "NFA.eval_append_singleton", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.eval_nil", "module": "Mathlib.Computability.NFA"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.ext_iff", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "eq_iff_iff", "module": "Init.Core"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "ite_cond_eq_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}, {"name": "bisim_comp", "content": "lemma bisim_comp (m : RawCNFA A) :\n    m.Sim M₁ → M₁.Bisim M₂ → m.Sim M₂"}, {"name": "bisimul_comp", "content": "lemma bisimul_comp {m : RawCNFA A} :\n    m.Simul M₁ R₁ ⊤ ∅ → M₁.Bisimul R₂ M₂ →\n    m.Simul M₂ (R₁.comp R₂) ⊤ ∅"}, {"name": "CNFA.bv_recognizes_equiv", "content": "lemma CNFA.bv_recognizes_equiv {m : CNFA n} :\n    m.bv_recognizes L ↔ ∃ (M : NFA' n), m.Sim M ∧ M.accepts = L"}, {"name": "List.Vector.append_get_ge", "content": "@[simp]\nlemma List.Vector.append_get_ge {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: n ≤ i) :\n    (x ++ y).get i = y.get ((i.cast (Nat.add_comm n m) |>.subNat n hlt))"}, {"name": "List.Vector.append_get_lt", "content": "@[simp]\nlemma List.Vector.append_get_lt {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: i < n) :\n    (x ++ y).get i = x.get (i.castLT hlt)"}, {"name": "BitVecs.transport_getElem", "content": "@[simp]\nlemma BitVecs.transport_getElem {bvs : BitVecs m} (f : Fin n → Fin m) (i : Fin n) :\n    (bvs.transport f).bvs.get i = bvs.bvs.get (f i)"}, {"name": "CNFA.minimize_bv_language", "content": "lemma CNFA.minimize_bv_language {m : CNFA n} :\n    m.bv_recognizes L → m.minimize.bv_recognizes L"}, {"name": "CNFA.minimize_language", "content": "lemma CNFA.minimize_language {m : CNFA n} :\n    m.recognizes L → m.minimize.recognizes L"}, {"name": "CNFA.reverse_language", "content": "lemma CNFA.reverse_language {m : CNFA n} (hl : m.recognizes L) : m.reverse.recognizes L.reverse"}, {"name": "CNFA.reverse_spec", "content": "lemma CNFA.reverse_spec {m : CNFA n} : m.reverse.Sim m.toNFA'.reverse"}, {"name": "RawCNFA.reverse_spec", "content": "lemma RawCNFA.reverse_spec {m : RawCNFA A} (hwf : m.WF) :\n    let m'"}, {"name": "RawCNFA.reverse_spec_procesState", "content": "lemma RawCNFA.reverse_spec_procesState {m : RawCNFA A} (hwf : m.WF) s₀ a₀ ss' (hs₀ : s₀ ∈ m.states) :\n    let motive m' ss'"}, {"name": "CNFA.determinize_language", "content": "lemma CNFA.determinize_language {m : CNFA n} :\n    m.recognizes L → m.determinize.recognizes L"}, {"name": "CNFA.lift_bv_language", "content": "@[simp]\nlemma CNFA.lift_bv_language {m : CNFA n1} {f : Fin n1 → Fin n2} :\n    m.bv_recognizes L → (m.lift f |>.bv_recognizes (BitVecs.transport f ⁻¹' L))"}, {"name": "CNFA.lift_spec", "content": "lemma CNFA.lift_spec (m : CNFA n1) (f : Fin n1 → Fin n2) {M : NFA' n1} :\n    m.Sim M → (m.lift f |>.Sim (M.lift f))"}, {"name": "CNFA.proj_bv_language", "content": "lemma CNFA.proj_bv_language {m : CNFA n2} {f : Fin n1 → Fin n2} :\n    m.bv_recognizes L → (m.proj f |>.bv_recognizes (BitVecs.transport f '' L))"}], "used_local_defs": [{"name": "NFA.sa", "content": "def NFA.sa (_ : NFA α σ) := σ → Language α"}, {"name": "NFA.correct", "content": "structure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q"}, {"name": "BVNRel", "content": "abbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop"}, {"name": "NFA'.sa", "content": "def NFA'.sa (M : NFA' n) := M.σ → BVNRel n"}, {"name": "langRel", "content": "def langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }"}, {"name": "NFA'.correct", "content": "structure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))"}, {"name": "NFA'.correct2", "content": "structure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)"}, {"name": "Alphabet", "content": "abbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)"}, {"name": "finFunToBitVec", "content": "def finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)"}, {"name": "bitVecToFinFun", "content": "def bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]"}, {"name": "NFA.ofFSM", "content": "def NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }"}, {"name": "inFSMRel", "content": "@[simp]\nabbrev inFSMRel (p : FSM arity) {w} (bvn : List.Vector (BitVec w) _) :=\n  bvn.get (Fin.last (FinEnum.card arity)) = p.evalBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))"}, {"name": "NFA'.ofFSM_sa", "content": "def NFA'.ofFSM_sa (p : FSM arity) : (NFA'.ofFSM' p).sa := fun q _ bvn =>\n    inFSMRel p bvn ∧ q = p.carryBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))"}, {"name": "NFA'.ofFSM_correct", "content": "def NFA'.ofFSM_correct (p : FSM arity) :\n    (NFA'.ofFSM' p).correct (ofFSM_sa p) (fun _ bvn => inFSMRel p bvn) :="}, {"name": "CNFA.ofFSM", "content": "def CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where"}, {"name": "NFA.msbState", "content": "inductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype"}, {"name": "liftUnop", "content": "def liftUnop n : Fin (n + 1) → Fin (n + 2) :=\n  fun k =>\n    if k = n then Fin.last (n+1) else k.castLE (by admit /- proof elided -/\n    )"}, {"name": "TermBinop", "content": "inductive TermBinop where\n| and | or | xor | add | sub"}, {"name": "TermBinop.subst", "content": "def TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂"}, {"name": "TermBinop.openTerm", "content": "def TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)"}, {"name": "TermBinop.termGadget", "content": "def TermBinop.termGadget (t : TermBinop) : CNFA 3 :=\n  match t with\n  | .and => FSM.ofTerm (.and (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .or => FSM.ofTerm (.or (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .xor => FSM.ofTerm (.xor (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .add => FSM.ofTerm (.add (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .sub => FSM.ofTerm (.sub (.var 0) (.var 1)) |> CNFA.ofFSM"}, {"name": "TermUnop", "content": "inductive TermUnop where\n| neg | not | shiftL (k : Nat)"}, {"name": "TermUnop.openTerm", "content": "def TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k"}, {"name": "TermUnop.openTerm_arity'", "content": "@[simp]\ndef TermUnop.openTerm_arity' (op : TermUnop) : op.openTerm.arity + 1 = 2 :="}, {"name": "TermUnop.subst", "content": "def TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k"}, {"name": "TermUnop.termGadget", "content": "def TermUnop.termGadget (t : TermUnop) : CNFA 2 :=\n  match t with\n  | .neg => FSM.ofTerm (.neg (.var 0)) |> CNFA.ofFSM\n  | .not => FSM.ofTerm (.not (.var 0)) |> CNFA.ofFSM\n  | .shiftL k => FSM.ofTerm (.shiftL (.var 0) k) |> CNFA.ofFSM"}, {"name": "autOfTermUnop", "content": "def autOfTermUnop (op : TermUnop) (m : CNFA (n + 1)) : CNFA (n + 1) :=\n  let mop : CNFA 2 := op.termGadget\n  let mop : CNFA (n + 2) := mop.lift (λ i ↦ i.natAdd n)\n  let m : CNFA (n + 2) := m.lift (λ i ↦ i.castLE (by admit /- proof elided -/\n  ))\n  let m := CNFA.inter m mop\n  let mfinal := m.proj (liftUnop n)\n  mfinal.minimize"}, {"name": "swapLastTwo", "content": "def swapLastTwo (x : Fin (n + 2)) : Fin (n + 2) :=\n  if x = Fin.last (n + 1) then n else if x = n then Fin.last (n + 1) else x"}], "used_local_lemmas": [{"name": "NFA.correct_spec", "content": "lemma NFA.correct_spec {M : NFA α σ} {ζ : M.sa} {L : Language α} :\n    M.correct ζ L → M.accepts = L"}, {"name": "in_enc", "content": "@[simp]\nlemma in_enc : x ∈ enc '' S ↔ dec x ∈ S"}, {"name": "dec_snoc_in_langRel", "content": "@[simp]\nlemma dec_snoc_in_langRel {n} {R : BVNRel n} {w : BitVecs' n} {a : BitVec n} :\n    dec (w ++ [a]) ∈ langRel R ↔\n      R (List.Vector.ofFn fun k => .cons (a.getLsbD k) ((dec w).bvs.get k))"}, {"name": "NFA'.correct_spec", "content": "lemma NFA'.correct_spec {M : NFA' n} {ζ : M.sa} {L : BVNRel n} :\n    M.correct ζ L → M.accepts = langRel L"}, {"name": "NFA'.ofFSM_spec", "content": "@[simp]\nlemma NFA'.ofFSM_spec (t : Term) :\n    (ofFSM (FSM.ofTerm t)).accepts = t.language"}, {"name": "CNFA.ofFSM_spec", "content": "lemma CNFA.ofFSM_spec (p : FSM arity) :\n    (CNFA.ofFSM p).Sim (NFA'.ofFSM p)"}, {"name": "CNFA.ofFSM_bv_language", "content": "lemma CNFA.ofFSM_bv_language :\n    (CNFA.ofFSM (FSM.ofTerm t)).bv_recognizes t.language"}, {"name": "TermBinop.subst_arity'", "content": "lemma TermBinop.subst_arity' {op : TermBinop} : (op.subst t₁ t₂).arity + 1= t₁.arity ⊔ t₂.arity + 1"}, {"name": "BitVecs.cast_eq", "content": "@[simp]\nlemma BitVecs.cast_eq  (x : BitVecs n) (h : n = n') : h ▸ x = x.cast h"}, {"name": "Fin.natAdd_zero'", "content": "lemma Fin.natAdd_zero' [h : NeZero m] : Fin.natAdd (m := m) n 0 = n"}, {"name": "TermUnop.subst_arity'", "content": "@[simp]\nlemma TermUnop.subst_arity' {op : TermUnop} : (op.subst t).arity + 1 = t.arity + 1"}, {"name": "TermUnop.alt_lang", "content": "lemma TermUnop.alt_lang {t : Term} (op : TermUnop) :\n  (op.subst_arity' ▸ (op.subst t).language) =\n    let lop : Set (BitVecs 2) := op.openTerm_arity' ▸ op.openTerm.language\n    let lop' : Set (BitVecs (t.arity + 2)) := lop.lift (λ i ↦ i.natAdd t.arity)\n    let lt : Set (BitVecs (t.arity + 2)) := t.language.lift (λ i ↦ i.castLE (by omega))\n    let l := lt ∩ lop'\n    l.proj (liftUnop t.arity)"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\ndef NFA.sa (_ : NFA α σ) := σ → Language α\n\nstructure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q\n\nabbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop\n\ndef NFA'.sa (M : NFA' n) := M.σ → BVNRel n\n\ndef langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }\n\nstructure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))\n\nstructure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)\n\nsection fsm\n\nabbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)\n\nvariable {arity : Type} [FinEnum arity]\n\ndef finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)\n\ndef bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]\n\ndef NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }\n\n@[simp]\nabbrev inFSMRel (p : FSM arity) {w} (bvn : List.Vector (BitVec w) _) :=\n  bvn.get (Fin.last (FinEnum.card arity)) = p.evalBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))\n\ndef NFA'.ofFSM_sa (p : FSM arity) : (NFA'.ofFSM' p).sa := fun q _ bvn =>\n    inFSMRel p bvn ∧ q = p.carryBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))\n\ndef NFA'.ofFSM_correct (p : FSM arity) :\n    (NFA'.ofFSM' p).correct (ofFSM_sa p) (fun _ bvn => inFSMRel p bvn) :=\n\nopen BitStream in\n\ndef CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where\n\nend fsm\n\nsection nfas_relations\n\ninductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype\n\nend nfas_relations\n\ndef liftUnop n : Fin (n + 1) → Fin (n + 2) :=\n  fun k =>\n    if k = n then Fin.last (n+1) else k.castLE (by admit /- proof elided -/\n    )\n\ninductive TermBinop where\n| and | or | xor | add | sub\n\ndef TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂\n\ndef TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)\n\ndef TermBinop.termGadget (t : TermBinop) : CNFA 3 :=\n  match t with\n  | .and => FSM.ofTerm (.and (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .or => FSM.ofTerm (.or (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .xor => FSM.ofTerm (.xor (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .add => FSM.ofTerm (.add (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .sub => FSM.ofTerm (.sub (.var 0) (.var 1)) |> CNFA.ofFSM\n\ninductive TermUnop where\n| neg | not | shiftL (k : Nat)\n\ndef TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k\n\n@[simp]\ndef TermUnop.openTerm_arity' (op : TermUnop) : op.openTerm.arity + 1 = 2 :=\n\ndef TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k\n\ndef TermUnop.termGadget (t : TermUnop) : CNFA 2 :=\n  match t with\n  | .neg => FSM.ofTerm (.neg (.var 0)) |> CNFA.ofFSM\n  | .not => FSM.ofTerm (.not (.var 0)) |> CNFA.ofFSM\n  | .shiftL k => FSM.ofTerm (.shiftL (.var 0) k) |> CNFA.ofFSM\n\ndef autOfTermUnop (op : TermUnop) (m : CNFA (n + 1)) : CNFA (n + 1) :=\n  let mop : CNFA 2 := op.termGadget\n  let mop : CNFA (n + 2) := mop.lift (λ i ↦ i.natAdd n)\n  let m : CNFA (n + 2) := m.lift (λ i ↦ i.castLE (by admit /- proof elided -/\n  ))\n  let m := CNFA.inter m mop\n  let mfinal := m.proj (liftUnop n)\n  mfinal.minimize\n\ndef swapLastTwo (x : Fin (n + 2)) : Fin (n + 2) :=\n  if x = Fin.last (n + 1) then n else if x = n then Fin.last (n + 1) else x", "target_theorem": "lemma autOfTermUnop_bv_language op {t : Term} (m : CNFA (t.arity + 1)) :\n    m.bv_recognizes t.language →\n    (autOfTermUnop op m |>.bv_recognizes (op.subst_arity' ▸ (op.subst t).language)) :=", "ground_truth_proof": ":= by\n  rintro hrec\n  rw [TermUnop.alt_lang]\n  simp only [autOfTermUnop]\n  simp\n  apply CNFA.minimize_bv_language\n  apply CNFA.proj_bv_language\n  apply CNFA.inter_bv_language\n  · apply CNFA.lift_bv_language; assumption\n  · apply CNFA.lift_bv_language\n    simp [TermUnop.openTerm, TermUnop.termGadget]\n    rcases op <;> apply CNFA.ofFSM_bv_language", "nesting_depth": 11, "transitive_dep_count": 289, "subset_aristotle": false, "category": "Compiler"}
{"id": 342, "thm_name": "Std.HashMap.fold_induction", "thm_stmt": "theorem Std.HashMap.fold_induction [BEq α] [LawfulBEq α] [DecidableEq α] [Hashable α]\n  {f : γ → α → β → γ} {m : HashMap α β} {motive : γ → (α → Option β) → Prop} :\n    motive b (λ _ ↦ none) →\n    (∀ b x y m, m x = none → motive b m → motive (f b x y) (Function.update m x y)) →\n    motive (m.fold f b) m.toPFun", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/ForMathlib.lean", "imports": ["import Mathlib.Data.Rel", "import Mathlib.Data.FinEnum", "import Mathlib.Data.Vector.Basic", "import Blase.AutoStructs.ForLean", "import Mathlib.Computability.NFA"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Function.update", "module": "Mathlib.Logic.Function.Basic"}, {"name": "LawfulBEq", "module": "Init.Core"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.find?", "module": "Init.Data.List.Basic"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}, {"name": "Option.map", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.snd", "module": "Init.Prelude"}, {"name": "Std.HashMap.map", "module": "Std.Data.HashMap.AdditionalOperations"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Set.ext", "module": "Mathlib.Data.Set.Defs"}, {"name": "Std.HashMap.contains_eq_isSome_getElem?", "module": "Std.Data.HashMap.Lemmas"}, {"name": "Std.HashMap.contains_eq_isSome_getKey?", "module": "Std.Data.HashMap.Lemmas"}, {"name": "Std.HashMap.find?_toList_eq_some_iff_getKey?_eq_some_and_getElem?_eq_some", "module": "Std.Data.HashMap.Lemmas"}, {"name": "Function.update_apply", "module": "Mathlib.Logic.Function.Basic"}, {"name": "List.find?_some", "module": "Init.Data.List.Find"}, {"name": "List.mem_of_find?_eq_some", "module": "Init.Data.List.Find"}, {"name": "List.pairwise_append", "module": "Init.Data.List.Pairwise"}, {"name": "Std.HashMap.fold_eq_foldl_toList", "module": "Std.Data.HashMap.Lemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Std.HashMap.toPFun", "content": "def Std.HashMap.toPFun [BEq α] [Hashable α] (m : HashMap α β) (x : α) : Option β := m[x]?"}], "used_local_lemmas": [{"name": "Std.HashMap.toPFun_toList", "content": "theorem Std.HashMap.toPFun_toList[BEq α] [LawfulBEq α] [Hashable α] (m : HashMap α β) :\n    m.toPFun = λ k ↦ m.toList.find? (λ x ↦ x.1 == k) |>.map Prod.snd"}], "local_ctx": "import Mathlib.Computability.NFA\n\nimport Mathlib.Data.FinEnum\n\nimport Mathlib.Data.Rel\n\nimport Mathlib.Data.Vector.Basic\n\nimport Blase.AutoStructs.ForLean\n\nopen Set\n\nopen Mathlib\n\nopen SetRel\n\nnamespace NFA\n\nend NFA\n\ndef Std.HashMap.toPFun [BEq α] [Hashable α] (m : HashMap α β) (x : α) : Option β := m[x]?", "target_theorem": "theorem Std.HashMap.fold_induction [BEq α] [LawfulBEq α] [DecidableEq α] [Hashable α]\n  {f : γ → α → β → γ} {m : HashMap α β} {motive : γ → (α → Option β) → Prop} :\n    motive b (λ _ ↦ none) →\n    (∀ b x y m, m x = none → motive b m → motive (f b x y) (Function.update m x y)) →\n    motive (m.fold f b) m.toPFun :=", "ground_truth_proof": ":= by\n  rintro hemp hind\n  rw [Std.HashMap.fold_eq_foldl_toList, toPFun_toList]\n  have := m.distinct_keys_toList\n  revert this\n  induction m.toList using List.reverseRecOn\n  case nil =>\n    simp_all\n  case append_singleton xs xy ih =>\n    rcases xy with ⟨x, y⟩\n    rintro hd\n    simp_all\n    simp [List.pairwise_append] at hd\n    rcases hd with ⟨hd, hnew⟩\n    let f := fun k => Option.map Prod.snd (List.find? (fun x => x.1 == k) xs)\n    specialize ih (by simp [hd])\n    have hnewf : f x = none := by\n      simp [f]; grind\n    specialize hind _ x y _ hnewf ih\n    convert hind using 1\n    ext a b; simp\n    rw [Function.update_apply]\n    split_ifs with hcond\n    · subst hcond; constructor\n      · rintro ⟨a', hf | hf⟩\n        · obtain h := List.find?_some hf\n          simp at h; subst h\n          grind [List.mem_of_find?_eq_some]\n        · grind\n      · simp only [Option.some.injEq, true_and]\n        rintro rfl\n        aesop\n    · simp [f]; grind", "nesting_depth": 2, "transitive_dep_count": 26, "subset_aristotle": false, "category": "Compiler"}
{"id": 343, "thm_name": "Lets.getPureExpr_var_appendInr", "thm_stmt": "@[simp] theorem Lets.getPureExpr_var_appendInr (lets : Lets d Γ_in eff Γ_out) (e : Expr d Γ_out _ ty₁)\n    (v : Var Γ_out ty₂):\n    getPureExpr (lets.var e) v.appendInr\n    = (fun ⟨_, w, e'⟩ => ⟨_, w,  e'.changeVars <| e.contextHom⟩) <$> (getPureExpr lets v)", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Framework/Basic.lean", "imports": ["import LeanMLIR.HVector", "import LeanMLIR.ErasedContext", "import SSA/Projects/CIRCT/HSxComb/HSxCombFunctor.lean", "import SSA/Projects/CIRCT/DCxComb/DCxCombFunctor.lean", "import SSA/Projects/Tensor2D/Tensor2D.lean", "import SSA/Projects/RISCV64/Base.lean", "import SSA/Projects/ModArith/Basic.lean", "import SSA/Projects/Scf/ScfFunctor.lean", "import Mathlib.Data.Finset.Union", "import LeanMLIR.Framework.Dialect", "import LeanMLIR/LeanMLIR/Transforms/CSE.lean", "import LeanMLIR/LeanMLIR/Examples.lean", "import LeanMLIR/LeanMLIR/Transforms/DCE.lean", "import LeanMLIR.LeanMLIR.HVector", "import SSA/Projects/FullyHomomorphicEncryption/Basic.lean", "import LeanMLIR/LeanMLIR/Dialects/LLVM/Basic.lean", "import SSA/Projects/Tensor1D/Tensor1D.lean", "import LeanMLIR/LeanMLIR/Framework/Macro.lean", "import SSA/Projects/LLVMRiscV/LLVMAndRiscv.lean", "import LeanMLIR.EffectKind"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "decidable_of_iff", "module": "Init.PropLemmas"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Valuation.mk", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "Sigma.mk", "module": "Init.Core"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.matchAlts", "module": "Lean.Parser.Term"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Stream'", "module": "Mathlib.Data.Stream.Defs"}], "used_repo_defs": [{"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "map", "content": "def map (f : Ty₁ → Ty₂) : Ctxt Ty₁ → Ctxt Ty₂ :=\n  ofList ∘ (List.map f) ∘ toList"}, {"name": "Hom", "content": "abbrev Hom (Γ Γ' : Ctxt Ty) := ⦃t : Ty⦄ → Γ.Var t → Γ'.Var t"}, {"name": "dropUntil", "content": "def dropUntil : Ctxt Ty :=\n  ⟨Γ.toList.drop (v.val + 1)⟩"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "Hom.castCodomain", "content": "def Hom.castCodomain (h : Δ = Δ') (f : Γ.Hom Δ) : Γ.Hom Δ' :=\n  fun _t v => (f v).castCtxt h"}, {"name": "appendInl", "content": "def appendInl (v : Γ.Var t) : (Γ ++ Δ).Var t :=\n  ⟨v.val, by admit /- proof elided -/\n  ⟩"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "dropUntilHom", "content": "abbrev dropUntilHom : Hom (Γ.dropUntil v) Γ := dropUntilDiff.toHom"}, {"name": "dropUntilDiff", "content": "def dropUntilDiff : Diff (Γ.dropUntil v) Γ :=\n  ⟨v.val+1, by admit /- proof elided -/\n  ⟩"}, {"name": "emptyElim", "content": "def emptyElim {α : Sort _} {t : Ty} : Ctxt.Var ∅ t → α :=\n  fun ⟨_, h⟩ => by admit /- proof elided -/"}, {"name": "cons", "content": "@[match_pattern]\ndef cons (hd : Ty) : Ctxt Ty → Ctxt Ty\n| ⟨tl⟩ => ⟨hd :: tl⟩"}, {"name": "Diff", "content": "def Diff (Γ₁ Γ₂ : Ctxt Ty) : Type :=\n  {d : Nat // Diff.Valid Γ₁ Γ₂ d}"}, {"name": "Diff.Valid", "content": "@[simp]\nabbrev Diff.Valid (Γ₁ Γ₂ : Ctxt Ty) (d : Nat) : Prop :=\n  ∀ {i t}, Γ₁[i]? = some t → Γ₂[i+d]? = some t"}, {"name": "Hom.id", "content": "@[simp] abbrev Hom.id {Γ : Ctxt Ty} : Γ.Hom Γ :=\n  fun _ v => v"}, {"name": "appendInr", "content": "def appendInr (v : Var Δ t) : (Γ ++ Δ).Var t :=\n  ⟨v.val + Γ.length, by admit /- proof elided -/\n  ⟩"}, {"name": "length", "content": "@[grind=]\ndef length (Γ : Ctxt Ty) : Nat := Γ.toList.length"}, {"name": "map", "content": "def map (f : ∀ (a : α), A a → B a) :\n    ∀ {l : List α}, HVector A l → HVector B l\n  | [],   .nil        => .nil\n  | t::_, .cons a as  => .cons (f t a) (map f as)"}, {"name": "HVectorLiteral", "content": "structure HVectorLiteral where\n  u : Level\n  v : Level\n  α : Q(Type $u)\n  A : Q($α → Type $v)\n  elems : Array ((a : Q($α)) × Q($A $a))\n\n      instance : DialectSignature $dialect where\n        signature := fun op => match op with $matchAlts:matchAlts\n    )"}, {"name": "(q", "content": "noncomputable instance (q : ℕ) [Fact (q > 1)] : DialectDenote (ModArith q) where\ndenote\n  | .add, arg, _ =>\n      \n      (fun args : R q × R q => args.1 + args.2) arg.toPair\n  | .sub, arg, _ =>\n      \n      (fun args : R q × R q => args.1 - args.2) arg.toPair\n  | .mul, arg, _ =>\n      \n      (fun args : R q × R q => args.1 * args.2) arg.toPair\n  | .const _ c, _, _ =>\n      \n      c"}, {"name": "Op.signature", "content": "@[simp, reducible]\ndef Op.signature : Op q n → Signature (Ty q n) :=\n  fun o => {sig := Op.sig o, returnTypes := [Op.outTy o], regSig := []}"}, {"name": "Op.sig", "content": "@[simp, reducible]\ndef Op.sig : Op q n → List (Ty q n)\n| Op.add => [Ty.polynomialLike, Ty.polynomialLike]\n| Op.sub => [Ty.polynomialLike, Ty.polynomialLike]\n| Op.mul => [Ty.polynomialLike, Ty.polynomialLike]\n| Op.mul_constant => [Ty.polynomialLike, Ty.integer]\n| Op.leading_term => [Ty.polynomialLike]\n| Op.monomial => [Ty.integer, Ty.index]\n| Op.monomial_mul => [Ty.polynomialLike, Ty.index]\n| Op.from_tensor => [Ty.tensor]\n| Op.to_tensor => [Ty.polynomialLike]\n| Op.const _ => []\n| Op.const_int _ => []\n| Op.const_idx _ => []"}, {"name": "Op", "content": "inductive Op (q : Nat) (n : Nat)\n  | add : Op q n\n  | sub : Op q n\n  | mul : Op q n\n  | mul_constant : Op q n\n    \n    \n  | leading_term : Op q n\n  | monomial : Op q n\n  | monomial_mul : Op q n\n  | from_tensor : Op q n\n  | to_tensor : Op q n\n  | const (c : R q n) : Op q n\n  | const_int (c : Int) : Op q n\n  | const_idx (i : Nat) : Op q n"}, {"name": "Op.outTy", "content": "@[simp, reducible]\ndef Op.outTy : Op q n → Ty q n\n| Op.add | Op.sub | Op.mul | Op.mul_constant | Op.leading_term | Op.monomial\n| Op.monomial_mul | Op.from_tensor | Op.const _  => Ty.polynomialLike\n| Op.to_tensor => Ty.tensor\n| Op.const_int _ => Ty.integer\n| Op.const_idx _ => Ty.index"}, {"name": "Op.regSig", "content": "@[reducible, simp]\ndef Op.regSig : Op → RegionSignature Ty\n  | .map2d => [([Ty.int], [.int])]\n  | _ => []"}, {"name": "", "content": "instance : DialectSignature Ex where\n  signature\n    | .add    => ⟨[.nat, .nat], [], [.nat], .pure⟩\n    | .beq    => ⟨[.nat, .nat], [], [.bool], .pure⟩\n    | .cst _  => ⟨[], [], [.nat], .pure⟩"}, {"name": "ExOp", "content": "inductive ExOp :  Type\n  | add : ExOp\n  | beq : ExOp\n  | cst : ℕ → ExOp\n  deriving DecidableEq"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  | bool\n  deriving DecidableEq"}, {"name": "add", "content": "def add {Γ : Ctxt _} (e₁ e₂ : Ctxt.Var Γ .nat) : Expr Ex Γ .pure [.nat] :=\n  Expr.mk\n    (op := .add)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "Ex", "content": "abbrev Ex : Dialect where\n  Op := ExOp\n  Ty := ExTy"}, {"name": "cst", "content": "def cst {Γ : Ctxt _} (n : ℕ) : Expr Ex Γ .pure [.nat]  :=\n  Expr.mk\n    (op := .cst n)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "Dialect", "content": "structure Dialect where\n  (Op : Type)\n  (Ty : Type)\n  (m : Type → Type := Id)"}, {"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "", "content": "@[reducible]\ninstance : DialectSignature Tensor2D where\n  signature op := { sig := op.sig, regSig := op.regSig, returnTypes := [op.outTy] }"}, {"name": "", "content": "instance : DialectSignature Tensor1D where\n  signature op := { sig := op.sig, regSig := op.regSig, returnTypes := [op.outTy], effectKind := .pure }"}, {"name": "", "content": "instance : DialectSignature RV64 where\n  signature o := {sig := Op.sig o, returnTypes := [Op.outTy o], regSig := []}"}, {"name": "Op.sig", "content": "@[simp, reducible]\ndef Op.sig : Op → List Ty\n  | .li _ => []\n  | .mulh  => [Ty.bv, Ty.bv]\n  | .mulhu  => [Ty.bv, Ty.bv]\n  | .mulhsu  => [Ty.bv, Ty.bv]\n  | .divu =>  [Ty.bv, Ty.bv]\n  | .remuw  => [Ty.bv, Ty.bv]\n  | .remu  =>  [Ty.bv, Ty.bv]\n  | .addiw (_imm : BitVec 12) => [Ty.bv]\n  | .lui (_imm : BitVec 20) => [Ty.bv]\n  | .auipc (_imm : BitVec 20)  => [Ty.bv]\n  | .slliw (_shamt : BitVec 5)  => [Ty.bv]\n  | .srliw (_shamt : BitVec 5) => [Ty.bv]\n  | .sraiw (_shamt : BitVec 5) => [Ty.bv]\n  | .slli (_shamt : BitVec 6) => [Ty.bv]\n  | .srli (_shamt : BitVec 6) => [Ty.bv]\n  | .srai (_shamt : BitVec 6) => [Ty.bv]\n  | .addw => [Ty.bv, Ty.bv]\n  | .subw => [Ty.bv, Ty.bv]\n  | .sllw => [Ty.bv, Ty.bv]\n  | .srlw => [Ty.bv, Ty.bv]\n  | .sraw => [Ty.bv, Ty.bv]\n  | .add => [Ty.bv, Ty.bv]\n  | .slt => [Ty.bv, Ty.bv]\n  | .sltu => [Ty.bv, Ty.bv]\n  | .and => [Ty.bv, Ty.bv]\n  | .or => [Ty.bv, Ty.bv]\n  | .xor => [Ty.bv, Ty.bv]\n  | .sll => [Ty.bv, Ty.bv]\n  | .srl => [Ty.bv, Ty.bv]\n  | .sub => [Ty.bv, Ty.bv]\n  | .sra => [Ty.bv, Ty.bv]\n  | .remw  => [Ty.bv, Ty.bv]\n  | .rem  =>  [Ty.bv, Ty.bv]\n  | .mul => [Ty.bv, Ty.bv]\n  | .mulw => [Ty.bv, Ty.bv]\n  | .div  =>  [Ty.bv, Ty.bv]\n  | .divw  =>  [Ty.bv, Ty.bv]\n  | .divuw  =>  [Ty.bv, Ty.bv]\n  | .addi (_imm : BitVec 12) => [Ty.bv]\n  | .slti (_imm : BitVec 12) => [Ty.bv]\n  | .sltiu (_imm : BitVec 12) => [Ty.bv]\n  | .andi (_imm : BitVec 12) => [Ty.bv]\n  | .ori (_imm : BitVec 12) => [Ty.bv]\n  | .xori (_imm : BitVec 12) => [Ty.bv]\n  | .bclr => [Ty.bv, Ty.bv]\n  | .bext => [Ty.bv, Ty.bv]\n  | .binv => [Ty.bv, Ty.bv]\n  | .bset  => [Ty.bv, Ty.bv]\n  | .bclri (_shamt : BitVec 6) => [Ty.bv]\n  | .bexti (_shamt : BitVec 6) => [Ty.bv]\n  | .binvi (_shamt : BitVec 6) => [Ty.bv]\n  | .bseti (_shamt : BitVec 6) => [Ty.bv]\n  | .adduw => [Ty.bv, Ty.bv]\n  | .sh1adduw => [Ty.bv, Ty.bv]\n  | .sh2adduw => [Ty.bv, Ty.bv]\n  | .sh3adduw => [Ty.bv, Ty.bv]\n  | .sh1add => [Ty.bv, Ty.bv]\n  | .sh2add => [Ty.bv, Ty.bv]\n  | .sh3add => [Ty.bv, Ty.bv]\n  | .slliuw (_shamt : BitVec 6) => [Ty.bv]\n  | .andn => [Ty.bv, Ty.bv]\n  | .orn => [Ty.bv, Ty.bv]\n  | .xnor => [Ty.bv, Ty.bv]\n  | .clz\n  | .clzw\n  | .ctz\n  | .ctzw\n  | .max => [Ty.bv, Ty.bv]\n  | .maxu => [Ty.bv, Ty.bv]\n  | .min  => [Ty.bv, Ty.bv]\n  | .minu  => [Ty.bv, Ty.bv]\n  | .sextb => [Ty.bv]\n  | .sexth => [Ty.bv]\n  | .zexth => [Ty.bv]\n  | .rol => [Ty.bv, Ty.bv]\n  | .rolw => [Ty.bv, Ty.bv]\n  | .ror => [Ty.bv, Ty.bv]\n  | .rori (_shamt : BitVec 6) =>[Ty.bv]\n  | .roriw (_shamt : BitVec 5) =>[Ty.bv]\n  | .rorw => [Ty.bv, Ty.bv]\n  | .pack => [Ty.bv, Ty.bv]\n  | .packh => [Ty.bv, Ty.bv]\n  | .packw => [Ty.bv, Ty.bv]\n  | .mv => [Ty.bv]\n  | .not => [Ty.bv]\n  | .neg => [Ty.bv]\n  | .negw => [Ty.bv]\n  | .sextw => [Ty.bv]\n  | .zextb => [Ty.bv]\n  | .zextw => [Ty.bv]\n  | .seqz => [Ty.bv]\n  | .snez => [Ty.bv]\n  | .sltz => [Ty.bv]\n  | .sgtz => [Ty.bv]"}, {"name": "Op", "content": "inductive Op\n   \n  | li : (val : BitVec 64) → Op\n  | lui (imm : BitVec 20)\n  | auipc (imm : BitVec 20)\n  | addi (imm : BitVec 12)\n  | andi (imm : BitVec 12)\n  | ori (imm : BitVec 12)\n  | xori (imm : BitVec 12)\n  | addiw (imm : BitVec 12)\n  | add\n  | slli (shamt : BitVec 6)\n  | sub\n  | and\n  | or\n  | xor\n  | sll\n  | srl\n  | sra\n  | addw\n  | subw\n  | sllw\n  | srlw\n  | sraw\n  | slti (imm : BitVec 12)\n  | sltiu (imm : BitVec 12)\n  | srli (shamt : BitVec 6)\n  | srai (shamt : BitVec 6)\n  | slliw (shamt : BitVec 5)\n  | srliw (shamt : BitVec 5)\n  | sraiw (shamt : BitVec 5)\n  | slt\n  | sltu\n   \n  | mul\n  | mulw\n  | mulh\n  | mulhu\n  | mulhsu\n  | divw\n  | divuw\n  | div\n  | divu\n  | remw\n  | rem\n  | remuw\n  | remu\n   \n   \n  | adduw\n  | sh1adduw\n  | sh2adduw\n  | sh3adduw\n  | sh1add\n  | sh2add\n  | sh3add\n  | slliuw (shamt : BitVec 6)\n   \n  | andn\n  | orn\n  | xnor\n  | clz\n  | clzw\n  | ctz\n  | ctzw\n  | max\n  | maxu\n  | min\n  | minu\n  | sextb\n  | sexth\n  | zexth\n  | rol\n  | rolw\n  | ror\n  | rori (_shamt : BitVec 6)\n  | roriw (_shamt : BitVec 5)\n  | rorw\n   \n  | bclr\n  | bclri (shamt : BitVec 6)\n  | bext\n  | bexti (shamt : BitVec 6)\n  | binv\n  | binvi (shamt : BitVec 6)\n  | bset\n  | bseti (shamt : BitVec 6)\n   \n  | pack\n  | packh\n  | packw\n   \n  | mv\n  | not\n  | neg\n  | negw\n  | sextw\n  | zextb\n  | zextw\n  | seqz\n  | snez\n  | sltz\n  | sgtz\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "Ty", "content": "inductive Ty\n  | bv : Ty\n  deriving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "", "content": "instance : DialectSignature (FHE q n) := ⟨Op.signature⟩"}, {"name": "", "content": "instance : DialectSignature LLVM where\n  signature op := ⟨op.sig, [], [op.outTy], .pure⟩"}, {"name": "", "content": "instance : DialectSignature HSxComb where\n  signature := fun op =>\n    match op with\n    | .comb o => liftSig (signature o) \n    \n    \n    | .hs o => MLIR2Handshake.instDialectSignatureHandshake.signature o"}, {"name": "Op", "content": "inductive Op : Type _\n  | comb (o : MLIR2Comb.Comb.Op)\n  | hs (o : MLIR2Handshake.Handshake.Op)\n  deriving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "liftSig", "content": "def liftSig (sig : Signature MLIR2Comb.Ty) : Signature MLIR2Handshake.Ty :=\n  Signature.mk (sig.sig.map liftTy) [] (liftTy sig.outTy)"}, {"name": "liftTy", "content": "def liftTy : MLIR2Comb.Ty → MLIR2Handshake.Ty\n| .bitvec w => .stream (.bitvec w)"}, {"name": "Ty", "content": "inductive Ty\n| stream (ty2 : Ty2) : Ty \n| stream2 (ty2 : Ty2) : Ty \n| stream2token (ty2 : Ty2) : Ty \nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "Ty", "content": "inductive Ty\n| bitvec (w : Nat) : Ty \nderiving DecidableEq, Repr, ToExpr"}, {"name": "Ty2", "content": "inductive Ty2\n  | bitvec (w : Nat) : Ty2\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "map", "content": "def map {α β : Type} (s : Stream α) (f : α → β) : Stream β :=\n  fun i => (s i).map f"}, {"name": "Stream", "content": "def Stream (β : Type) := Stream' (Option β)"}, {"name": "", "content": "instance : DialectSignature DCxComb where\n  signature := fun op =>\n    match op with\n    | .comb o => liftSig (signature o) \n    \n    \n    | .dc o => MLIR2DC.instDialectSignatureDC.signature o"}, {"name": "Op", "content": "inductive Op : Type _\n  | comb (o : MLIR2Comb.Comb.Op)\n  | dc (o : MLIR2DC.DC.Op)\n  deriving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "liftSig", "content": "def liftSig (sig : Signature MLIR2Comb.Ty) : Signature MLIR2DC.Ty :=\n  Signature.mk (sig.sig.map liftTy) [] (liftTy sig.outTy)"}, {"name": "liftTy", "content": "def liftTy : MLIR2Comb.Ty → MLIR2DC.Ty\n| .bitvec w => .valuestream w"}, {"name": "Ty", "content": "inductive Ty\n| tokenstream : Ty\n| tokenstream2 : Ty\n| valuestream (w : Nat) : Ty \n| valuestream2 (w : Nat) : Ty \n| valuetokenstream (w : Nat) : Ty \n| variadicvaluetokenstream (w : Nat) : Ty \nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "[SIG", "content": "instance [SIG : DialectSignature d] [DENOTE : DialectDenote d] {Γ : Ctxt d.Ty} {t}\n    (com : Com d Γ .pure t) : Inhabited (DCEType com) where\n  default :=\n    ⟨Γ, Hom.id, com, by admit /- proof elided -/\n    ⟩"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  | bool\n  deriving DecidableEq, Repr"}, {"name": "cst", "content": "def cst {Γ : Ctxt _} (n : ℕ) : Expr Ex Γ .pure [.nat] :=\n  Expr.mk\n    (op := .cst n)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "LLVMPlusRiscVSignature", "content": "@[simp]\ninstance LLVMPlusRiscVSignature : DialectSignature LLVMPlusRiscV where\n  signature\n  | .llvm llvmOp => .llvm <$> DialectSignature.signature llvmOp\n  | .riscv riscvOp => .riscv <$> DialectSignature.signature riscvOp\n  | .castRiscv w =>\n      {sig := [Ty.riscv .bv], returnTypes := [Ty.llvm (.bitvec w)], regSig := []}\n  | .castLLVM w =>\n      {sig := [Ty.llvm (.bitvec w)], returnTypes := [Ty.riscv .bv], regSig := []}"}, {"name": "Op", "content": "inductive Op where\n  | llvm : LLVM.Op -> Op\n  | riscv : RISCV64.RV64.Op -> Op\n  | castRiscv : Nat → Op\n  | castLLVM : Nat → Op\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "Ty", "content": "inductive Ty where\n  | llvm : LLVM.Ty -> Ty\n  | riscv : RISCV64.RV64.Ty -> Ty\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "", "content": "instance : DialectSignature ExOp ExTy where\n  signature\n    | .add    => ⟨[.nat, .nat], [], .nat, .pure⟩\n    | .beq    => ⟨[.nat, .nat], [], .bool, .pure⟩\n    | .cst _  => ⟨[], [], .nat, .pure⟩"}, {"name": "ExOp", "content": "inductive ExOp :  Type\n  | add : ExOp\n  | beq : ExOp\n  | cst : ℕ → ExOp\n  deriving DecidableEq, Repr"}, {"name": "add", "content": "def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .nat) : Expr Γ .nat :=\n  Expr.mk\n    (op := .add)\n    (ty_eq := rfl)\n    (eff_le := EffectKind.le_refl _)\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  deriving DecidableEq, Repr"}, {"name": "Expr", "content": "abbrev Expr (Γ) (ty) := _root_.Expr ExOp Γ .pure ty"}, {"name": "cst", "content": "def cst {Γ : Ctxt _} (n : ℕ) : Expr Γ .nat  :=\n  Expr.mk\n    (op := .cst n)\n    (ty_eq := rfl)\n    (eff_le := EffectKind.le_refl _)\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "", "content": "instance : DialectSignature ExOp ExTy where\n  signature\n  | .add    => ⟨[.nat, .nat], [], .nat, .pure⟩\n  | .runK _ => ⟨[.nat], [([.nat], .nat)], .nat, .pure⟩"}, {"name": "ExOp", "content": "inductive ExOp :  Type\n  | add : ExOp\n  | runK : ℕ → ExOp\n  deriving DecidableEq, Repr"}, {"name": "add", "content": "def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .nat) : Expr Γ .nat :=\n  Expr.mk\n    (op := .add)\n    (ty_eq := rfl)\n    (eff_le := EffectKind.pure_le _)\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "[TyDenote", "content": "@[reducible]\ninstance [TyDenote d.Ty] [DialectSignature d] [DialectDenote d]\n    [B : HasBool d] [N : HasNat d] [I : HasInt d] : DialectSignature (Scf d) where\n   signature\n   | .coe o => signature (d:=d) o\n    | .if t t' => ⟨[B.ty, t], [(⟨[t]⟩, [t']), (⟨[t]⟩, [t'])], [t'], .impure⟩\n      \n      \n      \n      \n      \n    | .for t => ⟨[ I.ty,  I.ty,  N.ty, t], [(⟨[I.ty, t]⟩, [t])], [t], .impure⟩\n    | .run t => ⟨[t], [(⟨[t]⟩, [t])], [t], .impure⟩\n    | .iterate _k => ⟨[I.ty], [(⟨[I.ty]⟩, [I.ty])], [I.ty], .impure⟩"}, {"name": "HasTy", "content": "class HasTy (d : Dialect) (DenotedTy : Type) [TyDenote d.Ty] [DialectSignature d] where\n    ty : d.Ty\n    denote_eq : toType ty = DenotedTy := by admit /- proof elided -/"}, {"name": "Scf.Op", "content": "inductive Scf.Op (Op' Ty' : Type) (m') [TyDenote Ty'] [DialectSignature ⟨Op', Ty', m'⟩]\n    [DialectDenote ⟨Op', Ty', m'⟩] : Type _\n  | coe (o : Op')\n  | iterate (k : ℕ) \n  | run (inputty : Ty')  \n  | if (inputty retty' : Ty')  \n  | for (ty : Ty')\n  deriving DecidableEq, Repr"}, {"name": "iterate", "content": "@[simp_denote] def iterate {Γ : Ctxt _} (k : Nat) (input : Var Γ Arith.Ty.int)\n    (body : Com ScfArith ⟨[.int]⟩ .impure .int) : Expr ScfArith Γ .impure .int :=\n  Expr.mk\n    (op := .iterate k)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons input .nil)\n    (regArgs := HVector.cons body HVector.nil)"}, {"name": "ScfArith", "content": "abbrev ScfArith := Scf Arith"}, {"name": "Scf", "content": "def Scf (d : Dialect) [TyDenote d.Ty] [DialectSignature d] [DialectDenote d] : Dialect where\n  Op := Scf.Op d.Op d.Ty d.m\n  Ty := d.Ty\n  m  := d.m"}, {"name": "Op", "content": "inductive Op\n  | add : Op  \n  | add_nat : Op  \n  | axpy : Op  \n  | neg : Op  \n  | const : (val : ℤ) → Op\n  | const_nat : (val : ℕ) → Op"}, {"name": "run", "content": "@[simp_denote]\ndef run {Γ : Ctxt _} {t : Arith.Ty} (v : Var Γ t) (body : Com ScfArith ⟨[t]⟩ .impure t) :\n    Expr ScfArith Γ .impure t :=\n  Expr.mk\n    (op := .run t)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons v .nil)\n    (regArgs := HVector.cons body <| HVector.nil)"}, {"name": "Ty", "content": "inductive Ty\n| int\n| bool\n| nat\n deriving DecidableEq, Repr"}, {"name": "Valuation.instAppendHVector", "content": "@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V"}, {"name": "neg", "content": "@[simp_denote] def neg {Γ : Ctxt _} (a : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .neg)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "axpy", "content": "@[simp_denote] def axpy {Γ : Ctxt _} (a : Var Γ .int) (x : Var Γ .nat) (b: Var Γ .int) :\n    Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .axpy)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons x <| .cons b .nil)\n    (regArgs := .nil)"}, {"name": "add_nat", "content": "@[simp_denote] def add_nat (e₁ e₂ : Var Γ .nat) : Expr ScfArith Γ .pure .nat :=\n  Expr.mk\n    (op := .coe <| .add_nat)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "Ty", "content": "inductive Ty\n  | int\n   \n  | int2\n  deriving DecidableEq, Lean.ToExpr"}, {"name": "Op", "content": "inductive Op\n  | noop\n  | mkPair\n  | unPair\n  deriving Lean.ToExpr"}, {"name": "Arith", "content": "abbrev Arith : Dialect := {Op, Ty}"}, {"name": "", "content": "@[reducible]\ninstance : DialectSignature Arith where\n  signature\n    | .axpy => ⟨[.int, .nat, .int], [], [.int], .pure⟩\n    | .neg => ⟨[.int], [], [.int], .pure⟩\n    | .const _ => ⟨[], [], [.int], .pure⟩\n    | .const_nat _ => ⟨[], [], [.nat], .pure⟩\n    | .add   => ⟨[.int, .int], [], [.int], .pure⟩\n    | .add_nat   => ⟨[.nat, .nat], [], [.nat], .pure⟩"}, {"name": "add", "content": "@[simp_denote] def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .add)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "TyDenote.toType", "content": "notation \"⟦\" x \"⟧\" => TyDenote.toType x"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Option.map_none", "module": "Init.Data.Option.Basic"}, {"name": "Option.map_some", "module": "Init.Data.Option.Basic"}, {"name": "cast_eq_iff_heq", "module": "Batteries.Logic"}, {"name": "Function.comp_apply", "module": "Init.Core"}, {"name": "Option.map_eq_map", "module": "Init.Data.Option.Lemmas"}, {"name": "Option.map_map", "module": "Init.Data.Option.Lemmas"}, {"name": "heq_eq_eq", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "map_map", "content": "theorem map_map {A B C : α → Type*} {l : List α} (t : HVector A l)\n    (f : ∀ a, A a → B a) (g : ∀ a, B a → C a) :\n    (t.map f).map g = t.map (fun a v => g a (f a v))"}], "used_local_defs": [{"name": "RegionSignature", "content": "abbrev RegionSignature Ty := List (Ctxt Ty × List Ty)"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "Signature.mk", "content": "abbrev Signature.mk (sig : List Ty) (regSig : RegionSignature Ty) (returnTypes : List Ty) : Signature Ty :=\n { sig, regSig, returnTypes }"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "DialectSignature.sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "DialectSignature.regSig", "content": "def regSig       := Signature.regSig ∘ s.signature"}, {"name": "DialectSignature.returnTypes", "content": "def returnTypes  := Signature.returnTypes ∘ s.signature"}, {"name": "DialectSignature.effectKind", "content": "def effectKind   := Signature.effectKind ∘ s.signature"}, {"name": "DialectDenote", "content": "class DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))"}, {"name": "Expr", "content": "inductive Expr : (Γ : Ctxt d.Ty) → (eff : EffectKind) → (ty : List d.Ty) → Type where\n  | mk {Γ} {ty} (op : d.Op)\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) <| DialectSignature.sig op)\n     \n    (regArgs : HVector (fun t : Ctxt d.Ty × List d.Ty => Com t.1 .impure t.2)\n      (DialectSignature.regSig op)) : Expr Γ eff ty"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "Regions", "content": "abbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "HVector", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "Expr", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Com.decidableEq", "content": "protected instance Com.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty]\n    {Γ : Ctxt d.Ty} {eff : EffectKind} {tys : List d.Ty} : DecidableEq (Com d Γ eff tys)\n  | .rets v₁, .rets v₂ => decidable_of_iff (v₁ = v₂) (by admit /- proof elided -/\n  )\n  | .var (ty := ty₁) e₁ body₁, .var (ty := ty₂) e₂ body₂ =>\n    if hα : ty₁ = ty₂\n    then by\n      subst hα\n      letI := Expr.decidableEq e₁ e₂\n      letI := Com.decidableEq body₁ body₂\n      exact decidable_of_iff (e₁ = e₂ ∧ body₁ = body₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )\n  | .rets _, .var _ _ => isFalse (fun h => Com.noConfusion h)\n  | .var _ _, .rets _ => isFalse (fun h => Com.noConfusion h)"}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "Expr.regArgs", "content": "def Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r)"}, {"name": "Expr.contextHom", "content": "abbrev Expr.contextHom (e : Expr d Γ eff ts) : Γ.Hom e.outContext :=\n  Hom.id.appendCodomain"}, {"name": "Expr.changeVars", "content": "def Expr.changeVars (varsMap : Γ.Hom Γ') {ty} (e : Expr d Γ eff ty) :\n    Expr d Γ' eff ty :=\n  ⟨e.op, e.ty_eq, e.eff_le, e.args.map varsMap, e.regArgs⟩"}, {"name": "FlatCom", "content": "structure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts"}, {"name": "Lets.getPureExprAux", "content": "def Lets.getPureExprAux {Γ₁ Γ₂ : Ctxt d.Ty} {t} : Lets d Γ₁ eff Γ₂ → (v : Var Γ₂ t) →\n    Option (Σ ts, (Var ⟨ts⟩ t) × Expr d (Γ₂.dropUntil v) .pure ts)\n  | .nil, _ => none\n  | .var (Γ_out := Γ_out) (t := t) lets e, v => by admit /- proof elided -/\n    | right v =>\n        apply cast ?_ <| Lets.getPureExprAux lets v\n        simp\n    | left v =>\n        have h : (Ctxt.dropUntil t v) ++ Γ_out = e.outContext.dropUntil v.appendInl := by admit /- proof elided -/"}, {"name": "Lets.getPureExpr", "content": "def Lets.getPureExpr {Γ₁ Γ₂ : Ctxt d.Ty} (lets : Lets d Γ₁ eff Γ₂) {t : d.Ty} (v : Var Γ₂ t) :\n    Option (Σ ts, (Var ⟨ts⟩ t) × Expr d Γ₂ .pure ts) :=\n  (getPureExprAux lets v).map fun ⟨_, v, e⟩ =>\n    ⟨_, v, e.changeVars Ctxt.dropUntilHom⟩"}], "used_local_lemmas": [{"name": "Expr.changeVars_changeVars", "content": "@[simp] theorem Expr.changeVars_changeVars (e : Expr d Γ eff ty) (f : Γ.Hom Δ) (g : Δ.Hom Ξ) :\n    (e.changeVars f).changeVars g = e.changeVars (f.comp g)"}, {"name": "Expr.changeVars_castCodomain", "content": "theorem Expr.changeVars_castCodomain (e : Expr d Γ eff t)\n    (f : Hom Γ Δ) (h : Δ = Δ') :\n    e.changeVars (f.castCodomain h) = cast (by simp [h]) (e.changeVars f)"}, {"name": "Lets.getPureExprAux_var_appendInr", "content": "@[simp] theorem Lets.getPureExprAux_var_appendInr (lets : Lets d Γ_in eff Γ_out)\n    (e : Expr d Γ_out eff ty₁) (v : Var Γ_out ty₂) :\n    getPureExprAux (lets.var e) v.appendInr\n    = (getPureExprAux lets v).map fun ⟨_, w, e⟩ =>\n        ⟨_, w, e.changeVars <| Hom.id.castCodomain (by simp)⟩"}], "local_ctx": "import LeanMLIR.ErasedContext\n\nimport LeanMLIR.HVector\n\nimport LeanMLIR.EffectKind\n\nimport LeanMLIR.Framework.Dialect\n\nimport Mathlib.Data.Finset.Union\n\nopen Ctxt (Var VarSet Valuation Hom)\n\nopen TyDenote (toType)\n\nabbrev RegionSignature Ty := List (Ctxt Ty × List Ty)\n\nstructure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure\n\nabbrev Signature.mk (sig : List Ty) (regSig : RegionSignature Ty) (returnTypes : List Ty) : Signature Ty :=\n { sig, regSig, returnTypes }\n\nclass DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty\n\nnamespace DialectSignature\n\nvariable {d} [s : DialectSignature d]\n\ndef sig          := Signature.sig ∘ s.signature\n\ndef regSig       := Signature.regSig ∘ s.signature\n\ndef returnTypes  := Signature.returnTypes ∘ s.signature\n\ndef effectKind   := Signature.effectKind ∘ s.signature\n\nend DialectSignature\n\nclass DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))\n\nsection DataStructures\n\nvariable (d : Dialect) [DialectSignature d]\n\ninductive Expr : (Γ : Ctxt d.Ty) → (eff : EffectKind) → (ty : List d.Ty) → Type where\n  | mk {Γ} {ty} (op : d.Op)\n    (ty_eq : ty = DialectSignature.returnTypes op)\n    (eff_le : DialectSignature.effectKind op ≤ eff)\n    (args : HVector (Var Γ) <| DialectSignature.sig op)\n     \n    (regArgs : HVector (fun t : Ctxt d.Ty × List d.Ty => Com t.1 .impure t.2)\n      (DialectSignature.regSig op)) : Expr Γ eff ty\n\ninductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β\n\nend\n\nabbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ\n\nabbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig\n\ninductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext\n\nvariable {d} [DialectSignature d]\n\nprotected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )\n\nprotected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )\n\nprotected instance Com.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty]\n    {Γ : Ctxt d.Ty} {eff : EffectKind} {tys : List d.Ty} : DecidableEq (Com d Γ eff tys)\n  | .rets v₁, .rets v₂ => decidable_of_iff (v₁ = v₂) (by admit /- proof elided -/\n  )\n  | .var (ty := ty₁) e₁ body₁, .var (ty := ty₂) e₂ body₂ =>\n    if hα : ty₁ = ty₂\n    then by\n      subst hα\n      letI := Expr.decidableEq e₁ e₂\n      letI := Com.decidableEq body₁ body₂\n      exact decidable_of_iff (e₁ = e₂ ∧ body₁ = body₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )\n  | .rets _, .var _ _ => isFalse (fun h => Com.noConfusion h)\n  | .var _ _, .rets _ => isFalse (fun h => Com.noConfusion h)\n\nend -- decEq\n\nend DataStructures\n\nvariable {d : Dialect} [DialectSignature d]\n\nsection Rec\n\nvariable {eff t} {motive : ∀ {Γ}, Com d Γ eff t → Sort u}\n          (rets : ∀ {Γ : Ctxt _} , (v : HVector Γ.Var t) → motive (Com.rets v))\n          (var : ∀ {Γ} {u},\n            (e : Expr d Γ eff u) → (body : Com d e.outContext eff t) →\n              motive body → motive (Com.var e body))\n\ndef Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl\n\nvariable {rets} {var} {Γ : Ctxt _}\n\nend Rec\n\ndef Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)\n\ndef Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)\n\nsection Lemmas\n\nnamespace Com\n\nend Com\n\nend Lemmas\n\ndef Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) \n\nabbrev Expr.contextHom (e : Expr d Γ eff ts) : Γ.Hom e.outContext :=\n  Hom.id.appendCodomain\n\nsection Lemmas\n\nend Lemmas\n\nvariable [TyDenote d.Ty] [DialectDenote d] [DecidableEq d.Ty] [Monad d.m] [LawfulMonad d.m]\n\nend\n\nsection Unfoldings\n\nopen EffectKind (liftEffect)\n\nend Unfoldings\n\nsection Lemmas\n\nend Lemmas\n\ndef Expr.changeVars (varsMap : Γ.Hom Γ') {ty} (e : Expr d Γ eff ty) :\n    Expr d Γ' eff ty :=\n  ⟨e.op, e.ty_eq, e.eff_le, e.args.map varsMap, e.regArgs⟩\n\nsection Lemmas\n\nvariable {Γ Γ' : Ctxt d.Ty} {t} (f : Γ.Hom Γ') (e : Expr d Γ eff t) (V : Γ'.Valuation)\n\nend Lemmas\n\nstructure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts\n\nsection Lemmas\n\nend Lemmas\n\nsection toPureLemmas\n\nvariable {Γ eff ty} {e : Expr d Γ eff ty} (h : e.HasPureOp)\n\nend toPureLemmas\n\nsection DenoteInsert\n\nend DenoteInsert\n\ndef Lets.getPureExprAux {Γ₁ Γ₂ : Ctxt d.Ty} {t} : Lets d Γ₁ eff Γ₂ → (v : Var Γ₂ t) →\n    Option (Σ ts, (Var ⟨ts⟩ t) × Expr d (Γ₂.dropUntil v) .pure ts)\n  | .nil, _ => none\n  | .var (Γ_out := Γ_out) (t := t) lets e, v => by admit /- proof elided -/\n    | right v =>\n        apply cast ?_ <| Lets.getPureExprAux lets v\n        simp\n    | left v =>\n        have h : (Ctxt.dropUntil t v) ++ Γ_out = e.outContext.dropUntil v.appendInl := by admit /- proof elided -/\n\ndef Lets.getPureExpr {Γ₁ Γ₂ : Ctxt d.Ty} (lets : Lets d Γ₁ eff Γ₂) {t : d.Ty} (v : Var Γ₂ t) :\n    Option (Σ ts, (Var ⟨ts⟩ t) × Expr d Γ₂ .pure ts) :=\n  (getPureExprAux lets v).map fun ⟨_, v, e⟩ =>\n    ⟨_, v, e.changeVars Ctxt.dropUntilHom⟩", "target_theorem": "@[simp] theorem Lets.getPureExpr_var_appendInr (lets : Lets d Γ_in eff Γ_out) (e : Expr d Γ_out _ ty₁)\n    (v : Var Γ_out ty₂):\n    getPureExpr (lets.var e) v.appendInr\n    = (fun ⟨_, w, e'⟩ => ⟨_, w,  e'.changeVars <| e.contextHom⟩) <$> (getPureExpr lets v) :=", "ground_truth_proof": ":= by\n  simp only [getPureExpr, getPureExprAux_var_appendInr, Option.map_eq_map, Option.map_map]\n  congr 1\n  funext ⟨_, e⟩\n  simp only [Function.comp_apply, Expr.changeVars_changeVars, Sigma.mk.injEq, heq_eq_eq, true_and]\n  congr 2\n  funext t v\n  apply Subtype.ext\n  simp [Hom.castCodomain]\n  grind", "nesting_depth": 6, "transitive_dep_count": 76, "subset_aristotle": false, "category": "Compiler"}
{"id": 344, "thm_name": "mem_matchArg", "thm_stmt": "theorem mem_matchArg {Δ_out}\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}\n    {l : List d.Ty} {argsₗ : HVector (Var Γ_out) l}\n    {argsᵣ : HVector (Var Δ_out) l} {ma : Mapping Δ_in Γ_out}\n    {varMap : Mapping Δ_in Γ_out}\n    (hvarMap : ((), varMap) ∈ matchArg lets matchLets argsₗ argsᵣ ma)\n    {t' v'} : ⟨t', v'⟩ ∈ matchLets.varsOfVec argsᵣ → ⟨t', v'⟩ ∈ varMap :=\n  match l, argsₗ, argsᵣ/- , ma, varMap, hvarMap -/ with\n  | .nil, .nil, .nil /- , _, varMap, _ -/ => by simp [Lets.varsOfVec]\n  | .cons t ts, .cons vₗ argsₗ, .cons vᵣ args /-, ma, varMap, h -/ => by\n    simp only [matchArg, bind, Option.mem_def, StateT.bind, Option.bind_eq_some_iff] at hvarMap\n    rcases hvarMap with ⟨ma', h₁, h₂⟩\n    simp only [HVector.vars_cons, Finset.biUnion_insert, Finset.mem_union,\n      Finset.mem_biUnion, Sigma.exists, Lets.varsOfVec]\n    rintro (h | ⟨a, b, hab⟩)\n    · exact AList.keys_subset_keys_of_entries_subset_entries\n        (isMonotone_matchArg _ _ h₂)\n        (mem_matchVar (matchLets := matchLets) h₁ h)\n    · apply mem_matchArg h₂\n      unfold Lets.varsOfVec\n      apply Finset.mem_biUnion.mpr ⟨_, hab.1, hab.2⟩", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Transforms/Rewrite/Match.lean", "imports": ["import LeanMLIR.Framework", "import LeanMLIR.LeanMLIR.Framework.Basic", "import LeanMLIR.Transforms.Rewrite.Mapping"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "AList.insert", "module": "Mathlib.Data.List.AList"}, {"name": "String", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Valuation.map", "module": "Mathlib.RingTheory.Valuation.Basic"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "StateT.bind", "module": "Init.Control.State"}, {"name": "Sigma.mk", "module": "Init.Core"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.Subset", "module": "Mathlib.Data.Set.Defs"}, {"name": "List.Subset", "module": "Init.Data.List.Basic"}, {"name": "reduceCtorEq", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}], "used_repo_defs": [{"name": "Mapping", "content": "abbrev Mapping (Γ Δ : Ctxt Ty) : Type :=\n  @AList (Σ t, Var Γ t) (fun x => Var Δ x.1)"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "last", "content": "@[match_pattern]\ndef last (Γ : Ctxt Ty) (t : Ty) : Ctxt.Var (Ctxt.cons t Γ) t :=\n  ⟨0, by admit /- proof elided -/\n  ⟩"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "Lets.varsOfVec", "content": "def Lets.varsOfVec (lets : Lets d Γ_in eff Γ_out) (vs : HVector Γ_out.Var ts) :\n    VarSet Γ_in :=\n  (vs.vars).biUnion (fun v => lets.vars v.2)"}, {"name": "Lets.vars", "content": "def Lets.vars : Lets d Γ_in eff Γ_out → Var Γ_out t → VarSet Γ_in\n  | .nil, v => VarSet.ofVar v\n  | .var lets e, v => by admit /- proof elided -/\n      | right v => exact lets.vars v\n      | left _ => exact lets.varsOfVec e.args"}, {"name": "Com.vars", "content": "def Com.vars (com : Com d Γ eff ts) : VarSet Γ :=\n  com.toLets.varsOfVec com.returnVars"}, {"name": "Expr.returnVars", "content": "def Expr.returnVars (e : Expr d Γ eff tys) : HVector e.outContext.Var tys :=\n  .ofFn _ _ <| fun i => (Var.ofFin i).appendInl"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) "}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "ofFin", "content": "def ofFin (i : Fin Γ.length) : Γ.Var (Γ[i]) :=\n  ⟨i.val, by admit /- proof elided -/\n  ⟩"}, {"name": "Com.returnVars", "content": "def Com.returnVars : (com : Com d Γ eff ts) → HVector (Var com.outContext) ts\n  | .rets vs => vs\n  | .var _ body => body.returnVars"}, {"name": "Com.toLets", "content": "def Com.toLets (com : Com d Γ eff t) : Lets d Γ eff com.outContext :=\n  Lets.nil.addComToEnd com"}, {"name": "VarSet", "content": "abbrev VarSet (Γ : Ctxt Ty) : Type :=\n  Finset (Σ t, Γ.Var t)"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "HVector.vars", "content": "def HVector.vars {l : List d.Ty} (T : HVector (Var Γ) l) : VarSet Γ :=\n  T.foldl (fun _ s a => insert ⟨_, a⟩ s) ∅"}, {"name": "ofVar", "content": "@[simp]\ndef ofVar {Γ : Ctxt Ty} (v : Γ.Var t) : VarSet Γ :=\n  {⟨_, v⟩}"}, {"name": "foldl", "content": "def foldl {B : Type*} (f : ∀ (a : α), B → A a → B) :\n    ∀ {l : List α}, B → HVector A l → B\n  | [],   b, .nil       => b\n  | t::_, b, .cons a as => foldl f (f t b a) as"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "IsEmpty.exists_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "Option.isSome_none", "module": "Init.Data.Option.Basic"}, {"name": "iff_false", "module": "Init.SimpLemmas"}, {"name": "Option.bind_eq_some_iff", "module": "Init.Data.Option.Lemmas"}, {"name": "AList.entries_insert_of_notMem", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup_eq_none", "module": "Mathlib.Data.List.AList"}, {"name": "List.subset_cons_of_subset", "module": "Init.Data.List.Sublist"}, {"name": "Option.mem_def", "module": "Init.Data.Option.Instances"}, {"name": "IsEmpty.forall_iff", "module": "Mathlib.Logic.IsEmpty"}, {"name": "forall_eq'", "module": "Init.PropLemmas"}, {"name": "iff_true", "module": "Init.SimpLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "AList.keys_subset_keys_of_entries_subset_entries", "module": "Mathlib.Data.List.AList"}, {"name": "Finset.biUnion_insert", "module": "Mathlib.Data.Finset.Union"}, {"name": "Finset.mem_biUnion", "module": "Mathlib.Data.Finset.Union"}, {"name": "Finset.mem_union", "module": "Mathlib.Data.Finset.Lattice.Basic"}, {"name": "AList.lookup_isSome", "module": "Mathlib.Data.List.AList"}, {"name": "Finset.mem_singleton", "module": "Mathlib.Data.Finset.Insert"}, {"name": "Sigma.mk.inj_iff", "module": "Mathlib.Data.Sigma.Basic"}, {"name": "and_imp", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "appendCases_appendInl", "content": "@[simp] theorem appendCases_appendInl (v : Γ.Var t) :\n    appendCases (motive := motive) left right v.appendInl = (left v)"}, {"name": "HVector.vars_cons", "content": "@[simp] theorem HVector.vars_cons {t  : d.Ty} {l : List d.Ty}\n    (v : Var Γ t) (T : HVector (Var Γ) l) :\n    (HVector.cons v T).vars = insert ⟨_, v⟩ T.vars"}, {"name": "appendCases_appendInr", "content": "@[simp] theorem appendCases_appendInr (v : Γ.Var t) :\n    appendCases (motive := motive) left right v.appendInr = (right v)"}], "used_local_defs": [{"name": "MatchVarM", "content": "abbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)"}, {"name": "MatchVar", "content": "abbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit"}, {"name": "MatchVarM.unifyVars", "content": "def MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)"}, {"name": "matchArg", "content": "def matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)"}, {"name": "matchVar", "content": "def matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/"}, {"name": "MatchVar.IsMonotone", "content": "def MatchVar.IsMonotone (f : MatchVar Δ Γ) : Prop :=\n    ∀ mapIn, ∀ mapOut ∈ f mapIn,\n      mapIn.entries ⊆ mapOut.2.entries"}], "used_local_lemmas": [{"name": "unifyVars_eq_some_iff", "content": "@[simp]\ntheorem unifyVars_eq_some_iff :\n    unifyVars w v mapIn = some ((), mapOut)\n    ↔ ( mapIn.lookup ⟨t, w⟩ = none ∧ mapIn.insert ⟨t, w⟩ v = mapOut\n        ∨ mapIn.lookup ⟨t, w⟩ = v ∧ mapIn = mapOut\n      )"}, {"name": "MatchVar.liftM_bind_eq_some_iff", "content": "@[simp]\ntheorem MatchVar.liftM_bind_eq_some_iff (x? : Option α)\n    (f : α → MatchVarM Δ Γ β) :\n    ((liftM x? >>= f) mapIn = some mapOut)\n    ↔ ( ∃ h : x?.isSome,\n        f (x?.get h) mapIn = some mapOut )"}, {"name": "MatchVar.isMonotone_bind", "content": "@[simp]\ntheorem MatchVar.isMonotone_bind {f : MatchVar Δ Γ} {g : Unit → MatchVar Δ Γ} :\n    f.IsMonotone → (g ()).IsMonotone → IsMonotone (f >>= g)"}, {"name": "MatchVar.isMonotone_bind_liftM", "content": "@[simp]\ntheorem MatchVar.isMonotone_bind_liftM {x? : Option α} {g : α → MatchVar Δ Γ} :\n    IsMonotone (liftM x? >>= g) ↔ (∀ x ∈ x?, (g x).IsMonotone)"}, {"name": "MatchVar.isMonotone_none", "content": "@[simp] theorem MatchVar.isMonotone_none : IsMonotone (none : MatchVar Δ Γ)"}, {"name": "MatchVar.isMonotone_unifyVars", "content": "theorem MatchVar.isMonotone_unifyVars  : IsMonotone (unifyVars w v)"}, {"name": "isMonotone_matchVarArg_aux", "content": "theorem isMonotone_matchVarArg_aux (lets : Lets d Γ_in eff Γ_out) :\n    (\n     ∀  (Δ_out : Ctxt d.Ty)\n        (matchLets : Lets d Δ_in EffectKind.pure Δ_out) (l : List d.Ty)\n        (argsl : HVector Γ_out.Var l) (argsr : HVector Δ_out.Var l),\n        (matchArg lets matchLets argsl argsr).IsMonotone\n    )\n    ∧ (\n      ∀ (Δ_out : Ctxt d.Ty) (t : d.Ty) (v : Γ_out.Var t)\n        (matchLets : Lets d Δ_in EffectKind.pure Δ_out)\n        (w : Var Δ_out t),\n        (matchVar lets v matchLets w).IsMonotone\n    )"}, {"name": "isMonotone_matchArg", "content": "theorem isMonotone_matchArg [DecidableEq d.Op]\n    {Γ_out Δ_in Δ_out : Ctxt d.Ty}\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}\n    {l : List d.Ty}\n    {argsl : HVector (Var Γ_out) l}\n    {argsr : HVector (Var Δ_out) l} :\n    (matchArg lets matchLets argsl argsr).IsMonotone"}], "local_ctx": "import LeanMLIR.Framework\n\nimport LeanMLIR.Transforms.Rewrite.Mapping\n\nopen Ctxt (Var VarSet Valuation Hom)\n\nvariable {d} [DialectSignature d] [DecidableEq d.Ty]\n\nvariable {Γ : Ctxt d.Ty} {ty : d.Ty}\n\nabbrev MatchVarM (Δ Γ : Ctxt d.Ty) := (StateT (Mapping Δ Γ) Option)\n\nabbrev MatchVar (Δ Γ : Ctxt d.Ty)  := MatchVarM Δ Γ Unit\n\ndef MatchVarM.unifyVars {Δ Γ : Ctxt d.Ty} (v : Δ.Var t) (w : Γ.Var t) : MatchVar Δ Γ :=\n  fun ma =>\n    match ma.lookup ⟨_, v⟩ with\n    | some v =>\n      if v = w then\n        some ((), ma)\n      else\n        none\n    | none =>\n      some ((), AList.insert ⟨_, v⟩ w ma)\n\nopen MatchVarM\n\nvariable [DecidableEq d.Op]\n\ndef matchArg [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (matchLets : Lets d Δ_in .pure Δ_out) :\n    {l : List d.Ty} → HVector (Var Γ_out) l → HVector (Var Δ_out) l →\n    MatchVar Δ_in Γ_out\n  | _, .nil, .nil => return\n  | t::l, .cons vₗ vsₗ, .cons vᵣ vsᵣ => do\n      matchVar (t := t) lets vₗ matchLets vᵣ\n      matchArg lets matchLets vsₗ vsᵣ\n  termination_by l => (sizeOf matchLets, l.length + 1)\n\ndef matchVar {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty} {t : d.Ty} [DecidableEq d.Op]\n    (lets : Lets d Γ_in eff Γ_out) (v : Var Γ_out t) :\n    (matchLets : Lets d Δ_in .pure Δ_out) →\n    (w : Var Δ_out t) →\n    MatchVar Δ_in Γ_out\n   \n  | @Lets.var _ _ _ _ Δ_out ts matchLets matchExpr, w => by admit /- proof elided -/\n      | right w =>\n        exact matchVar lets v matchLets w\n      | left w => exact do\n        let ⟨ts', w', ie⟩ ← lets.getPureExpr v\n        if hs : ∃ h : ie.op = matchExpr.op, ie.regArgs = (h ▸ matchExpr.regArgs) then\n          have hts : Ctxt.ofList ts' = ts := by admit /- proof elided -/\n\nend\n\nsection MatchVar\n\nvariable [DecidableEq d.Op] {Γ_in Γ_out Δ_in Δ_out t te}\n          {lets : Lets d Γ_in eff Γ_out} {v : Var Γ_out t}\n          {matchLets : Lets d Δ_in .pure Δ_out}\n          {matchExpr : Expr d Δ_out .pure te}\n\nvariable (lets v matchLets w) (mapIn : Mapping _ _) in\n\nvariable (lets matchLets) {tys} (vs ws : HVector _ tys) (mapIn : Mapping _ _) in\n\nnamespace MatchVarResult\n\nvariable [TyDenote d.Ty] [∀ (t : d.Ty), Inhabited ⟦t⟧] in\n\nsection Left\n\nvariable {w : Δ_out.Var t}\n\nvariable {mapIn} (mapOut : MatchVarResult lets v (.var matchLets matchExpr) w.appendInr mapIn)\n\nend Left\n\nvariable {w : Var ⟨te⟩ _} {mapIn}\n\nend MatchVarResult\n\nend MatchVar\n\nsection SubsetEntries\n\ndef MatchVar.IsMonotone (f : MatchVar Δ Γ) : Prop :=\n    ∀ mapIn, ∀ mapOut ∈ f mapIn,\n      mapIn.entries ⊆ mapOut.2.entries\n\nopen MatchVar\n\nsection UnifyVars\n\nvariable {Δ Γ : Ctxt d.Ty} {t} (w : Δ.Var t) (v : Γ.Var t)\n\nend UnifyVars\n\nvariable [DecidableEq d.Op]\n\nend SubsetEntries\n\nnamespace MatchArgResult\n\nvariable [DecidableEq d.Op] {Γ_in Γ_out Δ_in Δ_out te}\n          {lets : Lets d Γ_in eff Γ_out}\n          {matchLets : Lets d Δ_in .pure Δ_out}\n          {matchExpr : Expr d Δ_out .pure te}\n          {u us}\n          {v : Γ_out.Var u} {vs : HVector Γ_out.Var us}\n          {w : Δ_out.Var u} {ws : HVector Δ_out.Var us}\n          {mapIn : Mapping _ _}\n          (mapOut : MatchArgResult lets matchLets (v ::ₕ vs) (w ::ₕ ws) mapIn)\n\nend MatchArgResult\n\nsection DenoteLemmas\n\nvariable [TyDenote d.Ty] [DecidableEq d.Op]\n\nvariable [∀ (t : d.Ty), Inhabited ⟦t⟧]\n\nvariable [Monad d.m] [LawfulMonad d.m] [DialectDenote d]\n\nsection DenoteIntoSubtype\n\nend DenoteIntoSubtype\n\nvariable {Γ_in Γ_out Δ_in Δ_out : Ctxt d.Ty}\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}", "target_theorem": "theorem mem_matchArg {Δ_out}\n    {lets : Lets d Γ_in eff Γ_out}\n    {matchLets : Lets d Δ_in .pure Δ_out}\n    {l : List d.Ty} {argsₗ : HVector (Var Γ_out) l}\n    {argsᵣ : HVector (Var Δ_out) l} {ma : Mapping Δ_in Γ_out}\n    {varMap : Mapping Δ_in Γ_out}\n    (hvarMap : ((), varMap) ∈ matchArg lets matchLets argsₗ argsᵣ ma)\n    {t' v'} : ⟨t', v'⟩ ∈ matchLets.varsOfVec argsᵣ → ⟨t', v'⟩ ∈ varMap :=", "ground_truth_proof": ":=\n  match l, argsₗ, argsᵣ/- , ma, varMap, hvarMap -/ with\n  | .nil, .nil, .nil /- , _, varMap, _ -/ => by simp [Lets.varsOfVec]\n  | .cons t ts, .cons vₗ argsₗ, .cons vᵣ args /-, ma, varMap, h -/ => by\n    simp only [matchArg, bind, Option.mem_def, StateT.bind, Option.bind_eq_some_iff] at hvarMap\n    rcases hvarMap with ⟨ma', h₁, h₂⟩\n    simp only [HVector.vars_cons, Finset.biUnion_insert, Finset.mem_union,\n      Finset.mem_biUnion, Sigma.exists, Lets.varsOfVec]\n    rintro (h | ⟨a, b, hab⟩)\n    · exact AList.keys_subset_keys_of_entries_subset_entries\n        (isMonotone_matchArg _ _ h₂)\n        (mem_matchVar (matchLets := matchLets) h₁ h)\n    · apply mem_matchArg h₂\n      unfold Lets.varsOfVec\n      apply Finset.mem_biUnion.mpr ⟨_, hab.1, hab.2⟩", "nesting_depth": 7, "transitive_dep_count": 104, "subset_aristotle": false, "category": "Compiler"}
{"id": 345, "thm_name": "transBV_spec", "thm_stmt": "@[simp]\nlemma transBV_spec {m : CNFA n} {res} {s : m.m.states} :\n    s' ∈ bv_to_set (m.m.transBV' res s a) ↔\n      (s' ∈ bv_to_set res ∨ s' ∈ m.m.tr s a)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/Constructions.lean", "imports": ["import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.Worklist", "import Mathlib.Tactic.ApplyFun", "import Mathlib.Data.Fintype.Prod"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "LawfulBEq", "module": "Init.Core"}], "used_repo_defs": [{"name": "State", "content": "abbrev State := Nat"}, {"name": "RawCNFA.states", "content": "def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax"}, {"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "CNFA", "content": "structure CNFA (n : Nat) where\n  m : RawCNFA (BitVec n)\n  wf : m.WF"}, {"name": "Std.HashSet.toSet", "content": "def Std.HashSet.toSet [BEq α] [Hashable α] (m : HashSet α) : Set α := { x | x ∈ m }"}], "lib_lemmas": [{"name": "Finset.mem_range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Nat.lt_of_le_of_lt", "module": "Init.Prelude"}, {"name": "Set.mem_insert_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "Set.union_singleton", "module": "Mathlib.Data.Set.Insert"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}], "repo_lemmas": [{"name": "Std.HashSet.fold_induction", "content": "theorem Std.HashSet.fold_induction [BEq α] [LawfulBEq α] [Hashable α]\n  {f : β → α → β} {m : HashSet α} {motive : β → Set α → Prop} :\n    motive b ∅ →\n    (∀ b x s, x ∉ s → motive b s → motive (f b x) (s ∪ {x})) →\n    motive (m.fold f b) m.toSet"}, {"name": "Std.HashSet.toSet_toList[BEq", "content": "theorem Std.HashSet.toSet_toList[BEq α] [LawfulBEq α] [Hashable α] (m : HashSet α) : m.toSet = { x | x ∈ m.toList }"}], "used_local_defs": [{"name": "bv_to_set", "content": "private def bv_to_set (bv : BitVec w) : Set State :=\n  { s | bv.getLsbD s }"}], "used_local_lemmas": [{"name": "bv_to_set_or", "content": "@[simp]\nlemma bv_to_set_or {m : CNFA n} (x y : BitVec m.m.stateMax) :\n    (s ∈ bv_to_set (x ||| y)) ↔ (s ∈ bv_to_set x ∨ s ∈ bv_to_set y)"}, {"name": "bv_to_set_shift", "content": "@[simp]\nlemma bv_to_set_shift (x s : Nat) :\n    (s ∈ bv_to_set (1#w <<< x)) ↔ (s = x ∧ x < w)"}], "local_ctx": "import Mathlib.Data.Fintype.Prod\n\nimport Blase.AutoStructs.Worklist\n\nimport Mathlib.Tactic.ApplyFun\n\nopen SetRel\n\nsection sink\n\nvariable {A : Type} [BEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nend sink\n\nsection generic_prod\n\nvariable {α} [BEq α] [Hashable α] [LawfulBEq α]\n\nvariable {β} [BEq β] [Hashable β] [LawfulBEq β]\n\nvariable {S₁ : Finset α} {S₂ : Finset β}\n\nvariable {γ} (f : S₁ → S₂ → γ) (hinj : Function.Injective2 f)\n\nvariable {m₁ : Std.HashSet α} (hm₁ : ∀ s₁ ∈ m₁, s₁ ∈ S₁)\n\nvariable {m₂ : Std.HashSet β} (hm₂ : ∀ s₂ ∈ m₂, s₂ ∈ S₂)\n\nend generic_prod\n\nsection product\n\nvariable {A : Type} [BEq A] [LawfulBEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nend product\n\nsection determinization\n\nvariable {A : Type} [BEq A] [LawfulBEq A] [Hashable A] [DecidableEq A] [FinEnum A]\n\nprivate def bv_to_set (bv : BitVec w) : Set State :=\n  { s | bv.getLsbD s }", "target_theorem": "@[simp]\nlemma transBV_spec {m : CNFA n} {res} {s : m.m.states} :\n    s' ∈ bv_to_set (m.m.transBV' res s a) ↔\n      (s' ∈ bv_to_set res ∨ s' ∈ m.m.tr s a) :=", "ground_truth_proof": ":= by\n  let motive (bv : BitVec m.m.stateMax) (X : Set State) :=\n    ∀ s' (hlt : s' < m.m.stateMax),\n      s' ∈ bv_to_set bv ↔ (s' ∈ bv_to_set res ∨ s' ∈ X)\n  suffices h : motive (m.m.transBV' res s a) (m.m.tr s a).toSet by\n    by_cases hlt : s' < m.m.stateMax\n    · simp_all [motive]\n    · constructor\n      · simp_all [bv_to_set]\n      · rintro (h | h); simp_all [bv_to_set]\n        apply m.wf.trans_tgt_lt at h; simp_all only [RawCNFA.states, not_lt, Finset.mem_range]\n        suffices _ : m.m.stateMax < m.m.stateMax by simp_all\n        exact Nat.lt_of_le_of_lt hlt h\n  apply Std.HashSet.fold_induction\n  · simp [motive]\n  rintro bv s S hnin ih s' hlt\n  simp only [bv_to_set_or, bv_to_set_shift, Set.union_singleton, Set.mem_insert_iff]\n  constructor\n  · rintro (hold | ⟨rfl, hin⟩)\n    · apply ih _ hlt |>.mp at hold; tauto\n    · tauto\n  · rintro (hold | rfl | hS)\n    · left; apply (ih _ hlt).mpr; tauto\n    · tauto\n    · left; apply (ih _ hlt).mpr; tauto", "nesting_depth": 2, "transitive_dep_count": 28, "subset_aristotle": false, "category": "Compiler"}
{"id": 346, "thm_name": "autOfRelation_accepts", "thm_stmt": "@[simp]\nlemma autOfRelation_accepts (r : Relation) :\n    r.absAutOfRelation.accepts = r.language", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "BitVec", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "cmp", "module": "Mathlib.Data.Ordering.Basic"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "NFA.stepSet", "module": "Mathlib.Computability.NFA"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "Language", "module": "Mathlib.Computability.Language"}, {"name": "List.Vector.ofFn", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.replicate", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "BitVec.ofFin", "module": "Init.Prelude"}, {"name": "BitVec.ule", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.ult", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.sle", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.slt", "module": "Init.Data.BitVec.Basic"}], "used_repo_defs": [{"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "RelationOrdering", "content": "inductive RelationOrdering\n| lt | le | gt | ge\nderiving Repr, Fintype"}, {"name": "NFA'", "content": "structure NFA' (n : Nat) where\n  σ : Type\n  M : NFA (BitVec n) σ"}, {"name": "Relation", "content": "inductive Relation\n| eq\n| signed (ord : RelationOrdering)\n| unsigned (ord : RelationOrdering)\nderiving Repr"}, {"name": "evalRelation", "content": "def evalRelation (rel : Relation) {w} (bv1 bv2 : BitVec w) : Prop :=\n  match rel with\n  | .eq => bv1 = bv2\n  | .signed .lt => bv1.slt bv2\n  | .signed .le => bv1.sle  bv2\n  | .signed .gt => bv2.slt bv1\n  | .signed .ge => bv2.sle bv1\n  | .unsigned .lt => bv1.ult bv2\n  | .unsigned .le => bv1.ule bv2\n  | .unsigned .gt => bv2.ult bv1\n  | .unsigned .ge => bv2.ule bv1"}, {"name": "bv2", "content": "def bv2 : BitVec 4 := BitVec.ofNat 4 1 "}, {"name": "bv1", "content": "def bv1 : BitVec 4 := BitVec.ofNat 4 5 "}, {"name": "accepts", "content": "def accepts (M : NFA' n) : Set (BitVecs n) := dec '' M.accepts'"}, {"name": "accepts'", "content": "def accepts' (M : NFA' n) : Set (BitVecs' n) := M.M.accepts"}, {"name": "BitVecs'", "content": "abbrev BitVecs' (n : Nat) := List (BitVec n)"}, {"name": "dec", "content": "@[simps]\ndef dec (bvs' : BitVecs' n) : BitVecs n where\n  w := bvs'.length\n  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "enc", "content": "def enc (bvs : BitVecs n) : BitVecs' n :=\n  (List.finRange bvs.w).map (fun i =>\n    BitVec.ofFn (fun (k : Fin n) => (bvs.bvs.get k)[i]))"}, {"name": "instFinEnumBV", "content": "instance instFinEnumBV : FinEnum (BitVec w) where\n  card := 2^w\n  equiv := {\n    toFun := fun x => x.toFin\n    invFun := fun x => BitVec.ofFin x\n    left_inv := by admit /- proof elided -/"}], "lib_lemmas": [{"name": "NFA.eval_append_singleton", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.eval_nil", "module": "Mathlib.Computability.NFA"}, {"name": "BitVec.toNat_eq", "module": "Init.Data.BitVec.Lemmas"}, {"name": "le_iff_lt_or_eq", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Nat.le_antisymm", "module": "Init.Prelude"}, {"name": "BitVec.toInt_inj", "module": "Init.Data.BitVec.Lemmas"}], "repo_lemmas": [{"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}], "used_local_defs": [{"name": "NFA.sa", "content": "def NFA.sa (_ : NFA α σ) := σ → Language α"}, {"name": "NFA.correct", "content": "structure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q"}, {"name": "BVRel", "content": "abbrev BVRel := ∀ ⦃w⦄, BitVec w → BitVec w → Prop"}, {"name": "BVNRel", "content": "abbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop"}, {"name": "NFA'.sa", "content": "def NFA'.sa (M : NFA' n) := M.σ → BVNRel n"}, {"name": "NFA'.sa2", "content": "def NFA'.sa2 (M : NFA' 2) := M.σ → BVRel"}, {"name": "langRel", "content": "def langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }"}, {"name": "langRel2", "content": "def langRel2 (R : BVRel) : Set (BitVecs 2) :=\n  { bvs | R (bvs.bvs.get 0) (bvs.bvs.get 1) }"}, {"name": "NFA'.correct", "content": "structure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))"}, {"name": "NFA'.correct2", "content": "structure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)"}, {"name": "NFA.autEq", "content": "def NFA.autEq : NFA (BitVec 2) Unit :=\n  { start := ⊤, accept := ⊤, step _ a := { _s' | if a = 0 ∨ a = 3 then true else false }}"}, {"name": "NFA'.autEq", "content": "def NFA'.autEq : NFA' 2 :=\n  ⟨Unit, NFA.autEq⟩"}, {"name": "NFA'.eqRel", "content": "def NFA'.eqRel : BVRel := fun _ x y => x = y"}, {"name": "NFA.unsignedCmpState", "content": "inductive NFA.unsignedCmpState : Type where\n| eq | gt | lt\nderiving Fintype, DecidableEq"}, {"name": "NFA.unsignedCmpStep", "content": "def NFA.unsignedCmpStep (q : NFA.unsignedCmpState) (a : BitVec 2) : List NFA.unsignedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [ .eq ] | .eq, 1 => [ .gt ] | .eq, 2 => [ .lt ]\n  | .gt, 0 => [ .gt ] | .gt, 1 => [ .gt ] | .gt, 3 => [ .gt ] | .gt, 2 => [ .lt ]\n  | .lt, 0 => [ .lt ] | .lt, 1 => [ .gt ] | .lt, 2 => [ .lt ] | .lt, 3 => [ .lt ]"}, {"name": "NFA.autUnsignedCmp", "content": "def NFA.autUnsignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) unsignedCmpState where\n  step s a := { s' | s' ∈ unsignedCmpStep s a }\n  start := {s | s = .eq }\n  accept := { s | s ∈ match cmp with | .lt => [unsignedCmpState.lt] | .le => [.lt, .eq] | .gt => [.gt] | .ge => [.gt, .eq] }"}, {"name": "NFA'.autUnsignedCmp", "content": "def NFA'.autUnsignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autUnsignedCmp cmp⟩"}, {"name": "RelationOrdering.urel", "content": "def RelationOrdering.urel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .le => fun _ bv1 bv2 => bv1.ule bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ge => fun _ bv1 bv2 => bv2.ule bv1"}, {"name": "NFA'.autUnsignedCmpSA", "content": "def NFA'.autUnsignedCmpSA (q : NFA.unsignedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1"}, {"name": "NFA.signedCmpState", "content": "inductive NFA.signedCmpState : Type where\n| eq | gt | lt | ltfin | gtfin\nderiving DecidableEq, Fintype"}, {"name": "NFA.signedCmpStep", "content": "def NFA.signedCmpStep (q : NFA.signedCmpState) (a : BitVec 2) : List NFA.signedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [.eq] | .eq, 1 => [.gt, .ltfin] | .eq, 2 => [ .lt, .gtfin ]\n  | .gt, 0 => [ .gt, .gtfin ] | .gt, 1 => [ .gt, .ltfin ] | .gt, 3 => [ .gt, .gtfin ] | .gt, 2 => [ .lt, .gtfin ]\n  | .lt, 0 => [ .lt, .ltfin ] | .lt, 1 => [ .gt, .ltfin ] | .lt, 2 => [ .lt, .gtfin ] | .lt, 3 => [ .lt, .ltfin ]\n  | .gtfin, _ => ∅\n  | .ltfin, _ => ∅"}, {"name": "NFA.autSignedCmp", "content": "def NFA.autSignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) signedCmpState where\n  step s a := { s' | s' ∈ signedCmpStep s a }\n  start := { s | s = signedCmpState.eq }\n  accept := { s | s ∈ match cmp with | .lt => [NFA.signedCmpState.ltfin] | .le => [.ltfin, .eq] | .gt => [.gtfin] | .ge => [.gtfin, .eq] }"}, {"name": "NFA'.autSignedCmp", "content": "def NFA'.autSignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autSignedCmp cmp⟩"}, {"name": "RelationOrdering.srel", "content": "def RelationOrdering.srel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.slt bv2\n  | .le => fun _ bv1 bv2 => bv1.sle bv2\n  | .gt => fun _ bv1 bv2 => bv2.slt bv1\n  | .ge => fun _ bv1 bv2 => bv2.sle bv1"}, {"name": "NFA'.autSignedCmpSA", "content": "def NFA'.autSignedCmpSA (q : NFA.signedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ltfin => fun _ bv1 bv2 => bv1.slt bv2\n  | .gtfin => fun _ bv1 bv2 => bv2.slt bv1"}, {"name": "Relation.absAutOfRelation", "content": "def Relation.absAutOfRelation (rel : Relation) : NFA' 2 :=\n  match rel with\n  | .eq => NFA'.autEq\n  | .unsigned cmp => NFA'.autUnsignedCmp cmp\n  | .signed cmp => NFA'.autSignedCmp cmp"}], "used_local_lemmas": [{"name": "NFA.correct_spec", "content": "lemma NFA.correct_spec {M : NFA α σ} {ζ : M.sa} {L : Language α} :\n    M.correct ζ L → M.accepts = L"}, {"name": "in_enc", "content": "@[simp]\nlemma in_enc : x ∈ enc '' S ↔ dec x ∈ S"}, {"name": "dec_snoc_in_langRel", "content": "@[simp]\nlemma dec_snoc_in_langRel {n} {R : BVNRel n} {w : BitVecs' n} {a : BitVec n} :\n    dec (w ++ [a]) ∈ langRel R ↔\n      R (List.Vector.ofFn fun k => .cons (a.getLsbD k) ((dec w).bvs.get k))"}, {"name": "NFA'.correct_spec", "content": "lemma NFA'.correct_spec {M : NFA' n} {ζ : M.sa} {L : BVNRel n} :\n    M.correct ζ L → M.accepts = langRel L"}, {"name": "NFA'.correct2_spec", "content": "lemma NFA'.correct2_spec {M : NFA' 2} {ζ : M.sa2} {L : BVRel} :\n    M.correct2 ζ L → M.accepts = langRel2 L"}, {"name": "NFA'.autEq_correct", "content": "lemma NFA'.autEq_correct : autEq.correct2 (fun _ => eqRel) eqRel"}, {"name": "BitVec.ule_iff_ult_or_eq", "content": "lemma BitVec.ule_iff_ult_or_eq {w : ℕ} (bv1 bv2 : BitVec w):\n    (bv1.ule bv2) = true ↔ (bv1.ult bv2) = true ∨ bv1 = bv2"}, {"name": "ucmp_tricho", "content": "@[simp]\nlemma ucmp_tricho {bv1 bv2 : BitVec w} : (bv2.ult bv1) = false → (bv1.ult bv2) = false → bv1 = bv2"}, {"name": "NFA'.autUnsignedCmp_correct", "content": "lemma NFA'.autUnsignedCmp_correct cmp : autUnsignedCmp cmp |>.correct2 autUnsignedCmpSA cmp.urel"}, {"name": "BitVec.sle_iff_slt_or_eq", "content": "private lemma BitVec.sle_iff_slt_or_eq {w : ℕ} (bv1 bv2 : BitVec w):\n    (bv1.sle bv2) = true ↔ (bv1.slt bv2) = true ∨ bv1 = bv2"}, {"name": "NFA'.autSignedCmp_correct", "content": "lemma NFA'.autSignedCmp_correct cmp : autSignedCmp cmp |>.correct2 autSignedCmpSA cmp.srel"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\ndef NFA.sa (_ : NFA α σ) := σ → Language α\n\nstructure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q\n\nabbrev BVRel := ∀ ⦃w⦄, BitVec w → BitVec w → Prop\n\nabbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop\n\ndef NFA'.sa (M : NFA' n) := M.σ → BVNRel n\n\ndef NFA'.sa2 (M : NFA' 2) := M.σ → BVRel\n\ndef langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }\n\ndef langRel2 (R : BVRel) : Set (BitVecs 2) :=\n  { bvs | R (bvs.bvs.get 0) (bvs.bvs.get 1) }\n\nstructure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))\n\nstructure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)\n\nsection fsm\n\nvariable {arity : Type} [FinEnum arity]\n\nopen BitStream in\n\nend fsm\n\nsection nfas_relations\n\ndef NFA.autEq : NFA (BitVec 2) Unit :=\n  { start := ⊤, accept := ⊤, step _ a := { _s' | if a = 0 ∨ a = 3 then true else false }}\n\ndef NFA'.autEq : NFA' 2 :=\n  ⟨Unit, NFA.autEq⟩\n\ndef NFA'.eqRel : BVRel := fun _ x y => x = y\n\ninductive NFA.unsignedCmpState : Type where\n| eq | gt | lt\nderiving Fintype, DecidableEq\n\ndef NFA.unsignedCmpStep (q : NFA.unsignedCmpState) (a : BitVec 2) : List NFA.unsignedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [ .eq ] | .eq, 1 => [ .gt ] | .eq, 2 => [ .lt ]\n  | .gt, 0 => [ .gt ] | .gt, 1 => [ .gt ] | .gt, 3 => [ .gt ] | .gt, 2 => [ .lt ]\n  | .lt, 0 => [ .lt ] | .lt, 1 => [ .gt ] | .lt, 2 => [ .lt ] | .lt, 3 => [ .lt ]\n\ndef NFA.autUnsignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) unsignedCmpState where\n  step s a := { s' | s' ∈ unsignedCmpStep s a }\n  start := {s | s = .eq }\n  accept := { s | s ∈ match cmp with | .lt => [unsignedCmpState.lt] | .le => [.lt, .eq] | .gt => [.gt] | .ge => [.gt, .eq] }\n\ndef NFA'.autUnsignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autUnsignedCmp cmp⟩\n\ndef RelationOrdering.urel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .le => fun _ bv1 bv2 => bv1.ule bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ge => fun _ bv1 bv2 => bv2.ule bv1\n\ndef NFA'.autUnsignedCmpSA (q : NFA.unsignedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n\ninductive NFA.signedCmpState : Type where\n| eq | gt | lt | ltfin | gtfin\nderiving DecidableEq, Fintype\n\ndef NFA.signedCmpStep (q : NFA.signedCmpState) (a : BitVec 2) : List NFA.signedCmpState :=\n  match q, a with\n  | .eq, 0 => [ .eq ] | .eq, 3 => [.eq] | .eq, 1 => [.gt, .ltfin] | .eq, 2 => [ .lt, .gtfin ]\n  | .gt, 0 => [ .gt, .gtfin ] | .gt, 1 => [ .gt, .ltfin ] | .gt, 3 => [ .gt, .gtfin ] | .gt, 2 => [ .lt, .gtfin ]\n  | .lt, 0 => [ .lt, .ltfin ] | .lt, 1 => [ .gt, .ltfin ] | .lt, 2 => [ .lt, .gtfin ] | .lt, 3 => [ .lt, .ltfin ]\n  | .gtfin, _ => ∅\n  | .ltfin, _ => ∅\n\ndef NFA.autSignedCmp (cmp: RelationOrdering) : NFA (BitVec 2) signedCmpState where\n  step s a := { s' | s' ∈ signedCmpStep s a }\n  start := { s | s = signedCmpState.eq }\n  accept := { s | s ∈ match cmp with | .lt => [NFA.signedCmpState.ltfin] | .le => [.ltfin, .eq] | .gt => [.gtfin] | .ge => [.gtfin, .eq] }\n\ndef NFA'.autSignedCmp (cmp: RelationOrdering) : NFA' 2 :=\n  ⟨_, NFA.autSignedCmp cmp⟩\n\ndef RelationOrdering.srel (cmp : RelationOrdering) : BVRel :=\n  match cmp with\n  | .lt => fun _ bv1 bv2 => bv1.slt bv2\n  | .le => fun _ bv1 bv2 => bv1.sle bv2\n  | .gt => fun _ bv1 bv2 => bv2.slt bv1\n  | .ge => fun _ bv1 bv2 => bv2.sle bv1\n\ndef NFA'.autSignedCmpSA (q : NFA.signedCmpState) : BVRel :=\n  match q with\n  | .eq => fun _ bv1 bv2 => bv1 = bv2\n  | .lt => fun _ bv1 bv2 => bv1.ult bv2\n  | .gt => fun _ bv1 bv2 => bv2.ult bv1\n  | .ltfin => fun _ bv1 bv2 => bv1.slt bv2\n  | .gtfin => fun _ bv1 bv2 => bv2.slt bv1\n\nend nfas_relations\n\ndef Relation.absAutOfRelation (rel : Relation) : NFA' 2 :=\n  match rel with\n  | .eq => NFA'.autEq\n  | .unsigned cmp => NFA'.autUnsignedCmp cmp\n  | .signed cmp => NFA'.autSignedCmp cmp", "target_theorem": "@[simp]\nlemma autOfRelation_accepts (r : Relation) :\n    r.absAutOfRelation.accepts = r.language :=", "ground_truth_proof": ":= by\n  simp [Relation.absAutOfRelation]\n  rcases r with ⟨⟩ | ⟨cmp⟩ | ⟨cmp⟩ <;> simp\n  · rw [NFA'.correct2_spec NFA'.autEq_correct]\n    simp [langRel2, NFA'.eqRel, evalRelation]\n  · rw [NFA'.correct2_spec (NFA'.autSignedCmp_correct cmp)]\n    simp [langRel2, evalRelation, RelationOrdering.srel]\n    cases cmp <;> simp\n  · rw [NFA'.correct2_spec (NFA'.autUnsignedCmp_correct cmp)]\n    simp [langRel2, evalRelation, RelationOrdering.urel]\n    cases cmp <;> simp", "nesting_depth": 6, "transitive_dep_count": 90, "subset_aristotle": false, "category": "Compiler"}
{"id": 347, "thm_name": "ScfFunctor.ForAddToMul.correct", "thm_stmt": "theorem correct : Com.denote (lhs v0) = Com.denote (rhs v0)", "lean_root": "lean-mlir", "rel_path": "SSA/Projects/Scf/ScfFunctor.lean", "imports": ["import LeanMLIR.Util", "import LeanMLIR.Framework", "import LeanMLIR.ErasedContext", "import Mathlib.Tactic.Linarith", "import Mathlib.Logic.Function.Iterate", "import LeanMLIR.Tactic"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "macro \"simp_peephole\" loc:(location)? : tactic =>", "content": "macro \"simp_peephole\" loc:(location)? : tactic =>\n  `(tactic|(\n      \n      first\n      | rw [funext_iff (α := Ctxt.Valuation _)] $[$loc]?\n      | change ∀ (_ : Ctxt.Valuation _), _ $[$loc]?\n      | skip\n\n      \n      simp (config := {failIfUnchanged := false}) only\n        [Expr.denote_castPureToEff, simp_denote] $[$loc]?\n      \n      \n      \n  ))"}, {"name": "cons", "content": "@[match_pattern]\ndef cons (hd : Ty) : Ctxt Ty → Ctxt Ty\n| ⟨tl⟩ => ⟨hd :: tl⟩"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "FlatCom", "content": "structure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "DialectDenote", "content": "class DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))"}, {"name": "sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "regSig", "content": "def regSig       := Signature.regSig ∘ s.signature"}, {"name": "RegionSignature", "content": "abbrev RegionSignature Ty := List (Ctxt Ty × List Ty)"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "effectKind", "content": "def effectKind   := Signature.effectKind ∘ s.signature"}, {"name": "returnTypes", "content": "def returnTypes  := Signature.returnTypes ∘ s.signature"}, {"name": "Dialect", "content": "structure Dialect where\n  (Op : Type)\n  (Ty : Type)\n  (m : Type → Type := Id)"}, {"name": "Op", "content": "inductive Op (q : Nat) (n : Nat)\n  | add : Op q n\n  | sub : Op q n\n  | mul : Op q n\n  | mul_constant : Op q n\n    \n    \n  | leading_term : Op q n\n  | monomial : Op q n\n  | monomial_mul : Op q n\n  | from_tensor : Op q n\n  | to_tensor : Op q n\n  | const (c : R q n) : Op q n\n  | const_int (c : Int) : Op q n\n  | const_idx (i : Nat) : Op q n"}, {"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "Valuation.instAppendHVector", "content": "@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V"}, {"name": "Ty", "content": "inductive Ty\n  | int\n   \n  | int2\n  deriving DecidableEq, Lean.ToExpr"}, {"name": "Op", "content": "inductive Op\n  | noop\n  | mkPair\n  | unPair\n  deriving Lean.ToExpr"}, {"name": "Com.ret", "content": "def Com.ret {Γ : Ctxt d.Ty} {ty : d.Ty} {eff : EffectKind} : Γ.Var ty → Com d Γ eff [ty] :=\n  (Com.rets [·]ₕ)"}, {"name": "Com.letPure", "content": "def Com.letPure (e : Expr d Γ .pure t) (body : Com d (e.outContext) eff u) : Com d Γ eff u :=\n  body.var (e.castPureToEff eff)"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "Expr.castPureToEff", "content": "def Expr.castPureToEff (eff : EffectKind) : Expr d Γ .pure t → Expr d Γ eff t :=\n  changeEffect (EffectKind.pure_le eff)"}, {"name": "Expr.changeEffect", "content": "def Expr.changeEffect {eff₁ eff₂ : EffectKind} (h : eff₁ ≤ eff₂) :\n    Expr d Γ eff₁ t → Expr d Γ eff₂ t\n  | Expr.mk op ty_eq eff_le args regArgs =>\n    have heff : DialectSignature.effectKind op ≤ eff₂ := by admit /- proof elided -/"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "Expr.regArgs", "content": "def Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)"}, {"name": "Regions", "content": "abbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig"}, {"name": "Expr.op", "content": "def Expr.op {Γ : Ctxt d.Ty} {eff : EffectKind} {ty} (e : Expr d Γ eff ty) : d.Op :=\n  Expr.casesOn e (fun op _ _ _ _ => op)"}, {"name": "Com.denote", "content": "def Com.denote : Com d Γ eff ty → (Γv : Valuation Γ) →\n    eff.toMonad d.m (HVector toType ty)\n  | .rets vs, Γv     => pure (vs.map Γv)\n  | .var e body, V => e.denote V >>= body.denote"}, {"name": "Lets.denote", "content": "def Lets.denote [DialectSignature d] [DialectDenote d] {Γ₂}\n    (lets : Lets d Γ₁ eff Γ₂) (V : Valuation Γ₁) : (eff.toMonad d.m <| Valuation Γ₂) :=\n  match lets with\n  | .nil          => return V\n  | .var lets' e  => lets'.denote V >>= e.denote"}, {"name": "HVector.denote", "content": "def HVector.denote :\n    {l : RegionSignature d.Ty} → (T : HVector (fun t => Com d t.1 .impure t.2) l) →\n    HVector (fun t => t.1.Valuation → EffectKind.impure.toMonad d.m (HVector toType t.2)) l\n  | _, .nil => HVector.nil\n  | _, .cons v vs => HVector.cons (v.denote) (HVector.denote vs)"}, {"name": "FlatCom.denote", "content": "@[simp] abbrev FlatCom.denote [DialectDenote d]\n    (flatCom : FlatCom d Γ eff Γ_out ts)\n    (V : Γ.Valuation) : eff.toMonad d.m (HVector toType ts) :=\n  flatCom.lets.denote V >>= (return flatCom.rets.map ·)"}, {"name": "RegionSignature.map", "content": "def RegionSignature.map (f : Ty → Ty') : RegionSignature Ty → RegionSignature Ty' :=\n  List.map fun ⟨Γ, ty⟩ => (Γ.map f, ty.map f)"}, {"name": "Signature.map", "content": "def Signature.map (f : Ty → Ty') : Signature Ty → Signature Ty' :=\n  fun sig => {\n    sig         := sig.sig.map f\n    regSig      := sig.regSig.map f\n    returnTypes := sig.returnTypes.map f\n  }"}, {"name": "map", "content": "def map (f : Ty₁ → Ty₂) : Ctxt Ty₁ → Ctxt Ty₂ :=\n  ofList ∘ (List.map f) ∘ toList"}, {"name": "Expr.denote", "content": "def Expr.denote {ty} (e : Expr d Γ eff ty) (V : Valuation Γ) :\n    eff.toMonad d.m (e.outContext.Valuation) :=\n  match e with\n  | ⟨op, ty_eq, heff, args, regArgs⟩ => do\n      let argsDenote := args.map V\n      let val ← EffectKind.liftEffect heff <| DialectDenote.denote op argsDenote regArgs.denote\n      return (val ++ V).cast (by admit /- proof elided -/\n      )"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) "}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "liftEffect", "content": "def liftEffect [Pure m] {e1 e2 : EffectKind} {α : Type}\n    (hle : e1 ≤ e2) (v1 : e1.toMonad m α) : e2.toMonad m α :=\n  match e1, e2, hle with\n    | .pure, .pure, _ | .impure, .impure, _ => v1\n    | .pure, .impure, _ => Pure.pure v1"}, {"name": "toMonad", "content": "def toMonad (e : EffectKind) (m : Type → Type) : Type → Type :=\n  match e with\n  | pure => Id\n  | impure => m"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "Com.ty", "content": "def Com.ty : Com d Γ eff [t] → d.Ty := fun _ => t"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "TyDenote.toType", "content": "notation \"⟦\" x \"⟧\" => TyDenote.toType x"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "Function.comp_apply", "module": "Init.Core"}, {"name": "Function.iterate_succ", "module": "Mathlib.Logic.Function.Iterate"}, {"name": "Nat.cast_add", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_one", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "Int.mul_comm", "module": "Init.Data.Int.Lemmas"}, {"name": "add_left_iterate", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "nsmul_eq_mul", "module": "Mathlib.Algebra.Ring.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "ScfFunctor.HasTy", "content": "class HasTy (d : Dialect) (DenotedTy : Type) [TyDenote d.Ty] [DialectSignature d] where\n    ty : d.Ty\n    denote_eq : toType ty = DenotedTy := by admit /- proof elided -/"}, {"name": "ScfFunctor.Scf.Op", "content": "inductive Scf.Op (Op' Ty' : Type) (m') [TyDenote Ty'] [DialectSignature ⟨Op', Ty', m'⟩]\n    [DialectDenote ⟨Op', Ty', m'⟩] : Type _\n  | coe (o : Op')\n  | iterate (k : ℕ) \n  | run (inputty : Ty')  \n  | if (inputty retty' : Ty')  \n  | for (ty : Ty')\n  deriving DecidableEq, Repr"}, {"name": "ScfFunctor.Scf", "content": "def Scf (d : Dialect) [TyDenote d.Ty] [DialectSignature d] [DialectDenote d] : Dialect where\n  Op := Scf.Op d.Op d.Ty d.m\n  Ty := d.Ty\n  m  := d.m"}, {"name": "ScfFunctor.Scf.LoopBody", "content": "abbrev LoopBody (t : Type) : Type := Int → t → t"}, {"name": "ScfFunctor.Scf.LoopBody.counterDecorator", "content": "def counterDecorator (δ : Int) (f : LoopBody α) : Int × α → Int × α :=\n  fun (i, v) => (i + δ, f i v)"}, {"name": "ScfFunctor.Scf.LoopBody.IndexInvariant", "content": "def IndexInvariant (f : LoopBody t) : Prop :=\n  ∀ (i j : Int) (v : t), f i v = f j v"}, {"name": "ScfFunctor.Scf.LoopBody.atZero", "content": "def atZero (f : LoopBody t) : t → t := fun v => f 0 v"}, {"name": "ScfFunctor.Arith.Ty", "content": "inductive Ty\n| int\n| bool\n| nat\n deriving DecidableEq, Repr"}, {"name": "ScfFunctor.Arith.Op", "content": "inductive Op\n  | add : Op  \n  | add_nat : Op  \n  | axpy : Op  \n  | neg : Op  \n  | const : (val : ℤ) → Op\n  | const_nat : (val : ℕ) → Op"}, {"name": "ScfFunctor.Arith.Arith", "content": "abbrev Arith : Dialect := {Op, Ty}"}, {"name": "ScfFunctor.ScfArith", "content": "abbrev ScfArith := Scf Arith"}, {"name": "ScfFunctor.cst", "content": "@[simp_denote] def cst (n : ℤ) : Expr ScfArith Γ .pure .int  :=\n  Expr.mk\n    (op := .coe <| .const n)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "ScfFunctor.add", "content": "@[simp_denote] def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .add)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "ScfFunctor.add_nat", "content": "@[simp_denote] def add_nat (e₁ e₂ : Var Γ .nat) : Expr ScfArith Γ .pure .nat :=\n  Expr.mk\n    (op := .coe <| .add_nat)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)"}, {"name": "ScfFunctor.axpy", "content": "@[simp_denote] def axpy {Γ : Ctxt _} (a : Var Γ .int) (x : Var Γ .nat) (b: Var Γ .int) :\n    Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .axpy)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons x <| .cons b .nil)\n    (regArgs := .nil)"}, {"name": "ScfFunctor.neg", "content": "@[simp_denote] def neg {Γ : Ctxt _} (a : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .neg)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "ScfFunctor.iterate", "content": "@[simp_denote] def iterate {Γ : Ctxt _} (k : Nat) (input : Var Γ Arith.Ty.int)\n    (body : Com ScfArith ⟨[.int]⟩ .impure .int) : Expr ScfArith Γ .impure .int :=\n  Expr.mk\n    (op := .iterate k)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons input .nil)\n    (regArgs := HVector.cons body HVector.nil)"}, {"name": "ScfFunctor.run", "content": "@[simp_denote]\ndef run {Γ : Ctxt _} {t : Arith.Ty} (v : Var Γ t) (body : Com ScfArith ⟨[t]⟩ .impure t) :\n    Expr ScfArith Γ .impure t :=\n  Expr.mk\n    (op := .run t)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons v .nil)\n    (regArgs := HVector.cons body <| HVector.nil)"}, {"name": "ScfFunctor.for_", "content": "@[simp_denote] def for_ {Γ : Ctxt Arith.Ty} {t : Arith.Ty}\n    (start step : Var Γ Arith.Ty.int)\n    (niter : Var Γ Arith.Ty.nat) (v : Var Γ t) (body : Com ScfArith ⟨[.int, t]⟩ .impure t) :\n      Expr ScfArith Γ .impure t :=\n  Expr.mk\n    (op := .for t)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons start <| .cons step <| .cons niter <| .cons v .nil)\n    (regArgs := HVector.cons body <| HVector.nil)"}, {"name": "ScfFunctor.ForAddToMul.lhs", "content": "def lhs (vincrement : ℤ) : Com ScfArith ⟨[  .nat,   .int]⟩ .impure .int :=\n    Com.letPure (cst 0) <|\n    Com.letPure  (cst 1) <|\n    Com.var (for_ (t := .int)\n                        ⟨  1, rfl⟩\n                        ⟨  0, rfl⟩\n                        ⟨  2, rfl⟩\n                        ⟨  3, rfl⟩ (\n      Com.letPure (cst vincrement) <|\n      Com.letPure (add ⟨0, rfl⟩ ⟨2, rfl⟩) \n      <| Com.ret ⟨0, rfl⟩)) <|\n  Com.ret ⟨0, rfl⟩"}, {"name": "ScfFunctor.ForAddToMul.rhs", "content": "def rhs (vincrement : ℤ) : Com ScfArith ⟨[  .nat,   .int]⟩ .pure .int :=\n  Com.var (cst vincrement) <|\n  Com.var (axpy ⟨0, rfl⟩ ⟨1, rfl⟩ ⟨2, rfl⟩) <|\n  Com.ret ⟨0, rfl⟩"}], "used_local_lemmas": [{"name": "ScfFunctor.Scf.LoopBody.eq_invariant_fn", "content": "theorem eq_invariant_fn\n    (f : LoopBody t) (g : t → t) (hf : ∀ (i : Int) (v : t), f i v = g v) :\n    LoopBody.IndexInvariant f ∧ atZero f = g"}, {"name": "ScfFunctor.Scf.LoopBody.IndexInvariant.eval'", "content": "@[simp]\ntheorem eval' {f : LoopBody t} (hf : LoopBody.IndexInvariant f) (i : Int) (v : t) :\n    f i v = f.atZero v"}], "local_ctx": "import Mathlib.Logic.Function.Iterate\n\nimport Mathlib.Tactic.Linarith\n\nimport LeanMLIR.Framework\n\nimport LeanMLIR.Tactic\n\nimport LeanMLIR.ErasedContext\n\nimport LeanMLIR.Util\n\nopen LeanMLIR\n\nopen Ctxt(Var)\n\nnamespace ScfFunctor\n\nopen TyDenote\n\nclass HasTy (d : Dialect) (DenotedTy : Type) [TyDenote d.Ty] [DialectSignature d] where\n    ty : d.Ty\n    denote_eq : toType ty = DenotedTy := by admit /- proof elided -/\n\ninductive Scf.Op (Op' Ty' : Type) (m') [TyDenote Ty'] [DialectSignature ⟨Op', Ty', m'⟩]\n    [DialectDenote ⟨Op', Ty', m'⟩] : Type _\n  | coe (o : Op')\n  | iterate (k : ℕ) \n  | run (inputty : Ty')  \n  | if (inputty retty' : Ty')  \n  | for (ty : Ty')\n  deriving DecidableEq, Repr\n\ndef Scf (d : Dialect) [TyDenote d.Ty] [DialectSignature d] [DialectDenote d] : Dialect where\n  Op := Scf.Op d.Op d.Ty d.m\n  Ty := d.Ty\n  m  := d.m\n\nnamespace Scf\n\nsection InheritedInstances\n\nvariable {d : Dialect} [TyDenote d.Ty] [DialectSignature d] [DialectDenote d]\n\nend InheritedInstances\n\nabbrev LoopBody (t : Type) : Type := Int → t → t\n\nnamespace LoopBody\n\ndef counterDecorator (δ : Int) (f : LoopBody α) : Int × α → Int × α :=\n  fun (i, v) => (i + δ, f i v)\n\ndef atZero (f : LoopBody t) : t → t := fun v => f 0 v\n\nend LoopBody\n\nnamespace LoopBody.IndexInvariant\n\nend LoopBody.IndexInvariant\n\nnamespace LoopBody.counterDecorator\n\nend LoopBody.counterDecorator\n\nvariable [TyDenote d.Ty] [DialectSignature d] [DialectDenote d]\n  [B : HasBool d] [N : HasNat d] [Z : HasInt d]\n\nopen Ctxt (Valuation) in\n\nend Scf\n\nnamespace Arith\n\ninductive Ty\n| int\n| bool\n| nat\n deriving DecidableEq, Repr\n\ninductive Op\n  | add : Op  \n  | add_nat : Op  \n  | axpy : Op  \n  | neg : Op  \n  | const : (val : ℤ) → Op\n  | const_nat : (val : ℕ) → Op\n\nabbrev Arith : Dialect := {Op, Ty}\n\nend Arith\n\nabbrev ScfArith := Scf Arith\n\nopen LeanMLIR.SingleReturnCompat (Com Expr)\n\n@[simp_denote] def cst (n : ℤ) : Expr ScfArith Γ .pure .int  :=\n  Expr.mk\n    (op := .coe <| .const n)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)\n\n@[simp_denote] def add {Γ : Ctxt _} (e₁ e₂ : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .add)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)\n\n@[simp_denote] def add_nat (e₁ e₂ : Var Γ .nat) : Expr ScfArith Γ .pure .nat :=\n  Expr.mk\n    (op := .coe <| .add_nat)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons e₁ <| .cons e₂ .nil)\n    (regArgs := .nil)\n\n@[simp_denote] def axpy {Γ : Ctxt _} (a : Var Γ .int) (x : Var Γ .nat) (b: Var Γ .int) :\n    Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .axpy)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons x <| .cons b .nil)\n    (regArgs := .nil)\n\n@[simp_denote] def neg {Γ : Ctxt _} (a : Var Γ .int) : Expr ScfArith Γ .pure .int :=\n  Expr.mk\n    (op := .coe <| .neg)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)\n\n@[simp_denote] def iterate {Γ : Ctxt _} (k : Nat) (input : Var Γ Arith.Ty.int)\n    (body : Com ScfArith ⟨[.int]⟩ .impure .int) : Expr ScfArith Γ .impure .int :=\n  Expr.mk\n    (op := .iterate k)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons input .nil)\n    (regArgs := HVector.cons body HVector.nil)\n\n@[simp_denote]\ndef run {Γ : Ctxt _} {t : Arith.Ty} (v : Var Γ t) (body : Com ScfArith ⟨[t]⟩ .impure t) :\n    Expr ScfArith Γ .impure t :=\n  Expr.mk\n    (op := .run t)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons v .nil)\n    (regArgs := HVector.cons body <| HVector.nil)\n\n@[simp_denote] def for_ {Γ : Ctxt Arith.Ty} {t : Arith.Ty}\n    (start step : Var Γ Arith.Ty.int)\n    (niter : Var Γ Arith.Ty.nat) (v : Var Γ t) (body : Com ScfArith ⟨[.int, t]⟩ .impure t) :\n      Expr ScfArith Γ .impure t :=\n  Expr.mk\n    (op := .for t)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons start <| .cons step <| .cons niter <| .cons v .nil)\n    (regArgs := HVector.cons body <| HVector.nil)\n\nnamespace ForAddToMul\n\ndef lhs (vincrement : ℤ) : Com ScfArith ⟨[  .nat,   .int]⟩ .impure .int :=\n    Com.letPure (cst 0) <|\n    Com.letPure  (cst 1) <|\n    Com.var (for_ (t := .int)\n                        ⟨  1, rfl⟩\n                        ⟨  0, rfl⟩\n                        ⟨  2, rfl⟩\n                        ⟨  3, rfl⟩ (\n      Com.letPure (cst vincrement) <|\n      Com.letPure (add ⟨0, rfl⟩ ⟨2, rfl⟩) \n      <| Com.ret ⟨0, rfl⟩)) <|\n  Com.ret ⟨0, rfl⟩\n\ndef rhs (vincrement : ℤ) : Com ScfArith ⟨[  .nat,   .int]⟩ .pure .int :=\n  Com.var (cst vincrement) <|\n  Com.var (axpy ⟨0, rfl⟩ ⟨1, rfl⟩ ⟨2, rfl⟩) <|\n  Com.ret ⟨0, rfl⟩\n\nopen Scf in\n\nopen Arith in", "target_theorem": "theorem correct : Com.denote (lhs v0) = Com.denote (rhs v0) :=", "ground_truth_proof": ":= by\n  unfold lhs rhs\n  simp_peephole\n  intros A B\n  rw [Scf.LoopBody.counterDecorator.const_index_fn_iterate (f' := fun v => v0 + v)] <;> try rfl\n  simp only [add_left_iterate, nsmul_eq_mul, Int.mul_comm]", "nesting_depth": 8, "transitive_dep_count": 102, "subset_aristotle": false, "category": "Compiler"}
{"id": 348, "thm_name": "Zipper.denote_insertPureCom_eq_of", "thm_stmt": "theorem denote_insertPureCom_eq_of [LawfulMonad d.m]\n    {zip : Zipper d Γ_in eff tys} {vs}\n    {newCom : Com d zip.Γ_mid .pure newTys} {V_in : Valuation Γ_in}\n    (h : ∀ V : zip.top.ValidDenotation,\n        newCom.denote V.val = vs.map V.val) :\n    (zip.insertPureCom vs newCom).denote V_in = zip.denote V_in", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/Framework/Zipper.lean", "imports": ["import LeanMLIR.Transforms.Rewrite.Match", "import LeanMLIR.LeanMLIR.Transforms.Rewrite.Match", "import LeanMLIR.LeanMLIR.HVector", "import LeanMLIR.LeanMLIR.Framework.Basic", "import LeanMLIR.Framework.Basic"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "Subtype.mk", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "Sigma.mk", "module": "Init.Core"}, {"name": "Subtype.val", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"llvm.and\"     : MLIR.Pretty.uniform_op", "content": "syntax \"llvm.and\"     : MLIR.Pretty.uniform_op\n\nsyntax \"llvm.ashr\"    : MLIR.Pretty.exact_op\n\nsyntax \"llvm.add\"     : MLIR.Pretty.overflow_op\n\nsyntax \"llvm.return\"  : MLIR.Pretty.uniform_op"}, {"name": "notation:50 x \" ≤ₛ \" y => BitVec.sle x y", "content": "notation:50 x \" ≤ₛ \" y => BitVec.sle x y"}, {"name": "notation:50 x \" >ᵤ \" y => BitVec.ult y x", "content": "notation:50 x \" >ᵤ \" y => BitVec.ult y x"}, {"name": "notation:50 x \" ≥ᵤ \" y => BitVec.ule y x", "content": "notation:50 x \" ≥ᵤ \" y => BitVec.ule y x"}, {"name": "notation:50 x \" <ᵤ \" y => BitVec.ult x y", "content": "notation:50 x \" <ᵤ \" y => BitVec.ult x y"}, {"name": "notation:50 x \" ≥ₛ \" y => BitVec.sle y x", "content": "notation:50 x \" ≥ₛ \" y => BitVec.sle y x"}, {"name": "notation:50 x \" <ₛ \" y => BitVec.slt x y", "content": "notation:50 x \" <ₛ \" y => BitVec.slt x y"}, {"name": "notation:50 x \" >ₛ \" y => BitVec.slt y x", "content": "notation:50 x \" >ₛ \" y => BitVec.slt y x"}, {"name": "notation:50 x \" ≤ᵤ \" y => BitVec.ule x y", "content": "notation:50 x \" ≤ᵤ \" y => BitVec.ule x y"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $resName:mlir_op_operand = $name:InstCombine.cmp_op_name $x, $y $[: $t]?) => do\n    let some opName := extractOpName name.raw\n      | Macro.throwUnsupported\n    let t ← t.getDM `(mlir_type| _)\n    `(mlir_op| $resName:mlir_op_operand = $opName ($x, $y) : ($t, $t) -> (i1) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $resName:mlir_op_operand = $name:InstCombine.int_cast_op $x : $t to $t') => do\n    let some opName := extractOpName name.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $resName:mlir_op_operand = $opName ($x) : ($t) -> $t')"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant( $x $[: $inner_type]?)\n      $[: $outer_type]? ) => do\n       \n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      let inner_type := inner_type.getD outer_type\n      `(mlir_op| $res:mlir_op_operand = \"llvm.mlir.constant\"()\n          {value = $x:neg_num : $inner_type} : () -> ($outer_type) )\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant( ${ $x:term }) $[: $t]?) => do\n      let t ← t.getDM `(mlir_type| _)\n      let x ← `(MLIR.AST.AttrValue.int $x [mlir_type| $t])\n      `(mlir_op| $res:mlir_op_operand = \"llvm.mlir.constant\"() {value = $$($x) } : () -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (true) $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (1 : i1) : i1)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (false) $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant (0 : i1) : i1)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant $x $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant($x $[: $t]?) $[: $t]?)\n  | `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant ${ $x:term } $[: $t]?) =>\n      `(mlir_op| $res:mlir_op_operand = llvm.mlir.constant($$($x) $[: $t]?) $[: $t]?)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.icmp $p $x, $y $[: $t]?) => do\n    let t ← t.getDM `(mlir_type| _)\n    match p.getString with\n      | \"eq\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.eq\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ne\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ne\" ($x, $y) : ($t, $t) -> (i1))\n      | \"slt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.slt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sle\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sle\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sgt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sgt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"sge\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.sge\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ult\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ult\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ule\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ule\" ($x, $y) : ($t, $t) -> (i1))\n      | \"ugt\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.ugt\" ($x, $y) : ($t, $t) -> (i1))\n      | \"uge\" => `(mlir_op| $res:mlir_op_operand = \"llvm.icmp.uge\" ($x, $y) : ($t, $t) -> (i1))\n      | _ => Macro.throwErrorAt p s!\"unexpected predicate {p.getString}\""}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = llvm.select $c, $x, $y $[: $t]?) => do\n    let t ← t.getDM `(mlir_type| _)\n    `(mlir_op| $res:mlir_op_operand = \"llvm.select\" ($c, $x, $y) : (i1, $t, $t) -> ($t))"}, {"name": "Lets", "content": "inductive Lets (Γ_in : Ctxt d.Ty) (eff : EffectKind) :\n    (Γ_out : Ctxt d.Ty) → Type where\n  | nil : Lets Γ_in eff Γ_in\n  | var (body : Lets Γ_in eff Γ_out) (e : Expr d Γ_out eff t) : Lets Γ_in eff e.outContext"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "EffectKind", "content": "inductive EffectKind\n| pure \n| impure \nderiving Repr, DecidableEq, Lean.ToExpr"}, {"name": "Com", "content": "inductive Com : Ctxt d.Ty → EffectKind → List d.Ty → Type where\n  | rets {Γ} {tys} {eff : EffectKind} (vs : HVector Γ.Var tys) : Com Γ eff tys\n  | var (e : Expr Γ eff ty) (body : Com (ty ++ Γ) eff β) : Com Γ eff β"}, {"name": "FlatCom", "content": "structure FlatCom (d : Dialect) [DialectSignature d]  (Γ_in : Ctxt d.Ty) (eff : EffectKind)\n    (Γ_out : Ctxt d.Ty) (ts : List d.Ty) where\n  lets : Lets d Γ_in eff Γ_out\n  rets : HVector Γ_out.Var ts"}, {"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "Hom.with", "content": "def Hom.with [DecidableEq Ty] {Γ₁ Γ₂ : Ctxt Ty} (f : Γ₁.Hom Γ₂) {ts}\n    (v₁ : HVector Γ₁.Var ts) (v₂ : HVector Γ₂.Var ts) : Γ₁.Hom Γ₂ :=\n  fun _ w =>\n    match v₁.idxOf? w with\n    | none => f w\n    | some ⟨i, h⟩ => (v₂.get i).cast h"}, {"name": "Hom", "content": "abbrev Hom (Γ Γ' : Ctxt Ty) := ⦃t : Ty⦄ → Γ.Var t → Γ'.Var t"}, {"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "Valuation.instAppendHVector", "content": "@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V"}, {"name": "map", "content": "def map (f : Ty₁ → Ty₂) : Ctxt Ty₁ → Ctxt Ty₂ :=\n  ofList ∘ (List.map f) ∘ toList"}, {"name": "Com.outContextHom", "content": "def Com.outContextHom (com : Com d Γ eff t) : Γ.Hom com.outContext :=\n  com.outContextDiff.toHom"}, {"name": "Com.outContext", "content": "def Com.outContext {Γ} : Com d Γ eff ts → Ctxt d.Ty :=\n  Com.rec' (motive := fun _ => Ctxt d.Ty)\n    (@fun Γ _ => Γ) \n    (fun _ _ r => r) "}, {"name": "Com.rec'", "content": "def Com.rec' {Γ} (com : Com d Γ eff t) : motive com :=\n   \n  Com.rec\n    (motive_1 := fun _ _ _ _ => PUnit)\n    (motive_2 := fun _ eff' t' c =>\n      (h₁ : eff = eff') → (h₂ : t = t') → motive (h₁ ▸ h₂ ▸ c))\n    (motive_3 := fun _ _ => PUnit)\n    (fun _ _ _ _ _ _ => ⟨⟩) \n    (fun v h₁ h₂ => \n      cast (by admit /- proof elided -/\n      ) <| rets (h₂ ▸ v))\n    (fun e' body' _ r' h₁ h₂ => \n      let e := h₁ ▸ e'\n      let body : Com _ _ eff t := cast (by admit /- proof elided -/\n      ) body'\n      let r : motive body := cast (by admit /- proof elided -/\n      ) (r' h₁ h₂)\n      cast (by admit /- proof elided -/\n      ) <| var e body r)\n    ⟨⟩\n    (fun _ _ _ _ => ⟨⟩)\n    com\n    rfl\n    rfl"}, {"name": "Valuation.cast", "content": "def Valuation.cast {Γ Δ : Ctxt Ty} (h : Γ = Δ) (V : Valuation Γ) : Valuation Δ :=\n  fun _ v => V <| v.castCtxt h.symm"}, {"name": "Com.outContextDiff", "content": "def Com.outContextDiff (com : Com d Γ eff ts) : Γ.Diff com.outContext :=\n  ⟨com.bvars, by admit /- proof elided -/\n      ⟩"}, {"name": "Expr.outContext", "content": "abbrev Expr.outContext (_ : Expr d Γ eff ts) : Ctxt d.Ty :=\n  ts ++ Γ"}, {"name": "Expr.bvars", "content": "@[simp, grind=] def Expr.bvars (e : Expr d Γ eff Δ) : Nat :=\n  (DialectSignature.returnTypes e.op).length"}, {"name": "returnTypes", "content": "def returnTypes  := Signature.returnTypes ∘ s.signature"}, {"name": "Signature", "content": "structure Signature (Ty : Type) where\n  mkEffectful ::\n  sig         : List Ty\n  regSig      : RegionSignature Ty\n  returnTypes : List Ty\n  effectKind  : EffectKind := .pure"}, {"name": "DialectSignature", "content": "class DialectSignature (d : Dialect) where\n  signature : d.Op → Signature d.Ty"}, {"name": "Com.bvars", "content": "def Com.bvars : Com d Γ eff t → Nat :=\n  Com.rec'\n           (fun _ => 0)\n      (fun e _body bodySize => e.bvars + bodySize)"}, {"name": "Diff", "content": "def Diff (Γ₁ Γ₂ : Ctxt Ty) : Type :=\n  {d : Nat // Diff.Valid Γ₁ Γ₂ d}"}, {"name": "Diff.Valid", "content": "@[simp]\nabbrev Diff.Valid (Γ₁ Γ₂ : Ctxt Ty) (d : Nat) : Prop :=\n  ∀ {i t}, Γ₁[i]? = some t → Γ₂[i+d]? = some t"}, {"name": "toHom", "content": "def toHom (d : Diff Γ₁ Γ₂) : Hom Γ₁ Γ₂ :=\n  fun _ v => ⟨v.val + d.val, d.property v.property⟩"}, {"name": "castCtxt", "content": "def castCtxt (h_eq : Γ = Δ) : Γ.Var ty → Δ.Var ty\n  | ⟨i, h⟩ => ⟨i, h_eq ▸ h⟩"}, {"name": "Com.denoteLets", "content": "def Com.denoteLets : (com : Com d Γ eff ty) → (Γv : Valuation Γ) →\n    eff.toMonad d.m (com.outContext.Valuation)\n  | .rets _, V => pure V\n  | .var e body, V =>\n      e.denote V >>= body.denoteLets >>= fun V =>\n        return V.cast (by admit /- proof elided -/\n        )"}, {"name": "DialectDenote", "content": "class DialectDenote (d : Dialect) [TyDenote d.Ty] [DialectSignature d] where\n  denote : (op : d.Op) → HVector toType (DialectSignature.sig op) →\n    (HVector (fun t : Ctxt d.Ty × List d.Ty =>\n      t.1.Valuation\n      → EffectKind.impure.toMonad d.m (HVector toType t.2))\n            (DialectSignature.regSig op)) →\n    ((DialectSignature.effectKind op).toMonad d.m\n      (HVector toType <| DialectSignature.returnTypes op))"}, {"name": "Lets.denote", "content": "def Lets.denote [DialectSignature d] [DialectDenote d] {Γ₂}\n    (lets : Lets d Γ₁ eff Γ₂) (V : Valuation Γ₁) : (eff.toMonad d.m <| Valuation Γ₂) :=\n  match lets with\n  | .nil          => return V\n  | .var lets' e  => lets'.denote V >>= e.denote"}, {"name": "sig", "content": "def sig          := Signature.sig ∘ s.signature"}, {"name": "regSig", "content": "def regSig       := Signature.regSig ∘ s.signature"}, {"name": "RegionSignature", "content": "abbrev RegionSignature Ty := List (Ctxt Ty × List Ty)"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "effectKind", "content": "def effectKind   := Signature.effectKind ∘ s.signature"}, {"name": "Dialect", "content": "structure Dialect where\n  (Op : Type)\n  (Ty : Type)\n  (m : Type → Type := Id)"}, {"name": "Op", "content": "inductive Op (q : Nat) (n : Nat)\n  | add : Op q n\n  | sub : Op q n\n  | mul : Op q n\n  | mul_constant : Op q n\n    \n    \n  | leading_term : Op q n\n  | monomial : Op q n\n  | monomial_mul : Op q n\n  | from_tensor : Op q n\n  | to_tensor : Op q n\n  | const (c : R q n) : Op q n\n  | const_int (c : Int) : Op q n\n  | const_idx (i : Nat) : Op q n"}, {"name": "HVector.denote", "content": "def HVector.denote :\n    {l : RegionSignature d.Ty} → (T : HVector (fun t => Com d t.1 .impure t.2) l) →\n    HVector (fun t => t.1.Valuation → EffectKind.impure.toMonad d.m (HVector toType t.2)) l\n  | _, .nil => HVector.nil\n  | _, .cons v vs => HVector.cons (v.denote) (HVector.denote vs)"}, {"name": "FlatCom.denote", "content": "@[simp] abbrev FlatCom.denote [DialectDenote d]\n    (flatCom : FlatCom d Γ eff Γ_out ts)\n    (V : Γ.Valuation) : eff.toMonad d.m (HVector toType ts) :=\n  flatCom.lets.denote V >>= (return flatCom.rets.map ·)"}, {"name": "RegionSignature.map", "content": "def RegionSignature.map (f : Ty → Ty') : RegionSignature Ty → RegionSignature Ty' :=\n  List.map fun ⟨Γ, ty⟩ => (Γ.map f, ty.map f)"}, {"name": "Signature.map", "content": "def Signature.map (f : Ty → Ty') : Signature Ty → Signature Ty' :=\n  fun sig => {\n    sig         := sig.sig.map f\n    regSig      := sig.regSig.map f\n    returnTypes := sig.returnTypes.map f\n  }"}, {"name": "Expr.denote", "content": "def Expr.denote {ty} (e : Expr d Γ eff ty) (V : Valuation Γ) :\n    eff.toMonad d.m (e.outContext.Valuation) :=\n  match e with\n  | ⟨op, ty_eq, heff, args, regArgs⟩ => do\n      let argsDenote := args.map V\n      let val ← EffectKind.liftEffect heff <| DialectDenote.denote op argsDenote regArgs.denote\n      return (val ++ V).cast (by admit /- proof elided -/\n      )"}, {"name": "Expr.op", "content": "def Expr.op {Γ : Ctxt d.Ty} {eff : EffectKind} {ty} (e : Expr d Γ eff ty) : d.Op :=\n  Expr.casesOn e (fun op _ _ _ _ => op)"}, {"name": "liftEffect", "content": "def liftEffect [Pure m] {e1 e2 : EffectKind} {α : Type}\n    (hle : e1 ≤ e2) (v1 : e1.toMonad m α) : e2.toMonad m α :=\n  match e1, e2, hle with\n    | .pure, .pure, _ | .impure, .impure, _ => v1\n    | .pure, .impure, _ => Pure.pure v1"}, {"name": "toMonad", "content": "def toMonad (e : EffectKind) (m : Type → Type) : Type → Type :=\n  match e with\n  | pure => Id\n  | impure => m"}, {"name": "Com.denote", "content": "def Com.denote : Com d Γ eff ty → (Γv : Valuation Γ) →\n    eff.toMonad d.m (HVector toType ty)\n  | .rets vs, Γv     => pure (vs.map Γv)\n  | .var e body, V => e.denote V >>= body.denote"}, {"name": "Com.ty", "content": "def Com.ty : Com d Γ eff [t] → d.Ty := fun _ => t"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "cast", "content": "def cast (h_eq : ty₁ = ty₂) : Γ.Var ty₁ → Γ.Var ty₂\n  | ⟨i, h⟩ => ⟨i, h_eq ▸ h⟩"}, {"name": "cast", "content": "def cast (h₁ : Γ = Γ') (h₂ : Δ = Δ') : Diff Γ Δ → Diff Γ' Δ'\n  | ⟨n, h⟩ => ⟨n, by admit /- proof elided -/\n  ⟩"}, {"name": "Com.castPureToEff", "content": "def Com.castPureToEff (eff : EffectKind) : Com d Γ .pure t → Com d Γ eff t :=\n  changeEffect (EffectKind.pure_le eff)"}, {"name": "Com.changeEffect", "content": "def Com.changeEffect {eff₁ eff₂ : EffectKind} (h : eff₁ ≤ eff₂) :\n    Com d Γ eff₁ t → Com d Γ eff₂ t := fun com =>\n  Com.rec' (motive := @fun Γ _ => eff₁ ≤ eff₂ → Com d Γ eff₂ t)\n            (fun v _h               => rets v)\n      (fun e _body castBody h => var (e.changeEffect h) (castBody h))\n    com h"}, {"name": "Expr.changeEffect", "content": "def Expr.changeEffect {eff₁ eff₂ : EffectKind} (h : eff₁ ≤ eff₂) :\n    Expr d Γ eff₁ t → Expr d Γ eff₂ t\n  | Expr.mk op ty_eq eff_le args regArgs =>\n    have heff : DialectSignature.effectKind op ≤ eff₂ := by admit /- proof elided -/"}, {"name": "Expr.args", "content": "def Expr.args {Γ ts} (e : Expr d Γ eff ts) :\n    HVector (Var Γ) (DialectSignature.sig e.op) :=\n  Expr.casesOn e (fun _ _ _ args _ => args)"}, {"name": "Expr.regArgs", "content": "def Expr.regArgs {Γ ts} (e : Expr d Γ eff ts) :\n    Regions d (DialectSignature.regSig e.op) :=\n  Expr.casesOn e (fun _ _ _ _ regArgs => regArgs)"}, {"name": "Regions", "content": "abbrev Regions (regSig : RegionSignature d.Ty) : Type :=\n  HVector (fun t => Com d t.1 .impure t.2) regSig"}, {"name": "com", "content": "def com := mkCom (d := InstCombine.MetaLLVM 0) bb0 |>.toOption |>.get (by admit /- proof elided -/\n)"}, {"name": "bb0", "content": "def bb0 : Region 0 := [mlir_region|\n{\n  ^bb0(%arg0: i32):\n    %0 = llvm.mlir.constant(8) : i32\n    %1 = llvm.mlir.constant(31) : i32\n    %2 = llvm.ashr %arg0, %1 : i32\n    %3 = llvm.and %2, %0 : i32\n    %4 = llvm.add %3, %2 : i32\n    llvm.return %4 : i32\n  }]"}, {"name": "Region", "content": "structure Region where\n  (name: String)\n  (args: List <| TypedSSAVal φ)\n  (ops: List Op)"}, {"name": "MetaLLVM", "content": "abbrev MetaLLVM (φ : Nat) : Dialect where\n  Op := MOp φ\n  Ty := MTy φ"}, {"name": "Ty", "content": "@[deprecated \"Use `LLVM.Ty` instead\" (since:=\"2025-04-30\")] abbrev Ty := LLVM.Ty"}, {"name": "Op", "content": "@[deprecated \"Use `LLVM.Op` instead\" (since:=\"2025-04-30\")] abbrev Op := LLVM.Op"}, {"name": "MOp", "content": "inductive MOp (φ : Nat) : Type\n  | unary   (w : Width φ) (op : MOp.UnaryOp φ) :  MOp φ\n  | binary  (w : Width φ) (op : MOp.BinaryOp) :  MOp φ\n  | select  (w : Width φ) : MOp φ\n  | icmp    (c : IntPred) (w : Width φ) : MOp φ\n   \n  | const (w : Width φ) (val : ℤ) : MOp φ\nderiving Repr, DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "binary", "content": "@[match_pattern] abbrev binary (w : Nat) (op : MOp.BinaryOp) : LLVM.Op :=\n  MOp.binary (.concrete w) op"}, {"name": "MOp.BinaryOp", "content": "inductive MOp.BinaryOp : Type\n  | and\n  | or   (disjoint : DisjointFlag := {disjoint := false} )\n  | xor\n  | shl  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | lshr (exact : ExactFlag := {exact := false} )\n  | ashr (exact : ExactFlag := {exact := false} )\n  | urem\n  | srem\n  | add  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | mul  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | sub  (nswnuw : NoWrapFlags := {nsw := false, nuw := false} )\n  | sdiv (exact : ExactFlag := {exact := false} )\n  | udiv (exact : ExactFlag := {exact := false} )\nderiving DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "LLVM", "content": "def LLVM : Dialect where\n  Op := MOp 0\n  Ty := MTy 0"}, {"name": "MTy", "content": "inductive MTy (φ : Nat)\n  | bitvec (w : Width φ) : MTy φ\n  deriving DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "Width", "content": "abbrev Width φ := ConcreteOrMVar Nat φ"}, {"name": "ConcreteOrMVar", "content": "inductive ConcreteOrMVar (α : Type u) (φ : Nat)\n  | concrete (a : α)\n  | mvar (i : Fin φ)\n  deriving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "const", "content": "@[match_pattern] abbrev const (w : Nat) (val : ℤ) : LLVM.Op        := MOp.const (.concrete w) val"}, {"name": "MOp.UnaryOp", "content": "inductive MOp.UnaryOp (φ : Nat) : Type\n  | neg\n  | not\n  | copy\n  | freeze\n  | trunc (w' : Width φ) (noWrapFlags : NoWrapFlags := {nsw := false, nuw := false} )\n  | zext  (w' : Width φ) (nneg : NonNegFlag := {nneg := false} )\n  | sext  (w' : Width φ)\nderiving Repr, DecidableEq, Inhabited, Lean.ToExpr"}, {"name": "select", "content": "@[simp_llvm_option]\ndef select {w : Nat} (c? : IntW 1) (x? y? : IntW w ) : IntW w := do\n  let c ← c?\n  if c = 1#1 then x? else y?"}, {"name": "IntW", "content": "def IntW w := PoisonOr <| BitVec w"}, {"name": "PoisonOr", "content": "structure PoisonOr (α : Type) where\n  val : α\n  poisonous : Bool\nderiving Inhabited, DecidableEq"}, {"name": "icmp", "content": "@[simp_llvm_option]\ndef icmp {w : Nat} (c : IntPred) (x y : IntW w) : IntW 1 := do\n  let x' ← x\n  let y' ← y\n  icmp? c x' y'"}, {"name": "icmp?", "content": "@[simp_llvm]\ndef icmp? {w : Nat} (c : IntPred) (x y : BitVec w) : IntW 1 :=\n  .value ↑(icmp' c x y)"}, {"name": "IntPred", "content": "inductive IntPred where\n  | eq\n  | ne\n  | ugt\n  | uge\n  | ult\n  | ule\n  | sgt\n  | sge\n  | slt\n  | sle\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "icmp'", "content": "@[simp_llvm]\ndef icmp' {w : Nat} (c : IntPred) (x y : BitVec w) : Bool :=\n  match c with\n    | .eq => (x == y)\n    | .ne => (x != y)\n    | .sgt => (x >ₛ y)\n    | .sge => (x ≥ₛ y)\n    | .slt => (x <ₛ y)\n    | .sle => (x ≤ₛ y)\n    | .ugt => (x >ᵤ y)\n    | .uge => (x ≥ᵤ y)\n    | .ult => (x <ᵤ y)\n    | .ule => (x ≤ᵤ y)"}, {"name": "mkCom", "content": "def mkCom [TransformTy d φ] [TransformExpr d φ] [TransformReturn d φ]\n  (reg : MLIR.AST.Region φ) :\n  ExceptM d (Σ (Γ : Ctxt d.Ty) (eff : EffectKind) (ty : _), Com d Γ eff ty) :=\n  match reg.ops with\n  | [] => throw <| .generic \"Ill-formed region (empty)\"\n  | coms => BuilderM.runWithEmptyMapping <| do\n    let Γ ← declareBindings ∅ reg.args\n    let com ← mkComHelper Γ coms\n    return ⟨Γ, com⟩"}, {"name": "FlatCom.denoteLets", "content": "def FlatCom.denoteLets (flatCom : FlatCom d Γ eff Γ_out t) (Γv : Γ.Valuation) :\n    eff.toMonad d.m <| Γ_out.Valuation :=\n  flatCom.lets.denote Γv"}, {"name": "Com.toLets", "content": "def Com.toLets (com : Com d Γ eff t) : Lets d Γ eff com.outContext :=\n  Lets.nil.addComToEnd com"}, {"name": "Lets.castPureToEff", "content": "def Lets.castPureToEff (eff : EffectKind) : Lets d Γ_in .pure Γ_out → Lets d Γ_in eff Γ_out\n  | .nil => .nil\n  | .var body e => .var (body.castPureToEff eff) (e.castPureToEff eff)"}, {"name": "Expr.castPureToEff", "content": "def Expr.castPureToEff (eff : EffectKind) : Expr d Γ .pure t → Expr d Γ eff t :=\n  changeEffect (EffectKind.pure_le eff)"}, {"name": "Expr.returnVars", "content": "def Expr.returnVars (e : Expr d Γ eff tys) : HVector e.outContext.Var tys :=\n  .ofFn _ _ <| fun i => (Var.ofFin i).appendInl"}, {"name": "ofFin", "content": "def ofFin (i : Fin Γ.length) : Γ.Var (Γ[i]) :=\n  ⟨i.val, by admit /- proof elided -/\n  ⟩"}, {"name": "Com.returnVars", "content": "def Com.returnVars : (com : Com d Γ eff ts) → HVector (Var com.outContext) ts\n  | .rets vs => vs\n  | .var _ body => body.returnVars"}, {"name": "Valuation.comap", "content": "def Valuation.comap {Γi Γo : Ctxt Ty} (Γiv: Γi.Valuation) (hom : Ctxt.Hom Γo Γi) : Γo.Valuation :=\n  fun _to vo => Γiv (hom vo)"}, {"name": "map", "content": "def map (f : ∀ (a : α), A a → B a) :\n    ∀ {l : List α}, HVector A l → HVector B l\n  | [],   .nil        => .nil\n  | t::_, .cons a as  => .cons (f t a) (map f as)"}, {"name": "HVectorLiteral", "content": "structure HVectorLiteral where\n  u : Level\n  v : Level\n  α : Q(Type $u)\n  A : Q($α → Type $v)\n  elems : Array ((a : Q($α)) × Q($A $a))"}, {"name": "Lets.ValidDenotation", "content": "def Lets.ValidDenotation (lets : Lets d Γ_in eff Γ_out) :=\n  { V // ∀ {t ts} {v : Var _ t} {w : Var ⟨ts⟩ t} {e} ,\n          lets.getPureExpr v = some ⟨ts, w, e⟩\n          → (e.pdenoteOp V)[w] = V v }"}, {"name": "Lets.denoteIntoSubtype", "content": "def Lets.denoteIntoSubtype (lets : Lets d Γ_in eff Γ_out) (Γv : Valuation Γ_in) :\n    eff.toMonad d.m lets.ValidDenotation :=\n  match lets with\n    | .nil => return ⟨Γv, by admit /- proof elided -/\n    ⟩\n    | @Lets.var _ _ _ _ Γ_out eTy body e => do\n        let ⟨Vout, h⟩ ← body.denoteIntoSubtype Γv\n        let Ve ← e.denoteOpIntoSubtype Vout\n        return ⟨Ve.val ++ Vout, by admit /- proof elided -/\n        ⟩"}, {"name": "Expr.denoteOpIntoSubtype", "content": "def Expr.denoteOpIntoSubtype (e : Expr d Γ_in eff tys) (Γv : Valuation Γ_in) :\n    eff.toMonad d.m {x // e.IsDenotationForPureE Γv x} :=\n  match h_pure : e.toPure? with\n    | some ePure => pure ⟨ePure.denoteOp Γv, by admit /- proof elided -/\n    ⟩\n    | none => (Subtype.mk · (by admit /- proof elided -/\n    )) <$> (e.denoteOp Γv)"}, {"name": "Expr.IsDenotationForPureE", "content": "abbrev Expr.IsDenotationForPureE (e : Expr d Γ eff tys) (Γv : Valuation Γ)\n    (x : HVector toType tys) : Prop :=\n  ∀ (ePure : Expr d Γ .pure tys), e.toPure? = some ePure → ePure.denoteOp Γv = x"}, {"name": "appendCases", "content": "@[elab_as_elim]\ndef appendCases\n    {motive : (Γ ++ Δ).Var t → Sort u}\n    (left : (v : Var Γ t) → motive (appendInl v))\n    (right : (v : Var Δ t) → motive (appendInr v)) :\n    (v : (Γ ++ Δ).Var t) → motive v\n  | ⟨idx, h⟩ =>\n    if hv : idx < Γ.length then\n      left ⟨idx, by admit /- proof elided -/\n      ⟩\n    else\n      let v' : Var _ _ := ⟨idx - Γ.length, by admit /- proof elided -/\n      ⟩\n      have eq : v'.appendInr = ⟨idx, h⟩ := by admit /- proof elided -/"}, {"name": "Expr.toPure?", "content": "def Expr.toPure? (e : Expr d Γ eff ty) : Option (Expr d Γ .pure ty) :=\n  if h : e.HasPureOp then\n    some <| e.toPure h\n  else\n    none"}, {"name": "Expr.toPure", "content": "def Expr.toPure (e : Expr d Γ eff ty) (h : e.HasPureOp) : Expr d Γ .pure ty :=\n  ⟨e.op, e.ty_eq, EffectKind.le_of_eq h, e.args, e.regArgs⟩"}, {"name": "Expr.ty", "content": "def Expr.ty : Expr d Γ eff [t] → d.Ty := fun _ => t"}, {"name": "Expr.HasPureOp", "content": "def Expr.HasPureOp (e : Expr d Γ eff ty) : Prop :=\n  DialectSignature.effectKind e.op = .pure"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}, {"name": "TyDenote.toType", "content": "notation \"⟦\" x \"⟧\" => TyDenote.toType x"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Com.denoteLets_eq", "content": "theorem Com.denoteLets_eq {com : Com d Γ eff t} : com.denoteLets = com.toLets.denote"}, {"name": "Lets.denote_var", "content": "@[simp] theorem Lets.denote_var {lets : Lets d Γ_in eff Γ_out} {e : Expr d Γ_out eff t} :\n    (lets.var e).denote = fun V_in => lets.denote V_in >>= e.denote"}, {"name": "castCtxt_rfl", "content": "@[simp, grind=] theorem castCtxt_rfl (h : Γ = Γ) : v.castCtxt h = v"}, {"name": "Com.returnVars_castPureToEff", "content": "@[simp] theorem Com.returnVars_castPureToEff (eff : _) (com : Com d Γ .pure tys) :\n    (com.castPureToEff eff).returnVars = com.returnVars.map (fun _ v => v.castCtxt (by simp))"}, {"name": "Valuation.comap_apply", "content": "@[simp] theorem Valuation.comap_apply {Γi Γo : Ctxt Ty}\n    (V : Γi.Valuation) (f : Ctxt.Hom Γo Γi) (v : Γo.Var t) :\n    V.comap f v = V (f v)"}, {"name": "Com.denoteLets_castPureToEff", "content": "@[simp] theorem Com.denoteLets_castPureToEff {com : Com d Γ .pure ty} :\n    denoteLets (com.castPureToEff eff)\n    = fun V => pure (com.denoteLets V |>.comap fun _ v => v.castCtxt (by simp))"}, {"name": "Com.denoteLets_returnVars", "content": "@[simp] theorem Com.denoteLets_returnVars (c : Com d Γ .pure tys) (V : Valuation Γ) :\n    c.returnVars.map (c.denoteLets V) = c.denote V"}, {"name": "Id.bind_eq'", "content": "theorem Id.bind_eq' (x : Id α) (f : α → id β) : x >>= f = f x"}, {"name": "Id.pure_eq'", "content": "theorem Id.pure_eq' (a : α) : (pure a : Id α) = a"}, {"name": "Ctxt.Valuation.comap_outContextHom_denoteLets", "content": "@[simp] theorem Ctxt.Valuation.comap_outContextHom_denoteLets {com : Com d Γ .pure ty} {V} :\n    Valuation.comap (com.denoteLets V) com.outContextHom = V"}, {"name": "Com.bvars_castPureToEff", "content": "@[simp] theorem Com.bvars_castPureToEff {com : Com d Γ .pure ty} :\n    (com.castPureToEff eff).bvars = com.bvars"}, {"name": "Valuation.comap_with", "content": "@[simp] theorem Valuation.comap_with [DecidableEq Ty] {Γ Δ : Ctxt Ty}\n    {V : Valuation Γ} {map : Δ.Hom Γ} {vs : HVector Δ.Var ty} {ws : HVector Γ.Var ty} :\n    V.comap (map.with vs ws) = (V.comap map).reassignVars vs (ws.map V)"}, {"name": "map_map", "content": "theorem map_map {A B C : α → Type*} {l : List α} (t : HVector A l)\n    (f : ∀ a, A a → B a) (g : ∀ a, B a → C a) :\n    (t.map f).map g = t.map (fun a v => g a (f a v))"}, {"name": "castCtxt_castCtxt", "content": "@[simp, grind=] theorem castCtxt_castCtxt (h₁ : Γ = Δ) (h₂ : Δ = Ξ) :\n    (v.castCtxt h₁).castCtxt h₂ = v.castCtxt (by simp [*])"}, {"name": "Lets.denote_eq_denoteIntoSubtype", "content": "theorem Lets.denote_eq_denoteIntoSubtype (lets : Lets d Γ_in eff Γ_out) (Γv : Valuation Γ_in) :\n    lets.denote Γv = Subtype.val <$> (lets.denoteIntoSubtype Γv)"}, {"name": "Expr.denoteOp_eq_denoteOpIntoSubtype", "content": "theorem Expr.denoteOp_eq_denoteOpIntoSubtype (e : Expr d Γ eff tys) (V : Valuation Γ) :\n    e.denoteOp V = Subtype.val <$> e.denoteOpIntoSubtype V"}], "used_local_defs": [{"name": "Zipper", "content": "structure Zipper (Γ_in : Ctxt d.Ty) (eff : EffectKind) (tys : List d.Ty) where\n   \n  {Γ_mid : Ctxt d.Ty}\n   \n  top : Lets d Γ_in eff Γ_mid\n   \n  bot : Com d Γ_mid eff tys"}, {"name": "Zipper.denote", "content": "def denote (zip : Zipper d Γ_in eff tys) (V_in : Valuation Γ_in) :\n    eff.toMonad d.m (HVector toType tys) :=\n  (zip.top.denote V_in) >>= zip.bot.denote"}, {"name": "Zipper.insertCom", "content": "def insertCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy) (newCom : Com d zip.Γ_mid eff newTy) :\n    Zipper d Γ_in eff ty :=\n  let top := zip.top.addComToEnd newCom\n  \n  let bot := zip.bot.changeVars <| newCom.outContextHom.with vs newCom.returnVars\n  \n  \n  { top, bot }"}, {"name": "Zipper.insertPureCom", "content": "def insertPureCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy)\n    (newCom : Com d zip.Γ_mid .pure newTy) : Zipper d Γ_in eff ty :=\n  zip.insertCom vs (newCom.castPureToEff eff)"}], "used_local_lemmas": [{"name": "Zipper.denote_insertCom", "content": "theorem denote_insertCom {zip : Zipper d Γ_in eff t₁} [LawfulMonad d.m]\n    {newCom : Com d zip.Γ_mid eff newTys} {vs : HVector zip.Γ_mid.Var newTys} :\n    (zip.insertCom vs newCom).denote = (fun (V_in : Valuation Γ_in) => do\n      let V_mid ← zip.top.denote V_in\n      let V_newMid ← newCom.denoteLets V_mid\n      zip.bot.denote\n        (V_newMid.comap <| newCom.outContextHom.with vs newCom.returnVars)\n      )"}, {"name": "Zipper.denote_insertPureCom", "content": "theorem denote_insertPureCom {zip : Zipper d Γ_in eff t₁} [LawfulMonad d.m]\n    {newCom : Com d zip.Γ_mid .pure newTys} {vs : HVector zip.Γ_mid.Var newTys} :\n    (zip.insertPureCom vs newCom).denote = (fun (V_in : Valuation Γ_in) => do\n      let V_mid ← zip.top.denote V_in\n      zip.bot.denote\n        ((Com.denoteLets newCom V_mid).comap <| newCom.outContextHom.with vs newCom.returnVars)\n      )"}], "local_ctx": "import LeanMLIR.Framework.Basic\n\nimport LeanMLIR.Transforms.Rewrite.Match\n\nopen Ctxt (Valuation Var Hom)\n\nvariable (d : Dialect) [DialectSignature d]\n\nstructure Zipper (Γ_in : Ctxt d.Ty) (eff : EffectKind) (tys : List d.Ty) where\n   \n  {Γ_mid : Ctxt d.Ty}\n   \n  top : Lets d Γ_in eff Γ_mid\n   \n  bot : Com d Γ_mid eff tys\n\nnamespace Zipper\n\nvariable {d}\n\nsection Denote\n\nvariable [TyDenote d.Ty] [DialectDenote d] [Monad d.m]\n\ndef denote (zip : Zipper d Γ_in eff tys) (V_in : Valuation Γ_in) :\n    eff.toMonad d.m (HVector toType tys) :=\n  (zip.top.denote V_in) >>= zip.bot.denote\n\nend Denote\n\nsection ToCom\n\nvariable {Γ_mid}\n\nvariable [TyDenote d.Ty] [DialectDenote d] [Monad d.m]\n\nend ToCom\n\nsection InsertCom\n\nvariable [DecidableEq d.Ty]\n\ndef insertCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy) (newCom : Com d zip.Γ_mid eff newTy) :\n    Zipper d Γ_in eff ty :=\n  let top := zip.top.addComToEnd newCom\n  \n  let bot := zip.bot.changeVars <| newCom.outContextHom.with vs newCom.returnVars\n  \n  \n  { top, bot }\n\ndef insertPureCom (zip : Zipper d Γ_in eff ty)\n    (vs : HVector zip.Γ_mid.Var newTy)\n    (newCom : Com d zip.Γ_mid .pure newTy) : Zipper d Γ_in eff ty :=\n  zip.insertCom vs (newCom.castPureToEff eff)\n\nsection Lemmas\n\nvariable [TyDenote d.Ty] [DialectDenote d] [Monad d.m]", "target_theorem": "theorem denote_insertPureCom_eq_of [LawfulMonad d.m]\n    {zip : Zipper d Γ_in eff tys} {vs}\n    {newCom : Com d zip.Γ_mid .pure newTys} {V_in : Valuation Γ_in}\n    (h : ∀ V : zip.top.ValidDenotation,\n        newCom.denote V.val = vs.map V.val) :\n    (zip.insertPureCom vs newCom).denote V_in = zip.denote V_in :=", "ground_truth_proof": ":= by\n  simp only [denote_insertPureCom, Valuation.comap_with,\n  Valuation.comap_outContextHom_denoteLets, Com.denoteLets_returnVars]\n  unfold Zipper.denote\n  simp [Lets.denote_eq_denoteIntoSubtype, h]", "nesting_depth": 12, "transitive_dep_count": 152, "subset_aristotle": false, "category": "Compiler"}
{"id": 349, "thm_name": "R.coeff_fromTensor", "thm_stmt": "theorem R.coeff_fromTensor (tensor : List Int)\n    (htensorlen : tensor.length < 2^n) :\n    (R.fromTensor (q := q) (n := n) tensor).coeff i = (tensor.getD i 0)", "lean_root": "lean-mlir", "rel_path": "SSA/Projects/FullyHomomorphicEncryption/Basic.lean", "imports": ["import Mathlib.Data.List.Basic", "import Mathlib.Data.List.ToFinsupp", "import Mathlib.RingTheory.Polynomial.Quotient", "import Mathlib.Data.ZMod.Defs", "import Mathlib.RingTheory.Ideal.Defs", "import Mathlib.RingTheory.Ideal.Basic", "import Mathlib.Data.ZMod.Basic", "import Mathlib.Algebra.MonoidAlgebra.Basic", "import LeanMLIR.Framework", "import Mathlib.Tactic.Cases", "import Mathlib.Algebra.Polynomial.RingDivision", "import Mathlib.Data.Finset.Sort"], "used_lib_defs": [{"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Ideal", "module": "Mathlib.RingTheory.Ideal.Defs"}, {"name": "Ideal.Quotient.mk", "module": "Mathlib.RingTheory.Ideal.Quotient.Defs"}, {"name": "Ideal.span", "module": "Mathlib.RingTheory.Ideal.Span"}, {"name": "Polynomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.monomial", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Function.surjInv", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Polynomial.coeff", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Int.cast", "module": "Init.Data.Int.Basic"}, {"name": "List.toFinsupp", "module": "Mathlib.Data.List.ToFinsupp"}, {"name": "Polynomial.ofFinsupp", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "Polynomial.degree", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "WithBot", "module": "Mathlib.Order.TypeTags"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}, {"name": "List.zipIdx", "module": "Init.Data.List.Basic"}, {"name": "Polynomial.Monic", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "List.getD", "module": "Init.Data.List.BasicAux"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "OfNat", "module": "Init.Prelude"}, {"name": "OfNat.ofNat", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Polynomial.toFinsupp", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Semiring", "module": "Mathlib.Algebra.Ring.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Nat.cast", "module": "Init.Data.Cast"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.length_map", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_cons", "module": "Init.Data.List.Basic"}, {"name": "List.map_nil", "module": "Init.Data.List.Basic"}, {"name": "List.toFinsupp_concat_eq_toFinsupp_add_single", "module": "Mathlib.Data.List.ToFinsupp"}, {"name": "Polynomial.ofFinsupp_add", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Polynomial.ofFinsupp_single", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "List.foldl_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.foldl_cons", "module": "Init.Data.List.Basic"}, {"name": "List.foldl_nil", "module": "Init.Data.List.Basic"}, {"name": "List.zipIdx_append", "module": "Init.Data.List.Nat.Range"}, {"name": "List.zipIdx_cons", "module": "Init.Data.List.Basic"}, {"name": "List.zipIdx_nil", "module": "Init.Data.List.Basic"}, {"name": "map_add", "module": "Mathlib.Algebra.Group.Hom.Defs"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "Nat.one_le_two_pow", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.sub_add_cancel", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.monic_X_pow_add", "module": "Mathlib.Algebra.Polynomial.Monic"}, {"name": "Function.surjInv_eq", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Ideal.mem_span_singleton", "module": "Mathlib.RingTheory.Ideal.Span"}, {"name": "Polynomial.modByMonic_eq_of_dvd_sub", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "Polynomial.modByMonic_eq_self_iff", "module": "Mathlib.Algebra.Polynomial.Div"}, {"name": "List.getD_map", "module": "Mathlib.Data.List.GetD"}, {"name": "List.toFinsupp_apply", "module": "Mathlib.Data.List.ToFinsupp"}, {"name": "Polynomial.coeff_ofFinsupp", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "Fin.pos'", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.cast_ofNat", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_pow", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Polynomial.degree_X_pow", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Polynomial.degree_add_eq_left_of_degree_lt", "module": "Mathlib.Algebra.Polynomial.Degree.Operations"}, {"name": "Polynomial.degree_one", "module": "Mathlib.Algebra.Polynomial.Degree.Definitions"}, {"name": "Finset.max_le", "module": "Mathlib.Data.Finset.Max"}, {"name": "Finset.mem_filter", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Finset.mem_insert", "module": "Mathlib.Data.Finset.Insert"}, {"name": "Finset.mem_range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Finset.range_add_one", "module": "Mathlib.Data.Finset.Range"}, {"name": "List.length_cons", "module": "Init.Data.List.Basic"}, {"name": "Nat.cast_add", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_one", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.le_add_one_iff", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.le_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "Polynomial.support_ofFinsupp", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "WithBot.coe_le_coe", "module": "Mathlib.Order.WithBot"}, {"name": "Nat.lt_of_le_of_lt", "module": "Init.Prelude"}, {"name": "WithBot.coe_strictMono", "module": "Mathlib.Order.WithBot"}, {"name": "WithBot.lt_def", "module": "Mathlib.Order.WithBot"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}, {"name": "R.fromPoly", "content": "abbrev R.fromPoly {q n : Nat} : (ZMod q)[X] →+* R q n := Ideal.Quotient.mk (Ideal.span {f q n})"}, {"name": "R.representative'", "content": "private noncomputable def R.representative' :\n    R q n → (ZMod q)[X] := Function.surjInv (R.surjective_fromPoly q n)"}, {"name": "R.representative", "content": "noncomputable def R.representative :\n    R q n → (ZMod q)[X] := fun x => R.representative' q n x %ₘ (f q n)"}, {"name": "R.coeff", "content": "noncomputable def R.coeff {q n} (a : R q n) (i : Nat) : ZMod q :=\n  Polynomial.coeff a.representative i"}, {"name": "R.monomial", "content": "noncomputable def R.monomial {q n : Nat} (c : ZMod q) (i : Nat): R q n :=\n  R.fromPoly (Polynomial.monomial i c)"}, {"name": "R.fromTensor", "content": "noncomputable def R.fromTensor {q n} (coeffs : List Int) : R q n :=\n  coeffs.zipIdx.foldl (init := 0) fun res (c, i) =>\n    res + R.monomial ↑c i"}, {"name": "R.fromTensorFinsupp", "content": "noncomputable def R.fromTensorFinsupp (q : Nat) (coeffs : List Int) : (ZMod q)[X] :=\n  Polynomial.ofFinsupp (List.toFinsupp (coeffs.map Int.cast))"}], "used_local_lemmas": [{"name": "f_deg_eq", "content": "theorem f_deg_eq : (f q n).degree = 2^n"}, {"name": "f_monic", "content": "theorem f_monic : Monic (f q n)"}, {"name": "R.fromPoly_rep'_eq_ideal", "content": "theorem R.fromPoly_rep'_eq_ideal :\n    forall a : (ZMod q)[X],\n      ∃ i ∈ Ideal.span {f q n}, (R.fromPoly (n:=n) a).representative' = a + i"}, {"name": "R.representative_fromPoly", "content": "theorem R.representative_fromPoly :\n    forall a : (ZMod q)[X], (R.fromPoly (n:=n) a).representative = a %ₘ (f q n)"}, {"name": "R.representative_fromPoly_eq", "content": "@[simp]\ntheorem R.representative_fromPoly_eq (x : (ZMod q)[X]) (DEGREE: x.degree < (f q n).degree) :\n   R.representative q n (R.fromPoly (n:=n) x) = x"}, {"name": "Polynomial.degree_toFinsupp", "content": "theorem Polynomial.degree_toFinsupp [Semiring M] [DecidableEq M]\n  (xs : List M) :\n  degree { toFinsupp := List.toFinsupp (l := xs) } ≤ List.length xs"}, {"name": "R.fromTensorFinsupp_degree", "content": "theorem R.fromTensorFinsupp_degree (q : Nat) (coeffs : List Int):\n  (R.fromTensorFinsupp q coeffs).degree ≤ coeffs.length"}, {"name": "R.fromTensorFinsupp_coeffs", "content": "theorem R.fromTensorFinsupp_coeffs (coeffs : List Int) :\n  Polynomial.coeff (fromTensorFinsupp q coeffs) i = ↑(List.getD coeffs i 0)"}, {"name": "R.fromTensorFinsupp_concat_monomial", "content": "theorem R.fromTensorFinsupp_concat_monomial (c : Int) (cs : List Int) :\n    (R.fromTensorFinsupp q (cs ++ [c])) =\n      (R.fromTensorFinsupp q cs) +\n        (Polynomial.monomial cs.length (Int.cast c : (ZMod q)))"}, {"name": "R.fromTensor_eq_fromTensorFinsupp_fromPoly", "content": "theorem R.fromTensor_eq_fromTensorFinsupp_fromPoly {coeffs : List Int} :\n    R.fromTensor (q := q) (n := n) coeffs =\n  R.fromPoly (q := q) (n := n) (R.fromTensorFinsupp q coeffs)"}], "local_ctx": "import Mathlib.RingTheory.Polynomial.Quotient\n\nimport Mathlib.RingTheory.Ideal.Defs\n\nimport Mathlib.RingTheory.Ideal.Basic\n\nimport Mathlib.Data.ZMod.Defs\n\nimport Mathlib.Data.ZMod.Basic\n\nimport Mathlib.Algebra.MonoidAlgebra.Basic\n\nimport Mathlib.Algebra.Polynomial.RingDivision\n\nimport Mathlib.Data.Finset.Sort\n\nimport Mathlib.Data.List.ToFinsupp\n\nimport Mathlib.Data.List.Basic\n\nimport Mathlib.Tactic.Cases\n\nimport LeanMLIR.Framework\n\nopen Polynomial -- for R[X] notation\n\nsection CommRing\n\nvariable (q t : Nat) [Fact (q > 1)] (n : Nat)\n\nnoncomputable def f : (ZMod q)[X] := X^(2^n) + 1\n\nabbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})\n\nabbrev R.fromPoly {q n : Nat} : (ZMod q)[X] →+* R q n := Ideal.Quotient.mk (Ideal.span {f q n})\n\nend CommRing\n\nsection Representative\n\nvariable (q t n : Nat)\n\nprivate noncomputable def R.representative' :\n    R q n → (ZMod q)[X] := Function.surjInv (R.surjective_fromPoly q n)\n\nnoncomputable def R.representative :\n    R q n → (ZMod q)[X] := fun x => R.representative' q n x %ₘ (f q n)\n\nvariable [Fact (q > 1)]\n\nend Representative\n\nsection Coeff\n\nnoncomputable def R.coeff {q n} (a : R q n) (i : Nat) : ZMod q :=\n  Polynomial.coeff a.representative i\n\nnoncomputable def R.monomial {q n : Nat} (c : ZMod q) (i : Nat): R q n :=\n  R.fromPoly (Polynomial.monomial i c)\n\nnoncomputable def R.fromTensor {q n} (coeffs : List Int) : R q n :=\n  coeffs.zipIdx.foldl (init := 0) fun res (c, i) =>\n    res + R.monomial ↑c i\n\nend Coeff\n\nsection FinnSupp\n\nvariable {q n : Nat}\n\nnoncomputable def R.fromTensorFinsupp (q : Nat) (coeffs : List Int) : (ZMod q)[X] :=\n  Polynomial.ofFinsupp (List.toFinsupp (coeffs.map Int.cast))\n\nend FinnSupp\n\nsection Tensor\n\nvariable {q n : Nat} [Fact (q > 1)]", "target_theorem": "theorem R.coeff_fromTensor (tensor : List Int)\n    (htensorlen : tensor.length < 2^n) :\n    (R.fromTensor (q := q) (n := n) tensor).coeff i = (tensor.getD i 0) :=", "ground_truth_proof": ":= by\n  rw [fromTensor_eq_fromTensorFinsupp_fromPoly]\n  have hfromTensorFinsuppDegree := fromTensorFinsupp_degree q tensor\n  rw [coeff, representative_fromPoly_eq]\n  apply fromTensorFinsupp_coeffs\n  case DEGREE =>\n    generalize htensor_degree : degree (fromTensorFinsupp q tensor) = tensor_degree\n    rw [f_deg_eq]\n    cases tensor_degree\n    case bot =>\n      rw [WithBot.lt_def]\n      simp\n      exists 2^n\n    norm_cast\n    case coe tensor_degree =>\n      /- I hate this coercion stuff -/\n      apply WithBot.coe_strictMono\n      norm_cast\n      have htrans : tensor_degree  ≤ List.length tensor := by\n        rw [htensor_degree] at hfromTensorFinsuppDegree\n        rw [← WithBot.coe_le_coe]\n        assumption\n      apply Nat.lt_of_le_of_lt htrans htensorlen", "nesting_depth": 4, "transitive_dep_count": 95, "subset_aristotle": false, "category": "Compiler"}
{"id": 350, "thm_name": "BitStream.ofBitVec_add", "thm_stmt": "theorem ofBitVec_add : ofBitVecSext (x + y) ≈ʷ (ofBitVecSext x) + (ofBitVecSext y)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/Fast/BitStream.lean", "imports": ["import Mathlib.Logic.Function.Iterate", "import Mathlib.Tactic.NormNum"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Add", "module": "Init.Prelude"}, {"name": "Add.add", "module": "Init.Prelude"}, {"name": "BitVec.carry", "module": "Init.Data.BitVec.Bitblast"}, {"name": "BitVec.getLsbD", "module": "Init.Data.BitVec.Basic"}, {"name": "HAdd", "module": "Init.Prelude"}, {"name": "HAdd.hAdd", "module": "Init.Prelude"}, {"name": "Int.succ", "module": "Mathlib.Data.Int.Init"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.add_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_two_eq_zero_or_one", "module": "Init.Data.Nat.Lemmas"}, {"name": "BitVec.add_eq", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.carry_succ", "module": "Init.Data.BitVec.Bitblast"}, {"name": "BitVec.getElem_add", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Nat.add_eq", "module": "Init.Data.Nat.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "BitStream.ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}, {"name": "BitStream.addAux", "content": "def addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "BitStream.add", "content": "def add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1"}, {"name": "BitStream.zero", "content": "abbrev zero   : BitStream := fun _ => false"}], "used_local_lemmas": [{"name": "BitStream.two_le_add_iff_odd_and_odd", "content": "private theorem two_le_add_iff_odd_and_odd (n m : Nat) :\n    2 ≤ n % 2 + m % 2 ↔ n % 2 = 1 ∧ m % 2 = 1"}, {"name": "BitStream.add_odd_iff_neq", "content": "private theorem add_odd_iff_neq (n m : Nat) :\n    (n + m) % 2 = 1 ↔ (n % 2 = 1) ≠ (m % 2 = 1)"}], "local_ctx": "import Mathlib.Tactic.NormNum\n\nimport Mathlib.Logic.Function.Iterate\n\nsection UpStream\n\nnamespace Int\n\nend Int\n\nend UpStream\n\ndef BitStream : Type := Nat → Bool\n\nnamespace BitStream\n\nsection Basic\n\nsection Lemmas\n\nend Lemmas\n\nend Basic\n\nsection OfNat\n\nend OfNat\n\nsection ToBitVec\n\nabbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb\n\nsection Lemmas\n\nend Lemmas\n\nend ToBitVec\n\nsection BitwiseOps\n\nsection Lemmas\n\nvariable {w : Nat}\n\nvariable (x y : BitStream) (i : Nat)\n\nvariable (x y : BitVec (w+1))\n\nend Lemmas\n\nend BitwiseOps\n\nsection Scan\n\nend Scan\n\nsection FindIndex\n\nsection Arith\n\ndef addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)\n\ndef add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1\n\nabbrev zero   : BitStream := fun _ => false\n\nsection Lemmas\n\nvariable {w : Nat} {x y : BitVec w} {a b a' b' : BitStream}\n\nlocal infix:20 \" ≈ʷ \" => EqualUpTo w", "target_theorem": "theorem ofBitVec_add : ofBitVecSext (x + y) ≈ʷ (ofBitVecSext x) + (ofBitVecSext y) :=", "ground_truth_proof": ":= by\n  intros n a\n  have add_lemma : ⟨(x + y).getLsbD n, BitVec.carry (n + 1) x y false ⟩ = (ofBitVecSext x).addAux (ofBitVecSext y) n := by\n    induction n\n    case zero =>\n      simp [addAux, BitVec.adcb, BitVec.carry, BitVec.getLsbD, a,\n        two_le_add_iff_odd_and_odd, add_odd_iff_neq]\n      bv_decide\n    case succ i ih =>\n      simp [addAux, ← ih (by omega), BitVec.adcb, a, BitVec.carry_succ, BitVec.getElem_add];\n  simp [HAdd.hAdd, Add.add, BitStream.add, ← add_lemma, a, -BitVec.add_eq, -Nat.add_eq]", "nesting_depth": 3, "transitive_dep_count": 24, "subset_aristotle": false, "category": "Compiler"}
{"id": 351, "thm_name": "Predicate.evalUlt_denote_true_iff", "thm_stmt": "theorem Predicate.evalUlt_denote_true_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalUlt (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = true ↔\n      (Term.denote w b vars) ≤ (Term.denote w a vars)", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/Fast/Defs.lean", "imports": ["import Blase.SingleWidth.Defs", "import Mathlib.Data.Fin.Basic", "import Mathlib.Data.Bool.Basic", "import Blase.Fast.BitStream", "import Blase.Blase.Fast.BitStream"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "BitVec.ult", "module": "Init.Data.BitVec.Basic"}, {"name": "BitVec.carry", "module": "Init.Data.BitVec.Bitblast"}], "used_repo_defs": [{"name": "syntax \"slt\" : MLIR.Pretty.uniform_op", "content": "syntax \"slt\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "borrow", "content": "def borrow (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).2"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "nxor", "content": "def nxor (a b : BitStream) : BitStream := fun i => a i == b i"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "falseIffNeq", "content": "abbrev falseIffNeq (n : Nat) : BitStream := fun i => decide (i == n)"}, {"name": "falseIffLt", "content": "abbrev falseIffLt (n : Nat) : BitStream := fun i => decide (i ≥ n)"}, {"name": "falseIffGe", "content": "abbrev falseIffGe (n : Nat) : BitStream := fun i => decide (i < n)"}, {"name": "falseIffEq", "content": "abbrev falseIffEq (n : Nat) : BitStream := fun i => decide (i != n)"}, {"name": "falseIffGt", "content": "abbrev falseIffGt (n : Nat) : BitStream := fun i => decide (i ≤ n)"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "falseIffLe", "content": "abbrev falseIffLe (n : Nat) : BitStream := fun i => decide (i > n)"}, {"name": "negOne", "content": "abbrev negOne : BitStream := fun _ => true"}, {"name": "shiftLeft", "content": "def shiftLeft (x : BitStream) (k : Nat) : BitStream :=\n  fun i => if i < k then false else x (i - k) "}, {"name": "ofNat", "content": "def ofNat (x : Nat) : BitStream :=\n  Nat.testBit x"}, {"name": "one", "content": "abbrev one    : BitStream := (· == 0)"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "Term.denote", "content": "def Term.denote (w : Nat) (t : Term) (vars : List (BitVec w)) : BitVec w :=\n  match t with\n  | ofNat n => BitVec.ofNat w n\n  | var n => vars.getD n default\n  | zero => 0#w\n  | negOne => -1#w\n  | one  => 1#w\n  | and a b => (a.denote w vars) &&& (b.denote w vars)\n  | or a b => (a.denote w vars) ||| (b.denote w vars)\n  | xor a b => (a.denote w vars) ^^^ (b.denote w vars)\n  | not a => ~~~ (a.denote w vars)\n  | add a b => (a.denote w vars) + (b.denote w vars)\n  | sub a b => (a.denote w vars) - (b.denote w vars)\n  | neg a => - (a.denote w vars)\n  \n  \n  | shiftL a n => (a.denote w vars) <<< n"}, {"name": "Predicate.denote", "content": "def Predicate.denote (p : Predicate) (w : Nat) (vars : List (BitVec w)) : Prop :=\n  match p with\n  | .width .ge k => k ≤ w \n  | .width .gt k => k < w \n  | .width .le k => w ≤ k\n  | .width .lt k => w < k\n  | .width .neq k => w ≠ k\n  | .width .eq k => w = k\n  | .binary .eq t₁ t₂ => t₁.denote w vars = t₂.denote w vars\n  | .binary .neq t₁ t₂ => t₁.denote w vars ≠ t₂.denote w vars\n  | .binary .sle  t₁ t₂ => ((t₁.denote w vars).sle (t₂.denote w vars)) = true\n  | .binary .slt  t₁ t₂ => ((t₁.denote w vars).slt (t₂.denote w vars)) = true\n  | .binary .ule  t₁ t₂ => ((t₁.denote w vars).ule (t₂.denote w vars)) = true\n  | .binary .ult  t₁ t₂ => (t₁.denote w vars).ult (t₂.denote w vars) = true\n  | .land  p q => p.denote w vars ∧ q.denote w vars\n  | .lor  p q => p.denote w vars ∨ q.denote w vars"}, {"name": "Predicate", "content": "inductive Predicate : Type where\n \n| width (wp : WidthPredicate) (n : Nat) : Predicate\n| binary (p : BinaryPredicate) (t₁ t₂ : Term)\n| land  (p q : Predicate) : Predicate\n| lor (p q : Predicate) : Predicate\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "WidthPredicate", "content": "inductive WidthPredicate\n| eq\n| neq\n| lt\n| le\n| gt\n| ge\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}, {"name": "toBitVec", "content": "def toBitVec (w : Nat) (x : BitStream) : BitVec w :=\n  match w with\n  | 0   => 0#0\n  | w+1 => (x.toBitVec w).cons (x w)"}], "lib_lemmas": [{"name": "BitVec.lt_def", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.of_length_zero", "module": "Init.Data.BitVec.Lemmas"}, {"name": "BitVec.ult_eq_not_carry", "module": "Init.Data.BitVec.Bitblast"}], "repo_lemmas": [{"name": "subAux_eq_BitVec_carry", "content": "@[simp] theorem subAux_eq_BitVec_carry (a b : BitStream) (w i : Nat) (hi : i < w) :\n    (a.subAux b i).2 = !(BitVec.carry (i + 1) (a.toBitVec w) ((~~~b).toBitVec w) true)"}], "used_local_defs": [{"name": "Term.eval", "content": "def Term.eval (t : Term) (vars : List BitStream) : BitStream :=\n  match t with\n  | var n       => vars.getD n default\n  | zero        => BitStream.zero\n  | one         => BitStream.one\n  | negOne      => BitStream.negOne\n  | ofNat n     => BitStream.ofNat n\n  | and t₁ t₂   => (t₁.eval vars) &&& (t₂.eval vars)\n  | or t₁ t₂    => (t₁.eval vars) ||| (t₂.eval vars)\n  | xor t₁ t₂   => (t₁.eval vars) ^^^ (t₂.eval vars)\n  | not t       => ~~~(t.eval vars)\n  | add t₁ t₂   => (Term.eval t₁ vars) + (Term.eval t₂ vars)\n  | sub t₁ t₂   => (Term.eval t₁ vars) - (Term.eval t₂ vars)\n  | neg t       => -(Term.eval t vars)\n\n\n  | shiftL t n  => BitStream.shiftLeft (Term.eval t vars) n"}, {"name": "Predicate.evalEq", "content": "def Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr"}, {"name": "Predicate.evalNeq", "content": "def Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd"}, {"name": "Predicate.evalLor", "content": "def Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)"}, {"name": "Predicate.evalLand", "content": "def Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)"}, {"name": "Predicate.evalUlt", "content": "def Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true"}, {"name": "Predicate.evalMsbEq", "content": "def Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false"}, {"name": "Predicate.evalSlt", "content": "def Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))"}, {"name": "Predicate.eval", "content": "def Predicate.eval (p : Predicate) (vars : List BitStream) : BitStream :=\n  match p with\n  | .width .eq n => BitStream.falseIffEq n\n  | .width .neq n => BitStream.falseIffNeq n\n  | .width .lt n => BitStream.falseIffLt n\n  | .width .le n => BitStream.falseIffLe n\n  | .width .gt n => BitStream.falseIffGt n\n  | .width .ge n => BitStream.falseIffGe n\n  | lor p q => Predicate.evalLor (p.eval vars) (q.eval vars)\n  | land p q => Predicate.evalLand (p.eval vars) (q.eval vars)\n  | binary .eq t₁ t₂ => Predicate.evalEq (t₁.eval vars) (t₂.eval vars)\n   \n  | binary .neq t1 t2 => Predicate.evalNeq (t1.eval vars) (t2.eval vars)\n  | binary .ult t₁ t₂ => Predicate.evalUlt (t₁.eval vars) (t₂.eval vars)\n  | binary .ule t₁ t₂ =>\n     Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalUlt (t₁.eval vars) (t₂.eval vars))\n  | binary .slt t₁ t₂ => Predicate.evalSlt (t₁.eval vars) (t₂.eval vars)\n  | binary .sle t₁ t₂ => Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalSlt (t₁.eval vars) (t₂.eval vars))"}], "used_local_lemmas": [{"name": "BitVec.lt_eq_decide_ult", "content": "private theorem BitVec.lt_eq_decide_ult {x y : BitVec w} : (x < y) = decide (x.ult y)"}, {"name": "Predicate.evalUlt_denote_false_iff", "content": "theorem Predicate.evalUlt_denote_false_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalUlt (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = false ↔\n    (Term.denote w a vars < Term.denote w b vars)"}], "local_ctx": "import Mathlib.Data.Bool.Basic\n\nimport Mathlib.Data.Fin.Basic\n\nimport Blase.Fast.BitStream\n\nimport Blase.SingleWidth.Defs\n\nopen Term\n\nopen BitStream in\n\ndef Term.eval (t : Term) (vars : List BitStream) : BitStream :=\n  match t with\n  | var n       => vars.getD n default\n  | zero        => BitStream.zero\n  | one         => BitStream.one\n  | negOne      => BitStream.negOne\n  | ofNat n     => BitStream.ofNat n\n  | and t₁ t₂   => (t₁.eval vars) &&& (t₂.eval vars)\n  | or t₁ t₂    => (t₁.eval vars) ||| (t₂.eval vars)\n  | xor t₁ t₂   => (t₁.eval vars) ^^^ (t₂.eval vars)\n  | not t       => ~~~(t.eval vars)\n  | add t₁ t₂   => (Term.eval t₁ vars) + (Term.eval t₂ vars)\n  | sub t₁ t₂   => (Term.eval t₁ vars) - (Term.eval t₂ vars)\n  | neg t       => -(Term.eval t vars)\n\n\n  | shiftL t n  => BitStream.shiftLeft (Term.eval t vars) n\n\ndef Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr\n\ndef Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd\n\ndef Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)\n\ndef Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)\n\ndef Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true\n\ndef Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false\n\ndef Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))\n\nopen BitStream in\n\ndef Predicate.eval (p : Predicate) (vars : List BitStream) : BitStream :=\n  match p with\n  | .width .eq n => BitStream.falseIffEq n\n  | .width .neq n => BitStream.falseIffNeq n\n  | .width .lt n => BitStream.falseIffLt n\n  | .width .le n => BitStream.falseIffLe n\n  | .width .gt n => BitStream.falseIffGt n\n  | .width .ge n => BitStream.falseIffGe n\n  | lor p q => Predicate.evalLor (p.eval vars) (q.eval vars)\n  | land p q => Predicate.evalLand (p.eval vars) (q.eval vars)\n  | binary .eq t₁ t₂ => Predicate.evalEq (t₁.eval vars) (t₂.eval vars)\n   \n  | binary .neq t1 t2 => Predicate.evalNeq (t1.eval vars) (t2.eval vars)\n  | binary .ult t₁ t₂ => Predicate.evalUlt (t₁.eval vars) (t₂.eval vars)\n  | binary .ule t₁ t₂ =>\n     Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalUlt (t₁.eval vars) (t₂.eval vars))\n  | binary .slt t₁ t₂ => Predicate.evalSlt (t₁.eval vars) (t₂.eval vars)\n  | binary .sle t₁ t₂ => Predicate.evalLor\n       (Predicate.evalEq (t₁.eval vars) (t₂.eval vars))\n       (Predicate.evalSlt (t₁.eval vars) (t₂.eval vars))\n\nsection Predicate\n\nend Predicate", "target_theorem": "theorem Predicate.evalUlt_denote_true_iff {w : Nat} (a b : Term) (vars : List (BitVec w)) :\n    evalUlt (a.eval (List.map .ofBitVecSext vars)) (b.eval (List.map .ofBitVecSext vars)) w = true ↔\n      (Term.denote w b vars) ≤ (Term.denote w a vars) :=", "ground_truth_proof": ":= by\n  obtain ⟨h₁, h₂⟩ := evalUlt_denote_false_iff a b vars\n  constructor\n  · intros h\n    by_contra h'\n    simp at h'\n    specialize (h₂ h')\n    simp [h₂] at h\n  · intros h\n    by_contra h'\n    simp at h'\n    specialize (h₁ h')\n    bv_omega", "nesting_depth": 6, "transitive_dep_count": 50, "subset_aristotle": false, "category": "Compiler"}
{"id": 352, "thm_name": "Ctxt.Valuation.reassignVars_eq", "thm_stmt": "@[simp] theorem Valuation.reassignVars_eq [DecidableEq Ty] (V : Γ.Valuation) :\n    V.reassignVars vs (vs.map V) = V", "lean_root": "lean-mlir", "rel_path": "LeanMLIR/LeanMLIR/ErasedContext.lean", "imports": ["import LeanMLIR.HVector", "import LeanMLIR/LeanMLIR/Tests/Tactic/ElimValuation.lean", "import SSA/Projects/PaperExamples/VariadicExample.lean", "import SSA/Projects/Tensor2D/Tensor2D.lean", "import Mathlib.Data.Fintype.Basic", "import SSA/Projects/RISCV64/Base.lean", "import SSA/Tests/Core/Print.lean", "import SSA/Projects/ModArith/Basic.lean", "import SSA/Projects/PaperExamples/PaperExamples.lean", "import SSA/Projects/Scf/ScfFunctor.lean", "import SSA/Projects/ISL/Explicit/Base.lean", "import SSA/Projects/CIRCT/Handshake/Handshake.lean", "import SSA/Projects/CIRCT/DC/DCSync.lean", "import SSA/Projects/CIRCT/Comb/Comb.lean", "import LeanMLIR/LeanMLIR/Transforms/CSE.lean", "import SSA/Projects/CIRCT/DCPlus/DCPlus.lean", "import LeanMLIR/LeanMLIR/Examples.lean", "import LeanMLIR/LeanMLIR/Transforms/DCE.lean", "import LeanMLIR.LeanMLIR.HVector", "import LeanMLIR/LeanMLIR/Dialects/LLVM/Basic.lean", "import SSA/Projects/Tensor1D/Tensor1D.lean", "import SSA/Projects/LLVMRiscV/LLVMAndRiscv.lean", "import SSA/Projects/SLLVM/Dialect/Basic.lean"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "StateM", "module": "Init.Control.State"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.Expr", "module": "Lean.Expr"}, {"name": "Stream'", "module": "Mathlib.Data.Stream.Defs"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Stream'.tail", "module": "Mathlib.Data.Stream.Defs"}, {"name": "Id.run", "module": "Init.Control.Id"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "List.replicate", "module": "Init.Data.List.Basic"}, {"name": "BitVec.ofInt", "module": "Init.Data.BitVec.Basic"}, {"name": "inferInstanceAs", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}], "used_repo_defs": [{"name": "HVector.decidableEqReg", "content": "protected instance HVector.decidableEqReg [DecidableEq d.Op] [DecidableEq d.Ty] :\n    ∀ {l : RegionSignature d.Ty}, DecidableEq (HVector (fun t => Com d t.1 .impure t.2) l)\n  | _, .nil, .nil => isTrue rfl\n  | _, .cons x₁ v₁, .cons x₂ v₂ =>\n    letI := HVector.decidableEqReg v₁ v₂\n    letI := Com.decidableEq x₁ x₂\n    decidable_of_iff (x₁ = x₂ ∧ v₁ = v₂) (by admit /- proof elided -/\n    )"}, {"name": "idxOf?", "content": "def idxOf? (x : A a) {as} [DecidableEq α] [∀ a, DecidableEq (A a)] :\n    HVector A as → Option { i : Fin <| as.length // as.get i = a }\n  | .nil => none\n  | .cons (a:=b) y ys =>\n      if h : ∃ h : a = b, x = h ▸ y then\n        some ⟨(0 : Fin <| _ + 1), h.1 ▸ rfl⟩\n      else\n        (ys.idxOf? x).map fun ⟨i, h⟩ =>\n          ⟨i.succ, by admit /- proof elided -/\n          ⟩"}, {"name": "map", "content": "def map (f : ∀ (a : α), A a → B a) :\n    ∀ {l : List α}, HVector A l → HVector B l\n  | [],   .nil        => .nil\n  | t::_, .cons a as  => .cons (f t a) (map f as)"}, {"name": "HVectorLiteral", "content": "structure HVectorLiteral where\n  u : Level\n  v : Level\n  α : Q(Type $u)\n  A : Q($α → Type $v)\n  elems : Array ((a : Q($α)) × Q($A $a))"}, {"name": "", "content": "instance : TyDenote Unit where toType := fun _ => Unit"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}, {"name": "Ty", "content": "inductive Ty (q : Nat) (n : Nat)\n  | index : Ty q n\n  | integer : Ty q n\n  | tensor : Ty q n\n  | polynomialLike : Ty q n\n  deriving DecidableEq, Repr"}, {"name": "", "content": "instance : TyDenote ISL.Ty where\n  toType := fun\n    | .regIndex => RegIndex\n    | .bits w => BitVec w\n\n \ndef_denote for ISL\n  | .regConst r   => [r]ₕ\n  | .regRead      => fun r regFile => ([regFile.read r]ₕ, regFile)\n  | .regWrite     => fun r v regFile => ([]ₕ, regFile.write r v)\n  | .bitsConst x  => [x]ₕ\n  | .bitsAdd _    => fun (x y : BitVec _) => [x + y]ₕ"}, {"name": "ExplicitISL", "content": "def ExplicitISL : Dialect where\n  Op := ISLOp\n  Ty := ExpTy\n  m := Id "}, {"name": "ExpTy", "content": "inductive ExpTy where\n  | isl : ISL.Ty → ExpTy\n  | regFile"}, {"name": "ISL", "content": "def ISL : Dialect where\n  Ty := ISLTy\n  Op := ISLOp\n  m := StateM RegFile"}, {"name": "ISLTy", "content": "inductive ISLTy\n  | regIndex\n  | bits (w : Nat)\n  deriving DecidableEq, ToExpr, Repr"}, {"name": "ISLOp", "content": "inductive ISLOp\n  | regConst (r : RegIndex)\n  | regRead\n  | regWrite\n  | bitsConst {w : Nat} (x : BitVec w)\n  | bitsAdd (w : Nat)\n  deriving DecidableEq, ToExpr, Repr"}, {"name": "RegIndex", "content": "def RegIndex := Fin RegFile.numRegisters"}, {"name": "RegFile.numRegisters", "content": "def RegFile.numRegisters := 32"}, {"name": "Dialect", "content": "structure Dialect where\n  (Op : Type)\n  (Ty : Type)\n  (m : Type → Type := Id)"}, {"name": "Op", "content": "inductive Op (q : Nat) (n : Nat)\n  | add : Op q n\n  | sub : Op q n\n  | mul : Op q n\n  | mul_constant : Op q n\n    \n    \n  | leading_term : Op q n\n  | monomial : Op q n\n  | monomial_mul : Op q n\n  | from_tensor : Op q n\n  | to_tensor : Op q n\n  | const (c : R q n) : Op q n\n  | const_int (c : Int) : Op q n\n  | const_idx (i : Nat) : Op q n"}, {"name": "RegFile", "content": "structure RegFile where\n  regs : Vector (BitVec RegFile.registerWidth) RegFile.numRegisters"}, {"name": "RegFile.registerWidth", "content": "def RegFile.registerWidth := 64"}, {"name": "read", "content": "def read (self : RegFile) (r : RegIndex) : BitVec registerWidth :=\n  self.regs[r.val]"}, {"name": "write", "content": "def write (self : RegFile) (r : RegIndex) (v : BitVec registerWidth) : RegFile :=\n  RegFile.mk <| self.regs.set r.val v"}, {"name": "instDCTyDenote", "content": "instance instDCTyDenote : TyDenote Ty where\ntoType := fun\n| Ty.tokenstream => CIRCTStream.DCPlusOp.TokenStream\n| Ty.tokenstream2 => CIRCTStream.DCPlusOp.TokenStream × CIRCTStream.DCPlusOp.TokenStream\n| Ty.valuestream w => CIRCTStream.DCPlusOp.ValueStream (BitVec w)\n| Ty.valuestream2 w => CIRCTStream.DCPlusOp.ValueStream (BitVec w) × CIRCTStream.DCPlusOp.ValueStream (BitVec w)\n| Ty.valuetokenstream w => CIRCTStream.DCPlusOp.ValueStream (BitVec w) × CIRCTStream.DCPlusOp.TokenStream\n| Ty.variadicvaluetokenstream w => CIRCTStream.DCPlusOp.VariadicValueStream w × CIRCTStream.DCPlusOp.TokenStream\n\n\ndef_denote for DCPlus where\n  | .fst => fun s => [s.fst]ₕ\n  | .fstVal _ => fun s => [s.fst]ₕ\n  | .fstValPure _ => fun s => [s.fst]ₕ\n  | .fstVal' _ => fun s => [s.fst.mapOpt (·[0]?)]ₕ\n  | .snd => fun s => [s.snd]ₕ\n  | .sndValPure _ => fun s => [s.snd]ₕ\n  | .pair _ => fun s₁ s₂ => [(s₁, s₂)]ₕ\n  | .sndVal _ => fun s => [s.snd]ₕ\n  | .sndVal' _ => fun s => [s.fst.mapOpt (·[0]?)]ₕ\n  | .tokVal' _ => fun s => [s.snd]ₕ\n  | .merge => fun s₁ s₂ => [CIRCTStream.DCPlusOp.merge s₁ s₂]ₕ\n  | .branch => fun s₁ s₂ => [CIRCTStream.DCPlusOp.branch s₁ s₂]ₕ\n  | .fork => fun s => [CIRCTStream.DCPlusOp.fork s]ₕ\n  | .forkVal => fun s => [CIRCTStream.DCPlusOp.forkVal s]ₕ\n  | .join => fun s₁ s₂ => [CIRCTStream.DCPlusOp.join s₁ s₂]ₕ\n  | .mux => fun s₁ s₂ c => [CIRCTStream.DCPlusOp.mux s₁ s₂ c]ₕ\n  | .muxVal => fun s₁ s₂ c => [CIRCTStream.DCPlusOp.muxVal s₁ s₂ c]ₕ\n  | .sink => fun s => [CIRCTStream.DCPlusOp.sink s]ₕ\n  | .source => [CIRCTStream.DCPlusOp.source]ₕ\n  | .sourceOnes => [CIRCTStream.DCPlusOp.sourceOnes]ₕ\n  | .cMerge => fun s₁ s₂ => [CIRCTStream.DCPlusOp.cMerge s₁ s₂]ₕ\n  | .supp => fun s₁ s₂ => [CIRCTStream.DCPlusOp.supp s₁ s₂]ₕ\n  | .not => fun s₁ => [CIRCTStream.DCPlusOp.not s₁]ₕ"}, {"name": "DCPlus", "content": "abbrev DCPlus : Dialect where\n  Op := Op\n  Ty := Ty"}, {"name": "def_signature for DCPlus where", "content": "def_signature for DCPlus where\n  | .fst => (Ty.tokenstream2) → (Ty.tokenstream)\n  | .fstVal t => (Ty.valuetokenstream t) → Ty.valuestream t\n  | .fstVal' t => (Ty.variadicvaluetokenstream t) → Ty.valuestream t\n  | .fstValPure t => (Ty.valuestream2 t) → Ty.valuestream t\n  | .snd => (Ty.tokenstream2) → (Ty.tokenstream)\n  | .sndValPure t => (Ty.valuestream2 t) → Ty.valuestream t\n  | .pair w => (Ty.valuestream w, Ty.valuestream w) → Ty.valuestream2 w\n  | .sndVal t => (Ty.valuetokenstream t) → Ty.tokenstream\n  | .sndVal' t => (Ty.variadicvaluetokenstream t) → Ty.valuestream t\n  | .tokVal' t => (Ty.variadicvaluetokenstream t) → Ty.tokenstream\n  | .merge => (Ty.tokenstream, Ty.tokenstream) → Ty.valuestream 1\n  | .branch => (Ty.valuestream 1, Ty.tokenstream) → Ty.tokenstream2\n  | .fork => (Ty.tokenstream) → Ty.tokenstream2\n  | .forkVal => (Ty.valuestream 1) → Ty.valuestream2 1\n  | .join => (Ty.tokenstream, Ty.tokenstream) → Ty.tokenstream\n  | .mux => (Ty.tokenstream, Ty.tokenstream, Ty.valuestream 1) → Ty.tokenstream\n  | .muxVal => (Ty.valuestream 1, Ty.valuestream 1, Ty.valuestream 1) → Ty.valuestream 1\n  | .sink => (Ty.tokenstream) → Ty.tokenstream\n  | .source => () → Ty.tokenstream\n  | .sourceOnes => () → Ty.valuestream 1\n  | .cMerge => (Ty.tokenstream, Ty.tokenstream) → Ty.valuetokenstream 1\n  | .supp => (Ty.valuestream 1, Ty.tokenstream) → Ty.tokenstream\n  | .not => (Ty.valuestream 1) → Ty.valuestream 1"}, {"name": "sndVal", "content": "def sndVal {Γ} (a : Γ.Var (MLIR2DCPlus.Ty.valuetokenstream r)) : Expr (DCPlus) Γ .pure (.tokenstream) :=\n    Expr.mk\n    (op := .sndVal r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "Ty", "content": "inductive Ty\n| tokenstream : Ty\n| tokenstream2 : Ty\n| valuestream (w : Nat) : Ty \n| valuestream2 (w : Nat) : Ty \n| valuetokenstream (w : Nat) : Ty \n| variadicvaluetokenstream (w : Nat) : Ty \nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "Context", "content": "def Context (Ty : Type) : Type :=\n  List (VarName × Ty)"}, {"name": "VarName", "content": "def VarName : Type := String"}, {"name": "hasType", "content": "def hasType (Γ : Context Ty) (v : VarName) (ty : Ty) : Prop :=\n  Γ.lookup v = some ty"}, {"name": "Expr.decidableEq", "content": "protected instance Expr.decidableEq [DecidableEq d.Op] [DecidableEq d.Ty] :\n    {Γ : Ctxt d.Ty} → {ty : List d.Ty} → DecidableEq (Expr d Γ eff ty)\n  | Γ, _, .mk op₁ rfl eff_le₁ arg₁ regArgs₁, .mk op₂ eq eff_le₂ arg₂ regArgs₂ =>\n    if ho : op₁ = op₂ then by\n      subst ho\n      letI := HVector.decidableEq arg₁ arg₂\n      letI := HVector.decidableEqReg regArgs₁ regArgs₂\n      exact decidable_of_iff (arg₁ = arg₂ ∧ regArgs₁ = regArgs₂) (by admit /- proof elided -/\n      )\n    else isFalse (by admit /- proof elided -/\n    )"}, {"name": "AffineExpr", "content": "inductive AffineExpr\n  | Var: String -> AffineExpr\n  deriving DecidableEq, Repr"}, {"name": "mux", "content": "def mux {Γ : Ctxt _}  (a b : Γ.Var (MLIR2DCPlus.Ty.tokenstream)) (c : Γ.Var (MLIR2DCPlus.Ty.valuestream 1)) : Expr (DCPlus) Γ .pure (.tokenstream) :=\n  Expr.mk\n    (op := .mux)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .cons c <| .nil)\n    (regArgs := .nil)"}, {"name": "branch", "content": "def branch {Γ : Ctxt _} (c : Γ.Var (MLIR2DCPlus.Ty.valuestream 1)) (a : Γ.Var (MLIR2DCPlus.Ty.tokenstream)) : Expr (DCPlus) Γ .pure (.tokenstream2) :=\n  Expr.mk\n    (op := .branch)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons c <| .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "merge", "content": "def merge {Γ : Ctxt _}  (a b : Γ.Var (MLIR2DCPlus.Ty.tokenstream)) : Expr (DCPlus) Γ .pure (.valuestream 1) :=\n  Expr.mk\n    (op := .merge)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "cMerge", "content": "def cMerge {Γ : Ctxt _} (a b : Γ.Var (MLIR2DCPlus.Ty.tokenstream)) : Expr (DCPlus) Γ .pure (.valuetokenstream 1) :=\n  Expr.mk\n    (op := .cMerge)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "fork", "content": "def fork {Γ : Ctxt _}  (a : Γ.Var (MLIR2DCPlus.Ty.tokenstream)) : Expr (DCPlus) Γ .pure (.tokenstream2) :=\n  Expr.mk\n    (op := .fork)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "fstVal", "content": "def fstVal (a : Γ.Var (MLIR2DCPlus.Ty.valuetokenstream r)) : Expr (DCPlus) Γ .pure (.valuestream r) :=\n    Expr.mk\n    (op := .fstVal r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "forkVal", "content": "def forkVal {Γ : Ctxt _}  (a : Γ.Var (MLIR2DCPlus.Ty.valuestream 1)) : Expr (DCPlus) Γ .pure (.valuestream2 1) :=\n  Expr.mk\n    (op := .forkVal)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "join", "content": "def join {Γ : Ctxt _}  (a b : Γ.Var (MLIR2DCPlus.Ty.tokenstream)) : Expr (DCPlus) Γ .pure (.tokenstream) :=\n  Expr.mk\n    (op := .join)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "Op", "content": "inductive Op\n| fst\n| snd\n| pair (w : Nat)\n| fstVal (w : Nat)\n| fstValPure (w : Nat)\n| sndVal (w : Nat)\n| sndValPure (w : Nat)\n| fstVal' (w : Nat)\n| sndVal' (w : Nat)\n| tokVal' (w : Nat)\n| fork\n| forkVal\n| join\n| merge\n| mux\n| muxVal\n| cMerge\n| branch\n| source\n| sourceOnes\n| sink\n| supp\n| not\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "fstValPure", "content": "def fstValPure (a : Γ.Var (MLIR2DCPlus.Ty.valuestream2 r)) : Expr (DCPlus) Γ .pure (.valuestream r) :=\n    Expr.mk\n    (op := .fstValPure r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "sink", "content": "def sink {Γ : Ctxt _} (a : Γ.Var (MLIR2DCPlus.Ty.tokenstream)) : Expr (DCPlus) Γ .pure (.tokenstream) :=\n  Expr.mk\n    (op := .sink)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "supp", "content": "def supp {Γ : Ctxt _} (a : Γ.Var (MLIR2DCPlus.Ty.tokenstream)) (c : Γ.Var (MLIR2DCPlus.Ty.valuestream 1)) : Expr (DCPlus) Γ .pure (.tokenstream) :=\n  Expr.mk\n    (op := .supp)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons c <| .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "muxVal", "content": "def muxVal {Γ : Ctxt _}  (a b c : Γ.Var (MLIR2DCPlus.Ty.valuestream 1)) : Expr (DCPlus) Γ .pure (.valuestream 1) :=\n  Expr.mk\n    (op := .muxVal)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .cons c <| .nil)\n    (regArgs := .nil)"}, {"name": "sndValPure", "content": "def sndValPure (a : Γ.Var (MLIR2DCPlus.Ty.valuestream2 r)) : Expr (DCPlus) Γ .pure (.valuestream r) :=\n    Expr.mk\n    (op := .sndValPure r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "sourceOnes", "content": "def sourceOnes : Expr (DCPlus) Γ .pure (.valuestream 1) :=\n  Expr.mk\n    (op := .sourceOnes)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "snd", "content": "def snd {Γ} (a : Γ.Var (MLIR2DCPlus.Ty.tokenstream2)) : Expr (DCPlus) Γ .pure (.tokenstream) :=\n    Expr.mk\n    (op := .snd)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "source", "content": "def source : Expr (DCPlus) Γ .pure (.tokenstream) :=\n  Expr.mk\n    (op := .source)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "fst", "content": "def fst (a : Γ.Var (MLIR2DCPlus.Ty.tokenstream2)) : Expr (DCPlus) Γ .pure (.tokenstream) :=\n    Expr.mk\n    (op := .fst)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "TokenStream", "content": "def TokenStream := Stream Unit"}, {"name": "Stream", "content": "def Stream (β : Type) := Stream' (Option β)"}, {"name": "ValueStream", "content": "def ValueStream := Stream"}, {"name": "VariadicValueStream", "content": "def VariadicValueStream (w : Nat) := CIRCTStream.Stream (List (BitVec w))"}, {"name": "branch", "content": "def branch (c : ValueStream (BitVec 1)) (x : TokenStream) : TokenStream × TokenStream  :=\n  Stream.corec₂ (β := ValueStream (BitVec 1) × TokenStream) (c, x) fun ⟨c, x⟩ =>\n    Id.run <| do\n      match c 0 with\n        | none => (none, none, (c.tail, x))\n        | some x₀ =>\n          if x₀.msb then\n            (some (), none, (c.tail, x.tail))\n          else\n            (none, some (), (c.tail, x.tail))"}, {"name": "tail", "content": "def tail : Stream α → Stream α := Stream'.tail"}, {"name": "corec₂", "content": "def corec₂ {β} (s0 : β) (f : β → (Option α × Option γ × β)) : Stream α × Stream γ :=\n  let f' := fun b =>\n    let x := f b\n    (x.fst, x.snd.fst)\n  let g := (f · |>.snd.snd)\n  let x := Stream'.corec f' g s0\n  (\n    fun i => (x i).fst,\n    fun i => (x i).snd,\n  )"}, {"name": "cMerge", "content": "def cMerge (x y : TokenStream) : ValueStream (BitVec 1) × TokenStream :=\n  Stream.corec₂ (β := TokenStream × TokenStream) (x, y) fun ⟨x, y⟩ =>\n    match x 0, y 0 with\n    | some x', some _ => (some 1, some x', (x.tail, y))\n    | some x', none => (some 1, some x', (x.tail, y.tail))\n    | none, some y' => (some 0, some y', (x.tail, y.tail))\n    | none, none => (none, none, (x.tail, y.tail))"}, {"name": "fork", "content": "def fork (x : TokenStream) : TokenStream × TokenStream  :=\n  Stream.corec₂ (β := TokenStream) x\n    fun x => Id.run <| do\n      (x 0, x 0, x.tail)"}, {"name": "forkVal", "content": "def forkVal (x : ValueStream (BitVec 1)) : ValueStream (BitVec 1) × ValueStream (BitVec 1)  :=\n  Stream.corec₂ (β := ValueStream (BitVec 1)) x\n    fun x => Id.run <| do\n      (x 0, x 0, x.tail)"}, {"name": "join", "content": "def join (x y : TokenStream) : TokenStream  :=\n  Stream.corec (β := TokenStream × TokenStream) (x, y) fun ⟨x, y⟩ =>\n    match x 0, y 0 with\n    | some _, some _ => (some (), (x.tail, y.tail))\n    | some _, none => (none, (x, y.tail))\n    | none, some _ => (none, (x.tail, y))\n    | none, none => (none, (x.tail, y.tail))"}, {"name": "corec", "content": "def corec {α} {β} (s0 : β) (f : β → (Option α × β)) : Stream α :=\n  Stream'.corec (f · |>.fst) (f · |>.snd) s0"}, {"name": "merge", "content": "def merge (x y : TokenStream) : ValueStream (BitVec 1) :=\n  Stream.corec (β := TokenStream × TokenStream) (x, y) fun ⟨x, y⟩ =>\n    match x 0, y 0 with\n    | some _, some _ => (some 1, (x.tail, y))\n    | some _, none => (some 1, (x.tail, y.tail))\n    | none, some _ => (some 0, (x.tail, y.tail))\n    | none, none => (none, (x.tail, y.tail))"}, {"name": "mux", "content": "def mux (x y : TokenStream) (c : ValueStream (BitVec 1)): TokenStream :=\n  Stream.corec (β := TokenStream × TokenStream × ValueStream (BitVec 1)) (x, y, c)\n  fun ⟨x, y, c⟩ =>\n    match (c 0) with\n    | none => (none, x, y, c.tail) \n    | some 1#1 =>\n      match (x 0) with\n      | none => (none, x.tail, y, c) \n      | some _ => (some (), x.tail, y, c.tail) \n    | some 0#1 =>\n      match (y 0) with\n      | none => (none, x, y.tail, c) \n      | some _ => (some (), x, y.tail, c.tail) "}, {"name": "muxVal", "content": "def muxVal (x y c : ValueStream (BitVec 1)): ValueStream (BitVec 1) :=\n  Stream.corec (β := ValueStream (BitVec 1) × ValueStream (BitVec 1) × ValueStream (BitVec 1)) (x, y, c)\n  fun ⟨x, y, c⟩ =>\n    match (c 0) with\n    | none => (none, x, y, c.tail) \n    | some 1#1 =>\n      match (x 0) with\n      | none => (none, x.tail, y, c) \n      | some e => (some e, x.tail, y, c.tail) \n    | some 0#1 =>\n      match (y 0) with\n      | none => (none, x, y.tail, c) \n      | some e => (some e, x, y.tail, c.tail) "}, {"name": "not", "content": "def not (c : ValueStream (BitVec 1)) : (ValueStream (BitVec 1)) :=\n  Stream.corec (β := ValueStream (BitVec 1)) c fun c =>\n  match c 0 with\n  | some 1 => (some 0, c.tail)\n  | some 0 => (some 1, c.tail)\n  | _ => (none, c.tail)"}, {"name": "sink", "content": "def sink (x : TokenStream) : TokenStream :=\n  Stream.corec (β := TokenStream) x fun x => (none, x.tail)"}, {"name": "source", "content": "def source : TokenStream :=\n  Stream.corec () fun () => (some (), ())"}, {"name": "sourceOnes", "content": "def sourceOnes : ValueStream (BitVec 1) :=\n  Stream.corec () fun () => (1#1, ())"}, {"name": "supp", "content": "def supp (c : ValueStream (BitVec 1)) (x : TokenStream) : TokenStream := (branch c x).snd"}, {"name": "mapOpt", "content": "def mapOpt {α β : Type} (s : Stream α) (f : α → (Option β)) : Stream β :=\n  fun i => (s i).bind f"}, {"name": "", "content": "instance : TyDenote Ty where toType\n  | .felt => Fin BabyBear"}, {"name": "BabyBear", "content": "def BabyBear := 2^31 - 2^27 + 1"}, {"name": "Ty", "content": "inductive Ty\n  | felt"}, {"name": "instDCTyDenote", "content": "instance instDCTyDenote : TyDenote Ty where\ntoType := fun\n| Ty.tokenstream => CIRCTStream.DCOp.TokenStream\n| Ty.tokenstream2 => CIRCTStream.DCOp.TokenStream × CIRCTStream.DCOp.TokenStream\n| Ty.valuestream w => CIRCTStream.DCOp.ValueStream (BitVec w)\n| Ty.valuestream2 w => CIRCTStream.DCOp.ValueStream (BitVec w) × CIRCTStream.DCOp.ValueStream (BitVec w)\n| Ty.valuetokenstream w => CIRCTStream.DCOp.ValueStream (BitVec w) × CIRCTStream.DCOp.TokenStream\n| Ty.variadicvaluetokenstream w => CIRCTStream.DCOp.VariadicValueStream w × CIRCTStream.DCOp.TokenStream\n\n\ndef_denote for DC where\n  | .fst => fun s => [s.fst]ₕ\n  | .fstVal _ => fun s => [s.fst]ₕ\n  | .fstVal' _ => fun s => [s.fst.mapOpt (·[0]?)]ₕ\n  | .snd => fun s => [s.snd]ₕ\n  | .pair _ => fun s₁ s₂ => [(s₁, s₂)]ₕ\n  | .sndVal _ => fun s => [s.snd]ₕ\n  | .sndVal' _ => fun s => [s.fst.mapOpt (·[0]?)]ₕ\n  | .tokVal' _ => fun s => [s.snd]ₕ\n  | .merge => fun s₁ s₂ => [CIRCTStream.DCOp.merge s₁ s₂]ₕ\n  | .branch => fun s => [CIRCTStream.DCOp.branch s]ₕ\n  | .fork => fun s => [CIRCTStream.DCOp.fork s]ₕ\n  | .join => fun s₁ s₂ => [CIRCTStream.DCOp.join s₁ s₂]ₕ\n  | .select => fun s₁ s₂ c => [CIRCTStream.DCOp.select s₁ s₂ c]ₕ\n  | .sink => fun s => [CIRCTStream.DCOp.sink s]ₕ\n  | .source => [CIRCTStream.DCOp.source]ₕ\n  | .pack _ => fun s₁ s₂ => [CIRCTStream.DCOp.pack s₁ s₂]ₕ\n  | .pack2 _ => fun s₁ => [CIRCTStream.DCOp.pack2 s₁]ₕ\n  | .unpack _ => fun s => [CIRCTStream.DCOp.unpack s]ₕ\n  | .unpack2 _ => fun s₁ s₂ => [CIRCTStream.DCOp.unpack2 s₁ s₂]ₕ"}, {"name": "branch", "content": "def branch (x : ValueStream (BitVec 1)): TokenStream × TokenStream  :=\n  Stream.corec₂ (β := ValueStream (BitVec 1)) x fun x =>\n    Id.run <| do\n      match x 0 with\n        | none => (none, none, (x.tail))\n        | some x₀ =>\n          if x₀.msb then\n            (some (), none, (x.tail))\n          else\n            (none, some (), (x.tail))"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "pack", "content": "def pack (x : ValueStream α) (y : TokenStream) : ValueStream α :=\n  Stream.corec (β := ValueStream α × TokenStream) (x, y) fun ⟨x, y⟩ =>\n    match x 0, y 0 with\n    | some x₀, some _ => (x₀, (x.tail, y.tail))\n    | some _, none => (none, (x, y.tail)) \n    | none, some _ => (none, (x.tail, y)) \n    | none, none => (none, (x.tail, y.tail))"}, {"name": "select", "content": "def select (x y : TokenStream) (c : ValueStream (BitVec 1)): TokenStream :=\n  Stream.corec (β := TokenStream × TokenStream × Stream (BitVec 1)) (x, y, c)\n  fun ⟨x, y, c⟩ =>\n    match (c 0) with\n    | none => (none, x, y, c.tail) \n    | some 1#1 =>\n      match (x 0) with\n      | none => (none, x.tail, y, c) \n      | some _ => (some (), x.tail, y, c.tail) \n    | some 0#1 =>\n      match (y 0) with\n      | none => (none, x, y.tail, c) \n      | some _ => (some (), x, y.tail, c.tail) "}, {"name": "unpack", "content": "def unpack (x : ValueStream (BitVec w)) : ValueStream (BitVec w) × TokenStream :=\n  Stream.corec₂ (β := Stream (BitVec w)) (x)\n    fun x => Id.run <| do\n      match x 0 with\n      | some _ => return (x 0, some (), x.tail)\n      | none => return (none, none, x.tail)"}, {"name": "DC", "content": "abbrev DC : Dialect where\n  Op := Op\n  Ty := Ty"}, {"name": "def_signature for DC where", "content": "def_signature for DC where\n  | .fst => (Ty.tokenstream2) → (Ty.tokenstream)\n  | .fstVal t => (Ty.valuetokenstream t) → Ty.valuestream t\n  | .fstVal' t => (Ty.variadicvaluetokenstream t) → Ty.valuestream t\n  | .snd => (Ty.tokenstream2) → (Ty.tokenstream)\n  | .pair w => (Ty.valuestream w, Ty.valuestream w) → Ty.valuestream2 w\n  | .sndVal t => (Ty.valuetokenstream t) → Ty.tokenstream\n  | .sndVal' t => (Ty.variadicvaluetokenstream t) → Ty.valuestream t\n  | .tokVal' t => (Ty.variadicvaluetokenstream t) → Ty.tokenstream\n  | .merge => (Ty.tokenstream, Ty.tokenstream) → Ty.valuestream 1\n  | .branch => (Ty.valuestream 1) → Ty.tokenstream2\n  | .fork => (Ty.tokenstream) → Ty.tokenstream2\n  | .join => (Ty.tokenstream, Ty.tokenstream) → Ty.tokenstream\n  | .select => (Ty.tokenstream, Ty.tokenstream, Ty.valuestream 1) → Ty.tokenstream\n  | .sink => (Ty.tokenstream) → Ty.tokenstream\n  | .source => () → Ty.tokenstream\n  | .pack t => (Ty.valuestream t, Ty.tokenstream) → Ty.valuestream t\n  | .pack2 t => (Ty.variadicvaluetokenstream t) → Ty.valuestream2 t\n  | .unpack t => (Ty.valuestream t) → Ty.valuetokenstream t\n  | .unpack2 t => (Ty.valuestream t, Ty.valuestream t) → Ty.variadicvaluetokenstream t"}, {"name": "Op", "content": "inductive Op\n| fst\n| snd\n| pair (w : Nat)\n| fstVal (w : Nat)\n| sndVal (w : Nat)\n| fstVal' (w : Nat)\n| sndVal' (w : Nat)\n| tokVal' (w : Nat)\n| merge\n| branch\n| fork\n| join\n| select\n| sink\n| source\n| pack (w : Nat)\n| pack2 (w : Nat)\n| unpack (w : Nat)\n| unpack2 (w : Nat)\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "branch", "content": "def branch {Γ : Ctxt _} (a : Γ.Var (.valuestream 1)) : Expr (DC) Γ .pure (.tokenstream2) :=\n  Expr.mk\n    (op := .branch)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "fork", "content": "def fork (a : Γ.Var (.tokenstream)) : Expr (DC) Γ .pure (.tokenstream2) :=\n  Expr.mk\n    (op := .fork)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "unpack2", "content": "def unpack2 {r} {Γ : Ctxt _} (a : Γ.Var (.valuestream r)) (b : Γ.Var (.valuestream r)) : Expr (DC) Γ .pure (.variadicvaluetokenstream r) :=\n  Expr.mk\n    (op := .unpack2 r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "fst", "content": "def fst {Γ : Ctxt _} (a : Γ.Var (.tokenstream2)) : Expr (DC) Γ .pure (.tokenstream)  :=\n  Expr.mk\n    (op := .fst)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "sink", "content": "def sink {Γ : Ctxt _} (a : Γ.Var (.tokenstream)) : Expr (DC) Γ .pure (.tokenstream) :=\n  Expr.mk\n    (op := .sink)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "merge", "content": "def merge {Γ : Ctxt _} (a b : Γ.Var (.tokenstream)) : Expr (DC) Γ .pure (.valuestream 1) :=\n  Expr.mk\n    (op := .merge)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "pack", "content": "def pack {r} {Γ : Ctxt _} (a : Γ.Var (.valuestream r)) (b : Γ.Var (.tokenstream)) : Expr (DC) Γ .pure (.valuestream r) :=\n  Expr.mk\n    (op := .pack r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "select", "content": "def select {Γ : Ctxt _} (a b : Γ.Var (.tokenstream)) (c : Γ.Var (.valuestream 1)) : Expr (DC) Γ .pure (.tokenstream) :=\n  Expr.mk\n    (op := .select)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .cons c <| .nil)\n    (regArgs := .nil)"}, {"name": "source", "content": "def source : Expr (DC) Γ .pure (.tokenstream) :=\n  Expr.mk\n    (op := .source)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .nil)\n    (regArgs := .nil)"}, {"name": "join", "content": "def join {Γ : Ctxt _} (a b : Γ.Var (.tokenstream)) : Expr (DC) Γ .pure (.tokenstream) :=\n  Expr.mk\n    (op := .join)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "pack2", "content": "def pack2 {r} {Γ : Ctxt _} (a : Γ.Var (.variadicvaluetokenstream r)) : Expr (DC) Γ .pure (.valuestream2 r) :=\n  Expr.mk\n    (op := .pack2 r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a  <| .nil)\n    (regArgs := .nil)"}, {"name": "fstVal", "content": "def fstVal {r} {Γ : Ctxt _} (a : Γ.Var (.valuetokenstream r))  : Expr (DC) Γ .pure (.valuestream r)  :=\n  Expr.mk\n    (op := .fstVal r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "snd", "content": "def snd {Γ : Ctxt _} (a : Γ.Var (.tokenstream2)) : Expr (DC) Γ .pure (.tokenstream)  :=\n  Expr.mk\n    (op := .snd)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "unpack", "content": "def unpack {r} {Γ : Ctxt _} (a : Γ.Var (.valuestream r)) : Expr (DC) Γ .pure (.valuetokenstream r) :=\n  Expr.mk\n    (op := .unpack r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "sndVal", "content": "def sndVal {r} {Γ : Ctxt _} (a : Γ.Var (.valuetokenstream r))  : Expr (DC) Γ .pure (.tokenstream)  :=\n  Expr.mk\n    (op := .sndVal r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "pair", "content": "def pair {r} {Γ : Ctxt _} (a b: Γ.Var (.valuestream r)) : Expr (DC) Γ .pure (.valuestream2 r)  :=\n  Expr.mk\n    (op := .pair r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "pack2", "content": "def pack2 (x : VariadicValueStream α × TokenStream) : (ValueStream (BitVec α)) × (ValueStream (BitVec α))  :=\n  Stream.corec₂ (β := VariadicValueStream α × TokenStream) (x) fun ⟨x, y⟩ =>\n    match x 0, y 0 with\n    | some x', some _ => (x'[0]?, x'[1]?, (x.tail, y.tail))\n    | some _, none => (none, none, (x, y.tail))\n    | none, some _ => (none, none, (x.tail, y)) \n    | none, none => (none, none, (x.tail, y.tail)) "}, {"name": "unpack2", "content": "def unpack2 (x : ValueStream (BitVec w)) (y : ValueStream (BitVec w)) : VariadicValueStream w × TokenStream :=\n  Stream.corec₂ (β := CIRCTStream.Stream (BitVec w) × CIRCTStream.Stream (BitVec w)) (x, y)\n    fun (x, y) => Id.run <| do\n      match x 0, y 0 with\n      | some x', some y' => return (some [x', y'], some .unit, (x.tail, y.tail))\n      | some _, none => return (none, none, (x, y.tail))\n      | none, some _ => return (none, none, (x.tail, y)) \n      | none, none => return (none, none, (x.tail, y.tail))"}, {"name": "", "content": "@[reducible]\ninstance : TyDenote ExTy where\n  toType\n    | .nat => Nat\n    | .bool => Bool"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  | bool\n  deriving DecidableEq"}, {"name": "", "content": "instance : TyDenote Ty where toType := Ty.toType"}, {"name": "Ty.toType", "content": "def Ty.toType : Ty → Type\n| .int => Int\n| .ix => Index\n| .tensor2d => Tensor2d' Int "}, {"name": "", "content": "instance : TyDenote Ty where\n  toType\n  | .int => Int\n  | .ix => Index\n  | .tensor1d => Tensor1d Int"}, {"name": "Index", "content": "abbrev Index := ℕ"}, {"name": "Tensor1d", "content": "structure Tensor1d (α : Type) [Inhabited α] where\n  size : Index\n  val :  Index → α\n  spec : ∀ (ix: Index), ix >= size -> val ix = default"}, {"name": "Tensor2d", "content": "structure Tensor2d (α : Type) where\n  size0 : Nat\n  size1 : Nat\n  val :  Fin size0 → Fin size1 → α"}, {"name": "", "content": "instance : TyDenote TestDialect.Ty where toType\n  | .int => Int\n  | .int2 => Int × Int"}, {"name": "def_signature for TestDialect", "content": "def_signature for TestDialect\n  | .noop => () -> []\n  | .mkPair => (.int, .int) -> .int2\n  | .unPair => (.int2) -> [.int, .int]"}, {"name": "def_denote for TestDialect", "content": "def_denote for TestDialect\n  | .noop => []ₕ\n  | .unPair => fun (x, y) => [x, y]ₕ\n  | .mkPair => fun x y => [(x, y)]ₕ"}, {"name": "Op", "content": "inductive Op\n  | noop\n  | mkPair\n  | unPair\n  deriving Lean.ToExpr"}, {"name": "TestDialect", "content": "def TestDialect : Dialect where\n  Ty := Ty\n  Op := Op"}, {"name": "Ty", "content": "inductive Ty\n  | int\n   \n  | int2\n  deriving DecidableEq, Lean.ToExpr"}, {"name": "", "content": "instance : TyDenote Ty where\n  toType\n  | Ty.bv => BitVec 64"}, {"name": "Ty", "content": "inductive Ty\n  | bv : Ty\n  deriving DecidableEq, Repr, Inhabited, Lean.ToExpr"}, {"name": "", "content": "instance : TyDenote LLVM.Ty where\n  toType := fun\n    | bitvec w => LLVM.IntW w"}, {"name": "IntW", "content": "def IntW w := PoisonOr <| BitVec w"}, {"name": "PoisonOr", "content": "structure PoisonOr (α : Type) where\n  val : α\n  poisonous : Bool\nderiving Inhabited, DecidableEq"}, {"name": "", "content": "@[reducible]\ninstance : TyDenote Ty where\n  toType\n    | .int => BitVec 32"}, {"name": "SimpleReg.int", "content": "abbrev SimpleReg.int : SimpleReg.Ty := .int"}, {"name": "", "content": "instance : TyDenote Ty where toType\n  | .int => BitVec 32"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  | bool\n  deriving DecidableEq, Repr"}, {"name": "", "content": "instance : TyDenote Ty2 where\ntoType := fun\n|  Ty2.bitvec w => BitVec w"}, {"name": "Ty2", "content": "inductive Ty2\n  | bitvec (w : Nat) : Ty2\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "instHandshakeTyDenote", "content": "instance instHandshakeTyDenote : TyDenote Ty where\ntoType := fun\n| Ty.stream ty2 => CIRCTStream.Stream (TyDenote.toType ty2)\n| Ty.stream2 ty2 => CIRCTStream.Stream (TyDenote.toType ty2) × CIRCTStream.Stream (TyDenote.toType ty2)\n| Ty.stream2token ty2 => CIRCTStream.Stream (TyDenote.toType ty2) × CIRCTStream.Stream (TyDenote.toType (Ty2.bitvec 1))"}, {"name": "def_denote for Handshake where", "content": "def_denote for Handshake where\n| .fst _ => fun s => [s.fst]ₕ\n| .snd _ => fun s => [s.snd]ₕ\n| .branch _ => fun s c => [HandshakeOp.branch s c]ₕ\n| .merge _ => fun s₁ s₂ => [HandshakeOp.merge s₁ s₂]ₕ\n| .altMerge _ => fun s₁ s₂ => [HandshakeOp.altMerge s₁ s₂]ₕ\n| .fork _ => fun s => [HandshakeOp.fork s]ₕ\n| .controlMerge _ => fun s₁ s₂ => [HandshakeOp.controlMerge s₁ s₂]ₕ\n| .join _ => fun s₁ s₂ => [HandshakeOp.join s₁ s₂]ₕ\n| .mux _ => fun s₁ s₂ c => [HandshakeOp.mux s₁ s₂ c]ₕ\n| .sink _ => fun s => [HandshakeOp.sink s]ₕ\n| .sync _ => fun s₁ s₂ => [HandshakeOp.sync s₁ s₂]ₕ\n| .supp _ => fun s₁ s₂ => [HandshakeOp.supp s₁ s₂]ₕ\n| .not => fun s₁ => [HandshakeOp.not s₁]ₕ"}, {"name": "Handshake", "content": "abbrev Handshake : Dialect where\n  Op := Op\n  Ty := Ty"}, {"name": "def_signature for Handshake where", "content": "def_signature for Handshake where\n| .fst t => (Ty.stream2 t) → Ty.stream t\n| .snd t => (Ty.stream2 t) → Ty.stream t\n| .branch t => (Ty.stream t, Ty.stream (Ty2.bitvec 1)) → Ty.stream2 t\n| .merge t => (Ty.stream t, Ty.stream t) → Ty.stream t\n| .altMerge t => (Ty.stream t, Ty.stream t) → Ty.stream t\n| .fork t => (Ty.stream t) → Ty.stream2 t \n| .controlMerge t => (Ty.stream t, Ty.stream t) → (Ty.stream2token t)\n| .join t => (Ty.stream t, Ty.stream t) → (Ty.stream (Ty2.bitvec 1))\n| .mux t => (Ty.stream t, Ty.stream t, Ty.stream (Ty2.bitvec 1)) → Ty.stream t\n| .sink t => (Ty.stream t) → (Ty.stream (Ty2.bitvec 1))\n| .sync t => (Ty.stream t, Ty.stream t) → Ty.stream2 t\n| .supp t => (Ty.stream t, Ty.stream (Ty2.bitvec 1)) → Ty.stream t\n| .not => (Ty.stream (Ty2.bitvec 1)) → Ty.stream (Ty2.bitvec 1)"}, {"name": "Ty", "content": "inductive Ty\n| stream (ty2 : Ty2) : Ty \n| stream2 (ty2 : Ty2) : Ty \n| stream2token (ty2 : Ty2) : Ty \nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "sync", "content": "def sync {Γ : Ctxt _} (a b : Var Γ (.stream r)) : Expr (Handshake) Γ .pure (.stream2 r)  :=\n  Expr.mk\n    (op := .sync r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "snd", "content": "def snd {Γ : Ctxt _} (a : Var Γ (.stream2 r)) : Expr (Handshake) Γ .pure (.stream r)  :=\n  Expr.mk\n    (op := .snd r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "Op", "content": "inductive Op\n| fst (t : Ty2)\n| snd (t : Ty2)\n| branch (t : Ty2)\n| merge (t : Ty2)\n| altMerge (t : Ty2)\n| fork (t : Ty2)\n| controlMerge (t : Ty2)\n| join (t : Ty2)\n| mux (t : Ty2)\n| sink (t : Ty2)\n| sync (t : Ty2)\n| supp (t : Ty2)\n| not\nderiving Inhabited, DecidableEq, Repr, Lean.ToExpr"}, {"name": "merge", "content": "def merge {Γ : Ctxt _} (a b : Var Γ (.stream r)) : Expr (Handshake) Γ .pure (.stream r)  :=\n  Expr.mk\n    (op := .merge r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "supp", "content": "def supp {Γ : Ctxt _} (a : Var Γ (.stream r)) (b : Var Γ (.stream (.bitvec 1))) : Expr (Handshake) Γ .pure (.stream r)  :=\n  Expr.mk\n    (op := .supp r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons b <| .nil)\n    (regArgs := .nil)"}, {"name": "branch", "content": "def branch {r} {Γ : Ctxt _} (a : Var Γ (.stream r)) (c : Var Γ (.stream (.bitvec 1))) : Expr (Handshake) Γ .pure (.stream2 r) :=\n  Expr.mk\n    (op := .branch r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .cons c <| .nil)\n    (regArgs := .nil)"}, {"name": "fst", "content": "def fst {Γ : Ctxt _} (a : Var Γ (.stream2 r)) : Expr (Handshake) Γ .pure (.stream r)  :=\n  Expr.mk\n    (op := .fst r)\n    (ty_eq := rfl)\n    (eff_le := by admit /- proof elided -/\n    )\n    (args := .cons a <| .nil)\n    (regArgs := .nil)"}, {"name": "altMerge", "content": "def altMerge (x y : Stream α) : Stream α :=\n  Stream.corec (β := Stream α × Stream α × ConsumeFrom) (x, y, .left) fun ⟨x, y, consume⟩ =>\n    match consume with\n      | .left  =>\n        let x0 := x.head\n        let x := x.tail\n        let nextConsume := match x0 with\n          | some _ => .right\n          | none   => .left\n        (x0, x, y, nextConsume)\n      | .right =>\n        let y0 := y.head\n        let y := y.tail\n        let nextConsume := match y0 with\n          | some _ => .left\n          | none   => .right\n        (y0, x, y, nextConsume)"}, {"name": "branch", "content": "def branch (x : Stream α) (c : Stream (BitVec 1)) : Stream α × Stream α :=\n  Stream.corec₂ (β := Stream α × Stream (BitVec 1)) (x, c)\n    fun ⟨x, c⟩ => Id.run <| do\n      let c₀ := c 0\n      let c' := c.tail\n      let x₀ := x 0\n      let x' := x.tail\n      match c₀, x₀ with\n        | none, _ => (none, none, (x, c'))\n        | _, none => (none, none, (x', c))\n        | some c₀, some x₀ =>\n          if c₀ = 1 then\n            (some x₀, none, (x', c'))\n          else\n            (none, some x₀, (x', c'))"}, {"name": "controlMerge", "content": "def controlMerge (x y : Stream α) : Stream α × Stream (BitVec 1) :=\n  Stream.corec₂ (β := Stream α × Stream α) (x, y) fun ⟨x, y⟩ =>\n    match x 0, y 0 with\n    | some x', some _ => (some x', some 1, (x.tail, y))\n    | some x', none => (some x', some 1, (x.tail, y.tail))\n    | none, some y' => (some y', some 0, (x.tail, y.tail))\n    | none, none => (none, none, (x.tail, y.tail))"}, {"name": "fork", "content": "def fork (x : Stream α) : Stream α × Stream α :=\n  Stream.corec₂ (β := Stream α) x\n    fun x => Id.run <| do\n      let x0 := x 0\n      let x' := x.tail\n      (x0, x0, x')"}, {"name": "join", "content": "def join (x y : Stream α) : Stream (BitVec 1) :=\n  Stream.corec (β := Stream α × Stream α) (x, y) fun ⟨x, y⟩ =>\n    match x 0, y 0 with\n    | some _, some _ => (some 1, (x.tail, y.tail))\n    | some _, none   => (none, (x, y.tail))\n    | none, some _   => (none, (x.tail, y))\n    | none, none     => (none, (x.tail, y.tail))"}, {"name": "merge", "content": "def merge (x y : Stream α) : Stream α :=\n  Stream.corec (β := Stream α × Stream α) (x, y) fun ⟨x, y⟩ =>\n    match x 0, y 0 with\n    | some x', some _ => (some x', (x.tail, y))\n    | some x', none => (some x', (x.tail, y.tail))\n    | none, some y' => (some y', (x.tail, y.tail))\n    | none, none => (none, (x.tail, y.tail))"}, {"name": "mux", "content": "def mux (x y : Stream α) (c : Stream (BitVec 1)) : Stream α :=\n  Stream.corec (β := Stream α × Stream α × Stream (BitVec 1)) (x, y, c) fun ⟨x, y, c⟩ => Id.run <| do\n    match x 0, y 0, c 0 with\n      | none, _, some 1 => (none, (x.tail, y, c)) \n      | some _, _, some 1 => (x 0, (x.tail, y, c.tail)) \n      | _, none, some 0 => (none, (x, y.tail, c)) \n      | _, some _, some 0 => (y 0, (x, y.tail, c.tail)) \n      | _, _, none => (none, (x, y, c.tail)) "}, {"name": "not", "content": "def not (x : Stream (BitVec 1)) : Stream (BitVec 1) :=\n  Stream.corec (β := Stream (BitVec 1)) x fun x =>\n    match x 0 with\n    | some 1 => (some 0, (x.tail))\n    | some 0 => (some 0,  (x.tail))\n    | none => (none, (x.tail))"}, {"name": "sink", "content": "def sink (x : Stream α) : Stream (BitVec 1) :=\n  Stream.corec (β := Stream α) (x) fun (x) => (none, x.tail)"}, {"name": "supp", "content": "def supp (x : Stream α) (c : Stream (BitVec 1)) : Stream α := (branch x c).snd"}, {"name": "sync", "content": "def sync (x y : Stream α) : Stream α × Stream α :=\n  Stream.corec₂ (β := Stream α × Stream α) (x, y) fun ⟨x, y⟩ =>\n    match x 0, y 0 with\n    | some x', some y' => (some x', some y', (x.tail, y.tail))\n    | some _, none => (none, none, (x, y.tail))\n    | none, some _ => (none, none, (x.tail, y))\n    | none, none => (none, none, (x.tail, y.tail))"}, {"name": "", "content": "@[simp]\ninstance : TyDenote LLVMPlusRiscV.Ty where\n  toType := fun\n    | .llvm llvmTy => TyDenote.toType llvmTy\n    | .riscv riscvTy => TyDenote.toType riscvTy"}, {"name": "Op", "content": "inductive Op where\n  | llvm : LLVM.Op -> Op\n  | riscv : RISCV64.RV64.Op -> Op\n  | castRiscv : Nat → Op\n  | castLLVM : Nat → Op\n  deriving DecidableEq, Repr, Lean.ToExpr"}, {"name": "ExTy", "content": "inductive ExTy\n  | nat\n  deriving DecidableEq, Repr"}, {"name": "", "content": "@[reducible]\ninstance : TyDenote ExTy where\n  toType\n    | .nat => Nat"}, {"name": "", "content": "instance : TyDenote (Dialect.Ty VariadicDialect) where\n  toType := fun | .int => BitVec 32"}, {"name": "def_denote for VariadicDialect where", "content": "def_denote for VariadicDialect where\n  | .const z => BitVec.ofInt _ z\n  | .add _   => fun xs => xs.foldl (fun ⟨⟩ => (· + ·)) 0#32"}, {"name": "Op", "content": "inductive Op : Type\n| add (n : Nat) : Op\n| const : (val : ℤ) → Op\nderiving DecidableEq, Repr"}, {"name": "VariadicDialect", "content": "def VariadicDialect : Dialect where\n  Op := Op\n  Ty := Ty"}, {"name": "def_signature for VariadicDialect where", "content": "def_signature for VariadicDialect where\n  | .const _  => () -> .int\n  | .add n    => ${List.replicate n .int} → .int"}, {"name": "Ty", "content": "inductive Ty\n| int\nderiving DecidableEq, Repr"}, {"name": "", "content": "instance : TyDenote (Dialect.Ty Comb) where\n  toType := fun\n  | .bitvec w => BitVec w"}, {"name": "", "content": "instance : TyDenote (Scf d).Ty := inferInstanceAs (TyDenote d.Ty)"}, {"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "", "content": "instance : TyDenote Ty where\n  toType\n    | .int => ℤ\n    | .bool => Bool\n    | .nat => Nat"}, {"name": "Ty", "content": "inductive Ty\n| int\n| bool\n| nat\n deriving DecidableEq, Repr"}, {"name": "", "content": "instance : TyDenote SLLVM.Ty where\n  toType\n  | .arith t => ⟦t⟧\n  | .ptr => SLLVM.Ptr\n  | .mem => MemorySSAState"}, {"name": "SLLVMOp", "content": "inductive SLLVMOp where\n  | arith (o : LLVM.Op)\n  | ptradd\n  | load (w : Nat)\n  | store (w : Nat)\n  | alloca (w : Nat)\n  | loadPure (w : Nat)\n  | storePure (w : Nat)\n  deriving DecidableEq, Lean.ToExpr"}, {"name": "Ty.mem", "content": "@[match_pattern] abbrev Ty.mem : SLLVM.Ty := .mem\n\n@[match_pattern] nonrec abbrev Op.arith : LLVM.Op → SLLVM.Op := .arith\n\n@[match_pattern] nonrec abbrev Op.neg (w : Nat) : SLLVM.Op := arith <| Op.neg w\n@[match_pattern] nonrec abbrev Op.not (w : Nat) : SLLVM.Op := arith <| Op.not w\n@[match_pattern] nonrec abbrev Op.copy (w : Nat) : SLLVM.Op := arith <| Op.copy w\n@[match_pattern] nonrec abbrev Op.freeze (w : Nat) : SLLVM.Op := arith <| Op.freeze w\n@[match_pattern] nonrec abbrev Op.sext (w w' : Nat) : SLLVM.Op := arith <| Op.sext w w'\n@[match_pattern] nonrec abbrev Op.zext (w w' : Nat) (flag : LLVM.NonNegFlag := { }) : SLLVM.Op := arith <| Op.zext w w' flag\n@[match_pattern] nonrec abbrev Op.trunc (w w' : Nat) (flags : LLVM.NoWrapFlags := { }) : SLLVM.Op := arith <| Op.trunc w w' flags\n\n@[match_pattern] nonrec abbrev Op.and (w : Nat) : SLLVM.Op := arith <| Op.and w\n@[match_pattern] nonrec abbrev Op.or (w : Nat) (flag : LLVM.DisjointFlag := { }) : SLLVM.Op := arith <| Op.or w flag\n@[match_pattern] nonrec abbrev Op.xor (w : Nat) : SLLVM.Op := arith <| Op.xor w\n@[match_pattern] nonrec abbrev Op.shl (w : Nat) (flags : LLVM.NoWrapFlags := { }) : SLLVM.Op := arith <| Op.shl w flags\n@[match_pattern] nonrec abbrev Op.lshr (w : Nat) (flag : LLVM.ExactFlag := { }) : SLLVM.Op := arith <| Op.lshr w flag\n@[match_pattern] nonrec abbrev Op.ashr (w : Nat) (flag : LLVM.ExactFlag := { }) : SLLVM.Op := arith <| Op.ashr w flag\n@[match_pattern] nonrec abbrev Op.add (w : Nat) (flags : LLVM.NoWrapFlags := { }) : SLLVM.Op := arith <| Op.add w flags\n@[match_pattern] nonrec abbrev Op.mul (w : Nat) (flags : LLVM.NoWrapFlags := { }) : SLLVM.Op := arith <| Op.mul w flags\n@[match_pattern] nonrec abbrev Op.sub (w : Nat) (flags : LLVM.NoWrapFlags := { }) : SLLVM.Op := arith <| Op.sub w flags\n\n@[match_pattern] nonrec abbrev Op.icmp (c : LLVM.IntPred) (w : Nat) : SLLVM.Op := arith <| Op.icmp c w\n@[match_pattern] nonrec abbrev Op.const (w : Nat) (val : Int) : SLLVM.Op := arith <| Op.const w val\n@[match_pattern] nonrec abbrev Op.select (w : Nat) : SLLVM.Op := arith <| Op.select w\n\n@[match_pattern] nonrec abbrev Op.udiv (w : Nat) (flag : LLVM.ExactFlag := { }) : SLLVM.Op := arith <| Op.udiv w flag\n@[match_pattern] nonrec abbrev Op.sdiv (w : Nat) (flag : LLVM.ExactFlag := { }) : SLLVM.Op := arith <| Op.sdiv w flag\n@[match_pattern] nonrec abbrev Op.urem : Nat → SLLVM.Op := arith ∘ Op.urem\n@[match_pattern] nonrec abbrev Op.srem : Nat → SLLVM.Op := arith ∘ Op.srem"}, {"name": "Ty.ptr", "content": "@[match_pattern] abbrev Ty.ptr : SLLVM.Ty := .ptr"}, {"name": "MemorySSAState", "content": "def MemorySSAState := PoisonOr MemoryState"}, {"name": "PoisonOr", "content": "structure PoisonOr (α : Type) where\n  ofOption :: toOption : Option α\n  deriving DecidableEq"}, {"name": "MemoryState", "content": "structure MemoryState where\n  mem : Std.HashMap BlockId Block\n  deriving Inhabited"}, {"name": "GlobalState", "content": "structure GlobalState where\n  alloc : AllocState\n  mem : MemoryState"}, {"name": "BlockId", "content": "structure BlockId where\n  id : Nat\n  deriving DecidableEq, Hashable, Inhabited"}, {"name": "Block", "content": "inductive Block where\n   \n  | dead\n   \n  | live (b : LiveBlock)"}, {"name": "LiveBlock", "content": "structure LiveBlock where\n  (length : Nat)\n  (bytes : BitVec (8 * length))"}, {"name": "Ptr", "content": "def Ptr : Type := PoisonOr Pointer"}, {"name": "infixr:50 \"::ₕ\" => HVector.cons", "content": "infixr:50 \"::ₕ\" => HVector.cons"}], "lib_lemmas": [{"name": "List.get_eq_getElem", "module": "Init.Data.List.Lemmas"}, {"name": "List.length_cons", "module": "Init.Data.List.Basic"}], "repo_lemmas": [{"name": "map_cons", "content": "@[simp] theorem map_cons {A B : α → Type u} {as : List α} {f : (a : α) → A a → B a}\n    {x : A a} {xs : HVector A as} :\n    map f (cons x xs) = cons (f _ x) (map f xs)"}], "used_local_defs": [{"name": "TyDenote", "content": "class TyDenote (β : Type) : Type 1 where\n  toType : β → Type"}, {"name": "Ctxt", "content": "structure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq"}, {"name": "Ctxt.cons", "content": "@[match_pattern]\ndef cons (hd : Ty) : Ctxt Ty → Ctxt Ty\n| ⟨tl⟩ => ⟨hd :: tl⟩"}, {"name": "Ctxt.Var", "content": "def Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }"}, {"name": "Ctxt.Valuation", "content": "def Valuation (Γ : Ctxt Ty) : Type :=\n  ⦃t : Ty⦄ → Γ.Var t → (toType t)"}, {"name": "Ctxt.Valuation.nil", "content": "def Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim"}, {"name": "Ctxt.Valuation", "content": "@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V"}, {"name": "Ctxt.Valuation.reassignVars", "content": "def Valuation.reassignVars [DecidableEq Ty] {ts : List Ty} {Γ : Ctxt Ty}\n    (V : Γ.Valuation) (var : HVector Γ.Var ts) (val : HVector toType ts) : Γ.Valuation :=\n  fun _ vneedle =>\n    match var.idxOf? vneedle with\n    | none => V vneedle\n    | some ⟨i, h⟩ => h ▸ val.get i"}], "used_local_lemmas": [], "local_ctx": "import Mathlib.Data.Fintype.Basic\n\nimport LeanMLIR.HVector\n\nclass TyDenote (β : Type) : Type 1 where\n  toType : β → Type\n\nnotation \"⟦\" x \"⟧\" => TyDenote.toType x\n\nstructure Ctxt (Ty : Type) : Type where\n  ofList :: toList : List Ty\n  \n  deriving Repr, Lean.ToExpr, DecidableEq\n\nvariable {Ty : Type} {Γ Δ : Ctxt Ty}\n\nnamespace Ctxt\n\nsection Instances\n\nopen Lean in\n\nend Instances\n\n@[match_pattern]\ndef cons (hd : Ty) : Ctxt Ty → Ctxt Ty\n| ⟨tl⟩ => ⟨hd :: tl⟩\n\nsection GetElemLemmas\n\nend GetElemLemmas\n\nsection Lemmas\n\nvariable (Γ : Ctxt Ty) (ts us : List Ty)\n\nvariable {m} [Monad m] [LawfulMonad m] (t u : m _) in\n\nsection Lemmas\n\nvariable {Γ Δ : Ctxt Ty} {tys : List Ty}\n\nend Lemmas\n\nend Lemmas\n\nsection Rec\n\nend Rec\n\ndef Var (Γ : Ctxt Ty) (t : Ty) : Type :=\n  { i : Nat // Γ[i]? = some t }\n\nnamespace Var\n\nsection Lemmas\n\nvariable {Γ : Ctxt Ty} {t : Ty}\n\nend Lemmas\n\nsection Lemmas\n\nvariable {t} (v : Var Γ t)\n\nend Lemmas\n\nsection Lemmas\n\nvariable {t : Ty} {v : Γ.Var t}\n\nend Lemmas\n\nsection Lemmas\n\nend Lemmas\n\nend Var\n\nend Ctxt\n\nopen Ctxt\n\nnamespace HVector\n\nvariable {A : α → _} {as : List α} (xs : HVector A as) {Γ : Ctxt α}\n\nend HVector\n\nnamespace Ctxt\n\nsection Comp\n\nvariable {Γ Δ Ξ : Ctxt Ty} (f : Hom Γ Δ) (g : Hom Δ Ξ)\n\nend Comp\n\nsection Lemmas\n\nend Lemmas\n\nvariable {Γ Δ Δ' : Ctxt Ty} in\n\nsection Valuation\n\nvariable [TyDenote Ty]\n\ndef Valuation (Γ : Ctxt Ty) : Type :=\n  ⦃t : Ty⦄ → Γ.Var t → (toType t)\n\ndef Valuation.nil : Ctxt.Valuation (∅ : Ctxt Ty) := fun _ v => v.emptyElim\n\ninfixr:67 \"::ᵥ\" => Valuation.cons\n\nvariable {m} [Monad m] [LawfulMonad m] in\n\nvariable {V : Γ.Valuation} {W : Δ.Valuation}\n\n@[simp]\ninstance Valuation.instAppendHVector (Γ : Ctxt Ty) (ts : List Ty) :\n    HAppend (HVector toType ts) (Valuation Γ) (Valuation <| ⟨ts⟩ ++ Γ) where\n  hAppend vals V :=\n    (Valuation.ofHVector vals) ++ V\n\nsection Lemmas\n\nvariable {ts : List Ty}\n\nend Lemmas\n\ndef Valuation.reassignVars [DecidableEq Ty] {ts : List Ty} {Γ : Ctxt Ty}\n    (V : Γ.Valuation) (var : HVector Γ.Var ts) (val : HVector toType ts) : Γ.Valuation :=\n  fun _ vneedle =>\n    match var.idxOf? vneedle with\n    | none => V vneedle\n    | some ⟨i, h⟩ => h ▸ val.get i", "target_theorem": "@[simp] theorem Valuation.reassignVars_eq [DecidableEq Ty] (V : Γ.Valuation) :\n    V.reassignVars vs (vs.map V) = V :=", "ground_truth_proof": ":= by\n  funext t w\n  unfold reassignVars\n  induction vs\n  case nil => rfl\n  case cons v vs ih =>\n    by_cases h_eq : w.eq v\n    · have := h_eq.ty_eq\n      subst this\n      have := h_eq.to_eq\n      subst this\n      simp\n    · unfold Var.eq at h_eq\n      simp only [HVector.idxOf?, h_eq, ↓reduceDIte, List.get_eq_getElem, List.length_cons,\n        HVector.map_cons]\n      split at ih <;> simp_all", "nesting_depth": 6, "transitive_dep_count": 23, "subset_aristotle": false, "category": "Compiler"}
{"id": 353, "thm_name": "autOfTermBinop_bv_language", "thm_stmt": "lemma autOfTermBinop_bv_language op {t₁ t₂ : Term} (m₁ : CNFA (t₁.arity + 1)) (m₂ : CNFA (t₂.arity + 1)) :\n    m₁.bv_recognizes t₁.language →\n    m₂.bv_recognizes t₂.language →\n    (autOfTermBinop op m₁ m₂ |>.bv_recognizes (op.subst_arity' ▸ (op.subst t₁ t₂).language))", "lean_root": "lean-mlir", "rel_path": "Blase/Blase/AutoStructs/FormulaToAuto.lean", "imports": ["import Blase.SingleWidth.Defs", "import Blase.Blase.AutoStructs.Basic", "import Blase.AutoStructs.Constructions", "import Blase.Blase.Fast.BitStream", "import Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use", "import Blase.Blase.AutoStructs.ForMathlib", "import Blase.AutoStructs.Defs", "import Mathlib.Tactic.FinCases", "import Mathlib.Data.BitVec", "import Mathlib.Tactic.Ring", "import Blase.Blase.AutoStructs.Constructions", "import Blase.AutoStructs.FiniteStateMachine", "import Batteries.Data.Fin.Lemmas", "import Batteries.Data.Fin.Basic"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BitVec.adcb", "module": "Init.Data.BitVec.Bitblast"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "BitVec.iunfoldr", "module": "Init.Data.BitVec.Folds"}, {"name": "FinEnum", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.card", "module": "Mathlib.Data.FinEnum"}, {"name": "Polynomial.X", "module": "Mathlib.Algebra.Polynomial.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Array.foldl", "module": "Init.Data.Array.Basic"}, {"name": "Std.HashMap.emptyWithCapacity", "module": "Std.Data.HashMap.Basic"}, {"name": "Array.size", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Field", "module": "Mathlib.Algebra.Field.Defs"}, {"name": "Int.xor", "module": "Mathlib.Data.Int.Bitwise"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Empty", "module": "Init.Prelude"}, {"name": "Empty.elim", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Fin.castLE", "module": "Init.Data.Fin.Basic"}, {"name": "cond", "module": "Init.Prelude"}, {"name": "Nat.testBit", "module": "Init.Data.Nat.Bitwise.Basic"}, {"name": "Fin.last", "module": "Init.Data.Fin.Basic"}, {"name": "Array.emptyWithCapacity", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Array.empty", "module": "Init.Prelude"}, {"name": "FinEnum.toList", "module": "Mathlib.Data.FinEnum"}, {"name": "FinEnum.equiv", "module": "Mathlib.Data.FinEnum"}, {"name": "NFA", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.accept", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.start", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.step", "module": "Mathlib.Computability.NFA"}, {"name": "List.Vector", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "NFA.stepSet", "module": "Mathlib.Computability.NFA"}, {"name": "Subsingleton", "module": "Init.Core"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "BitVec.ofNat", "module": "Init.Prelude"}, {"name": "BitVec.zero", "module": "Init.Data.BitVec.Basic"}, {"name": "Language", "module": "Mathlib.Computability.Language"}, {"name": "BitVec.cons", "module": "Init.Data.BitVec.Basic"}, {"name": "List.Vector.ofFn", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.replicate", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.finRange", "module": "Init.Data.List.FinRange"}, {"name": "List.reverseRecOn", "module": "Mathlib.Data.List.Induction"}, {"name": "SetRel", "module": "Mathlib.Data.Rel"}, {"name": "Array.back?", "module": "Init.Data.Array.Basic"}, {"name": "Array.isEmpty", "module": "Init.Data.Array.Basic"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}, {"name": "L", "module": "Archive.Hairer"}, {"name": "Fin.mk", "module": "Init.Prelude"}, {"name": "Fin.cast", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.castLT", "module": "Init.Data.Fin.Basic"}, {"name": "Fin.subNat", "module": "Init.Data.Fin.Basic"}, {"name": "List.Vector.get", "module": "Mathlib.Data.Vector.Defs"}, {"name": "List.Vector.nil", "module": "Mathlib.Data.Vector.Defs"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "DFA", "module": "Mathlib.Computability.DFA"}, {"name": "NFA.toDFA", "module": "Mathlib.Computability.NFA"}, {"name": "List.range", "module": "Init.Data.List.Basic"}, {"name": "Vector.ofFn", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "syntax \"max\" : MLIR.Pretty.uniform_op", "content": "syntax \"max\" : MLIR.Pretty.uniform_op\n\nsyntax \"slt\" : MLIR.Pretty.uniform_op\n\nsyntax \"xor\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}, {"name": "carry", "content": "def carry (initCarry : Bool) (x y : BitStream) : BitStream :=\n  fun n => (addAux' initCarry x y n).2"}, {"name": "BitStream", "content": "def BitStream : Type := Nat → Bool"}, {"name": "addAux'", "content": "def addAux' (carryIn : Bool) (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => carryIn\n    | i + 1 => (addAux' carryIn x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "BitVec.ofFn", "content": "def BitVec.ofFn {w : Nat} (f : Fin w → Bool) : BitVec w :=\n  BitVec.iunfoldr (fun i _ => ((), f i)) () |>.2"}, {"name": "f", "content": "noncomputable def f : (ZMod q)[X] := X^(2^n) + 1"}, {"name": "worklistRun", "content": "def worklistRun (final : S → Bool) (inits : Array S)\n    (hinits : inits.toList.Nodup) (f : S → Array (BitVec n × S)) : CNFA n :=\n  ⟨worklistRun' _ S final inits hinits f, worklistRun'_wf (BitVec n) S⟩"}, {"name": "worklistRun'", "content": "def worklistRun' (final : S → Bool) (inits : Array S) (hinits : inits.toList.Nodup) (f : S → Array (A × S)) : RawCNFA A :=\n  let st0 := worklist.initState _ _ inits hinits final\n  go st0\nwhere go (st0 : worklist.St A S) : RawCNFA A :=\n  if hemp : st0.worklist.isEmpty then st0.m else\n  let sa? := st0.worklist.back?\n  match heq : sa? with\n  | some sa =>\n    let wl := st0.worklist.pop\n    let st1 := { st0 with worklist := wl,\n                          worklist_nodup := by admit /- proof elided -/"}, {"name": "worklist.St", "content": "structure worklist.St where\n  m : RawCNFA A\n  map : Std.HashMap S State := ∅\n  worklist : Array S := ∅\n  worklist_nodup : worklist.toList.Nodup\n  worklist_incl : ∀ sa ∈ worklist, sa ∈ map"}, {"name": "worklist.initState", "content": "def worklist.initState (inits : Array S) (hinits : inits.toList.Nodup) (final? : S → Bool) : worklist.St A S :=\n  let m := RawCNFA.empty (A := A)\n  let mapm := inits.foldl (init := (Std.HashMap.emptyWithCapacity, m)) fun (map, m) sa =>\n    let (s, m) := m.newState\n    let m := m.addInitial s\n    let m := if final? sa then m.addFinal s else m\n    (map.insert sa s, m)\n  let map := mapm.1\n  let m := mapm.2\n  let worklist_incl : ∀ sa ∈ inits, sa ∈ map :="}, {"name": "RawCNFA.statesFinset", "content": "instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort"}, {"name": "State", "content": "abbrev State := Nat"}, {"name": "RawCNFA.empty", "content": "def RawCNFA.empty : RawCNFA A := {\n  stateMax := 0\n  initials := ∅\n  finals := ∅\n  trans := ∅\n}"}, {"name": "processOneElem", "content": "def processOneElem (final : S → Bool) (s : State) (st : worklist.St A S) : A × S → worklist.St A S :=\n  fun (a', sa') =>\n    let (s', st') := st.addOrCreateState _ _ (final sa') sa'\n    let m := st'.m.addTrans a' s s'\n    { st' with m }"}, {"name": "worklist.St.addOrCreateState", "content": "def worklist.St.addOrCreateState (st : worklist.St A S) (final? : Bool) (sa : S) : State × worklist.St A S :=\n  match heq : st.map[sa]? with\n  | some s => (s, st)\n  | none =>\n    let (s, m) := st.m.newState\n    let m := if final? then m.addFinal s else m\n    let map := st.map.insert sa s\n    let worklist := st.worklist.push sa\n    have worklist_nodup : worklist.toList.Nodup := by admit /- proof elided -/"}, {"name": "CNFA", "content": "structure CNFA (n : Nat) where\n  m : RawCNFA (BitVec n)\n  wf : m.WF"}, {"name": "FSM", "content": "structure FSM (arity : Type) : Type 1 where\n   \n  ( α  : Type )\n  [ i : FinEnum α ]\n  [ h : Hashable α ]\n  [ dec_eq : DecidableEq α ]\n   \n  ( initCarry : α → Bool )\n   \n  outputCirc : Circuit (α ⊕ arity)\n  nextStateCirc : α → Circuit (α ⊕ arity)"}, {"name": "Circuit", "content": "inductive Circuit (α : Type u) : Type u\n  | tru : Circuit α\n  | fals : Circuit α\n   \n  | var : (positive: Bool) → α → Circuit α\n  | and : Circuit α → Circuit α → Circuit α\n  | or : Circuit α → Circuit α → Circuit α\n  | xor : Circuit α → Circuit α → Circuit α\nderiving Repr, DecidableEq"}, {"name": "Var", "content": "def Var (Γ : Context Ty) (ty : Ty) : Type := { v : VarName // Γ.hasType v ty }"}, {"name": "sub", "content": "def sub (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).1"}, {"name": "subAux", "content": "def subAux (x y : BitStream) : Nat → Bool × Bool\n  | 0 => (xor (x 0) (y 0), !(x 0) && y 0)\n  | n+1 =>\n    let borrow := (subAux x y n).2\n    let a := x (n + 1)\n    let b := y (n + 1)\n    (xor a (xor b borrow), !a && b || ((!(xor a b)) && borrow))"}, {"name": "ofTerm", "content": "abbrev ofTerm (t : Term) : FSM (Fin t.arity) := termEvalEqFSM t |>.toFSM"}, {"name": "Term", "content": "inductive Term : Type\n| var : Nat → Term\n \n| zero : Term\n \n| negOne : Term\n \n| one : Term\n \n| ofNat (n : Nat) : Term\n \n| and : Term → Term → Term\n \n| or : Term → Term → Term\n \n| xor : Term → Term → Term\n \n| not : Term → Term\n \n| add : Term → Term → Term\n \n| sub : Term → Term → Term\n \n| neg : Term → Term\n\n\n \n| shiftL : Term → Nat → Term\n\n\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "termEvalEqFSM", "content": "def termEvalEqFSM : ∀ (t : Term), FSMTermSolution t\n  | ofNat n =>\n    { toFSM := FSM.ofNat n,\n      good := by admit /- proof elided -/"}, {"name": "or", "content": "def or : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ||| Circuit.var true (inr false),\n    nextStateCirc := fun a => a.elim\n  }"}, {"name": "shiftLeft", "content": "def shiftLeft (n : Nat) : FSM Unit :=\n  match n with\n  | 0 => FSM.id\n  | n + 1 => composeUnaryAux (FSM.ls false) (shiftLeft n)"}, {"name": "id", "content": "def id : FSM Unit := {\n α := Empty,\n initCarry := Empty.elim,\n outputCirc := Circuit.var true (inr ()),\n nextStateCirc := Empty.elim\n}"}, {"name": "ls", "content": "def ls (b : Bool) : FSM Unit :=\n  { α := Unit,\n    initCarry := fun _ => b,\n    nextStateCirc := fun () => Circuit.var true (inr ()),\n    outputCirc := Circuit.var true (inl ())\n  }"}, {"name": "composeUnaryAux", "content": "def composeUnaryAux\n    (p : FSM Unit)\n    (q : FSM arity) :\n    FSM arity :=\n  p.compose\n    arity\n    _\n    (λ _ => id)\n    (λ _ => q)"}, {"name": "compose", "content": "def compose [FinEnum arity] [DecidableEq arity] [Hashable arity]\n    (new_arity : Type)        \n    (q_arity : arity → Type)  \n    (vars : ∀ (a : arity), q_arity a → new_arity)\n    \n    \n    (q : ∀ (a : arity), FSM (q_arity a)) : \n    FSM new_arity :=\n  { α := p.α ⊕ (Σ a, (q a).α),\n    i := by admit /- proof elided -/"}, {"name": "and", "content": "def and : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := fun a => a.elim,\n    outputCirc := Circuit.var true (inr true) &&& Circuit.var true (inr false),\n  }"}, {"name": "xor", "content": "def xor : FSM Bool :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    outputCirc := Circuit.var true (inr true) ^^^ Circuit.var true (inr false),\n    nextStateCirc := Empty.elim\n  }"}, {"name": "neg", "content": "def neg : FSM Unit :=\n  { α := Unit,\n    i := by admit /- proof elided -/"}, {"name": "composeBinary", "content": "def composeBinary\n    (p : FSM Bool)\n    {t₁ t₂ : Term}\n    (q₁ : FSMTermSolution t₁)\n    (q₂ : FSMTermSolution t₂) :\n    FSM (Fin (max t₁.arity t₂.arity)) := composeBinaryAux p q₁.toFSM q₂.toFSM"}, {"name": "composeBinaryAux", "content": "def composeBinaryAux\n    (p : FSM Bool)\n    (q₁ : FSM (Fin a₁))\n    (q₂ : FSM (Fin a₂)) :\n    FSM (Fin (max a₁ a₂)) :=\n  p.compose (Fin (max a₁ a₂))\n    (λ b => Fin (cond b a₁ a₂))\n    (λ b i => Fin.castLE (by admit /- proof elided -/\n    ) i)\n    (λ b => match b with\n      | true => q₁\n      | false => q₂)"}, {"name": "FSMTermSolution", "content": "structure FSMTermSolution (t : Term) extends FSM (Fin t.arity) where\n  ( good : t.evalFin = toFSM.eval )"}, {"name": "Term.evalFin", "content": "@[simp] def Term.evalFin (t : Term) (vars : Fin (arity t) → BitStream) : BitStream :=\n  match t with\n  | var n => vars (Fin.last n)\n  | zero    => BitStream.zero\n  | one     => BitStream.one\n  | negOne  => BitStream.negOne\n  | ofNat n => BitStream.ofNat n\n  | and t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | or t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | xor t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | not t     => ~~~(t.evalFin vars)\n  | add t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | sub t₁ t₂ =>\n      let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | neg t       => -(Term.evalFin t vars)\n \n \n  | shiftL t n  => BitStream.shiftLeft (Term.evalFin t vars) n"}, {"name": "Predicate.evalFin", "content": "@[simp] def Predicate.evalFin (p : Predicate) (vars : Fin (arity p) → BitStream) : BitStream :=\nmatch p with\n| .width .eq n => BitStream.falseIffEq n\n| .width .neq n => BitStream.falseIffNeq n\n| .width .lt n => BitStream.falseIffLt n\n| .width .le n => BitStream.falseIffLe n\n| .width .gt n => BitStream.falseIffGt n\n| .width .ge n => BitStream.falseIffGe n\n| .binary .eq t₁ t₂ =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalEq x₁ x₂\n| .binary .neq t₁ t₂  =>\n    let x₁ := t₁.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    let x₂ := t₂.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n    ) i))\n    Predicate.evalNeq x₁ x₂\n| .land p q =>\n  \n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLand x₁ x₂\n| .lor p q =>\n  \n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor x₁ x₂\n| .binary .slt p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalSlt x₁ x₂\n| .binary .sle p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalSlt x₁ x₂) (Predicate.evalEq x₁ x₂)\n| .binary .ult p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  (Predicate.evalUlt x₁ x₂)\n| .binary .ule p q =>\n  let x₁ := p.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  let x₂ := q.evalFin (fun i => vars (Fin.castLE (by admit /- proof elided -/\n  ) i))\n  Predicate.evalLor (Predicate.evalUlt x₁ x₂) (Predicate.evalEq x₁ x₂)"}, {"name": "Predicate.evalUlt", "content": "def Predicate.evalUlt (t₁ t₂ : BitStream) : BitStream := (~~~ (t₁.borrow t₂)).concat true"}, {"name": "borrow", "content": "def borrow (x y : BitStream) : BitStream :=\n  fun n => (subAux x y n).2"}, {"name": "Predicate.evalLor", "content": "def Predicate.evalLor (t₁ t₂ : BitStream) : BitStream := (t₁ &&& t₂)"}, {"name": "Predicate.evalSlt", "content": "def Predicate.evalSlt (t₁ t₂ : BitStream) : BitStream :=\n    (((Predicate.evalUlt t₁ t₂)) ^^^ (Predicate.evalMsbEq t₁ t₂))"}, {"name": "Predicate.evalMsbEq", "content": "def Predicate.evalMsbEq (t₁ t₂ : BitStream) : BitStream :=\n  (t₁ ^^^ t₂).concat false"}, {"name": "Predicate.evalLand", "content": "def Predicate.evalLand (t₁ t₂ : BitStream) : BitStream := (t₁ ||| t₂)"}, {"name": "Predicate.evalNeq", "content": "def Predicate.evalNeq (t₁ t₂ : BitStream) : BitStream := (t₁.nxor t₂).concat true |>.scanAnd"}, {"name": "nxor", "content": "def nxor (a b : BitStream) : BitStream := fun i => a i == b i"}, {"name": "scanAnd", "content": "def scanAnd (s : BitStream) : BitStream := scanl true Bool.and s"}, {"name": "scanl", "content": "abbrev scanl (init : Bool) (f : Bool → Bool → Bool) (s : BitStream) : BitStream :=\n  fun n => match n with\n    | 0 => f init (s 0)\n    | n+1 => f (scanl init f s n) (s (n + 1))"}, {"name": "Predicate.evalEq", "content": "def Predicate.evalEq (t₁ t₂ : BitStream) : BitStream := (t₁ ^^^ t₂).concat false |>.scanOr"}, {"name": "scanOr", "content": "def scanOr (s : BitStream) : BitStream := scanl false Bool.or s"}, {"name": "Predicate", "content": "inductive Predicate : Type where\n \n| width (wp : WidthPredicate) (n : Nat) : Predicate\n| binary (p : BinaryPredicate) (t₁ t₂ : Term)\n| land  (p q : Predicate) : Predicate\n| lor (p q : Predicate) : Predicate\nderiving Repr, Inhabited, Lean.ToExpr"}, {"name": "falseIffNeq", "content": "abbrev falseIffNeq (n : Nat) : BitStream := fun i => decide (i == n)"}, {"name": "falseIffLt", "content": "abbrev falseIffLt (n : Nat) : BitStream := fun i => decide (i ≥ n)"}, {"name": "falseIffLe", "content": "abbrev falseIffLe (n : Nat) : BitStream := fun i => decide (i > n)"}, {"name": "falseIffGe", "content": "abbrev falseIffGe (n : Nat) : BitStream := fun i => decide (i < n)"}, {"name": "falseIffEq", "content": "abbrev falseIffEq (n : Nat) : BitStream := fun i => decide (i != n)"}, {"name": "falseIffGt", "content": "abbrev falseIffGt (n : Nat) : BitStream := fun i => decide (i ≤ n)"}, {"name": "Term.width", "content": "def Term.width (t : Term) : WidthExpr :=\n  match t with\n\n  | .ofNat w _n => w\n  | .var _v w => w\n  | .add w _a _b => w\n  | .zext _a wnew => wnew\n  | .setWidth _a wnew => wnew\n  | .sext _a wnew => wnew\n  | .bor w _a _b => w\n  | .band w _a _b => w\n  | .bxor w _a _b => w\n  | .bnot w _a => w\n  | .boolVar _v => WidthExpr.const 1 \n  | .boolConst _b => WidthExpr.const 1\n  | .shiftl w _a _k => w\n  | .bvOfBool _b => WidthExpr.const 1\n  | binWidthRel _k wa wb => WidthExpr.const 0\n  | binRel _k w _a _b => w\n  | or _p1 _p2 => WidthExpr.const 0\n  | and _p1 _p2 => WidthExpr.const 0\n  | pvar _v => WidthExpr.const 0\n  | boolBinRel _k _a _b => WidthExpr.const 0"}, {"name": "Term.arity", "content": "@[simp] def Term.arity : Term → Nat\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| ofNat _ => 0\n| Term.and t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.or t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.xor t₁ t₂ => max (arity t₁) (arity t₂)\n| Term.not t => arity t\n| add t₁ t₂ => max (arity t₁) (arity t₂)\n| sub t₁ t₂ => max (arity t₁) (arity t₂)\n| neg t => arity t\n\n\n| shiftL t .. => arity t"}, {"name": "negOne", "content": "abbrev negOne : BitStream := fun _ => true"}, {"name": "shiftLeft", "content": "def shiftLeft (x : BitStream) (k : Nat) : BitStream :=\n  fun i => if i < k then false else x (i - k) "}, {"name": "ofNat", "content": "def ofNat (x : Nat) : BitStream :=\n  Nat.testBit x"}, {"name": "one", "content": "abbrev one    : BitStream := (· == 0)"}, {"name": "zero", "content": "abbrev zero   : BitStream := fun _ => false"}, {"name": "ofNat", "content": "def ofNat (n : Nat)  : FSM (Fin 0) :=\n  match hn : n with\n  | 0 => FSM.zero\n\n  | n' + 1 =>\n    let bit := n.testBit 0\n    let m := n / 2\n    have h : m < n := by admit /- proof elided -/"}, {"name": "zero", "content": "def zero : FSM (Fin 0) :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := Empty.elim,\n    outputCirc := Circuit.fals\n  }"}, {"name": "composeUnary", "content": "def composeUnary\n    (p : FSM Unit)\n    {t : Term}\n    (q : FSMTermSolution t) :\n    FSM (Fin t.arity) := composeUnaryAux p q.toFSM"}, {"name": "one", "content": "def one : FSM (Fin 0) :=\n  { α := Unit,\n    i := by admit /- proof elided -/"}, {"name": "var", "content": "def var (n : ℕ) : FSM (Fin (n+1)) :=\n  { α := Empty,\n    i := by admit /- proof elided -/"}, {"name": "add", "content": "def add : FSM Bool :=\n  { α := Unit,\n    initCarry := λ _ => false,\n    nextStateCirc := fun () =>\n      Circuit.var true (inr true) &&& Circuit.var true (inr false) |||\n      Circuit.var true (inr true) &&& Circuit.var true (inl ()) |||\n      Circuit.var true (inr false) &&& Circuit.var true (inl ()),\n    outputCirc := Circuit.var true (inr true) ^^^\n                  Circuit.var true (inr false) ^^^\n                  Circuit.var true (inl ()),\n  }"}, {"name": "negOne", "content": "def negOne : FSM (Fin 0) :=\n  { α := Empty,\n    i := by admit /- proof elided -/"}, {"name": "sub", "content": "def sub : FSM Bool :=\n  { α := Unit,\n    initCarry := fun _ => false,\n    outputCirc := Circuit.var true (inr true) ^^^\n                  Circuit.var true (inr false) ^^^\n                  Circuit.var true (inl ()),\n    nextStateCirc := fun _ =>\n      (Circuit.var false (inr true) &&& Circuit.var true (inr false)) |||\n      (Circuit.var false (inr true) ^^^ Circuit.var true (inr false)) &&&\n      (Circuit.var true (inl ()))\n  }"}, {"name": "not", "content": "def not : FSM Unit :=\n  { α := Empty,\n    initCarry := Empty.elim,\n    nextStateCirc := Empty.elim,\n    outputCirc := Circuit.var false (inr ())\n  }"}, {"name": "add", "content": "def add (x y : BitStream) : BitStream :=\n  fun n => (addAux x y n).1"}, {"name": "addAux", "content": "def addAux (x y : BitStream) (i : Nat) :  Bool × Bool :=\n  let carryIn : Bool := match i with\n    | 0 => false\n    | i + 1 => (addAux x y i).2\n  Prod.swap (BitVec.adcb (x i) (y i) carryIn)"}, {"name": "neg", "content": "def neg (x : BitStream) : BitStream :=\n  fun n => (negAux x n).1"}, {"name": "negAux", "content": "def negAux (x : BitStream) : Nat → Bool × Bool\n  | 0 => (x 0, !(x 0))\n  | n+1 =>\n    let borrow := (negAux x n).2\n    let a := x (n + 1)\n    (xor (!a) borrow, !a && borrow)"}, {"name": "liftMaxSuccSucc2", "content": "def liftMaxSuccSucc2 (n m : Nat) : Fin (m + 1) → Fin (max n m + 3) :=\n  fun k => if _ : k = Fin.last m then max n m + 1 else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftMaxSuccSucc1", "content": "def liftMaxSuccSucc1 (n m : Nat) : Fin (n + 1) → Fin (max n m + 3) :=\n  fun k => if _ : k = Fin.last n then (max n m).cast else k.castLE (by admit /- proof elided -/\n  )"}, {"name": "liftLast3", "content": "def liftLast3 n : Fin 3 → Fin (n + 3)\n| 0 => n\n| 1 => n + 1\n| 2 => Fin.last (n + 2)"}, {"name": "CNFA.inter", "content": "def CNFA.inter (m1 m2 : CNFA n) : CNFA n := product (fun b1 b2 => b1 && b2) m1 m2"}, {"name": "product", "content": "def product (final? : Bool → Bool → Bool) (m₁ m₂ : CNFA n) : CNFA n :=\n  worklistRun (m₁.m.states × m₂.m.states) final (product.inits m₁ m₂)\n    (by admit /- proof elided -/\n    ) f\nwhere final (ss : m₁.m.states × m₂.m.states) := final? (ss.1 ∈ m₁.m.finals) (ss.2 ∈ m₂.m.finals)\n      f (ss : m₁.m.states × m₂.m.states) :=\n        let (s1, s2) := ss\n        (FinEnum.toList (α := BitVec n)).foldl (init := Array.empty) fun as a =>\n          product.prodArray' (λ s₁ s₂ ↦ (a, (s₁, s₂)))\n            (fun s' => m₁.wf.trans_tgt_lt (s := s1) (a := a)) (fun s' => m₂.wf.trans_tgt_lt (s := s2) (a := a)) as"}, {"name": "product.prodArray'", "content": "@[inline]\ndef product.prodArray' (a : Array γ) :=\n  m₁.attachWith _ hm₁ |>.fold (init := a) fun is s1 =>\n    m₂.attachWith _ hm₂ |>.fold (init := is) fun is s2 =>\n      is.push (f s1 s2)"}, {"name": "product.inits_nodup", "content": "def product.inits_nodup : inits m₁ m₂ |>.toList.Nodup :="}, {"name": "product.inits", "content": "def product.inits (m₁ m₂ : CNFA n) :=\n  product.prodArray Prod.mk @m₁.wf.initials_lt @m₂.wf.initials_lt"}, {"name": "product.prodArray", "content": "@[inline]\ndef product.prodArray := prodArray' f hm₁ hm₂ (Array.emptyWithCapacity <| m₁.size * m₂.size)"}, {"name": "CNFA.inter_bv_language", "content": "def CNFA.inter_bv_language (m₁ m₂ : CNFA n) :\n    m₁.bv_recognizes L₁ →\n    m₂.bv_recognizes L₂ →\n    (m₁.inter m₂).bv_recognizes (L₁ ∩ L₂) :="}, {"name": "HashSet.inter", "content": "def HashSet.inter [BEq A] [Hashable A] (m1 m2 : Std.HashSet A) : Std.HashSet A :=\n  m1.fold (init := ∅) fun mi x => if m2.contains x then mi.insert x else mi"}, {"name": "NFA'", "content": "structure NFA' (n : Nat) where\n  σ : Type\n  M : NFA (BitVec n) σ"}, {"name": "BitVecs", "content": "structure BitVecs (n : Nat) where\n  w : Nat\n  bvs : List.Vector (BitVec w) n"}, {"name": "eval", "content": "def eval (x : arity → BitStream) : BitStream :=\n  fun n => (p.nextBit (p.carry x n) (fun i => x i n)).2"}, {"name": "nextBit", "content": "def nextBit : p.State → (arity → Bool) → p.State × Bool :=\n  fun carry inputBits =>\n    let input := Sum.elim carry inputBits\n    let newState : p.State  := fun (a : p.α) => (p.nextStateCirc a).eval input\n    let outBit : Bool       := (p.outputCirc).eval input\n    (newState, outBit)"}, {"name": "State", "content": "abbrev State : Type := p.α → Bool"}, {"name": "carry", "content": "def carry (x : arity → BitStream) : ℕ → p.State\n  | 0 => p.initCarry\n  | n+1 => (p.nextBit (carry x n) (fun i => x i n)).1"}, {"name": "carryBV", "content": "def carryBV (x : ar → BitVec w) : p.State :=\n  p.carry (fun ar => .ofBitVecSext (x ar)) w"}, {"name": "evalBV", "content": "def evalBV {w} (x : ar → BitVec w) : BitVec w :=\n  BitVec.ofFn fun k => p.eval (fun ar => .ofBitVecSext (x ar)) k"}, {"name": "ofBitVecSext", "content": "abbrev ofBitVecSext {w} (x : BitVec w) : BitStream :=\n  fun i => if i < w then x.getLsbD i else x.msb"}, {"name": "Term.language", "content": "def Term.language (t : Term) : Set (BitVecs (t.arity + 1)) :=\n  { bvs : BitVecs (t.arity + 1) | t.evalFinBV (fun n => bvs.bvs.get n) = bvs.bvs.get t.arity }"}, {"name": "Formula.arity", "content": "@[simp]\ndef Formula.arity : Formula → Nat\n| width _ _ => 0\n| atom _ t1 t2 => max t1.arity t2.arity\n| msbSet t => t.arity\n| unop _ φ => φ.arity\n| binop _ φ1 φ2 => max φ1.arity φ2.arity"}, {"name": "Term.evalFinBV", "content": "@[simp] def Term.evalFinBV (t : Term) (vars : Fin (arity t) → BitVec w) : BitVec w :=\n  match t with\n  | .var n => vars (Fin.last n)\n  | .zero    => BitVec.zero w\n  | .one     => 1\n  | .negOne  => -1\n  | .ofNat n => BitVec.ofNat _ n\n  | .and t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ &&& x₂\n  | .or t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ||| x₂\n  | .xor t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ ^^^ x₂\n  | .not t     => ~~~(t.evalFinBV vars)\n  \n  | .add t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ + x₂\n  | .sub t₁ t₂ =>\n      let x₁ := t₁.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      let x₂ := t₂.evalFinBV (fun i => vars (Fin.castLE (by admit /- proof elided -/\n      ) i))\n      x₁ - x₂\n  | .neg t       => -(t.evalFinBV vars)\n  | .shiftL a n => (a.evalFinBV vars) <<< n"}, {"name": "enc", "content": "def enc (bvs : BitVecs n) : BitVecs' n :=\n  (List.finRange bvs.w).map (fun i =>\n    BitVec.ofFn (fun (k : Fin n) => (bvs.bvs.get k)[i]))"}, {"name": "BitVecs'", "content": "abbrev BitVecs' (n : Nat) := List (BitVec n)"}, {"name": "dec", "content": "@[simps]\ndef dec (bvs' : BitVecs' n) : BitVecs n where\n  w := bvs'.length\n  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k"}, {"name": "accepts", "content": "def accepts (M : NFA' n) : Set (BitVecs n) := dec '' M.accepts'"}, {"name": "accepts'", "content": "def accepts' (M : NFA' n) : Set (BitVecs' n) := M.M.accepts"}, {"name": "worklistRun_spec", "content": "def worklistRun_spec : (worklistRun S final inits hinits f |>.Sim $ nfa' inits final f) :=\n  worklistRun'_spec inits final f"}, {"name": "nfa'", "content": "def nfa' : NFA' n :=\n  { σ := _, M := nfa inits final f }"}, {"name": "nfa", "content": "def nfa : NFA A S where\n  start := { sa | sa ∈ inits }\n  accept := { sa | final sa }\n  step sa a := { sa' | (a, sa') ∈ f sa }"}, {"name": "worklistRun'_spec", "content": "def worklistRun'_spec :\n    (worklistRun' A S final inits hinits f |>.Sim $ nfa inits final f) :="}, {"name": "StInv", "content": "structure StInv (m : RawCNFA A) (map : Std.HashMap S State) where\n  wf : m.WF\n  map_states : ∀ (sa : S) s, map[sa]? = some s → s ∈ m.states\n  map_surj : ∀ s : m.states, ∃ (sa : S), map[sa]? = some s.val\n  map_inj : ∀ {s} {sa sa' : S}, map[sa]? = some s → map[sa']? = some s → sa = sa'"}, {"name": "worklist.St.D", "content": "def worklist.St.D (st : worklist.St A S) : Set S := st.visited"}, {"name": "worklist.St.visited", "content": "def worklist.St.visited (st : worklist.St A S) : Set S := { s : S | s ∈ st.map ∧ s ∉ st.worklist }"}, {"name": "worklistGo_spec", "content": "def worklistGo_spec {st : worklist.St A S} (inv : StInv A S st.m st.map) :\n    st.sim inits final f ∅ →\n    (worklistRun'.go A S final f st |>.Sim $ nfa inits final f) :="}, {"name": "worklist.St.rel", "content": "def worklist.St.rel (st : worklist.St A S) : SetRel State S := {(s, sa) | st.map[sa]? = some s }"}, {"name": "processOneElem_mot", "content": "def processOneElem_mot (s : State) (sa : S) (n : ℕ) (st : worklist.St A S) : Prop :=\n  st.map[sa]? = some s ∧\n  sa ∈ st.visited ∧\n  StInv A S st.m st.map ∧\n  st.sim inits final f  {(sa1, a, sa') | sa1 = sa ∧ ∃ k ≥ n, (f sa)[k]? = some (a, sa') }"}, {"name": "worklist.St.sim", "content": "abbrev worklist.St.sim {st : worklist.St A S} (T : Set (S × A × S)) :=\n  st.m.Simul (nfa inits final f) st.rel st.D T"}, {"name": "RawCNFA.Sim", "content": "def RawCNFA.Sim (m : RawCNFA A) (A : NFA A S) := ∃ R, RawCNFA.Simul m A R ⊤ ∅"}, {"name": "RawCNFA.Simul", "content": "structure RawCNFA.Simul (m : RawCNFA A) (M : NFA A Q) (R : SetRel State Q) (D : Set Q) (T : Set (Q × A × Q)) where\n  accept {s q} : s ~[R] q → (s ∈ m.finals ↔ q ∈ M.accept)\n  initial₁ {s} : s ∈ m.initials → ∃ q ∈ M.start, s ~[R] q\n  initial₂ {q} : q ∈ M.start → ∃ s ∈ m.initials, s ~[R] q\n  trans_match₁ {s s' a q} : s ~[R] q → s' ∈ m.tr s a → ∃ q', q' ∈ M.step q a ∧ s' ~[R] q'\n  trans_match₂ {s a q q'} : s ~[R] q → q' ∈ M.step q a → q ∈ D → (q, a, q') ∉ T → ∃ s', s' ∈ m.tr s a ∧ s' ~[R] q'"}, {"name": "RawCNFA.SimulFun", "content": "structure RawCNFA.SimulFun (m : RawCNFA A) (M : NFA A Q) (f : m.states ≃ Q)  where\n  accept {q} : ((f.invFun q).val ∈ m.finals ↔ q ∈ M.accept)\n  initial {q} : q ∈ M.start ↔ (f.invFun q).val ∈ m.initials\n  trans_match {a q q'} : q' ∈ M.step q a ↔ (f.invFun q').val ∈ m.tr (f.invFun q) a"}, {"name": "RawCNFA.tr", "content": "@[inline]\ndef RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅"}, {"name": "R", "content": "abbrev R := (ZMod q)[X] ⧸ (Ideal.span {f q n})"}, {"name": "CNFA.Sim", "content": "def CNFA.Sim (m : CNFA n) (M : NFA' n) :=\n  m.m.Sim M.M"}, {"name": "CNFA.bv_recognizes", "content": "def CNFA.bv_recognizes (m : CNFA n) (L : Set (BitVecs n)) :=\n  ∃ L', m.recognizes L' ∧ L = dec '' L'"}, {"name": "RawCNFA.recognizes", "content": "def RawCNFA.recognizes (m : RawCNFA A) (L : Language A) :=\n  ∃ (σ : Type) (M : NFA A σ), m.Sim M ∧ M.accepts = L"}, {"name": "CNFA.recognizes", "content": "def CNFA.recognizes (m : CNFA n) (L : Language (BitVec n)) :=\n  ∃ (M : NFA' n), m.Sim M ∧ M.M.accepts = L"}, {"name": "BitVecs.cast", "content": "def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=\n  { w := bvs.w, bvs := h ▸ bvs.bvs }"}, {"name": "Valuation.cons", "content": "def Valuation.cons {Γ : Ctxt Ty} {t : Ty} (x : toType t) (V : Γ.Valuation) :\n    (Γ.cons t).Valuation :="}, {"name": "CNFA.minimize", "content": "def CNFA.minimize (m : CNFA n) : CNFA n :=\n  let mᵣ := m.reverse.determinize\n  mᵣ.reverse.determinize"}, {"name": "CNFA.determinize", "content": "def CNFA.determinize (m : CNFA n) : CNFA n :=\n  worklistRun (BitVec m.m.stateMax)\n    (fun ss => ss.any fun n b => b == true && n ∈ m.m.finals)\n    (determinize.inits m)\n    (by admit /- proof elided -/\n    )\n    f\nwhere\n  f := fun (ss : BitVec m.m.stateMax) =>\n        (FinEnum.toList (BitVec n)).foldl (init := Array.empty) fun ts a =>\n          let ss' := m.m.transSetBV ss a\n          ts.push (a, ss')"}, {"name": "CNFA.determinize.inits", "content": "def CNFA.determinize.inits (m : CNFA n) : Array (BitVec m.m.stateMax) :=\n  #[BitVec.ofFn (fun n => n ∈ m.m.initials)]"}, {"name": "CNFA.reverse", "content": "def CNFA.reverse (m : CNFA n) : CNFA n :=\n  ⟨m.m.reverse, RawCNFA.reverse_spec m.wf |>.1⟩"}, {"name": "RawCNFA.reverse", "content": "def RawCNFA.reverse (m : RawCNFA A) : RawCNFA A :=\n  let m' := { stateMax := m.stateMax, trans := Std.HashMap.emptyWithCapacity m.trans.size, initials := m.finals, finals := m.initials}\n  m.trans.fold (init := m') processState\nwhere\n  processState := fun m' (s, a) ss' =>\n    ss'.fold (init := m') fun m' s' => m'.addTrans a s' s"}, {"name": "CNFA.toNFA'", "content": "def CNFA.toNFA' (m : CNFA n) : NFA' n := ⟨_, m.toNFA⟩"}, {"name": "CNFA.toNFA", "content": "def CNFA.toNFA (m : CNFA n) : NFA (BitVec n) m.m.states where\n  start := { s | s.val ∈ m.m.initials }\n  accept := { s | s.val ∈ m.m.finals }\n  step s₁ a := { s₂ | s₂.val ∈ m.m.tr s₁.val a }"}, {"name": "RawCNFA.states", "content": "def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax"}, {"name": "reverse", "content": "def reverse (M : NFA' n) : NFA' n where\n  σ := _\n  M := M.M.reverse"}, {"name": "CNFA.determinize_spec", "content": "def CNFA.determinize_spec (m : CNFA n)\n  {M : NFA' n} (hsim : m.Sim M) :\n    m.determinize.Sim M.determinize :="}, {"name": "bv_to_set", "content": "private def bv_to_set (bv : BitVec w) : Set State :=\n  { s | bv.getLsbD s }"}, {"name": "_root_.SetRel.set_eq", "content": "structure _root_.SetRel.set_eq (R : SetRel α β) (A : Set α) (B : Set β) where\n  fwd : a ∈ A → ∃ b ∈ B, a ~[R] b\n  bwd : b ∈ B → ∃ a ∈ A, a ~[R] b"}, {"name": "RawCNFA.lift", "content": "@[inline]\ndef RawCNFA.lift (m₁: RawCNFA (BitVec n1)) (f : Fin n1 → Fin n2) : RawCNFA (BitVec n2) :=\n  let trans := (List.range m₁.stateMax).foldl (init := ∅) fun m2 s => processState m2 s\n  { m₁ with trans }\nwhere"}, {"name": "CNFA.lift", "content": "@[inline]\ndef CNFA.lift (m: CNFA n1) (f : Fin n1 → Fin n2) : CNFA n2 :=\n  ⟨m.m.lift f, m.m.lift_wf m.wf⟩"}, {"name": "BitVecs.transport", "content": "def BitVecs.transport (f : Fin n → Fin m) (bvs : BitVecs m) : BitVecs n :=\n  { w := bvs.w, bvs := bvs.bvs.transport f }"}, {"name": "BitVec.transport", "content": "def BitVec.transport (f : Fin n2 → Fin n1) (bv : BitVec n1) : BitVec n2 :=\n  BitVec.ofFn fun i => bv.getLsbD (f i)"}, {"name": "List.Vector.transport", "content": "def List.Vector.transport (v : Vector α m) (f : Fin n → Fin m) : Vector α n :=\n  Vector.ofFn fun i => v.get (f i)"}, {"name": "BitVecs'.transport", "content": "def BitVecs'.transport (f : Fin n → Fin m) (bvs' : BitVecs' m): BitVecs' n :=\n  bvs'.map fun bv => bv.transport f"}, {"name": "RawCNFA.proj", "content": "@[inline]\ndef RawCNFA.proj (m1: RawCNFA (BitVec n1)) (f : Fin n2 → Fin n1) : RawCNFA (BitVec n2) :=\n  let trans := m1.trans.keysArray.foldl (init := Std.HashMap.emptyWithCapacity) process\n  { m1 with trans }\nwhere"}, {"name": "CNFA.proj_spec", "content": "def CNFA.proj_spec (m : CNFA n2) (f : Fin n1 → Fin n2) {M : NFA' n2} :\n    m.Sim M → (m.proj f |>.Sim (M.proj f)) :="}, {"name": "CNFA.proj", "content": "@[inline]\ndef CNFA.proj (m: CNFA n2) (f : Fin n1 → Fin n2) : CNFA n1 :=\n  ⟨m.m.proj f, m.m.proj_wf m.wf⟩"}, {"name": "infixr:67 \"::ᵥ\" => Valuation.cons", "content": "infixr:67 \"::ᵥ\" => Valuation.cons"}], "lib_lemmas": [{"name": "List.nodup_singleton", "module": "Mathlib.Data.List.Nodup"}, {"name": "NFA.eval_append_singleton", "module": "Mathlib.Computability.NFA"}, {"name": "NFA.eval_nil", "module": "Mathlib.Computability.NFA"}, {"name": "Set.mem_setOf_eq", "module": "Mathlib.Data.Set.Operations"}, {"name": "Nat.add_comm", "module": "Init.Data.Nat.Basic"}, {"name": "Fin.add_def", "module": "Init.Data.Fin.Lemmas"}, {"name": "Fin.castLE_castLE", "module": "Init.Data.Fin.Lemmas"}, {"name": "Nat.le_of_eq", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}], "repo_lemmas": [{"name": "ext", "content": "@[ext]\ntheorem ext {x y : BitStream} (h : ∀ i, x i = y i) : x = y"}, {"name": "bisim_comp", "content": "lemma bisim_comp (m : RawCNFA A) :\n    m.Sim M₁ → M₁.Bisim M₂ → m.Sim M₂"}, {"name": "bisimul_comp", "content": "lemma bisimul_comp {m : RawCNFA A} :\n    m.Simul M₁ R₁ ⊤ ∅ → M₁.Bisimul R₂ M₂ →\n    m.Simul M₂ (R₁.comp R₂) ⊤ ∅"}, {"name": "CNFA.bv_recognizes_equiv", "content": "lemma CNFA.bv_recognizes_equiv {m : CNFA n} :\n    m.bv_recognizes L ↔ ∃ (M : NFA' n), m.Sim M ∧ M.accepts = L"}, {"name": "List.Vector.append_get_ge", "content": "@[simp]\nlemma List.Vector.append_get_ge {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: n ≤ i) :\n    (x ++ y).get i = y.get ((i.cast (Nat.add_comm n m) |>.subNat n hlt))"}, {"name": "List.Vector.append_get_lt", "content": "@[simp]\nlemma List.Vector.append_get_lt {x : List.Vector α n} {y : List.Vector α m} {i : Fin (n+m)} (hlt: i < n) :\n    (x ++ y).get i = x.get (i.castLT hlt)"}, {"name": "CNFA.minimize_bv_language", "content": "lemma CNFA.minimize_bv_language {m : CNFA n} :\n    m.bv_recognizes L → m.minimize.bv_recognizes L"}, {"name": "CNFA.minimize_language", "content": "lemma CNFA.minimize_language {m : CNFA n} :\n    m.recognizes L → m.minimize.recognizes L"}, {"name": "CNFA.reverse_language", "content": "lemma CNFA.reverse_language {m : CNFA n} (hl : m.recognizes L) : m.reverse.recognizes L.reverse"}, {"name": "CNFA.reverse_spec", "content": "lemma CNFA.reverse_spec {m : CNFA n} : m.reverse.Sim m.toNFA'.reverse"}, {"name": "RawCNFA.reverse_spec", "content": "lemma RawCNFA.reverse_spec {m : RawCNFA A} (hwf : m.WF) :\n    let m'"}, {"name": "RawCNFA.reverse_spec_procesState", "content": "lemma RawCNFA.reverse_spec_procesState {m : RawCNFA A} (hwf : m.WF) s₀ a₀ ss' (hs₀ : s₀ ∈ m.states) :\n    let motive m' ss'"}, {"name": "CNFA.determinize_language", "content": "lemma CNFA.determinize_language {m : CNFA n} :\n    m.recognizes L → m.determinize.recognizes L"}, {"name": "CNFA.lift_bv_language", "content": "@[simp]\nlemma CNFA.lift_bv_language {m : CNFA n1} {f : Fin n1 → Fin n2} :\n    m.bv_recognizes L → (m.lift f |>.bv_recognizes (BitVecs.transport f ⁻¹' L))"}, {"name": "CNFA.lift_spec", "content": "lemma CNFA.lift_spec (m : CNFA n1) (f : Fin n1 → Fin n2) {M : NFA' n1} :\n    m.Sim M → (m.lift f |>.Sim (M.lift f))"}, {"name": "CNFA.proj_bv_language", "content": "lemma CNFA.proj_bv_language {m : CNFA n2} {f : Fin n1 → Fin n2} :\n    m.bv_recognizes L → (m.proj f |>.bv_recognizes (BitVecs.transport f '' L))"}], "used_local_defs": [{"name": "NFA.sa", "content": "def NFA.sa (_ : NFA α σ) := σ → Language α"}, {"name": "NFA.correct", "content": "structure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q"}, {"name": "BVNRel", "content": "abbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop"}, {"name": "NFA'.sa", "content": "def NFA'.sa (M : NFA' n) := M.σ → BVNRel n"}, {"name": "langRel", "content": "def langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }"}, {"name": "NFA'.correct", "content": "structure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))"}, {"name": "NFA'.correct2", "content": "structure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)"}, {"name": "Alphabet", "content": "abbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)"}, {"name": "finFunToBitVec", "content": "def finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)"}, {"name": "bitVecToFinFun", "content": "def bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]"}, {"name": "NFA.ofFSM", "content": "def NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }"}, {"name": "inFSMRel", "content": "@[simp]\nabbrev inFSMRel (p : FSM arity) {w} (bvn : List.Vector (BitVec w) _) :=\n  bvn.get (Fin.last (FinEnum.card arity)) = p.evalBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))"}, {"name": "NFA'.ofFSM_sa", "content": "def NFA'.ofFSM_sa (p : FSM arity) : (NFA'.ofFSM' p).sa := fun q _ bvn =>\n    inFSMRel p bvn ∧ q = p.carryBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))"}, {"name": "NFA'.ofFSM_correct", "content": "def NFA'.ofFSM_correct (p : FSM arity) :\n    (NFA'.ofFSM' p).correct (ofFSM_sa p) (fun _ bvn => inFSMRel p bvn) :="}, {"name": "CNFA.ofFSM", "content": "def CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where"}, {"name": "NFA.msbState", "content": "inductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype"}, {"name": "liftOp", "content": "def liftOp n : Fin (n + 1) → Fin (n + 3) :=\n  fun k =>\n    if k = n then Fin.last (n+2) else k.castLE (by admit /- proof elided -/\n    )"}, {"name": "liftOp_unchanged", "content": "@[simp]\ndef liftOp_unchanged (k : Fin n) : liftOp n k.castSucc = k.castLE (by simp) :="}, {"name": "TermBinop", "content": "inductive TermBinop where\n| and | or | xor | add | sub"}, {"name": "TermBinop.subst", "content": "def TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂"}, {"name": "TermBinop.openTerm", "content": "def TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)"}, {"name": "TermBinop.openTerm_arity", "content": "@[simp]\ndef TermBinop.openTerm_arity (op : TermBinop) : op.openTerm.arity + 1 = 3 :="}, {"name": "TermBinop.termGadget", "content": "def TermBinop.termGadget (t : TermBinop) : CNFA 3 :=\n  match t with\n  | .and => FSM.ofTerm (.and (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .or => FSM.ofTerm (.or (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .xor => FSM.ofTerm (.xor (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .add => FSM.ofTerm (.add (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .sub => FSM.ofTerm (.sub (.var 0) (.var 1)) |> CNFA.ofFSM"}, {"name": "autOfTermBinop", "content": "def autOfTermBinop (op : TermBinop) (m₁ : CNFA (n + 1)) (m₂ : CNFA (m + 1)) : CNFA ((n ⊔ m) + 1 ) :=\n  let mop : CNFA 3 := op.termGadget\n  let f₁ := liftMaxSuccSucc1 n m\n  let m1' := m₁.lift f₁\n  let f₂ := liftMaxSuccSucc2 n m\n  let m2' := m₂.lift f₂\n  let mop := mop.lift $ liftLast3 (max (FinEnum.card (Fin n)) (FinEnum.card (Fin m)))\n  let m := CNFA.inter m1' m2' |> CNFA.inter mop\n  let mfinal := m.proj (liftOp _)\n  mfinal.minimize"}, {"name": "swapLastTwoBlock", "content": "def swapLastTwoBlock (x : Fin (n + 3)) : Fin (n + 3) :=\n  if x = Fin.last (n+2) then n\n  else if x = n+1 then Fin.last (n + 2)\n  else if x = n then n + 1\n  else x"}, {"name": "TermUnop", "content": "inductive TermUnop where\n| neg | not | shiftL (k : Nat)"}, {"name": "TermUnop.openTerm", "content": "def TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k"}, {"name": "TermUnop.openTerm_arity", "content": "def TermUnop.openTerm_arity (op : TermUnop) : op.openTerm.arity = 1 :="}, {"name": "TermUnop.subst", "content": "def TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k"}], "used_local_lemmas": [{"name": "NFA.correct_spec", "content": "lemma NFA.correct_spec {M : NFA α σ} {ζ : M.sa} {L : Language α} :\n    M.correct ζ L → M.accepts = L"}, {"name": "in_enc", "content": "@[simp]\nlemma in_enc : x ∈ enc '' S ↔ dec x ∈ S"}, {"name": "dec_snoc_in_langRel", "content": "@[simp]\nlemma dec_snoc_in_langRel {n} {R : BVNRel n} {w : BitVecs' n} {a : BitVec n} :\n    dec (w ++ [a]) ∈ langRel R ↔\n      R (List.Vector.ofFn fun k => .cons (a.getLsbD k) ((dec w).bvs.get k))"}, {"name": "NFA'.correct_spec", "content": "lemma NFA'.correct_spec {M : NFA' n} {ζ : M.sa} {L : BVNRel n} :\n    M.correct ζ L → M.accepts = langRel L"}, {"name": "NFA'.ofFSM_spec", "content": "@[simp]\nlemma NFA'.ofFSM_spec (t : Term) :\n    (ofFSM (FSM.ofTerm t)).accepts = t.language"}, {"name": "CNFA.ofFSM_spec", "content": "lemma CNFA.ofFSM_spec (p : FSM arity) :\n    (CNFA.ofFSM p).Sim (NFA'.ofFSM p)"}, {"name": "CNFA.ofFSM_bv_language", "content": "lemma CNFA.ofFSM_bv_language :\n    (CNFA.ofFSM (FSM.ofTerm t)).bv_recognizes t.language"}, {"name": "TermBinop.subst_arity'", "content": "lemma TermBinop.subst_arity' {op : TermBinop} : (op.subst t₁ t₂).arity + 1= t₁.arity ⊔ t₂.arity + 1"}, {"name": "BitVecs.cast_eq", "content": "@[simp]\nlemma BitVecs.cast_eq  (x : BitVecs n) (h : n = n') : h ▸ x = x.cast h"}, {"name": "TermBinop.alt_lang", "content": "lemma TermBinop.alt_lang {t₁ t₂ : Term} (op : TermBinop) :\n  (op.subst_arity' ▸ (op.subst t₁ t₂).language) =\n    let lop : Set (BitVecs 3) := op.openTerm_arity ▸ op.openTerm.language\n    let lop' : Set (BitVecs ((t₁.arity ⊔ t₂.arity) + 3)) := lop.lift (liftLast3 (max t₁.arity t₂.arity))\n    let l₁ := t₁.language.lift (liftMaxSuccSucc1 t₁.arity t₂.arity)\n    let l₂ := t₂.language.lift (liftMaxSuccSucc2 t₁.arity t₂.arity)\n    let l := l₁ ∩ l₂ ∩ lop'\n    l.proj (liftOp _)"}, {"name": "TermUnop.subst_arity'", "content": "@[simp]\nlemma TermUnop.subst_arity' {op : TermUnop} : (op.subst t).arity + 1 = t.arity + 1"}], "local_ctx": "import Batteries.Data.Fin.Basic\n\nimport Batteries.Data.Fin.Lemmas\n\nimport Blase.SingleWidth.Defs\n\nimport Blase.AutoStructs.Constructions\n\nimport Blase.AutoStructs.Defs\n\nimport Blase.AutoStructs.FiniteStateMachine\n\nimport Mathlib.Tactic.Ring\n\nimport Mathlib.Data.Nat.Size -- TODO: remove and get rid of shiftLeft_eq_mul_pow use\n\nimport Mathlib.Data.BitVec\n\nimport Mathlib.Tactic.FinCases\n\nopen Fin.NatCast\n\nopen Mathlib\n\ndef NFA.sa (_ : NFA α σ) := σ → Language α\n\nstructure NFA.correct (M : NFA α σ) (ζ : M.sa) (L : Language α) where\n  cond1 : ∀ w, (w ∈ L ↔ ∃ q ∈ M.accept, w ∈ ζ q)\n  cond2 : ∀ w q, q ∈ M.eval w ↔ w ∈ ζ q\n\nabbrev BVNRel n := ∀ ⦃w⦄, List.Vector (BitVec w) n → Prop\n\ndef NFA'.sa (M : NFA' n) := M.σ → BVNRel n\n\ndef langRel (R : BVNRel n) : Set (BitVecs n) :=\n  { bvs | R bvs.bvs }\n\nstructure NFA'.correct (M : NFA' n) (ζ : M.sa) (L : BVNRel n) where\n  cond1 : ∀ ⦃w⦄ (bvn : List.Vector (BitVec w) n), (L bvn ↔ ∃ q ∈ M.M.accept, ζ q bvn)\n  cond2 q : q ∈ M.M.start ↔ ζ q (List.Vector.replicate n .nil)\n  cond3 q a {w} (bvn : List.Vector (BitVec w) n) : q ∈ M.M.stepSet { q | ζ q bvn } a ↔\n              ζ q (List.Vector.ofFn fun k => BitVec.cons (a.getLsbD k) (bvn.get k))\n\nstructure NFA'.correct2 (M : NFA' 2) (ζ : M.sa2) (L : BVRel) where\n  cond1 : ∀ (bv1 bv2 : BitVec w), (L bv1 bv2 ↔ ∃ q ∈ M.M.accept, ζ q bv1 bv2)\n  cond2 q : q ∈ M.M.start ↔ ζ q .nil .nil\n  cond3 q a w (bv1 bv2 : BitVec w) : q ∈ M.M.stepSet { q | ζ q bv1 bv2 } a ↔\n              ζ q (BitVec.cons (a.getLsbD 0) bv1) (BitVec.cons (a.getLsbD 1) bv2)\n\nsection fsm\n\nabbrev Alphabet (arity: Type) [FinEnum arity] := BitVec (FinEnum.card arity + 1)\n\nvariable {arity : Type} [FinEnum arity]\n\ndef finFunToBitVec [fe : FinEnum carry] (c : carry → Bool) : BitVec (FinEnum.card carry) :=\n  BitVec.ofFn fun i => c (fe.equiv.invFun i)\n\ndef bitVecToFinFun [FinEnum ar] (bv : BitVec $ FinEnum.card ar) : ar → Bool :=\n  fun c => bv[FinEnum.equiv.toFun c]\n\ndef NFA.ofFSM (p : FSM arity) : NFA (Alphabet arity) (p.α → Bool) where\n  start := { q | q = p.initCarry }\n  accept := ⊤\n  step s a := {s' |\n    let (s'', b) := p.nextBit s (bitVecToFinFun (a.truncate $ FinEnum.card arity))\n    s' = s'' ∧ a.msb = b }\n\n@[simp]\nabbrev inFSMRel (p : FSM arity) {w} (bvn : List.Vector (BitVec w) _) :=\n  bvn.get (Fin.last (FinEnum.card arity)) = p.evalBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))\n\ndef NFA'.ofFSM_sa (p : FSM arity) : (NFA'.ofFSM' p).sa := fun q _ bvn =>\n    inFSMRel p bvn ∧ q = p.carryBV (fun ar => bvn.get (FinEnum.equiv.toFun ar))\n\ndef NFA'.ofFSM_correct (p : FSM arity) :\n    (NFA'.ofFSM' p).correct (ofFSM_sa p) (fun _ bvn => inFSMRel p bvn) :=\n\nopen BitStream in\n\ndef CNFA.ofFSM (p : FSM arity) : CNFA (FinEnum.card arity + 1) :=\n  worklistRun (BitVec (FinEnum.card p.α))\n    (fun _ => true)\n    #[finFunToBitVec p.initCarry]\n    (by admit /- proof elided -/\n    )\n    f\n  where\n\nend fsm\n\nsection nfas_relations\n\ninductive NFA.msbState : Type where\n| i | f\nderiving DecidableEq, Fintype\n\nend nfas_relations\n\ndef liftOp n : Fin (n + 1) → Fin (n + 3) :=\n  fun k =>\n    if k = n then Fin.last (n+2) else k.castLE (by admit /- proof elided -/\n    )\n\n@[simp]\ndef liftOp_unchanged (k : Fin n) : liftOp n k.castSucc = k.castLE (by simp) :=\n\ninductive TermBinop where\n| and | or | xor | add | sub\n\ndef TermBinop.subst (op : TermBinop) (t₁ t₂ : Term) : Term :=\n  match op with\n  | .and => .and t₁ t₂\n  | .or => .or t₁ t₂\n  | .xor => .xor t₁ t₂\n  | .add => .add t₁ t₂\n  | .sub => .sub t₁ t₂\n\ndef TermBinop.openTerm (op : TermBinop) : Term := op.subst (.var 0) (.var 1)\n\n@[simp]\ndef TermBinop.openTerm_arity (op : TermBinop) : op.openTerm.arity + 1 = 3 :=\n\ndef TermBinop.termGadget (t : TermBinop) : CNFA 3 :=\n  match t with\n  | .and => FSM.ofTerm (.and (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .or => FSM.ofTerm (.or (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .xor => FSM.ofTerm (.xor (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .add => FSM.ofTerm (.add (.var 0) (.var 1)) |> CNFA.ofFSM\n  | .sub => FSM.ofTerm (.sub (.var 0) (.var 1)) |> CNFA.ofFSM\n\ndef autOfTermBinop (op : TermBinop) (m₁ : CNFA (n + 1)) (m₂ : CNFA (m + 1)) : CNFA ((n ⊔ m) + 1 ) :=\n  let mop : CNFA 3 := op.termGadget\n  let f₁ := liftMaxSuccSucc1 n m\n  let m1' := m₁.lift f₁\n  let f₂ := liftMaxSuccSucc2 n m\n  let m2' := m₂.lift f₂\n  let mop := mop.lift $ liftLast3 (max (FinEnum.card (Fin n)) (FinEnum.card (Fin m)))\n  let m := CNFA.inter m1' m2' |> CNFA.inter mop\n  let mfinal := m.proj (liftOp _)\n  mfinal.minimize\n\ndef swapLastTwoBlock (x : Fin (n + 3)) : Fin (n + 3) :=\n  if x = Fin.last (n+2) then n\n  else if x = n+1 then Fin.last (n + 2)\n  else if x = n then n + 1\n  else x\n\ninductive TermUnop where\n| neg | not | shiftL (k : Nat)\n\ndef TermUnop.openTerm (op : TermUnop) : Term :=\n  match op with\n  | .neg => .neg (.var 0)\n  | .not => .not (.var 0)\n  | .shiftL k => .shiftL (.var 0) k\n\ndef TermUnop.openTerm_arity (op : TermUnop) : op.openTerm.arity = 1 :=\n\ndef TermUnop.subst (op : TermUnop) (t : Term) : Term :=\n  match op with\n  | .neg => .neg t\n  | .not => .not t\n  | .shiftL k => .shiftL t k", "target_theorem": "lemma autOfTermBinop_bv_language op {t₁ t₂ : Term} (m₁ : CNFA (t₁.arity + 1)) (m₂ : CNFA (t₂.arity + 1)) :\n    m₁.bv_recognizes t₁.language →\n    m₂.bv_recognizes t₂.language →\n    (autOfTermBinop op m₁ m₂ |>.bv_recognizes (op.subst_arity' ▸ (op.subst t₁ t₂).language)) :=", "ground_truth_proof": ":= by\n  rintro hrec₁ hrec₂\n  simp [autOfTermBinop]\n  rw [TermBinop.alt_lang]\n  simp\n  apply CNFA.minimize_bv_language\n  apply CNFA.proj_bv_language\n  ac_nf\n  apply CNFA.inter_bv_language\n  · apply CNFA.lift_bv_language\n    rcases op <;> simp [TermBinop.termGadget, TermBinop.openTerm] <;> apply CNFA.ofFSM_bv_language\n  · apply CNFA.inter_bv_language\n    · apply CNFA.lift_bv_language; assumption\n    · apply CNFA.lift_bv_language; assumption", "nesting_depth": 10, "transitive_dep_count": 292, "subset_aristotle": false, "category": "Compiler"}
{"id": 354, "thm_name": "Tensor1D.extractslice_insertslice", "thm_stmt": "theorem extractslice_insertslice [Inhabited α]\n  (t: Tensor1d α)\n  (sliceix: Nat)\n  (slice: Tensor1d α)\n  (CORRECT: ((t.insertslice sliceix slice).extract sliceix slice.size).size ≠ 0)\n  : (t.insertslice sliceix slice).extract sliceix slice.size = slice", "lean_root": "lean-mlir", "rel_path": "SSA/Projects/Tensor1D/Tensor1D.lean", "imports": ["import LeanMLIR.Util", "import LeanMLIR.Framework", "import Mathlib.Tactic.Linarith"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"neg\" : MLIR.Pretty.uniform_op", "content": "syntax \"neg\" : MLIR.Pretty.uniform_op"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = const ($x)\n      $[: $outer_type]? ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _)\n      `(mlir_op| $res:mlir_op_operand = \"const\"()\n          {val = $x:num : $outer_type} : ($outer_type) -> ($outer_type) )"}, {"name": "macro_rules", "content": "macro_rules\n  | `(mlir_op| $res:mlir_op_operand = li ($x)\n     $[: $outer_type]?  ) => do\n      let outer_type ← outer_type.getDM `(mlir_type| _ )\n      `(mlir_op| $res:mlir_op_operand = \"li\"()\n          {imm = $x:num : $outer_type } : ($outer_type) -> ($outer_type))"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithImmediate $reg1 , $x : $t)  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {imm = $x:num : $t} : ($t) -> ($t) )"}, {"name": "macro_rules", "content": "macro_rules\n| `(mlir_op| $res:mlir_op_operand = $op1:MLIR.Pretty.RV.opWithShamt $reg1 , $x  : $t )  => do\n    let some opName := MLIR.EDSL.Pretty.extractOpName op1.raw\n      | Macro.throwUnsupported\n    `(mlir_op| $res:mlir_op_operand = $opName ($reg1) {shamt = $x:num : $t}  : ($t) -> ($t) )"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Tensor1D.Index", "content": "abbrev Index := ℕ"}, {"name": "Tensor1D.Tensor1d", "content": "structure Tensor1d (α : Type) [Inhabited α] where\n  size : Index\n  val :  Index → α\n  spec : ∀ (ix: Index), ix >= size -> val ix = default"}, {"name": "Tensor1D.Tensor1d.extract", "content": "def Tensor1d.extract [Inhabited α] (t: Tensor1d α)\n  (left: Index) (len: Index) : Tensor1d α :=\n  let right := if (left + len) < t.size then left + len else 0\n  let size := right - left\n  { size := size,\n    val := fun ix =>\n    if left + len < t.size\n    then if (ix < len) then t.val (ix + left) else default\n    else default,\n    spec := by admit /- proof elided -/"}, {"name": "Tensor1D.Tensor1d.insertslice", "content": "def Tensor1d.insertslice  [Inhabited α] (t: Tensor1d α)\n  (sliceix: Nat)\n  (slice : Tensor1d α) : Tensor1d α where\n  size := if sliceix > t.size then 0 else t.size + slice.size\n  val := fun ix =>\n    if sliceix > t.size then default \n    else if ix >= t.size + slice.size then default \n    else\n      let go (ix: Nat) : α :=\n        if ix < sliceix then t.val sliceix\n        else if ix < sliceix + slice.size then slice.val (ix - sliceix)\n        else t.val (ix - (sliceix + slice.size))\n      go ix\n  spec := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "Tensor1D.not_lt_is_geq", "content": "theorem not_lt_is_geq {a b: Nat} (NOT_LT: ¬ (a < b)): a >= b"}], "local_ctx": "import LeanMLIR.Framework\n\nimport LeanMLIR.Util\n\nimport Mathlib.Tactic.Linarith\n\nnamespace Tensor1D\n\nabbrev Index := ℕ\n\nstructure Tensor1d (α : Type) [Inhabited α] where\n  size : Index\n  val :  Index → α\n  spec : ∀ (ix: Index), ix >= size -> val ix = default\n\ndef Tensor1d.extract [Inhabited α] (t: Tensor1d α)\n  (left: Index) (len: Index) : Tensor1d α :=\n  let right := if (left + len) < t.size then left + len else 0\n  let size := right - left\n  { size := size,\n    val := fun ix =>\n    if left + len < t.size\n    then if (ix < len) then t.val (ix + left) else default\n    else default,\n    spec := by admit /- proof elided -/\n\ndef Tensor1d.insertslice  [Inhabited α] (t: Tensor1d α)\n  (sliceix: Nat)\n  (slice : Tensor1d α) : Tensor1d α where\n  size := if sliceix > t.size then 0 else t.size + slice.size\n  val := fun ix =>\n    if sliceix > t.size then default \n    else if ix >= t.size + slice.size then default \n    else\n      let go (ix: Nat) : α :=\n        if ix < sliceix then t.val sliceix\n        else if ix < sliceix + slice.size then slice.val (ix - sliceix)\n        else t.val (ix - (sliceix + slice.size))\n      go ix\n  spec := by admit /- proof elided -/", "target_theorem": "theorem extractslice_insertslice [Inhabited α]\n  (t: Tensor1d α)\n  (sliceix: Nat)\n  (slice: Tensor1d α)\n  (CORRECT: ((t.insertslice sliceix slice).extract sliceix slice.size).size ≠ 0)\n  : (t.insertslice sliceix slice).extract sliceix slice.size = slice :=", "ground_truth_proof": ":= by {\n    simp[Tensor1d.insertslice, Tensor1d.extract]\n    cases slice\n    simp;\n    rename_i slicesize sliceval spec\n    by_cases A : (t.size < sliceix) <;> simp[A]\n\n    case pos => simp[Tensor1d.insertslice, Tensor1d.extract, A] at CORRECT ;\n    case neg =>\n      have B : t.size >= sliceix := not_lt_is_geq A\n\n      by_cases C:(sliceix < t.size) <;> simp[C]\n      case neg => simp [Tensor1d.insertslice, Tensor1d.extract, A, C] at CORRECT\n      case pos =>\n          funext ix\n          by_cases D: (ix < slicesize) <;> simp[D]\n          case neg =>\n            -- here we fail, because we do not know that 'slice' behaves like a\n            -- real tensor that returns 'default' outside of its range.\n            -- This is something we need to add into the spec of a Tensor.\n            have E : ix >= slicesize := by simp[Index] at *; linarith\n            simp[spec _ E]\n          case pos =>\n            try simp\n            by_cases E:(t.size + slicesize <= ix + sliceix) <;> simp[E]\n            case pos =>\n              have CONTRA : False := by simp[Index] at *; linarith;\n              simp at CONTRA;\n            case neg =>\n              intros K\n              have CONTRA : False := by simp[Index] at *; linarith\n              simp at CONTRA\n}", "nesting_depth": 2, "transitive_dep_count": 8, "subset_aristotle": false, "category": "Compiler"}
{"id": 355, "thm_name": "TLA.wf1", "thm_stmt": "@[tla_derive]\ntheorem wf1 (p q : pred σ) (next a : action σ) :\n  ((p ∧ ⟨next⟩ ⇒ ◯ p ∨ ◯ q) ∧\n   (p ∧ ⟨next⟩ ∧ ⟨a⟩ ⇒ ◯ q) ∧\n   (p ⇒ Enabled a ∨ q) ∧\n   (□ ⟨next⟩ ∧ 𝒲ℱ a)) |-tla- (p ↝ q)", "lean_root": "Lentil", "rel_path": "Lentil/Rules/WF.lean", "imports": ["import Lentil.Tactics.Structural", "import Lentil.Gadgets.TheoremDeriving", "import Lentil.Rules.Basic"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Std.Associative", "module": "Init.Core"}, {"name": "Std.Commutative", "module": "Init.Core"}], "used_repo_defs": [{"name": "syntax:15 tlafml:16 \" → \" tlafml:15 : tlafml", "content": "syntax:15 tlafml:16 \" → \" tlafml:15 : tlafml\n\nsyntax:35 tlafml:36 \" ∧ \" tlafml:35 : tlafml\n\nsyntax:30 tlafml:31 \" ∨ \" tlafml:30 : tlafml\n\nsyntax:20 tlafml:21 \" ↝ \" tlafml:20 : tlafml\n\nsyntax:25 tlafml:26 \" 𝑈 \" tlafml:25 : tlafml\n\nsyntax:17 tlafml:18 \" ⇒ \" tlafml:17 : tlafml\n\nsyntax:max tlafml:max \" |-tla- \" tlafml:max : term\n\nsyntax:max tlafml:max \" =tla= \" tlafml:max : term\n\nsyntax \"∀ \" extBinder \", \" tlafml:51 : tlafml\n\nsyntax tlafml_heading_op := \"¬\" <|> \"□\" <|> \"◇\" <|> \"◯\"\n\nsyntax:max tlafml_heading_op tlafml:40 : tlafml\n\nsyntax \"[tlafml|\" tlafml \"]\" : term\n\nsyntax \"∃ \" extBinder \", \" tlafml:51 : tlafml\n\nsyntax:max \"Enabled\" term:40 : tlafml\n\nsyntax \"⟨ \" term \" ⟩\" : tlafml"}, {"name": "macro \"tla_unfold_simp\" : tactic => `(tactic| (simp [tlasimp", "content": "macro \"tla_unfold_simp\" : tactic => `(tactic| (simp [tlasimp_def] at *))\n\nsyntax:max \"|-tla- \" tlafml:max : term\n\nsyntax:arg \"𝒲ℱ\" term:max : tlafml\n\nsyntax (name := tlaDerive) \"tla_derive\" : attr"}, {"name": "macro \"tla_unfold_simp'\" : tactic => `(tactic| (tla_unfold_s", "content": "macro \"tla_unfold_simp'\" : tactic => `(tactic| (tla_unfold_simp ; (try simp only [execsimp] at *)))"}, {"name": "macro_rules", "content": "macro_rules\n  | `([tlafml| ( $f:tlafml ) ]) => `([tlafml| $f ])\n  | `([tlafml| ⌜ $t:term ⌝ ]) => `(TLA.state_pred $t)\n  | `([tlafml| ⌞ $t:term ⌟ ]) => `(TLA.pure_pred $t)\n  | `([tlafml| ⟨ $t:term ⟩ ]) => `(TLA.action_pred $t)\n  | `([tlafml| ⊤ ]) => `(TLA.tla_true)\n  | `([tlafml| ⊥ ]) => `(TLA.tla_false)\n  | `([tlafml| $op:tlafml_heading_op $f:tlafml ]) => do\n    let opterm ← match op with\n      | `(tlafml_heading_op|¬) => `(TLA.tla_not)\n      | `(tlafml_heading_op|□) => `(TLA.always)\n      | `(tlafml_heading_op|◇) => `(TLA.eventually)\n      | `(tlafml_heading_op|◯) => `(TLA.later)\n      | _ => Macro.throwUnsupported\n    `($opterm [tlafml| $f ])\n  | `([tlafml| Enabled $t:term ]) => `(TLA.tla_enabled $t)\n  | `([tlafml| $f1:tlafml → $f2:tlafml ]) => `(TLA.tla_implies [tlafml| $f1 ] [tlafml| $f2 ])\n  | `([tlafml| $f1:tlafml ∧ $f2:tlafml ]) => `(TLA.tla_and [tlafml| $f1 ] [tlafml| $f2 ])\n  | `([tlafml| $f1:tlafml ∨ $f2:tlafml ]) => `(TLA.tla_or [tlafml| $f1 ] [tlafml| $f2 ])\n  | `([tlafml| $f1:tlafml 𝑈 $f2:tlafml ]) => `(TLA.tla_until [tlafml| $f1 ] [tlafml| $f2 ])\n  | `([tlafml| ∀ $x:ident, $f:tlafml]) => `(TLA.tla_forall fun $x:ident => [tlafml| $f ])\n  | `([tlafml| ∀ $x:ident : $t, $f:tlafml]) => `(TLA.tla_forall fun $x:ident : $t => [tlafml| $f ])\n  | `([tlafml| ∃ $x:ident, $f:tlafml]) => `(TLA.tla_exists fun $x:ident => [tlafml| $f ])\n  | `([tlafml| ∃ $x:ident : $t, $f:tlafml]) => `(TLA.tla_exists fun $x:ident : $t => [tlafml| $f ])\n  | `([tlafml| $op:tlafml_bigop $x:binderIdent ∈ $l:term, $f:tlafml]) =>\n    \n    match op with\n    | `(tlafml_bigop|⋀ ) => do `(TLA.tla_bigwedge (fun $(← binderIdentToFunBinder x) => [tlafml| $f ]) $l)\n    | `(tlafml_bigop|⋁ ) => do `(TLA.tla_bigvee (fun $(← binderIdentToFunBinder x) => [tlafml| $f ]) $l)\n    | _ => Macro.throwUnsupported\n  | `([tlafml| $t:term ]) => `($t)"}, {"name": "macro_rules", "content": "macro_rules\n  | `([tlafml| $f1:tlafml ↝ $f2:tlafml ]) => `(TLA.leads_to [tlafml| $f1 ] [tlafml| $f2 ])\n  | `([tlafml| $f1:tlafml ⇒ $f2:tlafml ]) => `(TLA.always_implies [tlafml| $f1 ] [tlafml| $f2 ])\n  | `([tlafml| 𝒲ℱ $t:term ]) => `(TLA.weak_fairness $t)"}, {"name": "macro_rules", "content": "macro_rules\n  | `($f1:tlafml |-tla- $f2:tlafml) => `(TLA.pred_implies [tlafml| $f1 ] [tlafml| $f2 ])\n  | `(|-tla- $f1:tlafml) => `(TLA.valid [tlafml| $f1 ])\n  | `($f1:tlafml =tla= $f2:tlafml) => `([tlafml| $f1 ] = [tlafml| $f2 ])\n  | `($e:term |=tla= $f:tlafml) => `(TLA.exec.satisfies [tlafml| $f ] $e)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic| try_unfold_at_all $idt:ident ) => `(tactic| (try unfold $idt at *) )\n  | `(tactic| try_unfold_at_all $idt:ident $idts:ident* ) => `(tactic| (try unfold $idt at *) ; try_unfold_at_all $idts* )"}, {"name": "exec.drop", "content": "def exec.drop {α : Type u} (k : Nat) (σ : exec α) : exec α := λ n => σ (n + k)"}, {"name": "tla_true", "content": "def tla_true {α : Type u} : pred α := pure_pred True"}, {"name": "pred", "content": "def pred (σ : Type u) := exec σ → Prop"}, {"name": "valid", "content": "def valid {α : Type u} (p : pred α) : Prop := ∀ (σ : exec α), σ.satisfies p"}, {"name": "tla_until", "content": "def tla_until {α : Type u} (p q : pred α) : pred α := λ σ => ∃ i, (q <| σ.drop i) ∧ ∀ j < i, (p <| σ.drop j)"}, {"name": "tla_false", "content": "def tla_false {α : Type u} : pred α := pure_pred False"}, {"name": "pure_pred", "content": "def pure_pred {α : Type u} (p : Prop) : pred α := state_pred (fun _ => p)"}, {"name": "state_pred", "content": "def state_pred {σ : Type u} (f : σ → Prop) : pred σ :=\n  fun e => f (e 0)"}, {"name": "tla_and", "content": "def tla_and {α : Type u} (p q : pred α) : pred α := fun σ => p σ ∧ q σ"}, {"name": "tla_exists", "content": "def tla_exists {α : Sort u} {β : Type v} (p : α → pred β) : pred β := fun σ => ∃ x, p x σ"}, {"name": "tla_forall", "content": "def tla_forall {α : Sort u} {β : Type v} (p : α → pred β) : pred β := fun σ => ∀ x, p x σ"}, {"name": "TLA.always_implies", "content": "def TLA.always_implies {α : Type u} (p q : TLA.pred α) : TLA.pred α := [tlafml| □ (p → q) ]"}, {"name": "tla_bigwedge", "content": "def tla_bigwedge {α : Type u} {β : Type v} {c} [Foldable c] (f : β → pred α) (s : c β) : pred α :=\n  Foldable.fold tla_and tla_true f s"}, {"name": "TLA.weak_fairness", "content": "def TLA.weak_fairness {α : Type u} (a : action α) : pred α := [tlafml| □ ((□ (Enabled a)) → ◇ ⟨a⟩)]"}, {"name": "action", "content": "def action (σ : Type u) := σ → σ → Prop"}, {"name": "eventually", "content": "def eventually {α : Type u} (p : pred α) : pred α := λ σ => ∃ k, p <| σ.drop k"}, {"name": "tla_not", "content": "def tla_not {α : Type u} (p : pred α) : pred α := fun σ => ¬ p σ"}, {"name": "tla_or", "content": "def tla_or {α : Type u} (p q : pred α) : pred α := fun σ => p σ ∨ q σ"}, {"name": "pred_implies", "content": "def pred_implies {α : Type u} (p q : pred α) : Prop := ∀ (σ : exec α), σ.satisfies p → σ.satisfies q"}, {"name": "exec", "content": "def exec (σ : Type u) := Nat → σ"}, {"name": "exec.satisfies", "content": "def exec.satisfies {α : Type u} (p : pred α) (σ : exec α) : Prop := p σ"}, {"name": "tla_enabled", "content": "def tla_enabled {α : Type u} (a : action α) : pred α := state_pred (enabled a)"}, {"name": "enabled", "content": "def enabled {α : Type u} (a : action α) (s : α) : Prop := ∃ s', a s s'"}, {"name": "TLA.leads_to", "content": "def TLA.leads_to {α : Type u} (p q : TLA.pred α) : TLA.pred α := [tlafml| □ (p → ◇ q) ]"}, {"name": "action_pred", "content": "def action_pred {σ : Type u} (a : action σ) : pred σ :=\n  fun e => a (e 0) (e 1)"}, {"name": "tla_implies", "content": "def tla_implies {α : Type u} (p q : pred α) : pred α := fun σ => p σ → q σ"}, {"name": "later", "content": "def later {α : Type u} (p : pred α) : pred α := λ σ => p <| σ.drop 1"}, {"name": "always", "content": "def always {α : Type u} (p : pred α) : pred α := λ σ => ∀ k, p <| σ.drop k"}, {"name": "tla_bigvee", "content": "def tla_bigvee {α : Type u} {β : Type v} {c} [Foldable c] (f : β → pred α) (s : c β) : pred α :=\n  Foldable.fold tla_or tla_false f s"}, {"name": "Foldable", "content": "class Foldable (c : Type u → Type v) where\n  fold {α : Type u} {β : Type w} (op : β → β → β) [Std.Commutative op] [Std.Associative op]\n    (b : β) (f : α → β) (s : c α) : β"}], "lib_lemmas": [{"name": "Nat.add_assoc", "module": "Init.Data.Nat.Basic"}], "repo_lemmas": [{"name": "always_eventually_or_distrib", "content": "@[tladual]\ntheorem always_eventually_or_distrib : (□ ◇ (p ∨ q)) =tla= (□ ◇ p ∨ □ ◇ q)"}, {"name": "dual_lemma", "content": "theorem dual_lemma (p q : pred σ) : ¬ p =tla= ¬ q → (p) =tla= (q)"}, {"name": "eventually_always_and_distrib", "content": "theorem eventually_always_and_distrib : (◇ □ (p ∧ q)) =tla= (◇ □ p ∧ ◇ □ q)"}, {"name": "always_and", "content": "theorem always_and : (□ (p ∧ q)) =tla= (□ p ∧ □ q)"}, {"name": "pred_eq_iff_iff", "content": "theorem pred_eq_iff_iff (p q : pred σ) : (p) =tla= (q) ↔ (p) |-tla- (q) ∧ (q) |-tla- (p)"}, {"name": "eventually_and_split", "content": "@[tladual]\ntheorem eventually_and_split : (◇ (p ∧ q)) |-tla- (◇ p ∧ ◇ q)"}, {"name": "eventually_or", "content": "@[tladual]\ntheorem eventually_or : (◇ (p ∨ q)) =tla= (◇ p ∨ ◇ q)"}, {"name": "wf1'", "content": "theorem wf1' (p q init inv : σ → Prop) (next a : action σ)\n  (hpuntilq : ∀ s s', p s → next s s' → p s' ∨ q s')\n  (haq : ∀ s s', p s → next s s' → a s s' → q s')\n  (henable : ∀ s, inv s → p s → enabled a s ∨ q s)\n  (hinit_inv : ∀ s, init s → inv s)\n  (hnext_inv : ∀ s s', next s s' → inv s → inv s') :\n  (⌜ init ⌝ ∧ □ ⟨next⟩ ∧ 𝒲ℱ a) |-tla- (⌜ p ⌝ ↝ ⌜ q ⌝)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "TLA.wf_alt1", "content": "theorem wf_alt1 : (𝒲ℱ a) =tla= □ ◇ ((¬ Enabled a) ∨ □ ◇ ⟨a⟩)"}, {"name": "TLA.wf_alt1'", "content": "theorem wf_alt1' : (𝒲ℱ a) =tla= □ ◇ ((¬ Enabled a) ∨ ⟨a⟩)"}], "local_ctx": "import Lentil.Rules.Basic\n\nimport Lentil.Tactics.Structural\n\nimport Lentil.Gadgets.TheoremDeriving\n\nopen Classical\n\nnamespace TLA\n\nsection wf\n\nvariable {σ : Type u}\n\nsection wf_def\n\nvariable {a : action σ}\n\nend wf_def", "target_theorem": "@[tla_derive]\ntheorem wf1 (p q : pred σ) (next a : action σ) :\n  ((p ∧ ⟨next⟩ ⇒ ◯ p ∨ ◯ q) ∧\n   (p ∧ ⟨next⟩ ∧ ⟨a⟩ ⇒ ◯ q) ∧\n   (p ⇒ Enabled a ∨ q) ∧\n   (□ ⟨next⟩ ∧ 𝒲ℱ a)) |-tla- (p ↝ q) :=", "ground_truth_proof": ":= by\n  rw [wf_alt1']\n  intro e ⟨hpuntilq, haq, henable, hnext, hwf_alt⟩ k hp\n  specialize hwf_alt k ; rcases hwf_alt with ⟨k1, hwf_alt⟩\n  -- know that: either `q` holds between `k` and `k + k1`, or `p` holds at `k1`\n  -- use `henable` to know that if it is the latter case, then `q` must hold in the next step\n  have htmp : (∃ k' ≤ k1, q <| e.drop (k + k')) ∨ (p <| e.drop (k + k1)) := by\n    clear hwf_alt\n    induction k1 with\n    | zero => right ; assumption\n    | succ n ih => {\n      rw [← Nat.add_assoc]\n      rcases ih with ⟨k', hle, ih⟩ | ih\n      · left ; exists k' ; constructor ; omega ; apply ih\n      · specialize hpuntilq _ ⟨ih, (hnext _)⟩\n        rcases hpuntilq with hq | hq <;> tla_unfold_simp'\n        · right ; apply hq\n        · left ; exists (n + 1) ; aesop\n    }\n  rcases htmp with ⟨k', _, hq⟩ | hq <;> tla_unfold_simp'\n  · aesop\n  · rcases hwf_alt with hq2 | hq2\n    · specialize henable _ hq ; aesop\n    · exists (k1 + 1)\n      specialize haq (k + k1) hq ; rw [← Nat.add_assoc] ; apply haq <;> aesop", "nesting_depth": 27, "transitive_dep_count": 17, "subset_aristotle": false, "category": "Framework"}
{"id": 356, "thm_name": "simulation_step", "thm_stmt": "theorem simulation_step :\n  ∀ C impconf1 impconf2 machconf1,\n  step impconf1 impconf2 ->\n  match_config C impconf1 machconf1 ->\n  ∃ machconf2,\n      (plus (transition C) machconf1 machconf2\n       \\/ (star (transition C) machconf1 machconf2\n           /\\ (measure' impconf2 < measure' impconf1)))\n   /\\ match_config C impconf2 machconf2", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Compil.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import LeroyCompilerVerificationCourse.Sequences"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Exists", "module": "Init.Core"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "cont", "content": "@[grind] inductive cont where\n| Kstop\n| Kseq (c : com) (k : cont)\n| Kwhile (b : bexp) (c : com) (k : cont)"}, {"name": "store", "content": "def store : Type := ident → Int"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "star", "content": "@[grind] inductive star (R : α → α → Prop) : α → α → Prop where\n  | star_refl : ∀ x : α, star R x x\n  | star_step : ∀ {x y z}, R x y → star R y z → star R x z"}, {"name": "plus", "content": "@[grind cases]\ninductive plus (R : α → α → Prop) : α → α → Prop where\n| plus_left : ∀ {a b c}, R a b → star R b c → plus R a c\n\n\ngrind_pattern plus.plus_left => star R b c, plus R a c"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "step", "content": "inductive step : com × cont × store -> com × cont × store -> Prop where\n  | step_assign : ∀ x a k s,\n      step (.ASSIGN x a, k, s) (.SKIP, k, update x (aeval s a) s)\n    \n  | step_seq : ∀ c1 c2 s k,\n      step (.SEQ c1 c2, k, s) (c1, .Kseq c2 k, s)\n      \n  | step_ifthenelse : ∀ b c1 c2 k s,\n      step (.IFTHENELSE b c1 c2, k, s) ((if beval s b then c1 else c2), k, s)\n      \n  | step_while_done : ∀ b c k s,\n      beval s b = false ->\n      step (.WHILE b c, k, s) (.SKIP, k, s)\n      \n  | step_while_true : ∀ b c k s,\n      beval s b = true ->\n      step (.WHILE b c, k, s) (c, .Kwhile b c k, s)\n      \n  | step_skip_seq : ∀ c k s,\n      step (.SKIP, .Kseq c k, s) (c, k, s)\n      \n  | step_skip_while : ∀ b c k s,\n      step (.SKIP, .Kwhile b c k, s) (.WHILE b c, k, s)"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "Exists.elim", "module": "Init.Core"}, {"name": "Int.add_assoc", "module": "Init.Data.Int.Lemmas"}, {"name": "Or.intro_left", "module": "Init.Prelude"}, {"name": "Or.intro_right", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "star_one", "content": "@[grind] theorem star_one (R : α → α → Prop) {a b : α} (h : R a b) : star R a b"}, {"name": "plus_star", "content": "@[grind] theorem plus_star {a b} (h : plus R a b) : star R a b"}, {"name": "star_trans", "content": "@[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c"}, {"name": "plus_right", "content": "theorem plus_right : star R a b -> R b c -> plus R a c"}], "used_local_defs": [{"name": "instr", "content": "@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr"}, {"name": "codelen", "content": "@[grind] def codelen (c : List instr) : Int := c.length"}, {"name": "stack", "content": "def stack : Type := List Int"}, {"name": "config", "content": "def config : Type := Int × stack × store"}, {"name": "instr_at", "content": "@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)"}, {"name": "transition", "content": "@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)"}, {"name": "transitions", "content": "@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)"}, {"name": "compile_aexp", "content": "@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])"}, {"name": "compile_bexp", "content": "@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2"}, {"name": "compile_com", "content": "@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []"}, {"name": "code_at", "content": "@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2"}, {"name": "compile_cont", "content": "inductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc"}, {"name": "match_config", "content": "inductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)"}, {"name": "com_size", "content": "def com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)"}, {"name": "cont_size", "content": "def cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'"}, {"name": "measure'", "content": "def measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)"}], "used_local_lemmas": [{"name": "codelen_cons", "content": "@[grind =] theorem codelen_cons :\n  ∀ i c, codelen (i :: c) = codelen c + 1"}, {"name": "codelen_app", "content": "@[grind =] theorem codelen_app :\n  ∀ c1 c2, codelen (c1 ++ c2) = codelen c1 + codelen c2"}, {"name": "instr_a", "content": "@[grind =>] theorem instr_a : ∀ i c2 c1 pc,\n  pc = codelen c1 ->\n  instr_at (c1 ++ (i :: c2) ) pc = .some i"}, {"name": "code_at_app_right", "content": "@[grind] theorem code_at_app_right :\n  ∀ C pc C1 C2,\n  code_at C pc (C1 ++ C2) ->\n  code_at C (pc + codelen C1) C2"}, {"name": "code_at_to_instr_at", "content": "@[grind] theorem code_at_to_instr_at : code_at C pc (c1 ++ i :: c2) → instr_at C (pc + codelen c1) = .some i"}, {"name": "compile_aexp_correct", "content": "theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :\n  code_at C pc (compile_aexp a) →\n  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s)"}, {"name": "compile_bexp_correct", "content": "theorem compile_bexp_correct (C : List instr) (s : store) (b : bexp) (d1 d0 : Int) (pc : Int) (stk : stack) (h : code_at C pc (compile_bexp b d1 d0))  :\n   transitions C\n       (pc, stk, s)\n       (pc + codelen (compile_bexp b d1 d0) + (if beval s b then d1 else d0), stk, s)"}, {"name": "compile_cont_Kseq_inv", "content": "theorem compile_cont_Kseq_inv (C : List instr) (c : com) (k :cont) (pc : Int) (s : store) (H : compile_cont C (.Kseq c k) pc) :\n  ∃ pc',\n  star (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com c)\n  ∧ compile_cont C k (pc' + codelen (compile_com c))"}, {"name": "compile_cont_Kwhile_inv", "content": "theorem compile_cont_Kwhile_inv (C : List instr) (b : bexp) (c : com) (k : cont) (pc : Int) (s : store) (H : compile_cont C (.Kwhile b c k) pc) :\n  ∃ pc',\n  plus (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com (.WHILE b c))\n  ∧ compile_cont C k (pc' + codelen (compile_com (.WHILE b c)))"}, {"name": "match_config_skip", "content": "theorem match_config_skip (C : List instr) (k : cont) (s : store) (pc : Int) (H : compile_cont C k pc) :\n match_config C (.SKIP, k, s) (pc, [], s)"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\n@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr\n\n@[grind] def codelen (c : List instr) : Int := c.length\n\ndef stack : Type := List Int\n\ndef config : Type := Int × stack × store\n\n@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)\n\n@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n\n@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)\n\n@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])\n\n@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2\n\n@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []\n\n@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2\n\ninductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc\n\ninductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)\n\ndef com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)\n\ndef cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'\n\ndef measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)", "target_theorem": "theorem simulation_step :\n  ∀ C impconf1 impconf2 machconf1,\n  step impconf1 impconf2 ->\n  match_config C impconf1 machconf1 ->\n  ∃ machconf2,\n      (plus (transition C) machconf1 machconf2\n       \\/ (star (transition C) machconf1 machconf2\n           /\\ (measure' impconf2 < measure' impconf1)))\n   /\\ match_config C impconf2 machconf2 :=", "ground_truth_proof": ":= by\n    intro C impconf1 impconf2 matchconf1 STEP MATCH\n    cases MATCH\n    case match_config_intro c k st pc h₁ h₂ =>\n      rcases impconf2 with ⟨c' , k', s'⟩\n      cases STEP\n      next x a =>\n        constructor\n        constructor\n        case h.left =>\n          apply Or.intro_left\n          apply plus_right\n          apply compile_aexp_correct\n          rotate_left\n          exact a\n          rotate_left\n          · grind\n          · apply transition.trans_setvar\n            rotate_left\n            exact x\n            grind\n        apply match_config_skip\n        grind\n      next c2 =>\n        constructor\n        constructor\n        apply Or.intro_right\n        constructor\n        apply star.star_refl\n        simp [measure', com_size, cont_size]\n        grind\n        constructor\n        simp [compile_com] at h₁\n        grind\n        apply compile_cont.ccont_seq\n        · grind\n        rotate_right\n        · exact pc + codelen (compile_com c') + codelen (compile_com c2)\n        · rfl\n        · grind\n      next b c1 c2 =>\n        generalize h₃ : compile_com c1 = code1\n        generalize h₄ : compile_bexp b 0 (codelen code1 + 1) = codeb\n        generalize h₅ : compile_com c2 = code2\n        simp [compile_com, h₃, h₄, h₅] at h₁ h₂\n        constructor\n        constructor\n        apply Or.intro_right\n        constructor\n        · apply compile_bexp_correct\n          rotate_left\n          · exact b\n          · exact 0\n          · exact (codelen code1 + 1)\n          · grind\n        · simp [measure', com_size]\n          grind\n        · rw [h₄]\n          constructor\n          · by_cases beval st b = true\n            case pos isTrue =>\n              simp [isTrue] at *\n              grind\n            case neg isFalse =>\n              simp [isFalse] at *\n              rw [h₅]\n              have := @code_at_app_right C pc (codeb ++ code1 ++ [instr.Ibranch (codelen code2)]) code2 (by grind)\n              simp [codelen_cons, codelen_app] at this\n              simp [codelen] at *\n              grind\n          · by_cases beval st b = true\n            case pos isTrue =>\n              simp [isTrue] at *\n              apply compile_cont.ccont_branch\n              rotate_right\n              · exact (pc + codelen (codeb ++ (code1 ++ instr.Ibranch (codelen code2) :: code2)))\n              rotate_right\n              · exact codelen code2\n              any_goals grind\n            case neg isFalse =>\n              simp [isFalse] at *\n              grind\n      next b c isFalse =>\n        generalize h₃ : compile_com c = codec\n        generalize h₄ : (compile_bexp b 0 (codelen codec + 1)) = codeb\n        constructor\n        constructor\n        apply Or.intro_right\n        constructor\n        · apply compile_bexp_correct\n          rotate_left\n          · exact b\n          · exact 0\n          · exact (codelen codec + 1)\n          · grind\n        · simp [measure', com_size]\n          fun_induction com_size with grind\n        · simp [isFalse]\n          rw [h₄]\n          simp [compile_com] at h₂ h₁\n          rw [h₃, h₄] at h₂ h₁\n          constructor\n          · grind\n          · grind\n      next b isTrue =>\n        generalize h₃ : compile_com c' = codec\n        generalize h₄ : compile_bexp b 0 (codelen codec + 1) = codeb\n        constructor\n        constructor\n        · apply Or.intro_right\n          constructor\n          · apply compile_bexp_correct\n            rotate_left\n            · exact b\n            · exact 0\n            · exact (codelen codec + 1)\n            · grind\n          · simp [measure', cont_size, com_size]\n        · simp [isTrue]\n          rw [h₄]\n          constructor\n          · simp [compile_com] at h₁\n            rw [h₃, h₄] at h₁\n            grind\n          · simp [compile_com, h₃, h₄] at h₁ h₂\n            apply compile_cont.ccont_while\n            rotate_left 4\n            · exact h₂\n            · exact (-(codelen codeb + codelen codec + 1))\n            rotate_left 3\n            · simp [compile_com, h₃, h₄]\n              exact h₁\n            · grind\n            · grind\n            · grind\n      next =>\n        have := compile_cont_Kseq_inv C c' k' pc st (by simp [compile_com, codelen] at h₂; grind)\n        apply Exists.elim this\n        intro pc'\n        intro ⟨w₁, w₂⟩\n        exists (pc', [], st)\n        constructor\n        · apply Or.intro_right\n          constructor\n          · exact w₁\n          · simp [measure', cont_size, com_size]\n        · constructor\n          · exact w₂.1\n          · simp [compile_com, codelen] at h₂\n            grind\n      next b c =>\n        have := compile_cont_Kwhile_inv C b c k' pc st (by simp [compile_com, codelen] at h₂; grind)\n        apply Exists.elim this\n        intro pc'\n        intro ⟨ w₁, w₂ ⟩\n        exists (pc', [], st)\n        constructor\n        · apply Or.intro_left\n          · exact w₁\n        · constructor\n          · exact w₂.1\n          · exact w₂.2", "nesting_depth": 4, "transitive_dep_count": 64, "subset_aristotle": true, "category": "Compiler"}
{"id": 357, "thm_name": "compile_com_correct_terminating", "thm_stmt": "theorem compile_com_correct_terminating (s s' : store) (c : com) (h₁ : cexec s c s') :\n ∀ C pc stk, code_at C pc (compile_com c) →\n  transitions C\n      (pc, stk, s)\n      (pc + codelen (compile_com c), stk, s')", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Compil.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import LeroyCompilerVerificationCourse.Sequences"], "used_lib_defs": [{"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "store", "content": "def store : Type := ident → Int"}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "star", "content": "@[grind] inductive star (R : α → α → Prop) : α → α → Prop where\n  | star_refl : ∀ x : α, star R x x\n  | star_step : ∀ {x y z}, R x y → star R y z → star R x z"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "cexec", "content": "@[grind] inductive cexec : store → com → store → Prop where\n  | cexec_skip :\n      cexec s .SKIP s\n  | cexec_assign :\n      cexec s (.ASSIGN x a) (update x (aeval s a) s)\n  | cexec_seq :\n      cexec s c1 s' -> cexec s' c2 s'' ->\n      cexec s (.SEQ c1 c2) s''\n  | cexec_ifthenelse :\n      cexec s (if beval s b then c1 else c2) s' ->\n      cexec s (.IFTHENELSE b c1 c2) s'\n  | cexec_while_done :\n      beval s b = false ->\n      cexec s (.WHILE b c) s\n  | cexec_while_loop :\n      beval s b = true -> cexec s c s' -> cexec s' (.WHILE b c) s'' ->\n      cexec s (.WHILE b c) s''"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "star_one", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "Int.add_assoc", "module": "Init.Data.Int.Lemmas"}, {"name": "Int.add_zero", "module": "Init.Data.Int.Lemmas"}], "repo_lemmas": [{"name": "star_trans", "content": "@[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c"}], "used_local_defs": [{"name": "instr", "content": "@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr"}, {"name": "codelen", "content": "@[grind] def codelen (c : List instr) : Int := c.length"}, {"name": "stack", "content": "def stack : Type := List Int"}, {"name": "config", "content": "def config : Type := Int × stack × store"}, {"name": "instr_at", "content": "@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)"}, {"name": "transition", "content": "@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)"}, {"name": "transitions", "content": "@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)"}, {"name": "compile_aexp", "content": "@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])"}, {"name": "compile_bexp", "content": "@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2"}, {"name": "compile_com", "content": "@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []"}, {"name": "code_at", "content": "@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2"}], "used_local_lemmas": [{"name": "codelen_cons", "content": "@[grind =] theorem codelen_cons :\n  ∀ i c, codelen (i :: c) = codelen c + 1"}, {"name": "codelen_singleton", "content": "@[grind =] theorem codelen_singleton : codelen [i] = 1"}, {"name": "codelen_app", "content": "@[grind =] theorem codelen_app :\n  ∀ c1 c2, codelen (c1 ++ c2) = codelen c1 + codelen c2"}, {"name": "instr_a", "content": "@[grind =>] theorem instr_a : ∀ i c2 c1 pc,\n  pc = codelen c1 ->\n  instr_at (c1 ++ (i :: c2) ) pc = .some i"}, {"name": "code_at_app_right", "content": "@[grind] theorem code_at_app_right :\n  ∀ C pc C1 C2,\n  code_at C pc (C1 ++ C2) ->\n  code_at C (pc + codelen C1) C2"}, {"name": "code_at_to_instr_at", "content": "@[grind] theorem code_at_to_instr_at : code_at C pc (c1 ++ i :: c2) → instr_at C (pc + codelen c1) = .some i"}, {"name": "compile_aexp_correct", "content": "theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :\n  code_at C pc (compile_aexp a) →\n  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s)"}, {"name": "compile_bexp_correct", "content": "theorem compile_bexp_correct (C : List instr) (s : store) (b : bexp) (d1 d0 : Int) (pc : Int) (stk : stack) (h : code_at C pc (compile_bexp b d1 d0))  :\n   transitions C\n       (pc, stk, s)\n       (pc + codelen (compile_bexp b d1 d0) + (if beval s b then d1 else d0), stk, s)"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\n@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr\n\n@[grind] def codelen (c : List instr) : Int := c.length\n\ndef stack : Type := List Int\n\ndef config : Type := Int × stack × store\n\n@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)\n\n@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n\n@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)\n\n@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])\n\n@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2\n\n@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []\n\n@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2", "target_theorem": "theorem compile_com_correct_terminating (s s' : store) (c : com) (h₁ : cexec s c s') :\n ∀ C pc stk, code_at C pc (compile_com c) →\n  transitions C\n      (pc, stk, s)\n      (pc + codelen (compile_com c), stk, s') :=", "ground_truth_proof": ":= by\n  induction h₁\n  case cexec_skip =>\n    intro C pc stk h\n    unfold compile_com\n    dsimp [codelen]\n    simp only [Int.add_zero]\n    apply star.star_refl\n  case cexec_assign s' x a =>\n    intro C pc stk h\n    unfold compile_com\n    apply star_trans\n    · apply compile_aexp_correct (a := a)\n      grind\n    · apply star_one\n      · have := @transition.trans_setvar C\n        grind\n  case cexec_seq s'2 c1 s1 c2 s2 cexec1 cexec2 c1_ih c2_ih =>\n    intro C pc stk h\n    apply star_trans\n    · apply c1_ih\n      grind\n    · specialize c2_ih C (pc + codelen (compile_com c1)) stk\n      simp [compile_com, codelen_app]\n      simp [Int.add_assoc] at c2_ih\n      apply c2_ih\n      grind\n  case cexec_ifthenelse s b c1 c2 s' cexec_h ih =>\n    intro C pc stk\n    generalize heq1 : compile_com c1 = code1\n    generalize heq2 : compile_com c2 = code2\n    generalize heq3 : compile_bexp b 0 (codelen code1 + 1) = code3\n    simp [compile_com]\n    rw [heq1, heq2, heq3]\n    intro h\n    apply star_trans\n    · have := compile_bexp_correct C s b 0 (codelen code1 + 1) pc stk (by grind)\n      apply this\n    · by_cases beval s b = true\n      case pos isTrue =>\n        simp [isTrue]\n        apply star_trans\n        · apply ih\n          grind\n        · apply star_one\n          · apply transition.trans_branch (d := codelen code2) <;> grind\n      case neg isFalse =>\n        simp [isFalse]\n        rw [heq3]\n        specialize ih C (pc + codelen code3 + (codelen code1 + 1)) stk\n        simp [isFalse] at ih\n        suffices h2 : code_at C (pc + codelen code3 + (codelen code1 + 1)) (compile_com c2) from by\n          specialize ih h2\n          simp [codelen_app, codelen_cons]\n          have : (pc + codelen code3 + (codelen code1 + 1) + codelen (compile_com c2)) = (pc + (codelen code3 + (codelen code1 + (codelen code2 + 1)))) := by grind\n          rw [this] at ih\n          apply ih\n        have := @code_at_app_right C pc (code3 ++ code1 ++ [instr.Ibranch (codelen code2)]) code2 (by simp[h])\n        grind\n  case cexec_while_done s b c1 isFalse =>\n    intro C pc stk h\n    generalize heq1 : compile_com c1 = code_body\n    generalize heq2 : compile_bexp b 0 (codelen code_body + 1) = code_branch\n    generalize heq3 : - (codelen code_branch + codelen code_body + 1) = d\n    simp [compile_com]\n    rw [heq1, heq2, heq3]\n    simp [codelen_app, codelen_singleton]\n    apply star_trans\n    · apply compile_bexp_correct C s b 0 (codelen code_body + 1) pc stk (by grind)\n    · grind\n  case cexec_while_loop s b c1 s_intermediate s' isTrue cexec1 cexec2 ih1 ih2 =>\n    intro C pc stk\n    generalize heq1 : compile_com c1 = code_body\n    generalize heq2 : compile_bexp b 0 (codelen code_body + 1) = code_branch\n    generalize heq3 : - (codelen code_branch + codelen code_body + 1) = d\n    simp [compile_com]\n    rw [heq1, heq2, heq3]\n    intro h\n    apply star_trans\n    · apply compile_bexp_correct C s b 0 (codelen code_body + 1) pc stk (by grind)\n    · apply star_trans\n      · apply ih1\n        grind\n      · apply star_trans\n        · apply star_one\n          apply transition.trans_branch (d := d)\n          rotate_left\n          rotate_left\n          · exact (pc + codelen code_branch + codelen code_body + 1 + d)\n          · grind\n          · grind\n        · specialize ih2 C (pc + codelen code_branch + codelen code_body + 1 + d) stk\n          suffices h2 : code_at C (pc + codelen code_branch + codelen code_body + 1 + d) (compile_com (com.WHILE b c1)) from by\n            specialize ih2 h2\n            simp [compile_com] at ih2\n            rw [heq1, heq2, heq3] at ih2\n            simp [codelen_app]\n            simp [codelen_app] at ih2\n            have : (pc + codelen code_branch + codelen code_body + 1 + d +\n      (codelen code_branch + (codelen code_body + codelen [instr.Ibranch d])) ) = (pc + (codelen code_branch + (codelen code_body + codelen [instr.Ibranch d]))) := by grind\n            rw [←this]\n            apply ih2\n          grind", "nesting_depth": 5, "transitive_dep_count": 48, "subset_aristotle": true, "category": "Compiler"}
{"id": 358, "thm_name": "fixpoint_join_increasing", "thm_stmt": "theorem fixpoint_join_increasing (_ : Store) (F : Store → Store) (F_mon : ∀ x y, le x y → le (F x) (F y)) (S1 S2 : Store) : le S1 S2 → le (fixpoint_join' S1 F F_mon) (fixpoint_join' S2 F F_mon)", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Fixpoints.lean", "imports": ["import LeroyCompilerVerificationCourse.Constprop", "import LeroyCompilerVerificationCourse.Imp", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "WellFounded", "module": "Init.WF"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.Equiv", "module": "Std.Data.HashMap.Basic"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.add", "module": "Init.Data.Int.Basic"}, {"name": "Int.sub", "module": "Init.Data.Int.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "BitVec", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Store", "content": "def Store := Std.HashMap ident Int"}, {"name": "Join", "content": "@[grind] def Join (S1 S2 : Store) : Store :=\n  S1.filter (fun key _ => S2.get? key == S1.get? key)"}, {"name": "Le", "content": "@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n"}, {"name": "Equal", "content": "def Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2"}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "Beval", "content": "@[grind] def Beval (S : Store) (b : bexp) : Option Bool :=\n  match b with\n  | .TRUE => .some true\n  | .FALSE => .some false\n  | .EQUAL a1 a2 => lift2 (fun m n => m == n) (Aeval S a1) (Aeval S a2)\n  | .LESSEQUAL a1 a2 => lift2 (fun m n => m <= n) (Aeval S a1) (Aeval S a2)\n  | .NOT b1 => lift1 (fun m => !m) (Beval S b1)\n  | .AND b1 b2 => lift2 (fun m n => m && n) (Beval S b1) (Beval S b2)"}, {"name": "lift1", "content": "@[grind] def lift1 {A B : Type} (f : A -> B) (o : Option A) : Option B :=\n  match o with\n  | .some x => .some (f x)\n  | .none => .none"}, {"name": "Aeval", "content": "@[grind] def Aeval (S : Store) (a : aexp) : Option Int :=\n  match a with\n  | .CONST n => .some n\n  | .VAR x => S.get? x\n  | .PLUS a1 a2 => lift2 (Int.add) (Aeval S a1) (Aeval S a2)\n  | .MINUS a1 a2 => lift2 (Int.sub) (Aeval S a1) (Aeval S a2)"}, {"name": "lift2", "content": "@[grind] def lift2 {A B C : Type} (f : A -> B -> C) (o1 : Option A) (o2 : Option B) : Option C :=\n  match o1, o2 with\n  | .some x1, .some x2 => .some (f x1 x2) | _, _ => .none"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "Update", "content": "@[grind] def Update (x : ident) (N : Option Int) (S : Store) : Store :=\n  match N with\n  | .none => S.erase x\n  | .some n => S.insert x n"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "Std.HashMap.Equiv.getElem?_eq", "module": "Std.Data.HashMap.Lemmas"}], "repo_lemmas": [{"name": "Le_Join_l", "content": "theorem Le_Join_l : ∀ S1 S2, Le S1 (Join S1 S2)"}], "used_local_defs": [{"name": "OrderStruct", "content": "@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/"}, {"name": "Monotone", "content": "class Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)"}, {"name": "iterate", "content": "@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']"}, {"name": "Eq'", "content": "@[grind] def Eq' (S1 S2 : Store) : Prop := Equal S1 S2"}, {"name": "Eq'_sym", "content": "def Eq'_sym : ∀ S1 S2, Eq' S1 S2 → Eq' S2 S1 :="}, {"name": "_inst_OrderStruct", "content": "noncomputable instance : OrderStruct Store where\n  eq := Equal\n  le := Le\n  beq (S1 S2 : Store) :=  Decidable.decide (Equal S1 S2)\n  le_trans := Le_trans\n  gt_wf := Gt_wf"}, {"name": "_inst_Monotone", "content": "instance : Monotone Store (fun x => Join Init (F x)) where\n  F_mon := by admit /- proof elided -/"}, {"name": "fixpoint_join", "content": "noncomputable def fixpoint_join : Store :="}, {"name": "wrapper", "content": "noncomputable instance wrapper (F : Store → Store) (F_mon : ∀ x y, le x y → le (F x) (F y)) : Monotone Store F where\n  F_mon := by admit /- proof elided -/"}, {"name": "fixpoint_join'", "content": "noncomputable def fixpoint_join' (S : Store) (F : Store → Store) (F_mon : ∀ x y, le x y → le (F x) (F y)) :="}], "used_local_lemmas": [{"name": "iterate_correct", "content": "@[grind] theorem iterate_correct (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) (heq : y = iterate _ F x PRE SMALL ) : eq y (F y) ∧ ∀ z, le (F z) z → le y z"}, {"name": "Eq_Le", "content": "@[grind] theorem Eq_Le : ∀ S1 S2, Eq' S1 S2 → Le S1 S2"}, {"name": "Le_trans", "content": "@[grind] theorem Le_trans : ∀ S1 S2 S3, Le S1 S2 → Le S2 S3 → Le S1 S3"}, {"name": "fixpoint_join_eq", "content": "theorem fixpoint_join_eq : Eq' (Join Init (F (fixpoint_join Init F) )) (fixpoint_join Init F)"}, {"name": "fixpoint_join_smallest", "content": "theorem fixpoint_join_smallest :\n  ∀ S, Le (Join Init (F S)) S -> Le (fixpoint_join Init F) S"}, {"name": "Join_increasing", "content": "@[grind] theorem Join_increasing :\n  ∀ S1 S2 S3 S4,\n  Le S1 S2 -> Le S3 S4 -> Le (Join S1 S3) (Join S2 S4)"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport LeroyCompilerVerificationCourse.Constprop\n\nimport Batteries.Data.List.Perm\n\n@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/\n\nopen OrderStruct\n\nclass Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)\n\nopen Monotone\n\nsection FixpointExistence\n\nvariable (α : Sort u) (F : α → α) [OrderWithBot α]\n\nopen OrderStruct OrderWithBot\n\nend FixpointExistence\n\nsection Iterate\n\nvariable (α : Sort u) [inst : OrderStruct α] (F : α → α) [Monotone α F]\n\nopen OrderStruct\n\n@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']\n\nend Iterate\n\nsection Fixpoint\n\nopen OrderWithBot\n\nvariable {α : Sort u} [i : OrderWithBot α] (F : α → α) [Monotone α F]\n\nend Fixpoint\n\nsection Constprop\n\nopen Std.HashMap\n\n@[grind] def Eq' (S1 S2 : Store) : Prop := Equal S1 S2\n\ndef Eq'_sym : ∀ S1 S2, Eq' S1 S2 → Eq' S2 S1 :=\n\nopen OrderStruct\n\nnoncomputable instance : OrderStruct Store where\n  eq := Equal\n  le := Le\n  beq (S1 S2 : Store) :=  Decidable.decide (Equal S1 S2)\n  le_trans := Le_trans\n  gt_wf := Gt_wf\n\nend Constprop\n\nsection FixpointJoin\n\nvariable (Init : Store)\n\nvariable (F : Store → Store) [Monotone Store F]\n\ninstance : Monotone Store (fun x => Join Init (F x)) where\n  F_mon := by admit /- proof elided -/\n\nnoncomputable def fixpoint_join : Store :=\n\nend FixpointJoin\n\nnoncomputable instance wrapper (F : Store → Store) (F_mon : ∀ x y, le x y → le (F x) (F y)) : Monotone Store F where\n  F_mon := by admit /- proof elided -/\n\nnoncomputable def fixpoint_join' (S : Store) (F : Store → Store) (F_mon : ∀ x y, le x y → le (F x) (F y)) :=", "target_theorem": "theorem fixpoint_join_increasing (_ : Store) (F : Store → Store) (F_mon : ∀ x y, le x y → le (F x) (F y)) (S1 S2 : Store) : le S1 S2 → le (fixpoint_join' S1 F F_mon) (fixpoint_join' S2 F F_mon) :=", "ground_truth_proof": ":= by\n  intro hyp\n  apply @fixpoint_join_smallest S1 F (by grind [wrapper]) (fixpoint_join' S2 F F_mon)\n  generalize heq : fixpoint_join' S2 F F_mon = fix2\n  have : (Le (Join S2 (F fix2)) fix2) := by\n    apply Eq_Le\n    · have := @fixpoint_join_eq S2 F (by grind [wrapper])\n      rw [←heq]\n      apply this\n  apply Le_trans\n  rotate_left\n  · apply this\n  · apply Join_increasing\n    · exact hyp\n    · grind", "nesting_depth": 4, "transitive_dep_count": 27, "subset_aristotle": false, "category": "Compiler"}
{"id": 359, "thm_name": "cp_com_correct_terminating", "thm_stmt": "theorem cp_com_correct_terminating :\n  ∀ c s1 s2 S1,\n  cexec s1 c s2 -> matches' s1 S1 -> cexec s1 (cp_com S1 c) s2", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Constprop.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import Std.Data.HashMap"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Int.add", "module": "Init.Data.Int.Basic"}, {"name": "Int.sub", "module": "Init.Data.Int.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Std.HashMap.Equiv", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.emptyWithCapacity", "module": "Std.Data.HashMap.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "BitVec", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "store", "content": "def store : Type := ident → Int"}, {"name": "cexec", "content": "@[grind] inductive cexec : store → com → store → Prop where\n  | cexec_skip :\n      cexec s .SKIP s\n  | cexec_assign :\n      cexec s (.ASSIGN x a) (update x (aeval s a) s)\n  | cexec_seq :\n      cexec s c1 s' -> cexec s' c2 s'' ->\n      cexec s (.SEQ c1 c2) s''\n  | cexec_ifthenelse :\n      cexec s (if beval s b then c1 else c2) s' ->\n      cexec s (.IFTHENELSE b c1 c2) s'\n  | cexec_while_done :\n      beval s b = false ->\n      cexec s (.WHILE b c) s\n  | cexec_while_loop :\n      beval s b = true -> cexec s c s' -> cexec s' (.WHILE b c) s'' ->\n      cexec s (.WHILE b c) s''"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "Std.HashMap.Equiv.getElem?_eq", "module": "Std.Data.HashMap.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "mk_PLUS_CONST", "content": "@[grind] def mk_PLUS_CONST (a : aexp) (n : Int) : aexp :=\n  if n = 0 then a else\n  match a with\n  | .CONST m => .CONST (m + n)\n  | .PLUS a (.CONST m) => .PLUS a (.CONST (m + n))\n  | _ => .PLUS a (.CONST n)"}, {"name": "mk_PLUS", "content": "@[grind] def mk_PLUS (a1 a2 : aexp) : aexp :=\n  match a1, a2 with\n  | .CONST m, _ => mk_PLUS_CONST a2 m\n  | _, .CONST m => mk_PLUS_CONST a1 m\n  | .PLUS a1 (.CONST m1), .PLUS a2 (.CONST m2) => mk_PLUS_CONST (.PLUS a1 a2) (m1 + m2)\n  | .PLUS a1 (.CONST m1), _ => mk_PLUS_CONST (.PLUS a1 a2) m1\n  | _, .PLUS a2 (.CONST m2) => mk_PLUS_CONST (.PLUS a1 a2) m2\n  | _, _ => .PLUS a1 a2"}, {"name": "mk_MINUS", "content": "@[grind] def mk_MINUS (a1 a2 : aexp) : aexp :=\n  match a1, a2 with\n  | _, .CONST m => mk_PLUS_CONST a1 (-m)\n  | .PLUS a1 (.CONST m1), .PLUS a2 (.CONST m2) => mk_PLUS_CONST (.MINUS a1 a2) (m1 - m2)\n  | .PLUS a1 (.CONST m1), _ => mk_PLUS_CONST (.MINUS a1 a2) m1\n  | _, .PLUS a2 (.CONST m2) => mk_PLUS_CONST (.MINUS a1 a2) (-m2)\n  | _, _ => .MINUS a1 a2"}, {"name": "mk_EQUAL", "content": "@[grind] def mk_EQUAL (a1 a2 : aexp) : bexp :=\n  match a1, a2 with\n  | .CONST n1, .CONST n2 => if n1 = n2 then .TRUE else .FALSE\n  | .PLUS a1 (.CONST n1), .CONST n2 => .EQUAL a1 (.CONST (n2 - n1))\n  | _, _ => .EQUAL a1 a2"}, {"name": "mk_LESSEQUAL", "content": "@[grind] def mk_LESSEQUAL (a1 a2 : aexp) : bexp :=\n  match a1, a2 with\n  | .CONST n1, .CONST n2 => if n1 <= n2 then .TRUE else .FALSE\n  | .PLUS a1 (.CONST n1), .CONST n2 => .LESSEQUAL a1 (.CONST (n2 - n1))\n  | _, _ => .LESSEQUAL a1 a2"}, {"name": "mk_NOT", "content": "@[grind] def mk_NOT (b : bexp) : bexp :=\n  match b with\n  | .TRUE => .FALSE\n  | .FALSE => .TRUE\n  | .NOT b => b\n  | _ => .NOT b"}, {"name": "mk_AND", "content": "@[grind] def mk_AND (b1 b2 : bexp) : bexp :=\n  match b1, b2 with\n  | .TRUE, _ => b2\n  | _, .TRUE => b1\n  | .FALSE, _ => .FALSE\n  | _, .FALSE => .FALSE\n  | _, _ => .AND b1 b2"}, {"name": "mk_IFTHENELSE", "content": "@[grind] def mk_IFTHENELSE (b : bexp) (c1 c2 : com) : com :=\n  match b with\n  | .TRUE => c1\n  | .FALSE => c2\n  | _ => .IFTHENELSE b c1 c2"}, {"name": "mk_WHILE", "content": "@[grind] def mk_WHILE (b : bexp) (c : com) : com :=\n  match b with\n  | .FALSE => .SKIP\n  | _ => .WHILE b c"}, {"name": "Store", "content": "def Store := Std.HashMap ident Int"}, {"name": "matches'", "content": "@[grind] def matches' (s : store) (S : Store) : Prop :=\n  ∀ x n, S.get? x = .some n -> s x = n"}, {"name": "Le", "content": "@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n"}, {"name": "Top", "content": "@[grind] def Top : Store := Std.HashMap.emptyWithCapacity"}, {"name": "Join", "content": "@[grind] def Join (S1 S2 : Store) : Store :=\n  S1.filter (fun key _ => S2.get? key == S1.get? key)"}, {"name": "Equal", "content": "def Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2"}, {"name": "lift1", "content": "@[grind] def lift1 {A B : Type} (f : A -> B) (o : Option A) : Option B :=\n  match o with\n  | .some x => .some (f x)\n  | .none => .none"}, {"name": "lift2", "content": "@[grind] def lift2 {A B C : Type} (f : A -> B -> C) (o1 : Option A) (o2 : Option B) : Option C :=\n  match o1, o2 with\n  | .some x1, .some x2 => .some (f x1 x2) | _, _ => .none"}, {"name": "Aeval", "content": "@[grind] def Aeval (S : Store) (a : aexp) : Option Int :=\n  match a with\n  | .CONST n => .some n\n  | .VAR x => S.get? x\n  | .PLUS a1 a2 => lift2 (Int.add) (Aeval S a1) (Aeval S a2)\n  | .MINUS a1 a2 => lift2 (Int.sub) (Aeval S a1) (Aeval S a2)"}, {"name": "Beval", "content": "@[grind] def Beval (S : Store) (b : bexp) : Option Bool :=\n  match b with\n  | .TRUE => .some true\n  | .FALSE => .some false\n  | .EQUAL a1 a2 => lift2 (fun m n => m == n) (Aeval S a1) (Aeval S a2)\n  | .LESSEQUAL a1 a2 => lift2 (fun m n => m <= n) (Aeval S a1) (Aeval S a2)\n  | .NOT b1 => lift1 (fun m => !m) (Beval S b1)\n  | .AND b1 b2 => lift2 (fun m n => m && n) (Beval S b1) (Beval S b2)"}, {"name": "Update", "content": "@[grind] def Update (x : ident) (N : Option Int) (S : Store) : Store :=\n  match N with\n  | .none => S.erase x\n  | .some n => S.insert x n"}, {"name": "fixpoint_rec", "content": "@[grind] noncomputable def fixpoint_rec (F : Store -> Store) (fuel : Nat) (S : Store) : Store :=\n  match fuel with\n  | 0 => Top\n  | fuel + 1 =>\n      let S' := F S\n      if Equal S' S then S else fixpoint_rec F fuel S'"}, {"name": "num_iter", "content": "@[grind] def num_iter : Nat := 20"}, {"name": "fixpoint", "content": "@[grind] noncomputable def fixpoint (F : Store -> Store) (init_S : Store) : Store :=\n  fixpoint_rec F num_iter init_S"}, {"name": "Cexec", "content": "@[grind] noncomputable def Cexec (S : Store) (c : com) : Store :=\n  match c with\n  | .SKIP => S\n  | .ASSIGN x a => Update x (Aeval S a) S\n  | .SEQ c1 c2 => Cexec (Cexec S c1) c2\n  | .IFTHENELSE b c1 c2 =>\n      match Beval S b with\n      | .some true => Cexec S c1\n      | .some false => Cexec S c2\n      | .none => Join (Cexec S c1) (Cexec S c2)\n  | .WHILE _ c1 =>\n      fixpoint (fun x => Join S (Cexec x c1)) S"}, {"name": "cp_aexp", "content": "@[grind =] def cp_aexp (S : Store) (a : aexp) : aexp :=\n  match a with\n  | .CONST n => .CONST n\n  | .VAR x => match S.get? x with\n    | .some n => .CONST n\n    | .none => .VAR x\n  | .PLUS a1 a2 => mk_PLUS (cp_aexp S a1) (cp_aexp S a2)\n  | .MINUS a1 a2 => mk_MINUS (cp_aexp S a1) (cp_aexp S a2)"}, {"name": "cp_bexp", "content": "@[grind] def cp_bexp (S : Store) (b : bexp) : bexp :=\n  match b with\n  | .TRUE => .TRUE\n  | .FALSE => .FALSE\n  | .EQUAL a1 a2 => mk_EQUAL (cp_aexp S a1) (cp_aexp S a2)\n  | .LESSEQUAL a1 a2 => mk_LESSEQUAL (cp_aexp S a1) (cp_aexp S a2)\n  | .NOT b => mk_NOT (cp_bexp S b)\n  | .AND b1 b2 => mk_AND (cp_bexp S b1) (cp_bexp S b2)"}, {"name": "cp_com", "content": "@[grind] noncomputable def cp_com (S : Store) (c : com) : com :=\n  match c with\n  | .SKIP => .SKIP\n  | .ASSIGN x a =>\n      .ASSIGN x (cp_aexp S a)\n  | .SEQ c1 c2 =>\n      .SEQ (cp_com S c1) (cp_com (Cexec S c1) c2)\n  | .IFTHENELSE b c1 c2 =>\n      mk_IFTHENELSE (cp_bexp S b) (cp_com S c1) (cp_com S c2)\n  | .WHILE b c =>\n      let sfix := Cexec S (.WHILE b c)\n      mk_WHILE (cp_bexp sfix b) (cp_com sfix c)"}], "used_local_lemmas": [{"name": "mk_EQUAL_sound", "content": "theorem mk_EQUAL_sound :\n  ∀ s a1 a2, beval s (mk_EQUAL a1 a2) = (aeval s a1 = aeval s a2)"}, {"name": "mk_LESSEQUAL_sound", "content": "theorem mk_LESSEQUAL_sound :\n  ∀ s a1 a2, beval s (mk_LESSEQUAL a1 a2) = (aeval s a1 <= aeval s a2)"}, {"name": "mk_NOT_sound", "content": "theorem mk_NOT_sound :\n  ∀ s b, beval s (mk_NOT b) = ¬ (beval s b)"}, {"name": "mk_AND_sound", "content": "theorem mk_AND_sound :\n  ∀ s b1 b2, beval s (mk_AND b1 b2) = (beval s b1 ∧ beval s b2)"}, {"name": "cexec_mk_IFTHENELSE", "content": "theorem cexec_mk_IFTHENELSE : ∀ s1 b c1 c2 s2,\n  cexec s1 (if beval s1 b then c1 else c2) s2 ->\n  cexec s1 (mk_IFTHENELSE b c1 c2) s2"}, {"name": "cexec_mk_WHILE_done", "content": "theorem cexec_mk_WHILE_done : ∀ s1 b c,\n  beval s1 b = false ->\n  cexec s1 (mk_WHILE b c) s1"}, {"name": "cexec_mk_WHILE_loop", "content": "theorem cexec_mk_WHILE_loop : ∀ b c s1 s2 s3,\n  beval s1 b = true -> cexec s1 c s2 -> cexec s2 (mk_WHILE b c) s3 ->\n  cexec s1 (mk_WHILE b c) s3"}, {"name": "matches_Le", "content": "theorem matches_Le : ∀ s S1 S2, Le S1 S2 -> matches' s S1 -> matches' s S2"}, {"name": "Le_Join_l", "content": "theorem Le_Join_l : ∀ S1 S2, Le S1 (Join S1 S2)"}, {"name": "Le_Join_r", "content": "theorem  Le_Join_r : ∀ S1 S2, Le S2 (Join S1 S2)"}, {"name": "Equal_Le", "content": "theorem Equal_Le : ∀ S1 S2, Equal S1 S2 -> Le S1 S2"}, {"name": "Beval_sound", "content": "theorem Beval_sound :\n  ∀ s S, matches' s S ->\n  ∀ b n, Beval S b = .some n -> beval s b = n"}, {"name": "fixpoint_sound", "content": "theorem fixpoint_sound (F : Store → Store) (init_S : Store) (h : S = fixpoint F init_S) :\n  Le (F S) S"}, {"name": "Cexec_sound", "content": "@[grind] theorem Cexec_sound :\n  ∀ c s1 s2 S1,\n  cexec s1 c s2 -> matches' s1 S1 -> matches' s2 (Cexec S1 c)"}, {"name": "cp_bexp_sound", "content": "theorem cp_bexp_sound :\n  ∀ s S, matches' s S ->\n  ∀ b, beval s (cp_bexp S b) = beval s b"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport Std.Data.HashMap\n\nopen Classical in\n\n@[grind] def mk_PLUS_CONST (a : aexp) (n : Int) : aexp :=\n  if n = 0 then a else\n  match a with\n  | .CONST m => .CONST (m + n)\n  | .PLUS a (.CONST m) => .PLUS a (.CONST (m + n))\n  | _ => .PLUS a (.CONST n)\n\n@[grind] def mk_PLUS (a1 a2 : aexp) : aexp :=\n  match a1, a2 with\n  | .CONST m, _ => mk_PLUS_CONST a2 m\n  | _, .CONST m => mk_PLUS_CONST a1 m\n  | .PLUS a1 (.CONST m1), .PLUS a2 (.CONST m2) => mk_PLUS_CONST (.PLUS a1 a2) (m1 + m2)\n  | .PLUS a1 (.CONST m1), _ => mk_PLUS_CONST (.PLUS a1 a2) m1\n  | _, .PLUS a2 (.CONST m2) => mk_PLUS_CONST (.PLUS a1 a2) m2\n  | _, _ => .PLUS a1 a2\n\n@[grind] def mk_MINUS (a1 a2 : aexp) : aexp :=\n  match a1, a2 with\n  | _, .CONST m => mk_PLUS_CONST a1 (-m)\n  | .PLUS a1 (.CONST m1), .PLUS a2 (.CONST m2) => mk_PLUS_CONST (.MINUS a1 a2) (m1 - m2)\n  | .PLUS a1 (.CONST m1), _ => mk_PLUS_CONST (.MINUS a1 a2) m1\n  | _, .PLUS a2 (.CONST m2) => mk_PLUS_CONST (.MINUS a1 a2) (-m2)\n  | _, _ => .MINUS a1 a2\n\n@[grind] def mk_EQUAL (a1 a2 : aexp) : bexp :=\n  match a1, a2 with\n  | .CONST n1, .CONST n2 => if n1 = n2 then .TRUE else .FALSE\n  | .PLUS a1 (.CONST n1), .CONST n2 => .EQUAL a1 (.CONST (n2 - n1))\n  | _, _ => .EQUAL a1 a2\n\n@[grind] def mk_LESSEQUAL (a1 a2 : aexp) : bexp :=\n  match a1, a2 with\n  | .CONST n1, .CONST n2 => if n1 <= n2 then .TRUE else .FALSE\n  | .PLUS a1 (.CONST n1), .CONST n2 => .LESSEQUAL a1 (.CONST (n2 - n1))\n  | _, _ => .LESSEQUAL a1 a2\n\n@[grind] def mk_NOT (b : bexp) : bexp :=\n  match b with\n  | .TRUE => .FALSE\n  | .FALSE => .TRUE\n  | .NOT b => b\n  | _ => .NOT b\n\n@[grind] def mk_AND (b1 b2 : bexp) : bexp :=\n  match b1, b2 with\n  | .TRUE, _ => b2\n  | _, .TRUE => b1\n  | .FALSE, _ => .FALSE\n  | _, .FALSE => .FALSE\n  | _, _ => .AND b1 b2\n\n@[grind] def mk_IFTHENELSE (b : bexp) (c1 c2 : com) : com :=\n  match b with\n  | .TRUE => c1\n  | .FALSE => c2\n  | _ => .IFTHENELSE b c1 c2\n\n@[grind] def mk_WHILE (b : bexp) (c : com) : com :=\n  match b with\n  | .FALSE => .SKIP\n  | _ => .WHILE b c\n\ndef Store := Std.HashMap ident Int\n\n@[grind] def matches' (s : store) (S : Store) : Prop :=\n  ∀ x n, S.get? x = .some n -> s x = n\n\n@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n\n\n@[grind] def Top : Store := Std.HashMap.emptyWithCapacity\n\n@[grind] def Join (S1 S2 : Store) : Store :=\n  S1.filter (fun key _ => S2.get? key == S1.get? key)\n\ndef Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2\n\n@[grind] def lift1 {A B : Type} (f : A -> B) (o : Option A) : Option B :=\n  match o with\n  | .some x => .some (f x)\n  | .none => .none\n\n@[grind] def lift2 {A B C : Type} (f : A -> B -> C) (o1 : Option A) (o2 : Option B) : Option C :=\n  match o1, o2 with\n  | .some x1, .some x2 => .some (f x1 x2) | _, _ => .none\n\n@[grind] def Aeval (S : Store) (a : aexp) : Option Int :=\n  match a with\n  | .CONST n => .some n\n  | .VAR x => S.get? x\n  | .PLUS a1 a2 => lift2 (Int.add) (Aeval S a1) (Aeval S a2)\n  | .MINUS a1 a2 => lift2 (Int.sub) (Aeval S a1) (Aeval S a2)\n\n@[grind] def Beval (S : Store) (b : bexp) : Option Bool :=\n  match b with\n  | .TRUE => .some true\n  | .FALSE => .some false\n  | .EQUAL a1 a2 => lift2 (fun m n => m == n) (Aeval S a1) (Aeval S a2)\n  | .LESSEQUAL a1 a2 => lift2 (fun m n => m <= n) (Aeval S a1) (Aeval S a2)\n  | .NOT b1 => lift1 (fun m => !m) (Beval S b1)\n  | .AND b1 b2 => lift2 (fun m n => m && n) (Beval S b1) (Beval S b2)\n\n@[grind] def Update (x : ident) (N : Option Int) (S : Store) : Store :=\n  match N with\n  | .none => S.erase x\n  | .some n => S.insert x n\n\n@[grind] noncomputable def fixpoint_rec (F : Store -> Store) (fuel : Nat) (S : Store) : Store :=\n  match fuel with\n  | 0 => Top\n  | fuel + 1 =>\n      let S' := F S\n      if Equal S' S then S else fixpoint_rec F fuel S'\n\n@[grind] def num_iter : Nat := 20\n\n@[grind] noncomputable def fixpoint (F : Store -> Store) (init_S : Store) : Store :=\n  fixpoint_rec F num_iter init_S\n\n@[grind] noncomputable def Cexec (S : Store) (c : com) : Store :=\n  match c with\n  | .SKIP => S\n  | .ASSIGN x a => Update x (Aeval S a) S\n  | .SEQ c1 c2 => Cexec (Cexec S c1) c2\n  | .IFTHENELSE b c1 c2 =>\n      match Beval S b with\n      | .some true => Cexec S c1\n      | .some false => Cexec S c2\n      | .none => Join (Cexec S c1) (Cexec S c2)\n  | .WHILE _ c1 =>\n      fixpoint (fun x => Join S (Cexec x c1)) S\n\n@[grind =] def cp_aexp (S : Store) (a : aexp) : aexp :=\n  match a with\n  | .CONST n => .CONST n\n  | .VAR x => match S.get? x with\n    | .some n => .CONST n\n    | .none => .VAR x\n  | .PLUS a1 a2 => mk_PLUS (cp_aexp S a1) (cp_aexp S a2)\n  | .MINUS a1 a2 => mk_MINUS (cp_aexp S a1) (cp_aexp S a2)\n\n@[grind] def cp_bexp (S : Store) (b : bexp) : bexp :=\n  match b with\n  | .TRUE => .TRUE\n  | .FALSE => .FALSE\n  | .EQUAL a1 a2 => mk_EQUAL (cp_aexp S a1) (cp_aexp S a2)\n  | .LESSEQUAL a1 a2 => mk_LESSEQUAL (cp_aexp S a1) (cp_aexp S a2)\n  | .NOT b => mk_NOT (cp_bexp S b)\n  | .AND b1 b2 => mk_AND (cp_bexp S b1) (cp_bexp S b2)\n\n@[grind] noncomputable def cp_com (S : Store) (c : com) : com :=\n  match c with\n  | .SKIP => .SKIP\n  | .ASSIGN x a =>\n      .ASSIGN x (cp_aexp S a)\n  | .SEQ c1 c2 =>\n      .SEQ (cp_com S c1) (cp_com (Cexec S c1) c2)\n  | .IFTHENELSE b c1 c2 =>\n      mk_IFTHENELSE (cp_bexp S b) (cp_com S c1) (cp_com S c2)\n  | .WHILE b c =>\n      let sfix := Cexec S (.WHILE b c)\n      mk_WHILE (cp_bexp sfix b) (cp_com sfix c)", "target_theorem": "theorem cp_com_correct_terminating :\n  ∀ c s1 s2 S1,\n  cexec s1 c s2 -> matches' s1 S1 -> cexec s1 (cp_com S1 c) s2 :=", "ground_truth_proof": ":= by\n    intro c s1 s2 S1 EXEC AG\n    induction c generalizing s1 s2 S1\n    any_goals grind\n    case ASSIGN x a =>\n      cases EXEC\n      next =>\n        have := @cexec.cexec_assign s1 x (cp_aexp S1 a)\n        grind\n    case IFTHENELSE b c1 c2 c1_ih c2_ih =>\n      cases EXEC\n      next =>\n        apply cexec_mk_IFTHENELSE\n        grind [cp_bexp_sound]\n    case WHILE b c c_ih =>\n      generalize heq1 : com.WHILE b c = loop\n      generalize heq2 : Cexec S1 (.WHILE b c) = X\n      have INNER : ∀ s1 c1 s2,\n                 cexec s1 c1 s2 ->\n                 c1 = .WHILE b c ->\n                 matches' s1 X ->\n                 cexec s1 (mk_WHILE (cp_bexp X b) (cp_com X c)) s2 := by\n                  intro s1 c1\n                  induction c1 generalizing s1 c\n                  any_goals grind\n                  case WHILE b1 c1 c1_ih  =>\n                    intro s2 EXEC EQ AG1\n                    injections heq1' heq2'\n                    generalize heq : (com.WHILE b1 c1) = loop\n                    rw [heq] at EXEC\n                    induction EXEC\n                    any_goals grind\n                    case cexec_while_done isFalse =>\n                      apply cexec_mk_WHILE_done\n                      · grind [cp_bexp_sound]\n                    case cexec_while_loop s3 b' c' s4 s5 isTrue EXEC2 EXEC3 a_ih a_ih2 =>\n                      apply cexec_mk_WHILE_loop\n                      · grind [cp_bexp_sound]\n                      · apply c_ih\n                        · injections heq4 heq5\n                          rw [heq2'] at heq5\n                          rw [←heq5] at EXEC2\n                          exact EXEC2\n                        · exact AG1\n                      · apply a_ih2\n                        rotate_left\n                        · grind\n                        · apply matches_Le\n                          rw [←heq2]\n                          simp [Cexec]\n                          apply fixpoint_sound\n                          rotate_left\n                          · exact (fun x => Join S1 (Cexec x c))\n                          · exact S1\n                          rotate_left\n                          · grind\n                          · apply matches_Le\n                            · apply Le_Join_r\n                            · apply Cexec_sound\n                              · rw [←heq2']\n                                injections heq3 heq4\n                                rw [heq4]\n                                exact EXEC2\n                              · grind\n      rw [heq1] at EXEC\n      induction EXEC\n      any_goals grind\n      case cexec_while_loop s3 b' c' s4 s5 isTrue EXEC1 EXEC2 ih1 ih2 =>\n        injections heq3 heq4\n        simp [cp_com]\n        specialize INNER s3 (.WHILE b' c') s5\n        rw [←heq3, ←heq4, heq2]\n        apply INNER\n        any_goals grind\n        · rw [←heq2]\n          simp [Cexec]\n          apply matches_Le\n          · apply fixpoint_sound\n            rotate_left\n            · exact (fun x => Join S1 (Cexec x c))\n            · exact S1\n            · grind\n          · grind", "nesting_depth": 5, "transitive_dep_count": 71, "subset_aristotle": true, "category": "Compiler"}
{"id": 360, "thm_name": "Cexec_sound", "thm_stmt": "@[grind] theorem Cexec_sound :\n  ∀ c s1 s2 S1,\n  cexec s1 c s2 -> matches' s1 S1 -> matches' s2 (Cexec S1 c)", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Constprop.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import Std.Data.HashMap"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Int.add", "module": "Init.Data.Int.Basic"}, {"name": "Int.sub", "module": "Init.Data.Int.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Std.HashMap.Equiv", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.emptyWithCapacity", "module": "Std.Data.HashMap.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "BitVec", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "store", "content": "def store : Type := ident → Int"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "cexec", "content": "@[grind] inductive cexec : store → com → store → Prop where\n  | cexec_skip :\n      cexec s .SKIP s\n  | cexec_assign :\n      cexec s (.ASSIGN x a) (update x (aeval s a) s)\n  | cexec_seq :\n      cexec s c1 s' -> cexec s' c2 s'' ->\n      cexec s (.SEQ c1 c2) s''\n  | cexec_ifthenelse :\n      cexec s (if beval s b then c1 else c2) s' ->\n      cexec s (.IFTHENELSE b c1 c2) s'\n  | cexec_while_done :\n      beval s b = false ->\n      cexec s (.WHILE b c) s\n  | cexec_while_loop :\n      beval s b = true -> cexec s c s' -> cexec s' (.WHILE b c) s'' ->\n      cexec s (.WHILE b c) s''"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "Std.HashMap.Equiv.getElem?_eq", "module": "Std.Data.HashMap.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Store", "content": "def Store := Std.HashMap ident Int"}, {"name": "matches'", "content": "@[grind] def matches' (s : store) (S : Store) : Prop :=\n  ∀ x n, S.get? x = .some n -> s x = n"}, {"name": "Le", "content": "@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n"}, {"name": "Top", "content": "@[grind] def Top : Store := Std.HashMap.emptyWithCapacity"}, {"name": "Join", "content": "@[grind] def Join (S1 S2 : Store) : Store :=\n  S1.filter (fun key _ => S2.get? key == S1.get? key)"}, {"name": "Equal", "content": "def Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2"}, {"name": "lift1", "content": "@[grind] def lift1 {A B : Type} (f : A -> B) (o : Option A) : Option B :=\n  match o with\n  | .some x => .some (f x)\n  | .none => .none"}, {"name": "lift2", "content": "@[grind] def lift2 {A B C : Type} (f : A -> B -> C) (o1 : Option A) (o2 : Option B) : Option C :=\n  match o1, o2 with\n  | .some x1, .some x2 => .some (f x1 x2) | _, _ => .none"}, {"name": "Aeval", "content": "@[grind] def Aeval (S : Store) (a : aexp) : Option Int :=\n  match a with\n  | .CONST n => .some n\n  | .VAR x => S.get? x\n  | .PLUS a1 a2 => lift2 (Int.add) (Aeval S a1) (Aeval S a2)\n  | .MINUS a1 a2 => lift2 (Int.sub) (Aeval S a1) (Aeval S a2)"}, {"name": "Beval", "content": "@[grind] def Beval (S : Store) (b : bexp) : Option Bool :=\n  match b with\n  | .TRUE => .some true\n  | .FALSE => .some false\n  | .EQUAL a1 a2 => lift2 (fun m n => m == n) (Aeval S a1) (Aeval S a2)\n  | .LESSEQUAL a1 a2 => lift2 (fun m n => m <= n) (Aeval S a1) (Aeval S a2)\n  | .NOT b1 => lift1 (fun m => !m) (Beval S b1)\n  | .AND b1 b2 => lift2 (fun m n => m && n) (Beval S b1) (Beval S b2)"}, {"name": "Update", "content": "@[grind] def Update (x : ident) (N : Option Int) (S : Store) : Store :=\n  match N with\n  | .none => S.erase x\n  | .some n => S.insert x n"}, {"name": "fixpoint_rec", "content": "@[grind] noncomputable def fixpoint_rec (F : Store -> Store) (fuel : Nat) (S : Store) : Store :=\n  match fuel with\n  | 0 => Top\n  | fuel + 1 =>\n      let S' := F S\n      if Equal S' S then S else fixpoint_rec F fuel S'"}, {"name": "num_iter", "content": "@[grind] def num_iter : Nat := 20"}, {"name": "fixpoint", "content": "@[grind] noncomputable def fixpoint (F : Store -> Store) (init_S : Store) : Store :=\n  fixpoint_rec F num_iter init_S"}, {"name": "Cexec", "content": "@[grind] noncomputable def Cexec (S : Store) (c : com) : Store :=\n  match c with\n  | .SKIP => S\n  | .ASSIGN x a => Update x (Aeval S a) S\n  | .SEQ c1 c2 => Cexec (Cexec S c1) c2\n  | .IFTHENELSE b c1 c2 =>\n      match Beval S b with\n      | .some true => Cexec S c1\n      | .some false => Cexec S c2\n      | .none => Join (Cexec S c1) (Cexec S c2)\n  | .WHILE _ c1 =>\n      fixpoint (fun x => Join S (Cexec x c1)) S"}], "used_local_lemmas": [{"name": "matches_Le", "content": "theorem matches_Le : ∀ s S1 S2, Le S1 S2 -> matches' s S1 -> matches' s S2"}, {"name": "Le_Join_l", "content": "theorem Le_Join_l : ∀ S1 S2, Le S1 (Join S1 S2)"}, {"name": "Le_Join_r", "content": "theorem  Le_Join_r : ∀ S1 S2, Le S2 (Join S1 S2)"}, {"name": "Equal_Le", "content": "theorem Equal_Le : ∀ S1 S2, Equal S1 S2 -> Le S1 S2"}, {"name": "Beval_sound", "content": "theorem Beval_sound :\n  ∀ s S, matches' s S ->\n  ∀ b n, Beval S b = .some n -> beval s b = n"}, {"name": "fixpoint_sound", "content": "theorem fixpoint_sound (F : Store → Store) (init_S : Store) (h : S = fixpoint F init_S) :\n  Le (F S) S"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport Std.Data.HashMap\n\nopen Classical in\n\ndef Store := Std.HashMap ident Int\n\n@[grind] def matches' (s : store) (S : Store) : Prop :=\n  ∀ x n, S.get? x = .some n -> s x = n\n\n@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n\n\n@[grind] def Top : Store := Std.HashMap.emptyWithCapacity\n\n@[grind] def Join (S1 S2 : Store) : Store :=\n  S1.filter (fun key _ => S2.get? key == S1.get? key)\n\ndef Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2\n\n@[grind] def lift1 {A B : Type} (f : A -> B) (o : Option A) : Option B :=\n  match o with\n  | .some x => .some (f x)\n  | .none => .none\n\n@[grind] def lift2 {A B C : Type} (f : A -> B -> C) (o1 : Option A) (o2 : Option B) : Option C :=\n  match o1, o2 with\n  | .some x1, .some x2 => .some (f x1 x2) | _, _ => .none\n\n@[grind] def Aeval (S : Store) (a : aexp) : Option Int :=\n  match a with\n  | .CONST n => .some n\n  | .VAR x => S.get? x\n  | .PLUS a1 a2 => lift2 (Int.add) (Aeval S a1) (Aeval S a2)\n  | .MINUS a1 a2 => lift2 (Int.sub) (Aeval S a1) (Aeval S a2)\n\n@[grind] def Beval (S : Store) (b : bexp) : Option Bool :=\n  match b with\n  | .TRUE => .some true\n  | .FALSE => .some false\n  | .EQUAL a1 a2 => lift2 (fun m n => m == n) (Aeval S a1) (Aeval S a2)\n  | .LESSEQUAL a1 a2 => lift2 (fun m n => m <= n) (Aeval S a1) (Aeval S a2)\n  | .NOT b1 => lift1 (fun m => !m) (Beval S b1)\n  | .AND b1 b2 => lift2 (fun m n => m && n) (Beval S b1) (Beval S b2)\n\n@[grind] def Update (x : ident) (N : Option Int) (S : Store) : Store :=\n  match N with\n  | .none => S.erase x\n  | .some n => S.insert x n\n\n@[grind] noncomputable def fixpoint_rec (F : Store -> Store) (fuel : Nat) (S : Store) : Store :=\n  match fuel with\n  | 0 => Top\n  | fuel + 1 =>\n      let S' := F S\n      if Equal S' S then S else fixpoint_rec F fuel S'\n\n@[grind] def num_iter : Nat := 20\n\n@[grind] noncomputable def fixpoint (F : Store -> Store) (init_S : Store) : Store :=\n  fixpoint_rec F num_iter init_S\n\n@[grind] noncomputable def Cexec (S : Store) (c : com) : Store :=\n  match c with\n  | .SKIP => S\n  | .ASSIGN x a => Update x (Aeval S a) S\n  | .SEQ c1 c2 => Cexec (Cexec S c1) c2\n  | .IFTHENELSE b c1 c2 =>\n      match Beval S b with\n      | .some true => Cexec S c1\n      | .some false => Cexec S c2\n      | .none => Join (Cexec S c1) (Cexec S c2)\n  | .WHILE _ c1 =>\n      fixpoint (fun x => Join S (Cexec x c1)) S", "target_theorem": "@[grind] theorem Cexec_sound :\n  ∀ c s1 s2 S1,\n  cexec s1 c s2 -> matches' s1 S1 -> matches' s2 (Cexec S1 c) :=", "ground_truth_proof": ":= by\n    intro c\n    induction c\n    next =>\n      intro s1 s2 S1 EXEC\n      cases EXEC\n      grind\n    next x a =>\n      intro s1 s2 S1 EXEC\n      cases EXEC\n      grind\n    next c1 c2 c1_ih c2_ih =>\n      grind\n    next b c1 c2 c1_ih c2_ih =>\n      intro s1 s2 S1 EXEC\n      cases EXEC\n      next EXEC =>\n        by_cases beval s1 b\n        case pos h =>\n          unfold Cexec\n          intro h2\n          have := Beval_sound s1 S1 h2 b\n          split <;> grind\n        case neg h =>\n          simp [h] at EXEC\n          intro h2\n          unfold Cexec\n          have := Beval_sound s1 S1 h2 b\n          split <;> grind\n    case WHILE b c c1_ih =>\n      intro s1 s2 S1 EXEC MATCHES\n      generalize eq1 : (fun x => Join S1 (Cexec x c)) = F\n      generalize eq2 : fixpoint F S1 = X\n      have INNER : ∀ s1 c1 s2,\n                 cexec s1 c1 s2 ->\n                 c1 = .WHILE b c ->\n                 matches' s1 X ->\n                 matches' s2 X := by\n                  intro s3 c1 s4 EXEC2 EQ AG\n                  induction EXEC2\n                  any_goals grind\n                  case cexec_while_loop s' b' c' s5 s6 EXEC2 EXEC3 EXEC4 _ a_ih2 =>\n                    apply a_ih2\n                    · grind\n                    · apply matches_Le\n                      rotate_right\n                      · exact F X\n                      · exact @fixpoint_sound X F S1 (by grind)\n                      · rw [←eq2, ←eq1]\n                        simp\n                        apply matches_Le\n                        apply Le_Join_r\n                        rw [eq1, eq2]\n                        apply c1_ih\n                        injections EQ\n                        rename_i eq1 eq2\n                        rw [eq2] at EXEC3\n                        exact EXEC3\n                        exact AG\n      unfold Cexec\n      rw [eq1, eq2]\n      apply INNER\n      · apply EXEC\n      · rfl\n      · apply matches_Le\n        have := @fixpoint_sound X F\n        apply this\n        rotate_left\n        · exact S1\n        rotate_left\n        · grind\n        · rw [←eq1]\n          simp\n          apply matches_Le\n          · apply Le_Join_l\n          · exact MATCHES", "nesting_depth": 5, "transitive_dep_count": 50, "subset_aristotle": false, "category": "Compiler"}
{"id": 361, "thm_name": "compile_bexp_correct", "thm_stmt": "theorem compile_bexp_correct (C : List instr) (s : store) (b : bexp) (d1 d0 : Int) (pc : Int) (stk : stack) (h : code_at C pc (compile_bexp b d1 d0))  :\n   transitions C\n       (pc, stk, s)\n       (pc + codelen (compile_bexp b d1 d0) + (if beval s b then d1 else d0), stk, s)", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Compil.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import LeroyCompilerVerificationCourse.Sequences"], "used_lib_defs": [{"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "store", "content": "def store : Type := ident → Int"}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "star", "content": "@[grind] inductive star (R : α → α → Prop) : α → α → Prop where\n  | star_refl : ∀ x : α, star R x x\n  | star_step : ∀ {x y z}, R x y → star R y z → star R x z"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}], "lib_lemmas": [{"name": "star_one", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "Int.add_assoc", "module": "Init.Data.Int.Lemmas"}], "repo_lemmas": [{"name": "star_trans", "content": "@[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c"}], "used_local_defs": [{"name": "instr", "content": "@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr"}, {"name": "codelen", "content": "@[grind] def codelen (c : List instr) : Int := c.length"}, {"name": "stack", "content": "def stack : Type := List Int"}, {"name": "config", "content": "def config : Type := Int × stack × store"}, {"name": "instr_at", "content": "@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)"}, {"name": "transition", "content": "@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)"}, {"name": "transitions", "content": "@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)"}, {"name": "compile_aexp", "content": "@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])"}, {"name": "compile_bexp", "content": "@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2"}, {"name": "code_at", "content": "@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2"}], "used_local_lemmas": [{"name": "codelen_app", "content": "@[grind =] theorem codelen_app :\n  ∀ c1 c2, codelen (c1 ++ c2) = codelen c1 + codelen c2"}, {"name": "instr_a", "content": "@[grind =>] theorem instr_a : ∀ i c2 c1 pc,\n  pc = codelen c1 ->\n  instr_at (c1 ++ (i :: c2) ) pc = .some i"}, {"name": "code_at_to_instr_at", "content": "@[grind] theorem code_at_to_instr_at : code_at C pc (c1 ++ i :: c2) → instr_at C (pc + codelen c1) = .some i"}, {"name": "compile_aexp_correct", "content": "theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :\n  code_at C pc (compile_aexp a) →\n  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s)"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\n@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr\n\n@[grind] def codelen (c : List instr) : Int := c.length\n\ndef stack : Type := List Int\n\ndef config : Type := Int × stack × store\n\n@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)\n\n@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n\n@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)\n\n@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])\n\n@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2\n\n@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2", "target_theorem": "theorem compile_bexp_correct (C : List instr) (s : store) (b : bexp) (d1 d0 : Int) (pc : Int) (stk : stack) (h : code_at C pc (compile_bexp b d1 d0))  :\n   transitions C\n       (pc, stk, s)\n       (pc + codelen (compile_bexp b d1 d0) + (if beval s b then d1 else d0), stk, s) :=", "ground_truth_proof": ":= by\n    induction b generalizing d1 d0 pc\n    next =>\n      simp [compile_bexp, beval]\n      by_cases d1 = 0\n      case pos is_zero =>\n        simp [is_zero, codelen]\n        apply star.star_refl\n      case neg is_not_zero =>\n        apply star_one\n        grind\n    next =>\n      simp [compile_bexp, beval]\n      by_cases d0 = 0\n      case pos is_zero =>\n        simp [is_zero, codelen]\n        apply star.star_refl\n      case neg is_not_zero =>\n        apply star_one\n        simp [is_not_zero, codelen]\n        grind\n    next a1 a2 =>\n      simp [compile_bexp, beval]\n      apply star_trans\n      · apply compile_aexp_correct (a := a1)\n        grind\n      · apply star_trans\n        · apply compile_aexp_correct (a := a2)\n          grind\n        · apply star_one\n          · apply transition.trans_beq (d1 := d1) (d0 := d0) <;> grind\n    next a1 a2 =>\n      simp [compile_bexp, beval]\n      apply star_trans\n      · apply compile_aexp_correct (a := a1)\n        grind\n      · apply star_trans\n        · apply compile_aexp_correct (a := a2)\n          grind\n        · apply star_one\n          · apply transition.trans_ble (d1 := d1) (d0 := d0) <;> grind\n    next b1 ih =>\n      grind\n    next b1 b2 b1_ih b2_ih =>\n      generalize heq1 : compile_bexp b2 d1 d0 = code2\n      generalize heq2 : compile_bexp b1 0 (codelen code2 + d0) = code1\n      unfold compile_bexp\n      simp [heq1, heq2]\n      apply star_trans\n      · apply b1_ih (d1 := 0) (d0 := codelen code2 + d0)\n        grind\n      · by_cases beval s b1 = true\n        case pos isTrue =>\n          simp [isTrue]\n          rw [heq2]\n          simp [beval, isTrue]\n          specialize b2_ih d1 d0 (pc + codelen code1)\n          rw [heq1] at b2_ih\n          simp [compile_bexp, heq1, heq2] at h\n          specialize b2_ih (by grind)\n          simp [codelen_app]\n          simp [Int.add_assoc] at *\n          exact b2_ih\n        case neg isFalse =>\n          grind", "nesting_depth": 5, "transitive_dep_count": 34, "subset_aristotle": true, "category": "Compiler"}
{"id": 362, "thm_name": "dce_correct_terminating", "thm_stmt": "theorem dce_correct_terminating :\n  ∀ s c s', cexec s c s' ->\n  ∀ L s1, agree (live c L) s s1 ->\n  ∃ s1', cexec s1 (dce c L) s1' /\\ agree L s' s1'", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Deadcode.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import Std.Data.HashSet"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Std.HashSet", "module": "Std.Data.HashSet.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Std.HashSet.instSingleton", "module": "Std.Data.HashSet.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}], "used_repo_defs": [{"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "store", "content": "def store : Type := ident → Int"}, {"name": "cexec", "content": "@[grind] inductive cexec : store → com → store → Prop where\n  | cexec_skip :\n      cexec s .SKIP s\n  | cexec_assign :\n      cexec s (.ASSIGN x a) (update x (aeval s a) s)\n  | cexec_seq :\n      cexec s c1 s' -> cexec s' c2 s'' ->\n      cexec s (.SEQ c1 c2) s''\n  | cexec_ifthenelse :\n      cexec s (if beval s b then c1 else c2) s' ->\n      cexec s (.IFTHENELSE b c1 c2) s'\n  | cexec_while_done :\n      beval s b = false ->\n      cexec s (.WHILE b c) s\n  | cexec_while_loop :\n      beval s b = true -> cexec s c s' -> cexec s' (.WHILE b c) s'' ->\n      cexec s (.WHILE b c) s''"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "IdentSet", "content": "abbrev IdentSet := Std.HashSet ident"}, {"name": "fv_aexp", "content": "@[grind] def fv_aexp (a : aexp) : IdentSet :=\n  match a with\n  | .CONST _ => ∅\n  | .VAR v => Std.HashSet.instSingleton.singleton v\n  | .PLUS a1 a2 =>  (fv_aexp a1) ∪ (fv_aexp a2)\n  | .MINUS a1 a2 => (fv_aexp a1) ∪ (fv_aexp a2)"}, {"name": "fv_bexp", "content": "@[grind] def fv_bexp (b : bexp) : IdentSet :=\n  match b with\n  | .TRUE => ∅\n  | .FALSE => ∅\n  | .EQUAL a1 a2 => (fv_aexp a1) ∪ (fv_aexp a2)\n  | .LESSEQUAL a1 a2 => (fv_aexp a1) ∪ (fv_aexp a2)\n  | .NOT b1 => fv_bexp b1\n  | .AND b1 b2 => (fv_bexp b1) ∪ (fv_bexp b2)"}, {"name": "fv_com", "content": "@[grind] def fv_com (c : com) : IdentSet :=\n  match c with\n  | .SKIP => ∅\n  | .ASSIGN _ a => fv_aexp a\n  | .SEQ c1 c2 => (fv_com c1) ∪ (fv_com c2)\n  | .IFTHENELSE b c1 c2 => (fv_bexp b) ∪ ((fv_com c1) ∪ (fv_com c2))\n  | .WHILE b c => (fv_bexp b) ∪ (fv_com c)"}, {"name": "deadcode_fixpoint_rec", "content": "@[grind] noncomputable def deadcode_fixpoint_rec (F : IdentSet → IdentSet) (default : IdentSet) (fuel : Nat) (x : IdentSet) : IdentSet :=\n  match fuel with\n  | 0 => default\n  | fuel + 1 =>\n      let x' := F x\n      if x' ⊆ x then x else deadcode_fixpoint_rec F default fuel x'"}, {"name": "deadcode_fixpoint", "content": "@[grind] noncomputable def deadcode_fixpoint (F : IdentSet → IdentSet) (default : IdentSet) : IdentSet :=\n  deadcode_fixpoint_rec F default 20 ∅"}, {"name": "live", "content": "@[grind] noncomputable def live (c : com) (L : IdentSet) : IdentSet :=\n  match c with\n  | .SKIP => L\n  | .ASSIGN x a =>\n      if x ∈ L\n      then (L.erase x) ∪ (fv_aexp a)\n      else L\n  | .SEQ c1 c2 =>\n      live c1 (live c2 L)\n  | .IFTHENELSE b c1 c2 =>\n       (fv_bexp b) ∪ ((live c1 L) ∪ (live c2 L))\n  | .WHILE b c =>\n      let L' := (fv_bexp b) ∪  L\n      let default := (fv_com (.WHILE b c)) ∪ L\n      deadcode_fixpoint (fun x => L' ∪ (live c x)) default"}, {"name": "dce", "content": "@[grind] noncomputable def dce (c : com) (L : IdentSet) : com :=\n  match c with\n  | .SKIP => .SKIP\n  | .ASSIGN x a => if x ∈ L then .ASSIGN x a else .SKIP\n  | .SEQ c1 c2 => .SEQ (dce c1 (live c2 L)) (dce c2 L)\n  | .IFTHENELSE b c1 c2 => .IFTHENELSE b (dce c1 L) (dce c2 L)\n  | .WHILE b c => .WHILE b (dce c (live (.WHILE b c) L))"}, {"name": "agree", "content": "@[grind] def agree (L : IdentSet) (s1 s2 : store) : Prop :=\n  ∀ x, x  ∈ L -> s1 x = s2 x"}], "used_local_lemmas": [{"name": "subset_def", "content": "@[grind =] theorem subset_def (a b : IdentSet) : a ⊆ b ↔ ∀ x ∈ a, x ∈ b"}, {"name": "live_while_charact", "content": "theorem live_while_charact (b : bexp) (c : com) (L L' : IdentSet)\n  (eq : L' = live (.WHILE b c) L) :\n  (fv_bexp b) ⊆ L' ∧ L ⊆ L' ∧ (live c L') ⊆ L'"}, {"name": "beval_agree", "content": "theorem beval_agree :\n  ∀ L s1 s2, agree L s1 s2 ->\n  ∀ b, (fv_bexp b) ⊆ L -> beval s1 b = beval s2 b"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport Std.Data.HashSet\n\nabbrev IdentSet := Std.HashSet ident\n\n@[grind] def fv_aexp (a : aexp) : IdentSet :=\n  match a with\n  | .CONST _ => ∅\n  | .VAR v => Std.HashSet.instSingleton.singleton v\n  | .PLUS a1 a2 =>  (fv_aexp a1) ∪ (fv_aexp a2)\n  | .MINUS a1 a2 => (fv_aexp a1) ∪ (fv_aexp a2)\n\n@[grind] def fv_bexp (b : bexp) : IdentSet :=\n  match b with\n  | .TRUE => ∅\n  | .FALSE => ∅\n  | .EQUAL a1 a2 => (fv_aexp a1) ∪ (fv_aexp a2)\n  | .LESSEQUAL a1 a2 => (fv_aexp a1) ∪ (fv_aexp a2)\n  | .NOT b1 => fv_bexp b1\n  | .AND b1 b2 => (fv_bexp b1) ∪ (fv_bexp b2)\n\n@[grind] def fv_com (c : com) : IdentSet :=\n  match c with\n  | .SKIP => ∅\n  | .ASSIGN _ a => fv_aexp a\n  | .SEQ c1 c2 => (fv_com c1) ∪ (fv_com c2)\n  | .IFTHENELSE b c1 c2 => (fv_bexp b) ∪ ((fv_com c1) ∪ (fv_com c2))\n  | .WHILE b c => (fv_bexp b) ∪ (fv_com c)\n\n@[grind] noncomputable def deadcode_fixpoint_rec (F : IdentSet → IdentSet) (default : IdentSet) (fuel : Nat) (x : IdentSet) : IdentSet :=\n  match fuel with\n  | 0 => default\n  | fuel + 1 =>\n      let x' := F x\n      if x' ⊆ x then x else deadcode_fixpoint_rec F default fuel x'\n\n@[grind] noncomputable def deadcode_fixpoint (F : IdentSet → IdentSet) (default : IdentSet) : IdentSet :=\n  deadcode_fixpoint_rec F default 20 ∅\n\n@[grind] noncomputable def live (c : com) (L : IdentSet) : IdentSet :=\n  match c with\n  | .SKIP => L\n  | .ASSIGN x a =>\n      if x ∈ L\n      then (L.erase x) ∪ (fv_aexp a)\n      else L\n  | .SEQ c1 c2 =>\n      live c1 (live c2 L)\n  | .IFTHENELSE b c1 c2 =>\n       (fv_bexp b) ∪ ((live c1 L) ∪ (live c2 L))\n  | .WHILE b c =>\n      let L' := (fv_bexp b) ∪  L\n      let default := (fv_com (.WHILE b c)) ∪ L\n      deadcode_fixpoint (fun x => L' ∪ (live c x)) default\n\n@[grind] noncomputable def dce (c : com) (L : IdentSet) : com :=\n  match c with\n  | .SKIP => .SKIP\n  | .ASSIGN x a => if x ∈ L then .ASSIGN x a else .SKIP\n  | .SEQ c1 c2 => .SEQ (dce c1 (live c2 L)) (dce c2 L)\n  | .IFTHENELSE b c1 c2 => .IFTHENELSE b (dce c1 L) (dce c2 L)\n  | .WHILE b c => .WHILE b (dce c (live (.WHILE b c) L))\n\n@[grind] def agree (L : IdentSet) (s1 s2 : store) : Prop :=\n  ∀ x, x  ∈ L -> s1 x = s2 x", "target_theorem": "theorem dce_correct_terminating :\n  ∀ s c s', cexec s c s' ->\n  ∀ L s1, agree (live c L) s s1 ->\n  ∃ s1', cexec s1 (dce c L) s1' /\\ agree L s' s1' :=", "ground_truth_proof": ":= by\n    intro s c s' EXEC\n    induction EXEC\n    any_goals grind\n    case cexec_while_loop s1 b c1 s2 s3 isTrue EX1 EX2 a_ih a_ih2 =>\n      intro L s4 hyp\n      have ⟨t1, ht1, ht2⟩ :=  a_ih (live (.WHILE b c1) L) s4 (by grind)\n      have ⟨u1, hu1, hu2⟩ :=  a_ih2 L t1 ht2\n      exists u1\n      constructor\n      rotate_right\n      · exact hu2\n      · apply cexec.cexec_while_loop\n        · have := beval_agree (live (com.WHILE b c1) L) s1 s4\n          grind\n        · exact ht1\n        · grind\n    case cexec_assign s2 x a=>\n      intro L s3 AG\n      simp [live] at AG\n      by_cases x ∈ L\n      case neg notIn =>\n        exists s3\n        grind\n      case pos isIn =>\n        exists (update x (aeval s3 a) s3)\n        grind\n    case cexec_ifthenelse s2 b c1 c2 s3 EXEC ih =>\n      intro L s4 AG\n      simp [dce]\n      have EQ : beval s2 b = beval s4 b := by\n        apply beval_agree\n        · apply AG\n        · grind\n      by_cases beval s2 b = true\n      case pos isTrue =>\n        specialize ih L s4\n        grind\n      case neg isFalse =>\n        specialize ih L s4\n        grind\n    case cexec_while_done s2 b c isFalse =>\n      intro L s1 AG\n      have ⟨h1, h2, h3⟩ := live_while_charact b c L (live (com.WHILE b c) L) (by grind)\n      have EQ : beval s2 b = beval s1 b := by\n        apply beval_agree\n        · apply AG\n        · grind\n      exists s1\n      grind\n    case cexec_seq => grind [-subset_def] -- TODO: what does wrong?", "nesting_depth": 5, "transitive_dep_count": 38, "subset_aristotle": false, "category": "Compiler"}
{"id": 363, "thm_name": "compile_aexp_correct", "thm_stmt": "theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :\n  code_at C pc (compile_aexp a) →\n  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s)", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Compil.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import LeroyCompilerVerificationCourse.Sequences"], "used_lib_defs": [{"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "store", "content": "def store : Type := ident → Int"}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "star", "content": "@[grind] inductive star (R : α → α → Prop) : α → α → Prop where\n  | star_refl : ∀ x : α, star R x x\n  | star_step : ∀ {x y z}, R x y → star R y z → star R x z"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}], "lib_lemmas": [{"name": "star_one", "module": "Mathlib.Algebra.Star.Basic"}], "repo_lemmas": [{"name": "star_trans", "content": "@[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c"}], "used_local_defs": [{"name": "instr", "content": "@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr"}, {"name": "codelen", "content": "@[grind] def codelen (c : List instr) : Int := c.length"}, {"name": "stack", "content": "def stack : Type := List Int"}, {"name": "config", "content": "def config : Type := Int × stack × store"}, {"name": "instr_at", "content": "@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)"}, {"name": "transition", "content": "@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)"}, {"name": "transitions", "content": "@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)"}, {"name": "compile_aexp", "content": "@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])"}, {"name": "code_at", "content": "@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2"}], "used_local_lemmas": [{"name": "instr_a", "content": "@[grind =>] theorem instr_a : ∀ i c2 c1 pc,\n  pc = codelen c1 ->\n  instr_at (c1 ++ (i :: c2) ) pc = .some i"}, {"name": "code_at_to_instr_at", "content": "@[grind] theorem code_at_to_instr_at : code_at C pc (c1 ++ i :: c2) → instr_at C (pc + codelen c1) = .some i"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\n@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr\n\n@[grind] def codelen (c : List instr) : Int := c.length\n\ndef stack : Type := List Int\n\ndef config : Type := Int × stack × store\n\n@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)\n\n@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n\n@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)\n\n@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])\n\n@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2", "target_theorem": "theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :\n  code_at C pc (compile_aexp a) →\n  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s) :=", "ground_truth_proof": ":= by\n    induction a generalizing C pc stk\n    next =>\n      unfold transitions\n      grind\n    next =>\n      intro a\n      apply star_one\n      grind\n    next a1 a2 a1_ih a2_ih =>\n      simp [aeval, compile_aexp]\n      intro a\n      apply star_trans\n      · apply a1_ih\n        grind\n      · apply star_trans\n        · apply a2_ih\n          grind\n        · apply star_one\n          cases a\n          next c1 c3 a =>\n            have h1 := instr_a\n            have h2 := @transition.trans_add\n            grind\n    next a1 a2 a1_ih a2_ih =>\n      simp [aeval, compile_aexp]\n      intro a\n      apply star_trans\n      · apply a1_ih\n        grind\n      · apply star_trans\n        · apply a2_ih\n          grind\n        · apply star_trans\n          · apply star_one\n            · apply transition.trans_opp\n              grind\n          · apply star_one\n            · have := @code_at_to_instr_at C pc (compile_aexp a1 ++ compile_aexp a2 ++ [instr.Iopp])\n              have := @transition.trans_add\n              grind", "nesting_depth": 5, "transitive_dep_count": 27, "subset_aristotle": false, "category": "Compiler"}
{"id": 364, "thm_name": "cexec_to_reds", "thm_stmt": "theorem cexec_to_reds (s s' : store) (c : com) : cexec s c s' → star red (c, s) (.SKIP, s')", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Imp.lean", "imports": ["import LeroyCompilerVerificationCourse.Sequences"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}, {"name": "star", "content": "@[grind] inductive star (R : α → α → Prop) : α → α → Prop where\n  | star_refl : ∀ x : α, star R x x\n  | star_step : ∀ {x y z}, R x y → star R y z → star R x z"}], "lib_lemmas": [{"name": "star_one", "module": "Mathlib.Algebra.Star.Basic"}], "repo_lemmas": [{"name": "star_trans", "content": "@[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c"}], "used_local_defs": [{"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp)"}, {"name": "store", "content": "def store : Type := ident → Int"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "cexec", "content": "@[grind] inductive cexec : store → com → store → Prop where\n  | cexec_skip :\n      cexec s .SKIP s\n  | cexec_assign :\n      cexec s (.ASSIGN x a) (update x (aeval s a) s)\n  | cexec_seq :\n      cexec s c1 s' -> cexec s' c2 s'' ->\n      cexec s (.SEQ c1 c2) s''\n  | cexec_ifthenelse :\n      cexec s (if beval s b then c1 else c2) s' ->\n      cexec s (.IFTHENELSE b c1 c2) s'\n  | cexec_while_done :\n      beval s b = false ->\n      cexec s (.WHILE b c) s\n  | cexec_while_loop :\n      beval s b = true -> cexec s c s' -> cexec s' (.WHILE b c) s'' ->\n      cexec s (.WHILE b c) s''"}, {"name": "red", "content": "@[grind] inductive red : com × store → com × store → Prop where\n  | red_assign : ∀ x a s,\n      red (.ASSIGN x a, s) (.SKIP, update x (aeval s a) s)\n  | red_seq_done : ∀ c s,\n      red (.SEQ .SKIP c, s) (c, s)\n  | red_seq_step : ∀ c1 c s1 c2 s2,\n      red (c1, s1) (c2, s2) →\n      red (.SEQ c1 c, s1) (.SEQ c2 c, s2)\n  | red_ifthenelse : ∀ b c1 c2 s,\n      red (.IFTHENELSE b c1 c2, s) ((if beval s b then c1 else c2), s)\n  | red_while_done : ∀ b c s,\n      beval s b = false →\n      red (.WHILE b c, s) (.SKIP, s)\n  | red_while_loop : ∀ b c s,\n      beval s b = true →\n      red (.WHILE b c, s) (.SEQ c (.WHILE b c), s)"}], "used_local_lemmas": [{"name": "red_seq_steps", "content": "@[grind] theorem red_seq_steps (c2 c c' : com) (s s' : store) : star red (c, s) (c', s') → star red ((c;;c2), s) ((c';;c2), s')"}], "local_ctx": "import LeroyCompilerVerificationCourse.Sequences\n\ndef ident := String deriving BEq, Repr, Hashable\n\ninductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) \n\ndef store : Type := ident → Int\n\n@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2\n\ninductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       \n\n@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2\n\ninductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 \n\nnotation:10 l:10 \" ;; \" r:11 => com.SEQ l r\n\n@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y\n\n@[grind] inductive cexec : store → com → store → Prop where\n  | cexec_skip :\n      cexec s .SKIP s\n  | cexec_assign :\n      cexec s (.ASSIGN x a) (update x (aeval s a) s)\n  | cexec_seq :\n      cexec s c1 s' -> cexec s' c2 s'' ->\n      cexec s (.SEQ c1 c2) s''\n  | cexec_ifthenelse :\n      cexec s (if beval s b then c1 else c2) s' ->\n      cexec s (.IFTHENELSE b c1 c2) s'\n  | cexec_while_done :\n      beval s b = false ->\n      cexec s (.WHILE b c) s\n  | cexec_while_loop :\n      beval s b = true -> cexec s c s' -> cexec s' (.WHILE b c) s'' ->\n      cexec s (.WHILE b c) s''\n\n@[grind] inductive red : com × store → com × store → Prop where\n  | red_assign : ∀ x a s,\n      red (.ASSIGN x a, s) (.SKIP, update x (aeval s a) s)\n  | red_seq_done : ∀ c s,\n      red (.SEQ .SKIP c, s) (c, s)\n  | red_seq_step : ∀ c1 c s1 c2 s2,\n      red (c1, s1) (c2, s2) →\n      red (.SEQ c1 c, s1) (.SEQ c2 c, s2)\n  | red_ifthenelse : ∀ b c1 c2 s,\n      red (.IFTHENELSE b c1 c2, s) ((if beval s b then c1 else c2), s)\n  | red_while_done : ∀ b c s,\n      beval s b = false →\n      red (.WHILE b c, s) (.SKIP, s)\n  | red_while_loop : ∀ b c s,\n      beval s b = true →\n      red (.WHILE b c, s) (.SEQ c (.WHILE b c), s)", "target_theorem": "theorem cexec_to_reds (s s' : store) (c : com) : cexec s c s' → star red (c, s) (.SKIP, s') :=", "ground_truth_proof": ":= by\n  intro h\n  induction h\n  any_goals grind\n  case cexec_seq ih1 ih2  =>\n    apply star_trans\n    · apply red_seq_steps\n      exact ih1\n    · apply star.star_step\n      apply red.red_seq_done\n      grind\n  case cexec_while_loop ih1 ih2 =>\n    apply star_trans\n    · apply star_one\n      · apply red.red_while_loop\n        · grind\n    · apply star_trans\n      · apply red_seq_steps\n        · exact ih1\n      · apply star_trans\n        rotate_left\n        · apply ih2\n        · grind", "nesting_depth": 3, "transitive_dep_count": 20, "subset_aristotle": false, "category": "Compiler"}
{"id": 365, "thm_name": "compile_program_correct_terminating_2", "thm_stmt": "theorem compile_program_correct_terminating_2 :\n  ∀ c s s',\n  star step (c, .Kstop, s) (.SKIP, .Kstop, s') ->\n  machine_terminates (compile_program c) s s'", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Compil.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import LeroyCompilerVerificationCourse.Sequences"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Exists", "module": "Init.Core"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "store", "content": "def store : Type := ident → Int"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "star", "content": "@[grind] inductive star (R : α → α → Prop) : α → α → Prop where\n  | star_refl : ∀ x : α, star R x x\n  | star_step : ∀ {x y z}, R x y → star R y z → star R x z"}, {"name": "cont", "content": "@[grind] inductive cont where\n| Kstop\n| Kseq (c : com) (k : cont)\n| Kwhile (b : bexp) (c : com) (k : cont)"}, {"name": "step", "content": "inductive step : com × cont × store -> com × cont × store -> Prop where\n  | step_assign : ∀ x a k s,\n      step (.ASSIGN x a, k, s) (.SKIP, k, update x (aeval s a) s)\n    \n  | step_seq : ∀ c1 c2 s k,\n      step (.SEQ c1 c2, k, s) (c1, .Kseq c2 k, s)\n      \n  | step_ifthenelse : ∀ b c1 c2 k s,\n      step (.IFTHENELSE b c1 c2, k, s) ((if beval s b then c1 else c2), k, s)\n      \n  | step_while_done : ∀ b c k s,\n      beval s b = false ->\n      step (.WHILE b c, k, s) (.SKIP, k, s)\n      \n  | step_while_true : ∀ b c k s,\n      beval s b = true ->\n      step (.WHILE b c, k, s) (c, .Kwhile b c k, s)\n      \n  | step_skip_seq : ∀ c k s,\n      step (.SKIP, .Kseq c k, s) (c, k, s)\n      \n  | step_skip_while : ∀ b c k s,\n      step (.SKIP, .Kwhile b c k, s) (.WHILE b c, k, s)"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "plus", "content": "@[grind cases]\ninductive plus (R : α → α → Prop) : α → α → Prop where\n| plus_left : ∀ {a b c}, R a b → star R b c → plus R a c\n\n\ngrind_pattern plus.plus_left => star R b c, plus R a c"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "Exists.elim", "module": "Init.Core"}, {"name": "star_one", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "Int.add_assoc", "module": "Init.Data.Int.Lemmas"}, {"name": "Or.intro_left", "module": "Init.Prelude"}, {"name": "Or.intro_right", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "plus_star", "content": "@[grind] theorem plus_star {a b} (h : plus R a b) : star R a b"}, {"name": "star_trans", "content": "@[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c"}, {"name": "plus_right", "content": "theorem plus_right : star R a b -> R b c -> plus R a c"}], "used_local_defs": [{"name": "instr", "content": "@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr"}, {"name": "codelen", "content": "@[grind] def codelen (c : List instr) : Int := c.length"}, {"name": "stack", "content": "def stack : Type := List Int"}, {"name": "config", "content": "def config : Type := Int × stack × store"}, {"name": "instr_at", "content": "@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)"}, {"name": "transition", "content": "@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)"}, {"name": "transitions", "content": "@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)"}, {"name": "machine_terminates", "content": "def machine_terminates (C : List instr) (s_init : store) (s_final : store) : Prop :=\n  ∃ pc, transitions C (0, [], s_init) (pc, [], s_final)\n          ∧ instr_at C pc = .some .Ihalt"}, {"name": "compile_aexp", "content": "@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])"}, {"name": "compile_bexp", "content": "@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2"}, {"name": "compile_com", "content": "@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []"}, {"name": "compile_program", "content": "def compile_program (p : com) : List instr :=\n  compile_com p ++ .Ihalt :: []"}, {"name": "code_at", "content": "@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2"}, {"name": "compile_cont", "content": "inductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc"}, {"name": "match_config", "content": "inductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)"}, {"name": "com_size", "content": "def com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)"}, {"name": "cont_size", "content": "def cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'"}, {"name": "measure'", "content": "def measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)"}], "used_local_lemmas": [{"name": "codelen_cons", "content": "@[grind =] theorem codelen_cons :\n  ∀ i c, codelen (i :: c) = codelen c + 1"}, {"name": "codelen_app", "content": "@[grind =] theorem codelen_app :\n  ∀ c1 c2, codelen (c1 ++ c2) = codelen c1 + codelen c2"}, {"name": "instr_a", "content": "@[grind =>] theorem instr_a : ∀ i c2 c1 pc,\n  pc = codelen c1 ->\n  instr_at (c1 ++ (i :: c2) ) pc = .some i"}, {"name": "code_at_app_right", "content": "@[grind] theorem code_at_app_right :\n  ∀ C pc C1 C2,\n  code_at C pc (C1 ++ C2) ->\n  code_at C (pc + codelen C1) C2"}, {"name": "code_at_to_instr_at", "content": "@[grind] theorem code_at_to_instr_at : code_at C pc (c1 ++ i :: c2) → instr_at C (pc + codelen c1) = .some i"}, {"name": "compile_aexp_correct", "content": "theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :\n  code_at C pc (compile_aexp a) →\n  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s)"}, {"name": "compile_bexp_correct", "content": "theorem compile_bexp_correct (C : List instr) (s : store) (b : bexp) (d1 d0 : Int) (pc : Int) (stk : stack) (h : code_at C pc (compile_bexp b d1 d0))  :\n   transitions C\n       (pc, stk, s)\n       (pc + codelen (compile_bexp b d1 d0) + (if beval s b then d1 else d0), stk, s)"}, {"name": "compile_cont_Kstop_inv", "content": "theorem compile_cont_Kstop_inv (C : List instr) (pc : Int) (s : store) :\n  compile_cont C .Kstop pc →\n  ∃ pc',\n  star (transition C) (pc, [], s) (pc', [], s)\n  ∧ instr_at C pc' = .some .Ihalt"}, {"name": "compile_cont_Kseq_inv", "content": "theorem compile_cont_Kseq_inv (C : List instr) (c : com) (k :cont) (pc : Int) (s : store) (H : compile_cont C (.Kseq c k) pc) :\n  ∃ pc',\n  star (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com c)\n  ∧ compile_cont C k (pc' + codelen (compile_com c))"}, {"name": "compile_cont_Kwhile_inv", "content": "theorem compile_cont_Kwhile_inv (C : List instr) (b : bexp) (c : com) (k : cont) (pc : Int) (s : store) (H : compile_cont C (.Kwhile b c k) pc) :\n  ∃ pc',\n  plus (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com (.WHILE b c))\n  ∧ compile_cont C k (pc' + codelen (compile_com (.WHILE b c)))"}, {"name": "match_config_skip", "content": "theorem match_config_skip (C : List instr) (k : cont) (s : store) (pc : Int) (H : compile_cont C k pc) :\n match_config C (.SKIP, k, s) (pc, [], s)"}, {"name": "simulation_step", "content": "theorem simulation_step :\n  ∀ C impconf1 impconf2 machconf1,\n  step impconf1 impconf2 ->\n  match_config C impconf1 machconf1 ->\n  ∃ machconf2,\n      (plus (transition C) machconf1 machconf2\n       \\/ (star (transition C) machconf1 machconf2\n           /\\ (measure' impconf2 < measure' impconf1)))\n   /\\ match_config C impconf2 machconf2"}, {"name": "simulation_steps", "content": "theorem simulation_steps :\n  ∀ C impconf1 impconf2, star step impconf1 impconf2 ->\n  ∀ machconf1, match_config C impconf1 machconf1 ->\n  ∃ machconf2,\n     star (transition C) machconf1 machconf2\n  /\\ match_config C impconf2 machconf2"}, {"name": "match_initial_configs", "content": "theorem match_initial_configs :\n  ∀ c s,\n  match_config (compile_program c) (c, .Kstop, s) (0, [], s)"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\n@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr\n\n@[grind] def codelen (c : List instr) : Int := c.length\n\ndef stack : Type := List Int\n\ndef config : Type := Int × stack × store\n\n@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)\n\n@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n\n@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)\n\ndef machine_terminates (C : List instr) (s_init : store) (s_final : store) : Prop :=\n  ∃ pc, transitions C (0, [], s_init) (pc, [], s_final)\n          ∧ instr_at C pc = .some .Ihalt\n\n@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])\n\n@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2\n\n@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []\n\ndef compile_program (p : com) : List instr :=\n  compile_com p ++ .Ihalt :: []\n\n@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2\n\ninductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc\n\ninductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)\n\ndef com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)\n\ndef cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'\n\ndef measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)", "target_theorem": "theorem compile_program_correct_terminating_2 :\n  ∀ c s s',\n  star step (c, .Kstop, s) (.SKIP, .Kstop, s') ->\n  machine_terminates (compile_program c) s s' :=", "ground_truth_proof": ":= by\n    intro c s s' STAR\n    generalize heq : compile_program c = C\n    have ⟨ ms, A, B ⟩ := simulation_steps C (c, cont.Kstop, s) (com.SKIP, cont.Kstop, s') STAR (0, [], s) (by grind [match_initial_configs])\n    cases B\n    case match_config_intro pc w1 w2 =>\n      have ⟨pc', D, E ⟩ := compile_cont_Kstop_inv C (pc + codelen (compile_com com.SKIP)) s' w2\n      exists pc'\n      constructor\n      · apply star_trans\n        · exact A\n        · simp [compile_com, codelen] at D\n          exact D\n      · exact E", "nesting_depth": 5, "transitive_dep_count": 70, "subset_aristotle": false, "category": "Compiler"}
{"id": 366, "thm_name": "simulation_steps", "thm_stmt": "theorem simulation_steps :\n  ∀ C impconf1 impconf2, star step impconf1 impconf2 ->\n  ∀ machconf1, match_config C impconf1 machconf1 ->\n  ∃ machconf2,\n     star (transition C) machconf1 machconf2\n  /\\ match_config C impconf2 machconf2", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Compil.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import LeroyCompilerVerificationCourse.Sequences"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Exists", "module": "Init.Core"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "cont", "content": "@[grind] inductive cont where\n| Kstop\n| Kseq (c : com) (k : cont)\n| Kwhile (b : bexp) (c : com) (k : cont)"}, {"name": "store", "content": "def store : Type := ident → Int"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "star", "content": "@[grind] inductive star (R : α → α → Prop) : α → α → Prop where\n  | star_refl : ∀ x : α, star R x x\n  | star_step : ∀ {x y z}, R x y → star R y z → star R x z"}, {"name": "step", "content": "inductive step : com × cont × store -> com × cont × store -> Prop where\n  | step_assign : ∀ x a k s,\n      step (.ASSIGN x a, k, s) (.SKIP, k, update x (aeval s a) s)\n    \n  | step_seq : ∀ c1 c2 s k,\n      step (.SEQ c1 c2, k, s) (c1, .Kseq c2 k, s)\n      \n  | step_ifthenelse : ∀ b c1 c2 k s,\n      step (.IFTHENELSE b c1 c2, k, s) ((if beval s b then c1 else c2), k, s)\n      \n  | step_while_done : ∀ b c k s,\n      beval s b = false ->\n      step (.WHILE b c, k, s) (.SKIP, k, s)\n      \n  | step_while_true : ∀ b c k s,\n      beval s b = true ->\n      step (.WHILE b c, k, s) (c, .Kwhile b c k, s)\n      \n  | step_skip_seq : ∀ c k s,\n      step (.SKIP, .Kseq c k, s) (c, k, s)\n      \n  | step_skip_while : ∀ b c k s,\n      step (.SKIP, .Kwhile b c k, s) (.WHILE b c, k, s)"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "plus", "content": "@[grind cases]\ninductive plus (R : α → α → Prop) : α → α → Prop where\n| plus_left : ∀ {a b c}, R a b → star R b c → plus R a c\n\n\ngrind_pattern plus.plus_left => star R b c, plus R a c"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "Exists.elim", "module": "Init.Core"}, {"name": "star_one", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "Int.add_assoc", "module": "Init.Data.Int.Lemmas"}, {"name": "Or.intro_left", "module": "Init.Prelude"}, {"name": "Or.intro_right", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "plus_star", "content": "@[grind] theorem plus_star {a b} (h : plus R a b) : star R a b"}, {"name": "star_trans", "content": "@[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c"}, {"name": "plus_right", "content": "theorem plus_right : star R a b -> R b c -> plus R a c"}], "used_local_defs": [{"name": "instr", "content": "@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr"}, {"name": "codelen", "content": "@[grind] def codelen (c : List instr) : Int := c.length"}, {"name": "stack", "content": "def stack : Type := List Int"}, {"name": "config", "content": "def config : Type := Int × stack × store"}, {"name": "instr_at", "content": "@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)"}, {"name": "transition", "content": "@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)"}, {"name": "transitions", "content": "@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)"}, {"name": "compile_aexp", "content": "@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])"}, {"name": "compile_bexp", "content": "@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2"}, {"name": "compile_com", "content": "@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []"}, {"name": "code_at", "content": "@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2"}, {"name": "compile_cont", "content": "inductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc"}, {"name": "match_config", "content": "inductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)"}, {"name": "com_size", "content": "def com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)"}, {"name": "cont_size", "content": "def cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'"}, {"name": "measure'", "content": "def measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)"}], "used_local_lemmas": [{"name": "codelen_cons", "content": "@[grind =] theorem codelen_cons :\n  ∀ i c, codelen (i :: c) = codelen c + 1"}, {"name": "codelen_app", "content": "@[grind =] theorem codelen_app :\n  ∀ c1 c2, codelen (c1 ++ c2) = codelen c1 + codelen c2"}, {"name": "instr_a", "content": "@[grind =>] theorem instr_a : ∀ i c2 c1 pc,\n  pc = codelen c1 ->\n  instr_at (c1 ++ (i :: c2) ) pc = .some i"}, {"name": "code_at_app_right", "content": "@[grind] theorem code_at_app_right :\n  ∀ C pc C1 C2,\n  code_at C pc (C1 ++ C2) ->\n  code_at C (pc + codelen C1) C2"}, {"name": "code_at_to_instr_at", "content": "@[grind] theorem code_at_to_instr_at : code_at C pc (c1 ++ i :: c2) → instr_at C (pc + codelen c1) = .some i"}, {"name": "compile_aexp_correct", "content": "theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :\n  code_at C pc (compile_aexp a) →\n  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s)"}, {"name": "compile_bexp_correct", "content": "theorem compile_bexp_correct (C : List instr) (s : store) (b : bexp) (d1 d0 : Int) (pc : Int) (stk : stack) (h : code_at C pc (compile_bexp b d1 d0))  :\n   transitions C\n       (pc, stk, s)\n       (pc + codelen (compile_bexp b d1 d0) + (if beval s b then d1 else d0), stk, s)"}, {"name": "compile_cont_Kseq_inv", "content": "theorem compile_cont_Kseq_inv (C : List instr) (c : com) (k :cont) (pc : Int) (s : store) (H : compile_cont C (.Kseq c k) pc) :\n  ∃ pc',\n  star (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com c)\n  ∧ compile_cont C k (pc' + codelen (compile_com c))"}, {"name": "compile_cont_Kwhile_inv", "content": "theorem compile_cont_Kwhile_inv (C : List instr) (b : bexp) (c : com) (k : cont) (pc : Int) (s : store) (H : compile_cont C (.Kwhile b c k) pc) :\n  ∃ pc',\n  plus (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com (.WHILE b c))\n  ∧ compile_cont C k (pc' + codelen (compile_com (.WHILE b c)))"}, {"name": "match_config_skip", "content": "theorem match_config_skip (C : List instr) (k : cont) (s : store) (pc : Int) (H : compile_cont C k pc) :\n match_config C (.SKIP, k, s) (pc, [], s)"}, {"name": "simulation_step", "content": "theorem simulation_step :\n  ∀ C impconf1 impconf2 machconf1,\n  step impconf1 impconf2 ->\n  match_config C impconf1 machconf1 ->\n  ∃ machconf2,\n      (plus (transition C) machconf1 machconf2\n       \\/ (star (transition C) machconf1 machconf2\n           /\\ (measure' impconf2 < measure' impconf1)))\n   /\\ match_config C impconf2 machconf2"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\n@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr\n\n@[grind] def codelen (c : List instr) : Int := c.length\n\ndef stack : Type := List Int\n\ndef config : Type := Int × stack × store\n\n@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)\n\n@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n\n@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)\n\n@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])\n\n@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2\n\n@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []\n\n@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2\n\ninductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc\n\ninductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)\n\ndef com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)\n\ndef cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'\n\ndef measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)", "target_theorem": "theorem simulation_steps :\n  ∀ C impconf1 impconf2, star step impconf1 impconf2 ->\n  ∀ machconf1, match_config C impconf1 machconf1 ->\n  ∃ machconf2,\n     star (transition C) machconf1 machconf2\n  /\\ match_config C impconf2 machconf2 :=", "ground_truth_proof": ":= by\n      intro C impconf1 impconf2 STAR machconf1 MATCH\n      induction STAR generalizing machconf1\n      case star_refl x =>\n        exists machconf1\n        constructor\n        · apply star.star_refl\n        · exact MATCH\n      case star_step x y z STEP STAR ih =>\n        have ⟨ machconf2, steps2, match2 ⟩ := simulation_step C x y machconf1 STEP MATCH\n        specialize ih machconf2 match2\n        rcases ih with ⟨ machconf3, steps3, match3⟩\n        exists machconf3\n        have w : star (transition C) machconf1 machconf2 := by\n          cases steps2\n          case inl h =>\n            apply plus_star\n            exact h\n          case inr h =>\n            exact h.1\n        constructor\n        · apply star_trans\n          · exact w\n          · exact steps3\n        · exact match3", "nesting_depth": 5, "transitive_dep_count": 65, "subset_aristotle": false, "category": "Compiler"}
{"id": 367, "thm_name": "fixpoint_join_sound", "thm_stmt": "theorem fixpoint_join_sound : Le Init (fixpoint_join Init F) /\\ Le (F (fixpoint_join Init F)) (fixpoint_join Init F)", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Fixpoints.lean", "imports": ["import LeroyCompilerVerificationCourse.Constprop", "import LeroyCompilerVerificationCourse.Imp", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "outParam", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.Equiv", "module": "Std.Data.HashMap.Basic"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "WellFounded", "module": "Init.WF"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.add", "module": "Init.Data.Int.Basic"}, {"name": "Int.sub", "module": "Init.Data.Int.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "BitVec", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Join", "content": "@[grind] def Join (S1 S2 : Store) : Store :=\n  S1.filter (fun key _ => S2.get? key == S1.get? key)"}, {"name": "Store", "content": "def Store := Std.HashMap ident Int"}, {"name": "Le", "content": "@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n"}, {"name": "Equal", "content": "def Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2"}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "Beval", "content": "@[grind] def Beval (S : Store) (b : bexp) : Option Bool :=\n  match b with\n  | .TRUE => .some true\n  | .FALSE => .some false\n  | .EQUAL a1 a2 => lift2 (fun m n => m == n) (Aeval S a1) (Aeval S a2)\n  | .LESSEQUAL a1 a2 => lift2 (fun m n => m <= n) (Aeval S a1) (Aeval S a2)\n  | .NOT b1 => lift1 (fun m => !m) (Beval S b1)\n  | .AND b1 b2 => lift2 (fun m n => m && n) (Beval S b1) (Beval S b2)"}, {"name": "lift1", "content": "@[grind] def lift1 {A B : Type} (f : A -> B) (o : Option A) : Option B :=\n  match o with\n  | .some x => .some (f x)\n  | .none => .none"}, {"name": "Aeval", "content": "@[grind] def Aeval (S : Store) (a : aexp) : Option Int :=\n  match a with\n  | .CONST n => .some n\n  | .VAR x => S.get? x\n  | .PLUS a1 a2 => lift2 (Int.add) (Aeval S a1) (Aeval S a2)\n  | .MINUS a1 a2 => lift2 (Int.sub) (Aeval S a1) (Aeval S a2)"}, {"name": "lift2", "content": "@[grind] def lift2 {A B C : Type} (f : A -> B -> C) (o1 : Option A) (o2 : Option B) : Option C :=\n  match o1, o2 with\n  | .some x1, .some x2 => .some (f x1 x2) | _, _ => .none"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "Update", "content": "@[grind] def Update (x : ident) (N : Option Int) (S : Store) : Store :=\n  match N with\n  | .none => S.erase x\n  | .some n => S.insert x n"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "Std.HashMap.Equiv.getElem?_eq", "module": "Std.Data.HashMap.Lemmas"}], "repo_lemmas": [{"name": "Equal_Le", "content": "theorem Equal_Le : ∀ S1 S2, Equal S1 S2 -> Le S1 S2"}, {"name": "Le_Join_l", "content": "theorem Le_Join_l : ∀ S1 S2, Le S1 (Join S1 S2)"}, {"name": "Le_Join_r", "content": "theorem  Le_Join_r : ∀ S1 S2, Le S2 (Join S1 S2)"}], "used_local_defs": [{"name": "OrderStruct", "content": "@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/"}, {"name": "Monotone", "content": "class Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)"}, {"name": "iterate", "content": "@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']"}, {"name": "Eq'", "content": "@[grind] def Eq' (S1 S2 : Store) : Prop := Equal S1 S2"}, {"name": "Eq'_sym", "content": "def Eq'_sym : ∀ S1 S2, Eq' S1 S2 → Eq' S2 S1 :="}, {"name": "_inst_OrderStruct", "content": "noncomputable instance : OrderStruct Store where\n  eq := Equal\n  le := Le\n  beq (S1 S2 : Store) :=  Decidable.decide (Equal S1 S2)\n  le_trans := Le_trans\n  gt_wf := Gt_wf"}, {"name": "_inst_Monotone", "content": "instance : Monotone Store (fun x => Join Init (F x)) where\n  F_mon := by admit /- proof elided -/"}, {"name": "fixpoint_join", "content": "noncomputable def fixpoint_join : Store :="}], "used_local_lemmas": [{"name": "iterate_correct", "content": "@[grind] theorem iterate_correct (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) (heq : y = iterate _ F x PRE SMALL ) : eq y (F y) ∧ ∀ z, le (F z) z → le y z"}, {"name": "Eq_Le", "content": "@[grind] theorem Eq_Le : ∀ S1 S2, Eq' S1 S2 → Le S1 S2"}, {"name": "Le_trans", "content": "@[grind] theorem Le_trans : ∀ S1 S2 S3, Le S1 S2 → Le S2 S3 → Le S1 S3"}, {"name": "fixpoint_join_eq", "content": "theorem fixpoint_join_eq : Eq' (Join Init (F (fixpoint_join Init F) )) (fixpoint_join Init F)"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport LeroyCompilerVerificationCourse.Constprop\n\nimport Batteries.Data.List.Perm\n\n@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/\n\nopen OrderStruct\n\nclass Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)\n\nopen Monotone\n\nsection FixpointExistence\n\nvariable (α : Sort u) (F : α → α) [OrderWithBot α]\n\nopen OrderStruct OrderWithBot\n\nend FixpointExistence\n\nsection Iterate\n\nvariable (α : Sort u) [inst : OrderStruct α] (F : α → α) [Monotone α F]\n\nopen OrderStruct\n\n@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']\n\nend Iterate\n\nsection Fixpoint\n\nopen OrderWithBot\n\nvariable {α : Sort u} [i : OrderWithBot α] (F : α → α) [Monotone α F]\n\nend Fixpoint\n\nsection Constprop\n\nopen Std.HashMap\n\n@[grind] def Eq' (S1 S2 : Store) : Prop := Equal S1 S2\n\ndef Eq'_sym : ∀ S1 S2, Eq' S1 S2 → Eq' S2 S1 :=\n\nopen OrderStruct\n\nnoncomputable instance : OrderStruct Store where\n  eq := Equal\n  le := Le\n  beq (S1 S2 : Store) :=  Decidable.decide (Equal S1 S2)\n  le_trans := Le_trans\n  gt_wf := Gt_wf\n\nend Constprop\n\nsection FixpointJoin\n\nvariable (Init : Store)\n\nvariable (F : Store → Store) [Monotone Store F]\n\ninstance : Monotone Store (fun x => Join Init (F x)) where\n  F_mon := by admit /- proof elided -/\n\nnoncomputable def fixpoint_join : Store :=", "target_theorem": "theorem fixpoint_join_sound : Le Init (fixpoint_join Init F) /\\ Le (F (fixpoint_join Init F)) (fixpoint_join Init F) :=", "ground_truth_proof": ":= by\n  have LE : Le (Join Init (F (fixpoint_join Init F))) (fixpoint_join Init F) := by\n    apply Eq_Le\n    apply fixpoint_join_eq\n  constructor\n  · apply Le_trans\n    rotate_left\n    · exact LE\n    · apply Le_Join_l\n  · apply Le_trans\n    rotate_left\n    · exact LE\n    · apply Le_Join_r", "nesting_depth": 4, "transitive_dep_count": 22, "subset_aristotle": false, "category": "Compiler"}
{"id": 368, "thm_name": "compile_program_correct_diverging", "thm_stmt": "theorem compile_program_correct_diverging :\n  ∀ c s,\n  infseq step (c, .Kstop, s) ->\n  machine_diverges (compile_program c) s", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Compil.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import LeroyCompilerVerificationCourse.Sequences"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Exists", "module": "Init.Core"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "cont", "content": "@[grind] inductive cont where\n| Kstop\n| Kseq (c : com) (k : cont)\n| Kwhile (b : bexp) (c : com) (k : cont)"}, {"name": "store", "content": "def store : Type := ident → Int"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "infseq", "content": "def infseq {α} (R : α → α → Prop) : α → Prop :=\n  λ x : α => ∃ y, R x y ∧ infseq R y\n  coinductive_fixpoint"}, {"name": "step", "content": "inductive step : com × cont × store -> com × cont × store -> Prop where\n  | step_assign : ∀ x a k s,\n      step (.ASSIGN x a, k, s) (.SKIP, k, update x (aeval s a) s)\n    \n  | step_seq : ∀ c1 c2 s k,\n      step (.SEQ c1 c2, k, s) (c1, .Kseq c2 k, s)\n      \n  | step_ifthenelse : ∀ b c1 c2 k s,\n      step (.IFTHENELSE b c1 c2, k, s) ((if beval s b then c1 else c2), k, s)\n      \n  | step_while_done : ∀ b c k s,\n      beval s b = false ->\n      step (.WHILE b c, k, s) (.SKIP, k, s)\n      \n  | step_while_true : ∀ b c k s,\n      beval s b = true ->\n      step (.WHILE b c, k, s) (c, .Kwhile b c k, s)\n      \n  | step_skip_seq : ∀ c k s,\n      step (.SKIP, .Kseq c k, s) (c, k, s)\n      \n  | step_skip_while : ∀ b c k s,\n      step (.SKIP, .Kwhile b c k, s) (.WHILE b c, k, s)"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "plus", "content": "@[grind cases]\ninductive plus (R : α → α → Prop) : α → α → Prop where\n| plus_left : ∀ {a b c}, R a b → star R b c → plus R a c\n\n\ngrind_pattern plus.plus_left => star R b c, plus R a c"}, {"name": "star", "content": "@[grind] inductive star (R : α → α → Prop) : α → α → Prop where\n  | star_refl : ∀ x : α, star R x x\n  | star_step : ∀ {x y z}, R x y → star R y z → star R x z"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "Exists.elim", "module": "Init.Core"}, {"name": "star_one", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "Int.add_assoc", "module": "Init.Data.Int.Lemmas"}, {"name": "Or.intro_left", "module": "Init.Prelude"}, {"name": "Or.intro_right", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "plus_star", "content": "@[grind] theorem plus_star {a b} (h : plus R a b) : star R a b"}, {"name": "star_trans", "content": "@[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c"}, {"name": "plus_right", "content": "theorem plus_right : star R a b -> R b c -> plus R a c"}, {"name": "star_plus_trans", "content": "theorem star_plus_trans :\n  ∀ a b c, star R a b -> plus R b c -> plus R a c"}, {"name": "infseq_coinduction_principle_2", "content": "theorem infseq_coinduction_principle_2\n    (X : α → Prop) (h₁ : ∀ (a : α), X a → ∃ b, plus R a b ∧ X b) (a : α) (rel : X a) : infseq R a"}], "used_local_defs": [{"name": "instr", "content": "@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr"}, {"name": "codelen", "content": "@[grind] def codelen (c : List instr) : Int := c.length"}, {"name": "stack", "content": "def stack : Type := List Int"}, {"name": "config", "content": "def config : Type := Int × stack × store"}, {"name": "instr_at", "content": "@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)"}, {"name": "transition", "content": "@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)"}, {"name": "transitions", "content": "@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)"}, {"name": "machine_diverges", "content": "def machine_diverges (C : List instr) (s_init : store) : Prop :=\n  infseq (transition C) (0, [], s_init)"}, {"name": "compile_aexp", "content": "@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])"}, {"name": "compile_bexp", "content": "@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2"}, {"name": "compile_com", "content": "@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []"}, {"name": "compile_program", "content": "def compile_program (p : com) : List instr :=\n  compile_com p ++ .Ihalt :: []"}, {"name": "code_at", "content": "@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2"}, {"name": "compile_cont", "content": "inductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc"}, {"name": "match_config", "content": "inductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)"}, {"name": "com_size", "content": "def com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)"}, {"name": "cont_size", "content": "def cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'"}, {"name": "measure'", "content": "def measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)"}], "used_local_lemmas": [{"name": "codelen_cons", "content": "@[grind =] theorem codelen_cons :\n  ∀ i c, codelen (i :: c) = codelen c + 1"}, {"name": "codelen_app", "content": "@[grind =] theorem codelen_app :\n  ∀ c1 c2, codelen (c1 ++ c2) = codelen c1 + codelen c2"}, {"name": "instr_a", "content": "@[grind =>] theorem instr_a : ∀ i c2 c1 pc,\n  pc = codelen c1 ->\n  instr_at (c1 ++ (i :: c2) ) pc = .some i"}, {"name": "code_at_app_right", "content": "@[grind] theorem code_at_app_right :\n  ∀ C pc C1 C2,\n  code_at C pc (C1 ++ C2) ->\n  code_at C (pc + codelen C1) C2"}, {"name": "code_at_to_instr_at", "content": "@[grind] theorem code_at_to_instr_at : code_at C pc (c1 ++ i :: c2) → instr_at C (pc + codelen c1) = .some i"}, {"name": "compile_aexp_correct", "content": "theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :\n  code_at C pc (compile_aexp a) →\n  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s)"}, {"name": "compile_bexp_correct", "content": "theorem compile_bexp_correct (C : List instr) (s : store) (b : bexp) (d1 d0 : Int) (pc : Int) (stk : stack) (h : code_at C pc (compile_bexp b d1 d0))  :\n   transitions C\n       (pc, stk, s)\n       (pc + codelen (compile_bexp b d1 d0) + (if beval s b then d1 else d0), stk, s)"}, {"name": "compile_cont_Kseq_inv", "content": "theorem compile_cont_Kseq_inv (C : List instr) (c : com) (k :cont) (pc : Int) (s : store) (H : compile_cont C (.Kseq c k) pc) :\n  ∃ pc',\n  star (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com c)\n  ∧ compile_cont C k (pc' + codelen (compile_com c))"}, {"name": "compile_cont_Kwhile_inv", "content": "theorem compile_cont_Kwhile_inv (C : List instr) (b : bexp) (c : com) (k : cont) (pc : Int) (s : store) (H : compile_cont C (.Kwhile b c k) pc) :\n  ∃ pc',\n  plus (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com (.WHILE b c))\n  ∧ compile_cont C k (pc' + codelen (compile_com (.WHILE b c)))"}, {"name": "match_config_skip", "content": "theorem match_config_skip (C : List instr) (k : cont) (s : store) (pc : Int) (H : compile_cont C k pc) :\n match_config C (.SKIP, k, s) (pc, [], s)"}, {"name": "simulation_step", "content": "theorem simulation_step :\n  ∀ C impconf1 impconf2 machconf1,\n  step impconf1 impconf2 ->\n  match_config C impconf1 machconf1 ->\n  ∃ machconf2,\n      (plus (transition C) machconf1 machconf2\n       \\/ (star (transition C) machconf1 machconf2\n           /\\ (measure' impconf2 < measure' impconf1)))\n   /\\ match_config C impconf2 machconf2"}, {"name": "match_initial_configs", "content": "theorem match_initial_configs :\n  ∀ c s,\n  match_config (compile_program c) (c, .Kstop, s) (0, [], s)"}, {"name": "simulation_infseq_inv", "content": "theorem simulation_infseq_inv :\n  ∀ C n impconf1 machconf1,\n  infseq step impconf1 -> match_config C impconf1 machconf1 ->\n  (measure' impconf1 < n) ->\n  ∃ impconf2 machconf2,\n      infseq step impconf2\n   /\\ plus (transition C) machconf1 machconf2\n   /\\ match_config C impconf2 machconf2"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\n@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr\n\n@[grind] def codelen (c : List instr) : Int := c.length\n\ndef stack : Type := List Int\n\ndef config : Type := Int × stack × store\n\n@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)\n\n@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n\n@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)\n\ndef machine_diverges (C : List instr) (s_init : store) : Prop :=\n  infseq (transition C) (0, [], s_init)\n\n@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])\n\n@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2\n\n@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []\n\ndef compile_program (p : com) : List instr :=\n  compile_com p ++ .Ihalt :: []\n\n@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2\n\ninductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc\n\ninductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)\n\ndef com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)\n\ndef cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'\n\ndef measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)", "target_theorem": "theorem compile_program_correct_diverging :\n  ∀ c s,\n  infseq step (c, .Kstop, s) ->\n  machine_diverges (compile_program c) s :=", "ground_truth_proof": ":= by\n    intro c s H\n    generalize heq : compile_program c = C\n    unfold machine_diverges\n    apply infseq_coinduction_principle_2 (fun machconf => ∃ impconf, infseq step impconf /\\ match_config C impconf machconf)\n    rotate_left\n    · exists (c, .Kstop, s)\n      constructor\n      · exact H\n      · have := match_initial_configs c s\n        grind\n    · intro machconf ⟨ impconf , ⟨INFSEQ, MATCH ⟩⟩\n      have ⟨impconf2 , machconf2, INFSEQ2 , PLUS , MATCH2⟩  := simulation_infseq_inv C (measure' impconf +1) impconf machconf INFSEQ MATCH (by omega)\n      exists machconf2\n      constructor\n      · exact PLUS\n      · exists impconf2", "nesting_depth": 5, "transitive_dep_count": 72, "subset_aristotle": false, "category": "Compiler"}
{"id": 369, "thm_name": "simulation_infseq_inv", "thm_stmt": "theorem simulation_infseq_inv :\n  ∀ C n impconf1 machconf1,\n  infseq step impconf1 -> match_config C impconf1 machconf1 ->\n  (measure' impconf1 < n) ->\n  ∃ impconf2 machconf2,\n      infseq step impconf2\n   /\\ plus (transition C) machconf1 machconf2\n   /\\ match_config C impconf2 machconf2", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Compil.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import LeroyCompilerVerificationCourse.Sequences"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Exists", "module": "Init.Core"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "cont", "content": "@[grind] inductive cont where\n| Kstop\n| Kseq (c : com) (k : cont)\n| Kwhile (b : bexp) (c : com) (k : cont)"}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "store", "content": "def store : Type := ident → Int"}, {"name": "update", "content": "@[grind] def update (x : ident) (v : Int) (s : store) : store :=\n  fun y => if x == y then v else s y"}, {"name": "plus", "content": "@[grind cases]\ninductive plus (R : α → α → Prop) : α → α → Prop where\n| plus_left : ∀ {a b c}, R a b → star R b c → plus R a c\n\n\ngrind_pattern plus.plus_left => star R b c, plus R a c"}, {"name": "star", "content": "@[grind] inductive star (R : α → α → Prop) : α → α → Prop where\n  | star_refl : ∀ x : α, star R x x\n  | star_step : ∀ {x y z}, R x y → star R y z → star R x z"}, {"name": "infseq", "content": "def infseq {α} (R : α → α → Prop) : α → Prop :=\n  λ x : α => ∃ y, R x y ∧ infseq R y\n  coinductive_fixpoint"}, {"name": "step", "content": "inductive step : com × cont × store -> com × cont × store -> Prop where\n  | step_assign : ∀ x a k s,\n      step (.ASSIGN x a, k, s) (.SKIP, k, update x (aeval s a) s)\n    \n  | step_seq : ∀ c1 c2 s k,\n      step (.SEQ c1 c2, k, s) (c1, .Kseq c2 k, s)\n      \n  | step_ifthenelse : ∀ b c1 c2 k s,\n      step (.IFTHENELSE b c1 c2, k, s) ((if beval s b then c1 else c2), k, s)\n      \n  | step_while_done : ∀ b c k s,\n      beval s b = false ->\n      step (.WHILE b c, k, s) (.SKIP, k, s)\n      \n  | step_while_true : ∀ b c k s,\n      beval s b = true ->\n      step (.WHILE b c, k, s) (c, .Kwhile b c k, s)\n      \n  | step_skip_seq : ∀ c k s,\n      step (.SKIP, .Kseq c k, s) (c, k, s)\n      \n  | step_skip_while : ∀ b c k s,\n      step (.SKIP, .Kwhile b c k, s) (.WHILE b c, k, s)"}, {"name": "aeval", "content": "@[grind] def aeval (s : store) (a : aexp) : Int :=\n  match a with\n  | .CONST n => n\n  | .VAR x => s x\n  | .PLUS a1 a2 => aeval s a1 + aeval s a2\n  | .MINUS a1 a2 => aeval s a1 - aeval s a2"}, {"name": "beval", "content": "@[grind] def beval (s : store) (b : bexp) : Bool :=\n  match b with\n  | .TRUE => true\n  | .FALSE => false\n  | .EQUAL a1 a2 => aeval s a1 = aeval s a2\n  | .LESSEQUAL a1 a2 => aeval s a1 <= aeval s a2\n  | .NOT b1 =>  !(beval s b1)\n  | .AND b1 b2 => beval s b1 && beval s b2"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "Exists.elim", "module": "Init.Core"}, {"name": "star_one", "module": "Mathlib.Algebra.Star.Basic"}, {"name": "Int.add_assoc", "module": "Init.Data.Int.Lemmas"}, {"name": "Or.intro_left", "module": "Init.Prelude"}, {"name": "Or.intro_right", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "plus_star", "content": "@[grind] theorem plus_star {a b} (h : plus R a b) : star R a b"}, {"name": "star_trans", "content": "@[grind] theorem star_trans {α} (R : α → α → Prop) (a b : α) (sab : star R a b) : ∀ c : α, star R b c → star R a c"}, {"name": "plus_right", "content": "theorem plus_right : star R a b -> R b c -> plus R a c"}, {"name": "star_plus_trans", "content": "theorem star_plus_trans :\n  ∀ a b c, star R a b -> plus R b c -> plus R a c"}], "used_local_defs": [{"name": "instr", "content": "@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr"}, {"name": "codelen", "content": "@[grind] def codelen (c : List instr) : Int := c.length"}, {"name": "stack", "content": "def stack : Type := List Int"}, {"name": "config", "content": "def config : Type := Int × stack × store"}, {"name": "instr_at", "content": "@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)"}, {"name": "transition", "content": "@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)"}, {"name": "transitions", "content": "@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)"}, {"name": "compile_aexp", "content": "@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])"}, {"name": "compile_bexp", "content": "@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2"}, {"name": "compile_com", "content": "@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []"}, {"name": "code_at", "content": "@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2"}, {"name": "compile_cont", "content": "inductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc"}, {"name": "match_config", "content": "inductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)"}, {"name": "com_size", "content": "def com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)"}, {"name": "cont_size", "content": "def cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'"}, {"name": "measure'", "content": "def measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)"}], "used_local_lemmas": [{"name": "codelen_cons", "content": "@[grind =] theorem codelen_cons :\n  ∀ i c, codelen (i :: c) = codelen c + 1"}, {"name": "codelen_app", "content": "@[grind =] theorem codelen_app :\n  ∀ c1 c2, codelen (c1 ++ c2) = codelen c1 + codelen c2"}, {"name": "instr_a", "content": "@[grind =>] theorem instr_a : ∀ i c2 c1 pc,\n  pc = codelen c1 ->\n  instr_at (c1 ++ (i :: c2) ) pc = .some i"}, {"name": "code_at_app_right", "content": "@[grind] theorem code_at_app_right :\n  ∀ C pc C1 C2,\n  code_at C pc (C1 ++ C2) ->\n  code_at C (pc + codelen C1) C2"}, {"name": "code_at_to_instr_at", "content": "@[grind] theorem code_at_to_instr_at : code_at C pc (c1 ++ i :: c2) → instr_at C (pc + codelen c1) = .some i"}, {"name": "compile_aexp_correct", "content": "theorem compile_aexp_correct (C : List instr) (s : store) (a : aexp) (pc : Int) (stk : stack) :\n  code_at C pc (compile_aexp a) →\n  transitions C (pc, stk, s) (pc + codelen (compile_aexp a), aeval s a :: stk, s)"}, {"name": "compile_bexp_correct", "content": "theorem compile_bexp_correct (C : List instr) (s : store) (b : bexp) (d1 d0 : Int) (pc : Int) (stk : stack) (h : code_at C pc (compile_bexp b d1 d0))  :\n   transitions C\n       (pc, stk, s)\n       (pc + codelen (compile_bexp b d1 d0) + (if beval s b then d1 else d0), stk, s)"}, {"name": "compile_cont_Kseq_inv", "content": "theorem compile_cont_Kseq_inv (C : List instr) (c : com) (k :cont) (pc : Int) (s : store) (H : compile_cont C (.Kseq c k) pc) :\n  ∃ pc',\n  star (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com c)\n  ∧ compile_cont C k (pc' + codelen (compile_com c))"}, {"name": "compile_cont_Kwhile_inv", "content": "theorem compile_cont_Kwhile_inv (C : List instr) (b : bexp) (c : com) (k : cont) (pc : Int) (s : store) (H : compile_cont C (.Kwhile b c k) pc) :\n  ∃ pc',\n  plus (transition C) (pc, [], s) (pc', [], s)\n  ∧ code_at C pc' (compile_com (.WHILE b c))\n  ∧ compile_cont C k (pc' + codelen (compile_com (.WHILE b c)))"}, {"name": "match_config_skip", "content": "theorem match_config_skip (C : List instr) (k : cont) (s : store) (pc : Int) (H : compile_cont C k pc) :\n match_config C (.SKIP, k, s) (pc, [], s)"}, {"name": "simulation_step", "content": "theorem simulation_step :\n  ∀ C impconf1 impconf2 machconf1,\n  step impconf1 impconf2 ->\n  match_config C impconf1 machconf1 ->\n  ∃ machconf2,\n      (plus (transition C) machconf1 machconf2\n       \\/ (star (transition C) machconf1 machconf2\n           /\\ (measure' impconf2 < measure' impconf1)))\n   /\\ match_config C impconf2 machconf2"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\n@[grind] inductive instr : Type where\n  | Iconst (n : Int)             \n  | Ivar (x : ident)             \n  | Isetvar (x : ident)          \n  | Iadd                         \n  | Iopp                         \n  | Ibranch (d : Int)            \n  | Ibeq (d1 : Int) (d0 : Int)   \n  | Ible (d1 : Int) (d0 : Int)   \n  | Ihalt                        \n    deriving Repr\n\n@[grind] def codelen (c : List instr) : Int := c.length\n\ndef stack : Type := List Int\n\ndef config : Type := Int × stack × store\n\n@[grind] def instr_at (C : List instr) (pc : Int) : Option instr :=\n  match C with\n  | [] => .none\n  | i :: C' => if pc = 0 then .some i else instr_at C' (pc - 1)\n\n@[grind] inductive transition (C : List instr) : config → config → Prop where\n  | trans_const : ∀ pc stk s n,\n      instr_at C pc = .some (.Iconst n) →\n      transition C (pc    , stk     , s)\n                   (pc + 1, n :: stk, s)\n  | trans_var : ∀ pc stk s x,\n      instr_at C pc = .some (.Ivar x) ->\n      transition C (pc    , stk       , s)\n                   (pc + 1, s x :: stk, s)\n  | trans_setvar : ∀ pc stk s x n,\n      instr_at C pc = .some (.Isetvar x) ->\n      transition C (pc    , n :: stk, s)\n                   (pc + 1, stk     , update x n s)\n  | trans_add : ∀ pc stk s n1 n2,\n      instr_at C pc = .some (.Iadd) ->\n      transition C (pc    , n2 :: n1 :: stk , s)\n                   (pc + 1, (n1 + n2) :: stk, s)\n  | trans_opp : ∀ pc stk s n,\n      instr_at C pc = .some (.Iopp) ->\n      transition C (pc    , n :: stk    , s)\n                   (pc + 1, (- n) :: stk, s)\n  | trans_branch : ∀ pc stk s d pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      transition C (pc , stk, s)\n                   (pc', stk, s)\n  | trans_beq : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ibeq d1 d0) ->\n      pc' = pc + 1 + (if n1 = n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n  | trans_ble : ∀ pc stk s d1 d0 n1 n2 pc',\n      instr_at C pc = .some (.Ible d1 d0) ->\n      pc' = pc + 1 + (if n1 ≤ n2 then d1 else d0) ->\n      transition C (pc , n2 :: n1 :: stk, s)\n                   (pc', stk            , s)\n\n@[grind] def transitions (C : List instr) : config → config → Prop :=\n  star (transition C)\n\n@[grind] def compile_aexp (a : aexp) : List instr :=\n  match a with\n  | .CONST n => .Iconst n :: []\n  | .VAR x => .Ivar x :: []\n  | .PLUS a1 a2  => (compile_aexp a1) ++ (compile_aexp a2) ++ (.Iadd :: [])\n  | .MINUS a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ (.Iopp :: .Iadd :: [])\n\n@[grind] def compile_bexp (b : bexp) (d1 : Int) (d0 : Int) : List instr :=\n  match b with\n  | .TRUE => if d1 = 0 then [] else .Ibranch d1 :: []\n  | .FALSE => if d0 = 0 then [] else .Ibranch d0 :: []\n  | .EQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ibeq d1 d0 :: []\n  | .LESSEQUAL a1 a2 => compile_aexp a1 ++ compile_aexp a2 ++ .Ible d1 d0 :: []\n  | .NOT b1 => compile_bexp b1 d0 d1\n  | .AND b1 b2 =>\n      let code2 := compile_bexp b2 d1 d0\n      let code1 := compile_bexp b1 0 (codelen code2 + d0)\n      code1 ++ code2\n\n@[grind] def compile_com (c : com) : List instr :=\n  match c with\n  | .SKIP =>\n      []\n  | .ASSIGN x a =>\n      compile_aexp a ++ .Isetvar x :: []\n  | .SEQ c1 c2 =>\n      compile_com c1 ++ compile_com c2\n  | .IFTHENELSE b ifso ifnot =>\n      let code_ifso := compile_com ifso\n      let code_ifnot := compile_com ifnot\n      compile_bexp b 0 (codelen code_ifso + 1)\n      ++ code_ifso\n      ++ .Ibranch (codelen code_ifnot)\n      :: code_ifnot\n  | .WHILE b body =>\n      let code_body := compile_com body\n      let code_test := compile_bexp b 0 (codelen code_body + 1)\n      code_test\n      ++ code_body\n      ++ .Ibranch (- (codelen code_test + codelen code_body + 1)) :: []\n\n@[grind] inductive code_at : List instr → Int → List instr → Prop where\n  | code_at_intro : ∀ C1 C2 C3 pc,\n      pc = codelen C1 ->\n      code_at (C1 ++ C2 ++ C3) pc C2\n\ninductive compile_cont (C : List instr) : cont -> Int -> Prop where\n  | ccont_stop : ∀ pc,\n      instr_at C pc = .some .Ihalt ->\n      compile_cont C .Kstop pc\n  | ccont_seq : ∀ c k pc pc',\n      code_at C pc (compile_com c) ->\n      pc' = pc + codelen (compile_com c) ->\n      compile_cont C k pc' ->\n      compile_cont C (.Kseq c k) pc\n  | ccont_while : ∀ b c k pc d pc' pc'',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      code_at C pc' (compile_com (.WHILE b c)) ->\n      pc'' = pc' + codelen (compile_com (.WHILE b c)) ->\n      compile_cont C k pc'' ->\n      compile_cont C (.Kwhile b c k) pc\n  | ccont_branch : ∀ d k pc pc',\n      instr_at C pc = .some (.Ibranch d) ->\n      pc' = pc + 1 + d ->\n      compile_cont C k pc' ->\n      compile_cont C k pc\n\ninductive match_config (C : List instr) : com × cont × store -> config -> Prop where\n  | match_config_intro : ∀ c k st pc,\n      code_at C pc (compile_com c) ->\n      compile_cont C k (pc + codelen (compile_com c)) ->\n      match_config C (c, k, st) (pc, [], st)\n\ndef com_size (c : com) : Nat :=\n  match c with\n  | .SKIP => 1\n  | .ASSIGN _ _ => 1\n  | (c1 ;; c2) => (com_size c1 + com_size c2 + 1)\n  | .IFTHENELSE _ c1 c2 => (com_size c1 + com_size c2 + 1)\n  | .WHILE _ c1 => (com_size c1 + 1)\n\ndef cont_size (k : cont) : Nat :=\n  match k with\n  | .Kstop => 0\n  | .Kseq c k' => (com_size c + cont_size k')\n  | .Kwhile _ _ k' => cont_size k'\n\ndef measure' (impconf : com × cont × store) : Nat :=\n  match impconf with\n  | (c, k, _) => (com_size c + cont_size k)", "target_theorem": "theorem simulation_infseq_inv :\n  ∀ C n impconf1 machconf1,\n  infseq step impconf1 -> match_config C impconf1 machconf1 ->\n  (measure' impconf1 < n) ->\n  ∃ impconf2 machconf2,\n      infseq step impconf2\n   /\\ plus (transition C) machconf1 machconf2\n   /\\ match_config C impconf2 machconf2 :=", "ground_truth_proof": ":= by\n    intro C n impconf1 h1 h2 h3 h4\n    induction n generalizing impconf1 h1\n    case zero => contradiction\n    case succ n' ih =>\n      rw [infseq] at h2\n      rcases h2 with ⟨impconf2 , STEP, INFSEQ⟩\n      have ⟨ machconf2, h5 , h6 ⟩ := simulation_step C impconf1 impconf2 h1 STEP h3\n      cases h5\n      next PLUS =>\n        exists impconf2\n        exists machconf2\n      next w =>\n        rcases w with ⟨ STAR, MEASURE ⟩\n        specialize ih impconf2 machconf2 INFSEQ h6 (by omega)\n        rcases ih with ⟨ c1, m1, w⟩\n        exists c1\n        exists m1\n        constructor\n        · exact w.1\n        · constructor\n          · apply star_plus_trans\n            · exact STAR\n            · exact w.2.1\n          · exact w.2.2", "nesting_depth": 4, "transitive_dep_count": 67, "subset_aristotle": false, "category": "Compiler"}
{"id": 370, "thm_name": "fixpoint_sound", "thm_stmt": "theorem fixpoint_sound (F : Store → Store) (init_S : Store) (h : S = fixpoint F init_S) :\n  Le (F S) S", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Constprop.lean", "imports": ["import LeroyCompilerVerificationCourse.Imp", "import Std.Data.HashMap"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.Equiv", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.emptyWithCapacity", "module": "Std.Data.HashMap.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}], "lib_lemmas": [{"name": "Std.HashMap.Equiv.getElem?_eq", "module": "Std.Data.HashMap.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Store", "content": "def Store := Std.HashMap ident Int"}, {"name": "Le", "content": "@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n"}, {"name": "Top", "content": "@[grind] def Top : Store := Std.HashMap.emptyWithCapacity"}, {"name": "Equal", "content": "def Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2"}, {"name": "fixpoint_rec", "content": "@[grind] noncomputable def fixpoint_rec (F : Store -> Store) (fuel : Nat) (S : Store) : Store :=\n  match fuel with\n  | 0 => Top\n  | fuel + 1 =>\n      let S' := F S\n      if Equal S' S then S else fixpoint_rec F fuel S'"}, {"name": "num_iter", "content": "@[grind] def num_iter : Nat := 20"}, {"name": "fixpoint", "content": "@[grind] noncomputable def fixpoint (F : Store -> Store) (init_S : Store) : Store :=\n  fixpoint_rec F num_iter init_S"}], "used_local_lemmas": [{"name": "Equal_Le", "content": "theorem Equal_Le : ∀ S1 S2, Equal S1 S2 -> Le S1 S2"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport Std.Data.HashMap\n\nopen Classical in\n\ndef Store := Std.HashMap ident Int\n\n@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n\n\n@[grind] def Top : Store := Std.HashMap.emptyWithCapacity\n\ndef Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2\n\n@[grind] noncomputable def fixpoint_rec (F : Store -> Store) (fuel : Nat) (S : Store) : Store :=\n  match fuel with\n  | 0 => Top\n  | fuel + 1 =>\n      let S' := F S\n      if Equal S' S then S else fixpoint_rec F fuel S'\n\n@[grind] def num_iter : Nat := 20\n\n@[grind] noncomputable def fixpoint (F : Store -> Store) (init_S : Store) : Store :=\n  fixpoint_rec F num_iter init_S", "target_theorem": "theorem fixpoint_sound (F : Store → Store) (init_S : Store) (h : S = fixpoint F init_S) :\n  Le (F S) S :=", "ground_truth_proof": ":= by\n    have A : ∀ fuel S,\n             fixpoint_rec F fuel S = Top\n             \\/ Equal (F (fixpoint_rec F fuel S)) (fixpoint_rec F fuel S) := by\n      intro fuel\n      induction fuel\n      case zero => grind\n      case succ fuel' ih  =>\n        grind\n    have E : S = Top \\/ Equal (F S) S = true := by grind\n    cases E <;> grind [Equal_Le]", "nesting_depth": 5, "transitive_dep_count": 19, "subset_aristotle": false, "category": "Compiler"}
{"id": 371, "thm_name": "fixpoint_join_eq", "thm_stmt": "theorem fixpoint_join_eq : Eq' (Join Init (F (fixpoint_join Init F) )) (fixpoint_join Init F)", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Fixpoints.lean", "imports": ["import LeroyCompilerVerificationCourse.Constprop", "import LeroyCompilerVerificationCourse.Imp", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "outParam", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.Equiv", "module": "Std.Data.HashMap.Basic"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "WellFounded", "module": "Init.WF"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.add", "module": "Init.Data.Int.Basic"}, {"name": "Int.sub", "module": "Init.Data.Int.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "BitVec", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Equal", "content": "def Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2"}, {"name": "Store", "content": "def Store := Std.HashMap ident Int"}, {"name": "Join", "content": "@[grind] def Join (S1 S2 : Store) : Store :=\n  S1.filter (fun key _ => S2.get? key == S1.get? key)"}, {"name": "Le", "content": "@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n"}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "Beval", "content": "@[grind] def Beval (S : Store) (b : bexp) : Option Bool :=\n  match b with\n  | .TRUE => .some true\n  | .FALSE => .some false\n  | .EQUAL a1 a2 => lift2 (fun m n => m == n) (Aeval S a1) (Aeval S a2)\n  | .LESSEQUAL a1 a2 => lift2 (fun m n => m <= n) (Aeval S a1) (Aeval S a2)\n  | .NOT b1 => lift1 (fun m => !m) (Beval S b1)\n  | .AND b1 b2 => lift2 (fun m n => m && n) (Beval S b1) (Beval S b2)"}, {"name": "lift1", "content": "@[grind] def lift1 {A B : Type} (f : A -> B) (o : Option A) : Option B :=\n  match o with\n  | .some x => .some (f x)\n  | .none => .none"}, {"name": "Aeval", "content": "@[grind] def Aeval (S : Store) (a : aexp) : Option Int :=\n  match a with\n  | .CONST n => .some n\n  | .VAR x => S.get? x\n  | .PLUS a1 a2 => lift2 (Int.add) (Aeval S a1) (Aeval S a2)\n  | .MINUS a1 a2 => lift2 (Int.sub) (Aeval S a1) (Aeval S a2)"}, {"name": "lift2", "content": "@[grind] def lift2 {A B C : Type} (f : A -> B -> C) (o1 : Option A) (o2 : Option B) : Option C :=\n  match o1, o2 with\n  | .some x1, .some x2 => .some (f x1 x2) | _, _ => .none"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "Update", "content": "@[grind] def Update (x : ident) (N : Option Int) (S : Store) : Store :=\n  match N with\n  | .none => S.erase x\n  | .some n => S.insert x n"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Le_Join_l", "content": "theorem Le_Join_l : ∀ S1 S2, Le S1 (Join S1 S2)"}], "used_local_defs": [{"name": "OrderStruct", "content": "@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/"}, {"name": "Monotone", "content": "class Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)"}, {"name": "iterate", "content": "@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']"}, {"name": "Eq'", "content": "@[grind] def Eq' (S1 S2 : Store) : Prop := Equal S1 S2"}, {"name": "Eq'_sym", "content": "def Eq'_sym : ∀ S1 S2, Eq' S1 S2 → Eq' S2 S1 :="}, {"name": "_inst_OrderStruct", "content": "noncomputable instance : OrderStruct Store where\n  eq := Equal\n  le := Le\n  beq (S1 S2 : Store) :=  Decidable.decide (Equal S1 S2)\n  le_trans := Le_trans\n  gt_wf := Gt_wf"}, {"name": "_inst_Monotone", "content": "instance : Monotone Store (fun x => Join Init (F x)) where\n  F_mon := by admit /- proof elided -/"}, {"name": "fixpoint_join", "content": "noncomputable def fixpoint_join : Store :="}], "used_local_lemmas": [{"name": "iterate_correct", "content": "@[grind] theorem iterate_correct (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) (heq : y = iterate _ F x PRE SMALL ) : eq y (F y) ∧ ∀ z, le (F z) z → le y z"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport LeroyCompilerVerificationCourse.Constprop\n\nimport Batteries.Data.List.Perm\n\n@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/\n\nopen OrderStruct\n\nclass Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)\n\nopen Monotone\n\nsection FixpointExistence\n\nvariable (α : Sort u) (F : α → α) [OrderWithBot α]\n\nopen OrderStruct OrderWithBot\n\nend FixpointExistence\n\nsection Iterate\n\nvariable (α : Sort u) [inst : OrderStruct α] (F : α → α) [Monotone α F]\n\nopen OrderStruct\n\n@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']\n\nend Iterate\n\nsection Fixpoint\n\nopen OrderWithBot\n\nvariable {α : Sort u} [i : OrderWithBot α] (F : α → α) [Monotone α F]\n\nend Fixpoint\n\nsection Constprop\n\nopen Std.HashMap\n\n@[grind] def Eq' (S1 S2 : Store) : Prop := Equal S1 S2\n\ndef Eq'_sym : ∀ S1 S2, Eq' S1 S2 → Eq' S2 S1 :=\n\nopen OrderStruct\n\nnoncomputable instance : OrderStruct Store where\n  eq := Equal\n  le := Le\n  beq (S1 S2 : Store) :=  Decidable.decide (Equal S1 S2)\n  le_trans := Le_trans\n  gt_wf := Gt_wf\n\nend Constprop\n\nsection FixpointJoin\n\nvariable (Init : Store)\n\nvariable (F : Store → Store) [Monotone Store F]\n\ninstance : Monotone Store (fun x => Join Init (F x)) where\n  F_mon := by admit /- proof elided -/\n\nnoncomputable def fixpoint_join : Store :=", "target_theorem": "theorem fixpoint_join_eq : Eq' (Join Init (F (fixpoint_join Init F) )) (fixpoint_join Init F) :=", "ground_truth_proof": ":= by\n  generalize heq1 : fixpoint_join Init F = t\n  apply Eq'_sym\n  simp [fixpoint_join] at *\n  have := (@iterate_correct Store _ (fun x => Join Init (F x)) _ ?_ ?_ ?_ ?_ ?_ ).1\n  unfold Eq'\n  · exact this\n  · exact Init\n  · apply Le_Join_l\n  · intro z hyp x\n    specialize hyp x\n    grind\n  · rw [heq1]", "nesting_depth": 4, "transitive_dep_count": 17, "subset_aristotle": false, "category": "Compiler"}
{"id": 372, "thm_name": "Gt_wf", "thm_stmt": "theorem Gt_wf : WellFounded Gt", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Fixpoints.lean", "imports": ["import LeroyCompilerVerificationCourse.Constprop", "import LeroyCompilerVerificationCourse.Imp", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "outParam", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.Equiv", "module": "Std.Data.HashMap.Basic"}, {"name": "WellFounded", "module": "Init.WF"}, {"name": "InvImage", "module": "Init.Core"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.lt", "module": "Init.Prelude"}, {"name": "Nat.lt_wfRel", "module": "Init.WF"}, {"name": "Subrelation", "module": "Init.Core"}, {"name": "Equiv", "module": "Mathlib.Logic.Equiv.Defs"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.Subperm", "module": "Batteries.Data.List.Basic"}, {"name": "List.Pairwise", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "Equal", "content": "def Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2"}, {"name": "Store", "content": "def Store := Std.HashMap ident Int"}, {"name": "Le", "content": "@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n"}], "lib_lemmas": [{"name": "List.Pairwise.imp", "module": "Init.Data.List.Pairwise"}, {"name": "List.subperm_of_subset", "module": "Batteries.Data.List.Perm"}, {"name": "Std.HashMap.distinct_keys_toList", "module": "Std.Data.HashMap.Lemmas"}, {"name": "List.Subperm.length_le", "module": "Batteries.Data.List.Perm"}, {"name": "List.Subperm.perm_of_length_le", "module": "Batteries.Data.List.Perm"}, {"name": "Std.HashMap.Equiv.of_toList_perm", "module": "Std.Data.HashMap.Lemmas"}, {"name": "InvImage.wf", "module": "Init.WF"}, {"name": "Subrelation.wf", "module": "Init.WF"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Eq'", "content": "@[grind] def Eq' (S1 S2 : Store) : Prop := Equal S1 S2"}, {"name": "Gt", "content": "@[grind] def Gt (S1 S2 : Store) := Le S2 S1 ∧ ¬ Eq' S2 S1"}], "used_local_lemmas": [{"name": "hash_set_incl_size_leq", "content": "theorem hash_set_incl_size_leq (S1 S2 : Store) : Le S2 S1 → List.Subperm (S1.toList) (S2.toList)"}, {"name": "Le_cardinal", "content": "@[grind] theorem Le_cardinal :\n  ∀ S T : Store,\n  Le T S ->\n  S.size <= T.size ∧ (S.size = T.size → Equal S T)"}, {"name": "Gt_cardinal", "content": "@[grind] theorem Gt_cardinal :\n  ∀ S S', Gt S S' -> S.size < S'.size"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport LeroyCompilerVerificationCourse.Constprop\n\nimport Batteries.Data.List.Perm\n\nopen OrderStruct\n\nopen Monotone\n\nsection FixpointExistence\n\nvariable (α : Sort u) (F : α → α) [OrderWithBot α]\n\nopen OrderStruct OrderWithBot\n\nend FixpointExistence\n\nsection Iterate\n\nvariable (α : Sort u) [inst : OrderStruct α] (F : α → α) [Monotone α F]\n\nopen OrderStruct\n\nend Iterate\n\nsection Fixpoint\n\nopen OrderWithBot\n\nvariable {α : Sort u} [i : OrderWithBot α] (F : α → α) [Monotone α F]\n\nend Fixpoint\n\nsection Constprop\n\nopen Std.HashMap\n\n@[grind] def Eq' (S1 S2 : Store) : Prop := Equal S1 S2\n\n@[grind] def Gt (S1 S2 : Store) := Le S2 S1 ∧ ¬ Eq' S2 S1", "target_theorem": "theorem Gt_wf : WellFounded Gt :=", "ground_truth_proof": ":= by\n  have := @InvImage Store Nat Nat.lt fun x => x.size\n  have : ∀ (x y : Store), Gt x y → @InvImage Store Nat Nat.lt (fun x => x.size) x y := by\n    intro x y heq\n    unfold InvImage\n    simp\n    apply Gt_cardinal\n    exact heq\n  have subrel : Subrelation Gt (InvImage Nat.lt (fun x : Store => x.size)) := by\n    intro x y gt; grind\n  apply @Subrelation.wf Store (InvImage Nat.lt (fun x : Store => x.size)) Gt subrel\n  exact InvImage.wf (fun x : Store => x.size) (Nat.lt_wfRel.wf)", "nesting_depth": 5, "transitive_dep_count": 30, "subset_aristotle": false, "category": "Compiler"}
{"id": 373, "thm_name": "fixpoint_correct", "thm_stmt": "theorem fixpoint_correct :\n  eq (fixpoint' F) (F (fixpoint' F)) ∧ ∀ z : α, le (F z) z → le (fixpoint' F) z", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Fixpoints.lean", "imports": ["import LeroyCompilerVerificationCourse.Constprop", "import LeroyCompilerVerificationCourse.Imp", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "outParam", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.Equiv", "module": "Std.Data.HashMap.Basic"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "WellFounded", "module": "Init.WF"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.add", "module": "Init.Data.Int.Basic"}, {"name": "Int.sub", "module": "Init.Data.Int.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "BitVec", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Equal", "content": "def Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2"}, {"name": "Store", "content": "def Store := Std.HashMap ident Int"}, {"name": "Le", "content": "@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n"}, {"name": "Join", "content": "@[grind] def Join (S1 S2 : Store) : Store :=\n  S1.filter (fun key _ => S2.get? key == S1.get? key)"}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "Beval", "content": "@[grind] def Beval (S : Store) (b : bexp) : Option Bool :=\n  match b with\n  | .TRUE => .some true\n  | .FALSE => .some false\n  | .EQUAL a1 a2 => lift2 (fun m n => m == n) (Aeval S a1) (Aeval S a2)\n  | .LESSEQUAL a1 a2 => lift2 (fun m n => m <= n) (Aeval S a1) (Aeval S a2)\n  | .NOT b1 => lift1 (fun m => !m) (Beval S b1)\n  | .AND b1 b2 => lift2 (fun m n => m && n) (Beval S b1) (Beval S b2)"}, {"name": "lift1", "content": "@[grind] def lift1 {A B : Type} (f : A -> B) (o : Option A) : Option B :=\n  match o with\n  | .some x => .some (f x)\n  | .none => .none"}, {"name": "Aeval", "content": "@[grind] def Aeval (S : Store) (a : aexp) : Option Int :=\n  match a with\n  | .CONST n => .some n\n  | .VAR x => S.get? x\n  | .PLUS a1 a2 => lift2 (Int.add) (Aeval S a1) (Aeval S a2)\n  | .MINUS a1 a2 => lift2 (Int.sub) (Aeval S a1) (Aeval S a2)"}, {"name": "lift2", "content": "@[grind] def lift2 {A B C : Type} (f : A -> B -> C) (o1 : Option A) (o2 : Option B) : Option C :=\n  match o1, o2 with\n  | .some x1, .some x2 => .some (f x1 x2) | _, _ => .none"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "Update", "content": "@[grind] def Update (x : ident) (N : Option Int) (S : Store) : Store :=\n  match N with\n  | .none => S.erase x\n  | .some n => S.insert x n"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "OrderStruct", "content": "@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/"}, {"name": "OrderWithBot", "content": "class OrderWithBot (α : Sort u) extends OrderStruct α where\n  bot : α\n  bot_smallest : ∀ x, le bot x"}, {"name": "Monotone", "content": "class Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)"}, {"name": "iterate", "content": "@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']"}, {"name": "fixpoint'", "content": "@[grind] def fixpoint' : α := iterate α F bot (by admit /- proof elided -/\n) (by admit /- proof elided -/\n)"}], "used_local_lemmas": [{"name": "iterate_correct", "content": "@[grind] theorem iterate_correct (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) (heq : y = iterate _ F x PRE SMALL ) : eq y (F y) ∧ ∀ z, le (F z) z → le y z"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport LeroyCompilerVerificationCourse.Constprop\n\nimport Batteries.Data.List.Perm\n\n@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/\n\nopen OrderStruct\n\nclass OrderWithBot (α : Sort u) extends OrderStruct α where\n  bot : α\n  bot_smallest : ∀ x, le bot x\n\nclass Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)\n\nopen Monotone\n\nsection FixpointExistence\n\nvariable (α : Sort u) (F : α → α) [OrderWithBot α]\n\nopen OrderStruct OrderWithBot\n\nend FixpointExistence\n\nsection Iterate\n\nvariable (α : Sort u) [inst : OrderStruct α] (F : α → α) [Monotone α F]\n\nopen OrderStruct\n\n@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']\n\nend Iterate\n\nsection Fixpoint\n\nopen OrderWithBot\n\nvariable {α : Sort u} [i : OrderWithBot α] (F : α → α) [Monotone α F]\n\n@[grind] def fixpoint' : α := iterate α F bot (by admit /- proof elided -/\n) (by admit /- proof elided -/\n)", "target_theorem": "theorem fixpoint_correct :\n  eq (fixpoint' F) (F (fixpoint' F)) ∧ ∀ z : α, le (F z) z → le (fixpoint' F) z :=", "ground_truth_proof": ":= by\n    unfold fixpoint'\n    apply iterate_correct\n    rotate_left\n    · exact bot\n    · apply bot_smallest\n    · grind [bot_smallest]\n    · rfl", "nesting_depth": 4, "transitive_dep_count": 6, "subset_aristotle": false, "category": "Compiler"}
{"id": 374, "thm_name": "fixpoint_join_smallest", "thm_stmt": "theorem fixpoint_join_smallest :\n  ∀ S, Le (Join Init (F S)) S -> Le (fixpoint_join Init F) S", "lean_root": "LeroyCompilerVerificationCourse", "rel_path": "LeroyCompilerVerificationCourse/Fixpoints.lean", "imports": ["import LeroyCompilerVerificationCourse.Constprop", "import LeroyCompilerVerificationCourse.Imp", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "outParam", "module": "Init.Prelude"}, {"name": "k", "module": "QqTest.matching"}, {"name": "Std.HashMap", "module": "Std.Data.HashMap.Basic"}, {"name": "Std.HashMap.Equiv", "module": "Std.Data.HashMap.Basic"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "WellFounded", "module": "Init.WF"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Int.add", "module": "Init.Data.Int.Basic"}, {"name": "Int.sub", "module": "Init.Data.Int.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "BitVec", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Join", "content": "@[grind] def Join (S1 S2 : Store) : Store :=\n  S1.filter (fun key _ => S2.get? key == S1.get? key)"}, {"name": "Store", "content": "def Store := Std.HashMap ident Int"}, {"name": "Le", "content": "@[grind] def Le (S1 S2 : Store) : Prop :=\n  ∀ x n, S2.get? x = .some n -> S1.get? x = .some n"}, {"name": "Equal", "content": "def Equal (S1 S2 : Store) := Std.HashMap.Equiv S1 S2"}, {"name": "com", "content": "inductive com : Type where\n  | SKIP                                        \n  | ASSIGN (x : ident) (a : aexp)               \n  | SEQ (c1 : com) (c2 : com)                   \n  | IFTHENELSE (b : bexp) (c1 : com) (c2 : com) \n  | WHILE (b : bexp) (c1 : com)                 "}, {"name": "Beval", "content": "@[grind] def Beval (S : Store) (b : bexp) : Option Bool :=\n  match b with\n  | .TRUE => .some true\n  | .FALSE => .some false\n  | .EQUAL a1 a2 => lift2 (fun m n => m == n) (Aeval S a1) (Aeval S a2)\n  | .LESSEQUAL a1 a2 => lift2 (fun m n => m <= n) (Aeval S a1) (Aeval S a2)\n  | .NOT b1 => lift1 (fun m => !m) (Beval S b1)\n  | .AND b1 b2 => lift2 (fun m n => m && n) (Beval S b1) (Beval S b2)"}, {"name": "lift1", "content": "@[grind] def lift1 {A B : Type} (f : A -> B) (o : Option A) : Option B :=\n  match o with\n  | .some x => .some (f x)\n  | .none => .none"}, {"name": "Aeval", "content": "@[grind] def Aeval (S : Store) (a : aexp) : Option Int :=\n  match a with\n  | .CONST n => .some n\n  | .VAR x => S.get? x\n  | .PLUS a1 a2 => lift2 (Int.add) (Aeval S a1) (Aeval S a2)\n  | .MINUS a1 a2 => lift2 (Int.sub) (Aeval S a1) (Aeval S a2)"}, {"name": "lift2", "content": "@[grind] def lift2 {A B C : Type} (f : A -> B -> C) (o1 : Option A) (o2 : Option B) : Option C :=\n  match o1, o2 with\n  | .some x1, .some x2 => .some (f x1 x2) | _, _ => .none"}, {"name": "aexp", "content": "inductive aexp : Type where\n  | CONST (n : Int)               \n  | VAR (x : ident)               \n  | PLUS (a1 : aexp) (a2 : aexp)  \n  | MINUS (a1 : aexp) (s2 : aexp) "}, {"name": "ident", "content": "def ident := String deriving BEq, Repr, Hashable"}, {"name": "bexp", "content": "inductive bexp : Type where\n  | TRUE                              \n  | FALSE                             \n  | EQUAL (a1 : aexp) (a2 : aexp)     \n  | LESSEQUAL (a1 : aexp) (a2 : aexp) \n  | NOT (b1 : bexp)                   \n  | AND (b1 : bexp) (b2 : bexp)       "}, {"name": "Update", "content": "@[grind] def Update (x : ident) (N : Option Int) (S : Store) : Store :=\n  match N with\n  | .none => S.erase x\n  | .some n => S.insert x n"}, {"name": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r", "content": "notation:10 l:10 \" ;; \" r:11 => com.SEQ l r"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Le_Join_l", "content": "theorem Le_Join_l : ∀ S1 S2, Le S1 (Join S1 S2)"}], "used_local_defs": [{"name": "OrderStruct", "content": "@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/"}, {"name": "Monotone", "content": "class Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)"}, {"name": "iterate", "content": "@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']"}, {"name": "_inst_OrderStruct", "content": "noncomputable instance : OrderStruct Store where\n  eq := Equal\n  le := Le\n  beq (S1 S2 : Store) :=  Decidable.decide (Equal S1 S2)\n  le_trans := Le_trans\n  gt_wf := Gt_wf"}, {"name": "_inst_Monotone", "content": "instance : Monotone Store (fun x => Join Init (F x)) where\n  F_mon := by admit /- proof elided -/"}, {"name": "fixpoint_join", "content": "noncomputable def fixpoint_join : Store :="}], "used_local_lemmas": [{"name": "iterate_correct", "content": "@[grind] theorem iterate_correct (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) (heq : y = iterate _ F x PRE SMALL ) : eq y (F y) ∧ ∀ z, le (F z) z → le y z"}], "local_ctx": "import LeroyCompilerVerificationCourse.Imp\n\nimport LeroyCompilerVerificationCourse.Constprop\n\nimport Batteries.Data.List.Perm\n\n@[grind] class OrderStruct (α : Sort u) where\n  eq : α → α → Prop\n  le : α → α → Prop\n  beq : α → α → Bool\n  le_trans : ∀ x y z, le x y -> le y z -> le x z\n  beq_true' : ∀ x y : α, beq x y = true → eq x y := by admit /- proof elided -/\n\nopen OrderStruct\n\nclass Monotone (α : Sort u) (F : α → α) [OrderStruct α] where\n  F_mon : ∀ {x y : α}, le x y → le (F x) (F y)\n\nopen Monotone\n\nsection FixpointExistence\n\nvariable (α : Sort u) (F : α → α) [OrderWithBot α]\n\nopen OrderStruct OrderWithBot\n\nend FixpointExistence\n\nsection Iterate\n\nvariable (α : Sort u) [inst : OrderStruct α] (F : α → α) [Monotone α F]\n\nopen OrderStruct\n\n@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=\n  if beq x (F x) then x else iterate (F x) (by admit /- proof elided -/\n  ) (by admit /- proof elided -/\n  )\n  termination_by x\n  decreasing_by\n    grind [beq_false']\n\nend Iterate\n\nsection Fixpoint\n\nopen OrderWithBot\n\nvariable {α : Sort u} [i : OrderWithBot α] (F : α → α) [Monotone α F]\n\nend Fixpoint\n\nsection Constprop\n\nopen Std.HashMap\n\nopen OrderStruct\n\nnoncomputable instance : OrderStruct Store where\n  eq := Equal\n  le := Le\n  beq (S1 S2 : Store) :=  Decidable.decide (Equal S1 S2)\n  le_trans := Le_trans\n  gt_wf := Gt_wf\n\nend Constprop\n\nsection FixpointJoin\n\nvariable (Init : Store)\n\nvariable (F : Store → Store) [Monotone Store F]\n\ninstance : Monotone Store (fun x => Join Init (F x)) where\n  F_mon := by admit /- proof elided -/\n\nnoncomputable def fixpoint_join : Store :=", "target_theorem": "theorem fixpoint_join_smallest :\n  ∀ S, Le (Join Init (F S)) S -> Le (fixpoint_join Init F) S :=", "ground_truth_proof": ":= by\n    intro S LE\n    unfold fixpoint_join\n    have := (@iterate_correct Store _ (fun x => Join Init (F x)) _ (fixpoint_join Init F) Init (?_) ?_ ?_).2 S LE\n    exact this\n    · apply Le_Join_l\n    · intro z hyp x\n      specialize hyp x\n      grind\n    · unfold fixpoint_join\n      dsimp", "nesting_depth": 4, "transitive_dep_count": 11, "subset_aristotle": false, "category": "Compiler"}
{"id": 375, "thm_name": "spv_dot_pure_gen", "thm_stmt": "theorem spv_dot_pure_gen (spv1: SpV Int) (spv2: SpV Int) (n pnt1 pnt2: ℕ)\n  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n)\n  (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):\n  spv_dot spv1 spv2 pnt1 pnt2 =\n    ∑ i ∈ Finset.range n,\n      if max\n        (if spv1.size ≤ pnt1 then n else spv1.ind[pnt1]!)\n        (if spv2.size ≤ pnt2 then n else spv2.ind[pnt2]!) ≤ i then\n        spv1[i] * spv2[i]\n      else\n        0", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetExamples/SpMSpV_Example.lean", "imports": ["import Auto", "import CaseStudies.Velvet.Std", "import Mathlib.Algebra.BigOperators.Intervals", "import Loom.MonadAlgebras.WP.DoNames'", "import Loom.MonadAlgebras.WP.Tactic", "import Mathlib.Algebra.Ring.Int.Defs", "import Lean", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Array.replicate", "module": "Init.Data.Array.Basic"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.find?", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "syntax \"method\" ident leafny_binder* \"return\" \"(\" ident \":\" ", "content": "syntax \"method\" ident leafny_binder* \"return\" \"(\" ident \":\" term \")\"\n  (require_caluse )*\n  (ensures_caluse)* \"do\" doSeq\n  Termination.suffix : command\n\nsyntax \"ensures\" termBeforeReqEnsDo : ensures_caluse\n\nsyntax \"while_some\" term \":|\" termBeforeDo \"do\" doSeq : doElem\n\nsyntax \"while_some\" term \":|\" term\n  (invariantClause)+\n  (doneWith)?\n  \"do\" doSeq : doElem\n\nsyntax \"let\" term \":|\" term : doElem\n\nsyntax \"done_with\" termBeforeDo : doneWith\n\nsyntax \"invariant\" termBeforeDo linebreak : invariantClause\n\nsyntax \"while\" term\n  (invariantClause)*\n  (doneWith)?\n  (decreasingTerm)?\n  \"do\" doSeq : doElem\n\nsyntax \"(mut\" ident \":\" term \")\" : leafny_binder"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x:term :| $t) => `(doElem| let $x:term <- pickSuchThat _ (fun $x => type_with_name_prefix `choice $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t do $seq:doSeq) => do\n    let decr <- withRef (<- getRef) `(decreasing none)\n    let invs <- withRef (<- getRef) `(invariants [])\n    `(doElem|\n      for _ in Lean.Loop.mk do\n        $invs:term\n        onDoneGadget (with_name_prefix `done ¬$t:term)\n        $decr:term\n        if $t then\n          $seq:doSeq\n        else break)\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      let invd_some ← match inv_done with\n      | some invd_some => withRef invd_some ``($invd_some)\n      | none => ``(¬$t:term)\n      match measure with\n      | some measure_some =>\n        let decr <- withRef measure_some `(decreasing type_with_name_prefix `decreasing $measure_some)\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        let decr <- withRef (<- getRef) `(decreasing none)\n        let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n  | `(doElem| while_some $x:ident :| $t do $seq:doSeq) =>\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      `(doElem|\n        while ∃ $x:ident, $t do\n          let $x :| $t\n          $[$seq:doElem]*)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| while_some $x:ident :| $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]? do\n                $seq:doSeq) => do\n    let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n    let invd_some ← match inv_done with\n    | some invd_some => withRef invd_some ``($invd_some)\n    | none => ``(¬$t:term)\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      let decr <- withRef (<- getRef) `(decreasing none)\n      `(doElem|\n        for _ in Lean.Loop.mk do\n          $invs:term\n          onDoneGadget (with_name_prefix `done $invd_some:term)\n          $decr:term\n          if ∃ $x:ident, $t then\n            let $x :| $t\n            $[$seq:doElem]*\n          else break)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| for $x:ident in $t\n            $[invariant $inv:term\n            ]*\n            do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      match seq with\n      | `(doSeq| $[$seq:doElem]*)\n      | `(doSeq| $[$seq:doElem;]*)\n      | `(doSeq| { $[$seq:doElem]* }) =>\n        `(doElem|\n          for $x:ident in $t do\n            $invs:term\n            $[$seq:doElem]*)\n      | _ => Lean.Macro.throwError \"for expects a sequence of do-elements\""}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|loom_solver) =>\n    `(tactic|(\n      try simp at *\n      try aesop))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let balance := mkIdent `balance_name\n      let balanceType <- `(term| Bal)\n      let inv : Array Term <- inv.mapM fun (inv : Term) => withRef inv ``(fun ($(balance):ident : $balanceType)=> with_name_prefix `inv $inv)\n      let invd_some <- match inv_done with\n      | some invd_some => withRef invd_some ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done $invd_some)\n      | none => ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done ¬$t:term)\n      match measure with\n      | some measure_some =>\n        let measure_some ← withRef measure_some ``(type_with_name_prefix `decreasing ($measure_some:term))\n        do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget ($measure_some:term)\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget none\n            if $t then\n              $seq:doSeq\n            else break)"}, {"name": "macro_rules", "content": "macro_rules\n| `(doElem|balance_set $t) => do\n  let balId := mkIdent `balance\n  `(doElem|do\n    $balId:ident := $t\n    set $balId:ident\n    $balId:ident ← get)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem|$id:ident[$idx:term] := $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (fun _ => $val))\n  | `(doElem|$id:ident[$idx:term] += $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (· + $val))"}], "lib_lemmas": [{"name": "List.find?_eq_some_iff_getElem", "module": "Init.Data.List.Nat.Find"}, {"name": "List.find?_eq_none", "module": "Init.Data.List.Find"}, {"name": "List.mem_iff_get", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "em", "module": "Mathlib.Logic.Basic"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "le_iff_eq_or_lt", "module": "Mathlib.Order.Basic"}, {"name": "le_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_of_lt_of_le", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_or_gt_of_ne", "module": "Mathlib.Order.Defs.LinearOrder"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "SpV", "content": "structure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!"}, {"name": "spv_dot", "content": "def spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=\n  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then\n    0\n  else\n    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)\n    else\n      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n        spv_dot spv1 spv2 (pnt1 + 1) pnt2\n      else\n        spv_dot spv1 spv2 pnt1 (pnt2 + 1)\n  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)\n\n\nmethod SpVSpV\n  (mut out: Array Int)\n  (spv1: SpV Int)\n  (spv2: SpV Int) return (u: Unit)\n  ensures out.size = 1\n  ensures out[0]! = spv_dot spv1 spv2 0 0\n  do\n    out := Array.replicate 1 0\n    let mut pnt1 := 0\n    let mut pnt2 := 0\n    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size\n    invariant out.size = 1\n    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size\n    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0\n    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size\n    do\n      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!\n        pnt1 := pnt1 + 1\n        pnt2 := pnt2 + 1\n      else\n        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n          pnt1 := pnt1 + 1\n        else\n          pnt2 := pnt2 + 1\n    return\n\n\nmethod SpMSpV\n  (mut out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int) return (u: Unit)\n  ensures out.size = spm.size\n  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0\n  do\n    out := Array.replicate spm.size 0\n    let mut spmInd := Array.replicate spm.size 0\n    let mut spvInd := Array.replicate spm.size 0\n    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size\n    invariant spvInd.size = spm.size\n    invariant spmInd.size = spm.size\n    invariant out.size = spm.size\n    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size\n    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size\n    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0\n    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size\n    do\n      let ind_m := spmInd[i]!\n      let ind_v := spvInd[i]!\n      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then\n        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!\n        spmInd[i] += 1\n        spvInd[i] += 1\n      else\n        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then\n          spmInd[i] += 1\n        else\n          spvInd[i] += 1\n    return"}], "used_local_lemmas": [{"name": "getValSpV_eq", "content": "theorem getValSpV_eq (spv: SpV Int) (j: ℕ) (h_ind: j < spv.size): spv[spv.ind[j]!] = (spv.val)[j]!"}, {"name": "getValSpV_empty", "content": "theorem getValSpV_empty (spv: SpV Int) (j: ℕ) (h_empty: ∀ i < spv.size, spv.ind[i]! ≠ j): spv[j] = 0"}], "local_ctx": "import Auto\n\nimport Lean\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Algebra.Ring.Int.Defs\n\nimport Loom.MonadAlgebras.NonDetT.Extract\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.DoNames'\n\nimport CaseStudies.Velvet.Std\n\nsection SpMV\n\nstructure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!\n\ndef spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=\n  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then\n    0\n  else\n    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)\n    else\n      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n        spv_dot spv1 spv2 (pnt1 + 1) pnt2\n      else\n        spv_dot spv1 spv2 pnt1 (pnt2 + 1)\n  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)\n\n\nmethod SpVSpV\n  (mut out: Array Int)\n  (spv1: SpV Int)\n  (spv2: SpV Int) return (u: Unit)\n  ensures out.size = 1\n  ensures out[0]! = spv_dot spv1 spv2 0 0\n  do\n    out := Array.replicate 1 0\n    let mut pnt1 := 0\n    let mut pnt2 := 0\n    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size\n    invariant out.size = 1\n    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size\n    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0\n    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size\n    do\n      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!\n        pnt1 := pnt1 + 1\n        pnt2 := pnt2 + 1\n      else\n        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n          pnt1 := pnt1 + 1\n        else\n          pnt2 := pnt2 + 1\n    return\n\n\nmethod SpMSpV\n  (mut out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int) return (u: Unit)\n  ensures out.size = spm.size\n  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0\n  do\n    out := Array.replicate spm.size 0\n    let mut spmInd := Array.replicate spm.size 0\n    let mut spvInd := Array.replicate spm.size 0\n    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size\n    invariant spvInd.size = spm.size\n    invariant spmInd.size = spm.size\n    invariant out.size = spm.size\n    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size\n    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size\n    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0\n    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size\n    do\n      let ind_m := spmInd[i]!\n      let ind_v := spvInd[i]!\n      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then\n        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!\n        spmInd[i] += 1\n        spvInd[i] += 1\n      else\n        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then\n          spmInd[i] += 1\n        else\n          spvInd[i] += 1\n    return", "target_theorem": "theorem spv_dot_pure_gen (spv1: SpV Int) (spv2: SpV Int) (n pnt1 pnt2: ℕ)\n  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n)\n  (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):\n  spv_dot spv1 spv2 pnt1 pnt2 =\n    ∑ i ∈ Finset.range n,\n      if max\n        (if spv1.size ≤ pnt1 then n else spv1.ind[pnt1]!)\n        (if spv2.size ≤ pnt2 then n else spv2.ind[pnt2]!) ≤ i then\n        spv1[i] * spv2[i]\n      else\n        0 :=", "ground_truth_proof": ":= by\n  fun_induction spv_dot spv1 spv2 pnt1 pnt2 with\n  | case1 p1 p2 h =>\n    have all_zero: (∀ x ∈ Finset.range n, (if max (if spv1.size ≤ p1 then n else spv1.ind[p1]!) (if spv2.size ≤ p2 then n else spv2.ind[p2]!) ≤ x then spv1[x] * spv2[x] else 0) = 0) := by\n      intro x hx\n      simp\n      rcases h with ob1 | ob2\n      { simp [ob1]\n        simp at hx\n        simp [hx] }\n      simp [ob2]\n      simp at hx\n      simp [hx]\n    rw [Finset.sum_eq_zero all_zero]\n  | case2 p1 p2 h1 eq ih =>\n    rw [ih, ←getValSpV_eq spv1 p1 (by omega), ←getValSpV_eq spv2 p2 (by omega)]\n    have sum_eq_single:\n      ∑ i ∈ Finset.range n, (if i = spv1.ind[p1]! then spv1[spv1.ind[p1]!] * spv2[spv1.ind[p1]!] else 0) = (if spv1.ind[p1]! = spv1.ind[p1]! then spv1[spv1.ind[p1]!] * spv2[spv1.ind[p1]!] else 0) := by\n      have hb: ∀ i ∈ Finset.range n, i ≠ spv1.ind[p1]! → ((if i = spv1.ind[p1]! then spv1[spv1.ind[p1]!]! * spv2[spv1.ind[p1]!]! else 0) = 0) := by\n        intro i hi iq\n        simp at hi\n        simp [iq]\n      have hc: spv1.ind[p1]! ∉ Finset.range n → ((if spv1.ind[p1]! = spv1.ind[p1]! then spv1[spv1.ind[p1]!]! * spv2[spv1.ind[p1]!]! else 0) = 0) := by\n        intro nin\n        simp at nin\n        have bnd := sz1 p1 (by omega)\n        omega\n      apply Finset.sum_eq_single spv1.ind[p1]! hb hc\n    rw [if_pos] at sum_eq_single\n    rw [←eq, ←sum_eq_single]\n    rw [←Finset.sum_add_distrib]\n    apply Finset.sum_congr\n    { rfl }\n    intro x hx\n    simp at hx\n    have h2: ¬spv1.size ≤ p1 ∧ ¬spv2.size ≤ p2 := by omega\n    simp [h2]\n    have hnx: ¬(n ≤ x) := by simp [hx]\n    by_cases xeq: x = spv1.ind[p1]! <;> simp [xeq]\n    { intro in1 in2\n      by_cases edg1: spv1.size ≤ p1 + 1 <;> simp [edg1] at in1\n      { omega }\n      have p1lt := (spv1.inc) p1 (p1 + 1) (by omega) (by omega) (by simp)\n      -- simp [getElem] at in1\n      omega }\n    have outb_lemma (spv: SpV Int) (pos: ℕ) (hsz: 1 ≤ spv.size) (hb: spv.ind[spv.size - 1]! < pos): spv[pos] = 0 := by\n      by_cases ex: ∃ i < spv.size, spv.ind[i]! = pos\n      { rcases ex with ⟨i, hbi, hi⟩\n        by_cases heq: i = spv.size - 1\n        { simp [heq] at hi\n          omega }\n        have contra := (spv.inc) i (spv.size - 1) (by omega) (by omega) (by omega)\n        simp at hb\n        -- simp [getElem] at hi\n        omega }\n      simp at ex\n      simp [getValSpV_empty spv pos ex]\n    by_cases edg1: spv1.size ≤ p1 + 1 <;> simp [edg1]\n    { simp [hnx]\n      have p1eq : p1 = spv1.size - 1 := by omega\n      by_cases lex: spv1.ind[p1]! ≤ x <;> simp [lex]\n      have ltx: spv1.ind[p1]! < x := by omega\n      simp [p1eq] at ltx\n      simp [outb_lemma spv1 x (by omega) ltx] }\n    by_cases edg2: spv2.size ≤ p2 + 1 <;> simp [edg2]\n    { simp [hnx]\n      by_cases lex: spv1.ind[p1]! ≤ x <;> simp [lex]\n      have ltx: spv1.ind[p1]! < x := by omega\n      have p2eq: p2 = spv2.size - 1 := by omega\n      simp [eq, p2eq] at ltx\n      simp [outb_lemma spv2 x (by omega) ltx] }\n    by_cases val: spv1.ind[p1 + 1]! ≤ x ∧ spv2.ind[p2 + 1]! ≤ x <;> simp [val]\n    { have inc: spv1.ind[p1]! < spv1.ind[p1 + 1]! := (spv1.inc) p1 (p1 + 1) (by omega) (by omega) (by simp)\n      simp [le_of_lt (lt_of_lt_of_le inc val.left)] }\n    by_cases xb: spv1.ind[p1]! ≤ x <;> simp [xb]\n    have ltx: spv1.ind[p1]! < x := by omega\n    simp at val\n    have interm_lemma (spv: SpV Int) (pos idx: ℕ) (hsz: idx + 1 < spv.size) (inter: spv.ind[idx]! < pos ∧ pos < spv.ind[idx + 1]!): spv[pos] = 0 := by\n      by_cases ex: ∃ i < spv.size, spv.ind[i]! = pos\n      { rcases ex with ⟨i, hbi, hi⟩\n        have lt_lemma (i1 i2: ℕ) (hi1: i1 < spv.size) (hi2: i2 < spv.size): spv.ind[i1]! < spv.ind[i2]! → i1 < i2 := by\n          intro hlt\n          by_cases contra_lt: i2 < i1\n          { have inc := spv.inc i2 i1 hi2 hi1 contra_lt\n            simp at hlt\n            omega }\n          by_cases contra_eq: i1 = i2\n          { simp [contra_eq] at hlt }\n          omega\n        have left_x := lt_lemma idx i (by omega) (by omega) (by omega)\n        have right_x := lt_lemma i (idx + 1) (by omega) (by omega) (by omega)\n        omega }\n      simp at ex\n      simp [getValSpV_empty spv pos ex]\n    rcases (em (spv1.ind[p1 + 1]! ≤ x)) with c1 | c2\n    { simp [c1] at val\n      simp [interm_lemma spv2 x p2 (by omega) (by omega)] }\n    simp at c2\n    simp [interm_lemma spv1 x p1 (by omega) (by omega)]\n    rfl\n  | case3 p1 p2 h1 neq le ih =>\n    rw [ih]\n    apply Finset.sum_congr\n    { rfl }\n    intro x hx\n    by_cases edg: spv1.size = p1 + 1\n    { have sz_concl := sz1 p1 (by omega)\n      simp [edg]\n      simp at hx\n      have nx : ¬(n ≤ x) := by omega\n      by_cases b2: spv2.size ≤ p2 <;> simp [b2] <;> simp [nx]\n      intro ifc1 ifc2\n      by_cases ex: ∃ i < spv1.size, spv1.ind[i]! = x\n      { rcases ex with ⟨i, ib, hi⟩\n        by_cases il: p1 < i\n        { omega }\n        simp at il\n        rcases (le_iff_eq_or_lt.mp il) with ieq | ilt\n        { simp [ieq] at hi\n          omega }\n        have contra := spv1.inc i p1 ib (by omega) ilt\n        omega }\n      simp at ex\n      simp [getValSpV_empty spv1 x ex] }\n    have inb: ¬(spv1.size ≤ p1 + 1) ∧ ¬(spv1.size ≤ p1):= by omega\n    simp [inb]\n    by_cases edg2: spv2.size ≤ p2 <;> simp [edg2]\n    { simp at hx\n      have neg: ¬ (n ≤ x) := by omega\n      simp [neg] }\n    by_cases le2: spv2.ind[p2]! ≤ x <;> simp [le2]\n    by_cases upper: spv1.ind[p1 + 1]! ≤ x <;> simp [upper]\n    { simp at inb\n      simp [getElem]\n      simp [le_trans (le_of_lt (spv1.inc p1 (p1 + 1) inb.right inb.left (by simp))) upper] }\n    by_cases lower: spv1.ind[p1]! ≤ x <;> simp [lower]\n    by_cases ex: ∃ i < spv1.size, spv1.ind[i]! = x\n    { rcases ex with ⟨ind, indb, hind⟩\n      by_cases ind_x: ind = p1\n      { simp [ind_x] at hind\n        omega }\n      rcases (lt_or_gt_of_ne ind_x) with ltc | gtc\n      { have contra := spv1.inc ind p1 indb (by omega) ltc\n        -- simp [getElem] at lower\n        -- simp [getElem] at hind\n        omega }\n      by_cases indp1: ind = p1 + 1\n      { simp [indp1] at hind\n        omega }\n      have indlt: p1 + 1 < ind := by omega\n      have contra := spv1.inc (p1 + 1) ind (by omega) indb indlt\n      -- simp [getElem] at upper\n      -- simp [getElem] at hind\n      omega }\n    simp at ex\n    simp [getValSpV_empty spv1 x ex]\n  | case4 p1 p2 h1 neq nle ih =>\n    rw [ih]\n    apply Finset.sum_congr\n    { rfl }\n    intro x hx\n    by_cases edg: spv2.size = p2 + 1\n    { have sz_concl := sz2 p2 (by omega)\n      simp [edg]\n      simp at hx\n      have nx : ¬(n ≤ x) := by omega\n      by_cases b1: spv1.size ≤ p1 <;> simp [b1] <;> simp [nx]\n      intro ifc1 ifc2\n      by_cases ex: ∃ i < spv2.size, spv2.ind[i]! = x\n      { rcases ex with ⟨i, ib, hi⟩\n        by_cases il: p2 < i\n        { omega }\n        simp at il\n        rcases (le_iff_eq_or_lt.mp il) with ieq | ilt\n        { simp [ieq] at hi\n          omega }\n        have contra := (spv2.inc) i p2 ib (by omega) ilt\n        -- simp [getElem] at hi\n        -- simp [getElem] at ifc2\n        omega }\n      simp at ex\n      simp [getValSpV_empty spv2 x ex] }\n    have inb: ¬(spv2.size ≤ p2 + 1) ∧ ¬(spv2.size ≤ p2):= by omega\n    simp [inb]\n    by_cases edg1: spv1.size ≤ p1 <;> simp [edg1]\n    { simp at hx\n      have neg: ¬ (n ≤ x) := by omega\n      simp [neg] }\n    by_cases le1: spv1.ind[p1]! ≤ x <;> simp [le1]\n    by_cases upper: spv2.ind[p2 + 1]! ≤ x <;> simp [upper]\n    { simp at inb\n      simp [getElem]\n      simp [le_trans (le_of_lt ((spv2.inc) p2 (p2 + 1) inb.right inb.left (by simp))) upper] }\n    by_cases lower: spv2.ind[p2]! ≤ x <;> simp [lower]\n    by_cases ex: ∃ i < spv2.size, spv2.ind[i]! = x\n    { rcases ex with ⟨ind, indb, hind⟩\n      by_cases ind_x: ind = p2\n      { simp [ind_x] at hind\n        omega }\n      rcases (lt_or_gt_of_ne ind_x) with ltc | gtc\n      { have contra := (spv2.inc) ind p2 indb (by omega) ltc\n        -- simp [getElem] at lower\n        -- simp [getElem] at hind\n        omega }\n      by_cases indp2: ind = p2 + 1\n      { simp [indp2] at hind\n        omega }\n      have indlt: p2 + 1 < ind := by omega\n      have contra := (spv2.inc) (p2 + 1) ind (by omega) indb indlt\n      -- simp [getElem] at upper\n      -- simp [getElem] at hind\n      omega }\n    simp at ex\n    simp [getValSpV_empty spv2 x ex]", "nesting_depth": 2, "transitive_dep_count": 23, "subset_aristotle": false, "category": "Framework"}
{"id": 376, "thm_name": "VelvetM.total_decompose", "thm_stmt": "lemma VelvetM.total_decompose {α : Type} (x : VelvetM α) (post₁ post₂ : α -> Prop):\n  [totl| wp x post₁] ⊓ [part| wp x post₂] = [totl| wp x (post₁ ⊓ post₂)]", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetTheory.lean", "imports": ["import Loom.MonadAlgebras.NonDetT.Basic", "import Loom.MonadAlgebras.WP.Basic", "import Loom.MonadAlgebras.NonDetT'.Basic", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "ForInStep", "module": "Init.Core"}, {"name": "ForInStep.yield", "module": "Init.Core"}, {"name": "LE", "module": "Init.Prelude"}, {"name": "OrderBot", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "OrderTop", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "measure", "module": "Init.WF"}, {"name": "And", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.MonadEnv", "module": "Lean.Environment"}, {"name": "Lean.SimpleScopedEnvExtension", "module": "Lean.ScopedEnvExtension"}, {"name": "Lean.SimplePersistentEnvExtension", "module": "Lean.EnvExtension"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "open", "content": "notation \"[part|\" t \"]\" => open PartialCorrectness PartialCorrectness.DemonicChoice in t"}, {"name": "open", "content": "notation \"[part|\" t \"]\" => open ExceptionAsSuccess in t"}, {"name": "open", "content": "notation \"[totl|\" t \"]\" => open TotalCorrectness TotalCorrectness.DemonicChoice in t"}, {"name": "open", "content": "notation \"[totl|\" t \"]\" => open ExceptionAsFailure in t"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f => fun post => ⨆ a, ⌜p a⌝ ⊓ wp (f a) post\n  | _, .repeatCont init f cont => fun post => ⨆ (inv : ForInStep _ -> l),\n    ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓\n    spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)"}, {"name": "NonDetT", "content": "inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α"}, {"name": "spec", "content": "def spec (pre : l) (post : α -> l) : Cont l α :=\n  fun p => pre ⊓ ⌜post ≤ p⌝"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "NonDetT.wp", "content": "def   NonDetT.wp {l : Type u} {α : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m α -> Cont l α\n  | .pure ret => pure ret\n  | .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | .pickCont _ p f => fun post => ⨆ a, ⌜p a⌝ ⊓ wp (f a) post"}, {"name": "LE.pure", "content": "noncomputable def LE.pure {l : Type u} [inst: LE l] [OrderTop l] [OrderBot l] : Prop -> l := fun p =>\n  if p then ⊤ else ⊥"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f =>\n  fun post =>\n    let p : Set τ := p;\n    ⨅ a ∈ (p : Set τ), wp (f a) post\n  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l),\n      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓\n      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} {α : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m α -> Cont l α\n  | .pure ret => pure ret\n  | .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | .pickCont τ p f => fun post => let p : Set τ := p; ⨅ a ∈ (p : Set τ), wp (f a) post"}, {"name": "W", "content": "structure W (t : Type v) [Preorder t] (α : Type u) where\n  wp : Cont t α\n  wp_montone : wp.monotone"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f => fun post => let p : Set τ := p; ⌜∃ h, p h⌝ ⊓  ⨅ a ∈ (p : Set τ), wp (f a) post\n  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l) (measure : β -> Nat),\n      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) (fun | .yield b' => inv (.yield b') ⊓ ⌜ measure b' < measure b ⌝ | .done b' => inv (.done b'))⌝ ⊓\n      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)"}, {"name": "NonDetT.wp", "content": "def   NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ _ p _ f => fun post => ⨆ a, ⌜p a⌝ ⊓ wp (f a) post\n  | _, .repeatCont init f cont => fun post => ⨆ (inv : ForInStep _ -> l) (measure : _ -> Nat),\n    ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) (fun | .yield b' => inv (.yield b') ⊓ ⌜ measure b' < measure b ⌝ | .done b' => inv (.done b'))⌝ ⊓\n    spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)"}, {"name": "DivM", "content": "inductive DivM (α : Type u) where\n  | res (x : α)\n  | div"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "NonDetT.μ", "content": "def NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}, {"name": "NonDetT.bind", "content": "def NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f"}, {"name": "WPGen.bind", "content": "def WPGen.bind {x : m α} {f : α -> m β} (wpg : WPGen x) (wpgf : ∀ a, WPGen (f a)) :\n  WPGen (x >>= f) where\n  get := fun post => wpg.get (fun a => (wpgf a).get post)\n  prop := by admit /- proof elided -/"}, {"name": "_root_.Lean.SimpleScopedEnvExtension.get", "content": "private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "Context", "content": "structure Context where\n  ref         : Syntax\n   \n  m           : Syntax\n   \n  returnType  : Syntax\n  mutableVars : VarSet := {}\n  insideFor   : Bool := false"}, {"name": "_root_.Lean.SimplePersistentEnvExtension.get", "content": "private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "WPGen", "content": "structure WPGen (x : m α) where\n  get : Cont l α\n  \n  prop : ∀ post, get post <= wp x post"}, {"name": "_root_.Lean.EnvExtension.get", "content": "private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "CCPOBotLawful", "content": "class CCPOBotLawful (m : Type u -> Type v) [∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] where\n  prop {α} : CCPOBot.compBot (m := m) (α := α) = Lean.Order.bot"}], "lib_lemmas": [{"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "TotalCorrectness.DivM.wp_eq", "content": "lemma TotalCorrectness.DivM.wp_eq (α : Type) (x : DivM α) (post : α -> Prop) :\n  wp x post =\n    match x with"}, {"name": "PartialCorrectness.DivM.wp_eq", "content": "lemma PartialCorrectness.DivM.wp_eq (α : Type) (x : DivM α) (post : α -> Prop) :\n  wp x post =\n    match x with"}, {"name": "NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}, {"name": "NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α β : Type u} (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}, {"name": "NonDetT.wp_mono", "content": "lemma NonDetT.wp_mono [LawfulMonad m] {α : Type u} {l : Type u} [CompleteLattice l] [MAlgOrdered m l] (x : NonDetT m α) (f g : α -> l) :\n  (∀ a, f a <= g a) ->\n  NonDetT.wp x f <= NonDetT.wp x g"}, {"name": "spec_mono", "content": "lemma spec_mono {α : Type u} {l : Type u} [CompleteLattice l] (pre : l) (post : α -> l) (f g : α -> l) :\n  (∀ a, f a <= g a) ->\n  spec pre post f <= spec pre post g"}, {"name": "spec_mono", "content": "omit [MAlgOrdered m l] in\nlemma spec_mono {α : Type u}  {l : Type u} [CompleteLattice l] (pre : l) (post : α -> l) (f g : α -> l) :\n  (∀ a, f a <= g a) ->\n  spec pre post f <= spec pre post g"}, {"name": "NonDetT.wp_mono", "content": "lemma NonDetT.wp_mono  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α : Type u} (x : NonDetT m α) (f g : α -> l) :\n  (∀ a, f a <= g a) ->\n  NonDetT.wp x f <= NonDetT.wp x g"}], "used_local_defs": [], "used_local_lemmas": [{"name": "DivM.total_decompose", "content": "@[local simp]\nlemma DivM.total_decompose (α : Type) (x : DivM α) (post₁ post₂ : α -> Prop) :\n  ([totl| wp x post₁] ∧ [part| wp x post₂]) = [totl| wp x (post₁ ⊓ post₂)]"}, {"name": "wp_mono_part", "content": "@[local simp]\nlemma wp_mono_part (x : NonDetT DivM α) (post₁ post₂ : α -> Prop) :\n  (post₁ ≤ post₂) → ([totl|wp x post₁]) ≤ ([part| wp x post₂])"}], "local_ctx": "import Loom.MonadAlgebras.NonDetT.Extract\n\nnotation \"[totl|\" t \"]\" => open TotalCorrectness TotalCorrectness.DemonicChoice in t\n\nnotation \"[part|\" t \"]\" => open PartialCorrectness PartialCorrectness.DemonicChoice in t", "target_theorem": "lemma VelvetM.total_decompose {α : Type} (x : VelvetM α) (post₁ post₂ : α -> Prop):\n  [totl| wp x post₁] ⊓ [part| wp x post₂] = [totl| wp x (post₁ ⊓ post₂)] :=", "ground_truth_proof": ":= by\n    unhygienic induction x <;> try simp [loomLogicSimp]\n    { simp [DivM.total_decompose]\n      simp [[totl|DivM.wp_eq]]\n      split\n      { simp }\n      rename_i arg\n      have ind := f_ih arg post₁ post₂\n      simp at ind\n      rw [ind]\n      trivial }\n    { constructor <;> rintro hyp\n      { constructor; aesop\n        intro i hi\n        have hl := hyp.left.right i hi\n        have hr := hyp.right i hi\n        have ind := f_ih i post₁ post₂\n        simp [hl, hr] at ind\n        exact ind }\n      constructor; constructor; aesop\n      all_goals\n        intro i hi\n        have conj := hyp.right i hi\n        have ind := f_ih i post₁ post₂\n        simp [loomLogicSimp] at ind\n        rw [←ind] at conj\n        simp [conj] }\n    constructor\n    { intro conj\n      rcases conj with ⟨h, inv_spec, hspec⟩\n      rcases h with ⟨inv, x1, hinv⟩\n      rcases x1 with ⟨x, hx⟩\n      exists inv ⊓ inv_spec\n      constructor\n      { exists x\n        intro b hb\n        simp [←[totl| NonDetT.wp_eq_wp]]\n        simp [loomLogicSimp] at hb\n        have hxb := hx b hb.left\n        simp [←[totl| NonDetT.wp_eq_wp]] at hxb\n        simp [spec, LE.pure, loomLogicSimp, ←[part| NonDetT.wp_eq_wp]] at hspec\n        have hspecb := hspec.left b hb.right\n        have ind := f_ih b\n          (fun x_1 ↦\n          match x_1 with\n          | ForInStep.yield b' => inv (ForInStep.yield b') ∧ x b' < x b\n          | ForInStep.done b' => inv (ForInStep.done b'))\n          inv_spec\n        simp [loomLogicSimp] at ind\n        have indr := ind.mp (And.intro hxb hspecb)\n        have v1 := [totl| NonDetT.wp_mono\n          (f b)\n          (fun x_1 ↦\n            (match x_1 with\n              | ForInStep.yield b' => inv (ForInStep.yield b') ∧ x b' < x b\n              | ForInStep.done b' => inv (ForInStep.done b')) ∧\n              inv_spec x_1)\n          (fun x_1 ↦\n            match x_1 with\n            | ForInStep.yield b' => (inv (ForInStep.yield b') ∧ inv_spec (ForInStep.yield b')) ∧ x b' < x b\n            | ForInStep.done b' => inv (ForInStep.done b') ∧ inv_spec (ForInStep.done b'))\n          ]\n        simp [loomLogicSimp, ←[totl| NonDetT.wp_eq_wp]] at v1\n        exact v1\n          (fun x => by\n            match x with\n            | .yield b' => intro hb hspec; simp [hb, hspec]\n            | .done b' => intro hb hspec; simp [hb, hspec] )\n          indr }\n      simp [spec, LE.pure, loomLogicSimp, ←[totl| NonDetT.wp_eq_wp]] at hinv\n      simp [spec, LE.pure, loomLogicSimp, ←[part| NonDetT.wp_eq_wp]] at hspec\n      simp [spec, LE.pure, loomLogicSimp, ←[totl| NonDetT.wp_eq_wp]]\n      simp [hinv, hspec]\n      intro x inv_x inv_spec\n      have h₁ := hinv.right x inv_x\n      have h₂ := hspec.right.right x inv_spec\n      have cont_ind := cont_ih x post₁ post₂\n      simp [loomLogicSimp] at cont_ind\n      exact cont_ind.mp (And.intro h₁ h₂) }\n    intro hyp\n    rcases hyp with ⟨inv, x_ex, h_inv⟩\n    rcases x_ex with ⟨x, hx⟩\n    simp [spec]\n    simp [spec, LE.pure] at h_inv\n    constructor <;> exists inv <;> constructor\n    { exists x }\n    { simp [h_inv, LE.pure]\n      exact le_trans h_inv.right (by\n        simp [← [totl|NonDetT.wp_eq_wp]]\n        simp [loomLogicSimp]\n        intro x and_wp\n        have cont_ind := cont_ih x post₁ post₂\n        simp [loomLogicSimp, and_wp] at cont_ind\n        simp [cont_ind] ) }\n    { intro b hb\n      have hbx := hx b hb\n      simp [←[totl| NonDetT.wp_eq_wp]] at hbx\n      have hb_triv: True ≤ ([totl| wp (f b) fun x_1 ↦\n        match x_1 with\n        | ForInStep.yield b' => inv (ForInStep.yield b') ∧ x b' < x b\n        | ForInStep.done b' => inv (ForInStep.done b')]) := by\n        simp [loomLogicSimp]\n        exact hbx\n      have tr_intro: True ≤ ([part| NonDetT.wp (f b) inv]) := le_trans hb_triv (by\n        simp [loomLogicSimp]\n        intro wp_x\n        simp [←[part| NonDetT.wp_eq_wp]]\n        apply wp_mono_part (f b) (fun x_1 ↦\n          match x_1 with\n          | ForInStep.yield b' => inv (ForInStep.yield b') ∧ x b' < x b\n          | ForInStep.done b' => inv (ForInStep.done b')) (fun x => inv x)\n        { simp [loomLogicSimp]\n          intro x1\n          match x1 with\n          | ForInStep.yield b' => simp; intros; simp [*]\n          | ForInStep.done b' => simp }\n        exact wp_x)\n      simp at tr_intro\n      simp [tr_intro] }\n    simp [LE.pure, h_inv]\n    exact le_trans h_inv.right (by\n      simp [←[totl| NonDetT.wp_eq_wp], ←[part| NonDetT.wp_eq_wp]]\n      simp [loomLogicSimp])", "nesting_depth": 4, "transitive_dep_count": 56, "subset_aristotle": false, "category": "Framework"}
{"id": 377, "thm_name": "VSpV_correct_pure", "thm_stmt": "theorem VSpV_correct_pure (out: Array Int) (arr: Array Int)\n  (spv: SpV Int)\n  (h_b: ∀ i < spv.size, spv.ind[i]! < arr.size):\n  out.size = 1 → out[0]! = sumUpTo spv arr spv.size →\n    out[0]! = ∑ i ∈ Finset.range (arr.size), spv[i] * arr[i]!", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetExamples/SpMSpV_Example.lean", "imports": ["import Auto", "import CaseStudies.Velvet.Std", "import Mathlib.Algebra.BigOperators.Intervals", "import Loom.MonadAlgebras.WP.DoNames'", "import Loom.MonadAlgebras.WP.Tactic", "import Mathlib.Algebra.Ring.Int.Defs", "import Lean", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.find?", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.find?_eq_some_iff_getElem", "module": "Init.Data.List.Nat.Find"}, {"name": "List.find?_eq_none", "module": "Init.Data.List.Find"}, {"name": "List.mem_iff_get", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "Nat.lt_iff_le_and_ne", "module": "Init.Data.Nat.Basic"}, {"name": "add_left_cancel_iff", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "lt_or_gt_of_ne", "module": "Mathlib.Order.Defs.LinearOrder"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "SpV", "content": "structure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!"}, {"name": "sumUpTo", "content": "def sumUpTo\n  (spv : SpV Int)\n  (v : Array Int) (bound : ℕ) : Int := ∑ i ∈ Finset.range bound, ((spv.val)[i]! * v[(spv.ind)[i]!]!)"}], "used_local_lemmas": [{"name": "getValSpV_eq", "content": "theorem getValSpV_eq (spv: SpV Int) (j: ℕ) (h_ind: j < spv.size): spv[spv.ind[j]!] = (spv.val)[j]!"}, {"name": "getValSpV_empty", "content": "theorem getValSpV_empty (spv: SpV Int) (j: ℕ) (h_empty: ∀ i < spv.size, spv.ind[i]! ≠ j): spv[j] = 0"}], "local_ctx": "import Auto\n\nimport Lean\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Algebra.Ring.Int.Defs\n\nimport Loom.MonadAlgebras.NonDetT.Extract\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.DoNames'\n\nimport CaseStudies.Velvet.Std\n\nsection SpMV\n\nstructure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!\n\ndef sumUpTo\n  (spv : SpV Int)\n  (v : Array Int) (bound : ℕ) : Int := ∑ i ∈ Finset.range bound, ((spv.val)[i]! * v[(spv.ind)[i]!]!)", "target_theorem": "theorem VSpV_correct_pure (out: Array Int) (arr: Array Int)\n  (spv: SpV Int)\n  (h_b: ∀ i < spv.size, spv.ind[i]! < arr.size):\n  out.size = 1 → out[0]! = sumUpTo spv arr spv.size →\n    out[0]! = ∑ i ∈ Finset.range (arr.size), spv[i] * arr[i]! :=", "ground_truth_proof": ":= by\n      intro sz sum_eq\n      rw [sum_eq, sumUpTo]\n      have ind_lemma: ∀ k, k ≤ arr.size →\n        ∑ i ∈ Finset.range (spv.size), (if spv.ind[i]! < k then spv.val[i]! * arr[spv.ind[i]!]! else 0) =\n        ∑ i ∈ Finset.range k, spv[i] * arr[i]! := by\n          intro k\n          induction k with\n          | zero =>\n            simp\n          | succ m hm =>\n            intro lt_m\n            simp [Finset.sum_range_succ]\n            rw [←hm (by omega)]\n            have splitted_sum:\n              (∑ i ∈ Finset.range (spv.ind.size), if spv.ind[i]! < m then spv.val[i]! * arr[spv.ind[i]!]! else 0) +\n              (∑ i ∈ Finset.range (spv.ind.size), if spv.ind[i]! = m then spv.val[i]! * arr[spv.ind[i]!]! else 0) =\n              (∑ i ∈ Finset.range (spv.ind.size), if spv.ind[i]! < m + 1 then spv.val[i]! * arr[spv.ind[i]!]! else 0) := by\n                rw [←Finset.sum_add_distrib]\n                rw [Finset.sum_congr (by rfl)]\n                intro x hx\n                by_cases h_eq_m : spv.ind[x]! = m <;> simp [h_eq_m]\n                have miff : spv.ind[x]! < m ↔ spv.ind[x]! < m + 1 := by\n                  constructor <;> rintro h_lt <;> omega\n                simp [miff]\n            rw [←spv.size_eq.1, ←splitted_sum, add_left_cancel_iff.mpr]\n            by_cases exists_i: ∃ i < spv.size, spv.ind[i]! = m\n            { rcases exists_i with ⟨ind, h_ind⟩\n              rw [← h_ind.right]\n              have lemma_res := getValSpV_eq spv ind h_ind.left\n              simp at lemma_res\n              simp [lemma_res]\n              have almost_zero : ∀ i ∈ Finset.range (spv.ind.size), i ≠ ind →\n                ((if spv.ind[i]! = spv.ind[ind]! then spv.val[i]! * arr[spv.ind[i]!]! else 0) = 0) := by\n                  intro i i_inb i_not_ind\n                  by_cases vind_eq : spv.ind[i]! = spv.ind[ind]!\n                  { simp at i_inb\n                    have i_sz: i < spv.size := by\n                      rw [spv.size_eq.1] at i_inb\n                      exact i_inb\n                    rcases lt_or_gt_of_ne i_not_ind with i_lt | inb_lt\n                    { simp [getElem, Nat.lt_iff_le_and_ne.mp (spv.inc i ind i_sz h_ind.left i_lt)] }\n                    have lt := Nat.lt_iff_le_and_ne.mp (spv.inc ind i h_ind.left i_sz inb_lt)\n                    simp [getElem] at vind_eq\n                    simp [vind_eq] at lt }\n                  simp [vind_eq]\n              have ind_inb: ind ∉ Finset.range (spv.size) →\n                ((if spv.ind[ind]! = spv.ind[ind]! then spv.val[ind]! * arr[spv.ind[ind]!]! else 0) = 0) := by\n                  intro ind_not_inb\n                  simp at ind_not_inb\n                  omega\n              rw [←spv.size_eq.1] at ind_inb\n              simp [Finset.sum_eq_single ind almost_zero ind_inb] }\n            simp at exists_i\n            have h_getVal := getValSpV_empty spv m exists_i\n            simp at h_getVal\n            simp [h_getVal]\n            apply Finset.sum_eq_zero\n            rintro x hx\n            by_cases h_eq: spv.ind[x]! = m <;> simp [h_eq]\n            simp [←h_eq] at h_getVal\n            simp at hx\n            rw [spv.size_eq.1] at hx\n            simp [getValSpV_eq spv x hx] at h_getVal\n            simp [h_getVal]\n      have fin_lemma := ind_lemma (arr.size) (by rfl)\n      rw [←fin_lemma]\n      exact Finset.sum_congr (by rfl) fun i h_i => by aesop", "nesting_depth": 2, "transitive_dep_count": 19, "subset_aristotle": false, "category": "Framework"}
{"id": 378, "thm_name": "DemonicChoice.ExtractNonDet.extract_refines_wp_weak", "thm_stmt": "omit [MAlgDet m l] in\nlemma ExtractNonDet.extract_refines_wp_weak [∀ α, CCPO (m α)] [MAlgPartial m] [CCPOBotLawful m] (s : NonDetT m α) (inst : ExtractNonDet WeakFindable s) :\n  wp s post <= wp s.extractWeak post", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/NonDetT'/Extract.lean", "imports": ["import Loom.MonadAlgebras.WP.Gen", "import Loom.MonadAlgebras.WP.Liberal", "import Mathlib.Order.CompleteBooleanAlgebra", "import Mathlib.Logic.Function.Basic", "import Mathlib.Data.W.Basic", "import Loom.MonadAlgebras.NonDetT'.Basic", "import Loom.MonadAlgebras.WP.Basic", "import Loom.MonadAlgebras.Defs", "import Mathlib.Order.Lattice", "import Mathlib.Data.FinEnum", "import Mathlib.Order.Basic"], "used_lib_defs": [{"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "inline", "module": "Init.Core"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Encodable", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Encodable.decode", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Lean.Order.CCPO", "module": "Init.Internal.Order.Basic"}, {"name": "Lean.Order.bot", "module": "Init.Internal.Order.Basic"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "Lean.Order.CCPO.csup", "module": "Init.Internal.Order.Basic"}, {"name": "Lean.Order.chain", "module": "Init.Internal.Order.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.MonadEnv", "module": "Lean.Environment"}, {"name": "Lean.SimpleScopedEnvExtension", "module": "Lean.ScopedEnvExtension"}, {"name": "Lean.SimplePersistentEnvExtension", "module": "Lean.EnvExtension"}, {"name": "LE", "module": "Init.Prelude"}, {"name": "OrderBot", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "OrderTop", "module": "Mathlib.Order.BoundedOrder.Basic"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "MonadNonDet", "content": "class MonadNonDet (m : Type u → Type v) where\n  pick : (τ : Type u) → [Inhabited τ] → m τ\n  \n  pickSuchThat : (τ : Type u) → (p : τ → Prop) → [Findable p] → m τ\n  assume : (as : Prop) → [Decidable as] → m PUnit.{u+1}\n  \n  rep {α : Type u} : α → (α → m (ForInStep α)) → m α"}, {"name": "NonDetT", "content": "inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α"}, {"name": "CCPOBot", "content": "class CCPOBot (m : Type u -> Type v)  where\n  compBot {α} : m α"}, {"name": "CCPOBotLawful", "content": "class CCPOBotLawful (m : Type u -> Type v) [∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] where\n  prop {α} : CCPOBot.compBot (m := m) (α := α) = Lean.Order.bot"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "MAlgPartial", "content": "class MAlgPartial (m : Type u -> Type v) [Monad m] [∀ α, Lean.Order.CCPO (m α)]\n  [CompleteLattice l] [MAlgOrdered m l] where\n  csup_lift {α : Type u} (xc : Set (m α)) (post : α -> l) :\n    Lean.Order.chain xc ->\n    ⨅ x ∈ xc, MAlg.lift x post <= MAlg.lift (Lean.Order.CCPO.csup xc) post"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "MAlgDet", "content": "class MAlgDet (l : outParam (Type v)) [Monad m] [CompleteLattice l] [MAlgOrdered m l] where\n   \n  demonic {α ι : Type v} (c : m α) (p : ι -> α -> l) [Nonempty ι] :\n    ⨅ i, MAlg.lift c (p i) ≤ MAlg.lift c (fun x => ⨅ i, p i x)\n   \n  angelic {α ι : Type v} (c : m α) (p : ι -> α -> l) [Nonempty ι] :\n    ⨆ i, MAlg.lift c (p i) ≥ MAlg.lift c (fun x => ⨆ i, p i x)"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} {α : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m α -> Cont l α\n  | .pure ret => pure ret\n  | .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | .pickCont τ p f => fun post => let p : Set τ := p; ⨅ a ∈ (p : Set τ), wp (f a) post"}, {"name": "NonDetT.μ", "content": "def NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}, {"name": "NonDetT.bind", "content": "def NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f"}, {"name": "WPGen.bind", "content": "def WPGen.bind {x : m α} {f : α -> m β} (wpg : WPGen x) (wpgf : ∀ a, WPGen (f a)) :\n  WPGen (x >>= f) where\n  get := fun post => wpg.get (fun a => (wpgf a).get post)\n  prop := by admit /- proof elided -/"}, {"name": "_root_.Lean.SimpleScopedEnvExtension.get", "content": "private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "Context", "content": "structure Context where\n  ref         : Syntax\n   \n  m           : Syntax\n   \n  returnType  : Syntax\n  mutableVars : VarSet := {}\n  insideFor   : Bool := false"}, {"name": "_root_.Lean.SimplePersistentEnvExtension.get", "content": "private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "WPGen", "content": "structure WPGen (x : m α) where\n  get : Cont l α\n  \n  prop : ∀ post, get post <= wp x post"}, {"name": "_root_.Lean.EnvExtension.get", "content": "private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "LE.pure", "content": "noncomputable def LE.pure {l : Type u} [inst: LE l] [OrderTop l] [OrderBot l] : Prop -> l := fun p =>\n  if p then ⊤ else ⊥"}], "lib_lemmas": [{"name": "Set.mem_empty_iff_false", "module": "Mathlib.Data.Set.Basic"}, {"name": "eq_top_iff", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "le_iInf₂", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "le_trans'", "module": "Mathlib.Order.Basic"}, {"name": "ge_iff_le", "module": "Init.Core"}, {"name": "iInf_const", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "iInf_le_of_le", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "inf_comm", "module": "Mathlib.Order.Lattice"}, {"name": "le_top", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "monadLift_self", "module": "Init.Control.Lawful.Basic"}, {"name": "top_himp", "module": "Mathlib.Order.Heyting.Basic"}], "repo_lemmas": [{"name": "wp_pure", "content": "lemma wp_pure (x : α) (post : α -> l) : wp (m := m) (pure x) post = post x"}, {"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}, {"name": "NonDetT.wp_vis", "content": "@[simp]\nlemma NonDetT.wp_vis {β : Type u} (x : m β) (f : β → NonDetT m α) post :\n  _root_.wp (NonDetT.vis x f) post = _root_.wp x fun a => _root_.wp (f a) post"}, {"name": "NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}, {"name": "NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α β : Type u} (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}, {"name": "NonDetT.wp_pickCont", "content": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post"}, {"name": "NonDetT.wp_pickCont", "content": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⨆ a, ⌜p a⌝ ⊓ _root_.wp (f a) post"}, {"name": "wp_bind", "content": "lemma wp_bind {β} (x : m α) (f : α -> m β) (post : β -> l) :\n    wp (x >>= f) post = wp x (fun x => wp (f x) post)"}, {"name": "trueE", "content": "@[simp]\nlemma trueE (l : Type v) [inst: LE l] [OrderTop l] [OrderBot l] : ⌜True⌝ = (⊤ : l)"}], "used_local_defs": [{"name": "findNat", "content": "def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0"}, {"name": "find", "content": "def find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode"}, {"name": "WeakFindable", "content": "class WeakFindable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_some_p : find () = some x -> p x"}, {"name": "WeakFindable", "content": "instance WeakFindable.of_Findable {α : Type u} (p : α -> Prop) [Findable p] : WeakFindable p where\n  find := Findable.find p\n  find_some_p := Findable.find_some_p"}, {"name": "ExtractNonDet", "content": "inductive ExtractNonDet (findable : {τ : Type u} -> (τ -> Prop) -> Type u) {m} : {α : Type u} -> NonDetT m α -> Type _ where\n  | pure {α} : ∀ (x : α), ExtractNonDet findable (NonDetT.pure x)\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) :\n      (∀ y, ExtractNonDet findable (f y)) → ExtractNonDet findable (.vis x f)\n  | pickSuchThat {α} (τ : Type u) (p : τ -> Prop) (f : τ → NonDetT m α)\n    {_ : findable p}\n     : (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont τ p f)\n  | assume {α} (p : PUnit -> Prop) (f : PUnit → NonDetT m α) {_ : Decidable (p .unit)} :\n    (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont PUnit p f)"}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pure' : ExtractNonDet findable (Pure.pure (f := NonDetT m) x) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.liftM (x : m α) :\n  ExtractNonDet findable (liftM (n := NonDetT m) x) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.assume' {p : Prop} [Decidable p] : ExtractNonDet findable (MonadNonDet.assume (m :=  NonDetT m) p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pickSuchThat' {τ : Type u} (p : τ -> Prop) [Findable p] :\n  ExtractNonDet Findable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pickSuchThat_weak {τ : Type u} (p : τ -> Prop) [WeakFindable p] :\n  ExtractNonDet WeakFindable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.if {p : Prop} {dec : Decidable p} {x y : NonDetT m α}\n  (_ : ExtractNonDet findable x) (_ : ExtractNonDet findable y) :\n  ExtractNonDet findable (if p then x else y) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.ForIn_list {xs : List α} {init : β} {f : α → β → NonDetT m (ForInStep β)}\n  (_ : ∀ a b, ExtractNonDet findable (f a b)) :\n  ExtractNonDet findable (forIn xs init f) :="}, {"name": "NonDetT.extractGen", "content": "@[simp, inline]\ndef NonDetT.extractGen {findable : {τ : Type u} -> (τ -> Prop) -> Type u} {α : Type u}\n  (findOf : ∀ {τ : Type u} (p : τ -> Prop), findable p -> Unit -> Option τ)\n  : (s : NonDetT m α) -> (ex : ExtractNonDet findable s := by admit /- proof elided -/\n  ) -> m α\n  | .pure x, _ => Pure.pure x\n  | .vis x f, .vis _ _ _ => liftM x >>= (fun x => extractGen findOf (f x))\n  | .pickCont _ p f, .pickSuchThat _ _ _ _ =>\n    match findOf p ‹_› () with\n    | none => CCPOBot.compBot\n    | some x =>  extractGen findOf (f x)\n  | .pickCont _ p f, .assume _ _ _ =>\n    if p .unit then\n      extractGen findOf (f .unit)\n    else CCPOBot.compBot"}, {"name": "NonDetT.extractWeak", "content": "def NonDetT.extractWeak {α : Type u} (s : NonDetT m α) (ex : ExtractNonDet WeakFindable s := by admit /- proof elided -/\n) : m α :=\n  NonDetT.extractGen WeakFindable.find s"}], "used_local_lemmas": [{"name": "DemonicChoice.wp_csup", "content": "omit [CCPOBot m] [MAlgDet m l] [LawfulMonad m] in\nlemma wp_csup (xc : Set (m α)) (post : α -> l) [∀ α, CCPO (m α)] [MAlgPartial m]:\n  Lean.Order.chain xc ->\n  ⨅ c ∈ xc, wp c post ≤ wp (Lean.Order.CCPO.csup xc) post"}, {"name": "DemonicChoice.wp_bot", "content": "omit [CCPOBot m] [MAlgDet m l] [LawfulMonad m] in\nlemma wp_bot [∀ α, CCPO (m α)] [MAlgPartial m]:\n  wp (bot : m α) = fun _ => (⊤ : l)"}], "local_ctx": "import Mathlib.Logic.Function.Basic\n\nimport Mathlib.Order.CompleteBooleanAlgebra\n\nimport Mathlib.Order.Lattice\n\nimport Mathlib.Order.Basic\n\nimport Mathlib.Data.W.Basic\n\nimport Mathlib.Data.FinEnum\n\nimport Loom.MonadAlgebras.WP.Gen\n\nimport Loom.MonadAlgebras.WP.Liberal\n\nimport Loom.MonadAlgebras.NonDetT'.Basic\n\nopen Lean.Order\n\ndef findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0\n\ndef find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode\n\nclass WeakFindable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_some_p : find () = some x -> p x\n\ninstance WeakFindable.of_Findable {α : Type u} (p : α -> Prop) [Findable p] : WeakFindable p where\n  find := Findable.find p\n  find_some_p := Findable.find_some_p\n\ninductive ExtractNonDet (findable : {τ : Type u} -> (τ -> Prop) -> Type u) {m} : {α : Type u} -> NonDetT m α -> Type _ where\n  | pure {α} : ∀ (x : α), ExtractNonDet findable (NonDetT.pure x)\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) :\n      (∀ y, ExtractNonDet findable (f y)) → ExtractNonDet findable (.vis x f)\n  | pickSuchThat {α} (τ : Type u) (p : τ -> Prop) (f : τ → NonDetT m α)\n    {_ : findable p}\n     : (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont τ p f)\n  | assume {α} (p : PUnit -> Prop) (f : PUnit → NonDetT m α) {_ : Decidable (p .unit)} :\n    (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont PUnit p f)\n\ninstance ExtractNonDet.pure' : ExtractNonDet findable (Pure.pure (f := NonDetT m) x) :=\n\ninstance ExtractNonDet.liftM (x : m α) :\n  ExtractNonDet findable (liftM (n := NonDetT m) x) :=\n\ninstance ExtractNonDet.assume' {p : Prop} [Decidable p] : ExtractNonDet findable (MonadNonDet.assume (m :=  NonDetT m) p) :=\n\ninstance ExtractNonDet.pickSuchThat' {τ : Type u} (p : τ -> Prop) [Findable p] :\n  ExtractNonDet Findable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=\n\ninstance ExtractNonDet.pickSuchThat_weak {τ : Type u} (p : τ -> Prop) [WeakFindable p] :\n  ExtractNonDet WeakFindable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=\n\ninstance ExtractNonDet.if {p : Prop} {dec : Decidable p} {x y : NonDetT m α}\n  (_ : ExtractNonDet findable x) (_ : ExtractNonDet findable y) :\n  ExtractNonDet findable (if p then x else y) :=\n\ninstance ExtractNonDet.ForIn_list {xs : List α} {init : β} {f : α → β → NonDetT m (ForInStep β)}\n  (_ : ∀ a b, ExtractNonDet findable (f a b)) :\n  ExtractNonDet findable (forIn xs init f) :=\n\nvariable [Monad m] [CCPOBot m] [CompleteBooleanAlgebra l] [MAlgOrdered m l] [MAlgDet m l] [LawfulMonad m]\n\n@[simp, inline]\ndef NonDetT.extractGen {findable : {τ : Type u} -> (τ -> Prop) -> Type u} {α : Type u}\n  (findOf : ∀ {τ : Type u} (p : τ -> Prop), findable p -> Unit -> Option τ)\n  : (s : NonDetT m α) -> (ex : ExtractNonDet findable s := by admit /- proof elided -/\n  ) -> m α\n  | .pure x, _ => Pure.pure x\n  | .vis x f, .vis _ _ _ => liftM x >>= (fun x => extractGen findOf (f x))\n  | .pickCont _ p f, .pickSuchThat _ _ _ _ =>\n    match findOf p ‹_› () with\n    | none => CCPOBot.compBot\n    | some x =>  extractGen findOf (f x)\n  | .pickCont _ p f, .assume _ _ _ =>\n    if p .unit then\n      extractGen findOf (f .unit)\n    else CCPOBot.compBot\n\ndef NonDetT.extractWeak {α : Type u} (s : NonDetT m α) (ex : ExtractNonDet WeakFindable s := by admit /- proof elided -/\n) : m α :=\n  NonDetT.extractGen WeakFindable.find s\n\nnamespace DemonicChoice", "target_theorem": "omit [MAlgDet m l] in\nlemma ExtractNonDet.extract_refines_wp_weak [∀ α, CCPO (m α)] [MAlgPartial m] [CCPOBotLawful m] (s : NonDetT m α) (inst : ExtractNonDet WeakFindable s) :\n  wp s post <= wp s.extractWeak post :=", "ground_truth_proof": ":= by\n  unhygienic induction inst\n  { simp [wp_pure, NonDetT.extractWeak] }\n  { simp only [NonDetT.wp_vis, NonDetT.extractWeak, NonDetT.extractGen, monadLift_self, wp_bind]\n    apply wp_cons; aesop (add norm inf_comm) }\n  { simp only [NonDetT.wp_pickCont, NonDetT.extractWeak, NonDetT.extractGen]; split\n    simp only [CCPOBotLawful.prop, wp_bot, le_top]\n    rename_i y h\n    refine iInf_le_of_le y ?_\n    have := WeakFindable.find_some_p (p := p) (by assumption)\n    simp only [this, trueE, top_himp, ge_iff_le]; apply a_ih }\n  simp only [NonDetT.wp_pickCont, NonDetT.extractWeak, NonDetT.extractGen]\n  have : ∀ a : PUnit.{u_1 + 1}, a = .unit := by simp\n  simp [this, iInf_const]; split_ifs <;> simp [*, CCPOBotLawful.prop, wp_bot]\n  apply a_ih", "nesting_depth": 5, "transitive_dep_count": 82, "subset_aristotle": true, "category": "Framework"}
{"id": 379, "thm_name": "spv_dot_pure", "thm_stmt": "theorem spv_dot_pure (spv1 spv2: SpV Int) (n: ℕ)\n  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n) (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):\n    spv_dot spv1 spv2 0 0 = ∑ i ∈ Finset.range n, spv1[i] * spv2[i]", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetExamples/SpMSpV_Example.lean", "imports": ["import Auto", "import CaseStudies.Velvet.Std", "import Mathlib.Algebra.BigOperators.Intervals", "import Loom.MonadAlgebras.WP.DoNames'", "import Loom.MonadAlgebras.WP.Tactic", "import Mathlib.Algebra.Ring.Int.Defs", "import Lean", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Array.replicate", "module": "Init.Data.Array.Basic"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.find?", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "syntax \"method\" ident leafny_binder* \"return\" \"(\" ident \":\" ", "content": "syntax \"method\" ident leafny_binder* \"return\" \"(\" ident \":\" term \")\"\n  (require_caluse )*\n  (ensures_caluse)* \"do\" doSeq\n  Termination.suffix : command\n\nsyntax \"ensures\" termBeforeReqEnsDo : ensures_caluse\n\nsyntax \"while_some\" term \":|\" termBeforeDo \"do\" doSeq : doElem\n\nsyntax \"while_some\" term \":|\" term\n  (invariantClause)+\n  (doneWith)?\n  \"do\" doSeq : doElem\n\nsyntax \"let\" term \":|\" term : doElem\n\nsyntax \"done_with\" termBeforeDo : doneWith\n\nsyntax \"invariant\" termBeforeDo linebreak : invariantClause\n\nsyntax \"while\" term\n  (invariantClause)*\n  (doneWith)?\n  (decreasingTerm)?\n  \"do\" doSeq : doElem\n\nsyntax \"(mut\" ident \":\" term \")\" : leafny_binder"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x:term :| $t) => `(doElem| let $x:term <- pickSuchThat _ (fun $x => type_with_name_prefix `choice $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t do $seq:doSeq) => do\n    let decr <- withRef (<- getRef) `(decreasing none)\n    let invs <- withRef (<- getRef) `(invariants [])\n    `(doElem|\n      for _ in Lean.Loop.mk do\n        $invs:term\n        onDoneGadget (with_name_prefix `done ¬$t:term)\n        $decr:term\n        if $t then\n          $seq:doSeq\n        else break)\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      let invd_some ← match inv_done with\n      | some invd_some => withRef invd_some ``($invd_some)\n      | none => ``(¬$t:term)\n      match measure with\n      | some measure_some =>\n        let decr <- withRef measure_some `(decreasing type_with_name_prefix `decreasing $measure_some)\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        let decr <- withRef (<- getRef) `(decreasing none)\n        let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n  | `(doElem| while_some $x:ident :| $t do $seq:doSeq) =>\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      `(doElem|\n        while ∃ $x:ident, $t do\n          let $x :| $t\n          $[$seq:doElem]*)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| while_some $x:ident :| $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]? do\n                $seq:doSeq) => do\n    let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n    let invd_some ← match inv_done with\n    | some invd_some => withRef invd_some ``($invd_some)\n    | none => ``(¬$t:term)\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      let decr <- withRef (<- getRef) `(decreasing none)\n      `(doElem|\n        for _ in Lean.Loop.mk do\n          $invs:term\n          onDoneGadget (with_name_prefix `done $invd_some:term)\n          $decr:term\n          if ∃ $x:ident, $t then\n            let $x :| $t\n            $[$seq:doElem]*\n          else break)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| for $x:ident in $t\n            $[invariant $inv:term\n            ]*\n            do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      match seq with\n      | `(doSeq| $[$seq:doElem]*)\n      | `(doSeq| $[$seq:doElem;]*)\n      | `(doSeq| { $[$seq:doElem]* }) =>\n        `(doElem|\n          for $x:ident in $t do\n            $invs:term\n            $[$seq:doElem]*)\n      | _ => Lean.Macro.throwError \"for expects a sequence of do-elements\""}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|loom_solver) =>\n    `(tactic|(\n      try simp at *\n      try aesop))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let balance := mkIdent `balance_name\n      let balanceType <- `(term| Bal)\n      let inv : Array Term <- inv.mapM fun (inv : Term) => withRef inv ``(fun ($(balance):ident : $balanceType)=> with_name_prefix `inv $inv)\n      let invd_some <- match inv_done with\n      | some invd_some => withRef invd_some ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done $invd_some)\n      | none => ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done ¬$t:term)\n      match measure with\n      | some measure_some =>\n        let measure_some ← withRef measure_some ``(type_with_name_prefix `decreasing ($measure_some:term))\n        do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget ($measure_some:term)\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget none\n            if $t then\n              $seq:doSeq\n            else break)"}, {"name": "macro_rules", "content": "macro_rules\n| `(doElem|balance_set $t) => do\n  let balId := mkIdent `balance\n  `(doElem|do\n    $balId:ident := $t\n    set $balId:ident\n    $balId:ident ← get)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem|$id:ident[$idx:term] := $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (fun _ => $val))\n  | `(doElem|$id:ident[$idx:term] += $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (· + $val))"}], "lib_lemmas": [{"name": "List.find?_eq_none", "module": "Init.Data.List.Find"}, {"name": "List.mem_iff_get", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "List.find?_eq_some_iff_getElem", "module": "Init.Data.List.Nat.Find"}, {"name": "em", "module": "Mathlib.Logic.Basic"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "le_iff_eq_or_lt", "module": "Mathlib.Order.Basic"}, {"name": "le_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_of_lt_of_le", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_or_gt_of_ne", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "lt_iff_le_not_ge", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "SpV", "content": "structure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!"}, {"name": "spv_dot", "content": "def spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=\n  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then\n    0\n  else\n    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)\n    else\n      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n        spv_dot spv1 spv2 (pnt1 + 1) pnt2\n      else\n        spv_dot spv1 spv2 pnt1 (pnt2 + 1)\n  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)\n\n\nmethod SpVSpV\n  (mut out: Array Int)\n  (spv1: SpV Int)\n  (spv2: SpV Int) return (u: Unit)\n  ensures out.size = 1\n  ensures out[0]! = spv_dot spv1 spv2 0 0\n  do\n    out := Array.replicate 1 0\n    let mut pnt1 := 0\n    let mut pnt2 := 0\n    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size\n    invariant out.size = 1\n    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size\n    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0\n    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size\n    do\n      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!\n        pnt1 := pnt1 + 1\n        pnt2 := pnt2 + 1\n      else\n        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n          pnt1 := pnt1 + 1\n        else\n          pnt2 := pnt2 + 1\n    return\n\n\nmethod SpMSpV\n  (mut out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int) return (u: Unit)\n  ensures out.size = spm.size\n  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0\n  do\n    out := Array.replicate spm.size 0\n    let mut spmInd := Array.replicate spm.size 0\n    let mut spvInd := Array.replicate spm.size 0\n    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size\n    invariant spvInd.size = spm.size\n    invariant spmInd.size = spm.size\n    invariant out.size = spm.size\n    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size\n    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size\n    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0\n    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size\n    do\n      let ind_m := spmInd[i]!\n      let ind_v := spvInd[i]!\n      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then\n        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!\n        spmInd[i] += 1\n        spvInd[i] += 1\n      else\n        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then\n          spmInd[i] += 1\n        else\n          spvInd[i] += 1\n    return"}], "used_local_lemmas": [{"name": "getValSpV_eq", "content": "theorem getValSpV_eq (spv: SpV Int) (j: ℕ) (h_ind: j < spv.size): spv[spv.ind[j]!] = (spv.val)[j]!"}, {"name": "getValSpV_empty", "content": "theorem getValSpV_empty (spv: SpV Int) (j: ℕ) (h_empty: ∀ i < spv.size, spv.ind[i]! ≠ j): spv[j] = 0"}, {"name": "spv_dot_pure_gen", "content": "theorem spv_dot_pure_gen (spv1: SpV Int) (spv2: SpV Int) (n pnt1 pnt2: ℕ)\n  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n)\n  (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):\n  spv_dot spv1 spv2 pnt1 pnt2 =\n    ∑ i ∈ Finset.range n,\n      if max\n        (if spv1.size ≤ pnt1 then n else spv1.ind[pnt1]!)\n        (if spv2.size ≤ pnt2 then n else spv2.ind[pnt2]!) ≤ i then\n        spv1[i] * spv2[i]\n      else\n        0"}], "local_ctx": "import Auto\n\nimport Lean\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Algebra.Ring.Int.Defs\n\nimport Loom.MonadAlgebras.NonDetT.Extract\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.DoNames'\n\nimport CaseStudies.Velvet.Std\n\nsection SpMV\n\nstructure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!\n\ndef spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=\n  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then\n    0\n  else\n    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)\n    else\n      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n        spv_dot spv1 spv2 (pnt1 + 1) pnt2\n      else\n        spv_dot spv1 spv2 pnt1 (pnt2 + 1)\n  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)\n\n\nmethod SpVSpV\n  (mut out: Array Int)\n  (spv1: SpV Int)\n  (spv2: SpV Int) return (u: Unit)\n  ensures out.size = 1\n  ensures out[0]! = spv_dot spv1 spv2 0 0\n  do\n    out := Array.replicate 1 0\n    let mut pnt1 := 0\n    let mut pnt2 := 0\n    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size\n    invariant out.size = 1\n    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size\n    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0\n    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size\n    do\n      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!\n        pnt1 := pnt1 + 1\n        pnt2 := pnt2 + 1\n      else\n        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n          pnt1 := pnt1 + 1\n        else\n          pnt2 := pnt2 + 1\n    return\n\n\nmethod SpMSpV\n  (mut out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int) return (u: Unit)\n  ensures out.size = spm.size\n  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0\n  do\n    out := Array.replicate spm.size 0\n    let mut spmInd := Array.replicate spm.size 0\n    let mut spvInd := Array.replicate spm.size 0\n    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size\n    invariant spvInd.size = spm.size\n    invariant spmInd.size = spm.size\n    invariant out.size = spm.size\n    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size\n    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size\n    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0\n    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size\n    do\n      let ind_m := spmInd[i]!\n      let ind_v := spvInd[i]!\n      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then\n        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!\n        spmInd[i] += 1\n        spvInd[i] += 1\n      else\n        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then\n          spmInd[i] += 1\n        else\n          spvInd[i] += 1\n    return", "target_theorem": "theorem spv_dot_pure (spv1 spv2: SpV Int) (n: ℕ)\n  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n) (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):\n    spv_dot spv1 spv2 0 0 = ∑ i ∈ Finset.range n, spv1[i] * spv2[i] :=", "ground_truth_proof": ":= by\n      simp [spv_dot_pure_gen spv1 spv2 n 0 0 sz1 sz2]\n      apply Finset.sum_congr\n      { rfl }\n      intro x hx\n      simp at hx\n      by_cases em1: spv1.size = 0 <;> simp [em1]\n      { simp [lt_iff_le_not_ge.mp hx]\n        simp [getValSpV_empty spv1 x (by intro i hi; omega)] }\n      by_cases em2: spv2.size = 0 <;> simp [em2]\n      { intros\n        simp [getValSpV_empty spv2 x (by intro i hi; omega)] }\n      intro zer_ineq\n      have zer_lemma (spv: SpV Int) (i: ℕ) (hsz: spv.size ≠ 0): i < spv.ind[0]! → spv[i] = 0 := by\n        intro lt\n        have all_none: ∀ j < spv.size, spv.ind[j]! ≠ i := by\n          intro i1 hi1\n          by_cases i10 : i1 = 0\n          { simp [←i10] at lt\n            omega }\n          have contra := spv.inc 0 i1 (by omega) (by omega) (by omega)\n          simp at lt\n          omega\n        simp [getValSpV_empty spv i all_none]\n      by_cases sm1: spv1.ind[0]! ≤ x\n      { simp [sm1] at zer_ineq\n        simp [zer_lemma spv2 x em2 zer_ineq] }\n      simp at sm1\n      simp [zer_lemma spv1 x em1 sm1]", "nesting_depth": 3, "transitive_dep_count": 28, "subset_aristotle": false, "category": "Framework"}
{"id": 380, "thm_name": "array_extract_split_i_j_k", "thm_stmt": "lemma array_extract_split_i_j_k (arr : Array α) (i j k: Nat) :\n    i < j -> j < k -> k ≤ arr.size →\n    arr.extract i k = (arr.extract i j) ++ (arr.extract j k)", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetExamples/EncodeDecodeStr.lean", "imports": ["import Auto", "import CaseStudies.Velvet.Std", "import Loom.MonadAlgebras.WP.Tactic", "import CaseStudies.TestingUtil", "import Loom.MonadAlgebras.WP.DoNames'", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "Char", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Array", "module": "Init.Prelude"}, {"name": "Array.map", "module": "Init.Data.Array.Basic"}, {"name": "Array.replicate", "module": "Init.Data.Array.Basic"}, {"name": "Lean.Syntax", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Lean.Name", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.matchExpr", "module": "Lean.Parser.Term"}, {"name": "Lean.Parser.Term.optIdent", "module": "Lean.Parser.Term.Basic"}, {"name": "args", "module": "Auto.Parser.TPTP"}, {"name": "cond", "module": "Init.Prelude"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Lean.MonadQuotation", "module": "Init.Prelude"}, {"name": "Lean.MonadRef", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "Lean.withRef", "module": "Init.Prelude"}, {"name": "kind", "module": "Auto.Parser.TPTP"}, {"name": "Lean.MacroM", "module": "Init.Prelude"}, {"name": "ReaderT", "module": "Init.Prelude"}, {"name": "Bind", "module": "Init.Prelude"}, {"name": "Bind.bind", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.debugAssert", "module": "Lean.Parser.Term"}, {"name": "Lean.Parser.Term.doAssert", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doDbgTrace", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doDebugAssert", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.ident", "module": "Lean.Parser.Term"}, {"name": "Lean.Syntax.mkApp", "module": "Init.Meta.Defs"}, {"name": "Lean.mkIdentFrom", "module": "Init.Meta.Defs"}, {"name": "Lean.Parser.Term.let_delayed", "module": "Lean.Parser.Term"}, {"name": "Lean.mkIdentFromRef", "module": "Init.Meta.Defs"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "Lean.quoteNameMk", "module": "Init.Meta.Defs"}, {"name": "DoResultPR", "module": "Init.Core"}, {"name": "DoResultPRBC", "module": "Init.Core"}, {"name": "DoResultSBC", "module": "Init.Core"}, {"name": "ForInStep", "module": "Init.Core"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "DoResultBC", "module": "Init.Core"}, {"name": "ForInStep.yield", "module": "Init.Core"}, {"name": "Lean.Name.cmp", "module": "Lean.Data.Name"}, {"name": "Std.TreeMap", "module": "Std.Data.TreeMap.Basic"}, {"name": "Lean.Elab.Term.TermElabM", "module": "Lean.Elab.Term.TermElabM"}, {"name": "Lean.Parser.Term.doLetArrow", "module": "Lean.Parser.Do"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.doHave", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doLet", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doLetRec", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doReassign", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doReassignArrow", "module": "Lean.Parser.Do"}, {"name": "Lean.throwErrorAt", "module": "Lean.Exception"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.doSeqBracketed", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doSeqIndent", "module": "Lean.Parser.Do"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "Id.run", "module": "Init.Control.Id"}, {"name": "modify", "module": "Init.Prelude"}, {"name": "Lean.Elab.liftMacroM", "module": "Lean.Elab.Util"}, {"name": "Unit.unit", "module": "Init.Prelude"}, {"name": "term", "module": "Auto.Parser.TPTP"}, {"name": "type", "module": "Auto.Lib.Rebind"}, {"name": "Lean.Parser.Term.unreachable", "module": "Lean.Parser.Term"}, {"name": "Lean.throwError", "module": "Lean.Exception"}, {"name": "Lean.Elab.Term.Quotation.getPatternVars", "module": "Lean.Elab.Quotation.Util"}, {"name": "Lean.Elab.Term.getPatternVars", "module": "Lean.Elab.PatternVar"}, {"name": "PUnit.unit", "module": "Init.Prelude"}, {"name": "Stream", "module": "Init.Data.Stream"}, {"name": "Stream.next?", "module": "Init.Data.Stream"}, {"name": "Lean.mkNullNode", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.optType", "module": "Lean.Parser.Term.Basic"}, {"name": "Lean.Parser.Term.doCatch", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doCatchMatch", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.matchAlts", "module": "Lean.Parser.Term"}, {"name": "MonadExcept", "module": "Init.Prelude"}, {"name": "MonadExcept.tryCatch", "module": "Init.Prelude"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "tryCatchThe", "module": "Init.Prelude"}, {"name": "tryFinally", "module": "Init.Control.Except"}, {"name": "Lean.Parser.Term.doLetElse", "module": "Lean.Parser.Do"}, {"name": "Lean.mkNode", "module": "Init.Prelude"}, {"name": "Lean.HygieneInfo.mkIdent", "module": "Init.Meta.Defs"}, {"name": "Lean.Parser.Term.letId", "module": "Lean.Parser.Term"}, {"name": "Lean.hygieneInfoKind", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.letIdDecl", "module": "Lean.Parser.Term"}, {"name": "Lean.Parser.Term.letPatDecl", "module": "Lean.Parser.Term"}, {"name": "sequence", "module": "Mathlib.Control.Traversable.Basic"}, {"name": "Lean.Elab.Term.Quotation.getPatternsVars", "module": "Lean.Elab.Quotation.Util"}, {"name": "Lean.Elab.Term.getPatternsVars", "module": "Lean.Elab.PatternVar"}, {"name": "Lean.Elab.Term.expandMatchAlt", "module": "Lean.Elab.BindersUtil"}, {"name": "Lean.Parser.Term.letEqnsDecl", "module": "Lean.Parser.Term"}, {"name": "Lean.Parser.Term.letDecl", "module": "Lean.Parser.Term"}, {"name": "Lean.Parser.Term.doIdDecl", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doPatDecl", "module": "Lean.Parser.Do"}, {"name": "Lean.SyntaxNodeKind", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.termFor", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.termReturn", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.termTry", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.termUnless", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.liftMethod", "module": "Lean.Parser.Do"}, {"name": "termDepIfThenElse", "module": "Init.Notation"}, {"name": "termIfThenElse", "module": "Init.Notation"}, {"name": "Lean.Parser.Term.letrec", "module": "Lean.Parser.Term"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "Lean.Parser.Term.num", "module": "Lean.Parser.Term"}, {"name": "Lean.Syntax.node", "module": "Init.Prelude"}, {"name": "Lean.choiceKind", "module": "Init.Prelude"}, {"name": "error", "module": "Auto.Parser.TPTP"}, {"name": "id", "module": "Init.Prelude"}, {"name": "node", "module": "Test.SmtTranslation.Inductive"}, {"name": "Lean.MonadQuotation.addMacroScope", "module": "Init.Prelude"}, {"name": "Lean.Parser.Term.doBreak", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doContinue", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doExpr", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doFor", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doIf", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doMatch", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doMatchExpr", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doNested", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doReturn", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doTry", "module": "Lean.Parser.Do"}, {"name": "Lean.Parser.Term.doUnless", "module": "Lean.Parser.Do"}, {"name": "Lean.withIncRecDepth", "module": "Lean.Exception"}, {"name": "Lean.Elab.Term.expandOptType", "module": "Lean.Elab.BindersUtil"}, {"name": "Lean.Parser.Term.letRecDecls", "module": "Lean.Parser.Term"}, {"name": "Lean.Macro.throwErrorAt", "module": "Init.Prelude"}, {"name": "Lean.Meta.instantiateMVarsIfMVarApp", "module": "Lean.Meta.Basic"}, {"name": "Lean.Parser.Term.matchAlt", "module": "Lean.Parser.Term"}, {"name": "Lean.Parser.Term.matchExprAlt", "module": "Lean.Parser.Term"}, {"name": "Lean.Parser.Term.matchExprAlts", "module": "Lean.Parser.Term"}, {"name": "Lean.Parser.Term.matchExprElseAlt", "module": "Lean.Parser.Term"}, {"name": "Lean.Parser.Term.matchExprPat", "module": "Lean.Parser.Term"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Encodable", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Encodable.decode", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "inline", "module": "Init.Core"}, {"name": "Lean.Order.CCPO", "module": "Init.Internal.Order.Basic"}, {"name": "Lean.Order.MonoBind", "module": "Init.Internal.Order.Basic"}, {"name": "List.cons", "module": "Init.Prelude"}, {"name": "List.nil", "module": "Init.Prelude"}, {"name": "List.take", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "syntax \"let\" term \":|\" term : doElem", "content": "syntax \"let\" term \":|\" term : doElem\n\nsyntax \"method\" ident leafny_binder* \"return\" \"(\" ident \":\" term \")\"\n  (require_caluse )*\n  (ensures_caluse)* \"do\" doSeq\n  Termination.suffix : command"}, {"name": "macro \"extract_tactic\" : tactic =>", "content": "macro \"extract_tactic\" : tactic =>\n  `(tactic| repeat' (intros; extract_step <;> try dsimp))\n\nsyntax \"loom_solve\" : loom_solve_tactic\n\nsyntax \"ensures\" termBeforeReqEnsDo : ensures_caluse\n\nsyntax \"require\" termBeforeReqEnsDo : require_caluse\n\nsyntax \"prove_correct\" : prove_correct_command\n\nsyntax \"prove_correct\" ident Termination.suffix \"by\" tacticSeq : command\n\nsyntax \"done_with\" termBeforeDo : doneWith\n\nsyntax \"invariant\" termBeforeDo linebreak : invariantClause\n\nsyntax \"while\" term\n  (invariantClause)*\n  (doneWith)?\n  (decreasingTerm)?\n  \"do\" doSeq : doElem"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem|$id:ident[$idx:term] := $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (fun _ => $val))\n  | `(doElem|$id:ident[$idx:term] += $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (· + $val))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x:term :| $t) => `(doElem| let $x:term <- pickSuchThat _ (fun $x => type_with_name_prefix `choice $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t do $seq:doSeq) => do\n    let decr <- withRef (<- getRef) `(decreasing none)\n    let invs <- withRef (<- getRef) `(invariants [])\n    `(doElem|\n      for _ in Lean.Loop.mk do\n        $invs:term\n        onDoneGadget (with_name_prefix `done ¬$t:term)\n        $decr:term\n        if $t then\n          $seq:doSeq\n        else break)\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      let invd_some ← match inv_done with\n      | some invd_some => withRef invd_some ``($invd_some)\n      | none => ``(¬$t:term)\n      match measure with\n      | some measure_some =>\n        let decr <- withRef measure_some `(decreasing type_with_name_prefix `decreasing $measure_some)\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        let decr <- withRef (<- getRef) `(decreasing none)\n        let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n  | `(doElem| while_some $x:ident :| $t do $seq:doSeq) =>\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      `(doElem|\n        while ∃ $x:ident, $t do\n          let $x :| $t\n          $[$seq:doElem]*)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| while_some $x:ident :| $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]? do\n                $seq:doSeq) => do\n    let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n    let invd_some ← match inv_done with\n    | some invd_some => withRef invd_some ``($invd_some)\n    | none => ``(¬$t:term)\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      let decr <- withRef (<- getRef) `(decreasing none)\n      `(doElem|\n        for _ in Lean.Loop.mk do\n          $invs:term\n          onDoneGadget (with_name_prefix `done $invd_some:term)\n          $decr:term\n          if ∃ $x:ident, $t then\n            let $x :| $t\n            $[$seq:doElem]*\n          else break)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| for $x:ident in $t\n            $[invariant $inv:term\n            ]*\n            do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      match seq with\n      | `(doSeq| $[$seq:doElem]*)\n      | `(doSeq| $[$seq:doElem;]*)\n      | `(doSeq| { $[$seq:doElem]* }) =>\n        `(doElem|\n          for $x:ident in $t do\n            $invs:term\n            $[$seq:doElem]*)\n      | _ => Lean.Macro.throwError \"for expects a sequence of do-elements\""}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|loom_solver) =>\n    `(tactic|(\n      try simp at *\n      try aesop))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let balance := mkIdent `balance_name\n      let balanceType <- `(term| Bal)\n      let inv : Array Term <- inv.mapM fun (inv : Term) => withRef inv ``(fun ($(balance):ident : $balanceType)=> with_name_prefix `inv $inv)\n      let invd_some <- match inv_done with\n      | some invd_some => withRef invd_some ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done $invd_some)\n      | none => ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done ¬$t:term)\n      match measure with\n      | some measure_some =>\n        let measure_some ← withRef measure_some ``(type_with_name_prefix `decreasing ($measure_some:term))\n        do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget ($measure_some:term)\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget none\n            if $t then\n              $seq:doSeq\n            else break)"}, {"name": "macro_rules", "content": "macro_rules\n| `(doElem|balance_set $t) => do\n  let balId := mkIdent `balance\n  `(doElem|do\n    $balId:ident := $t\n    set $balId:ident\n    $balId:ident ← get)"}, {"name": "VelvetM.extract", "content": "def VelvetM.extract {α : Type} (x : VelvetM α)\n  [∀ α, Lean.Order.CCPO (DivM α)] [Lean.Order.MonoBind DivM] [Inhabited α] : α :=\n  x.run.run"}, {"name": "run", "content": "def run (code : Code) (m : Syntax) (returnType : Syntax) (uvars : Array Var := #[]) (kind := Kind.regular) : MacroM Syntax :=\n  toTerm code { m, returnType, kind, uvars }"}, {"name": "Kind", "content": "inductive Kind where\n  | regular\n  | forIn\n  | forInWithReturn\n  | nestedBC\n  | nestedPR\n  | nestedSBC\n  | nestedPRBC"}, {"name": "Var", "content": "abbrev Var := Syntax  "}, {"name": "Code", "content": "inductive Code where\n  | decl         (xs : Array Var) (doElem : Syntax) (k : Code)\n  | reassign     (xs : Array Var) (doElem : Syntax) (k : Code)\n   \n  | joinpoint    (name : Name) (params : Array (Var × Bool)) (body : Code) (k : Code)\n  | seq          (action : Syntax) (k : Code)\n  | action       (action : Syntax)\n  | break        (ref : Syntax)\n  | continue     (ref : Syntax)\n  | return       (ref : Syntax) (val : Syntax)\n   \n  | ite          (ref : Syntax) (h? : Option Var) (optIdent : Syntax) (cond : Syntax) (thenBranch : Code) (elseBranch : Code)\n  | match        (ref : Syntax) (gen : Syntax) (discrs : Syntax) (optMotive : Syntax) (alts : Array (Alt Code))\n  | matchExpr    (ref : Syntax) («meta» : Bool) (discr : Syntax) (alts : Array (AltExpr Code)) (elseBranch : Code)\n  | jmp          (ref : Syntax) (jpName : Name) (args : Array Syntax)\n  deriving Inhabited"}, {"name": "AltExpr", "content": "structure AltExpr (σ : Type) where\n  ref     : Syntax\n  var?    : Option Var\n  funName : Syntax\n  pvars   : Array Syntax\n  rhs     : σ\n  deriving Inhabited"}, {"name": "DoIfView", "content": "structure DoIfView where\n  ref        : Syntax\n  optIdent   : Syntax\n  cond       : Syntax\n  thenBranch : Syntax\n  elseBranch : Syntax"}, {"name": "Alt", "content": "structure Alt (σ : Type) where\n  ref      : Syntax\n  vars     : Array Var\n  patterns : Syntax\n  rhs      : σ\n  deriving Inhabited"}, {"name": "JPDecl", "content": "structure JPDecl where\n  name : Name\n  params : Array (Var × Bool)\n  body : Code"}, {"name": "toTerm", "content": "partial def toTerm (c : Code) : M Syntax := do\n  let term ← go c\n  if let some ref := c.getRef? then\n    annotate ref term\n  else\n    return term\nwhere\n  go (c : Code) : M Syntax := do\n    match c with\n    | .return ref val     => withRef ref <| returnToTerm val\n    | .continue ref       => withRef ref continueToTerm\n    | .break ref          => withRef ref breakToTerm\n    | .action e           => actionTerminalToTerm e\n    | .joinpoint j ps b k => mkJoinPoint j ps (← toTerm b) (← toTerm k)\n    | .jmp ref j args     => return mkJmp ref j args\n    | .decl _ stx k       => declToTerm stx (← toTerm k)\n    | .reassign _ stx k   => reassignToTerm stx (← toTerm k)\n    | .seq stx k          => seqToTerm stx (← toTerm k)\n    | .ite ref _ o c t e  => withRef ref <| do mkIte o c (← toTerm t) (← toTerm e)\n    | .match ref genParam discrs optMotive alts =>\n      let mut termAlts := #[]\n      for alt in alts do\n        let rhs ← toTerm alt.rhs\n        let termAlt := mkNode ``Parser.Term.matchAlt #[mkAtomFrom alt.ref \"|\", mkNullNode #[alt.patterns], mkAtomFrom alt.ref \"=>\", rhs]\n        termAlts := termAlts.push termAlt\n      let termMatchAlts := mkNode ``Parser.Term.matchAlts #[mkNullNode termAlts]\n      return mkNode ``Parser.Term.«match» #[mkAtomFrom ref \"match\", genParam, optMotive, discrs, mkAtomFrom ref \"with\", termMatchAlts]\n    | .matchExpr ref «meta» d alts elseBranch => withFreshMacroScope do\n      let d' ← `(discr)\n      let mut termAlts := #[]\n      for alt in alts do\n        let rhs ← `(($(← toTerm alt.rhs) : $((← read).m) _))\n        let optVar := if let some var := alt.var? then mkNullNode #[var, mkAtomFrom var \"@\"] else mkNullNode #[]\n        let pat := mkNode ``Parser.Term.matchExprPat #[optVar, alt.funName, mkNullNode alt.pvars]\n        let termAlt := mkNode ``Parser.Term.matchExprAlt #[mkAtomFrom alt.ref \"|\", pat, mkAtomFrom alt.ref \"=>\", rhs]\n        termAlts := termAlts.push termAlt\n      let elseBranch := mkNode ``Parser.Term.matchExprElseAlt #[mkAtomFrom ref \"|\", mkHole ref, mkAtomFrom ref \"=>\", (← toTerm elseBranch)]\n      let termMatchExprAlts := mkNode ``Parser.Term.matchExprAlts #[mkNullNode termAlts, elseBranch]\n      let body := mkNode ``Parser.Term.matchExpr #[mkAtomFrom ref \"match_expr\", d', mkAtomFrom ref \"with\", termMatchExprAlts]\n      if «meta» then\n        `(Bind.bind (instantiateMVarsIfMVarApp $d) fun discr => $body)\n      else\n        `(let discr := $d; $body)"}, {"name": "annotate", "content": "def annotate [Monad m] [MonadRef m] [MonadQuotation m] (ref : Syntax) (term : Syntax) : m Syntax :=\n  withRef term <| `(with_annotate_term $ref $term)"}, {"name": "Context", "content": "structure Context where\n  ref         : Syntax\n   \n  m           : Syntax\n   \n  returnType  : Syntax\n  mutableVars : VarSet := {}\n  insideFor   : Bool := false"}, {"name": "seqToTerm", "content": "def seqToTerm (action : Syntax) (k : Syntax) : M Syntax := withRef action <| withFreshMacroScope do\n  if action.getKind == ``Parser.Term.doDbgTrace then\n    let msg := action[1]\n    `(dbg_trace $msg; $k)\n  else if action.getKind == ``Parser.Term.doAssert then\n    let cond := action[1]\n    `(assert! $cond; $k)\n  else if action.getKind == ``Parser.Term.doDebugAssert then\n    let cond := action[1]\n    `(debugAssert| debug_assert! $cond; $k)\n  else\n    let action ← withRef action ``(($action : $((←read).m) PUnit))\n    ``(Bind.bind $action (fun (_ : PUnit) => $k))"}, {"name": "M", "content": "abbrev M := ReaderT Context MacroM"}, {"name": "Context", "content": "structure Context where\n   \n  m          : Syntax\n   \n  returnType : Syntax\n  uvars      : Array Var\n  kind       : Kind"}, {"name": "ToForInTermResult", "content": "structure ToForInTermResult where\n  uvars      : Array Var\n  term       : Syntax"}, {"name": "Code.getRef?", "content": "def Code.getRef? : Code → Option Syntax\n  | .decl _ doElem _     => doElem\n  | .reassign _ doElem _ => doElem\n  | .joinpoint ..        => none\n  | .seq a _             => a\n  | .action a            => a\n  | .break ref           => ref\n  | .continue ref        => ref\n  | .return ref _        => ref\n  | .ite ref ..          => ref\n  | .match ref ..        => ref\n  | .matchExpr ref ..    => ref\n  | .jmp ref ..          => ref"}, {"name": "mkIte", "content": "def mkIte (optIdent : Syntax) (cond : Syntax) (thenBranch : Syntax) (elseBranch : Syntax) : MacroM Syntax := do\n  if optIdent.isNone then\n    ``(if $cond then $thenBranch else $elseBranch)\n  else\n    let h := optIdent[0]\n    ``(if $h:ident : $cond then $thenBranch else $elseBranch)"}, {"name": "mkJmp", "content": "def mkJmp (ref : Syntax) (j : Name) (args : Array Syntax) : Syntax :=\n  Syntax.mkApp (mkIdentFrom ref j) args"}, {"name": "mkJoinPoint", "content": "def mkJoinPoint (j : Name) (ps : Array (Syntax × Bool)) (body : Syntax) (k : Syntax) : M Syntax := withRef body <| withFreshMacroScope do\n  let pTypes ← ps.mapM fun ⟨id, useTypeOf⟩ => do if useTypeOf then `(type_of% $id) else `(_)\n  let ps     := ps.map (·.1)\n   \n  `(let_delayed $(← mkIdentFromRef j):ident $[($ps : $pTypes)]* : $((← read).m) _ := $body; $k)"}, {"name": "returnToTerm", "content": "def returnToTerm (val : Syntax) : M Syntax := do\n  let ctx ← read\n  let u ← mkUVarTuple\n  match ctx.kind with\n  | .regular         => if ctx.uvars.isEmpty then ``(Pure.pure $val) else ``(Pure.pure (MProdWithNames.mk $val $u))\n  | .forIn           => ``(Pure.pure (ForInStep.done $u))\n  | .forInWithReturn => ``(Pure.pure (ForInStep.done (MProdWithNames.mk (some $val) $u)))\n  | .nestedBC        => unreachable!\n  | .nestedPR        => ``(Pure.pure (DoResultPR.«return» $val $u))\n  | .nestedSBC       => ``(Pure.pure (DoResultSBC.«pureReturn» $val $u))\n  | .nestedPRBC      => ``(Pure.pure (DoResultPRBC.«return» $val $u))"}, {"name": "mkUVarTuple", "content": "def mkUVarTuple : M Syntax := do\n  let ctx ← read\n  mkTuple ctx.uvars"}, {"name": "mkTuple", "content": "private def mkTuple (elems : Array Syntax) : MacroM Syntax := do\n  if elems.size = 0 then\n    mkUnit\n  else if h : elems.size = 1 then\n    ``(WithName.mk' $(elems[0]) $(Lean.quoteNameMk elems[0].getId))\n  else\n    let init <- ``(WithName.mk' $(elems.back!) $(Lean.quoteNameMk elems.back!.getId))\n    elems.extract 0 (elems.size - 1) |>.foldrM (init := init) fun elem tuple => do\n      let name := Lean.quoteNameMk elem.getId\n      ``(MProdWithNames.mk' $elem $tuple $name)"}, {"name": "MProdWithNames.mk'", "content": "abbrev MProdWithNames.mk' {α β : Type u} (a : α) (b : β)\n  (αName : Lean.Name := default) : MProdWithNames α β αName :=\n  @MProdWithNames.mk _ _ αName a b"}, {"name": "mkUnit", "content": "private def mkUnit : MacroM Syntax :=\n  ``((⟨⟩ : PUnit))"}, {"name": "WithName.mk'", "content": "abbrev WithName.mk' {α : Sort u} (a : α) (name : Lean.Name := default) : WithName α name :=\n  a"}, {"name": "continueToTerm", "content": "def continueToTerm : M Syntax := do\n  let ctx ← read\n  let u ← mkUVarTuple\n  match ctx.kind with\n  | .regular         => unreachable!\n  | .forIn           => ``(Pure.pure (ForInStep.yield $u))\n  | .forInWithReturn => ``(Pure.pure (ForInStep.yield (MProdWithNames.mk none $u)))\n  | .nestedBC        => ``(Pure.pure (DoResultBC.«continue» $u))\n  | .nestedPR        => unreachable!\n  | .nestedSBC       => ``(Pure.pure (DoResultSBC.«continue» $u))\n  | .nestedPRBC      => ``(Pure.pure (DoResultPRBC.«continue» $u))"}, {"name": "declToTerm", "content": "def declToTerm (decl : Syntax) (k : Syntax) : M Syntax := withRef decl <| withFreshMacroScope do\n  let kind := decl.getKind\n  if kind == ``Parser.Term.doLet then\n    let letDecl := decl[2]\n    `(let $letDecl:letDecl; $k)\n  else if kind == ``Parser.Term.doLetRec then\n    let letRecToken := decl[0]\n    let letRecDecls := decl[1]\n    return mkNode ``Parser.Term.letrec #[letRecToken, letRecDecls, mkNullNode, k]\n  else if kind == ``Parser.Term.doLetArrow then\n    let arg := decl[2]\n    if arg.getKind == ``Parser.Term.doIdDecl then\n      let id     := arg[0]\n      let type   := expandOptType id arg[1]\n      let doElem := arg[3]\n      \n      match isDoExpr? doElem with\n      | some action =>\n        let action ← withRef action `(($action : $((← read).m) $type))\n        ``(Bind.bind $action (fun ($id:ident : $type) => $k))\n      | none        => Macro.throwErrorAt decl \"unexpected kind of `do` declaration\"\n    else\n      Macro.throwErrorAt decl \"unexpected kind of `do` declaration\"\n  else if kind == ``Parser.Term.doHave then\n    \n    let args := decl.getArgs\n    let args := args ++ #[mkNullNode  , k]\n    return mkNode `Lean.Parser.Term.«have» args\n  else\n    Macro.throwErrorAt decl \"unexpected kind of `do` declaration\"\n\n  partial def doLetArrowToCode (doLetArrow : Syntax) (doElems : List Syntax) : M CodeBlock := do\n    let decl    := doLetArrow[2]\n    if decl.getKind == ``Parser.Term.doIdDecl then\n      let y := decl[0]\n      checkNotShadowingMutable #[y]\n      let doElem := decl[3]\n      let k ← withNewMutableVars #[y] (isMutableLet doLetArrow) (doSeqToCode doElems)\n      match isDoExpr? doElem with\n      | some _      => return mkVarDeclCore #[y] doLetArrow k\n      | none =>\n        checkLetArrowRHS doElem\n        let c ← doSeqToCode [doElem]\n        match doElems with\n        | []       => pure c\n        | kRef::_  => concat c kRef y k\n    else if decl.getKind == ``Parser.Term.doPatDecl then\n      let pattern := decl[0]\n      let doElem  := decl[2]\n      let optElse := decl[3]\n      if optElse.isNone then withFreshMacroScope do\n        let auxDo ← if isMutableLet doLetArrow then\n          `(do let%$doLetArrow __discr ← $doElem; let%$doLetArrow mut $pattern:term := __discr)\n        else\n          `(do let%$doLetArrow __discr ← $doElem; let%$doLetArrow $pattern:term := __discr)\n        doSeqToCode <| getDoSeqElems (getDoSeq auxDo) ++ doElems\n      else\n        let contSeq ← if isMutableLet doLetArrow then\n          let vars ← (← getPatternVarsEx pattern).mapM fun var => `(doElem| let mut $var := $var)\n          pure (vars ++ doElems.toArray)\n        else\n          pure doElems.toArray\n        let contSeq := mkDoSeq contSeq\n        let elseSeq := optElse[1]\n        let auxDo ← `(do let%$doLetArrow __discr ← $doElem; match%$doLetArrow __discr with | $pattern:term => $contSeq | _ => $elseSeq)\n        doSeqToCode <| getDoSeqElems (getDoSeq auxDo)\n    else\n      throwError \"unexpected kind of `do` declaration\""}, {"name": "checkLetArrowRHS", "content": "def checkLetArrowRHS (doElem : Syntax) : M Unit := do\n  let kind := doElem.getKind\n  if kind == ``Parser.Term.doLetArrow ||\n     kind == ``Parser.Term.doLet ||\n     kind == ``Parser.Term.doLetRec ||\n     kind == ``Parser.Term.doHave ||\n     kind == ``Parser.Term.doReassign ||\n     kind == ``Parser.Term.doReassignArrow then\n    throwErrorAt doElem \"invalid kind of value `{kind}` in an assignment\""}, {"name": "M", "content": "abbrev M := ReaderT Context TermElabM"}, {"name": "VarSet", "content": "abbrev VarSet := Std.TreeMap Name Syntax Name.cmp"}, {"name": "getDoSeq", "content": "private def getDoSeq (doStx : Syntax) : Syntax :=\n  doStx[1]\n\n  partial def doSeqToCode : List Syntax → M CodeBlock\n    | [] => do liftMacroM mkPureUnitAction\n    | doElem::doElems => withIncRecDepth <| withRef doElem do\n      checkSystem \"`do`-expander\"\n      match (← liftMacroM <| expandMacro? doElem) with\n      | some doElem => doSeqToCode (doElem::doElems)\n      | none =>\n      match (← liftMacroM <| expandDoIf? doElem) with\n      | some doElem => doSeqToCode (doElem::doElems)\n      | none =>\n      match (← liftMacroM <| expandDoLetExpr? doElem doElems) with\n      | some doElem => doSeqToCode [doElem]\n      | none =>\n        let (liftedDoElems, doElem) ← expandLiftMethod doElem\n        if !liftedDoElems.isEmpty then\n          doSeqToCode (liftedDoElems ++ [doElem] ++ doElems)\n        else\n          let ref := doElem\n          let k := doElem.getKind\n          if k == ``Parser.Term.doLet then\n            let vars ← getDoLetVars doElem\n            checkNotShadowingMutable vars\n            mkVarDeclCore vars doElem <$> withNewMutableVars vars (isMutableLet doElem) (doSeqToCode doElems)\n          else if k == ``Parser.Term.doHave then\n            let vars ← getDoHaveVars doElem\n            checkNotShadowingMutable vars\n            mkVarDeclCore vars doElem <$> (doSeqToCode doElems)\n          else if k == ``Parser.Term.doLetRec then\n            let vars ← getDoLetRecVars doElem\n            checkNotShadowingMutable vars\n            mkVarDeclCore vars doElem <$> (doSeqToCode doElems)\n          else if k == ``Parser.Term.doReassign then\n            let vars ← getDoReassignVars doElem\n            checkReassignable vars\n            let k ← doSeqToCode doElems\n            mkReassignCore vars doElem k\n          else if k == ``Parser.Term.doLetArrow then\n            doLetArrowToCode doElem doElems\n          else if k == ``Parser.Term.doLetElse then\n            doLetElseToCode doElem doElems\n          else if k == ``Parser.Term.doReassignArrow then\n            doReassignArrowToCode doElem doElems\n          else if k == ``Parser.Term.doIf then\n            doIfToCode doElem doElems\n          else if k == ``Parser.Term.doUnless then\n            doUnlessToCode doElem doElems\n          else if k == ``Parser.Term.doFor then withFreshMacroScope do\n            doForToCode doElem doElems\n          else if k == ``Parser.Term.doMatch then\n            doMatchToCode doElem doElems\n          else if k == ``Parser.Term.doMatchExpr then\n            doMatchExprToCode doElem doElems\n          else if k == ``Parser.Term.doTry then\n            doTryToCode doElem doElems\n          else if k == ``Parser.Term.doBreak then\n            ensureInsideFor\n            ensureEOS doElems\n            return mkBreak ref\n          else if k == ``Parser.Term.doContinue then\n            ensureInsideFor\n            ensureEOS doElems\n            return mkContinue ref\n          else if k == ``Parser.Term.doReturn then\n            doReturnToCode doElem doElems\n          else if k == ``Parser.Term.doDbgTrace then\n            return mkSeq doElem (← doSeqToCode doElems)\n          else if k == ``Parser.Term.doAssert then\n            return mkSeq doElem (← doSeqToCode doElems)\n          else if k == ``Parser.Term.doDebugAssert then\n            return mkSeq doElem (← doSeqToCode doElems)\n          else if k == ``Parser.Term.doNested then\n            let nestedDoSeq := doElem[1]\n            doSeqToCode (getDoSeqElems nestedDoSeq ++ doElems)\n          else if k == ``Parser.Term.doExpr then\n            let term := doElem[0]\n            if doElems.isEmpty then\n              return mkTerminalAction term\n            else\n              return mkSeq term (← doSeqToCode doElems)\n          else\n            throwError \"unexpected do-element of kind {doElem.getKind}:\\n{doElem}\""}, {"name": "getDoSeqElems", "content": "private def getDoSeqElems (doSeq : Syntax) : List Syntax :=\n  if doSeq.getKind == ``Parser.Term.doSeqBracketed then\n    doSeq[1].getArgs.toList.map fun arg => arg[0]\n  else if doSeq.getKind == ``Parser.Term.doSeqIndent then\n    doSeq[0].getArgs.toList.map fun arg => arg[0]\n  else\n    []\n\n  partial def doIfToCode (doIf : Syntax) (doElems : List Syntax) : M CodeBlock := do\n    let view := mkDoIfView doIf\n    let thenBranch ← doSeqToCode (getDoSeqElems view.thenBranch)\n    let elseBranch ← doSeqToCode (getDoSeqElems view.elseBranch)\n    let ite ← mkIte view.ref view.optIdent view.cond thenBranch elseBranch\n    concatWith ite doElems"}, {"name": "CodeBlock", "content": "structure CodeBlock where\n  code  : Code\n  uvars : VarSet := {} "}, {"name": "mkDoIfView", "content": "private def mkDoIfView (doIf : Syntax) : DoIfView := {\n  ref        := doIf\n  optIdent   := doIf[1][0]\n  cond       := doIf[1][1]\n  thenBranch := doIf[3]\n  elseBranch := doIf[5][1]\n}\n\n  partial def concatWith (c : CodeBlock) (doElems : List Syntax) : M CodeBlock :=\n    match doElems with\n    | [] => pure c\n    | nextDoElem :: _  => do\n      let k ← doSeqToCode doElems\n      let ref := nextDoElem\n      concat c ref none k"}, {"name": "concat", "content": "def concat (terminal : CodeBlock) (kRef : Syntax) (y? : Option Var) (k : CodeBlock) : TermElabM CodeBlock := do\n  unless hasTerminalAction terminal.code do\n    throwErrorAt kRef \"`do` element is unreachable\"\n  let (terminal, k) ← homogenize terminal k\n  let xs := varSetToArray k.uvars\n  let y ← match y? with | some y => pure y | none => `(y)\n  let ps := xs.map fun x => (x, true)\n  let ps := ps.push (y, false)\n  let jpDecl ← mkFreshJP ps k.code\n  let jp := jpDecl.name\n  let terminal ← liftMacroM <| convertTerminalActionIntoJmp terminal.code jp xs\n  return { code  := attachJP jpDecl terminal, uvars := k.uvars }"}, {"name": "homogenize", "content": "def homogenize (c₁ c₂ : CodeBlock) : TermElabM (CodeBlock × CodeBlock) := do\n  let ws := union c₁.uvars c₂.uvars\n  let c₁ ← extendUpdatedVars c₁ ws\n  let c₂ ← extendUpdatedVars c₂ ws\n  pure (c₁, c₂)"}, {"name": "extendUpdatedVars", "content": "partial def extendUpdatedVars (c : CodeBlock) (ws : VarSet) : TermElabM CodeBlock := do\n  if ws.any fun x _ => !c.uvars.contains x then\n    \n    pure { code := (← extendUpdatedVarsAux c.code ws), uvars := ws }\n  else\n    pure { c with uvars := ws }"}, {"name": "extendUpdatedVarsAux", "content": "partial def extendUpdatedVarsAux (c : Code) (ws : VarSet) : TermElabM Code :=\n  let rec update (c : Code) : TermElabM Code := do\n    match c with\n    | .joinpoint j ps b k    => return .joinpoint j ps (← update b) (← update k)\n    | .seq e k               => return .seq e (← update k)\n    | .match ref g ds t alts =>\n      if alts.any fun alt => alt.vars.any fun x => ws.contains x.getId then\n        \n        pullExitPoints c\n      else\n        return .match ref g ds t (← alts.mapM fun alt => do pure { alt with rhs := (← update alt.rhs) })\n    | .matchExpr ref «meta» d alts e =>\n      if alts.any fun alt => alt.vars.any fun x => ws.contains x.getId then\n        \n        pullExitPoints c\n      else\n        let alts ← alts.mapM fun alt => do pure { alt with rhs := (← update alt.rhs) }\n        let e ← update e\n        return .matchExpr ref «meta» d alts e\n    | .ite ref none o c t e => return .ite ref none o c (← update t) (← update e)\n    | .ite ref (some h) o cond t e =>\n      if ws.contains h.getId then\n        \n        pullExitPoints c\n      else\n        return Code.ite ref (some h) o cond (← update t) (← update e)\n    | .reassign xs stx k => return .reassign xs stx (← update k)\n    | .decl xs stx k => do\n      if xs.any fun x => ws.contains x.getId then\n        \n        pullExitPoints c\n      else\n        return .decl xs stx (← update k)\n    | c => return  c\n  update c"}, {"name": "AltExpr.vars", "content": "def AltExpr.vars (alt : AltExpr σ) : Array Var := Id.run do\n  let mut vars := #[]\n  if let some var := alt.var? then\n    vars := vars.push var\n  for pvar in alt.pvars do\n    match pvar with\n    | `(_) => pure ()\n    | _ => vars := vars.push pvar\n  return vars"}, {"name": "pullExitPoints", "content": "def pullExitPoints (c : Code) : TermElabM Code := do\n  if hasExitPoint c then\n    let (c, jpDecls) ← (pullExitPointsAux {} c).run #[]\n    return attachJPs jpDecls c\n  else\n    return c"}, {"name": "hasExitPoint", "content": "def hasExitPoint (c : Code) : Bool :=\n  hasExitPointPred c fun _ => true"}, {"name": "hasExitPointPred", "content": "partial def hasExitPointPred (c : Code) (p : Code → Bool) : Bool :=\n  let rec loop : Code → Bool\n    | .decl _ _ k             => loop k\n    | .reassign _ _ k         => loop k\n    | .joinpoint _ _ b k      => loop b || loop k\n    | .seq _ k                => loop k\n    | .ite _ _ _ _ t e        => loop t || loop e\n    | .match _ _ _ _ alts     => alts.any (loop ·.rhs)\n    | .matchExpr _ _ _ alts e => alts.any (loop ·.rhs) || loop e\n    | .jmp ..                 => false\n    | c                       => p c\n  loop c"}, {"name": "pullExitPointsAux", "content": "partial def pullExitPointsAux (rs : VarSet) (c : Code) : StateRefT (Array JPDecl) TermElabM Code := do\n  match c with\n  | .decl xs stx k         => return .decl xs stx (← pullExitPointsAux (eraseVars rs xs) k)\n  | .reassign xs stx k     => return .reassign xs stx (← pullExitPointsAux (insertVars rs xs) k)\n  | .joinpoint j ps b k    => return .joinpoint j ps (← pullExitPointsAux rs b) (← pullExitPointsAux rs k)\n  | .seq e k               => return .seq e (← pullExitPointsAux rs k)\n  | .ite ref x? o c t e    => return .ite ref x? o c (← pullExitPointsAux (eraseOptVar rs x?) t) (← pullExitPointsAux (eraseOptVar rs x?) e)\n  | .jmp ..                => return  c\n  | .break ref             => mkSimpleJmp ref rs (.break ref)\n  | .continue ref          => mkSimpleJmp ref rs (.continue ref)\n  | .return ref val        => mkJmp ref rs val (fun y => return .return ref y)\n  | .action e              =>\n    \n    mkAuxDeclFor e fun y =>\n      let ref := e\n      mkJmp ref rs y (fun yFresh => return .action (← ``(Pure.pure $yFresh)))\n  | .match ref g ds t alts =>\n    let alts ← alts.mapM fun alt => do pure { alt with rhs := (← pullExitPointsAux (eraseVars rs alt.vars) alt.rhs) }\n    return .match ref g ds t alts\n  | .matchExpr ref «meta» d alts e =>\n    let alts ← alts.mapM fun alt => do pure { alt with rhs := (← pullExitPointsAux (eraseVars rs alt.vars) alt.rhs) }\n    let e ← pullExitPointsAux rs e\n    return .matchExpr ref «meta» d alts e"}, {"name": "mkJmp", "content": "def mkJmp (ref : Syntax) (rs : VarSet) (val : Syntax) (mkJPBody : Syntax → MacroM Code) : StateRefT (Array JPDecl) TermElabM Code := do\n  let xs := varSetToArray rs\n  let args := xs.push val\n  let yFresh ← withRef ref `(y)\n  let ps := xs.map fun x => (x, true)\n  let ps := ps.push (yFresh, false)\n  let jpBody ← liftMacroM <| mkJPBody yFresh\n  let jp ← addFreshJP ps jpBody\n  return Code.jmp ref jp args"}, {"name": "varSetToArray", "content": "private def varSetToArray (s : VarSet) : Array Var :=\n  s.foldl (fun xs _ x => xs.push x) #[]"}, {"name": "addFreshJP", "content": "def addFreshJP (ps : Array (Var × Bool)) (body : Code) : StateRefT (Array JPDecl) TermElabM Name := do\n  let jp ← mkFreshJP ps body\n  modify fun (jps : Array JPDecl) => jps.push jp\n  pure jp.name"}, {"name": "mkFreshJP", "content": "def mkFreshJP (ps : Array (Var × Bool)) (body : Code) : TermElabM JPDecl := do\n  let ps ← if ps.isEmpty then\n    let y ← `(y)\n    pure #[(y.raw, false)]\n  else\n    pure ps\n  \n  \n  \n  let name ← mkFreshUserName `__do_jp\n  pure { name := name, params := ps, body := body }"}, {"name": "mkSimpleJmp", "content": "def mkSimpleJmp (ref : Syntax) (rs : VarSet) (c : Code) : StateRefT (Array JPDecl) TermElabM Code := do\n  let xs := varSetToArray rs\n  let jp ← addFreshJP (xs.map fun x => (x, true)) c\n  if xs.isEmpty then\n    let unit ← ``(Unit.unit)\n    return Code.jmp ref jp #[unit]\n  else\n    return Code.jmp ref jp xs"}, {"name": "eraseOptVar", "content": "def eraseOptVar (rs : VarSet) (x? : Option Var) : VarSet :=\n  match x? with\n  | none   => rs\n  | some x => rs.insert x.getId x"}, {"name": "mkAuxDeclFor", "content": "def mkAuxDeclFor {m} [Monad m] [MonadQuotation m] (e : Syntax) (mkCont : Syntax → m Code) : m Code := withRef e <| withFreshMacroScope do\n  let y ← `(y)\n  let doElem ← `(doElem| let y ← $e:term)\n  \n  let y ← `(ensure_expected_type% \"type mismatch, result value\" $y)\n  let k ← mkCont y\n  return .decl #[y] doElem k"}, {"name": "eraseVars", "content": "def eraseVars (rs : VarSet) (xs : Array Var) : VarSet :=\n  xs.foldl (·.erase ·.getId) rs"}, {"name": "insertVars", "content": "def insertVars (rs : VarSet) (xs : Array Var) : VarSet :=\n  xs.foldl (fun rs x => rs.insert x.getId x) rs"}, {"name": "attachJPs", "content": "def attachJPs (jpDecls : Array JPDecl) (k : Code) : Code :=\n  jpDecls.foldr attachJP k"}, {"name": "attachJP", "content": "def attachJP (jpDecl : JPDecl) (k : Code) : Code :=\n  Code.joinpoint jpDecl.name jpDecl.params jpDecl.body k"}, {"name": "union", "content": "private def union (s₁ s₂ : VarSet) : VarSet :=\n  s₁.foldl (·.insert ·) s₂"}, {"name": "convertTerminalActionIntoJmp", "content": "partial def convertTerminalActionIntoJmp (code : Code) (jp : Name) (xs : Array Var) : MacroM Code :=\n  let rec loop : Code → MacroM Code\n    | .decl xs stx k         => return .decl xs stx (← loop k)\n    | .reassign xs stx k     => return .reassign xs stx (← loop k)\n    | .joinpoint n ps b k    => return .joinpoint n ps (← loop b) (← loop k)\n    | .seq e k               => return .seq e (← loop k)\n    | .ite ref x? h c t e    => return .ite ref x? h c (← loop t) (← loop e)\n    | .action e              => mkAuxDeclFor e fun y =>\n      let ref := e\n      \n      let jmpArgs := xs.push y\n      return Code.jmp ref jp jmpArgs\n    | .match ref g ds t alts =>\n      return .match ref g ds t (← alts.mapM fun alt => do pure { alt with rhs := (← loop alt.rhs) })\n    | .matchExpr ref «meta» d alts e => do\n      let alts ← alts.mapM fun alt => do pure { alt with rhs := (← loop alt.rhs) }\n      let e ← loop e\n      return .matchExpr ref «meta» d alts e\n    | c => return c\n  loop code"}, {"name": "hasTerminalAction", "content": "def hasTerminalAction (c : Code) : Bool :=\n  hasExitPointPred c fun\n    | .action _ => true\n    | _ => false"}, {"name": "mkIte", "content": "def mkIte (ref : Syntax) (optIdent : Syntax) (cond : Syntax) (thenBranch : CodeBlock) (elseBranch : CodeBlock) : TermElabM CodeBlock := do\n  let x? := optIdent.getOptional?\n  let (thenBranch, elseBranch) ← homogenize thenBranch elseBranch\n  return {\n    code  := .ite ref x? optIdent cond thenBranch.code elseBranch.code,\n    uvars := thenBranch.uvars,\n  }"}, {"name": "ensureInsideFor", "content": "def ensureInsideFor : M Unit :=\n  unless (← read).insideFor do\n    throwError \"invalid `do` element, it must be inside `for`\"\n\n  partial def doForToCode (doFor : Syntax) (doElems : List Syntax) : M CodeBlock := do\n    let doForDecls := doFor[1].getSepArgs\n    if h : doForDecls.size > 1 then\n       \n      \n      let doForDecl := doForDecls[1]!\n      unless doForDecl[0].isNone do\n        throwErrorAt doForDecl[0] \"the proof annotation here has not been implemented yet\"\n      let y  := doForDecl[1]\n      let ys := doForDecl[3]\n      let doForDecls := doForDecls.eraseIdx 1\n      let body := doFor[3]\n      withFreshMacroScope do\n         \n        let toStreamApp ← withRef ys `(@toStream _ _ _ $ys)\n        let auxDo ←\n          `(do let mut s := $toStreamApp:term\n               for $doForDecls:doForDecl,* do\n                 match @Stream.next? _ _ _ s with\n                 | none => break\n                 | some ($y, s') =>\n                   s := s'\n                   do $body)\n        doSeqToCode (getDoSeqElems (getDoSeq auxDo) ++ doElems)\n    else withRef doFor do\n      let h?        := if doForDecls[0]![0].isNone then none else some doForDecls[0]![0][0]\n      let x         := doForDecls[0]![1]\n      withRef x <| checkNotShadowingMutable (← getPatternVarsEx x)\n      let xs        := doForDecls[0]![3]\n      let forElems  := getDoSeqElems doFor[3]\n      let forInBodyCodeBlock ← withFor (doSeqToCode forElems)\n      let ⟨uvars, forInBody⟩ ← mkForInBody x forInBodyCodeBlock\n      let ctx ← read\n      \n      \n      \n      let uvars := uvars.map fun v => ctx.mutableVars.getD v.getId v\n      let uvarsTuple ← liftMacroM do mkTuple uvars\n      let uvarsTuplePat ← liftMacroM do mkTuplePat uvars\n\n      if hasReturn forInBodyCodeBlock.code then\n        let forInBody ← liftMacroM <| destructTuple uvars (← `(r)) forInBody\n        let optType ← `(Option $((← read).returnType))\n        let forInTerm ← if let some h := h? then\n          annotate doFor\n            (← `(for_in'% $(xs) (MProdWithNames.mk (none : $optType) $uvarsTuple) fun $x $h (r : MProdWithNames $optType _) => let r := r.2; $forInBody))\n        else\n          annotate doFor\n            (← `(for_in% $(xs) (MProdWithNames.mk (none : $optType) $uvarsTuple) fun $x (r : MProdWithNames $optType _) => let r := r.2; $forInBody))\n        let auxDo ← `(do let r ← $forInTerm:term;\n                         $uvarsTuplePat:term := r.2;\n                         match r.1 with\n                         | none => Pure.pure (ensure_expected_type% \"type mismatch, `for`\" PUnit.unit)\n                         | some a => return ensure_expected_type% \"type mismatch, `for`\" a)\n        doSeqToCode (getDoSeqElems (getDoSeq auxDo) ++ doElems)\n      else\n        let forInBody ← liftMacroM <| destructTuple uvars (← `(r)) forInBody\n        let forInTerm ← if let some h := h? then\n          annotate doFor (← `(for_in'% $(xs) $uvarsTuple fun $x $h r => $forInBody))\n        else\n          annotate doFor (← `(for_in% $(xs) $uvarsTuple fun $x r => $forInBody))\n        if doElems.isEmpty then\n          let auxDo ← `(do let r ← $forInTerm:term;\n                           $uvarsTuplePat:term := r;\n                           Pure.pure (ensure_expected_type% \"type mismatch, `for`\" PUnit.unit))\n          doSeqToCode <| getDoSeqElems (getDoSeq auxDo)\n        else\n          let auxDo ← `(do let r ← $forInTerm:term; $uvarsTuplePat:term := r)\n          doSeqToCode <| getDoSeqElems (getDoSeq auxDo) ++ doElems"}, {"name": "destructTuple", "content": "private def destructTuple (uvars : Array Var) (x : Syntax) (body : Syntax) : MacroM Syntax := do\n  if uvars.size = 0 then\n    return body\n  else if h : uvars.size = 1 then\n    `(let $(uvars[0]):ident := WithName.erase $x; $body)\n  else\n    destruct uvars.toList x body\nwhere\n  destruct (as : List Var) (x : Syntax) (body : Syntax) : MacroM Syntax := do\n    match as with\n      | [a, b]  => `(let $a:ident := $x.1; let $b:ident := WithName.erase $x.2; $body)\n      | a :: as => withFreshMacroScope do\n        let rest ← destruct as (← `(x)) body\n        `(let $a:ident := $x.1; let x := $x.2; $rest)\n      | _ => unreachable!"}, {"name": "WithName.erase", "content": "abbrev WithName.erase {α : Type u} {name} (a : WithName α name) : α := a"}, {"name": "withFor", "content": "def withFor {α} (x : M α) : M α :=\n  withReader (fun ctx => { ctx with insideFor := true }) x"}, {"name": "checkNotShadowingMutable", "content": "def checkNotShadowingMutable (xs : Array Var) : M Unit := do\n  let throwInvalidShadowing (x : Name) : M Unit :=\n    throwError \"mutable variable `{x.simpMacroScopes}` cannot be shadowed\"\n  let ctx ← read\n  for x in xs do\n    if ctx.mutableVars.contains x.getId then\n      withRef x <| throwInvalidShadowing x.getId"}, {"name": "hasReturn", "content": "def hasReturn (c : Code) : Bool :=\n  hasExitPointPred c fun\n    | .return .. => true\n    | _ => false"}, {"name": "getPatternVarsEx", "content": "def getPatternVarsEx (pattern : Syntax) : TermElabM (Array Var) :=\n  getPatternVars pattern <|>\n  Quotation.getPatternVars pattern"}, {"name": "mkTuplePat", "content": "private def mkTuplePat (elems : Array Syntax) : MacroM Syntax := do\n  if elems.size = 0 then\n    mkUnit\n  else if h : elems.size = 1 then\n    return elems[0]\n  else\n    elems.extract 0 (elems.size - 1) |>.foldrM (init := elems.back!) fun elem tuple => do\n        ``(MProdWithNames.mk $elem $tuple)"}, {"name": "mkForInBody", "content": "def mkForInBody  (_ : Syntax) (forInBody : CodeBlock) : M ToForInTermResult := do\n  let ctx ← read\n  let uvars := forInBody.uvars\n  let uvars := varSetToArray uvars\n  let term ← liftMacroM <| ToTerm.run forInBody.code ctx.m ctx.returnType uvars (if hasReturn forInBody.code then ToTerm.Kind.forInWithReturn else ToTerm.Kind.forIn)\n  return ⟨uvars, term⟩"}, {"name": "mkBreak", "content": "def mkBreak (ref : Syntax) : CodeBlock :=\n  { code := .break ref }\n\n  partial def doUnlessToCode (doUnless : Syntax) (doElems : List Syntax) : M CodeBlock := withRef doUnless do\n    let cond  := doUnless[1]\n    let doSeq := doUnless[3]\n    let body ← doSeqToCode (getDoSeqElems doSeq)\n    let unlessCode ← liftMacroM <| mkUnless cond body\n    concatWith unlessCode doElems"}, {"name": "mkUnless", "content": "def mkUnless (cond : Syntax) (c : CodeBlock) : MacroM CodeBlock := do\n  let thenBranch ← mkPureUnitAction\n  return { c with code := .ite (← getRef) none mkNullNode cond thenBranch.code c.code }"}, {"name": "mkPureUnitAction", "content": "def mkPureUnitAction : MacroM CodeBlock := do\n  return mkTerminalAction (← mkPureUnit)"}, {"name": "mkPureUnit", "content": "private def mkPureUnit : MacroM Syntax :=\n  ``(pure PUnit.unit)"}, {"name": "mkTerminalAction", "content": "def mkTerminalAction (action : Syntax) : CodeBlock :=\n  { code := .action action }\n\n  partial def doTryToCode (doTry : Syntax) (doElems: List Syntax) : M CodeBlock := do\n    let tryCode ← doSeqToCode (getDoSeqElems doTry[1])\n    let optFinally := doTry[3]\n    let catches ← doTry[2].getArgs.mapM fun catchStx : Syntax => do\n      if catchStx.getKind == ``Parser.Term.doCatch then\n        let x       := catchStx[1]\n        if x.isIdent then\n          withRef x <| checkNotShadowingMutable #[x]\n        let optType := catchStx[2]\n        let c ← doSeqToCode (getDoSeqElems catchStx[4])\n        return { x := x, optType := optType, codeBlock := c : Catch }\n      else if catchStx.getKind == ``Parser.Term.doCatchMatch then\n        let matchAlts := catchStx[1]\n        let x ← `(ex)\n        let auxDo ← `(do match ex with $matchAlts)\n        let c ← doSeqToCode (getDoSeqElems (getDoSeq auxDo))\n        return { x := x, codeBlock := c, optType := mkNullNode : Catch }\n      else\n        throwError \"unexpected kind of `catch`\"\n    let finallyCode? ← if optFinally.isNone then pure none else some <$> doSeqToCode (getDoSeqElems optFinally[0][1])\n    if catches.isEmpty && finallyCode?.isNone then\n      throwError \"invalid `try`, it must have a `catch` or `finally`\"\n    let ctx ← read\n    let ws    := getTryCatchUpdatedVars tryCode catches finallyCode?\n    let uvars := varSetToArray ws\n    let a     := tryCatchPred tryCode catches finallyCode? hasTerminalAction\n    let r     := tryCatchPred tryCode catches finallyCode? hasReturn\n    let bc    := tryCatchPred tryCode catches finallyCode? hasBreakContinue\n    let toTerm (codeBlock : CodeBlock) : M Syntax := do\n      let codeBlock ← liftM $ extendUpdatedVars codeBlock ws\n      liftMacroM <| ToTerm.mkNestedTerm codeBlock.code ctx.m ctx.returnType uvars a r bc\n    let term ← toTerm tryCode\n    let term ← catches.foldlM (init := term) fun term «catch» => do\n      let catchTerm ← toTerm «catch».codeBlock\n      if catch.optType.isNone then\n        annotate doTry (← ``(MonadExcept.tryCatch $term (fun $(«catch».x):ident => $catchTerm)))\n      else\n        let type := «catch».optType[1]\n        annotate doTry (← ``(tryCatchThe $type $term (fun $(«catch».x):ident => $catchTerm)))\n    let term ← match finallyCode? with\n      | none             => pure term\n      | some finallyCode => withRef optFinally do\n        unless finallyCode.uvars.isEmpty do\n          throwError \"`finally` currently does not support reassignments\"\n        if hasBreakContinueReturn finallyCode.code then\n          throwError \"`finally` currently does `return`, `break`, nor `continue`\"\n        let finallyTerm ← liftMacroM <| ToTerm.run finallyCode.code ctx.m ctx.returnType {} ToTerm.Kind.regular\n        annotate doTry (← ``(tryFinally $term $finallyTerm))\n    let doElemsNew ← liftMacroM <| ToTerm.matchNestedTermResult term uvars a r bc\n    doSeqToCode (doElemsNew ++ doElems)"}, {"name": "tryCatchPred", "content": "def tryCatchPred (tryCode : CodeBlock) (catches : Array Catch) (finallyCode? : Option CodeBlock) (p : Code → Bool) : Bool :=\n  p tryCode.code ||\n  catches.any (fun «catch» => p «catch».codeBlock.code) ||\n  match finallyCode? with\n  | none => false\n  | some finallyCode => p finallyCode.code"}, {"name": "Catch", "content": "structure Catch where\n  x         : Syntax\n  optType   : Syntax\n  codeBlock : CodeBlock"}, {"name": "mkNestedTerm", "content": "def mkNestedTerm (code : Code) (m : Syntax) (returnType : Syntax) (uvars : Array Var) (a r bc : Bool) : MacroM Syntax := do\n  ToTerm.run code m returnType uvars (mkNestedKind a r bc)"}, {"name": "mkNestedKind", "content": "def mkNestedKind (a r bc : Bool) : Kind :=\n  match a, r, bc with\n  | true,  false, false => .regular\n  | false, true,  false => .regular\n  | false, false, true  => .nestedBC\n  | true,  true,  false => .nestedPR\n  | true,  false, true  => .nestedSBC\n  | false, true,  true  => .nestedSBC\n  | true,  true,  true  => .nestedPRBC\n  | false, false, false => unreachable!"}, {"name": "getTryCatchUpdatedVars", "content": "def getTryCatchUpdatedVars (tryCode : CodeBlock) (catches : Array Catch) (finallyCode? : Option CodeBlock) : VarSet :=\n  let ws := tryCode.uvars\n  let ws := catches.foldl (init := ws) fun ws alt => union alt.codeBlock.uvars ws\n  let ws := match finallyCode? with\n    | none   => ws\n    | some c => union c.uvars ws\n  ws"}, {"name": "hasBreakContinueReturn", "content": "def hasBreakContinueReturn (c : Code) : Bool :=\n  hasExitPointPred c fun\n    | .break _    => true\n    | .continue _ => true\n    | .return _ _ => true\n    | _ => false"}, {"name": "matchNestedTermResult", "content": "def matchNestedTermResult (term : Syntax) (uvars : Array Var) (a r bc : Bool) : MacroM (List Syntax) := do\n  let toDoElems (auxDo : Syntax) : List Syntax := getDoSeqElems (getDoSeq auxDo)\n  let u ← mkTuplePat uvars\n  match a, r, bc with\n  | true, false, false =>\n    if uvars.isEmpty then\n      return toDoElems (← `(do $term:term))\n    else\n      return toDoElems (← `(do let r ← $term:term; $u:term := r.2; pure r.1))\n  | false, true, false =>\n    if uvars.isEmpty then\n      return toDoElems (← `(do let r ← $term:term; return r))\n    else\n      return toDoElems (← `(do let r ← $term:term; $u:term := r.2; return r.1))\n  | false, false, true => toDoElems <$>\n    `(do let r ← $term:term;\n         match r with\n         | .break u => $u:term := u; break\n         | .continue u => $u:term := u; continue)\n  | true, true, false => toDoElems <$>\n    `(do let r ← $term:term;\n         match r with\n         | .pure a u => $u:term := u; pure a\n         | .return b u => $u:term := u; return b)\n  | true, false, true => toDoElems <$>\n    `(do let r ← $term:term;\n         match r with\n         | .pureReturn a u => $u:term := u; pure a\n         | .break u => $u:term := u; break\n         | .continue u => $u:term := u; continue)\n  | false, true, true => toDoElems <$>\n    `(do let r ← $term:term;\n         match r with\n         | .pureReturn a u => $u:term := u; return a\n         | .break u => $u:term := u; break\n         | .continue u => $u:term := u; continue)\n  | true, true, true => toDoElems <$>\n    `(do let r ← $term:term;\n         match r with\n         | .pure a u => $u:term := u; pure a\n         | .return a u => $u:term := u; return a\n         | .break u => $u:term := u; break\n         | .continue u => $u:term := u; continue)\n  | false, false, false => unreachable!"}, {"name": "hasBreakContinue", "content": "def hasBreakContinue (c : Code) : Bool :=\n  hasExitPointPred c fun\n    | .break _    => true\n    | .continue _ => true\n    | _ => false"}, {"name": "mkContinue", "content": "def mkContinue (ref : Syntax) : CodeBlock :=\n  { code := .continue ref }\n\n  partial def doLetElseToCode (doLetElse : Syntax) (doElems : List Syntax) : M CodeBlock := do\n    \n    let pattern := doLetElse[2]\n    let val     := doLetElse[4]\n    let elseSeq := doLetElse[6]\n    let contSeq ← if isMutableLet doLetElse then\n      let vars ← (← getPatternVarsEx pattern).mapM fun var => `(doElem| let mut $var := $var)\n      pure (vars ++ doElems.toArray)\n    else\n      pure doElems.toArray\n    let contSeq := mkDoSeq contSeq\n    let auxDo ← `(do match $val:term with | $pattern:term => $contSeq | _ => $elseSeq)\n    doSeqToCode <| getDoSeqElems (getDoSeq auxDo)"}, {"name": "isMutableLet", "content": "def isMutableLet (doElem : Syntax) : Bool :=\n  let kind := doElem.getKind\n  (kind == ``doLetArrow || kind == ``doLet || kind == ``doLetElse)\n  &&\n  !doElem[1].isNone"}, {"name": "mkDoSeq", "content": "def mkDoSeq (doElems : Array Syntax) : Syntax :=\n  mkNode `Lean.Parser.Term.doSeqIndent #[mkNullNode <| doElems.map fun doElem => mkNullNode #[doElem, mkNullNode]]"}, {"name": "mkSeq", "content": "def mkSeq (action : Syntax) (c : CodeBlock) : CodeBlock :=\n  { c with code := .seq action c.code }"}, {"name": "withNewMutableVars", "content": "def withNewMutableVars {α} (newVars : Array Var) (mutable : Bool) (x : M α) : M α :=\n  withReader (fun ctx => if mutable then { ctx with mutableVars := insertVars ctx.mutableVars newVars } else ctx) x"}, {"name": "mkReassignCore", "content": "def mkReassignCore (xs : Array Var) (stx : Syntax) (c : CodeBlock) : TermElabM CodeBlock := do\n  let us := c.uvars\n  let ws := insertVars us xs\n  \n  \n  let code ← if xs.any fun x => !us.contains x.getId then extendUpdatedVarsAux c.code ws else pure c.code\n  pure { code := .reassign xs stx code, uvars := ws }"}, {"name": "getDoReassignVars", "content": "def getDoReassignVars (doReassign : Syntax) : TermElabM (Array Var) := do\n  let arg := doReassign[0]\n  if arg.getKind == ``Parser.Term.letIdDecl then\n    return getLetIdDeclVars arg\n  else if arg.getKind == ``Parser.Term.letPatDecl then\n    getLetPatDeclVars arg\n  else\n    throwError \"unexpected kind of reassignment\""}, {"name": "getLetIdDeclVars", "content": "def getLetIdDeclVars (letIdDecl : Syntax) : Array Var :=\n  assert! letIdDecl.isOfKind ``Parser.Term.letIdDecl\n  \n  \n  getLetIdVars letIdDecl[0]"}, {"name": "getLetIdVars", "content": "def getLetIdVars (letId : Syntax) : Array Var :=\n  assert! letId.isOfKind ``Parser.Term.letId\n  \n  if letId[0].isIdent then\n    #[letId[0]]\n  else if letId[0].isOfKind hygieneInfoKind then\n    #[HygieneInfo.mkIdent letId[0] `this (canonical := true)]\n  else\n    #[]"}, {"name": "getLetPatDeclVars", "content": "def getLetPatDeclVars (letPatDecl : Syntax) : TermElabM (Array Var) := do\n  \n  let pattern := letPatDecl[0]\n  getPatternVarsEx pattern"}, {"name": "ensureEOS", "content": "def ensureEOS (doElems : List Syntax) : M Unit :=\n  unless doElems.isEmpty do\n    throwError \"must be last element in a `do` sequence\""}, {"name": "mkVarDeclCore", "content": "def mkVarDeclCore (xs : Array Var) (stx : Syntax) (c : CodeBlock) : CodeBlock := {\n  code := Code.decl xs stx c.code,\n  uvars := eraseVars c.uvars xs\n}\n\n  partial def doMatchToCode (doMatch : Syntax) (doElems: List Syntax) : M CodeBlock := do\n    let ref       := doMatch\n    let genParam  := doMatch[1]\n    let optMotive := doMatch[2]\n    let discrs    := doMatch[3]\n    let matchAlts := doMatch[5][0].getArgs \n    let matchAlts ← matchAlts.foldlM (init := #[]) fun result matchAlt => return result ++ (← liftMacroM <| expandMatchAlt matchAlt)\n    let alts ←  matchAlts.mapM fun matchAlt => do\n      let patterns := matchAlt[1][0]\n      let vars ← getPatternsVarsEx patterns.getSepArgs\n      withRef patterns <| checkNotShadowingMutable vars\n      let rhs  := matchAlt[3]\n      let rhs ← doSeqToCode (getDoSeqElems rhs)\n      pure { ref := matchAlt, vars := vars, patterns := patterns, rhs := rhs : Alt CodeBlock }\n    let matchCode ← mkMatch ref genParam discrs optMotive alts\n    concatWith matchCode doElems"}, {"name": "mkMatch", "content": "def mkMatch (ref : Syntax) (genParam : Syntax) (discrs : Syntax) (optMotive : Syntax) (alts : Array (Alt CodeBlock)) : TermElabM CodeBlock := do\n  \n  let ws := alts.foldl (union · ·.rhs.uvars) {}\n  let alts ← alts.mapM fun alt => do\n    let rhs ← extendUpdatedVars alt.rhs ws\n    return { ref := alt.ref, vars := alt.vars, patterns := alt.patterns, rhs := rhs.code : Alt Code }\n  return { code := .match ref genParam discrs optMotive alts, uvars := ws }"}, {"name": "getPatternsVarsEx", "content": "def getPatternsVarsEx (patterns : Array Syntax) : TermElabM (Array Var) :=\n  getPatternsVars patterns <|>\n  Quotation.getPatternsVars patterns"}, {"name": "doReturnToCode", "content": "def doReturnToCode (doReturn : Syntax) (doElems: List Syntax) : M CodeBlock := withRef doReturn do\n  ensureEOS doElems\n  let argOpt := doReturn[1]\n  let arg ← if argOpt.isNone then liftMacroM mkUnit else pure argOpt[0]\n  return mkReturn (← getRef) arg"}, {"name": "mkReturn", "content": "def mkReturn (ref : Syntax) (val : Syntax) : CodeBlock :=\n  { code := .return ref val }"}, {"name": "getDoHaveVars", "content": "def getDoHaveVars (doHave : Syntax) : TermElabM (Array Var) :=\n  \n  getLetDeclVars doHave[1]"}, {"name": "getLetDeclVars", "content": "def getLetDeclVars (letDecl : Syntax) : TermElabM (Array Var) := do\n  \n  let arg := letDecl[0]\n  if arg.getKind == ``Parser.Term.letIdDecl then\n    return getLetIdDeclVars arg\n  else if arg.getKind == ``Parser.Term.letPatDecl then\n    getLetPatDeclVars arg\n  else if arg.getKind == ``Parser.Term.letEqnsDecl then\n    return getLetEqnsDeclVars arg\n  else\n    throwError \"unexpected kind of let declaration\""}, {"name": "getLetEqnsDeclVars", "content": "def getLetEqnsDeclVars (letEqnsDecl : Syntax) : Array Var :=\n  assert! letEqnsDecl.isOfKind ``Parser.Term.letEqnsDecl\n  \n  \n  getLetIdVars letEqnsDecl[0]"}, {"name": "getDoLetRecVars", "content": "def getDoLetRecVars (doLetRec : Syntax) : TermElabM (Array Var) := do\n  \n  let letRecDecls := doLetRec[1][0].getSepArgs\n  let letDecls := letRecDecls.map fun p => p[2]\n  let mut allVars := #[]\n  for letDecl in letDecls do\n    let vars ← getLetDeclVars letDecl\n    allVars := allVars ++ vars\n  return allVars\n\n  partial def doReassignArrowToCode (doReassignArrow : Syntax) (doElems : List Syntax) : M CodeBlock := do\n    let decl := doReassignArrow[0]\n    if decl.getKind == ``Parser.Term.doIdDecl then\n      let doElem := decl[3]\n      let y      := decl[0]\n      let auxDo ← `(do let r ← $doElem; $y:ident := r)\n      doSeqToCode <| getDoSeqElems (getDoSeq auxDo) ++ doElems\n    else if decl.getKind == ``Parser.Term.doPatDecl then\n      let pattern := decl[0]\n      let doElem  := decl[2]\n      let optElse := decl[3]\n      if optElse.isNone then withFreshMacroScope do\n        let auxDo ← `(do let __discr ← $doElem; $pattern:term := __discr)\n        doSeqToCode <| getDoSeqElems (getDoSeq auxDo) ++ doElems\n      else\n        throwError \"reassignment with `|` (i.e., \\\"else clause\\\") is not currently supported\"\n    else\n      throwError \"unexpected kind of `do` reassignment\""}, {"name": "expandLiftMethod", "content": "def expandLiftMethod (doElem : Syntax) : M (List Syntax × Syntax) := do\n  if !hasLiftMethod doElem then\n    return ([], doElem)\n  else\n    let baseId ← withFreshMacroScope (MonadQuotation.addMacroScope `__do_lift)\n    let (doElem, doElemsNew) ← (expandLiftMethodAux baseId false false doElem).run []\n    return (doElemsNew, doElem)"}, {"name": "hasLiftMethod", "content": "private partial def hasLiftMethod : Syntax → Bool\n  | Syntax.node _ k args =>\n    if liftMethodDelimiter k then false\n    \n    \n    else if k == ``Parser.Term.liftMethod then true\n    \n    else if k == ``termDepIfThenElse || k == ``termIfThenElse then args.size >= 2 && hasLiftMethod args[1]!\n    else args.any hasLiftMethod\n  | _ => false"}, {"name": "liftMethodDelimiter", "content": "private def liftMethodDelimiter (k : SyntaxNodeKind) : Bool :=\n  k == ``Parser.Term.do ||\n  k == ``Parser.Term.doSeqIndent ||\n  k == ``Parser.Term.doSeqBracketed ||\n  k == ``Parser.Term.termReturn ||\n  k == ``Parser.Term.termUnless ||\n  k == ``Parser.Term.termTry ||\n  k == ``Parser.Term.termFor"}, {"name": "expandLiftMethodAux", "content": "private partial def expandLiftMethodAux (inQuot : Bool) (inBinder : Bool) : Syntax → StateT (List Syntax) M Syntax\n  | stx@(Syntax.node i k args) =>\n    if k == choiceKind then do\n      \n      let alts ← stx.getArgs.mapM (expandLiftMethodAux inQuot inBinder · |>.run [])\n      let (_, lifts) := alts[0]!\n      unless alts.all (·.2 == lifts) do\n        throwErrorAt stx \"cannot lift `(<- ...)` over inconsistent syntax variants, consider lifting out the binding manually\"\n      modify (· ++ lifts)\n      return .node i k (alts.map (·.1))\n    else if liftMethodDelimiter k then\n      return stx\n    \n    else if h : args.size >= 2 ∧ (k == ``termDepIfThenElse || k == ``termIfThenElse) then do\n      let inAntiquot := stx.isAntiquot && !stx.isEscapedAntiquot\n      let arg1 ← expandLiftMethodAux (inQuot && !inAntiquot || stx.isQuot) inBinder args[1]\n      let args := args.set! 1 arg1\n      return Syntax.node i k args\n    else if k == ``Parser.Term.liftMethod && !inQuot then withFreshMacroScope do\n      if inBinder then\n        throwErrorAt stx \"cannot lift `(<- ...)` over a binder, this error usually happens when you are trying to lift a method nested in a `fun`, `let`, or `match`-alternative, and it can often be fixed by adding a missing `do`\"\n      let term := args[1]!\n      let term ← expandLiftMethodAux inQuot inBinder term\n      \n      let id ← mkIdentFromRef (.num baseId (← get).length)\n      let auxDoElem : Syntax ← `(doElem| let $id:ident ← $term:term)\n      modify fun s => s ++ [auxDoElem]\n      return id\n    else do\n      let inAntiquot := stx.isAntiquot && !stx.isEscapedAntiquot\n      let inBinder   := inBinder || (!inQuot && liftMethodForbiddenBinder stx)\n      let args ← args.mapM (expandLiftMethodAux (inQuot && !inAntiquot || stx.isQuot) inBinder)\n      return Syntax.node i k args\n  | stx => return stx"}, {"name": "run", "content": "def run (doStx : Syntax) (m : Syntax) (returnType : Syntax) : TermElabM CodeBlock :=\n  (doSeqToCode <| getDoSeqElems <| getDoSeq doStx).run { ref := doStx, m, returnType }"}, {"name": "liftMethodForbiddenBinder", "content": "private def liftMethodForbiddenBinder (stx : Syntax) : Bool :=\n  let k := stx.getKind\n  \n  if k == ``Parser.Term.fun || k == ``Parser.Term.matchAlts ||\n     k == ``Parser.Term.doLetRec || k == ``Parser.Term.letrec then\n     \n    true\n  \n  else if k == ``Parser.Term.let then\n    letDeclHasBinders stx[1]\n  else if k == ``Parser.Term.doLet then\n    letDeclHasBinders stx[2]\n  else if k == ``Parser.Term.doLetArrow then\n    letDeclArgHasBinders stx[2]\n  else\n    false"}, {"name": "letDeclHasBinders", "content": "private def letDeclHasBinders (letDecl : Syntax) : Bool :=\n  letDeclArgHasBinders letDecl[0]"}, {"name": "letDeclArgHasBinders", "content": "private def letDeclArgHasBinders (letDeclArg : Syntax) : Bool :=\n  let k := letDeclArg.getKind\n  if k == ``Parser.Term.letPatDecl then\n    false\n  else if k == ``Parser.Term.letEqnsDecl then\n    true\n  else if k == ``Parser.Term.letIdDecl then\n    \n    let binders := letDeclArg[1]\n    binders.getNumArgs > 0\n  else\n    false"}, {"name": "checkReassignable", "content": "def checkReassignable (xs : Array Var) : M Unit := do\n  let throwInvalidReassignment (x : Name) : M Unit :=\n    throwError \"`{x.simpMacroScopes}` cannot be mutated, only variables declared using `let mut` can be mutated. If you did not intend to mutate but define `{x.simpMacroScopes}`, consider using `let {x.simpMacroScopes}` instead\"\n  let ctx ← read\n  for x in xs do\n    unless ctx.mutableVars.contains x.getId do\n      throwInvalidReassignment x.getId\n\n  partial def doMatchExprToCode (doMatchExpr : Syntax) (doElems: List Syntax) : M CodeBlock := do\n    let ref       := doMatchExpr\n    let «meta»    := doMatchExpr[1].isNone\n    let discr     := doMatchExpr[2]\n    let alts      := doMatchExpr[4][0].getArgs \n    let alts ← alts.mapM fun alt => do\n      let pat      := alt[1]\n      let var?     := if pat[0].isNone then none else some pat[0][0]\n      let funName  := pat[1]\n      let pvars    := pat[2].getArgs\n      let rhs      := alt[3]\n      let rhs ← doSeqToCode (getDoSeqElems rhs)\n      pure { ref, var?, funName, pvars, rhs }\n    let elseBranch ← doSeqToCode (getDoSeqElems doMatchExpr[4][1][3])\n    let matchCode ← mkMatchExpr ref «meta» discr alts elseBranch\n    concatWith matchCode doElems"}, {"name": "mkMatchExpr", "content": "def mkMatchExpr (ref : Syntax) («meta» : Bool) (discr : Syntax) (alts : Array (AltExpr CodeBlock)) (elseBranch : CodeBlock) : TermElabM CodeBlock := do\n  \n  let ws := alts.foldl (union · ·.rhs.uvars) {}\n  let ws := union ws elseBranch.uvars\n  let alts ← alts.mapM fun alt => do\n    let rhs ← extendUpdatedVars alt.rhs ws\n    return { alt with rhs := rhs.code : AltExpr Code }\n  let elseBranch ← extendUpdatedVars elseBranch ws\n  return { code := .matchExpr ref «meta» discr alts elseBranch.code, uvars := ws }"}, {"name": "getDoLetVars", "content": "def getDoLetVars (doLet : Syntax) : TermElabM (Array Var) :=\n  \n  getLetDeclVars doLet[2]"}, {"name": "isDoExpr?", "content": "def isDoExpr? (doElem : Syntax) : Option Syntax :=\n  if doElem.getKind == ``Parser.Term.doExpr then\n    some doElem[0]\n  else\n    none"}, {"name": "reassignToTerm", "content": "def reassignToTerm (reassign : Syntax) (k : Syntax) : MacroM Syntax := withRef reassign <| withFreshMacroScope do\n  match reassign with\n  | `(doElem| $x:ident := $rhs) => `(let $x:ident := ensure_type_of% $x $(quote \"invalid reassignment, value\") $rhs; $k)\n  | `(doElem| $e:term  := $rhs) => `(let $e:term  := ensure_type_of% $e $(quote \"invalid reassignment, value\") $rhs; $k)\n  | _ =>\n    \n    Macro.throwErrorAt reassign \"unexpected kind of `do` reassignment\""}, {"name": "actionTerminalToTerm", "content": "def actionTerminalToTerm (action : Syntax) : M Syntax := withRef action <| withFreshMacroScope do\n  let ctx ← read\n  let u ← mkUVarTuple\n  match ctx.kind with\n  | .regular         => if ctx.uvars.isEmpty then pure action else ``(Bind.bind $action fun y => Pure.pure (MProdWithNames.mk y $u))\n  | .forIn           => ``(Bind.bind $action fun (_ : PUnit) => Pure.pure (ForInStep.yield $u))\n  | .forInWithReturn => ``(Bind.bind $action fun (_ : PUnit) => Pure.pure (ForInStep.yield (MProdWithNames.mk none $u)))\n  | .nestedBC        => unreachable!\n  | .nestedPR        => ``(Bind.bind $action fun y => (Pure.pure (DoResultPR.«pure» y $u)))\n  | .nestedSBC       => ``(Bind.bind $action fun y => (Pure.pure (DoResultSBC.«pureReturn» y $u)))\n  | .nestedPRBC      => ``(Bind.bind $action fun y => (Pure.pure (DoResultPRBC.«pure» y $u)))"}, {"name": "breakToTerm", "content": "def breakToTerm : M Syntax := do\n  let ctx ← read\n  let u ← mkUVarTuple\n  match ctx.kind with\n  | .regular         => unreachable!\n  | .forIn           => ``(Pure.pure (ForInStep.done $u))\n  | .forInWithReturn => ``(Pure.pure (ForInStep.done (MProdWithNames.mk none $u)))\n  | .nestedBC        => ``(Pure.pure (DoResultBC.«break» $u))\n  | .nestedPR        => unreachable!\n  | .nestedSBC       => ``(Pure.pure (DoResultSBC.«break» $u))\n  | .nestedPRBC      => ``(Pure.pure (DoResultPRBC.«break» $u))"}, {"name": "DivM", "content": "inductive DivM (α : Type u) where\n  | res (x : α)\n  | div"}, {"name": "NonDetT.run", "content": "def NonDetT.run {α : Type u} :  NonDetT m α -> m α\n  | .pure x => Pure.pure x\n  | .vis x f => liftM x >>= (fun x => (f x).run)\n  | @NonDetT.pickCont _ _ _ p _ f =>\n    match Findable.find (p := p) () with\n    | none => CCPOBot.compBot\n    | some x =>  (f x).run\n  | .repeatCont init f cont =>\n    forIn Lean.Loop.mk init (fun _ x => (f x).run) >>= (fun x => (cont x).run)"}, {"name": "ExtractNonDet.ForIn_list", "content": "instance ExtractNonDet.ForIn_list {xs : List α} {init : β} {f : α → β → NonDetT m (ForInStep β)}\n  (_ : ∀ a b, ExtractNonDet findable (f a b)) :\n  ExtractNonDet findable (forIn xs init f) :="}, {"name": "MonadNonDet", "content": "class MonadNonDet (m : Type u → Type v) where\n  pick : (τ : Type u) → [Inhabited τ] → m τ\n  \n  pickSuchThat : (τ : Type u) → (p : τ → Prop) → [Findable p] → m τ\n  assume : (as : Prop) → [Decidable as] → m PUnit.{u+1}\n  \n  rep {α : Type u} : α → (α → m (ForInStep α)) → m α"}, {"name": "NonDetT", "content": "inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α"}, {"name": "Findable", "content": "class Findable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_none : (find ()).isNone -> ∀ x, ¬ p x\n  find_some_p : find () = some x -> p x"}, {"name": "find", "content": "def find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode"}, {"name": "findNat", "content": "def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0"}, {"name": "NonDetT.extract", "content": "@[inline]\ndef NonDetT.extract {α : Type u} (s : NonDetT m α) (ex : ExtractNonDet Findable s := by admit /- proof elided -/\n) : m α :=\n  NonDetT.extractGen Findable.find s"}, {"name": "NonDetT.extractGen", "content": "@[simp, inline]\ndef NonDetT.extractGen {findable : {τ : Type u} -> (τ -> Prop) -> Type u} {α : Type u}\n  (findOf : ∀ {τ : Type u} (p : τ -> Prop), findable p -> Unit -> Option τ)\n  : (s : NonDetT m α) -> (ex : ExtractNonDet findable s := by admit /- proof elided -/\n  ) -> m α\n  | .pure x, _ => Pure.pure x\n  | .vis x f, .vis _ _ _ => liftM x >>= (fun x => extractGen findOf (f x))\n  | .pickCont _ p f, .pickSuchThat _ _ _ _ =>\n    match findOf p ‹_› () with\n    | none => CCPOBot.compBot\n    | some x =>  extractGen findOf (f x)\n  | .pickCont _ p f, .assume _ _ _ =>\n    if p .unit then\n      extractGen findOf (f .unit)\n    else CCPOBot.compBot"}, {"name": "CCPOBot", "content": "class CCPOBot (m : Type u -> Type v)  where\n  compBot {α} : m α"}, {"name": "DivM.run", "content": "def DivM.run {α : Type u} [Inhabited α] : DivM α -> α\n  | DivM.res x => x\n  | DivM.div => default"}], "lib_lemmas": [{"name": "Array.append_right_inj", "module": "Init.Data.Array.Lemmas"}, {"name": "Array.push_eq_append", "module": "Init.Data.Array.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Encoding", "content": "structure Encoding where\n  cnt: Nat\n  c: Char\n  deriving Inhabited"}, {"name": "get_cnt_sum", "content": "def get_cnt_sum (l: List Encoding) :=\n  match l with\n  | List.nil => 0\n  | List.cons x xs => x.cnt + get_cnt_sum xs"}, {"name": "is_valid_run_sequence", "content": "@[reducible]\ndef is_valid_run_sequence (encoded_str: Array Encoding) :=\n    forall i, ( h: i < encoded_str.size ) -> (encoded_str[i]'h).cnt > 0\n\nmethod decodeStr' (encoded_str: Array Encoding)\n   return (res: Array Char)\n   require is_valid_run_sequence encoded_str\n   ensures (res.size = get_cnt_sum encoded_str.toList)\n     do\n       let mut decoded := Array.replicate 0 'x'\n       let mut i := 0\n       while i < encoded_str.size\n          invariant 0 <= i ∧ i <= encoded_str.size\n          invariant decoded.size = get_cnt_sum (encoded_str.extract 0 i).toList\n          done_with i = encoded_str.size\n          do\n            let elem := encoded_str[i]!\n            let elem_decoded := Array.replicate elem.cnt elem.c\n            decoded :=  decoded ++ elem_decoded\n            i := i + 1\n       return decoded\n\nprove_correct decodeStr' by\n  loom_solve\n  · simp[*] at *\n    have : decoded.size = get_cnt_sum (List.take i encoded_str.toList) := by admit /- proof elided -/"}, {"name": "decodeStrLean", "content": "@[grind]\ndef decodeStrLean (encoded_str: Array Encoding) : Array Char :=\n  let mp := Array.map (fun e => Array.replicate e.cnt e.c) encoded_str\n  mp.flatten"}], "used_local_lemmas": [{"name": "decodeStrLean_append", "content": "lemma decodeStrLean_append : forall arr1 arr2,\n  decodeStrLean (arr1 ++ arr2) = ( decodeStrLean arr1 ) ++ ( decodeStrLean arr2 )"}, {"name": "array_extract_split", "content": "lemma array_extract_split (arr : Array α) (i j : Nat) :\n    i < j → j ≤ arr.size →\n    arr.extract 0 j = (arr.extract 0 i) ++ (arr.extract i j)"}], "local_ctx": "import Auto\n\nimport Loom.MonadAlgebras.NonDetT.Extract\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.DoNames'\n\nimport CaseStudies.Velvet.Std\n\nimport CaseStudies.TestingUtil\n\nopen PartialCorrectness DemonicChoice\n\nsection RunLengthEncoding\n\nstructure Encoding where\n  cnt: Nat\n  c: Char\n  deriving Inhabited\n\ndef get_cnt_sum (l: List Encoding) :=\n  match l with\n  | List.nil => 0\n  | List.cons x xs => x.cnt + get_cnt_sum xs\n\n@[reducible]\ndef is_valid_run_sequence (encoded_str: Array Encoding) :=\n    forall i, ( h: i < encoded_str.size ) -> (encoded_str[i]'h).cnt > 0\n\nmethod decodeStr' (encoded_str: Array Encoding)\n   return (res: Array Char)\n   require is_valid_run_sequence encoded_str\n   ensures (res.size = get_cnt_sum encoded_str.toList)\n     do\n       let mut decoded := Array.replicate 0 'x'\n       let mut i := 0\n       while i < encoded_str.size\n          invariant 0 <= i ∧ i <= encoded_str.size\n          invariant decoded.size = get_cnt_sum (encoded_str.extract 0 i).toList\n          done_with i = encoded_str.size\n          do\n            let elem := encoded_str[i]!\n            let elem_decoded := Array.replicate elem.cnt elem.c\n            decoded :=  decoded ++ elem_decoded\n            i := i + 1\n       return decoded\n\nprove_correct decodeStr' by\n  loom_solve\n  · simp[*] at *\n    have : decoded.size = get_cnt_sum (List.take i encoded_str.toList) := by admit /- proof elided -/\n\n@[grind]\ndef decodeStrLean (encoded_str: Array Encoding) : Array Char :=\n  let mp := Array.map (fun e => Array.replicate e.cnt e.c) encoded_str\n  mp.flatten", "target_theorem": "lemma array_extract_split_i_j_k (arr : Array α) (i j k: Nat) :\n    i < j -> j < k -> k ≤ arr.size →\n    arr.extract i k = (arr.extract i j) ++ (arr.extract j k) :=", "ground_truth_proof": ":= by\n  -- 1. Introduce hypotheses into the context\n  intro h_lt h_le\n  simp[*]\n  grind\n\nprove_correct encodeStr by\n  loom_solve\n  · rw [array_extract_split str i j (by assumption) (by assumption)]\n    rw [Array.push_eq_append]\n    rw [decodeStrLean_append]\n    have : decodeStrLean encoding = str.extract 0 i := by trivial\n    rw [this]\n    refine (Array.append_right_inj (str.extract 0 i)).mpr ?_\n    repeat (unfold decodeStrLean ; simp_all)\n    apply Array.ext\n    grind\n    intros l hl hl2\n    have inv : ∀ (k : ℕ), i ≤ k → k < j → str[k]! = str[i] := by trivial\n    have inv' := inv (i+l)\n    grind", "nesting_depth": 17, "transitive_dep_count": 292, "subset_aristotle": false, "category": "Framework"}
{"id": 381, "thm_name": "TotalCorrectness.repeat_inv", "thm_stmt": "lemma repeat_inv (f : Unit -> β -> m (ForInStep β))\n  (inv : ForInStep β -> l) (measure : β -> Nat)\n  init :\n   (∀ b, triple (inv (.yield b)) (f () b) (fun | .yield b' => inv (.yield b') ⊓ ⌜ measure b' < measure b ⌝ | .done b' => inv (.done b'))) ->\n   triple (inv (.yield init)) (Loop.forIn.loop f init) (fun b => inv (.done b))", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/WP/Basic.lean", "imports": ["import Loom.MonadAlgebras.Instances.Gen", "import Loom.MonadAlgebras.Defs", "import Loom.MonadAlgebras.Instances.ReaderT", "import Loom.MonadAlgebras.Instances.ExceptT", "import Loom.MonadAlgebras.Instances.StateT", "import Loom.MonadAlgebras.Instances.Basic"], "used_lib_defs": [{"name": "ForInStep", "module": "Init.Core"}, {"name": "Lean.Order.CCPO", "module": "Init.Internal.Order.Basic"}, {"name": "Lean.Order.MonoBind", "module": "Init.Internal.Order.Basic"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "inline", "module": "Init.Core"}, {"name": "ForInStep.yield", "module": "Init.Core"}, {"name": "Lean.Loop", "module": "Init.While"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "Cont.monotone", "content": "@[simp]\ndef Cont.monotone {t : Type v} {α : Type u} [Preorder t] (wp : Cont t α) :=\n  ∀ (f f' : α -> t), (∀ a, f a ≤ f' a) → wp f ≤ wp f'"}, {"name": "Cont", "content": "abbrev Cont (t : Type v) (α : Type u) := (α -> t) -> t"}, {"name": "triple", "content": "notation \"{\" P \"}\" c \"{\" v \",\" Q \"}\" => triple P c (fun v => Q)"}], "lib_lemmas": [{"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "Cont.monotone_lift", "content": "lemma Cont.monotone_lift {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteLattice l] [MAlgOrdered m l] :\n  ∀ {α : Type u} (x : m α), MAlg.lift x |>.monotone"}, {"name": "triple_bind", "content": "lemma triple_bind {β} (pre : l) (x : m α) (cut : α -> l)\n  (f : α -> m β) (post : β -> l) :\n  triple pre x cut ->\n  (∀ y, triple (cut y) (f y) post) ->\n  triple pre (x >>= f) post"}, {"name": "triple_pure", "content": "lemma triple_pure (pre : l) (x : α) (post : α -> l) :\n  triple pre (pure (f := m) x) post <-> pre ≤ (post x)"}], "used_local_defs": [{"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}, {"name": "Loop.forIn.loop", "content": "@[specialize, inline]\ndef Loop.forIn.loop {m : Type u -> Type v} [Monad m] [∀ α, CCPO (m α)] [MonoBind m] (f : Unit → β → m (ForInStep β)) (b : β) : m β := do\n    match ← f () b with\n      | ForInStep.done b  => pure b\n      | ForInStep.yield b => loop f b\n  partial_fixpoint"}, {"name": "Loop.forIn", "content": "@[inline]\ndef Loop.forIn {β : Type u} [Monad m] [∀ α, CCPO (m α)] [MonoBind m]\n  (_ : Lean.Loop) (init : β) (f : Unit → β → m (ForInStep β)) : m β :=\n  Loop.forIn.loop f init"}], "used_local_lemmas": [{"name": "wp_pure", "content": "lemma wp_pure (x : α) (post : α -> l) : wp (m := m) (pure x) post = post x"}, {"name": "wp_bind", "content": "lemma wp_bind {β} (x : m α) (f : α -> m β) (post : β -> l) :\n    wp (x >>= f) post = wp x (fun x => wp (f x) post)"}, {"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}], "local_ctx": "import Loom.MonadAlgebras.Defs\n\nimport Loom.MonadAlgebras.Instances.Basic\n\nimport Loom.MonadAlgebras.Instances.ExceptT\n\nimport Loom.MonadAlgebras.Instances.StateT\n\nimport Loom.MonadAlgebras.Instances.ReaderT\n\nimport Loom.MonadAlgebras.Instances.Gen\n\nvariable {m : Type u -> Type v} [Monad m] [LawfulMonad m] {α : Type u} {l : Type u}\n\nsection\n\nvariable [CompleteLattice l]\n\nsection\n\nvariable [mprop : MAlgOrdered m l]\n\ndef wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post\n\ndef triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post\n\nend\n\nvariable [MAlgOrdered m l]\n\nend\n\nsection\n\nvariable [CompleteLattice l] [MAlgOrdered m l]\n\nnoncomputable\n\nend\n\nsection Determinism\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nend Determinism\n\nsection Loops\n\nopen Lean.Order\n\n@[specialize, inline]\ndef Loop.forIn.loop {m : Type u -> Type v} [Monad m] [∀ α, CCPO (m α)] [MonoBind m] (f : Unit → β → m (ForInStep β)) (b : β) : m β := do\n    match ← f () b with\n      | ForInStep.done b  => pure b\n      | ForInStep.yield b => loop f b\n  partial_fixpoint\n\n@[inline]\ndef Loop.forIn {β : Type u} [Monad m] [∀ α, CCPO (m α)] [MonoBind m]\n  (_ : Lean.Loop) (init : β) (f : Unit → β → m (ForInStep β)) : m β :=\n  Loop.forIn.loop f init\n\nvariable [inst: _root_.CompleteLattice l] [MAlgOrdered m l]\n\nnamespace PartialCorrectness\n\nvariable [∀ α, CCPO (m α)] [MonoBind m] [MAlgPartial m]\n\nend PartialCorrectness\n\nnamespace TotalCorrectness\n\nvariable [∀ α, CCPO (m α)] [MonoBind m]", "target_theorem": "lemma repeat_inv (f : Unit -> β -> m (ForInStep β))\n  (inv : ForInStep β -> l) (measure : β -> Nat)\n  init :\n   (∀ b, triple (inv (.yield b)) (f () b) (fun | .yield b' => inv (.yield b') ⊓ ⌜ measure b' < measure b ⌝ | .done b' => inv (.done b'))) ->\n   triple (inv (.yield init)) (Loop.forIn.loop f init) (fun b => inv (.done b)) :=", "ground_truth_proof": ":= by\n  intro hstep\n  have induc (C : β → Prop) (a : β) (h : ∀ x, (∀ y, measure y < measure x → C y) → C x): C a := by\n    have lem: ∀ n: Nat, ∀ x, measure x ≤ n →  C x := by\n      intro n\n      induction n with\n      | zero =>\n        intro x hx\n        exact h x (fun y => by omega)\n      | succ m ih =>\n        intro x hx\n        by_cases neq: measure x ≤ m\n        { exact ih x neq }\n        have eq: measure x = m + 1 := by omega\n        exact h x (fun y hy => ih y (by omega))\n    exact lem (measure a) a (by simp)\n  apply induc\n    (fun ini => triple (inv (.yield ini)) (Loop.forIn.loop f ( ini)) (fun b => inv (.done b))) init\n  intro b ih; unfold Loop.forIn.loop; simp [triple, wp_bind]; apply le_trans\n  apply hstep; apply wp_cons; rintro (_|_)\n  { simp [wp_pure] }\n  rename_i a\n  match ForInStep.yield a with\n  | .yield b' =>\n    simp; intro m_lt\n    have ihb := ih b' m_lt\n    simp [triple] at ihb\n    exact ihb\n  | .done b' =>\n    simp\n    rw [wp_pure b' (fun s: β => inv (ForInStep.done s))]", "nesting_depth": 4, "transitive_dep_count": 31, "subset_aristotle": true, "category": "Framework"}
{"id": 382, "thm_name": "SpMSpV_correct_triple", "thm_stmt": "theorem SpMSpV_correct_triple\n  (out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int)\n  (n: ℕ):\n  triple\n    (∀ i < spm.size, (∀ j < spm[i]!.size, spm[i]!.ind[j]! < n) ∧ (∀ j < spv.size, spv.ind[j]! < n))\n    (SpMSpV out spm spv)\n    fun ⟨_, outNew⟩ =>\n      outNew.size = spm.size ∧ ∀ i < spm.size, outNew[i]! = ∑ idx ∈ Finset.range n, spm[i]![idx]! * spv[idx]!", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetExamples/SpMSpV_Example.lean", "imports": ["import Auto", "import CaseStudies.Velvet.Std", "import Mathlib.Algebra.BigOperators.Intervals", "import Loom.MonadAlgebras.WP.DoNames'", "import Loom.MonadAlgebras.WP.Basic", "import Loom.MonadAlgebras.WP.Tactic", "import Mathlib.Algebra.Ring.Int.Defs", "import Lean", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Array.replicate", "module": "Init.Data.Array.Basic"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Lean.Name", "module": "Init.Prelude"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.find?", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "syntax \"method\" ident leafny_binder* \"return\" \"(\" ident \":\" ", "content": "syntax \"method\" ident leafny_binder* \"return\" \"(\" ident \":\" term \")\"\n  (require_caluse )*\n  (ensures_caluse)* \"do\" doSeq\n  Termination.suffix : command\n\nsyntax \"ensures\" termBeforeReqEnsDo : ensures_caluse\n\nsyntax \"while_some\" term \":|\" termBeforeDo \"do\" doSeq : doElem\n\nsyntax \"while_some\" term \":|\" term\n  (invariantClause)+\n  (doneWith)?\n  \"do\" doSeq : doElem\n\nsyntax \"let\" term \":|\" term : doElem\n\nsyntax \"done_with\" termBeforeDo : doneWith\n\nsyntax \"invariant\" termBeforeDo linebreak : invariantClause\n\nsyntax \"while\" term\n  (invariantClause)*\n  (doneWith)?\n  (decreasingTerm)?\n  \"do\" doSeq : doElem\n\nsyntax \"(mut\" ident \":\" term \")\" : leafny_binder"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x:term :| $t) => `(doElem| let $x:term <- pickSuchThat _ (fun $x => type_with_name_prefix `choice $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t do $seq:doSeq) => do\n    let decr <- withRef (<- getRef) `(decreasing none)\n    let invs <- withRef (<- getRef) `(invariants [])\n    `(doElem|\n      for _ in Lean.Loop.mk do\n        $invs:term\n        onDoneGadget (with_name_prefix `done ¬$t:term)\n        $decr:term\n        if $t then\n          $seq:doSeq\n        else break)\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      let invd_some ← match inv_done with\n      | some invd_some => withRef invd_some ``($invd_some)\n      | none => ``(¬$t:term)\n      match measure with\n      | some measure_some =>\n        let decr <- withRef measure_some `(decreasing type_with_name_prefix `decreasing $measure_some)\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        let decr <- withRef (<- getRef) `(decreasing none)\n        let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n  | `(doElem| while_some $x:ident :| $t do $seq:doSeq) =>\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      `(doElem|\n        while ∃ $x:ident, $t do\n          let $x :| $t\n          $[$seq:doElem]*)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| while_some $x:ident :| $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]? do\n                $seq:doSeq) => do\n    let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n    let invd_some ← match inv_done with\n    | some invd_some => withRef invd_some ``($invd_some)\n    | none => ``(¬$t:term)\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      let decr <- withRef (<- getRef) `(decreasing none)\n      `(doElem|\n        for _ in Lean.Loop.mk do\n          $invs:term\n          onDoneGadget (with_name_prefix `done $invd_some:term)\n          $decr:term\n          if ∃ $x:ident, $t then\n            let $x :| $t\n            $[$seq:doElem]*\n          else break)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| for $x:ident in $t\n            $[invariant $inv:term\n            ]*\n            do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      match seq with\n      | `(doSeq| $[$seq:doElem]*)\n      | `(doSeq| $[$seq:doElem;]*)\n      | `(doSeq| { $[$seq:doElem]* }) =>\n        `(doElem|\n          for $x:ident in $t do\n            $invs:term\n            $[$seq:doElem]*)\n      | _ => Lean.Macro.throwError \"for expects a sequence of do-elements\""}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|loom_solver) =>\n    `(tactic|(\n      try simp at *\n      try aesop))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let balance := mkIdent `balance_name\n      let balanceType <- `(term| Bal)\n      let inv : Array Term <- inv.mapM fun (inv : Term) => withRef inv ``(fun ($(balance):ident : $balanceType)=> with_name_prefix `inv $inv)\n      let invd_some <- match inv_done with\n      | some invd_some => withRef invd_some ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done $invd_some)\n      | none => ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done ¬$t:term)\n      match measure with\n      | some measure_some =>\n        let measure_some ← withRef measure_some ``(type_with_name_prefix `decreasing ($measure_some:term))\n        do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget ($measure_some:term)\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget none\n            if $t then\n              $seq:doSeq\n            else break)"}, {"name": "macro_rules", "content": "macro_rules\n| `(doElem|balance_set $t) => do\n  let balId := mkIdent `balance\n  `(doElem|do\n    $balId:ident := $t\n    set $balId:ident\n    $balId:ident ← get)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem|$id:ident[$idx:term] := $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (fun _ => $val))\n  | `(doElem|$id:ident[$idx:term] += $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (· + $val))"}, {"name": "WithName", "content": "abbrev WithName (α : Sort u) (name : Lean.Name := default) := α"}, {"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "triple", "content": "notation \"{\" P \"}\" c \"{\" v \",\" Q \"}\" => triple P c (fun v => Q)"}], "lib_lemmas": [{"name": "List.find?_eq_none", "module": "Init.Data.List.Find"}, {"name": "List.mem_iff_get", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "List.find?_eq_some_iff_getElem", "module": "Init.Data.List.Nat.Find"}, {"name": "em", "module": "Mathlib.Logic.Basic"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "le_iff_eq_or_lt", "module": "Mathlib.Order.Basic"}, {"name": "le_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_of_lt_of_le", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_or_gt_of_ne", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "lt_iff_le_not_ge", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}], "used_local_defs": [{"name": "SpV", "content": "structure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!"}, {"name": "spv_dot", "content": "def spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=\n  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then\n    0\n  else\n    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)\n    else\n      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n        spv_dot spv1 spv2 (pnt1 + 1) pnt2\n      else\n        spv_dot spv1 spv2 pnt1 (pnt2 + 1)\n  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)\n\n\nmethod SpVSpV\n  (mut out: Array Int)\n  (spv1: SpV Int)\n  (spv2: SpV Int) return (u: Unit)\n  ensures out.size = 1\n  ensures out[0]! = spv_dot spv1 spv2 0 0\n  do\n    out := Array.replicate 1 0\n    let mut pnt1 := 0\n    let mut pnt2 := 0\n    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size\n    invariant out.size = 1\n    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size\n    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0\n    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size\n    do\n      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!\n        pnt1 := pnt1 + 1\n        pnt2 := pnt2 + 1\n      else\n        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n          pnt1 := pnt1 + 1\n        else\n          pnt2 := pnt2 + 1\n    return\n\n\nmethod SpMSpV\n  (mut out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int) return (u: Unit)\n  ensures out.size = spm.size\n  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0\n  do\n    out := Array.replicate spm.size 0\n    let mut spmInd := Array.replicate spm.size 0\n    let mut spvInd := Array.replicate spm.size 0\n    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size\n    invariant spvInd.size = spm.size\n    invariant spmInd.size = spm.size\n    invariant out.size = spm.size\n    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size\n    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size\n    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0\n    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size\n    do\n      let ind_m := spmInd[i]!\n      let ind_v := spvInd[i]!\n      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then\n        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!\n        spmInd[i] += 1\n        spvInd[i] += 1\n      else\n        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then\n          spmInd[i] += 1\n        else\n          spvInd[i] += 1\n    return"}], "used_local_lemmas": [{"name": "getValSpV_eq", "content": "theorem getValSpV_eq (spv: SpV Int) (j: ℕ) (h_ind: j < spv.size): spv[spv.ind[j]!] = (spv.val)[j]!"}, {"name": "getValSpV_empty", "content": "theorem getValSpV_empty (spv: SpV Int) (j: ℕ) (h_empty: ∀ i < spv.size, spv.ind[i]! ≠ j): spv[j] = 0"}, {"name": "spv_dot_pure_gen", "content": "theorem spv_dot_pure_gen (spv1: SpV Int) (spv2: SpV Int) (n pnt1 pnt2: ℕ)\n  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n)\n  (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):\n  spv_dot spv1 spv2 pnt1 pnt2 =\n    ∑ i ∈ Finset.range n,\n      if max\n        (if spv1.size ≤ pnt1 then n else spv1.ind[pnt1]!)\n        (if spv2.size ≤ pnt2 then n else spv2.ind[pnt2]!) ≤ i then\n        spv1[i] * spv2[i]\n      else\n        0"}, {"name": "spv_dot_pure", "content": "theorem spv_dot_pure (spv1 spv2: SpV Int) (n: ℕ)\n  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n) (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):\n    spv_dot spv1 spv2 0 0 = ∑ i ∈ Finset.range n, spv1[i] * spv2[i]"}], "local_ctx": "import Auto\n\nimport Lean\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Algebra.Ring.Int.Defs\n\nimport Loom.MonadAlgebras.NonDetT.Extract\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.DoNames'\n\nimport CaseStudies.Velvet.Std\n\nsection SpMV\n\nstructure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!\n\ndef spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=\n  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then\n    0\n  else\n    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)\n    else\n      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n        spv_dot spv1 spv2 (pnt1 + 1) pnt2\n      else\n        spv_dot spv1 spv2 pnt1 (pnt2 + 1)\n  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)\n\n\nmethod SpVSpV\n  (mut out: Array Int)\n  (spv1: SpV Int)\n  (spv2: SpV Int) return (u: Unit)\n  ensures out.size = 1\n  ensures out[0]! = spv_dot spv1 spv2 0 0\n  do\n    out := Array.replicate 1 0\n    let mut pnt1 := 0\n    let mut pnt2 := 0\n    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size\n    invariant out.size = 1\n    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size\n    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0\n    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size\n    do\n      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!\n        pnt1 := pnt1 + 1\n        pnt2 := pnt2 + 1\n      else\n        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n          pnt1 := pnt1 + 1\n        else\n          pnt2 := pnt2 + 1\n    return\n\n\nmethod SpMSpV\n  (mut out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int) return (u: Unit)\n  ensures out.size = spm.size\n  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0\n  do\n    out := Array.replicate spm.size 0\n    let mut spmInd := Array.replicate spm.size 0\n    let mut spvInd := Array.replicate spm.size 0\n    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size\n    invariant spvInd.size = spm.size\n    invariant spmInd.size = spm.size\n    invariant out.size = spm.size\n    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size\n    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size\n    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0\n    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size\n    do\n      let ind_m := spmInd[i]!\n      let ind_v := spvInd[i]!\n      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then\n        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!\n        spmInd[i] += 1\n        spvInd[i] += 1\n      else\n        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then\n          spmInd[i] += 1\n        else\n          spvInd[i] += 1\n    return", "target_theorem": "theorem SpMSpV_correct_triple\n  (out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int)\n  (n: ℕ):\n  triple\n    (∀ i < spm.size, (∀ j < spm[i]!.size, spm[i]!.ind[j]! < n) ∧ (∀ j < spv.size, spv.ind[j]! < n))\n    (SpMSpV out spm spv)\n    fun ⟨_, outNew⟩ =>\n      outNew.size = spm.size ∧ ∀ i < spm.size, outNew[i]! = ∑ idx ∈ Finset.range n, spm[i]![idx]! * spv[idx]! :=", "ground_truth_proof": ":= by\n        simp [triple]\n        intro inb\n        apply wp_cons (SpMSpV out spm spv)\n          fun ⟨_, outNew⟩ =>\n            (∀ i < spm.size, outNew[i]! = spv_dot spm[i]! spv 0 0) ∧ outNew.size = spm.size\n        { rintro outNew; simp\n          intro sum_eq sz_eq\n          simp [sz_eq]\n          intro i ib\n          simp [←spv_dot_pure spm[i]! spv n (inb i ib).left (inb i ib).right]\n          exact sum_eq i ib }\n        simp\n        have triple_true := SpMSpV_correct out spm spv\n        simp [triple] at triple_true\n        simp [WithName] at triple_true\n        exact triple_true", "nesting_depth": 4, "transitive_dep_count": 37, "subset_aristotle": false, "category": "Framework"}
{"id": 383, "thm_name": "PartialCorrectness.repeat_inv", "thm_stmt": "lemma repeat_inv (f : Unit -> β -> m (ForInStep β))\n  (inv : ForInStep β -> l)\n  init :\n   (∀ b, triple (inv (.yield b)) (f () b) (inv)) ->\n   triple (inv (.yield init)) (Loop.forIn.loop f init) (fun b => inv (.done b))", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/WP/Basic.lean", "imports": ["import Loom.MonadAlgebras.Instances.Gen", "import Loom.MonadAlgebras.Defs", "import Loom.MonadAlgebras.Instances.ReaderT", "import Loom.MonadAlgebras.Instances.ExceptT", "import Loom.MonadAlgebras.Instances.StateT", "import Loom.MonadAlgebras.Instances.Basic"], "used_lib_defs": [{"name": "ForInStep", "module": "Init.Core"}, {"name": "Lean.Order.CCPO", "module": "Init.Internal.Order.Basic"}, {"name": "Lean.Order.MonoBind", "module": "Init.Internal.Order.Basic"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "inline", "module": "Init.Core"}, {"name": "ForInStep.yield", "module": "Init.Core"}, {"name": "Lean.Loop", "module": "Init.While"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Lean.Order.admissible", "module": "Init.Internal.Order.Basic"}, {"name": "Lean.Order.admissible_pi_apply", "module": "Init.Internal.Order.Basic"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "Lean.Order.CCPO.csup", "module": "Init.Internal.Order.Basic"}, {"name": "Lean.Order.chain", "module": "Init.Internal.Order.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}], "used_repo_defs": [{"name": "MAlgPartial", "content": "class MAlgPartial (m : Type u -> Type v) [Monad m] [∀ α, Lean.Order.CCPO (m α)]\n  [CompleteLattice l] [MAlgOrdered m l] where\n  csup_lift {α : Type u} (xc : Set (m α)) (post : α -> l) :\n    Lean.Order.chain xc ->\n    ⨅ x ∈ xc, MAlg.lift x post <= MAlg.lift (Lean.Order.CCPO.csup xc) post"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "Cont.monotone", "content": "@[simp]\ndef Cont.monotone {t : Type v} {α : Type u} [Preorder t] (wp : Cont t α) :=\n  ∀ (f f' : α -> t), (∀ a, f a ≤ f' a) → wp f ≤ wp f'"}, {"name": "Cont", "content": "abbrev Cont (t : Type v) (α : Type u) := (α -> t) -> t"}, {"name": "triple", "content": "notation \"{\" P \"}\" c \"{\" v \",\" Q \"}\" => triple P c (fun v => Q)"}], "lib_lemmas": [{"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans'", "module": "Mathlib.Order.Basic"}], "repo_lemmas": [{"name": "Cont.monotone_lift", "content": "lemma Cont.monotone_lift {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteLattice l] [MAlgOrdered m l] :\n  ∀ {α : Type u} (x : m α), MAlg.lift x |>.monotone"}], "used_local_defs": [{"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}, {"name": "Loop.forIn.loop", "content": "@[specialize, inline]\ndef Loop.forIn.loop {m : Type u -> Type v} [Monad m] [∀ α, CCPO (m α)] [MonoBind m] (f : Unit → β → m (ForInStep β)) (b : β) : m β := do\n    match ← f () b with\n      | ForInStep.done b  => pure b\n      | ForInStep.yield b => loop f b\n  partial_fixpoint"}, {"name": "Loop.forIn", "content": "@[inline]\ndef Loop.forIn {β : Type u} [Monad m] [∀ α, CCPO (m α)] [MonoBind m]\n  (_ : Lean.Loop) (init : β) (f : Unit → β → m (ForInStep β)) : m β :=\n  Loop.forIn.loop f init"}], "used_local_lemmas": [{"name": "wp_pure", "content": "lemma wp_pure (x : α) (post : α -> l) : wp (m := m) (pure x) post = post x"}, {"name": "wp_bind", "content": "lemma wp_bind {β} (x : m α) (f : α -> m β) (post : β -> l) :\n    wp (x >>= f) post = wp x (fun x => wp (f x) post)"}, {"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}, {"name": "PartialCorrectness.wp_csup", "content": "omit [MonoBind m] [LawfulMonad m] in\nlemma wp_csup (xc : Set (m α)) (post : α -> l) :\n  Lean.Order.chain xc ->\n  ⨅ c ∈ xc, wp c post ≤ wp (Lean.Order.CCPO.csup xc) post"}], "local_ctx": "import Loom.MonadAlgebras.Defs\n\nimport Loom.MonadAlgebras.Instances.Basic\n\nimport Loom.MonadAlgebras.Instances.ExceptT\n\nimport Loom.MonadAlgebras.Instances.StateT\n\nimport Loom.MonadAlgebras.Instances.ReaderT\n\nimport Loom.MonadAlgebras.Instances.Gen\n\nvariable {m : Type u -> Type v} [Monad m] [LawfulMonad m] {α : Type u} {l : Type u}\n\nsection\n\nvariable [CompleteLattice l]\n\nsection\n\nvariable [mprop : MAlgOrdered m l]\n\ndef wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post\n\ndef triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post\n\nend\n\nvariable [MAlgOrdered m l]\n\nend\n\nsection\n\nvariable [CompleteLattice l] [MAlgOrdered m l]\n\nnoncomputable\n\nend\n\nsection Determinism\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nend Determinism\n\nsection Loops\n\nopen Lean.Order\n\n@[specialize, inline]\ndef Loop.forIn.loop {m : Type u -> Type v} [Monad m] [∀ α, CCPO (m α)] [MonoBind m] (f : Unit → β → m (ForInStep β)) (b : β) : m β := do\n    match ← f () b with\n      | ForInStep.done b  => pure b\n      | ForInStep.yield b => loop f b\n  partial_fixpoint\n\n@[inline]\ndef Loop.forIn {β : Type u} [Monad m] [∀ α, CCPO (m α)] [MonoBind m]\n  (_ : Lean.Loop) (init : β) (f : Unit → β → m (ForInStep β)) : m β :=\n  Loop.forIn.loop f init\n\nvariable [inst: _root_.CompleteLattice l] [MAlgOrdered m l]\n\nnamespace PartialCorrectness\n\nvariable [∀ α, CCPO (m α)] [MonoBind m] [MAlgPartial m]", "target_theorem": "lemma repeat_inv (f : Unit -> β -> m (ForInStep β))\n  (inv : ForInStep β -> l)\n  init :\n   (∀ b, triple (inv (.yield b)) (f () b) (inv)) ->\n   triple (inv (.yield init)) (Loop.forIn.loop f init) (fun b => inv (.done b)) :=", "ground_truth_proof": ":= by\n  intro hstep\n  revert init\n  apply Loop.forIn.loop.fixpoint_induct (f := f) (motive :=\n    fun loop => ∀ init, triple (inv (.yield init)) (loop init) (fun b =>inv (.done b)))\n  { apply Lean.Order.admissible_pi_apply\n      (P := fun init loop => triple (inv (.yield init)) (loop) (fun b =>inv (.done b)))\n    simp [admissible, triple]; intro init loops cl h\n    apply le_trans'; apply wp_csup; solve_by_elim\n    simp; solve_by_elim }\n  intro loop ih init; simp [triple, wp_bind]; apply le_trans; apply hstep\n  apply wp_cons; rintro (_|_); simp [wp_pure]\n  apply ih", "nesting_depth": 4, "transitive_dep_count": 38, "subset_aristotle": false, "category": "Framework"}
{"id": 384, "thm_name": "SpVSpV_correct_triple", "thm_stmt": "theorem SpVSpV_correct_triple (out: Array Int) (spv1 spv2: SpV Int) (n: ℕ):\n  triple\n    ((∀ i < spv1.size, spv1.ind[i]! < n) ∧ (∀ i < spv2.size, spv2.ind[i]! < n))\n    (SpVSpV out spv1 spv2)\n    fun ⟨_, outNew⟩ =>\n      outNew[0]! = ∑ i ∈ Finset.range n, spv1[i]! * spv2[i]!", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetExamples/SpMSpV_Example.lean", "imports": ["import Auto", "import CaseStudies.Velvet.Std", "import Mathlib.Algebra.BigOperators.Intervals", "import Loom.MonadAlgebras.WP.DoNames'", "import Loom.MonadAlgebras.WP.Basic", "import Loom.MonadAlgebras.WP.Tactic", "import Mathlib.Algebra.Ring.Int.Defs", "import Lean", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Array.replicate", "module": "Init.Data.Array.Basic"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Lean.Name", "module": "Init.Prelude"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.find?", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "syntax \"method\" ident leafny_binder* \"return\" \"(\" ident \":\" ", "content": "syntax \"method\" ident leafny_binder* \"return\" \"(\" ident \":\" term \")\"\n  (require_caluse )*\n  (ensures_caluse)* \"do\" doSeq\n  Termination.suffix : command\n\nsyntax \"ensures\" termBeforeReqEnsDo : ensures_caluse\n\nsyntax \"while_some\" term \":|\" termBeforeDo \"do\" doSeq : doElem\n\nsyntax \"while_some\" term \":|\" term\n  (invariantClause)+\n  (doneWith)?\n  \"do\" doSeq : doElem\n\nsyntax \"let\" term \":|\" term : doElem\n\nsyntax \"done_with\" termBeforeDo : doneWith\n\nsyntax \"invariant\" termBeforeDo linebreak : invariantClause\n\nsyntax \"while\" term\n  (invariantClause)*\n  (doneWith)?\n  (decreasingTerm)?\n  \"do\" doSeq : doElem\n\nsyntax \"(mut\" ident \":\" term \")\" : leafny_binder"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| let $x:term :| $t) => `(doElem| let $x:term <- pickSuchThat _ (fun $x => type_with_name_prefix `choice $t))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t do $seq:doSeq) => do\n    let decr <- withRef (<- getRef) `(decreasing none)\n    let invs <- withRef (<- getRef) `(invariants [])\n    `(doElem|\n      for _ in Lean.Loop.mk do\n        $invs:term\n        onDoneGadget (with_name_prefix `done ¬$t:term)\n        $decr:term\n        if $t then\n          $seq:doSeq\n        else break)\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      let invd_some ← match inv_done with\n      | some invd_some => withRef invd_some ``($invd_some)\n      | none => ``(¬$t:term)\n      match measure with\n      | some measure_some =>\n        let decr <- withRef measure_some `(decreasing type_with_name_prefix `decreasing $measure_some)\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        let decr <- withRef (<- getRef) `(decreasing none)\n        let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            $invs:term\n            onDoneGadget (with_name_prefix `done $invd_some:term)\n            $decr:term\n            if $t then\n              $seq:doSeq\n            else break)\n  | `(doElem| while_some $x:ident :| $t do $seq:doSeq) =>\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      `(doElem|\n        while ∃ $x:ident, $t do\n          let $x :| $t\n          $[$seq:doElem]*)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| while_some $x:ident :| $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]? do\n                $seq:doSeq) => do\n    let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n    let invd_some ← match inv_done with\n    | some invd_some => withRef invd_some ``($invd_some)\n    | none => ``(¬$t:term)\n    match seq with\n    | `(doSeq| $[$seq:doElem]*)\n    | `(doSeq| $[$seq:doElem;]*)\n    | `(doSeq| { $[$seq:doElem]* }) =>\n      let decr <- withRef (<- getRef) `(decreasing none)\n      `(doElem|\n        for _ in Lean.Loop.mk do\n          $invs:term\n          onDoneGadget (with_name_prefix `done $invd_some:term)\n          $decr:term\n          if ∃ $x:ident, $t then\n            let $x :| $t\n            $[$seq:doElem]*\n          else break)\n    | _ => Lean.Macro.throwError \"while_some expects a sequence of do-elements\"\n  | `(doElem| for $x:ident in $t\n            $[invariant $inv:term\n            ]*\n            do $seq:doSeq) => do\n      let invs <- `(invariants [ $[(with_name_prefix `invariant $inv:term)],* ])\n      match seq with\n      | `(doSeq| $[$seq:doElem]*)\n      | `(doSeq| $[$seq:doElem;]*)\n      | `(doSeq| { $[$seq:doElem]* }) =>\n        `(doElem|\n          for $x:ident in $t do\n            $invs:term\n            $[$seq:doElem]*)\n      | _ => Lean.Macro.throwError \"for expects a sequence of do-elements\""}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic|loom_solver) =>\n    `(tactic|(\n      try simp at *\n      try aesop))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem| while $t\n              $[invariant $inv:term\n              ]*\n              $[done_with $inv_done]?\n              $[decreasing $measure]?\n              do $seq:doSeq) => do\n      let balance := mkIdent `balance_name\n      let balanceType <- `(term| Bal)\n      let inv : Array Term <- inv.mapM fun (inv : Term) => withRef inv ``(fun ($(balance):ident : $balanceType)=> with_name_prefix `inv $inv)\n      let invd_some <- match inv_done with\n      | some invd_some => withRef invd_some ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done $invd_some)\n      | none => ``(fun ($(balance):ident : $balanceType) => with_name_prefix `done ¬$t:term)\n      match measure with\n      | some measure_some =>\n        let measure_some ← withRef measure_some ``(type_with_name_prefix `decreasing ($measure_some:term))\n        do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget ($measure_some:term)\n            if $t then\n              $seq:doSeq\n            else break)\n      | none => do\n        `(doElem|\n          for _ in Lean.Loop.mk do\n            invariantGadget [ $[$inv:term],* ]\n            onDoneGadget ($invd_some:term)\n            decreasingGadget none\n            if $t then\n              $seq:doSeq\n            else break)"}, {"name": "macro_rules", "content": "macro_rules\n| `(doElem|balance_set $t) => do\n  let balId := mkIdent `balance\n  `(doElem|do\n    $balId:ident := $t\n    set $balId:ident\n    $balId:ident ← get)"}, {"name": "macro_rules", "content": "macro_rules\n  | `(doElem|$id:ident[$idx:term] := $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (fun _ => $val))\n  | `(doElem|$id:ident[$idx:term] += $val:term) =>\n    `(doElem| $id:term := ($id:term).modify $idx (· + $val))"}, {"name": "WithName", "content": "abbrev WithName (α : Sort u) (name : Lean.Name := default) := α"}, {"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "triple", "content": "notation \"{\" P \"}\" c \"{\" v \",\" Q \"}\" => triple P c (fun v => Q)"}], "lib_lemmas": [{"name": "List.find?_eq_none", "module": "Init.Data.List.Find"}, {"name": "List.mem_iff_get", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "List.find?_eq_some_iff_getElem", "module": "Init.Data.List.Nat.Find"}, {"name": "em", "module": "Mathlib.Logic.Basic"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "le_iff_eq_or_lt", "module": "Mathlib.Order.Basic"}, {"name": "le_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_of_lt_of_le", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_or_gt_of_ne", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "lt_iff_le_not_ge", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}], "used_local_defs": [{"name": "SpV", "content": "structure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!"}, {"name": "spv_dot", "content": "def spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=\n  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then\n    0\n  else\n    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)\n    else\n      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n        spv_dot spv1 spv2 (pnt1 + 1) pnt2\n      else\n        spv_dot spv1 spv2 pnt1 (pnt2 + 1)\n  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)\n\n\nmethod SpVSpV\n  (mut out: Array Int)\n  (spv1: SpV Int)\n  (spv2: SpV Int) return (u: Unit)\n  ensures out.size = 1\n  ensures out[0]! = spv_dot spv1 spv2 0 0\n  do\n    out := Array.replicate 1 0\n    let mut pnt1 := 0\n    let mut pnt2 := 0\n    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size\n    invariant out.size = 1\n    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size\n    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0\n    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size\n    do\n      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!\n        pnt1 := pnt1 + 1\n        pnt2 := pnt2 + 1\n      else\n        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n          pnt1 := pnt1 + 1\n        else\n          pnt2 := pnt2 + 1\n    return\n\n\nmethod SpMSpV\n  (mut out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int) return (u: Unit)\n  ensures out.size = spm.size\n  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0\n  do\n    out := Array.replicate spm.size 0\n    let mut spmInd := Array.replicate spm.size 0\n    let mut spvInd := Array.replicate spm.size 0\n    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size\n    invariant spvInd.size = spm.size\n    invariant spmInd.size = spm.size\n    invariant out.size = spm.size\n    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size\n    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size\n    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0\n    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size\n    do\n      let ind_m := spmInd[i]!\n      let ind_v := spvInd[i]!\n      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then\n        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!\n        spmInd[i] += 1\n        spvInd[i] += 1\n      else\n        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then\n          spmInd[i] += 1\n        else\n          spvInd[i] += 1\n    return"}], "used_local_lemmas": [{"name": "getValSpV_eq", "content": "theorem getValSpV_eq (spv: SpV Int) (j: ℕ) (h_ind: j < spv.size): spv[spv.ind[j]!] = (spv.val)[j]!"}, {"name": "getValSpV_empty", "content": "theorem getValSpV_empty (spv: SpV Int) (j: ℕ) (h_empty: ∀ i < spv.size, spv.ind[i]! ≠ j): spv[j] = 0"}, {"name": "spv_dot_pure_gen", "content": "theorem spv_dot_pure_gen (spv1: SpV Int) (spv2: SpV Int) (n pnt1 pnt2: ℕ)\n  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n)\n  (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):\n  spv_dot spv1 spv2 pnt1 pnt2 =\n    ∑ i ∈ Finset.range n,\n      if max\n        (if spv1.size ≤ pnt1 then n else spv1.ind[pnt1]!)\n        (if spv2.size ≤ pnt2 then n else spv2.ind[pnt2]!) ≤ i then\n        spv1[i] * spv2[i]\n      else\n        0"}, {"name": "spv_dot_pure", "content": "theorem spv_dot_pure (spv1 spv2: SpV Int) (n: ℕ)\n  (sz1: ∀ i < spv1.size, spv1.ind[i]! < n) (sz2: ∀ i < spv2.size, spv2.ind[i]! < n):\n    spv_dot spv1 spv2 0 0 = ∑ i ∈ Finset.range n, spv1[i] * spv2[i]"}], "local_ctx": "import Auto\n\nimport Lean\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Algebra.Ring.Int.Defs\n\nimport Loom.MonadAlgebras.NonDetT.Extract\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.DoNames'\n\nimport CaseStudies.Velvet.Std\n\nsection SpMV\n\nstructure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!\n\ndef spv_dot (spv1 spv2: SpV Int) (pnt1 pnt2: ℕ): Int :=\n  if (spv1.size) ≤ pnt1 ∨ (spv2.size) ≤ pnt2 then\n    0\n  else\n    if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n      (spv1.val)[pnt1]! * (spv2.val)[pnt2]! + spv_dot spv1 spv2 (pnt1 + 1) (pnt2 + 1)\n    else\n      if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n        spv_dot spv1 spv2 (pnt1 + 1) pnt2\n      else\n        spv_dot spv1 spv2 pnt1 (pnt2 + 1)\n  termination_by ((spv1.size) + (spv2.size) - pnt1 - pnt2)\n\n\nmethod SpVSpV\n  (mut out: Array Int)\n  (spv1: SpV Int)\n  (spv2: SpV Int) return (u: Unit)\n  ensures out.size = 1\n  ensures out[0]! = spv_dot spv1 spv2 0 0\n  do\n    out := Array.replicate 1 0\n    let mut pnt1 := 0\n    let mut pnt2 := 0\n    while pnt1 ≠ spv1.size ∧ pnt2 ≠ spv2.size\n    invariant out.size = 1\n    invariant pnt1 ≤ spv1.size ∧ pnt2 ≤ spv2.size\n    invariant out[0]! + spv_dot spv1 spv2 pnt1 pnt2 = spv_dot spv1 spv2 0 0\n    done_with pnt1 = spv1.size ∨ pnt2 = spv2.size\n    do\n      if (spv1.ind)[pnt1]! = (spv2.ind)[pnt2]! then\n        out[0] += (spv1.val)[pnt1]! * (spv2.val)[pnt2]!\n        pnt1 := pnt1 + 1\n        pnt2 := pnt2 + 1\n      else\n        if (spv1.ind)[pnt1]! < (spv2.ind)[pnt2]! then\n          pnt1 := pnt1 + 1\n        else\n          pnt2 := pnt2 + 1\n    return\n\n\nmethod SpMSpV\n  (mut out: Array Int)\n  (spm: Array (SpV Int))\n  (spv: SpV Int) return (u: Unit)\n  ensures out.size = spm.size\n  ensures ∀ i < spm.size, out[i]! = spv_dot spm[i]! spv 0 0\n  do\n    out := Array.replicate spm.size 0\n    let mut spmInd := Array.replicate spm.size 0\n    let mut spvInd := Array.replicate spm.size 0\n    while_some i :| i < spm.size ∧ spmInd[i]! < spm[i]!.size ∧ spvInd[i]! < spv.size\n    invariant spvInd.size = spm.size\n    invariant spmInd.size = spm.size\n    invariant out.size = spm.size\n    invariant ∀ i < spmInd.size, spmInd[i]! <= spm[i]!.size\n    invariant ∀ i < spvInd.size, spvInd[i]! <= spv.size\n    invariant ∀ i < spm.size, out[i]! + spv_dot spm[i]! spv spmInd[i]! spvInd[i]! = spv_dot spm[i]! spv 0 0\n    done_with ∀ i < spm.size, spmInd[i]! = spm[i]!.size ∨ spvInd[i]! = spv.size\n    do\n      let ind_m := spmInd[i]!\n      let ind_v := spvInd[i]!\n      if spm[i]!.ind[ind_m]! = spv.ind[ind_v]! then\n        out[i] += spm[i]!.val[ind_m]! * spv.val[ind_v]!\n        spmInd[i] += 1\n        spvInd[i] += 1\n      else\n        if spm[i]!.ind[ind_m]! < spv.ind[ind_v]! then\n          spmInd[i] += 1\n        else\n          spvInd[i] += 1\n    return", "target_theorem": "theorem SpVSpV_correct_triple (out: Array Int) (spv1 spv2: SpV Int) (n: ℕ):\n  triple\n    ((∀ i < spv1.size, spv1.ind[i]! < n) ∧ (∀ i < spv2.size, spv2.ind[i]! < n))\n    (SpVSpV out spv1 spv2)\n    fun ⟨_, outNew⟩ =>\n      outNew[0]! = ∑ i ∈ Finset.range n, spv1[i]! * spv2[i]! :=", "ground_truth_proof": ":= by\n      simp [triple]\n      intro b1 b2\n      apply wp_cons (SpVSpV out spv1 spv2)\n        fun ⟨_, outNew⟩ =>\n          outNew[0]! = spv_dot spv1 spv2 0 0 ∧ outNew.size = 1\n      { rintro outNew; simp\n        intro sum_eq sz_eq\n        simp [sum_eq]\n        exact spv_dot_pure spv1 spv2 n b1 b2 }\n      simp\n      have triple_true := SpVSpV_correct out spv1 spv2\n      simp [triple] at triple_true\n      simp [WithName] at triple_true\n      exact triple_true", "nesting_depth": 4, "transitive_dep_count": 37, "subset_aristotle": false, "category": "Framework"}
{"id": 385, "thm_name": "spmv_correct_triple", "thm_stmt": "theorem spmv_correct_triple (out: Array Int) (arr: Array Int) (spm: Array (SpV Int)):\n  triple\n    (∀ i < spm.size, ∀ j < spm[i]!.size, spm[i]!.ind[j]! < arr.size)\n    (spmv out arr spv spm)\n    fun ⟨_, outNew⟩ =>\n      (∀ j < outNew.size, outNew[j]! = ∑ i ∈ Finset.range (arr.size), spm[j]![i] * arr[i]!)", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetExamples/SpMSpV_Example.lean", "imports": ["import Auto", "import CaseStudies.Velvet.Std", "import Mathlib.Algebra.BigOperators.Intervals", "import Loom.MonadAlgebras.WP.DoNames'", "import Loom.MonadAlgebras.WP.Basic", "import Loom.MonadAlgebras.WP.Tactic", "import Mathlib.Algebra.Ring.Int.Defs", "import Lean", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Array.replicate", "module": "Init.Data.Array.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.find?", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "triple", "content": "notation \"{\" P \"}\" c \"{\" v \",\" Q \"}\" => triple P c (fun v => Q)"}], "lib_lemmas": [{"name": "List.find?_eq_some_iff_getElem", "module": "Init.Data.List.Nat.Find"}, {"name": "List.find?_eq_none", "module": "Init.Data.List.Find"}, {"name": "List.mem_iff_get", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "Nat.lt_iff_le_and_ne", "module": "Init.Data.Nat.Basic"}, {"name": "add_left_cancel_iff", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "lt_or_gt_of_ne", "module": "Mathlib.Order.Defs.LinearOrder"}], "repo_lemmas": [{"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}], "used_local_defs": [{"name": "SpV", "content": "structure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!"}, {"name": "sumUpTo", "content": "def sumUpTo\n  (spv : SpV Int)\n  (v : Array Int) (bound : ℕ) : Int := ∑ i ∈ Finset.range bound, ((spv.val)[i]! * v[(spv.ind)[i]!]!)"}], "used_local_lemmas": [{"name": "getValSpV_eq", "content": "theorem getValSpV_eq (spv: SpV Int) (j: ℕ) (h_ind: j < spv.size): spv[spv.ind[j]!] = (spv.val)[j]!"}, {"name": "getValSpV_empty", "content": "theorem getValSpV_empty (spv: SpV Int) (j: ℕ) (h_empty: ∀ i < spv.size, spv.ind[i]! ≠ j): spv[j] = 0"}, {"name": "VSpV_correct_pure", "content": "theorem VSpV_correct_pure (out: Array Int) (arr: Array Int)\n  (spv: SpV Int)\n  (h_b: ∀ i < spv.size, spv.ind[i]! < arr.size):\n  out.size = 1 → out[0]! = sumUpTo spv arr spv.size →\n    out[0]! = ∑ i ∈ Finset.range (arr.size), spv[i] * arr[i]!"}], "local_ctx": "import Auto\n\nimport Lean\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Algebra.Ring.Int.Defs\n\nimport Loom.MonadAlgebras.NonDetT.Extract\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.DoNames'\n\nimport CaseStudies.Velvet.Std\n\nsection SpMV\n\nstructure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!\n\ndef sumUpTo\n  (spv : SpV Int)\n  (v : Array Int) (bound : ℕ) : Int := ∑ i ∈ Finset.range bound, ((spv.val)[i]! * v[(spv.ind)[i]!]!)", "target_theorem": "theorem spmv_correct_triple (out: Array Int) (arr: Array Int) (spm: Array (SpV Int)):\n  triple\n    (∀ i < spm.size, ∀ j < spm[i]!.size, spm[i]!.ind[j]! < arr.size)\n    (spmv out arr spv spm)\n    fun ⟨_, outNew⟩ =>\n      (∀ j < outNew.size, outNew[j]! = ∑ i ∈ Finset.range (arr.size), spm[j]![i] * arr[i]!) :=", "ground_truth_proof": ":= by\n      simp [triple]\n      intro h_b\n      apply wp_cons\n        (spmv out arr spv spm)\n        fun ⟨_, outNew⟩ =>\n          ((∀ i < spm.size, outNew[i]! = sumUpTo spm[i]! arr spm[i]!.size) ∧ spm.size = outNew.size)\n      { simp; rintro outNew;\n        intro sum_eq sz_eq j h_j\n        have single_elem : (Array.replicate 1 outNew[j]!)[0]! = outNew[j]! := by\n          simp\n          -- simp [getElem!] at replicate_get\n          -- simp [getElem, replicate_get]\n        have single_th := VSpV_correct_pure\n          (Array.replicate 1 outNew[j]!)\n          arr\n          spm[j]!\n          (h_b j (by rw [←sz_eq] at h_j; exact h_j))\n          (by simp)\n          (by simp; exact sum_eq j (by rw [←sz_eq] at h_j; exact h_j))\n        simp at single_th\n        simp [←single_th] }\n      simp\n      have triple_true := spmv_correct out arr spv spm\n      simp [triple] at triple_true\n      exact triple_true", "nesting_depth": 3, "transitive_dep_count": 27, "subset_aristotle": false, "category": "Framework"}
{"id": 386, "thm_name": "VSpV_correct_triple", "thm_stmt": "theorem VSpV_correct_triple (out: Array Int) (arr: Array Int) (spv: SpV Int):\n  triple\n    (∀ i < spv.size, spv.ind[i]! < arr.size)\n    (VSpV out arr spv)\n    fun ⟨_, outNew⟩ =>\n      outNew[0]! = ∑ i ∈ Finset.range (arr.size), spv[i] * arr[i]!", "lean_root": "loom", "rel_path": "CaseStudies/Velvet/VelvetExamples/SpMSpV_Example.lean", "imports": ["import Auto", "import CaseStudies.Velvet.Std", "import Mathlib.Algebra.BigOperators.Intervals", "import Loom.MonadAlgebras.WP.DoNames'", "import Loom.MonadAlgebras.WP.Basic", "import Loom.MonadAlgebras.WP.Tactic", "import Mathlib.Algebra.Ring.Int.Defs", "import Lean", "import Loom.MonadAlgebras.NonDetT.Extract"], "used_lib_defs": [{"name": "Array", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.range", "module": "Mathlib.Data.Finset.Range"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.find?", "module": "Init.Data.List.Basic"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "triple", "content": "notation \"{\" P \"}\" c \"{\" v \",\" Q \"}\" => triple P c (fun v => Q)"}], "lib_lemmas": [{"name": "List.find?_eq_some_iff_getElem", "module": "Init.Data.List.Nat.Find"}, {"name": "List.find?_eq_none", "module": "Init.Data.List.Find"}, {"name": "List.mem_iff_get", "module": "Init.Data.List.Lemmas"}, {"name": "List.of_mem_zip", "module": "Init.Data.List.Zip"}, {"name": "Nat.lt_iff_le_and_ne", "module": "Init.Data.Nat.Basic"}, {"name": "add_left_cancel_iff", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "lt_or_gt_of_ne", "module": "Mathlib.Order.Defs.LinearOrder"}], "repo_lemmas": [{"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}], "used_local_defs": [{"name": "SpV", "content": "structure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!"}, {"name": "sumUpTo", "content": "def sumUpTo\n  (spv : SpV Int)\n  (v : Array Int) (bound : ℕ) : Int := ∑ i ∈ Finset.range bound, ((spv.val)[i]! * v[(spv.ind)[i]!]!)"}], "used_local_lemmas": [{"name": "getValSpV_eq", "content": "theorem getValSpV_eq (spv: SpV Int) (j: ℕ) (h_ind: j < spv.size): spv[spv.ind[j]!] = (spv.val)[j]!"}, {"name": "getValSpV_empty", "content": "theorem getValSpV_empty (spv: SpV Int) (j: ℕ) (h_empty: ∀ i < spv.size, spv.ind[i]! ≠ j): spv[j] = 0"}, {"name": "VSpV_correct_pure", "content": "theorem VSpV_correct_pure (out: Array Int) (arr: Array Int)\n  (spv: SpV Int)\n  (h_b: ∀ i < spv.size, spv.ind[i]! < arr.size):\n  out.size = 1 → out[0]! = sumUpTo spv arr spv.size →\n    out[0]! = ∑ i ∈ Finset.range (arr.size), spv[i] * arr[i]!"}], "local_ctx": "import Auto\n\nimport Lean\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Algebra.Ring.Int.Defs\n\nimport Loom.MonadAlgebras.NonDetT.Extract\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.DoNames'\n\nimport CaseStudies.Velvet.Std\n\nsection SpMV\n\nstructure SpV (valTyp : Type) where\n  ind: Array Nat\n  val: Array valTyp\n  size: ℕ\n  size_eq: ind.size = size ∧ val.size = size\n  inc: ∀ (i j: Nat), i < size → j < size →  i < j → ind[i]! < ind[j]!\n\ndef sumUpTo\n  (spv : SpV Int)\n  (v : Array Int) (bound : ℕ) : Int := ∑ i ∈ Finset.range bound, ((spv.val)[i]! * v[(spv.ind)[i]!]!)", "target_theorem": "theorem VSpV_correct_triple (out: Array Int) (arr: Array Int) (spv: SpV Int):\n  triple\n    (∀ i < spv.size, spv.ind[i]! < arr.size)\n    (VSpV out arr spv)\n    fun ⟨_, outNew⟩ =>\n      outNew[0]! = ∑ i ∈ Finset.range (arr.size), spv[i] * arr[i]! :=", "ground_truth_proof": ":= by\n      simp [triple]\n      intro h_b\n      apply wp_cons (VSpV out arr spv)\n        fun ⟨u, outNew⟩ =>\n          (outNew[0]! = sumUpTo spv arr spv.size ∧ outNew.size = 1)\n      { rintro ⟨u, outNew⟩; simp\n        intro sum_eq sz\n        exact VSpV_correct_pure outNew arr spv h_b sz sum_eq }\n      simp\n      have triple_true := VSpV_correct out arr spv\n      simp [triple] at triple_true\n      apply triple_true", "nesting_depth": 3, "transitive_dep_count": 26, "subset_aristotle": false, "category": "Framework"}
{"id": 387, "thm_name": "DemonicChoice.ExtractNonDet.extract_refines", "thm_stmt": "lemma ExtractNonDet.extract_refines (pre : l) (s : NonDetT m α) (inst : ExtractNonDet Findable s) :\n  triple pre s post ->\n  pre <= s.prop ⊤ ->\n  triple pre s.extract post", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/NonDetT'/Extract.lean", "imports": ["import Loom.MonadAlgebras.WP.Gen", "import Loom.MonadAlgebras.WP.Liberal", "import Mathlib.Order.CompleteBooleanAlgebra", "import Mathlib.Logic.Function.Basic", "import Mathlib.Data.W.Basic", "import Loom.MonadAlgebras.NonDetT'.Basic", "import Loom.MonadAlgebras.WP.Basic", "import Mathlib.Order.Lattice", "import Mathlib.Data.FinEnum", "import Mathlib.Order.Basic", "import Loom/MonadAlgebras/NonDetT/Findable.lean"], "used_lib_defs": [{"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Encodable", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Encodable.decode", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "BooleanAlgebra", "module": "Mathlib.Order.BooleanAlgebra.Defs"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "Decidable", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "inline", "module": "Init.Core"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "id", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.MonadEnv", "module": "Lean.Environment"}, {"name": "Lean.SimpleScopedEnvExtension", "module": "Lean.ScopedEnvExtension"}, {"name": "Lean.SimplePersistentEnvExtension", "module": "Lean.EnvExtension"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "MonadNonDet", "content": "class MonadNonDet (m : Type u → Type v) where\n  pick : (τ : Type u) → [Inhabited τ] → m τ\n  \n  pickSuchThat : (τ : Type u) → (p : τ → Prop) → [Findable p] → m τ\n  assume : (as : Prop) → [Decidable as] → m PUnit.{u+1}\n  \n  rep {α : Type u} : α → (α → m (ForInStep α)) → m α"}, {"name": "wlp", "content": "def wlp (c : m α) (post : α -> l) : l := iwp c post ⊔ wp c post"}, {"name": "iwp", "content": "abbrev iwp (c : m α) : Cont l α := Cont.inv (wp c)"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "Cont.inv", "content": "def Cont.inv {t : Type v} {α : Type u} [BooleanAlgebra t] (wp : Cont t α) : Cont t α :=\n  fun f => (wp fun x => (f x)ᶜ)ᶜ"}, {"name": "Cont", "content": "abbrev Cont (t : Type v) (α : Type u) := (α -> t) -> t"}, {"name": "NonDetT", "content": "inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α"}, {"name": "CCPOBot", "content": "class CCPOBot (m : Type u -> Type v)  where\n  compBot {α} : m α"}, {"name": "WPGen", "content": "structure WPGen (x : m α) where\n  get : Cont l α\n  \n  prop : ∀ post, get post <= wp x post"}, {"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}, {"name": "CCPOBotLawful", "content": "class CCPOBotLawful (m : Type u -> Type v) [∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] where\n  prop {α} : CCPOBot.compBot (m := m) (α := α) = Lean.Order.bot"}, {"name": "NonDetT.wp", "content": "def NonDetT.wp {l : Type u} {α : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m α -> Cont l α\n  | .pure ret => pure ret\n  | .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | .pickCont τ p f => fun post => let p : Set τ := p; ⨅ a ∈ (p : Set τ), wp (f a) post"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "NonDetT.μ", "content": "def NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "NonDetT.bind", "content": "def NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f"}, {"name": "WPGen.bind", "content": "def WPGen.bind {x : m α} {f : α -> m β} (wpg : WPGen x) (wpgf : ∀ a, WPGen (f a)) :\n  WPGen (x >>= f) where\n  get := fun post => wpg.get (fun a => (wpgf a).get post)\n  prop := by admit /- proof elided -/"}, {"name": "_root_.Lean.SimpleScopedEnvExtension.get", "content": "private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "Context", "content": "structure Context where\n  ref         : Syntax\n   \n  m           : Syntax\n   \n  returnType  : Syntax\n  mutableVars : VarSet := {}\n  insideFor   : Bool := false"}, {"name": "_root_.Lean.SimplePersistentEnvExtension.get", "content": "private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "_root_.Lean.EnvExtension.get", "content": "private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "NonDetT.pickSuchThat", "content": "def NonDetT.pickSuchThat (τ : Type u) (p : τ → Prop) [Findable p] : NonDetT m τ :=\n  NonDetT.pickCont τ p pure"}, {"name": "{l", "content": "instance {l σ : Type u} : MonadLift (Cont l) (Cont (σ -> l)) where\n  monadLift x := fun f s => x (f · s)"}, {"name": "triple", "content": "notation \"{\" P \"}\" c \"{\" v \",\" Q \"}\" => triple P c (fun v => Q)"}], "lib_lemmas": [{"name": "iInf_const", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "iInf_inf_eq", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "iInf_le_of_le", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "iSup_const", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "inf_assoc", "module": "Mathlib.Order.Lattice"}, {"name": "inf_comm", "module": "Mathlib.Order.Lattice"}, {"name": "inf_le_of_left_le", "module": "Mathlib.Order.Lattice"}, {"name": "le_trans'", "module": "Mathlib.Order.Basic"}], "repo_lemmas": [{"name": "wp_pure", "content": "lemma wp_pure (x : α) (post : α -> l) : wp (m := m) (pure x) post = post x"}, {"name": "wp_bind", "content": "lemma wp_bind {β} (x : m α) (f : α -> m β) (post : β -> l) :\n    wp (x >>= f) post = wp x (fun x => wp (f x) post)"}, {"name": "wlp_join_wp", "content": "lemma wlp_join_wp (c : m α) (post post' : α -> l) :\n  wlp c post ⊓ wp c post' = wp c (fun x => post x ⊓ post' x)"}, {"name": "wlp_himp", "content": "lemma wlp_himp (c : m α) (post post' : α -> l) :\n  wp c (fun x => post' x ⇨ post x) = wlp c post' ⇨ wp c post"}, {"name": "wp_wlp", "content": "omit [LawfulMonad m] in\nlemma wp_wlp (c : m α) (post : α -> l) :\n  wp c post <= wlp c post"}, {"name": "NonDetT.wp_vis", "content": "@[simp]\nlemma NonDetT.wp_vis {β : Type u} (x : m β) (f : β → NonDetT m α) post :\n  _root_.wp (NonDetT.vis x f) post = _root_.wp x fun a => _root_.wp (f a) post"}, {"name": "NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}, {"name": "NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α β : Type u} (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}, {"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}, {"name": "NonDetT.wp_pickCont", "content": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post"}, {"name": "NonDetT.wp_pickCont", "content": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⨆ a, ⌜p a⌝ ⊓ _root_.wp (f a) post"}, {"name": "meet_himp", "content": "lemma meet_himp (x x' y z : l) :\n  x = x' ->\n  (x ⇨ y) ⊓ (x' ⇨ z) = x ⇨ (y ⊓ z)"}], "used_local_defs": [{"name": "findNat", "content": "def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0"}, {"name": "find", "content": "def find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode"}, {"name": "WeakFindable", "content": "class WeakFindable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_some_p : find () = some x -> p x"}, {"name": "Findable", "content": "class Findable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_none : (find ()).isNone -> ∀ x, ¬ p x\n  find_some_p : find () = some x -> p x"}, {"name": "WeakFindable", "content": "instance WeakFindable.of_Findable {α : Type u} (p : α -> Prop) [Findable p] : WeakFindable p where\n  find := Findable.find p\n  find_some_p := Findable.find_some_p"}, {"name": "_inst_α", "content": "instance {p : α -> Prop} [Encodable α] [DecidablePred p] : Findable p where\n  find := fun _ => find p\n  find_none := find_none p\n  find_some_p := find_some_p p _"}, {"name": "_inst_α", "content": "@[instance high]\ninstance {p : α -> Prop} [FinEnum α] [DecidablePred p] : Findable p where\n  find := fun _ => FinEnum.toList α |>.find? p\n  find_none := by admit /- proof elided -/"}, {"name": "ExtractNonDet", "content": "inductive ExtractNonDet (findable : {τ : Type u} -> (τ -> Prop) -> Type u) {m} : {α : Type u} -> NonDetT m α -> Type _ where\n  | pure {α} : ∀ (x : α), ExtractNonDet findable (NonDetT.pure x)\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) :\n      (∀ y, ExtractNonDet findable (f y)) → ExtractNonDet findable (.vis x f)\n  | pickSuchThat {α} (τ : Type u) (p : τ -> Prop) (f : τ → NonDetT m α)\n    {_ : findable p}\n     : (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont τ p f)\n  | assume {α} (p : PUnit -> Prop) (f : PUnit → NonDetT m α) {_ : Decidable (p .unit)} :\n    (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont PUnit p f)"}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pure' : ExtractNonDet findable (Pure.pure (f := NonDetT m) x) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.liftM (x : m α) :\n  ExtractNonDet findable (liftM (n := NonDetT m) x) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.assume' {p : Prop} [Decidable p] : ExtractNonDet findable (MonadNonDet.assume (m :=  NonDetT m) p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pickSuchThat' {τ : Type u} (p : τ -> Prop) [Findable p] :\n  ExtractNonDet Findable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.pickSuchThat_weak {τ : Type u} (p : τ -> Prop) [WeakFindable p] :\n  ExtractNonDet WeakFindable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.if {p : Prop} {dec : Decidable p} {x y : NonDetT m α}\n  (_ : ExtractNonDet findable x) (_ : ExtractNonDet findable y) :\n  ExtractNonDet findable (if p then x else y) :="}, {"name": "ExtractNonDet", "content": "instance ExtractNonDet.ForIn_list {xs : List α} {init : β} {f : α → β → NonDetT m (ForInStep β)}\n  (_ : ∀ a b, ExtractNonDet findable (f a b)) :\n  ExtractNonDet findable (forIn xs init f) :="}, {"name": "NonDetT.extractGen", "content": "@[simp, inline]\ndef NonDetT.extractGen {findable : {τ : Type u} -> (τ -> Prop) -> Type u} {α : Type u}\n  (findOf : ∀ {τ : Type u} (p : τ -> Prop), findable p -> Unit -> Option τ)\n  : (s : NonDetT m α) -> (ex : ExtractNonDet findable s := by admit /- proof elided -/\n  ) -> m α\n  | .pure x, _ => Pure.pure x\n  | .vis x f, .vis _ _ _ => liftM x >>= (fun x => extractGen findOf (f x))\n  | .pickCont _ p f, .pickSuchThat _ _ _ _ =>\n    match findOf p ‹_› () with\n    | none => CCPOBot.compBot\n    | some x =>  extractGen findOf (f x)\n  | .pickCont _ p f, .assume _ _ _ =>\n    if p .unit then\n      extractGen findOf (f .unit)\n    else CCPOBot.compBot"}, {"name": "NonDetT.extract", "content": "@[inline]\ndef NonDetT.extract {α : Type u} (s : NonDetT m α) (ex : ExtractNonDet Findable s := by admit /- proof elided -/\n) : m α :=\n  NonDetT.extractGen Findable.find s"}, {"name": "NonDetT.prop", "content": "abbrev NonDetT.prop {α : Type u} : (s : NonDetT m α) -> Cont l α\n  | .pure x => Pure.pure x\n  | .vis x f => fun post => wlp x fun y => NonDetT.prop (f y) post\n  | .pickCont _ p f => fun post =>\n    (⨅ t, ⌜p t⌝ ⇨ NonDetT.prop (f t) post) ⊓ (⨆ t, ⌜p t⌝)"}, {"name": "Extractable", "content": "structure Extractable (x : NonDetT m α) where\n  cond : Cont l α\n  prop : ∀ post, cond post <= x.prop post"}, {"name": "ExtractNonDet.prop", "content": "def ExtractNonDet.prop {α : Type u} (s : NonDetT m α) :  ExtractNonDet WeakFindable s -> l\n  | .pure x => ⊤\n  | .vis x f ex => wlp x fun y => (ex y).prop\n  | .pickSuchThat _ p f ex => ⨅ t ∈ WeakFindable.find p (), (ex t).prop\n  | .assume p f ex =>\n      if p .unit then\n        (ex .unit).prop\n      else ⊤"}], "used_local_lemmas": [{"name": "DemonicChoice.ExtractNonDet.extract_refines_wp", "content": "lemma ExtractNonDet.extract_refines_wp (s : NonDetT m α) (inst : ExtractNonDet Findable s) :\n  wp s post ⊓ s.prop ⊤ <= wp s.extract post"}], "local_ctx": "import Mathlib.Logic.Function.Basic\n\nimport Mathlib.Order.CompleteBooleanAlgebra\n\nimport Mathlib.Order.Lattice\n\nimport Mathlib.Order.Basic\n\nimport Mathlib.Data.W.Basic\n\nimport Mathlib.Data.FinEnum\n\nimport Loom.MonadAlgebras.WP.Gen\n\nimport Loom.MonadAlgebras.WP.Liberal\n\nimport Loom.MonadAlgebras.NonDetT'.Basic\n\nopen Lean.Order\n\ndef findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0\n\ndef find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode\n\nclass WeakFindable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_some_p : find () = some x -> p x\n\nclass Findable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_none : (find ()).isNone -> ∀ x, ¬ p x\n  find_some_p : find () = some x -> p x\n\ninstance WeakFindable.of_Findable {α : Type u} (p : α -> Prop) [Findable p] : WeakFindable p where\n  find := Findable.find p\n  find_some_p := Findable.find_some_p\n\ninstance {p : α -> Prop} [Encodable α] [DecidablePred p] : Findable p where\n  find := fun _ => find p\n  find_none := find_none p\n  find_some_p := find_some_p p _\n\n@[instance high]\ninstance {p : α -> Prop} [FinEnum α] [DecidablePred p] : Findable p where\n  find := fun _ => FinEnum.toList α |>.find? p\n  find_none := by admit /- proof elided -/\n\ninductive ExtractNonDet (findable : {τ : Type u} -> (τ -> Prop) -> Type u) {m} : {α : Type u} -> NonDetT m α -> Type _ where\n  | pure {α} : ∀ (x : α), ExtractNonDet findable (NonDetT.pure x)\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) :\n      (∀ y, ExtractNonDet findable (f y)) → ExtractNonDet findable (.vis x f)\n  | pickSuchThat {α} (τ : Type u) (p : τ -> Prop) (f : τ → NonDetT m α)\n    {_ : findable p}\n     : (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont τ p f)\n  | assume {α} (p : PUnit -> Prop) (f : PUnit → NonDetT m α) {_ : Decidable (p .unit)} :\n    (∀ x, ExtractNonDet findable (f x)) → ExtractNonDet findable (.pickCont PUnit p f)\n\ninstance ExtractNonDet.pure' : ExtractNonDet findable (Pure.pure (f := NonDetT m) x) :=\n\ninstance ExtractNonDet.liftM (x : m α) :\n  ExtractNonDet findable (liftM (n := NonDetT m) x) :=\n\ninstance ExtractNonDet.assume' {p : Prop} [Decidable p] : ExtractNonDet findable (MonadNonDet.assume (m :=  NonDetT m) p) :=\n\ninstance ExtractNonDet.pickSuchThat' {τ : Type u} (p : τ -> Prop) [Findable p] :\n  ExtractNonDet Findable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=\n\ninstance ExtractNonDet.pickSuchThat_weak {τ : Type u} (p : τ -> Prop) [WeakFindable p] :\n  ExtractNonDet WeakFindable (MonadNonDet.pickSuchThat (m := NonDetT m) τ p) :=\n\ninstance ExtractNonDet.if {p : Prop} {dec : Decidable p} {x y : NonDetT m α}\n  (_ : ExtractNonDet findable x) (_ : ExtractNonDet findable y) :\n  ExtractNonDet findable (if p then x else y) :=\n\ninstance ExtractNonDet.ForIn_list {xs : List α} {init : β} {f : α → β → NonDetT m (ForInStep β)}\n  (_ : ∀ a b, ExtractNonDet findable (f a b)) :\n  ExtractNonDet findable (forIn xs init f) :=\n\nvariable [Monad m] [CCPOBot m] [CompleteBooleanAlgebra l] [MAlgOrdered m l] [MAlgDet m l] [LawfulMonad m]\n\n@[simp, inline]\ndef NonDetT.extractGen {findable : {τ : Type u} -> (τ -> Prop) -> Type u} {α : Type u}\n  (findOf : ∀ {τ : Type u} (p : τ -> Prop), findable p -> Unit -> Option τ)\n  : (s : NonDetT m α) -> (ex : ExtractNonDet findable s := by admit /- proof elided -/\n  ) -> m α\n  | .pure x, _ => Pure.pure x\n  | .vis x f, .vis _ _ _ => liftM x >>= (fun x => extractGen findOf (f x))\n  | .pickCont _ p f, .pickSuchThat _ _ _ _ =>\n    match findOf p ‹_› () with\n    | none => CCPOBot.compBot\n    | some x =>  extractGen findOf (f x)\n  | .pickCont _ p f, .assume _ _ _ =>\n    if p .unit then\n      extractGen findOf (f .unit)\n    else CCPOBot.compBot\n\n@[inline]\ndef NonDetT.extract {α : Type u} (s : NonDetT m α) (ex : ExtractNonDet Findable s := by admit /- proof elided -/\n) : m α :=\n  NonDetT.extractGen Findable.find s\n\nabbrev NonDetT.prop {α : Type u} : (s : NonDetT m α) -> Cont l α\n  | .pure x => Pure.pure x\n  | .vis x f => fun post => wlp x fun y => NonDetT.prop (f y) post\n  | .pickCont _ p f => fun post =>\n    (⨅ t, ⌜p t⌝ ⇨ NonDetT.prop (f t) post) ⊓ (⨆ t, ⌜p t⌝)\n\nstructure Extractable (x : NonDetT m α) where\n  cond : Cont l α\n  prop : ∀ post, cond post <= x.prop post\n\ndef ExtractNonDet.prop {α : Type u} (s : NonDetT m α) :  ExtractNonDet WeakFindable s -> l\n  | .pure x => ⊤\n  | .vis x f ex => wlp x fun y => (ex y).prop\n  | .pickSuchThat _ p f ex => ⨅ t ∈ WeakFindable.find p (), (ex t).prop\n  | .assume p f ex =>\n      if p .unit then\n        (ex .unit).prop\n      else ⊤\n\nnamespace DemonicChoice", "target_theorem": "lemma ExtractNonDet.extract_refines (pre : l) (s : NonDetT m α) (inst : ExtractNonDet Findable s) :\n  triple pre s post ->\n  pre <= s.prop ⊤ ->\n  triple pre s.extract post :=", "ground_truth_proof": ":= by\n  intro tr imp; apply le_trans'; apply ExtractNonDet.extract_refines_wp\n  simp; aesop", "nesting_depth": 5, "transitive_dep_count": 80, "subset_aristotle": false, "category": "Framework"}
{"id": 388, "thm_name": "StateT.wp_lift", "thm_stmt": "lemma StateT.wp_lift (c : m α) (post : α -> σ -> l) :\n  wp (liftM (n := StateT σ m) c) post = fun s => wp (m := m) c (post · s)", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/WP/Basic.lean", "imports": ["import Loom.MonadAlgebras.Instances.Gen", "import Loom.MonadAlgebras.Defs", "import Loom.MonadAlgebras.Instances.ReaderT", "import Loom.MonadAlgebras.Instances.ExceptT", "import Loom.MonadAlgebras.Instances.StateT", "import Loom.MonadAlgebras.Instances.Basic"], "used_lib_defs": [{"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "MonadLift", "module": "Init.Prelude"}, {"name": "MonadLift.monadLift", "module": "Init.Prelude"}, {"name": "StateT.lift", "module": "Init.Control.State"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}], "used_repo_defs": [{"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}], "lib_lemmas": [{"name": "StateT.ext", "module": "Init.Control.Lawful.Instances"}, {"name": "map_eq_pure_bind", "module": "Init.Control.Lawful.Basic"}], "repo_lemmas": [{"name": "MAlg.lift_StateT", "content": "lemma MAlg.lift_StateT [Monad m] [LawfulMonad m] [CompleteLattice l] [inst: MAlgOrdered m l] (x : StateT σ m α) :\n  MAlg.lift x post = fun s => MAlg.lift (x s) (fun xs => post xs.1 xs.2)"}], "used_local_defs": [{"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}], "used_local_lemmas": [{"name": "wp_pure", "content": "lemma wp_pure (x : α) (post : α -> l) : wp (m := m) (pure x) post = post x"}, {"name": "wp_bind", "content": "lemma wp_bind {β} (x : m α) (f : α -> m β) (post : β -> l) :\n    wp (x >>= f) post = wp x (fun x => wp (f x) post)"}], "local_ctx": "import Loom.MonadAlgebras.Defs\n\nimport Loom.MonadAlgebras.Instances.Basic\n\nimport Loom.MonadAlgebras.Instances.ExceptT\n\nimport Loom.MonadAlgebras.Instances.StateT\n\nimport Loom.MonadAlgebras.Instances.ReaderT\n\nimport Loom.MonadAlgebras.Instances.Gen\n\nvariable {m : Type u -> Type v} [Monad m] [LawfulMonad m] {α : Type u} {l : Type u}\n\nsection\n\nvariable [CompleteLattice l]\n\nsection\n\nvariable [mprop : MAlgOrdered m l]\n\ndef wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post\n\nend\n\nvariable [MAlgOrdered m l]\n\nend\n\nsection\n\nvariable [CompleteLattice l] [MAlgOrdered m l]\n\nnoncomputable\n\nend\n\nsection Determinism\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nend Determinism\n\nsection Loops\n\nopen Lean.Order\n\nvariable [inst: _root_.CompleteLattice l] [MAlgOrdered m l]\n\nnamespace PartialCorrectness\n\nvariable [∀ α, CCPO (m α)] [MonoBind m] [MAlgPartial m]\n\nend PartialCorrectness\n\nnamespace TotalCorrectness\n\nvariable [∀ α, CCPO (m α)] [MonoBind m]\n\nvariable [MAlgTotal m]\n\nend TotalCorrectness\n\nend Loops\n\nsection Lift\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nopen ExceptionAsSuccess in\n\nopen ExceptionAsFailure in\n\nopen TotalCorrectness in", "target_theorem": "lemma StateT.wp_lift (c : m α) (post : α -> σ -> l) :\n  wp (liftM (n := StateT σ m) c) post = fun s => wp (m := m) c (post · s) :=", "ground_truth_proof": ":= by\n  simp [wp, liftM, monadLift, MAlg.lift_StateT, MonadLift.monadLift, StateT.lift];\n  have liftE : ∀ α, MAlg.lift (m := m) (α := α) = wp := by intros; ext; simp [wp, liftM, monadLift]\n  ext s; rw [map_eq_pure_bind, liftE, liftE, wp_bind]; simp [wp_pure]", "nesting_depth": 2, "transitive_dep_count": 21, "subset_aristotle": false, "category": "Framework"}
{"id": 389, "thm_name": "Gen.wp_rand", "thm_stmt": "lemma Gen.wp_rand {α : Type} (c : Gen α) :\n  triple ⊤ c (fun _ => ⊤)", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/WP/Basic.lean", "imports": ["import Loom.MonadAlgebras.Instances.Gen", "import Loom.MonadAlgebras.Defs", "import Loom.MonadAlgebras.Instances.ReaderT", "import Loom.MonadAlgebras.Instances.ExceptT", "import Loom.MonadAlgebras.Instances.StateT", "import Loom.MonadAlgebras.Instances.Basic"], "used_lib_defs": [{"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Plausible.Gen", "module": "Plausible.Gen"}, {"name": "StdGen", "module": "Init.Data.Random"}, {"name": "ULift", "module": "Init.Prelude"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "ReaderT", "module": "Init.Prelude"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "inferInstanceAs", "module": "Init.Prelude"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}], "used_repo_defs": [{"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "MAlgGenInst", "content": "instance MAlgGenInst : MAlgOrdered Gen (ULift Nat -> ULift StdGen -> Prop) :=\n  inferInstanceAs\n    (MAlgOrdered\n      (ReaderT (ULift Nat)\n        (StateT (ULift StdGen) Id))\n      (ULift Nat ->\n        ULift StdGen -> Prop))"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "triple", "content": "notation \"{\" P \"}\" c \"{\" v \",\" Q \"}\" => triple P c (fun v => Q)"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "MAlg.lift_StateT", "content": "lemma MAlg.lift_StateT [Monad m] [LawfulMonad m] [CompleteLattice l] [inst: MAlgOrdered m l] (x : StateT σ m α) :\n  MAlg.lift x post = fun s => MAlg.lift (x s) (fun xs => post xs.1 xs.2)"}, {"name": "MAlg.lift_ReaderT", "content": "lemma MAlg.lift_ReaderT [Monad m] [LawfulMonad m] [CompleteLattice l] [inst: MAlgOrdered m l] (x : ReaderT σ m α) :\n  MAlg.lift x post = fun s => MAlg.lift (x s) (fun xs => post xs s)"}], "used_local_defs": [{"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}], "used_local_lemmas": [{"name": "StateT.wp_eq", "content": "lemma StateT.wp_eq (c : StateT σ m α) (post : α -> σ -> l) :\n  wp c post = fun s => wp (m := m) (c s) (fun xs => post xs.1 xs.2)"}, {"name": "ReaderT.wp_eq", "content": "lemma ReaderT.wp_eq (c : ReaderT σ m α) (post : α -> σ -> l) :\n  wp c post = fun s => wp (m := m) (c s) (post · s)"}], "local_ctx": "import Loom.MonadAlgebras.Defs\n\nimport Loom.MonadAlgebras.Instances.Basic\n\nimport Loom.MonadAlgebras.Instances.ExceptT\n\nimport Loom.MonadAlgebras.Instances.StateT\n\nimport Loom.MonadAlgebras.Instances.ReaderT\n\nimport Loom.MonadAlgebras.Instances.Gen\n\nvariable {m : Type u -> Type v} [Monad m] [LawfulMonad m] {α : Type u} {l : Type u}\n\nsection\n\nvariable [CompleteLattice l]\n\nsection\n\nvariable [mprop : MAlgOrdered m l]\n\ndef wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post\n\ndef triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post\n\nend\n\nvariable [MAlgOrdered m l]\n\nend\n\nsection\n\nvariable [CompleteLattice l] [MAlgOrdered m l]\n\nnoncomputable\n\nend\n\nsection Determinism\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nend Determinism\n\nsection Loops\n\nopen Lean.Order\n\nvariable [inst: _root_.CompleteLattice l] [MAlgOrdered m l]\n\nnamespace PartialCorrectness\n\nvariable [∀ α, CCPO (m α)] [MonoBind m] [MAlgPartial m]\n\nend PartialCorrectness\n\nnamespace TotalCorrectness\n\nvariable [∀ α, CCPO (m α)] [MonoBind m]\n\nvariable [MAlgTotal m]\n\nend TotalCorrectness\n\nend Loops\n\nsection Lift\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nopen ExceptionAsSuccess in\n\nopen ExceptionAsFailure in\n\nopen TotalCorrectness in\n\nend Lift\n\nsection ExceptT\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l] [IsHandler (ε := ε) hd]\n\nend ExceptT\n\nsection StateT\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nend StateT\n\nsection ReaderT\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nend ReaderT\n\nsection Gen\n\nopen Plausible", "target_theorem": "lemma Gen.wp_rand {α : Type} (c : Gen α) :\n  triple ⊤ c (fun _ => ⊤) :=", "ground_truth_proof": ":= by\n    simp [triple, MAlgGenInst, ReaderT.wp_eq, StateT.wp_eq]\n    simp [wp, liftM, monadLift, MAlg.lift, MAlgOrdered.μ]; rfl", "nesting_depth": 3, "transitive_dep_count": 26, "subset_aristotle": false, "category": "Framework"}
{"id": 390, "thm_name": "MAlgLift.wp_throw", "thm_stmt": "lemma MAlgLift.wp_throw\n  [Monad n] [CompleteLattice k] [MAlgOrdered n k] [MonadLiftT m n]\n  [MonadLiftT (ExceptT ε m) n]\n  [inst: MAlgLiftT (ExceptT ε m) l n k] :\n    wp (liftM (n := n) (throw (m := ExceptT ε m) e)) post = ⌜hd e⌝", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/WP/Basic.lean", "imports": ["import Loom.MonadAlgebras.Instances.Gen", "import Loom.MonadAlgebras.Defs", "import Loom.MonadAlgebras.Instances.ReaderT", "import Loom.MonadAlgebras.Instances.ExceptT", "import Loom.MonadAlgebras.Instances.StateT", "import Loom.MonadAlgebras.Instances.Basic"], "used_lib_defs": [{"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "MonadLiftT", "module": "Init.Prelude"}, {"name": "LE", "module": "Init.Prelude"}, {"name": "OrderBot", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "OrderTop", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "ExceptT", "module": "Init.Control.Except"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}, {"name": "ExceptT.mk", "module": "Init.Control.Except"}, {"name": "MonadExceptOf", "module": "Init.Prelude"}, {"name": "MonadExceptOf.throw", "module": "Init.Prelude"}, {"name": "throwThe", "module": "Init.Prelude"}, {"name": "error", "module": "Auto.Parser.TPTP"}, {"name": "Except", "module": "Init.Prelude"}, {"name": "Except.error", "module": "Init.Prelude"}, {"name": "Except.bind", "module": "Init.Control.Except"}, {"name": "ExceptT.bind", "module": "Init.Control.Except"}, {"name": "ExceptT.bindCont", "module": "Init.Control.Except"}, {"name": "ExceptT.instMonad", "module": "Init.Control.Except"}, {"name": "Function.comp", "module": "Init.Prelude"}, {"name": "Pi.hasLe", "module": "Mathlib.Order.Basic"}, {"name": "le", "module": "Test.SmtTranslation.Names"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "MAlgLiftT", "content": "class MAlgLiftT\n  (m : (Type u -> Type v)) (l : (Type u)) [Monad m] [CompleteLattice l] [MAlgOrdered m l]\n  (n : (Type u -> Type w)) (k : outParam (Type u)) [Monad n] [CompleteLattice k] [MAlgOrdered n k]\n  [MonadLiftT m n]\n  where\n    [cl : LogicLiftT l k]\n    μ_lift (x : m α) : MAlg.lift (liftM (n := n) x) f = liftM (n := Cont k) (MAlg.lift x) f"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "LogicLiftT", "content": "class LogicLiftT (l : (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLiftT (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "LE.pure", "content": "noncomputable def LE.pure {l : Type u} [inst: LE l] [OrderTop l] [OrderBot l] : Prop -> l := fun p =>\n  if p then ⊤ else ⊥"}, {"name": "IsHandler", "content": "class IsHandler {ε : Type*} (handler : outParam (ε -> Prop)) where"}, {"name": "Except.getD", "content": "abbrev Except.getD {ε α} (default : ε -> α)  : Except ε α -> α\n  | Except.ok p => p\n  | Except.error e => default e"}, {"name": "MAlgExcept", "content": "def MAlgExcept (ε : Type u) (df : ε -> Prop) (l : Type u) (m : Type u -> Type v)\n  [CompleteLattice l]\n  [Monad m] [LawfulMonad m] [inst: MAlgOrdered m l] : MAlgOrdered (ExceptT ε m) l where\n  μ := fun e => inst.μ $ Except.getD (⌜df ·⌝) <$> e\n  μ_ord_pure := by admit /- proof elided -/"}, {"name": "Except.bind'", "content": "abbrev Except.bind' {m : Type u -> Type v} {ε α β} [Monad m] : Except ε α -> (α -> ExceptT ε m β) -> ExceptT ε m β :=\n  fun x f => bind (m := ExceptT ε m) (pure (f := m) x) f"}, {"name": "OfHd", "content": "instance OfHd {hd : ε -> Prop} [hdInst : IsHandler hd]\n  [CompleteLattice l] [inst: MAlgOrdered m l] : MAlgOrdered (ExceptT ε m) l := MAlgExcept ε hd l m"}], "lib_lemmas": [{"name": "ExceptT.ext", "module": "Init.Control.Lawful.Instances"}], "repo_lemmas": [{"name": "MAlg.lift_ExceptT", "content": "lemma MAlg.lift_ExceptT ε (hd : ε -> Prop) [IsHandler hd] [CompleteLattice l] [inst: MAlgOrdered m l]\n   (c : ExceptT ε m α) post :\n  MAlg.lift c post = MAlg.lift (m := m) c (fun | .ok x => post x | .error e => ⌜hd e⌝)"}], "used_local_defs": [{"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}], "used_local_lemmas": [{"name": "wp_pure", "content": "lemma wp_pure (x : α) (post : α -> l) : wp (m := m) (pure x) post = post x"}, {"name": "wp_except_handler_eq", "content": "lemma wp_except_handler_eq ε (hd : ε -> Prop) [IsHandler hd] (c : ExceptT ε m α) post :\n  wp c post = wp (m := m) c (fun | .ok x => post x | .error e => ⌜hd e⌝)"}, {"name": "MAlgLift.wp_lift", "content": "omit [LawfulMonad m] in\nlemma MAlgLift.wp_lift [Monad n] [CompleteLattice k] [MAlgOrdered n k] [MonadLiftT m n]\n  -- [MonadLiftT (Cont l) (Cont k)]\n  [mAlgLift : MAlgLiftT m l n k] (c : m α):\n  wp (liftM (n := n) c) = fun (post : α -> k) => mAlgLift.cl.lift.monadLift (wp c) post"}, {"name": "ExceptT.wp_throw", "content": "lemma ExceptT.wp_throw (e : ε) :\n  wp (α := α) (throw (m := ExceptT ε m) e) = fun _ => ⌜hd e⌝"}], "local_ctx": "import Loom.MonadAlgebras.Defs\n\nimport Loom.MonadAlgebras.Instances.Basic\n\nimport Loom.MonadAlgebras.Instances.ExceptT\n\nimport Loom.MonadAlgebras.Instances.StateT\n\nimport Loom.MonadAlgebras.Instances.ReaderT\n\nimport Loom.MonadAlgebras.Instances.Gen\n\nvariable {m : Type u -> Type v} [Monad m] [LawfulMonad m] {α : Type u} {l : Type u}\n\nsection\n\nvariable [CompleteLattice l]\n\nsection\n\nvariable [mprop : MAlgOrdered m l]\n\ndef wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post\n\nend\n\nvariable [MAlgOrdered m l]\n\nend\n\nsection\n\nvariable [CompleteLattice l] [MAlgOrdered m l]\n\nnoncomputable\n\nend\n\nsection Determinism\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nend Determinism\n\nsection Loops\n\nopen Lean.Order\n\nvariable [inst: _root_.CompleteLattice l] [MAlgOrdered m l]\n\nnamespace PartialCorrectness\n\nvariable [∀ α, CCPO (m α)] [MonoBind m] [MAlgPartial m]\n\nend PartialCorrectness\n\nnamespace TotalCorrectness\n\nvariable [∀ α, CCPO (m α)] [MonoBind m]\n\nvariable [MAlgTotal m]\n\nend TotalCorrectness\n\nend Loops\n\nsection Lift\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l]\n\nopen ExceptionAsSuccess in\n\nopen ExceptionAsFailure in\n\nopen TotalCorrectness in\n\nend Lift\n\nsection ExceptT\n\nvariable [inst: CompleteLattice l] [MAlgOrdered m l] [IsHandler (ε := ε) hd]", "target_theorem": "lemma MAlgLift.wp_throw\n  [Monad n] [CompleteLattice k] [MAlgOrdered n k] [MonadLiftT m n]\n  [MonadLiftT (ExceptT ε m) n]\n  [inst: MAlgLiftT (ExceptT ε m) l n k] :\n    wp (liftM (n := n) (throw (m := ExceptT ε m) e)) post = ⌜hd e⌝ :=", "ground_truth_proof": ":= by\n    rw [MAlgLift.wp_lift, ExceptT.wp_throw]\n    simp only [LE.pure]; split <;> simp [inst.cl.lift_top, inst.cl.lift_bot]", "nesting_depth": 6, "transitive_dep_count": 45, "subset_aristotle": false, "category": "Framework"}
{"id": 391, "thm_name": "p_findNat_some", "thm_stmt": "lemma p_findNat_some (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  p i -> ∃ j, p j ∧ j <= i ∧ findNat p = some j", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/NonDetT/Findable.lean", "imports": ["import Mathlib.Order.Lattice", "import Mathlib.Data.FinEnum", "import Mathlib.Order.Basic", "import Mathlib.Order.CompleteBooleanAlgebra", "import Mathlib.Logic.Function.Basic", "import Mathlib.Data.W.Basic"], "used_lib_defs": [{"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Nat.decreasingInduction", "module": "Mathlib.Data.Nat.Init"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Option.isSome_iff_exists", "module": "Init.Data.Option.Lemmas"}, {"name": "Option.not_isSome_iff_eq_none", "module": "Init.Data.Option.Lemmas"}, {"name": "Option.isNone_iff_eq_none", "module": "Init.Data.Option.Instances"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "AccFrom", "content": "inductive AccFrom (p : Nat -> Prop) : Nat -> Prop\n    | now : p i -> AccFrom p i\n    | later : ¬ p i -> AccFrom p (i + 1) -> AccFrom p i"}, {"name": "findNat", "content": "def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0"}], "used_local_lemmas": [{"name": "AccFrom_findNat", "content": "lemma AccFrom_findNat (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  AccFrom p i -> (findNat.aux p i).isSome"}, {"name": "AccFrom_of_p", "content": "lemma AccFrom_of_p (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  p i -> ∀ j ≤ i, AccFrom p j"}, {"name": "exists_findNat", "content": "lemma exists_findNat (p : Nat -> Prop) [DecidablePred p] :\n  (∃ x, p x) ↔ (findNat p).isSome"}, {"name": "findNat_none", "content": "lemma findNat_none (p : Nat -> Prop) [DecidablePred p] :\n  (findNat p).isNone -> ∀ i, ¬ p i"}, {"name": "findNat_aux_some_le", "content": "lemma findNat_aux_some_le (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  findNat.aux p i = some j -> ∀ k, i <= k -> k < j -> ¬ p k"}, {"name": "findNat_some_p", "content": "lemma findNat_some_p (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  findNat p = some i -> p i"}], "local_ctx": "import Mathlib.Logic.Function.Basic\n\nimport Mathlib.Order.CompleteBooleanAlgebra\n\nimport Mathlib.Order.Lattice\n\nimport Mathlib.Order.Basic\n\nimport Mathlib.Data.W.Basic\n\nimport Mathlib.Data.FinEnum\n\ninductive AccFrom (p : Nat -> Prop) : Nat -> Prop\n    | now : p i -> AccFrom p i\n    | later : ¬ p i -> AccFrom p (i + 1) -> AccFrom p i\n\ndef findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0", "target_theorem": "lemma p_findNat_some (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  p i -> ∃ j, p j ∧ j <= i ∧ findNat p = some j :=", "ground_truth_proof": ":= by\n  intro pi;\n  have : (findNat p).isSome := by\n    false_or_by_contra; rename_i h\n    simp at h\n    rw [←Option.isNone_iff_eq_none] at h\n    have h := findNat_none _ h\n    aesop\n  revert this; simp [Option.isSome_iff_exists]\n  intro x h\n  have := findNat_aux_some_le p 0 h\n  exists x; repeat' constructor\n  { solve_by_elim [findNat_some_p] }\n  { have h := fun h₁ h₂ => this _ h₁ h₂ pi\n    simp at h; exact h }\n  solve_by_elim", "nesting_depth": 4, "transitive_dep_count": 15, "subset_aristotle": false, "category": "Framework"}
{"id": 392, "thm_name": "p_findNat_some", "thm_stmt": "lemma p_findNat_some (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  p i -> ∃ j, p j ∧ j <= i ∧ findNat p = some j", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/NonDetT'/Extract.lean", "imports": ["import Loom.MonadAlgebras.WP.Gen", "import Loom.MonadAlgebras.WP.Liberal", "import Mathlib.Order.CompleteBooleanAlgebra", "import Mathlib.Logic.Function.Basic", "import Mathlib.Data.W.Basic", "import Loom.MonadAlgebras.NonDetT'.Basic", "import Mathlib.Order.Lattice", "import Mathlib.Data.FinEnum", "import Mathlib.Order.Basic"], "used_lib_defs": [{"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat.decreasingInduction", "module": "Mathlib.Data.Nat.Init"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Option.isSome_iff_exists", "module": "Init.Data.Option.Lemmas"}, {"name": "forall_exists_index", "module": "Init.PropLemmas"}, {"name": "Option.not_isSome_iff_eq_none", "module": "Init.Data.Option.Lemmas"}, {"name": "Bool.not_eq_true", "module": "Init.SimpLemmas"}, {"name": "Option.isSome_eq_false_iff", "module": "Init.Data.Option.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "AccFrom", "content": "inductive AccFrom (p : Nat -> Prop) : Nat -> Prop\n    | now : p i -> AccFrom p i\n    | later : ¬ p i -> AccFrom p (i + 1) -> AccFrom p i"}, {"name": "findNat", "content": "def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0"}], "used_local_lemmas": [{"name": "AccFrom_findNat", "content": "lemma AccFrom_findNat (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  AccFrom p i -> (findNat.aux p i).isSome"}, {"name": "AccFrom_of_p", "content": "lemma AccFrom_of_p (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  p i -> ∀ j ≤ i, AccFrom p j"}, {"name": "exists_findNat", "content": "lemma exists_findNat (p : Nat -> Prop) [DecidablePred p] :\n  (∃ x, p x) ↔ (findNat p).isSome"}, {"name": "findNat_none", "content": "lemma findNat_none (p : Nat -> Prop) [DecidablePred p] :\n  (findNat p).isNone -> ∀ i, ¬ p i"}, {"name": "findNat_aux_some_le", "content": "lemma findNat_aux_some_le (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  findNat.aux p i = some j -> ∀ k, i <= k -> k < j -> ¬ p k"}, {"name": "findNat_some_p", "content": "lemma findNat_some_p (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  findNat p = some i -> p i"}], "local_ctx": "import Mathlib.Logic.Function.Basic\n\nimport Mathlib.Order.CompleteBooleanAlgebra\n\nimport Mathlib.Order.Lattice\n\nimport Mathlib.Order.Basic\n\nimport Mathlib.Data.W.Basic\n\nimport Mathlib.Data.FinEnum\n\nimport Loom.MonadAlgebras.WP.Gen\n\nimport Loom.MonadAlgebras.WP.Liberal\n\nimport Loom.MonadAlgebras.NonDetT'.Basic\n\nopen Lean.Order\n\ninductive AccFrom (p : Nat -> Prop) : Nat -> Prop\n    | now : p i -> AccFrom p i\n    | later : ¬ p i -> AccFrom p (i + 1) -> AccFrom p i\n\ndef findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0", "target_theorem": "lemma p_findNat_some (p : Nat -> Prop) [DecidablePred p] (i : Nat) :\n  p i -> ∃ j, p j ∧ j <= i ∧ findNat p = some j :=", "ground_truth_proof": ":= by\n  intro pi;\n  have : (findNat p).isSome := by\n    false_or_by_contra; rename_i h\n    simp only [Bool.not_eq_true, Option.isSome_eq_false_iff] at h\n    have h := findNat_none _ h\n    aesop\n  revert this; simp [Option.isSome_iff_exists]\n  intro x h\n  have := findNat_aux_some_le p 0 h\n  exists x; repeat' constructor\n  { solve_by_elim [findNat_some_p] }\n  { have h := fun h₁ h₂ => this _ h₁ h₂ pi\n    simp at h; exact h }\n  solve_by_elim", "nesting_depth": 4, "transitive_dep_count": 18, "subset_aristotle": true, "category": "Framework"}
{"id": 393, "thm_name": "PartialCorrectness.DemonicChoice.NonDetT.wp_pickCont", "thm_stmt": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) [Findable p] post :\n  _root_.wp (NonDetT.pickCont τ p f) post =  ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/NonDetT/Basic.lean", "imports": ["import Loom.MonadAlgebras.WP.Gen", "import Mathlib.Order.CompleteBooleanAlgebra", "import Mathlib.Logic.Function.Basic", "import Loom.MonadAlgebras.WP.Basic", "import Loom.MonadAlgebras.NonDetT.Findable", "import Loom.MonadAlgebras.Defs", "import Loom.MonadAlgebras.WP.Tactic", "import Mathlib.Order.Lattice", "import Mathlib.Order.Basic"], "used_lib_defs": [{"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "ForInStep", "module": "Init.Core"}, {"name": "ForInStep.yield", "module": "Init.Core"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Encodable", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Encodable.decode", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Membership", "module": "Init.Prelude"}, {"name": "Membership.mem", "module": "Init.Prelude"}, {"name": "Set.Mem", "module": "Mathlib.Data.Set.Defs"}, {"name": "id", "module": "Init.Prelude"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.MonadEnv", "module": "Lean.Environment"}, {"name": "Lean.SimpleScopedEnvExtension", "module": "Lean.ScopedEnvExtension"}, {"name": "Lean.SimplePersistentEnvExtension", "module": "Lean.EnvExtension"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}], "used_repo_defs": [{"name": "WPGen.assert", "content": "def WPGen.assert {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteBooleanAlgebra l] [MAlgOrdered m l] (h : l) : WPGen (assertGadget (m := m) h) where\n  get := fun post => h ⊓ (h ⇨ post .unit)\n  prop := by admit /- proof elided -/"}, {"name": "WPGen.if", "content": "def WPGen.if {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteBooleanAlgebra l] [MAlgOrdered m l]\n  {hd : Decidable h} {x y : m α}\n  (wpgx : h → WPGen x) (wpgy : ¬h → WPGen y)\n  : WPGen (if h then x else y) where\n  get := fun post =>\n    (⨅ hc : WithName h (Lean.Name.anonymous.mkStr \"if_pos\"), (wpgx hc).get post) ⊓\n    (⨅ hc : WithName (¬h) (Lean.Name.anonymous.mkStr \"if_neg\"), (wpgy hc).get post)\n  prop := by admit /- proof elided -/"}, {"name": "WPGen.let", "content": "def WPGen.let  {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteBooleanAlgebra l] [MAlgOrdered m l]\n  (y : β) {x : β -> m α} (wpgx : ∀ y, WPGen (x y)) : WPGen (let z := y; x z) where\n  get := fun post => ⨅ z, ⌜z = y⌝ ⇨ (wpgx z).get post\n  prop := by admit /- proof elided -/\nmacro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "LE.pure", "content": "noncomputable def LE.pure {l : Type u} [inst: LE l] [OrderTop l] [OrderBot l] : Prop -> l := fun p =>\n  if p then ⊤ else ⊥"}, {"name": "spec", "content": "def spec (pre : l) (post : α -> l) : Cont l α :=\n  fun p => pre ⊓ ⌜post ≤ p⌝"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "Findable", "content": "class Findable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_none : (find ()).isNone -> ∀ x, ¬ p x\n  find_some_p : find () = some x -> p x"}, {"name": "find", "content": "def find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode"}, {"name": "findNat", "content": "def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}, {"name": "spec", "content": "def spec (pre : l) (post : α -> l) : Cont l α :=\n  fun p => pre ⊓ ⌜post ≤ p⌝"}, {"name": "Loop.forIn.loop", "content": "@[specialize, inline]\ndef Loop.forIn.loop {m : Type u -> Type v} [Monad m] [∀ α, CCPO (m α)] [MonoBind m] (f : Unit → β → m (ForInStep β)) (b : β) : m β := do\n    match ← f () b with\n      | ForInStep.done b  => pure b\n      | ForInStep.yield b => loop f b\n  partial_fixpoint"}, {"name": "WPGen.bind", "content": "def WPGen.bind {x : m α} {f : α -> m β} (wpg : WPGen x) (wpgf : ∀ a, WPGen (f a)) :\n  WPGen (x >>= f) where\n  get := fun post => wpg.get (fun a => (wpgf a).get post)\n  prop := by admit /- proof elided -/"}, {"name": "_root_.Lean.SimpleScopedEnvExtension.get", "content": "private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "Context", "content": "structure Context where\n  ref         : Syntax\n   \n  m           : Syntax\n   \n  returnType  : Syntax\n  mutableVars : VarSet := {}\n  insideFor   : Bool := false"}, {"name": "_root_.Lean.SimplePersistentEnvExtension.get", "content": "private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "WPGen", "content": "structure WPGen (x : m α) where\n  get : Cont l α\n  \n  prop : ∀ post, get post <= wp x post"}, {"name": "_root_.Lean.EnvExtension.get", "content": "private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "CCPOBotLawful", "content": "class CCPOBotLawful (m : Type u -> Type v) [∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] where\n  prop {α} : CCPOBot.compBot (m := m) (α := α) = Lean.Order.bot"}], "lib_lemmas": [{"name": "iSup_le_iff", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "le_himp_iff", "module": "Mathlib.Order.Heyting.Basic"}, {"name": "map_eq_pure_bind", "module": "Init.Control.Lawful.Basic"}], "repo_lemmas": [{"name": "emp_sum", "content": "@[aesop safe]\nlemma emp_sum (x: Bal) (h: 0 ≤ x) : ∃ h: List Nat, h.sum ≤ x"}, {"name": "WPGen.intro", "content": "omit [LawfulMonad m] in\nlemma WPGen.intro (x : m α) (wpg : WPGen x) :\n  pre <= wpg.get post ->\n  -- pre <= wpg.sideCond ->\n  triple pre x post"}, {"name": "triple_forIn_deacreasing", "content": "theorem triple_forIn_deacreasing {β} {measure : β -> ℕ}\n  {init : β} {f : β → m (ForInStep β)}\n  (inv : β → l)\n  (hstep : ∀ b,\n    measure b <= measure init ->\n    triple\n      (inv b)\n      (f b)\n      (fun | .yield b' => inv b' ⊓ ⌜measure b' < measure b⌝ | .done b' => ⌜ measure b' = 0 ⌝ ⊓ inv b')) :\n  triple (inv init) (forIn [0:measure init] init (fun _ => f)) (fun b => inv b ⊓ ⌜measure b = 0⌝)"}, {"name": "trueE", "content": "@[simp]\nlemma trueE (l : Type v) [inst: LE l] [OrderTop l] [OrderBot l] : ⌜True⌝ = (⊤ : l)"}, {"name": "falseE", "content": "@[simp]\nlemma falseE (l : Type v) [inst: LE l] [OrderTop l] [OrderBot l] : ⌜False⌝ = (⊥ : l)"}, {"name": "LE.pure_imp", "content": "lemma LE.pure_imp {l : Type u} [inst: LE l] [OrderTop l] [OrderBot l]\n  (p₁ p₂ : Prop) : (p₁ -> p₂) -> ⌜p₁⌝ <= (⌜p₂⌝ : l)"}, {"name": "LE.pure_intro", "content": "@[simp]\nlemma LE.pure_intro {l : Type u} [inst: LE l] [OrderTop l] [OrderBot l]\n  (p : Prop) (h : l) : (⌜p⌝ <= h) = (p -> ⊤ <= h)"}, {"name": "pure_intro_l", "content": "@[simp]\nlemma pure_intro_l {l : Type u} [CompleteLattice l] (x y : l) :\n  (x ⊓ ⌜ p ⌝ <= y) = (p -> x <= y)"}, {"name": "pure_intro_r", "content": "@[simp]\nlemma pure_intro_r {l : Type u} [CompleteLattice l] (x y : l) :\n  (⌜ p ⌝ ⊓ x <= y) = (p -> x <= y)"}, {"name": "MAlgOrdered.bind", "content": "lemma MAlgOrdered.bind {α : Type u} {m} {l : Type u} [Monad m] [CompleteLattice l] [MAlgOrdered m l] :\n    ∀ (x : m α) (f g : α -> m l), μ ∘ f = μ ∘ g ->\n     μ (x >>= f) = μ (x >>= g)"}, {"name": "Cont.monotone_lift", "content": "lemma Cont.monotone_lift {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteLattice l] [MAlgOrdered m l] :\n  ∀ {α : Type u} (x : m α), MAlg.lift x |>.monotone"}, {"name": "MAlg.μ_eq", "content": "@[simp]\nlemma MAlg.μ_eq {m l} [Monad m] [CompleteLattice l] [MAlgOrdered m l] : MAlg.μ (m := m) = MAlgOrdered.μ (m := m)"}, {"name": "MAlg.monadLift_bind", "content": "lemma MAlg.monadLift_bind {α β} {l : Type u} {m : Type u -> Type v} [Monad m] [LawfulMonad m] [CompleteLattice l] [MAlgOrdered m l]\n  (x : m α) (f g : α -> Cont l β) :\n    f <= g ->\n    (lift x >>= f) ≤ (lift x >>= g)"}, {"name": "lift_cont_eq", "content": "@[simp]\nlemma lift_cont_eq {l σ : Type u} [CompleteLattice l] [CompleteLattice σ] (c : Cont l α) :\n  liftM (m := Cont l) (n := Cont (σ -> l)) c = fun post s => c (post · s)"}, {"name": "leE", "content": "@[loomLogicSimp]\nlemma leE (l : Type u) [PartialOrder l] (a b : α -> l) : a ≤ b ↔ ∀ x, a x ≤ b x"}, {"name": "lePropE", "content": "@[loomLogicSimp]\nlemma lePropE (a b : Prop) : (a ≤ b) = (a → b)"}, {"name": "pureE", "content": "@[loomLogicSimp]\nlemma pureE (l : Type u) [CompleteLattice l] (a : Prop) : (⌜a⌝ : α -> l) = fun _ => ⌜a⌝"}, {"name": "purePropE", "content": "@[loomLogicSimp]\nlemma purePropE  : (⌜a⌝ : Prop) = a"}, {"name": "infPropE", "content": "@[loomLogicSimp]\nlemma infPropE (a b : Prop) : (a ⊓ b) = (a ∧ b)"}, {"name": "infE", "content": "@[loomLogicSimp]\nlemma infE (l : Type u) [CompleteLattice l] (a b : α -> l) : (a ⊓ b) = fun x => a x ⊓ b x"}, {"name": "supE", "content": "@[loomLogicSimp]\nlemma supE (l : Type u) [CompleteLattice l] (a b : α -> l) : (a ⊔ b) = fun x => a x ⊔ b x"}, {"name": "supPropE", "content": "@[loomLogicSimp]\nlemma supPropE (a b : Prop) : (a ⊔ b) = (a ∨ b)"}, {"name": "iInfE", "content": "@[loomLogicSimp]\nlemma iInfE (l : Type u) [CompleteLattice l] (a : ι -> α -> Prop) : (⨅ i, a i) = fun x => ⨅ i, a i x"}, {"name": "iSupE", "content": "@[loomLogicSimp]\nlemma iSupE (l : Type u) [CompleteLattice l] (a : ι -> α -> Prop) : (⨆ i, a i) = fun x => ⨆ i, a i x"}, {"name": "himpE", "content": "@[loomLogicSimp]\nlemma himpE  (l : Type u) [CompleteBooleanAlgebra l] (a b : α -> l) :\n  (a ⇨ b) = fun x => a x ⇨ b x"}, {"name": "himpPureE", "content": "@[loomLogicSimp]\nlemma himpPureE (a b : Prop) :\n  (a ⇨ b) = (a -> b)"}, {"name": "topE", "content": "@[loomLogicSimp]\nlemma topE (l : Type u) [CompleteLattice l] : (⊤ : α -> l) = fun _ => ⊤"}, {"name": "topPureE", "content": "@[loomLogicSimp]\nlemma topPureE : (⊤ : Prop) = True"}, {"name": "List.sum_lt", "content": "@[aesop safe]\ntheorem List.sum_lt (x: Bal) : x < y -> x < ([Int.toNat y]).sum"}, {"name": "balance_lt", "content": "@[aesop safe]\ntheorem balance_lt (x: Bal) : x < x + 1"}, {"name": "Except.bind'_bind", "content": "lemma Except.bind'_bind {m : Type u -> Type v} {ε α β} [Monad m] [LawfulMonad m] (i : m (Except ε α)) (f : α -> ExceptT ε m β) :\n  (i >>= fun a => Except.bind' a f) = bind (m := ExceptT ε m) i f"}, {"name": "MAlg.lift_ExceptT", "content": "lemma MAlg.lift_ExceptT ε (hd : ε -> Prop) [IsHandler hd] [CompleteLattice l] [inst: MAlgOrdered m l]\n   (c : ExceptT ε m α) post :\n  MAlg.lift c post = MAlg.lift (m := m) c (fun | .ok x => post x | .error e => ⌜hd e⌝)"}, {"name": "lift_map", "content": "lemma lift_map {α : Type u} {β : Type u} (f : α -> β) (x : m α)\n  [Monad m] [Monad n] [LawfulMonad m] [LawfulMonad n] [MonadLiftT m n] [LawfulMonadLiftT m n] :\n  liftM (f <$> x) = f <$> liftM (n := n) x"}, {"name": "tail_length", "content": "@[aesop unsafe]\ntheorem tail_length : ∀ q : List Nat, q.nonEmpty → q.tail.length < q.length"}, {"name": "tail_sum", "content": "@[aesop norm]\ntheorem tail_sum (q: List Nat) (hnemp: q.nonEmpty): q.sum = q.tail.sum + q.head!"}, {"name": "non_zero_length", "content": "@[aesop unsafe]\ntheorem non_zero_length (q: List Nat) (hnemp: q.nonEmpty): 0 < q.length"}, {"name": "sum_zero", "content": "@[aesop norm]\ntheorem sum_zero : ∀ q : List Nat, ¬q.nonEmpty → q.sum = 0"}, {"name": "meet_himp", "content": "lemma meet_himp (x x' y z : l) :\n  x = x' ->\n  (x ⇨ y) ⊓ (x' ⇨ z) = x ⇨ (y ⊓ z)"}, {"name": "NonDetT.wp_mono", "content": "lemma NonDetT.wp_mono  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α : Type u} (x : NonDetT m α) (f g : α -> l) :\n  (∀ a, f a <= g a) ->\n  NonDetT.wp x f <= NonDetT.wp x g"}, {"name": "W_ext", "content": "@[ext]\nlemma W_ext (t : Type v) (α : Type u) [Preorder t] (w w' : W t α) :\n  w.wp = w'.wp → w = w'"}], "used_local_defs": [{"name": "NonDetT", "content": "inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α"}, {"name": "NonDetT.bind", "content": "def NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f"}, {"name": "PartialCorrectness.DemonicChoice.NonDetT.wp", "content": "def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f =>\n  fun post =>\n    let p : Set τ := p;\n    ⨅ a ∈ (p : Set τ), wp (f a) post\n  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l),\n      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓\n      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)"}, {"name": "PartialCorrectness.DemonicChoice.NonDetT.μ", "content": "def NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}], "used_local_lemmas": [{"name": "PartialCorrectness.DemonicChoice.NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α β : Type u} (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}, {"name": "PartialCorrectness.DemonicChoice.NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}], "local_ctx": "import Mathlib.Logic.Function.Basic\n\nimport Mathlib.Order.CompleteBooleanAlgebra\n\nimport Mathlib.Order.Lattice\n\nimport Mathlib.Order.Basic\n\nimport Loom.MonadAlgebras.WP.Basic\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.Gen\n\nimport Loom.MonadAlgebras.NonDetT.Findable\n\nsection NonDeterministicTransformer\n\ninductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α\n\nvariable {m : Type u -> Type v} {α β : Type u} [Monad m]\n\ndef NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f\n\nvariable [CompleteBooleanAlgebra l] [MAlgOrdered m l]\n\nnamespace PartialCorrectness\n\nnamespace DemonicChoice\n\n/- WP for NonDetT -/\nnoncomputable\n\ndef NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f =>\n  fun post =>\n    let p : Set τ := p;\n    ⨅ a ∈ (p : Set τ), wp (f a) post\n  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l),\n      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓\n      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)\n\ndef NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id\n\nvariable [LawfulMonad m]\n\n/- Ordered Monad Algebra instance for NonDetT -/\nnoncomputable\nscoped", "target_theorem": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) [Findable p] post :\n  _root_.wp (NonDetT.pickCont τ p f) post =  ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post :=", "ground_truth_proof": ":= by\n  simp [NonDetT.wp_eq_wp, NonDetT.wp]; congr; ext x\n  simp [Membership.mem, Set.Mem]\n  by_cases h: p x <;> simp [h]", "nesting_depth": 5, "transitive_dep_count": 48, "subset_aristotle": false, "category": "Framework"}
{"id": 394, "thm_name": "TotalCorrectness.DemonicChoice.NonDetT.wp_pickCont", "thm_stmt": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) [Findable p] post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⌜∃ h, p h⌝ ⊓ ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/NonDetT/Basic.lean", "imports": ["import Loom.MonadAlgebras.WP.Gen", "import Mathlib.Order.CompleteBooleanAlgebra", "import Mathlib.Logic.Function.Basic", "import Loom.MonadAlgebras.WP.Basic", "import Loom.MonadAlgebras.NonDetT.Findable", "import Loom.MonadAlgebras.Defs", "import Loom.MonadAlgebras.WP.Tactic", "import Mathlib.Order.Lattice", "import Mathlib.Order.Basic"], "used_lib_defs": [{"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "CompleteLattice", "module": "Mathlib.Order.CompleteLattice.Defs"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "outParam", "module": "Init.Prelude"}, {"name": "f", "module": "Test.SmtTranslation.Trigger"}, {"name": "ForInStep", "module": "Init.Core"}, {"name": "ForInStep.yield", "module": "Init.Core"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "measure", "module": "Init.WF"}, {"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Encodable", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Encodable.decode", "module": "Mathlib.Logic.Encodable.Basic"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Membership", "module": "Init.Prelude"}, {"name": "Membership.mem", "module": "Init.Prelude"}, {"name": "Set.Mem", "module": "Mathlib.Data.Set.Defs"}, {"name": "id", "module": "Init.Prelude"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.MonadEnv", "module": "Lean.Environment"}, {"name": "Lean.SimpleScopedEnvExtension", "module": "Lean.ScopedEnvExtension"}, {"name": "Lean.SimplePersistentEnvExtension", "module": "Lean.EnvExtension"}, {"name": "LawfulMonad", "module": "Init.Control.Lawful.Basic"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "spec", "content": "def spec (pre : l) (post : α -> l) : Cont l α :=\n  fun p => pre ⊓ ⌜post ≤ p⌝"}, {"name": "MAlgOrdered", "content": "class MAlgOrdered (l : outParam (Type v)) [Monad m] [CompleteLattice l] where\n  μ : m l -> l\n  μ_ord_pure : ∀ l, μ (pure l) = l\n  μ_ord_bind {α : Type v} :\n    ∀ (f g : α -> m l), μ ∘ f ≤ μ ∘ g ->\n      ∀ x : m α, μ (x >>= f) ≤ μ (x >>= g)"}, {"name": "Findable", "content": "class Findable {α : Type u} (p : α -> Prop) where\n  find : Unit -> Option α\n  find_none : (find ()).isNone -> ∀ x, ¬ p x\n  find_some_p : find () = some x -> p x"}, {"name": "find", "content": "def find [Encodable α] (p : α -> Prop) [DecidablePred p] : Option α :=\n  findNat (fun x => (Encodable.decode x).any (p ·)) |>.bind Encodable.decode"}, {"name": "findNat", "content": "def findNat (p : Nat -> Prop) [DecidablePred p] : Option Nat :=\n  let rec aux i :=\n    if p i then\n      some i\n    else\n      aux (i + 1)\n  partial_fixpoint\n  aux 0"}, {"name": "MAlg.lift", "content": "abbrev MAlg.lift {m : Type u -> Type v} {l : Type u} [Monad m] [MAlg m l] :\n  {α : Type u} -> m α -> Cont l α := fun x f => μ $ f <$> x"}, {"name": "MAlg", "content": "class MAlg [Monad m] (l : outParam (Type v)) where\n  μ : m l -> l\n  pure : ∀ l, μ (pure l) = l\n  bind : ∀ {α : Type v} (x : m α) (f g : α -> m l),\n    μ ∘ f = μ ∘ g ->\n    μ (x >>= f) = μ (x >>= g)"}, {"name": "WPGen.bind", "content": "def WPGen.bind {x : m α} {f : α -> m β} (wpg : WPGen x) (wpgf : ∀ a, WPGen (f a)) :\n  WPGen (x >>= f) where\n  get := fun post => wpg.get (fun a => (wpgf a).get post)\n  prop := by admit /- proof elided -/"}, {"name": "_root_.Lean.SimpleScopedEnvExtension.get", "content": "private def _root_.Lean.SimpleScopedEnvExtension.get [Inhabited σ] (ext : SimpleScopedEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "Context", "content": "structure Context where\n  ref         : Syntax\n   \n  m           : Syntax\n   \n  returnType  : Syntax\n  mutableVars : VarSet := {}\n  insideFor   : Bool := false"}, {"name": "_root_.Lean.SimplePersistentEnvExtension.get", "content": "private def _root_.Lean.SimplePersistentEnvExtension.get [Inhabited σ] (ext : SimplePersistentEnvExtension α σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "WPGen", "content": "structure WPGen (x : m α) where\n  get : Cont l α\n  \n  prop : ∀ post, get post <= wp x post"}, {"name": "_root_.Lean.EnvExtension.get", "content": "private def _root_.Lean.EnvExtension.get [Inhabited σ] (ext : EnvExtension σ)\n  [Monad m] [MonadEnv m] : m σ := do\n  return ext.getState (<- getEnv)"}, {"name": "CCPOBotLawful", "content": "class CCPOBotLawful (m : Type u -> Type v) [∀ α, Lean.Order.CCPO (m α)] [CCPOBot m] where\n  prop {α} : CCPOBot.compBot (m := m) (α := α) = Lean.Order.bot"}], "lib_lemmas": [{"name": "iSup_le_iff", "module": "Mathlib.Order.CompleteLattice.Basic"}, {"name": "le_himp_iff", "module": "Mathlib.Order.Heyting.Basic"}, {"name": "map_eq_pure_bind", "module": "Init.Control.Lawful.Basic"}], "repo_lemmas": [{"name": "MAlgOrdered.bind", "content": "lemma MAlgOrdered.bind {α : Type u} {m} {l : Type u} [Monad m] [CompleteLattice l] [MAlgOrdered m l] :\n    ∀ (x : m α) (f g : α -> m l), μ ∘ f = μ ∘ g ->\n     μ (x >>= f) = μ (x >>= g)"}], "used_local_defs": [{"name": "NonDetT", "content": "inductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α"}, {"name": "NonDetT.bind", "content": "def NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f"}, {"name": "PartialCorrectness.DemonicChoice.NonDetT.wp", "content": "def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f =>\n  fun post =>\n    let p : Set τ := p;\n    ⨅ a ∈ (p : Set τ), wp (f a) post\n  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l),\n      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓\n      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)"}, {"name": "PartialCorrectness.DemonicChoice.NonDetT.μ", "content": "def NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}, {"name": "PartialCorrectness.AngelicChoice.NonDetT.wp", "content": "def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f => fun post => ⨆ a, ⌜p a⌝ ⊓ wp (f a) post\n  | _, .repeatCont init f cont => fun post => ⨆ (inv : ForInStep _ -> l),\n    ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓\n    spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)"}, {"name": "PartialCorrectness.AngelicChoice.NonDetT.μ", "content": "def NonDetT.μ {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}, {"name": "TotalCorrectness.DemonicChoice.NonDetT.wp", "content": "def NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f => fun post => let p : Set τ := p; ⌜∃ h, p h⌝ ⊓  ⨅ a ∈ (p : Set τ), wp (f a) post\n  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l) (measure : β -> Nat),\n      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) (fun | .yield b' => inv (.yield b') ⊓ ⌜ measure b' < measure b ⌝ | .done b' => inv (.done b'))⌝ ⊓\n      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)"}, {"name": "TotalCorrectness.DemonicChoice.NonDetT.μ", "content": "def NonDetT.μ {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id"}], "used_local_lemmas": [{"name": "PartialCorrectness.DemonicChoice.NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] [LawfulMonad m] {α β : Type u} (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}, {"name": "PartialCorrectness.DemonicChoice.NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}, {"name": "PartialCorrectness.AngelicChoice.NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind [LawfulMonad m] {α β : Type u} {l : Type u} [CompleteLattice l] [MAlgOrdered m l] (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}, {"name": "PartialCorrectness.AngelicChoice.NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}, {"name": "TotalCorrectness.DemonicChoice.NonDetT.wp_bind", "content": "lemma NonDetT.wp_bind [LawfulMonad m] {α β : Type u} {l : Type u} [CompleteLattice l] [MAlgOrdered m l] (x : NonDetT m α) (f : α -> NonDetT m β)\n  (post : β -> l):\n  NonDetT.wp (x.bind f) post = NonDetT.wp x (fun x => NonDetT.wp (f x) post)"}, {"name": "TotalCorrectness.DemonicChoice.NonDetT.wp_eq_wp", "content": "lemma NonDetT.wp_eq_wp {α : Type u} (x : NonDetT m α) (post : α -> l) :\n  _root_.wp x post = NonDetT.wp x post"}], "local_ctx": "import Mathlib.Logic.Function.Basic\n\nimport Mathlib.Order.CompleteBooleanAlgebra\n\nimport Mathlib.Order.Lattice\n\nimport Mathlib.Order.Basic\n\nimport Loom.MonadAlgebras.WP.Basic\n\nimport Loom.MonadAlgebras.WP.Tactic\n\nimport Loom.MonadAlgebras.WP.Gen\n\nimport Loom.MonadAlgebras.NonDetT.Findable\n\nsection NonDeterministicTransformer\n\ninductive NonDetT (m : Type u -> Type v) : (α : Type u) -> Type _ where\n  | pure {α} (ret : α) : NonDetT m α\n  | vis {α} {β} (x : m β) (f : β → NonDetT m α) : NonDetT m α\n  | pickCont {α} (τ : Type u) (p : τ -> Prop) [Findable p] (f : τ → NonDetT m α) : NonDetT m α\n  | repeatCont {α} {β} (init : β) (f : β -> NonDetT m (ForInStep β)) (cont : β -> NonDetT m α) : NonDetT m α\n\nvariable {m : Type u -> Type v} {α β : Type u} [Monad m]\n\ndef NonDetT.bind (x : NonDetT m α) (f : α → NonDetT m β) : NonDetT m β :=\n  match x with\n  | pure ret => f ret\n  | vis x f' => vis x fun y => bind (f' y) f\n  | pickCont τ p f' => pickCont τ p fun t => bind (f' t) f\n  | repeatCont init f' cont => repeatCont init f' fun t => bind (cont t) f\n\nvariable [CompleteBooleanAlgebra l] [MAlgOrdered m l]\n\nnamespace PartialCorrectness\n\nnamespace DemonicChoice\n\n/- WP for NonDetT -/\nnoncomputable\n\ndef NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f =>\n  fun post =>\n    let p : Set τ := p;\n    ⨅ a ∈ (p : Set τ), wp (f a) post\n  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l),\n      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓\n      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)\n\ndef NonDetT.μ  {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id\n\nvariable [LawfulMonad m]\n\n/- Ordered Monad Algebra instance for NonDetT -/\nnoncomputable\nscoped\n\nend DemonicChoice\n\nnamespace AngelicChoice\n\n/- WP for NonDetT -/\nnoncomputable\n\ndef NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f => fun post => ⨆ a, ⌜p a⌝ ⊓ wp (f a) post\n  | _, .repeatCont init f cont => fun post => ⨆ (inv : ForInStep _ -> l),\n    ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) inv⌝ ⊓\n    spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)\n\ndef NonDetT.μ {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id\n\nvariable [LawfulMonad m]\n\n/- Ordered Monad Algebra instance for NonDetT -/\nnoncomputable\nscoped\n\nend AngelicChoice\n\nend PartialCorrectness\n\nnamespace TotalCorrectness\n\nnamespace DemonicChoice\n\n/- WP for NonDetT -/\nnoncomputable\n\ndef NonDetT.wp {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : {α : Type u} -> NonDetT m α -> Cont l α\n  | _, .pure ret => pure ret\n  | _, .vis x f => fun post => _root_.wp x fun a => wp (f a) post\n  | _, @NonDetT.pickCont _ _ τ p _ f => fun post => let p : Set τ := p; ⌜∃ h, p h⌝ ⊓  ⨅ a ∈ (p : Set τ), wp (f a) post\n  | _, @NonDetT.repeatCont _ _ β init f cont => fun post => ⨆ (inv : ForInStep β -> l) (measure : β -> Nat),\n      ⌜ ∀ b, (inv (ForInStep.yield b)) <= wp (f b) (fun | .yield b' => inv (.yield b') ⊓ ⌜ measure b' < measure b ⌝ | .done b' => inv (.done b'))⌝ ⊓\n      spec (inv (.yield init)) (fun b => inv (.done b)) (fun b => wp (cont b) post)\n\ndef NonDetT.μ {l : Type u} [CompleteLattice l] [MAlgOrdered m l] : NonDetT m l -> l := fun x => NonDetT.wp x id\n\nvariable [LawfulMonad m]\n\n/- Ordered Monad Algebra instance for NonDetT -/\nnoncomputable\nscoped", "target_theorem": "@[simp]\nlemma NonDetT.wp_pickCont {τ : Type u} p (f : τ → NonDetT m α) [Findable p] post :\n  _root_.wp (NonDetT.pickCont τ p f) post = ⌜∃ h, p h⌝ ⊓ ⨅ a, ⌜p a⌝ ⇨ _root_.wp (f a) post :=", "ground_truth_proof": ":= by\n  simp [NonDetT.wp_eq_wp, NonDetT.wp]; congr; ext x\n  simp [Membership.mem, Set.Mem]\n  by_cases h: p x <;> simp [h]", "nesting_depth": 5, "transitive_dep_count": 50, "subset_aristotle": false, "category": "Framework"}
{"id": 395, "thm_name": "Tm.sub_den_eq", "thm_stmt": "theorem Tm.sub_den_eq (e : Γ ⊢ τ) : ∀ {Δ}, (σ : Sb Γ Δ) → ⟦e.sub σ⟧ = (⟦e⟧) ∘' (⟦σ⟧)", "lean_root": "pcf-lean", "rel_path": "PCF/Denotation.lean", "imports": ["import PCF.Utility", "import «PCF».Flat", "import PCF.Domain", "import «PCF».Context"], "used_lib_defs": [{"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.fst", "module": "Init.Prelude"}, {"name": "Prod.snd", "module": "Init.Prelude"}, {"name": "Trans.trans", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}, {"name": "Con", "module": "Mathlib.GroupTheory.Congruence.Defs"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Eq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:max \"⟦\" τ \" ty⟧\" => Ty.den τ", "content": "notation:max \"⟦\" τ \" ty⟧\" => Ty.den τ"}, {"name": "notation:max \"⟦\" Γ \" cx⟧\" => Ev Γ", "content": "notation:max \"⟦\" Γ \" cx⟧\" => Ev Γ"}, {"name": "notation:100 \"⟦\" t \"⟧\" => Tm.den t", "content": "notation:100 \"⟦\" t \"⟧\" => Tm.den t"}, {"name": "notation:100 \"⟦\" r \"⟧\" => Ren.den r", "content": "notation:100 \"⟦\" r \"⟧\" => Ren.den r"}, {"name": "notation:100 \"⟦\" σ \"⟧\" => Sb.den σ", "content": "notation:100 \"⟦\" σ \"⟧\" => Sb.den σ"}, {"name": "notation:100 \"⟦\" C \" con⟧\" => Con.den C", "content": "notation:100 \"⟦\" C \" con⟧\" => Con.den C"}, {"name": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x", "content": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x"}, {"name": "notation:max \"⨆\" => Domain.sup", "content": "notation:max \"⨆\" => Domain.sup"}, {"name": "notation:max \"⋆\" => Order.refl", "content": "notation:max \"⋆\" => Order.refl"}, {"name": "notation:max \"⊥\" => Domain.bot", "content": "notation:max \"⊥\" => Domain.bot"}, {"name": "Mono", "content": "structure Mono (α) (β) [Order α] [Order β] where\n  act : α → β\n  act' : is_monotone act"}, {"name": "Tm", "content": "inductive Tm : Cx → Ty → Type\n  | var : ∀ τ, Γ ∋ τ → Tm Γ τ\n  | true : Tm Γ .bool\n  | false : Tm Γ .bool\n  | zero : Tm Γ .nat\n  | succ : Tm Γ .nat → Tm Γ .nat\n  | pred : Tm Γ .nat → Tm Γ .nat\n  | zero? : Tm Γ .nat → Tm Γ .bool\n  | cond : Tm Γ .bool → Tm Γ τ → Tm Γ τ → Tm Γ τ\n  | fn : Tm (Γ ∷ τ) υ → Tm Γ (τ ⇒ υ)\n  | app : Tm Γ (τ ⇒ υ) → Tm Γ τ → Tm Γ υ\n  | fix : Tm Γ (τ ⇒ τ) → Tm Γ τ"}, {"name": "Cont.eval", "content": "def Cont.eval {α : Type i} {β : Type j} [Order α] [Order β] [Domain α] [Domain β]\n  : Cont (Cont α β × α) β := ⟨\n    Mono.eval_cont,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Cont.fst", "content": "def Cont.fst [Order α] [Order β] [Domain α] [Domain β] : Cont (α × β) α :=\n  ⟨⟨Prod.fst, And.left⟩, Domain.sup_is_mono (fun _ ↦ ⋆)⟩"}, {"name": "Cont", "content": "structure Cont (α) (β) [Order α] [Order β] [Domain α] [Domain β] where\n  fn : Mono α β\n  sub : ∀ {c : Chain α}, fn.act (⨆ c) ⊑ ⨆ (fn ∘ c)"}, {"name": "Domain", "content": "class Domain (α) [Order α] where\n  bot : α\n  sup : (c : Chain α) → α\n  is_bot {x} : bot ⊑ x\n  is_bound (c) (n): c.act n ⊑ sup c\n  is_least (c) {d} : ({n : _} → c.act n ⊑ d) → sup c ⊑ d"}, {"name": "Order", "content": "class Order (α) where\n  R : α → α → Prop\n  refl {x} : R x x\n  trans {x y z} : R x y → R y z → R x z\n  anti {x y} : R x y → R y x → x = y"}, {"name": "Chain", "content": "def Chain (α : Type i) [Order α] := Mono Nat α"}, {"name": "is_monotone", "content": "def is_monotone [Order α] [Order β] (f : α → β) := ∀ {x y : α}, x ⊑ y → f x ⊑ f y"}, {"name": "Cont.snd", "content": "def Cont.snd [Order α] [Order β] [Domain α] [Domain β] : Cont (α × β) β :=\n  ⟨⟨Prod.snd, And.right⟩, Domain.sup_is_mono (fun _ ↦ ⋆)⟩"}, {"name": "Chain.apply", "content": "def Chain.apply [Order α] [Order β] [Domain α] [Domain β] (c : Chain (Cont α β)) (a : α) : Chain β\n  := Mono.apply c a"}, {"name": "Mono.apply", "content": "def Mono.apply [Order α] [Order β] [Domain α] [Domain β] (c : Mono Nat (Cont α β)) (a : α) : Chain β where\n  act := fun n ↦ (c n) a\n  act' := fun a_b ↦ (c • a_b) _"}, {"name": "Mono.eval_cont", "content": "def Mono.eval_cont {α : Type i} {β : Type j} [Order α] [Order β] [Domain α] [Domain β]\n  : Mono (Cont α β × α) β :=\n  ⟨fun x ↦ x.fst x.snd, fun {x y} p ↦ (x.fst • p.right) ⬝ (p.left y.snd)⟩"}, {"name": "Mono.from_cont", "content": "def Mono.from_cont [Order α] [Order β] [Domain α] [Domain β] : Mono (Cont α β) (Mono α β) :=\n  ⟨Cont.fn, fun p a ↦ p a⟩"}, {"name": "Mono.sup", "content": "def Mono.sup [Order α] [Domain α] : Mono (Chain α) α :=\n  ⟨⨆, Domain.sup_is_mono⟩"}, {"name": "Cont.comp'", "content": "def Cont.comp' [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] (f : Cont β γ) (g : Cont α β)\n  : Cont α γ\n  := ⟨\n    ⟨fun x ↦ f (g x), fun x_y ↦ f • g • x_y⟩,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.eval", "content": "def Mono.eval {α : Type i} {β : Type j} [Order α] [Order β] : Mono (Mono α β × α) β :=\n  ⟨fun x ↦ x.fst x.snd, fun {x y} p ↦ (x.fst • p.right) ⬝ (p.left y.snd)⟩"}, {"name": "Chain.fst", "content": "def Chain.fst [Order α] [Order β] (c : Chain (α × β)) : Chain α :=\n  ⟨fun n ↦ (c n).fst, fun p ↦ by admit /- proof elided -/\n  ⟩"}, {"name": "Chain.snd", "content": "def Chain.snd [Order α] [Order β] (c : Chain (α × β)) : Chain β :=\n  ⟨fun n ↦ (c n).snd, fun p ↦ by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.const", "content": "def Mono.const [Order α] [Order β] (b : β) : Mono α β := ⟨fun _ ↦ b, fun _ ↦ ⋆⟩"}, {"name": "Mono.comp", "content": "def Mono.comp {α : Type i} {β : Type j} {γ : Type k} [Order α] [Order β] [Order γ]\n  : Mono (Mono β γ × Mono α β) (Mono α γ) := ⟨\n    fun h ↦ ⟨fun x ↦ h.fst (h.snd x), fun x_y ↦ h.fst • (h.snd • x_y)⟩,\n    fun {h₀ h₁} h a ↦ (h₀.fst • h.right a) ⬝ (h.left (h₁.snd a))\n  ⟩"}, {"name": "Mono.pair", "content": "def Mono.pair [Order α] [Order β] [Order γ]\n  (f : Mono γ α) (g : Mono γ β) : Mono γ (α × β) :=\n    ⟨fun c ↦ ⟨f c, g c⟩, fun p ↦ ⟨f • p, g • p⟩⟩"}, {"name": "Cont.const", "content": "def Cont.const [Order α] [Order β] [Domain α] [Domain β] (b : β) : Cont α β := ⟨Mono.const b, fun {c} ↦ by admit /- proof elided -/\n  ⟩"}, {"name": "Domain.sup_of_const", "content": "def Domain.sup_of_const [Order α] [Domain α] (a : α) : ⨆ (Mono.const a) = a :=\n  (by admit /- proof elided -/\n  ) ⇄! (Domain.is_bound (Mono.const a) 0)"}, {"name": "Cont.flat", "content": "def Cont.flat (f : α → β) : (Cont (Flat α) (Flat β)) := (Mono.flat f).promote_trivial"}, {"name": "Flat", "content": "inductive Flat (α : Type) : Type where\n  | none : Flat α\n  | some : α → Flat α"}, {"name": "Mono.flat", "content": "def Mono.flat (f : α → β) : (Mono (Flat α) (Flat β)) := ⟨\n    lift_flat f,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "lift_flat", "content": "def lift_flat (f : α → β) : Flat α → Flat β\n| .none => .none\n| .some x => .some (f x)"}, {"name": "Cont.cond", "content": "def Cont.cond [Order α] [Domain α] : Cont (Flat Bool) (Cont (α × α) α) := ⟨\n  cond',\n  by admit /- proof elided -/\n⟩"}, {"name": "cond'", "content": "def cond' [Order α] [Domain α] : Mono (Flat Bool) (Cont (α × α) α) := ⟨\n  fun b ↦ (\n    match b with\n    | .none => Cont.const ⊥\n    | .some true => Cont.fst\n    | .some false => Cont.snd\n  ),\n  by admit /- proof elided -/\n⟩"}, {"name": "Cont.pred", "content": "def Cont.pred : Cont (Flat Nat) (Flat Nat) := Mono.pred.promote_trivial"}, {"name": "Mono.pred", "content": "def Mono.pred : Mono (Flat Nat) (Flat Nat) := ⟨\n    Nat.partial_pred,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Nat.partial_pred", "content": "def Nat.partial_pred : Flat Nat → Flat Nat :=\n  fun n ↦ match n with\n  | .some (.succ n) => .some n\n  | _ => .none"}, {"name": "Cont.curry", "content": "def Cont.curry {α : Type i} {β : Type j}\n  [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont (α × β) γ) : Cont α (Cont β γ) := ⟨\n    ⟨\n      fun a ↦ ⟨\n        ⟨\n          fun b ↦ f (a, b),\n          fun b' ↦ f • ⟨⋆, b'⟩\n        ⟩,\n        by admit /- proof elided -/\n      ⟩,\n      fun a' b ↦ f • ⟨a', ⋆⟩\n    ⟩,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Cont.pair", "content": "def Cont.pair [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont γ α) (g : Cont γ β) : Cont γ (α × β) := ⟨\n    ⟨fun c ↦ ⟨f c, g c⟩, fun p ↦ ⟨f • p, g • p⟩⟩,\n    ⟨f.sub ⬝ Domain.sup_is_mono (fun _ ↦ ⋆), g.sub ⬝ Domain.sup_is_mono (fun _ ↦ ⋆)⟩\n  ⟩"}, {"name": "Cont.uncurry", "content": "def Cont.uncurry {α : Type i} {β : Type j}\n  [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont α (Cont β γ)) : Cont (α × β) γ  := ⟨\n    Mono.uncurry_cont f,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.uncurry_cont", "content": "def Mono.uncurry_cont {α : Type i} {β : Type j}\n  [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont α (Cont β γ)) : Mono (α × β) γ := ⟨\n    fun ⟨a, b⟩ ↦ (f a) b,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Cont.fix'", "content": "def Cont.fix' [Order α] [Domain α] : Cont (Cont α α) α := ⟨\n    fix_mono,\n    by\n      intro f\n      apply fix_is_least_prefixed\n      calc ⨆ f (⨆ (fix_mono ∘ f))\n        _ = ⨆ (f.apply (⨆ (fix_mono ∘ f))) := rfl\n        _ ⊑ ⨆ (Mono.sup ∘ Mono.comp ∘ Mono.pair (Mono.from_cont ∘ f) (Mono.const (fix_mono ∘ f))) :="}, {"name": "fix_is_prefixed", "content": "def fix_is_prefixed [Order α] [Domain α] (f : Cont α α) : is_prefixed f (⨆ f.iterations) :="}, {"name": "sup_succ", "content": "def sup_succ [Order α] [Domain α] {c : Chain α} : ⨆ (c ∘ Mono.succ) ⊑ ⨆ c :="}, {"name": "Mono.succ", "content": "def Mono.succ : Mono Nat Nat := ⟨Nat.succ, Nat.succ_le_succ⟩"}, {"name": "Cont.iterations", "content": "def Cont.iterations [Order α] [Domain α] (f : Cont α α) : Chain α := ⟨\n    fun n ↦ Cont.iter n f ⊥,\n    increasing_implies_monotone (fun n ↦ iter n f ⊥) (by admit /- proof elided -/\n    )\n  ⟩"}, {"name": "Cont.iter", "content": "def Cont.iter [Order α] [Domain α] : Nat → Cont α α → Cont α α\n| 0 => fun _ ↦ Cont.id\n| .succ n => fun f ↦ f ∘ iter n f"}, {"name": "Cont.id", "content": "def Cont.id [Order α] [Domain α] : Cont α α := ⟨Mono.id, ⋆⟩"}, {"name": "Mono.id", "content": "def Mono.id [Order α] : Mono α α\n  := ⟨Function.id, Function.id⟩"}, {"name": "Function.id", "content": "@[inline] def Function.id {α : Sort u} (a : α) : α := a"}, {"name": "increasing_implies_monotone", "content": "def increasing_implies_monotone [Order α] (f : Nat → α) : (∀ n, f n ⊑ f n.succ) → is_monotone f :="}, {"name": "is_prefixed", "content": "def is_prefixed [Order α] [Domain α] (f : Cont α α) (a : α) := f a ⊑ a"}, {"name": "Cont.fix", "content": "def Cont.fix [Order α] [Domain α] (f : Cont α α) := ⨆ f.iterations"}, {"name": "fix_is_least_prefixed", "content": "def fix_is_least_prefixed [Order α] [Domain α] (f : Cont α α) (a : α) (h : is_prefixed f a)\n  : f.fix ⊑ a :="}, {"name": "Cont.fix_mono", "content": "def Cont.fix_mono [Order α] [Domain α] : Mono (Cont α α) α := ⟨\n    Cont.fix,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Nat.zero?", "content": "def Nat.zero? : Nat → Bool\n| .zero => true\n| _ => false"}, {"name": "Cont.swap", "content": "def Cont.swap [Order α] [Domain α] [Order β] [Domain β] : Cont (α × β) (β × α) := ⟨\n  Mono.swap,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.swap", "content": "def Mono.swap [Order α] [Order β] : Mono (α × β) (β × α) := ⟨\n    fun p ↦ ⟨p.snd, p.fst⟩,\n    fun ⟨a', b'⟩ ↦ ⟨b', a'⟩\n  ⟩"}, {"name": "Cont.assoc_swap_assoc", "content": "def Cont.assoc_swap_assoc {α : Type i} {β : Type j}\n  [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] : Cont ((α × β) × γ) ((α × γ) × β) := ⟨\n    Mono.assoc_swap_assoc,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Mono.assoc_swap_assoc", "content": "def Mono.assoc_swap_assoc {α : Type i} {β : Type j}\n  [Order α] [Order β] [Order γ] : Mono ((α × β) × γ) ((α × γ) × β) := ⟨\n    fun p ↦ ⟨⟨p.fst.fst, p.snd⟩, p.fst.snd⟩,\n    fun ⟨⟨a', b'⟩, c'⟩ ↦ ⟨⟨a', c'⟩, b'⟩\n  ⟩"}, {"name": "Ty", "content": "inductive Ty\n  | bool\n  | nat\n  | pow : Ty → Ty → Ty"}, {"name": "DomainType", "content": "structure DomainType : Type (i + 1) :=\n  carrier : Type i\n  order : Order carrier\n  domain : Domain carrier"}, {"name": "Cx", "content": "inductive Cx\n  | nil\n  | cons : Cx -> Ty -> Cx"}, {"name": "Ren.weak", "content": "def Ren.weak {τ  : Ty} : Ren Γ (Γ ∷ τ) := Var.s"}, {"name": "Var", "content": "inductive Var : Cx → Ty → Type\n  | z : ∀ {Γ : Cx}, Var (Γ ∷ τ) τ\n  | s : ∀ {Γ : Cx} {υ : Ty} τ, Var Γ τ → Var (Γ ∷ υ) τ"}, {"name": "Sb", "content": "def Sb Γ Δ := ∀ τ, Γ ∋ τ → Δ ⊢ τ"}, {"name": "infixr:100 \" ⇒ \" => Ty.pow", "content": "infixr:100 \" ⇒ \" => Ty.pow"}, {"name": "infixl:70 \" ∷ \"  => Cx.cons", "content": "infixl:70 \" ∷ \"  => Cx.cons"}, {"name": "infix:70 \" ∋ \" => Var", "content": "infix:70 \" ∋ \" => Var"}, {"name": "infix:70 \" ⊢ \" => Tm", "content": "infix:70 \" ⊢ \" => Tm"}, {"name": "infix:100 \" ⊑ \" => Order.R", "content": "infix:100 \" ⊑ \" => Order.R"}, {"name": "notation:max \"⋆\" => Order.refl", "content": "notation:max \"⋆\" => Order.refl"}, {"name": "infix:100 \" ⇄! \" => Order.anti", "content": "infix:100 \" ⇄! \" => Order.anti"}, {"name": "infixl:100 \" • \" => Mono.act'", "content": "infixl:100 \" • \" => Mono.act'"}, {"name": "notation:max \"⊥\" => Domain.bot", "content": "notation:max \"⊥\" => Domain.bot"}, {"name": "notation:max \"⨆\" => Domain.sup", "content": "notation:max \"⨆\" => Domain.sup"}, {"name": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x", "content": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x"}, {"name": "infix:100 \" ∘ \" => Cont.comp'", "content": "infix:100 \" ∘ \" => Cont.comp'"}, {"name": "infixr:100 \" ∘' \" => Cont.comp'", "content": "infixr:100 \" ∘' \" => Cont.comp'"}], "lib_lemmas": [{"name": "congrArg", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "Cont.ext", "content": "@[ext] theorem Cont.ext [Order α] [Order β] [Domain α] [Domain β]\n  {f g : Cont α β} (p : f.fn.act = g.fn.act) : f = g"}, {"name": "congrArg2", "content": "theorem congrArg2\n  {α₀ : Sort u₀} {α₁ : Sort u₁} {β : Sort v} {a₀ a₀' : α₀} {a₁ a₁' : α₁}\n  (f : α₀ → α₁ → β) (h₀ : Eq a₀ a₀') (h₁ : Eq a₁ a₁') : Eq (f a₀ a₁) (f a₀' a₁')"}, {"name": "Cont.pair_after", "content": "theorem Cont.pair_after [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] [Order δ] [Domain δ]\n  (f : Cont γ α) (g : Cont γ β) (h : Cont δ γ) : (f ∘' h).pair (g ∘' h) = (f.pair g) ∘' h"}], "used_local_defs": [{"name": "Ty.den", "content": "noncomputable def Ty.den : Ty → DomainType\n  | .bool => ⟨Flat Bool, _, inferInstance⟩\n  | .nat => ⟨Flat Nat, _, inferInstance⟩\n  | .pow T₀ T₁ => by admit /- proof elided -/"}, {"name": "Ev", "content": "def Ev (Γ : Cx) : Type := ∀ τ, Var Γ τ → ↑⟦τ ty⟧"}, {"name": "Ev.push", "content": "def Ev.push {Γ : Cx} (ρ : ⟦Γ cx⟧) {τ : Ty} (d : ↑⟦τ ty⟧) : ⟦Γ ∷ τ cx⟧ :=\n  fun {τ} x ↦ match x with\n  | .z => d\n  | .s τ x => ρ τ x"}, {"name": "Ev.from", "content": "def Ev.from {Γ : Cx} {τ : Ty} : Cont (⟦Γ cx⟧ × ⟦τ ty⟧) (⟦Γ ∷ τ cx⟧) := ⟨\n  ⟨\n    fun ⟨ρ, d⟩ υ x ↦ ρ.push d υ x,\n    by admit /- proof elided -/\n  ⟩,\n  by admit /- proof elided -/\n⟩"}, {"name": "Tm.den", "content": "noncomputable def Tm.den : (Γ ⊢ τ) → Cont (⟦Γ cx⟧) (⟦τ ty⟧)\n  | .var τ x => ⟨⟨fun ρ ↦ ρ τ x, fun ρ₀_ρ₁ ↦ ρ₀_ρ₁ τ x⟩, ⋆⟩\n  | .true => Cont.const (.some .true)\n  | .false => Cont.const (.some .false)\n  | .zero => Cont.const (.some 0)\n  | .succ e => Cont.flat (Nat.succ) ∘ e.den\n  | .pred e => Cont.pred ∘ e.den\n  | .zero? e => Cont.flat (Nat.zero?) ∘ e.den\n  | .cond s t f  => Cont.uncurry (Cont.cond) ∘ Cont.pair s.den (Cont.pair t.den f.den)\n  | .fn e  => Cont.curry (e.den ∘ Ev.from)\n  | .app f e => Cont.eval ∘ (Cont.pair f.den e.den)\n  | .fix f => Cont.fix' ∘ f.den"}, {"name": "Ren.den", "content": "noncomputable def Ren.den (r : Ren Γ Δ) : Cont (⟦Δ cx⟧) (⟦Γ cx⟧) :=\n  ⟨⟨fun ρ _ x ↦ (⟦(x.ren r).tm⟧) ρ, fun ρ' _ x ↦ (⟦(x.ren r).tm⟧) • ρ'⟩, fun _ x ↦ (⟦(x.ren r).tm⟧).sub⟩"}, {"name": "Sb.den", "content": "noncomputable def Sb.den (σ : Sb Γ Δ) : Cont (⟦Δ cx⟧) (⟦Γ cx⟧) :=\n  ⟨⟨fun ρ _ x ↦ (⟦x.sub σ⟧) ρ, fun ρ' _ x ↦ (⟦x.sub σ⟧) • ρ'⟩, fun _ x ↦ (⟦x.sub σ⟧).sub⟩"}, {"name": "Con.den", "content": "noncomputable def Con.den : Con Δ υ Γ τ → Cont (⟦Γ cx⟧ × Cont (⟦Δ cx⟧) (⟦υ ty⟧)) ⟦τ ty⟧\n  | id         => Cont.uncurry Cont.id ∘' Cont.swap\n  | comp C₀ C₁ => Cont.uncurry (Cont.curry (C₁.den ∘' Cont.swap)\n                             ∘' Cont.curry (C₀.den ∘' Cont.swap)) ∘' Cont.swap\n  | sub C σ => Cont.uncurry ((Cont.curry C.den) ∘' (⟦σ⟧))\n  | succ C => Cont.flat (Nat.succ) ∘' C.den\n  | pred C => Cont.pred ∘' C.den\n  | zero? C => Cont.flat (Nat.zero?) ∘' C.den\n  | fn C   => Cont.curry ((Cont.uncurry (Cont.curry C.den ∘' Ev.from)) ∘' Cont.assoc_swap_assoc)\n  | cond_s C t f => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair C.den (Cont.pair ((⟦t⟧) ∘' Cont.fst) ((⟦f⟧) ∘' Cont.fst))\n  | cond_t s C f => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair ((⟦s⟧) ∘' Cont.fst) (Cont.pair C.den ((⟦f⟧) ∘' Cont.fst))\n  | cond_f s t C => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair ((⟦s⟧) ∘' Cont.fst) (Cont.pair ((⟦t⟧) ∘' Cont.fst) C.den)\n  | app_f C a => Cont.eval ∘' (Cont.pair C.den ((⟦a⟧) ∘' Cont.fst))\n  | app_a f C => Cont.eval ∘' (Cont.pair ((⟦f⟧) ∘' Cont.fst) C.den)\n  | fix C => Cont.fix' ∘' C.den"}], "used_local_lemmas": [{"name": "Ren.weak_den_eq", "content": "theorem Ren.weak_den_eq : (⟦Ren.weak⟧) (Ev.from (ρ, d)) = ρ"}], "local_ctx": "import «PCF».Flat\n\nimport «PCF».Context\n\nnoncomputable def Ty.den : Ty → DomainType\n  | .bool => ⟨Flat Bool, _, inferInstance⟩\n  | .nat => ⟨Flat Nat, _, inferInstance⟩\n  | .pow T₀ T₁ => by admit /- proof elided -/\n\nnotation:max \"⟦\" τ \" ty⟧\" => Ty.den τ\n\ndef Ev (Γ : Cx) : Type := ∀ τ, Var Γ τ → ↑⟦τ ty⟧\n\nnotation:max \"⟦\" Γ \" cx⟧\" => Ev Γ\n\ndef Ev.push {Γ : Cx} (ρ : ⟦Γ cx⟧) {τ : Ty} (d : ↑⟦τ ty⟧) : ⟦Γ ∷ τ cx⟧ :=\n  fun {τ} x ↦ match x with\n  | .z => d\n  | .s τ x => ρ τ x\n\ndef Ev.from {Γ : Cx} {τ : Ty} : Cont (⟦Γ cx⟧ × ⟦τ ty⟧) (⟦Γ ∷ τ cx⟧) := ⟨\n  ⟨\n    fun ⟨ρ, d⟩ υ x ↦ ρ.push d υ x,\n    by admit /- proof elided -/\n  ⟩,\n  by admit /- proof elided -/\n⟩\n\nnoncomputable def Tm.den : (Γ ⊢ τ) → Cont (⟦Γ cx⟧) (⟦τ ty⟧)\n  | .var τ x => ⟨⟨fun ρ ↦ ρ τ x, fun ρ₀_ρ₁ ↦ ρ₀_ρ₁ τ x⟩, ⋆⟩\n  | .true => Cont.const (.some .true)\n  | .false => Cont.const (.some .false)\n  | .zero => Cont.const (.some 0)\n  | .succ e => Cont.flat (Nat.succ) ∘ e.den\n  | .pred e => Cont.pred ∘ e.den\n  | .zero? e => Cont.flat (Nat.zero?) ∘ e.den\n  | .cond s t f  => Cont.uncurry (Cont.cond) ∘ Cont.pair s.den (Cont.pair t.den f.den)\n  | .fn e  => Cont.curry (e.den ∘ Ev.from)\n  | .app f e => Cont.eval ∘ (Cont.pair f.den e.den)\n  | .fix f => Cont.fix' ∘ f.den\n\nnotation:100 \"⟦\" t \"⟧\" => Tm.den t\n\nnoncomputable def Ren.den (r : Ren Γ Δ) : Cont (⟦Δ cx⟧) (⟦Γ cx⟧) :=\n  ⟨⟨fun ρ _ x ↦ (⟦(x.ren r).tm⟧) ρ, fun ρ' _ x ↦ (⟦(x.ren r).tm⟧) • ρ'⟩, fun _ x ↦ (⟦(x.ren r).tm⟧).sub⟩\n\nnotation:100 \"⟦\" r \"⟧\" => Ren.den r\n\nnoncomputable def Sb.den (σ : Sb Γ Δ) : Cont (⟦Δ cx⟧) (⟦Γ cx⟧) :=\n  ⟨⟨fun ρ _ x ↦ (⟦x.sub σ⟧) ρ, fun ρ' _ x ↦ (⟦x.sub σ⟧) • ρ'⟩, fun _ x ↦ (⟦x.sub σ⟧).sub⟩\n\nnotation:100 \"⟦\" σ \"⟧\" => Sb.den σ\n\nnoncomputable def Con.den : Con Δ υ Γ τ → Cont (⟦Γ cx⟧ × Cont (⟦Δ cx⟧) (⟦υ ty⟧)) ⟦τ ty⟧\n  | id         => Cont.uncurry Cont.id ∘' Cont.swap\n  | comp C₀ C₁ => Cont.uncurry (Cont.curry (C₁.den ∘' Cont.swap)\n                             ∘' Cont.curry (C₀.den ∘' Cont.swap)) ∘' Cont.swap\n  | sub C σ => Cont.uncurry ((Cont.curry C.den) ∘' (⟦σ⟧))\n  | succ C => Cont.flat (Nat.succ) ∘' C.den\n  | pred C => Cont.pred ∘' C.den\n  | zero? C => Cont.flat (Nat.zero?) ∘' C.den\n  | fn C   => Cont.curry ((Cont.uncurry (Cont.curry C.den ∘' Ev.from)) ∘' Cont.assoc_swap_assoc)\n  | cond_s C t f => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair C.den (Cont.pair ((⟦t⟧) ∘' Cont.fst) ((⟦f⟧) ∘' Cont.fst))\n  | cond_t s C f => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair ((⟦s⟧) ∘' Cont.fst) (Cont.pair C.den ((⟦f⟧) ∘' Cont.fst))\n  | cond_f s t C => Cont.uncurry (Cont.cond)\n    ∘' Cont.pair ((⟦s⟧) ∘' Cont.fst) (Cont.pair ((⟦t⟧) ∘' Cont.fst) C.den)\n  | app_f C a => Cont.eval ∘' (Cont.pair C.den ((⟦a⟧) ∘' Cont.fst))\n  | app_a f C => Cont.eval ∘' (Cont.pair ((⟦f⟧) ∘' Cont.fst) C.den)\n  | fix C => Cont.fix' ∘' C.den\n\nnotation:100 \"⟦\" C \" con⟧\" => Con.den C", "target_theorem": "theorem Tm.sub_den_eq (e : Γ ⊢ τ) : ∀ {Δ}, (σ : Sb Γ Δ) → ⟦e.sub σ⟧ = (⟦e⟧) ∘' (⟦σ⟧) :=", "ground_truth_proof": ":= by\n  induction e with\n  | fn e Φ =>\n    intro _ σ\n    calc ⟦e.fn.sub σ⟧\n      _ = Cont.curry ((⟦e.sub (σ.keep _)⟧) ∘ Ev.from) := rfl\n      _ = Cont.curry (((⟦e⟧) ∘' ⟦σ.keep _⟧) ∘ Ev.from) := by rw [Φ (σ.keep _)]\n      _ = (⟦e.fn⟧) ∘' ⟦σ⟧ := by {\n        apply Cont.ext ∘ funext\n        intro ρ\n        apply Cont.ext ∘ funext\n        intro d\n        have p : (⟦σ.keep _⟧) (Ev.from (ρ, d)) = Ev.from ((⟦σ⟧) ρ, d) := by {\n          funext τ x\n          cases x with\n          | z => rfl\n          | s τ x =>\n            calc (⟦σ.keep _⟧) (Ev.from (ρ, d)) τ x.succ\n              _ = (⟦(x.sub σ).ren Ren.weak⟧) (Ev.from (ρ, d)) := rfl\n              _ = ((⟦x.sub σ⟧) ∘' ⟦Ren.weak⟧) (Ev.from (ρ, d)) := by rw [(x.sub σ).ren_den_eq]\n              _ = (⟦x.sub σ⟧) ((⟦Ren.weak⟧) (Ev.from (ρ, d))) := rfl\n              _ = (⟦x.sub σ⟧) (ρ) := by rw [Ren.weak_den_eq]\n              _ = Ev.from ((⟦σ⟧) ρ, d) τ x.s := rfl\n        }\n        calc ((((⟦e⟧) ∘' ⟦σ.keep _⟧) ∘' Ev.from).curry ρ) d\n          _ = (⟦e⟧) ((⟦σ.keep _⟧) (Ev.from (ρ, d))) := rfl\n          _ = (⟦e⟧) (Ev.from ((⟦σ⟧) ρ, d)) := by rw [p]\n          _ = ((⟦e.fn⟧) ((⟦σ⟧) ρ)) d := rfl\n      }\n  | var | true | false | zero => intros; rfl\n  | succ _ Φ | pred _ Φ | zero? _ Φ | fix _ Φ => intro _ σ; exact congrArg _ (Φ σ)\n  | app _ _ Φf Φa =>\n    intro _ σ; exact congrArg2 (fun f a ↦ Cont.eval ∘' Cont.pair f a) (Φf σ) (Φa σ)\n  | cond s t f Φs Φt Φf =>\n    intro _ σ\n    calc ⟦(s.cond t f).sub σ⟧\n      _ = _ ∘' Cont.pair (⟦s.sub σ⟧) (Cont.pair (⟦t.sub σ⟧) (⟦f.sub σ⟧))          := rfl\n      _ = _ ∘' Cont.pair ((⟦s⟧) ∘' ⟦σ⟧) (Cont.pair ((⟦t⟧) ∘' ⟦σ⟧) ((⟦f⟧) ∘' ⟦σ⟧)) := by rw [Φs, Φt, Φf]\n      _ = _ ∘' Cont.pair (⟦s⟧) ((Cont.pair (⟦t⟧) (⟦f⟧))) ∘' ⟦σ⟧\n        := by rw [Cont.pair_after (⟦t⟧) (⟦f⟧) (⟦σ⟧), Cont.pair_after (⟦s⟧) _ (⟦σ⟧)]\n      _ = (⟦s.cond t f⟧) ∘' ⟦σ⟧ := rfl", "nesting_depth": 12, "transitive_dep_count": 91, "subset_aristotle": false, "category": "Semantics"}
{"id": 396, "thm_name": "Flat.sup_some", "thm_stmt": "theorem Flat.sup_some {c : Chain _} {a : α} : (∃ k, c.act k = .some a) ↔ (flat_sup c = .some a)", "lean_root": "pcf-lean", "rel_path": "PCF/Flat.lean", "imports": ["import «PCF».Domain"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Chain", "content": "def Chain (α : Type i) [Order α] := Mono Nat α"}, {"name": "Order", "content": "class Order (α) where\n  R : α → α → Prop\n  refl {x} : R x x\n  trans {x y z} : R x y → R y z → R x z\n  anti {x y} : R x y → R y x → x = y"}, {"name": "Mono", "content": "structure Mono (α) (β) [Order α] [Order β] where\n  act : α → β\n  act' : is_monotone act"}, {"name": "is_monotone", "content": "def is_monotone [Order α] [Order β] (f : α → β) := ∀ {x y : α}, x ⊑ y → f x ⊑ f y"}, {"name": "infix:100 \" ⊑ \" => Order.R", "content": "infix:100 \" ⊑ \" => Order.R"}, {"name": "infix:100 \" ⇄! \" => Order.anti", "content": "infix:100 \" ⇄! \" => Order.anti"}, {"name": "infixl:100 \" • \" => Mono.act'", "content": "infixl:100 \" • \" => Mono.act'"}, {"name": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x", "content": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x"}], "lib_lemmas": [{"name": "Classical.em", "module": "Init.Classical"}, {"name": "Nat.gt_of_not_le", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.le_of_lt", "module": "Init.Data.Nat.Basic"}, {"name": "dif_neg", "module": "Init.Core"}, {"name": "dif_pos", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Flat", "content": "inductive Flat (α : Type) : Type where\n  | none : Flat α\n  | some : α → Flat α"}, {"name": "flat_sup", "content": "noncomputable def flat_sup (c : Chain (Flat α)) : Flat α :=\n  if p : ∃ a n, c.act n = .some a then .some p.choose else .none"}], "used_local_lemmas": [{"name": "Flat.chain_some", "content": "theorem Flat.chain_some {c : Chain (Flat _)} {a b : α}\n  (p : ∃ k, c.act k = .some a) (q : ∃ k, c.act k = .some b) : a = b"}], "local_ctx": "import «PCF».Domain\n\ninductive Flat (α : Type) : Type where\n  | none : Flat α\n  | some : α → Flat α\n\nopen Classical\n\nnoncomputable def flat_sup (c : Chain (Flat α)) : Flat α :=\n  if p : ∃ a n, c.act n = .some a then .some p.choose else .none", "target_theorem": "theorem Flat.sup_some {c : Chain _} {a : α} : (∃ k, c.act k = .some a) ↔ (flat_sup c = .some a) :=", "ground_truth_proof": ":= by\n  constructor\n  case mp =>\n    intro h;\n    have p : ∃ a n, c.act n = .some a := ⟨a, h⟩\n    rw [Flat.chain_some h p.choose_spec]\n    exact dif_pos p\n  case mpr =>\n    intro h;\n    if p : ∃ a n, c.act n = .some a\n    then\n      have q : flat_sup c = .some _ := dif_pos p\n      rw [← h, q]\n      exact p.choose_spec\n    else\n      have q : flat_sup c = .none := dif_neg p\n      rw [q] at h\n      exact Flat.noConfusion h", "nesting_depth": 4, "transitive_dep_count": 14, "subset_aristotle": true, "category": "Semantics"}
{"id": 397, "thm_name": "Ren.ren_comp_eq", "thm_stmt": "theorem Ren.ren_comp_eq {t : Γ₀ ⊢ τ}\n  : ∀ {Γ₁ Γ₂} {σ₀₁ : Ren Γ₀ Γ₁} {σ₁₂ : Ren Γ₁ Γ₂}, t.ren (σ₀₁ ⬝ σ₁₂) = (t.ren σ₀₁).ren σ₁₂", "lean_root": "pcf-lean", "rel_path": "PCF/Substitution.lean", "imports": ["import PCF.Utility", "import «PCF».Syntax"], "used_lib_defs": [{"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Trans.trans", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Var", "content": "inductive Var : Cx → Ty → Type\n  | z : ∀ {Γ : Cx}, Var (Γ ∷ τ) τ\n  | s : ∀ {Γ : Cx} {υ : Ty} τ, Var Γ τ → Var (Γ ∷ υ) τ"}, {"name": "Cx", "content": "inductive Cx\n  | nil\n  | cons : Cx -> Ty -> Cx"}, {"name": "Ty", "content": "inductive Ty\n  | bool\n  | nat\n  | pow : Ty → Ty → Ty"}, {"name": "Tm", "content": "inductive Tm : Cx → Ty → Type\n  | var : ∀ τ, Γ ∋ τ → Tm Γ τ\n  | true : Tm Γ .bool\n  | false : Tm Γ .bool\n  | zero : Tm Γ .nat\n  | succ : Tm Γ .nat → Tm Γ .nat\n  | pred : Tm Γ .nat → Tm Γ .nat\n  | zero? : Tm Γ .nat → Tm Γ .bool\n  | cond : Tm Γ .bool → Tm Γ τ → Tm Γ τ → Tm Γ τ\n  | fn : Tm (Γ ∷ τ) υ → Tm Γ (τ ⇒ υ)\n  | app : Tm Γ (τ ⇒ υ) → Tm Γ τ → Tm Γ υ\n  | fix : Tm Γ (τ ⇒ τ) → Tm Γ τ"}, {"name": "infixl:70 \" ∷ \"  => Cx.cons", "content": "infixl:70 \" ∷ \"  => Cx.cons"}, {"name": "infix:70 \" ∋ \" => Var", "content": "infix:70 \" ∋ \" => Var"}, {"name": "infix:70 \" ⊢ \" => Tm", "content": "infix:70 \" ⊢ \" => Tm"}], "lib_lemmas": [{"name": "congrArg", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "congrArg3", "content": "theorem congrArg3\n  {α₀ : Sort u₀} {α₁ : Sort u₁} {α₂ : Sort u₂} {β : Sort v} {a₀ a₀' : α₀} {a₁ a₁' : α₁} {a₂ a₂' : α₂}\n  (f : α₀ → α₁ → α₂ → β) (h₀ : Eq a₀ a₀') (h₁ : Eq a₁ a₁') (h₂ : Eq a₂ a₂') : Eq (f a₀ a₁ a₂) (f a₀' a₁' a₂')"}, {"name": "congrArg2", "content": "theorem congrArg2\n  {α₀ : Sort u₀} {α₁ : Sort u₁} {β : Sort v} {a₀ a₀' : α₀} {a₁ a₁' : α₁}\n  (f : α₀ → α₁ → β) (h₀ : Eq a₀ a₀') (h₁ : Eq a₁ a₁') : Eq (f a₀ a₁) (f a₀' a₁')"}], "used_local_defs": [{"name": "Ren", "content": "def Ren Γ Δ := ∀ τ, Γ ∋ τ → Δ ∋ τ"}, {"name": "Var.ren", "content": "def Var.ren (v : Γ ∋ τ) (r : Ren Γ Δ) := r τ v"}, {"name": "Ren.keep", "content": "def Ren.keep (r : Ren Γ Δ) (τ  : Ty) : Ren (Γ ∷ τ) (Δ ∷ τ) :=\n  fun υ v => match v with\n  | .z     => .z\n  | .s _ x => (x.ren r).succ"}, {"name": "Tm.ren", "content": "def Tm.ren (t : Γ ⊢ τ) (r : Ren Γ Δ) : Δ ⊢ τ :=\n  match t with\n  | .var τ x    => (x.ren r).tm\n  | .true       => .true\n  | .false      => .false\n  | .zero       => .zero\n  | .succ e     => (e.ren r).succ\n  | .pred e     => (e.ren r).pred\n  | .zero? e    => (e.ren r).zero?\n  | .cond s t f => (s.ren r).cond (t.ren r) (f.ren r)\n  | .fn e       => (e.ren (r ∷ᵣ _)).fn\n  | .app f a    => (f.ren r).app (a.ren r)\n  | .fix f      => (f.ren r).fix"}], "used_local_lemmas": [{"name": "Ren.keep_comp", "content": "theorem Ren.keep_comp {r₀₁ : Ren Γ₀ Γ₁} {r₁₂ : Ren Γ₁ Γ₂}\n  : (r₀₁ ⬝ r₁₂) ∷ᵣ τ = (r₀₁ ∷ᵣ τ) ⬝ (r₁₂ ∷ᵣ τ)"}], "local_ctx": "import «PCF».Syntax\n\ndef Ren Γ Δ := ∀ τ, Γ ∋ τ → Δ ∋ τ\n\ndef Var.ren (v : Γ ∋ τ) (r : Ren Γ Δ) := r τ v\n\ndef Ren.keep (r : Ren Γ Δ) (τ  : Ty) : Ren (Γ ∷ τ) (Δ ∷ τ) :=\n  fun υ v => match v with\n  | .z     => .z\n  | .s _ x => (x.ren r).succ\n\ninfixl:70 \" ∷ᵣ \"  => Ren.keep\n\ninfixl:70 \" ++ᵣ \"  => Ren.keeps\n\ndef Tm.ren (t : Γ ⊢ τ) (r : Ren Γ Δ) : Δ ⊢ τ :=\n  match t with\n  | .var τ x    => (x.ren r).tm\n  | .true       => .true\n  | .false      => .false\n  | .zero       => .zero\n  | .succ e     => (e.ren r).succ\n  | .pred e     => (e.ren r).pred\n  | .zero? e    => (e.ren r).zero?\n  | .cond s t f => (s.ren r).cond (t.ren r) (f.ren r)\n  | .fn e       => (e.ren (r ∷ᵣ _)).fn\n  | .app f a    => (f.ren r).app (a.ren r)\n  | .fix f      => (f.ren r).fix", "target_theorem": "theorem Ren.ren_comp_eq {t : Γ₀ ⊢ τ}\n  : ∀ {Γ₁ Γ₂} {σ₀₁ : Ren Γ₀ Γ₁} {σ₁₂ : Ren Γ₁ Γ₂}, t.ren (σ₀₁ ⬝ σ₁₂) = (t.ren σ₀₁).ren σ₁₂ :=", "ground_truth_proof": ":= by\n  induction t with\n  | @fn _ τ υ e Φ =>\n    intro _ _ r₀₁ r₁₂\n    calc (e.ren ((r₀₁ ⬝ r₁₂) ∷ᵣ τ)).fn\n      _ = (e.ren ((r₀₁ ∷ᵣ τ) ⬝ (r₁₂ ∷ᵣ τ))).fn   := by rw [Ren.keep_comp]\n      _ = ((e.ren (r₀₁ ∷ᵣ τ)).ren (r₁₂ ∷ᵣ τ)).fn := by rw [Φ]\n  | var | true | false | zero => intros; rfl\n  | succ _ Φ | pred _ Φ | zero? _ Φ | fix _ Φ => exact congrArg _ Φ\n  | app _ _ Φf Φa       => exact congrArg2 _ Φf Φa\n  | cond _ _ _ Φs Φt Φf => exact congrArg3 _ Φs Φt Φf", "nesting_depth": 3, "transitive_dep_count": 20, "subset_aristotle": false, "category": "Semantics"}
{"id": 398, "thm_name": "Flat.under_eq", "thm_stmt": "theorem Flat.under_eq {x : Flat α} : x ⊑ .some a → x ⊑ .some b → a ≠ b → x = .none", "lean_root": "pcf-lean", "rel_path": "PCF/Flat.lean", "imports": ["import «PCF».Domain"], "used_lib_defs": [{"name": "Trans.trans", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Order", "content": "class Order (α) where\n  R : α → α → Prop\n  refl {x} : R x x\n  trans {x y z} : R x y → R y z → R x z\n  anti {x y} : R x y → R y x → x = y"}, {"name": "infix:100 \" ⊑ \" => Order.R", "content": "infix:100 \" ⊑ \" => Order.R"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import «PCF».Domain\n\nopen Classical\n\ninductive Flat (α : Type) : Type where\n  | none : Flat α\n  | some : α → Flat α\n\ninstance (a : Flat α) : Decidable (∃ k, a = .some k) :=\n  match a with\n  | .none => isFalse (fun p ↦ p.elim (fun _ y ↦ by injection y))\n  | .some a => isTrue (.intro a rfl)\n\ninstance [DecidableEq α] : DecidableEq (Flat α) := fun a b ↦\n  match a with\n  | .none => match b with\n    | .none => isTrue rfl\n    | .some _ => isFalse Flat.noConfusion\n  | .some a => match b with\n    | .none => isFalse Flat.noConfusion\n    | .some b => if p : a = b then isTrue (by rw [p]) else isFalse (fun q ↦ p (by injection q))\n\ninstance [DecidableEq α] : Order (Flat α) where\n  R := fun x y ↦ (x ≠ .none) → x = y\n  refl := fun _ ↦ rfl\n  trans {x y z} p q :=\n    if h : x = .none\n    then fun a ↦ (a h).elim\n    else fun a ↦ by rw [p h]; rw [p h] at h; rw [q h]\n  anti {x y} p q :=\n    if i : x = .none\n    then if j : y = .none then by rw [i, j] else by rw [q j]\n    else by rw [p i]\n\nnoncomputable def flat_sup (c : Chain (Flat α)) : Flat α :=\n  if p : ∃ a n, c.act n = .some a then .some p.choose else .none\n\nnoncomputable instance : Domain (Flat α) where\n  bot := .none\n  sup := flat_sup\n  is_bot := by admit /- proof elided -/\n  is_bound := by admit /- proof elided -/\n  is_least := by admit /- proof elided -/", "target_theorem": "theorem Flat.under_eq {x : Flat α} : x ⊑ .some a → x ⊑ .some b → a ≠ b → x = .none :=", "ground_truth_proof": ":= by\n  intro under_a under_b a_neq_b\n  by_cases x = none\n  case pos => assumption\n  case neg h => exfalso; exact a_neq_b (by injection (under_a h).symm ⬝ (under_b h))", "nesting_depth": 1, "transitive_dep_count": 2, "subset_aristotle": false, "category": "Semantics"}
{"id": 399, "thm_name": "Cont.pair_after", "thm_stmt": "theorem Cont.pair_after [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] [Order δ] [Domain δ]\n  (f : Cont γ α) (g : Cont γ β) (h : Cont δ γ) : (f ∘' h).pair (g ∘' h) = (f.pair g) ∘' h", "lean_root": "pcf-lean", "rel_path": "PCF/Domain.lean", "imports": ["import PCF/Denotation.lean", "import «PCF».Order", "import PCF/Flat.lean", "import PCF.Order"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Trans.trans", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x", "content": "notation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x"}, {"name": "notation:max \"⨆\" => Domain.sup", "content": "notation:max \"⨆\" => Domain.sup"}, {"name": "Order", "content": "class Order (α) where\n  R : α → α → Prop\n  refl {x} : R x x\n  trans {x y z} : R x y → R y z → R x z\n  anti {x y} : R x y → R y x → x = y"}, {"name": "Chain", "content": "def Chain (α : Type i) [Order α] := Mono Nat α"}, {"name": "Mono", "content": "structure Mono (α) (β) [Order α] [Order β] where\n  act : α → β\n  act' : is_monotone act"}, {"name": "is_monotone", "content": "def is_monotone [Order α] [Order β] (f : α → β) := ∀ {x y : α}, x ⊑ y → f x ⊑ f y"}, {"name": "(Γ", "content": "noncomputable instance (Γ : Cx) : Domain (⟦Γ cx⟧) where\n  bot := fun _ _ ↦ ⊥\n  sup := fun c _ x ↦ ⨆ ⟨fun n ↦ c.act n _ x, fun i_j ↦ c.act' i_j _ x⟩\n  is_bot := fun _ _ ↦ Domain.is_bot\n  is_bound := fun c {n} {_} x ↦ Domain.is_bound ⟨fun n ↦ c.act n _ x, fun i_j ↦ c.act' i_j _ x⟩ n\n  is_least := fun c _ p {_} x ↦ Domain.is_least ⟨fun n ↦ c.act n _ x, fun i_j ↦ c.act' i_j _ x⟩\n    (fun {_} ↦ p _ x)"}, {"name": "[Order", "content": "instance [Order α] [Order β] [Domain α] [Domain β] : Domain (Cont α β) where\n  bot := ⟨⟨fun _ ↦ ⊥, fun _ ↦ Domain.is_bot⟩, Domain.is_bot⟩\n  sup := fun c ↦ ⟨Mono.sup_cont c, by admit /- proof elided -/\n  ⟩\n  is_bot := fun _ ↦ Domain.is_bot\n  is_bound := by admit /- proof elided -/"}, {"name": "flat_sup", "content": "noncomputable def flat_sup (c : Chain (Flat α)) : Flat α :=\n  if p : ∃ a n, c.act n = .some a then .some p.choose else .none"}, {"name": "Flat", "content": "inductive Flat (α : Type) : Type where\n  | none : Flat α\n  | some : α → Flat α"}, {"name": "infix:100 \" ⊑ \" => Order.R", "content": "infix:100 \" ⊑ \" => Order.R"}, {"name": "notation:max \"⋆\" => Order.refl", "content": "notation:max \"⋆\" => Order.refl"}, {"name": "infix:100 \" ⇄! \" => Order.anti", "content": "infix:100 \" ⇄! \" => Order.anti"}, {"name": "infixl:100 \" • \" => Mono.act'", "content": "infixl:100 \" • \" => Mono.act'"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "Mono.ext", "content": "@[ext] theorem Mono.ext [Order α] [Order β] {f g : Mono α β} (p : f.act = g.act) : f = g"}], "used_local_defs": [{"name": "Domain", "content": "class Domain (α) [Order α] where\n  bot : α\n  sup : (c : Chain α) → α\n  is_bot {x} : bot ⊑ x\n  is_bound (c) (n): c.act n ⊑ sup c\n  is_least (c) {d} : ({n : _} → c.act n ⊑ d) → sup c ⊑ d"}, {"name": "DomainType", "content": "structure DomainType : Type (i + 1) :=\n  carrier : Type i\n  order : Order carrier\n  domain : Domain carrier"}, {"name": "_inst_DomainType", "content": "instance (τ : DomainType) : Order (τ) := τ.order"}, {"name": "_inst_DomainType", "content": "instance (τ : DomainType) : Domain (τ) := τ.domain"}, {"name": "Cont", "content": "structure Cont (α) (β) [Order α] [Order β] [Domain α] [Domain β] where\n  fn : Mono α β\n  sub : ∀ {c : Chain α}, fn.act (⨆ c) ⊑ ⨆ (fn ∘ c)"}, {"name": "Cont.comp'", "content": "def Cont.comp' [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] (f : Cont β γ) (g : Cont α β)\n  : Cont α γ\n  := ⟨\n    ⟨fun x ↦ f (g x), fun x_y ↦ f • g • x_y⟩,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Cont.pair", "content": "def Cont.pair [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont γ α) (g : Cont γ β) : Cont γ (α × β) := ⟨\n    ⟨fun c ↦ ⟨f c, g c⟩, fun p ↦ ⟨f • p, g • p⟩⟩,\n    ⟨f.sub ⬝ Domain.sup_is_mono (fun _ ↦ ⋆), g.sub ⬝ Domain.sup_is_mono (fun _ ↦ ⋆)⟩\n  ⟩"}], "used_local_lemmas": [{"name": "Cont.ext", "content": "@[ext] theorem Cont.ext [Order α] [Order β] [Domain α] [Domain β]\n  {f g : Cont α β} (p : f.fn.act = g.fn.act) : f = g"}], "local_ctx": "import «PCF».Order\n\nclass Domain (α) [Order α] where\n  bot : α\n  sup : (c : Chain α) → α\n  is_bot {x} : bot ⊑ x\n  is_bound (c) (n): c.act n ⊑ sup c\n  is_least (c) {d} : ({n : _} → c.act n ⊑ d) → sup c ⊑ d\n\nnotation:max \"⊥\" => Domain.bot\n\nnotation:max \"⨆\" => Domain.sup\n\nstructure DomainType : Type (i + 1) :=\n  carrier : Type i\n  order : Order carrier\n  domain : Domain carrier\n\ninstance (τ : DomainType) : Order (τ) := τ.order\n\ninstance (τ : DomainType) : Domain (τ) := τ.domain\n\nstructure Cont (α) (β) [Order α] [Order β] [Domain α] [Domain β] where\n  fn : Mono α β\n  sub : ∀ {c : Chain α}, fn.act (⨆ c) ⊑ ⨆ (fn ∘ c)\n\nnotation:101 f \" • \" x:100 => Mono.act' (Cont.fn f) x\n\ndef Cont.comp' [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] (f : Cont β γ) (g : Cont α β)\n  : Cont α γ\n  := ⟨\n    ⟨fun x ↦ f (g x), fun x_y ↦ f • g • x_y⟩,\n    by admit /- proof elided -/\n  ⟩\n\ninfix:100 \" ∘ \" => Cont.comp'\n\ninfixr:100 \" ∘' \" => Cont.comp'\n\ndef Cont.pair [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ]\n  (f : Cont γ α) (g : Cont γ β) : Cont γ (α × β) := ⟨\n    ⟨fun c ↦ ⟨f c, g c⟩, fun p ↦ ⟨f • p, g • p⟩⟩,\n    ⟨f.sub ⬝ Domain.sup_is_mono (fun _ ↦ ⋆), g.sub ⬝ Domain.sup_is_mono (fun _ ↦ ⋆)⟩\n  ⟩", "target_theorem": "theorem Cont.pair_after [Order α] [Domain α] [Order β] [Domain β] [Order γ] [Domain γ] [Order δ] [Domain δ]\n  (f : Cont γ α) (g : Cont γ β) (h : Cont δ γ) : (f ∘' h).pair (g ∘' h) = (f.pair g) ∘' h :=", "ground_truth_proof": ":= by\n  apply Cont.ext ∘ funext\n  intro x\n  rfl", "nesting_depth": 3, "transitive_dep_count": 12, "subset_aristotle": false, "category": "Semantics"}
{"id": 400, "thm_name": "Flat.leq_none", "thm_stmt": "theorem Flat.leq_none {a : Flat α} : a ⊑ .none → a = .none", "lean_root": "pcf-lean", "rel_path": "PCF/Flat.lean", "imports": ["import «PCF».Domain"], "used_lib_defs": [{"name": "Trans.trans", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Order", "content": "class Order (α) where\n  R : α → α → Prop\n  refl {x} : R x x\n  trans {x y z} : R x y → R y z → R x z\n  anti {x y} : R x y → R y x → x = y"}, {"name": "infix:100 \" ⊑ \" => Order.R", "content": "infix:100 \" ⊑ \" => Order.R"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Flat", "content": "inductive Flat (α : Type) : Type where\n  | none : Flat α\n  | some : α → Flat α"}], "used_local_lemmas": [{"name": "Flat.invert", "content": "theorem Flat.invert {x : Flat α} (p : x ≠ .none) : ∃ k, x = .some k"}], "local_ctx": "import «PCF».Domain\n\nopen Classical", "target_theorem": "theorem Flat.leq_none {a : Flat α} : a ⊑ .none → a = .none :=", "ground_truth_proof": ":= by\n  intro a_bf_n\n  by_cases a = none\n  case pos => assumption\n  case neg h =>\n    have ⟨n, a_eq_sn⟩ := Flat.invert h\n    exact a_bf_n (by intro a_eq_n; injection a_eq_n.symm ⬝ a_eq_sn)", "nesting_depth": 1, "transitive_dep_count": 3, "subset_aristotle": false, "category": "Semantics"}
{"id": 401, "thm_name": "evalExact_frame", "thm_stmt": "lemma evalExact_frame (h1 h2 : state) t (Q : val → hProp) :\n  evalExact h1 t (ofhProp Q) →\n  Finmap.Disjoint h1 h2 →\n  evalExact (h1 ∪ h2) t (Q ∗ (tohProp (fun h ↦ h = h2)))", "lean_root": "splean", "rel_path": "SPLean/Theories/SepLog.lean", "imports": ["import SPLean.Theories.Lang", "import Mathlib.Data.Multiset.Nodup", "import SPLean.Theories.XSimp", "import Mathlib.Data.Finset.Basic", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "Int.natAbs", "module": "Init.Data.Int.Basic"}, {"name": "Finmap.lookup", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "seq", "module": "Talk.DemoLeanSSR"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "propext", "module": "Init.Core"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "sdone", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "Iff", "module": "Init.Core"}, {"name": "Finmap.insert", "module": "Mathlib.Data.Finmap"}], "used_repo_defs": [{"name": "syntax \"fun\" ident+ \" => \" lang : lang", "content": "syntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" <= \" : bop\n\nsyntax \" >= \" : bop\n\nsyntax \"not\" : uop\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"ref\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"fix\" ident ident+ \" => \" lang : lang\n\nsyntax \"for\" ident \" in \" \"[\" lang \" : \" lang \"]\" \" {\"  (ppLine lang) ( \" }\") : lang\n\nsyntax \"while\" lang  \" {\"  (ppLine lang) ( \" }\") : lang\n\nsyntax \"alloc\" lang \" as \" ident \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "make_list", "content": "def make_list {A} (n : Nat) (v : A) : List A :=\n  match n with\n  | 0      => []\n  | n' + 1 => v :: make_list n' v"}, {"name": "trm_is_val", "content": "abbrev trm_is_val : trm → Prop\n  | trm_val _ => true\n  | _         => false"}, {"name": "conseq", "content": "def conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=\n  match vs with\n  | [] => ∅\n  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))"}, {"name": "null", "content": "def null : loc := 0"}, {"name": "evalbinop", "content": "inductive evalbinop : val → val → val → (val->Prop) → Prop where\n  | evalbinop_eq : forall v1 v2,\n      evalbinop val_eq v1 v2 (fun v => v = val_bool (is_true (v1 = v2)))\n  | evalbinop_neq : forall v1 v2,\n      evalbinop val_neq v1 v2 (fun v => v = val_bool (is_true (v1 ≠ v2)))\n  | evalbinop_add : forall n1 n2,\n      evalbinop val_add (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 + n2))\n  | evalbinop_addr : forall r₁ r₂,\n      evalbinop val_add (val_real r₁) (val_real r₂)\n        (fun v => v = val_real (r₁ + r₂))\n  | evalbinop_sub : forall n1 n2,\n      evalbinop val_sub (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 - n2))\n  | evalbinop_subr : forall r1 r2,\n      evalbinop val_sub (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 - r2))\n  | evalbinop_mul : forall n1 n2,\n      evalbinop val_mul (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 * n2))\n  | evalbinop_mulr : forall r1 r2,\n      evalbinop val_mul (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 * r2))\n  | evalbinop_div : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_div (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 / n2))\n  | evalbinop_divr : forall r1 r2,\n      ¬(r2 = 0) →\n      evalbinop val_div (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 / r2))\n  | evalbinop_mod : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_mod (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 % n2))\n  | evalbinop_le : forall n1 n2,\n      evalbinop val_le (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 <= n2))\n  | evalbinop_ler : forall r1 r2,\n      evalbinop val_le (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 <= r2))\n  | evalbinop_lt : forall n1 n2,\n      evalbinop val_lt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 < n2))\n  | evalbinop_ltr : forall r1 r2,\n      evalbinop val_lt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 < r2))\n  | evalbinop_ge : forall n1 n2,\n      evalbinop val_ge (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 >= n2))\n  | evalbinop_ger : forall r1 r2,\n      evalbinop val_ge (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 >= r2))\n  | evalbinop_gt : forall n1 n2,\n      evalbinop val_gt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 > n2))\n  | evalbinop_gtr : forall r1 r2,\n      evalbinop val_gt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 > r2))\n\n  \n  \n  \n  \n  | evalbinop_ptr_add : forall (p1 : loc) (p2 : Int) n,\n      p2 = p1 + n ->\n      evalbinop val_ptr_add (val_loc p1) (val_int n)\n        (fun v => v = val_loc (Int.natAbs p2))"}, {"name": "is_true", "content": "noncomputable def is_true (P : Prop) : Bool :=\n  if P then true else false"}, {"name": "evalunop", "content": "inductive evalunop : prim → val → (val → Prop) → Prop where\n  | evalunop_neg : forall b1,\n      evalunop val_neg (val_bool b1) (fun v => v = val_bool (¬ b1))\n  | evalunop_opp : forall n1,\n      evalunop val_opp (val_int n1) (fun v => v = val_int (- n1))\n  | evalunop_oppr : forall r1,\n      evalunop val_opp (val_real r1) (fun v => v = val_real (- r1))"}, {"name": "purepost", "content": "def purepost (s : state) (P : val → Prop) : val → state → Prop :=\n  fun v s' => P v ∧ s' = s"}, {"name": "read_state", "content": "def read_state (p : loc) (h : state) :=\n  match Finmap.lookup p h with\n  | some v => v\n  | none   => default"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "intersect", "content": "def intersect (s1 s2 : state) := s1 \\ (s1 \\ s2)"}, {"name": "hProp.Disjoint", "content": "def hProp.Disjoint (H₁ H₂ : hProp) :=\n  forall h1 h2, H₁ h1 -> H₂ h2 -> h1.Disjoint h2"}, {"name": "qstar", "content": "def qstar {A} (Q : A → hProp) (H : hProp) : A → hProp :=\n  fun x => hstar (Q x) H"}, {"name": "hstar", "content": "def hstar (H1 H2 : hProp) : hProp :=\n  fun h => exists h1 h2,\n    H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}], "lib_lemmas": [{"name": "Finmap.insert_union", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.Disjoint.symm_iff", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.disjoint_union_left", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.mem_iff", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.union_assoc", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.union_comm_of_disjoint", "module": "Mathlib.Data.Finmap"}], "repo_lemmas": [{"name": "in_read_union_l", "content": "lemma in_read_union_l (h1 h2 : state) (x : loc) :\n  x ∈ h1 → read_state x (h1 ∪ h2) = read_state x h1"}, {"name": "disjoint_insert_l", "content": "lemma disjoint_insert_l (h1 h2 : state) (x : loc) (v : val) :\n  Finmap.Disjoint h1 h2 →\n  x ∈ h1 →\n  Finmap.Disjoint (Finmap.insert x v h1) h2"}, {"name": "union_diff_disjoint_r", "content": "lemma union_diff_disjoint_r (h₁ h₂ h₃ : state) :\n  h₂.Disjoint h₃ →\n  (h₁ ∪ h₂) \\ h₃ = (h₁ \\ h₃) ∪ h₂"}, {"name": "lookup_diff", "content": "lemma lookup_diff (h₁ h₂ : state) :\n  p ∉ h₂ →\n  (h₁ \\ h₂).lookup p = h₁.lookup p"}, {"name": "lookup_diff_none", "content": "lemma lookup_diff_none (h₁ h₂ : state) :\n  p ∈ h₂ →\n  (h₁ \\ h₂).lookup p = none"}, {"name": "diff_non_mem", "content": "theorem diff_non_mem (h₁ h₂ : state) :\n  p ∈ h₂ → p ∉ h₁ \\ h₂"}, {"name": "union_monotone_r", "content": "lemma union_monotone_r (s₃ s₁ s₂ : state) :\n  s₁ = s₂ →\n  s₁ ∪ s₃ = s₂ ∪ s₃"}, {"name": "disjoint_update_not_r", "content": "lemma disjoint_update_not_r (h1 h2 : state) (x : loc) (v: val) :\n  Finmap.Disjoint h1 h2 →\n  x ∉ h2 →\n  Finmap.Disjoint (Finmap.insert x v h1) h2"}, {"name": "evalExact_post_eq", "content": "lemma evalExact_post_eq :\n  Q = Q' →\n  evalExact s t Q →\n  evalExact s t Q'"}, {"name": "erase_of_non_mem", "content": "lemma erase_of_non_mem (h : state) :\n  p ∉ h →\n  h.erase p = h"}, {"name": "diff_insert_intersect_id", "content": "lemma diff_insert_intersect_id (s₁ s₂ : state) :\n  (s₁ \\ s₂) ∪ (intersect s₁ s₂) = s₁"}, {"name": "lookup_intersect", "content": "lemma lookup_intersect (s₁ s₂ : state) :\n  p ∈ s₁ ∧ p ∈ s₂ →\n  (intersect s₁ s₂).lookup p = s₁.lookup p"}, {"name": "intersect_disjoint_cancel", "content": "lemma intersect_disjoint_cancel (s₁ s₂ s₃ : state) :\n  s₁.Disjoint s₃ →\n  (s₁ ∪ intersect s₂ s₃) \\ s₃ = s₁"}, {"name": "remove_not_in_r", "content": "lemma remove_not_in_r (h1 h2 : state) (p : loc) :\n  p ∉ h2 →\n  (h1 ∪ h2).erase p = h1.erase p ∪ h2"}, {"name": "disjoint_intersect_r", "content": "lemma disjoint_intersect_r (s₁ s₂ s₃ : state) :\n  s₂.Disjoint s₃ →\n  (intersect s₁ s₂).Disjoint s₃"}, {"name": "insert_union", "content": "lemma insert_union (h1 h2 : state) (p : loc) (v : val) :\n  p ∉ h1 ∪ h2 →\n  (h1 ∪ h2).insert p v = (h1.insert p v) ∪ h2"}, {"name": "reinsert_erase_union", "content": "lemma reinsert_erase_union (h1 h2 h3 : state) :\n  h3.lookup p = some v →\n  p ∉ h2 →\n  h3.erase p = h1 ∪ h2 →\n  h3 = (h1.insert p v) ∪ h2"}, {"name": "insert_delete_id", "content": "lemma insert_delete_id (h : state) (p : loc) :\n  p ∉ h →\n  h = (h.insert p v).erase p"}, {"name": "disjoint_disjoint_diff", "content": "lemma disjoint_disjoint_diff (h₁ h₂ h₃ : state) :\n  h₁.Disjoint h₂ →\n  (h₁ \\ h₃).Disjoint h₂"}, {"name": "erase_disjoint", "content": "lemma erase_disjoint (h1 h2 : state) (p : loc) :\n  h1.Disjoint h2 →\n  (h1.erase p).Disjoint h2"}], "used_local_defs": [{"name": "tohProp", "content": "abbrev tohProp (h : heap -> Prop) : hProp := h"}, {"name": "ofhProp", "content": "abbrev ofhProp (h : val -> hProp) : val -> heap -> Prop := h"}], "used_local_lemmas": [{"name": "frame_eq_rw", "content": "lemma frame_eq_rw :\n  s.Disjoint h2 →\n  (fun v' s' ↦ v' = v ∧ s' = s ∪ h2) =\n  (qstar (fun v' s' ↦ v' = v ∧ s' = s) (tohProp (fun h ↦ h = h2)))"}, {"name": "evalExact_frame_val", "content": "lemma evalExact_frame_val (v : val) (s h2 : state) :\n  s.Disjoint h2 →\n  evalExact (s ∪ h2) t (fun v' s' ↦ v' = v ∧ s' = s ∪ h2) →\n  evalExact (s ∪ h2) t\n    (qstar (fun v' s' ↦ v' = v ∧ s' = s) (tohProp (fun h ↦ h = h2)))"}, {"name": "purepost_frame", "content": "lemma purepost_frame :\n  s.Disjoint h2 →\n  (purepost (s ∪ h2) P) =\n  (qstar (purepost s P) (tohProp fun h ↦ h = h2))"}, {"name": "evalExact_frame_unop_binop", "content": "lemma evalExact_frame_unop_binop :\n  s.Disjoint h2 →\n  evalExact (s ∪ h2) t (purepost (s ∪ h2) P) →\n  evalExact (s ∪ h2) t (qstar (purepost s P) (tohProp fun h ↦ h = h2))"}, {"name": "read_state_frame", "content": "lemma read_state_frame :\n  s.Disjoint h2 →\n  p ∈ s →\n  (fun v' s' ↦ v' = read_state p (s ∪ h2) ∧ s' = s ∪ h2 ) =\n  (qstar (fun v' s' ↦ v' = read_state p s ∧ s' = s) (tohProp fun h ↦ h = h2))"}, {"name": "evalExact_frame_get", "content": "lemma evalExact_frame_get :\n  s.Disjoint h2 →\n  p ∈ s →\n  evalExact (s ∪ h2) t (fun v' s' ↦ v' = read_state p (s ∪ h2) ∧ s' = s ∪ h2 ) →\n  evalExact (s ∪ h2) t\n    (qstar (fun v' s' ↦ v' = read_state p s ∧ s' = s) (tohProp fun h ↦ h = h2))"}, {"name": "insert_frame", "content": "lemma insert_frame :\n  s.Disjoint h2 →\n  p ∈ s →\n  fun v'' s' ↦ v'' = val_unit ∧ s' = Finmap.insert p v' (s ∪ h2) =\n  (qstar (fun v'' s' ↦ v'' = val_unit ∧ s' = Finmap.insert p v' s) (tohProp fun h ↦ h = h2))"}, {"name": "evalExact_frame_set", "content": "lemma evalExact_frame_set :\n  s.Disjoint h2 →\n  p ∈ s →\n  evalExact (s ∪ h2) t\n    (fun v'' s' ↦ v'' = val_unit ∧ s' = Finmap.insert p v' (s ∪ h2)) →\n  evalExact (s ∪ h2) t\n    (qstar (fun v'' s' ↦ v'' = val_unit ∧ s' = Finmap.insert p v' s) (tohProp fun h ↦ h = h2))"}], "local_ctx": "import Mathlib.Data.Finmap\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Multiset.Nodup\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nopen trm val prim\n\nnotation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)\n\nsection evalProp\n\nabbrev tohProp (h : heap -> Prop) : hProp := h\n\nabbrev ofhProp (h : val -> hProp) : val -> heap -> Prop := h", "target_theorem": "lemma evalExact_frame (h1 h2 : state) t (Q : val → hProp) :\n  evalExact h1 t (ofhProp Q) →\n  Finmap.Disjoint h1 h2 →\n  evalExact (h1 ∪ h2) t (Q ∗ (tohProp (fun h ↦ h = h2))) :=", "ground_truth_proof": ":=\nby\n  simp [ofhProp]\n  move=> /== heval\n  elim: heval h2\n  { move=> > *\n    sby apply evalExact_frame_val }\n  { move=> > *\n    sby apply evalExact_frame_val }\n  { move=> > *\n    sby apply evalExact_frame_val }\n  { move=> ???????? ih1 ?? /ih1 ? ; constructor=>//\n    sby move=> ?? ![] }\n  { move=> ???????? ih1 ?? /ih1 ? ; apply evalExact.app_arg2=>//\n    sby move=> ?? ![] }\n  { sby move=> * ; apply evalExact.app_fun }\n  { sby move=> * ; apply evalExact.app_fix }\n  { move=> ??????? ih1 ih2 ? /ih1 ? ; apply evalExact.seq=>//\n    move=> ? s2 ![??? hQ2 *] ; subst s2 hQ2\n    sby apply ih2 }\n  { move=> ???????? ih1 ih2 ? /ih1 ? ; apply evalExact.let=>//\n    move=> ?? ![??? hQ2 ? hU] ; subst hU hQ2\n    sby apply ih2}\n  { sby move=> * }\n  { move=> > ? > *\n    apply evalExact_frame_unop_binop=> //\n    sby apply evalExact.unop }\n  { move=> > ? > *\n    apply evalExact_frame_unop_binop=> //\n    sby apply evalExact.binop }\n  { move=> > ; unfold tohProp\n    move=> _ _ ih1 ih2 > /ih1 {}ih1\n    apply evalExact.ref\n    { apply ih1 }\n    move=> {ih1} > ![>] hQ₁ /= -> ? -> p ?\n    have eqn:(p ∉ w) := by sdone\n    have eqn':((w.insert p v1).Disjoint h2) := by sby apply disjoint_update_not_r\n    move: hQ₁ eqn eqn'=> /ih2 /[apply] /[apply] {ih2}\n    srw insert_union=> // hq\n    apply evalExact_post_eq ; rotate_left ; apply hq\n    apply funext=> v ; apply funext=> h ; apply propext=> ⟨|⟩\n    { move=> ![>] /= ? -> ? ->\n      exists (w_2.erase p), h2=> ⟨//|/==⟩ ⟨|⟩\n      apply erase_disjoint=> //\n      sby srw remove_not_in_r }\n    move=> ![>] /= ? -> ?\n    scase: [p ∈ h]\n    { move=> ? ; srw erase_of_non_mem=> // []\n      exists w_2, h2=> /== ⟨|⟩ //\n      sby srw erase_of_non_mem }\n    move=> /Finmap.mem_iff [v'] /reinsert_erase_union heq herase\n    srw (heq w_2 h2)=> // {heq}\n    exists (w_2.insert p v'), h2=> /== ⟨|⟩\n    { srw -insert_delete_id=> //\n      have eqn:(p ∉ h.erase p) := by apply Finmap.not_mem_erase_self\n      move: eqn\n      sby srw herase }\n    sby apply disjoint_update_not_r }\n  { move=> > ? > *\n    apply evalExact_frame_get=> //\n    sby apply evalExact.get }\n  { move=> > [] ? > * * ;\n    apply evalExact_frame_set=> //\n    sby apply evalExact.set }\n  -- { move=> * ; apply eval.eval_free=>//\n  --   srw remove_disjoint_union_l ; apply hstar_intro=>//\n  --   sby apply disjoint_remove_l }\n  { move=> > ??? ih1 ih2 > /ih1 {ih1} ?\n    apply evalExact.alloc_arg=> // >\n    sby move=> ![>] }\n  { unfold tohProp=> > ?? ih > ? ; apply evalExact.alloc=> // >\n    move=> /ih /[apply] {}ih /Finmap.disjoint_union_left [] /[dup] /ih {}ih ?\n    srw Finmap.Disjoint.symm_iff -Finmap.union_assoc=> ?\n    have eqn:((sb ∪ sa).Disjoint h2) := by\n      sby srw Finmap.disjoint_union_left\n    apply ih in eqn=> {ih} hq ; apply evalExact_post_eq ; rotate_left ; apply hq\n    apply funext=> v ; apply funext=> h ; apply propext=> ⟨|⟩\n    { move=> ![>] /= ? -> ? ->\n      exists (w \\ sb), h2=> /== ⟨|⟩ // ⟨|⟩\n      { sby apply disjoint_disjoint_diff }\n      apply union_diff_disjoint_r\n      sby apply Finmap.Disjoint.symm }\n    move=> ![>] /= ? -> ? /[dup] heq\n    have eqn:((w ∪ h2).Disjoint sb) := by\n      { srw -heq ; unfold Finmap.Disjoint=> /== }\n    move: eqn=> /Finmap.disjoint_union_left [ ? _]\n    move=> /(union_monotone_r (intersect h sb))\n    srw diff_insert_intersect_id Finmap.union_assoc [2]Finmap.union_comm_of_disjoint\n    rotate_left\n    { apply Finmap.Disjoint.symm ; sby apply disjoint_intersect_r }\n    srw -Finmap.union_assoc=> ?\n    exists (w ∪ intersect h sb), h2=> //== ⟨|⟩\n    { sby srw intersect_disjoint_cancel }\n    constructor=> //\n    srw Finmap.disjoint_union_left ; constructor=> //\n    sby apply disjoint_intersect_r }\n  { move=> // }\n  move=> > ?? ih₁ ih₂ ??; econstructor\n  { apply ih₁=> // }\n  sby move=> > ![]", "nesting_depth": 6, "transitive_dep_count": 82, "subset_aristotle": false, "category": "Framework"}
{"id": 402, "thm_name": "Theories.eval_like_trm_apps_funs_pre", "thm_stmt": "lemma eval_like_trm_apps_funs_pre (heqv0 : v0 = trm_funs xs t1) :\n  eval_like t (trm_apps (val_funs xs t1) ts) ∧  -- NOTE: this part do not require `xs.Nodup`, but anyway\n  eval_like (isubst (xs.mkAlist vs) t1) t", "lean_root": "splean", "rel_path": "SPLean/Theories/WP1.lean", "imports": ["import SPLean.Theories.XChange", "import SPLean.Theories.Lang", "import Mathlib.Data.List.Indexes", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Lean", "import SPLean.Theories.WPUtil"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "AList.erase", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup", "module": "Mathlib.Data.List.AList"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "And", "module": "Init.Prelude"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sdone", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "AList.entries", "module": "Mathlib.Data.List.AList"}, {"name": "List.NodupKeys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.keys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}, {"name": "List.zipWith", "module": "Init.Data.List.Basic"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "AList.insert", "module": "Mathlib.Data.List.AList"}, {"name": "List.kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kinsert", "module": "Mathlib.Data.List.Sigma"}, {"name": "AList.keys", "module": "Mathlib.Data.List.AList"}], "used_repo_defs": [{"name": "syntax \"fun\" ident+ \" => \" lang : lang", "content": "syntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \" ++ \" : bop\n\nsyntax \"not\" : uop"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "val_funs", "content": "def val_funs (xs:List var) (t:trm) : val :=\n  match xs with\n  | [] => panic! \"function with zero argumets!\"\n  | x1::xs' => val_fun x1 (trm_funs xs' t)"}, {"name": "trm_funs", "content": "def trm_funs (xs:List var) (t:trm) : trm :=\n  match xs with\n  | [] => t\n  | x1::xs' => trm_fun x1 (trm_funs xs' t)"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "trm_apps", "content": "def trm_apps (f:trm) (ts:List trm) : trm :=\n  match ts with\n  | [] => f\n  | ti::ts' => trm_apps (trm_app f ti) ts'"}, {"name": "eval_like", "content": "def eval_like (t1 t2:trm) : Prop :=\n  forall s Q, eval s t1 Q -> eval s t2 Q"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "trm_is_val", "content": "abbrev trm_is_val : trm → Prop\n  | trm_val _ => true\n  | _         => false"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "AList.keys_nodup", "module": "Mathlib.Data.List.AList"}, {"name": "List.dlookup_dedupKeys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.lookup_ext", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.perm_dlookup", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.perm_nodupKeys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.map_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_zipWith", "module": "Init.Data.List.Zip"}, {"name": "List.nodup_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.perm_append_singleton", "module": "Init.Data.List.Perm"}, {"name": "List.zip_append", "module": "Init.Data.List.Zip"}, {"name": "List.kerase_cons_ne", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kerase_kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "AList.perm_erase", "module": "Mathlib.Data.List.AList"}, {"name": "AList.perm_lookup", "module": "Mathlib.Data.List.AList"}, {"name": "AList.mem_keys", "module": "Mathlib.Data.List.AList"}, {"name": "List.eraseP_of_forall_not", "module": "Init.Data.List.Erase"}, {"name": "AList.erase_erase", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup_erase_ne", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup_insert_ne", "module": "Mathlib.Data.List.AList"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "AList.toAList_cons", "module": "Mathlib.Data.List.AList"}, {"name": "List.nodup_middle", "module": "Mathlib.Data.List.Nodup"}], "repo_lemmas": [{"name": "eval_app_arg1'", "content": "lemma eval_app_arg1' s1 t1 t2 Q1 Q :\n  eval s1 t1 Q1 ->\n  (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n  eval s1 (trm_app t1 t2) Q"}], "used_local_defs": [{"name": "Theories.trms_to_vals", "content": "@[simp]\ndef trms_to_vals (ts:List trm) : Option (List val) := do\n  match ts with\n  | [] => return []\n  | (trm_val v) :: ts' => v :: (<- trms_to_vals ts')\n  | _ => failure"}, {"name": "Theories.ctx", "content": "abbrev ctx := AList (fun _ : var ↦ val)"}, {"name": "Theories.isubst", "content": "def isubst (E : ctx) (t : trm) : trm :=\n  match t with\n  | trm_val v =>\n      v\n  | trm_var x =>\n      match lookup x E with\n      | none   => t\n      | some v => v\n  | trm_fun x t1 =>\n      trm_fun x (isubst (erase x E) t1)\n  | trm_fix f x t1 =>\n      trm_fix f x (isubst (erase x (erase f E)) t1)\n  | trm_if t0 t1 t2 =>\n      trm_if (isubst E t0) (isubst E t1) (isubst E t2)\n  | trm_seq t1 t2 =>\n      trm_seq (isubst E t1) (isubst E t2)\n  | trm_let x t1 t2 =>\n      trm_let x (isubst E t1) (isubst (erase x E) t2)\n  | trm_ref x t1 t2 =>\n      trm_ref x (isubst E t1) (isubst (erase x E) t2)\n  | trm_alloc x t1 t2 =>\n      trm_alloc x (isubst E t1) (isubst (erase x E) t2)\n  | trm_app t1 t2 =>\n      trm_app (isubst E t1) (isubst E t2)\n  | trm_for x n1 n2 t =>\n      trm_for x (isubst E n1) (isubst E n2) (isubst (erase x E) t)\n  | trm_while c t =>\n      trm_while (isubst E c) (isubst E t)"}], "used_local_lemmas": [{"name": "Theories.trms_to_vals_some_equiv", "content": "lemma trms_to_vals_some_equiv ts vs : trms_to_vals ts = some vs → ts = vs.map trm_val"}, {"name": "Theories.List.toAList_perm", "content": "lemma List.toAList_perm {α : Type u} {β : α → Type v} [DecidableEq α]\n  (es es' : List (Sigma β)) (hnodup : es.NodupKeys) (hp : es.Perm es') :\n  es.toAList.entries.Perm es'.toAList.entries"}, {"name": "Theories.List.mkAlist_snoc_to_cons", "content": "lemma List.mkAlist_snoc_to_cons [DecidableEq α] (xs : List α) (vs : List β)\n    (x : α) (v : β) : x ∉ xs → xs.length = vs.length → xs.Nodup →\n    ((xs ++ [x]).mkAlist (vs ++ [v])).entries.Perm (((x :: xs).mkAlist (v :: vs)).entries)"}, {"name": "Theories.AList.erase_insert_cancel", "content": "lemma AList.erase_insert_cancel {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) (l : AList β) :\n    (AList.erase a (AList.insert a b l)).entries.Perm (AList.erase a l).entries"}, {"name": "Theories.AList.erase_insert_swap", "content": "lemma AList.erase_insert_swap {α : Type u} {β : α → Type v} [DecidableEq α] (a a' : α) (b : β a) (l : AList β) :\n  a ≠ a' → (AList.erase a' (AList.insert a b l)).entries.Perm (AList.insert a b (AList.erase a' l)).entries"}, {"name": "Theories.AList.erase_noop", "content": "lemma AList.erase_noop {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : AList β) :\n    a ∉ l → (AList.erase a l).entries.Perm l.entries"}, {"name": "Theories.AList.erase_twice", "content": "lemma AList.erase_twice {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : AList β) :\n    (AList.erase a (AList.erase a l)).entries.Perm (AList.erase a l).entries"}, {"name": "Theories.AList.erase_empty", "content": "lemma AList.erase_empty {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) :\n  AList.erase a (∅ : AList β) = ∅"}, {"name": "Theories.isubst_empty", "content": "lemma isubst_empty t : isubst ∅ t = t"}, {"name": "Theories.isubst_perm", "content": "lemma isubst_perm {al al'} t (hp : al.entries.Perm al'.entries) :\n  isubst al t = isubst al' t"}, {"name": "Theories.isubst_insert", "content": "lemma isubst_insert (al : ctx) x v t :\n  isubst (al.insert x v) t = subst x v (isubst (al.erase x) t)"}, {"name": "Theories.isubst_single", "content": "lemma isubst_single x v t : isubst (List.mkAlist [x] [v]) t = subst x v t"}, {"name": "Theories.trm_apps2", "content": "lemma trm_apps2 :\n  trm_apps (trm_app t1 t2) ts = trm_apps t1 (t2::ts)"}, {"name": "Theories.trm_apps_app", "content": "lemma trm_apps_app :\n  trm_apps t1 (ts ++ ts') = trm_apps (trm_apps t1 ts) ts'"}, {"name": "Theories.trm_funs_app", "content": "lemma trm_funs_app :\n  trm_funs (xs ++ xs') t1 = trm_funs xs (trm_funs xs' t1)"}, {"name": "Theories.eval_like_trm_fun_val_fun", "content": "lemma eval_like_trm_fun_val_fun x t : eval_like (trm_fun x t) (val_fun x t)"}, {"name": "Theories.eval_like_val_fun_trm_fun", "content": "lemma eval_like_val_fun_trm_fun x t : eval_like (val_fun x t) (trm_fun x t)"}, {"name": "Theories.eval_like_trm_app_left", "content": "lemma eval_like_trm_app_left t1 t1' t2 (hsat : ∃ s Q, eval s t1 Q) : eval_like t1 t1' → eval_like (trm_app t1 t2) (trm_app t1' t2)"}, {"name": "Theories.eval_like_trm_fun_val_fun_app_left", "content": "lemma eval_like_trm_fun_val_fun_app_left (x : var) (t1 t2 : trm) :\n  eval_like (trm_app (trm_fun x t1) t2) (trm_app (val_fun x t1) t2)"}, {"name": "Theories.eval_like_val_fun_trm_fun_app_left", "content": "lemma eval_like_val_fun_trm_fun_app_left (x : var) (t1 t2 : trm) :\n  eval_like (trm_app (val_fun x t1) t2) (trm_app (trm_fun x t1) t2)"}, {"name": "Theories.val_funs_snoc", "content": "lemma val_funs_snoc (xs : List var) (x : var) (h : xs ≠ []) (t : trm) :\n  val_funs (xs ++ [x]) t = val_funs xs (trm_fun x t)"}, {"name": "Theories.List.not_nil_snoc", "content": "lemma List.not_nil_snoc {α : Type u} (l : List α) : l ≠ [] → ∃ l' x, l = l' ++ [x]"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Indexes\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WPUtil\n\nopen trm val prim\n\nnamespace Theories\n\nsection tactics\n\nopen Lean Elab Tactic\n\nsection xapp\n\nend xapp\n\nend tactics\n\n@[simp]\ndef trms_to_vals (ts:List trm) : Option (List val) := do\n  match ts with\n  | [] => return []\n  | (trm_val v) :: ts' => v :: (<- trms_to_vals ts')\n  | _ => failure\n\nopen AList\n\nabbrev ctx := AList (fun _ : var ↦ val)\n\ndef isubst (E : ctx) (t : trm) : trm :=\n  match t with\n  | trm_val v =>\n      v\n  | trm_var x =>\n      match lookup x E with\n      | none   => t\n      | some v => v\n  | trm_fun x t1 =>\n      trm_fun x (isubst (erase x E) t1)\n  | trm_fix f x t1 =>\n      trm_fix f x (isubst (erase x (erase f E)) t1)\n  | trm_if t0 t1 t2 =>\n      trm_if (isubst E t0) (isubst E t1) (isubst E t2)\n  | trm_seq t1 t2 =>\n      trm_seq (isubst E t1) (isubst E t2)\n  | trm_let x t1 t2 =>\n      trm_let x (isubst E t1) (isubst (erase x E) t2)\n  | trm_ref x t1 t2 =>\n      trm_ref x (isubst E t1) (isubst (erase x E) t2)\n  | trm_alloc x t1 t2 =>\n      trm_alloc x (isubst E t1) (isubst (erase x E) t2)\n  | trm_app t1 t2 =>\n      trm_app (isubst E t1) (isubst E t2)\n  | trm_for x n1 n2 t =>\n      trm_for x (isubst E n1) (isubst E n2) (isubst (erase x E) t)\n  | trm_while c t =>\n      trm_while (isubst E c) (isubst E t)\n\nsection funs_fixs_eval_like\n\nvariable (xs : List var) (vs : List val) (t : trm) (v0 : trm)\n  (heqt : t = trm_apps v0 ts)\n  (hconv : trms_to_vals ts = vs)\n  (hform : var_funs xs vs.length)   -- NOTE: can be relaxed to `vs.length ≤ xs.length`", "target_theorem": "lemma eval_like_trm_apps_funs_pre (heqv0 : v0 = trm_funs xs t1) :\n  eval_like t (trm_apps (val_funs xs t1) ts) ∧  -- NOTE: this part do not require `xs.Nodup`, but anyway\n  eval_like (isubst (xs.mkAlist vs) t1) t :=", "ground_truth_proof": ":= by\n  apply trms_to_vals_some_equiv at hconv ; subst_eqs\n  move: hform=> /== hnodup hlen hnotempty\n  move: hnodup vs hlen t1\n  induction xs using List.list_reverse_induction with\n  | base => sdone\n  | ind xs x ih =>\n    move=> { hnotempty } /(List.nodup_middle (l₂ := [])) /== hnotin hnodup vs hlen t1\n    by_cases hvs : vs = []\n    { subst vs ; simp [trm_apps]=> ⟨|//⟩ ; apply eval_like_trm_funs_val_funs=> // }\n    move: hvs=> /List.not_nil_snoc [vs [v ?]] /[tac subst_eqs]\n    simp at hlen ; simp [trm_apps_app, trm_apps]\n    by_cases hxs : xs = []\n    { subst xs ; simp [val_funs, trm_funs] at *\n      cases vs=> //= { hlen } ; simp [trm_apps]\n      apply And.intro\n      { apply eval_like_trm_fun_val_fun_app_left=> // }\n      -- single subst\n      srw isubst_single\n      trans\n      on_goal 2=> apply eval_like_val_fun_trm_fun_app_left\n      move=> ??? ; apply eval.eval_app_fun=> // }\n    specialize @ih hxs hnodup _ hlen (trm_fun x t1) ; rcases ih with ⟨ih1, ih2⟩\n    apply And.intro\n    { srw val_funs_snoc // trm_funs_app ; simp [trm_funs]\n      trans\n      on_goal 2=> apply eval_like_trm_app_left\n      on_goal 3=> apply ih1\n      on_goal 2=> exists ∅, (fun _ _ ↦ True) ; apply ih2=> //   -- that's why we want to prove things together\n      move=> ??=> // }\n    srw trm_funs_app ; simp [trm_funs, trm_apps]\n    -- ih\n    trans\n    on_goal 2=> apply eval_like_trm_app_left\n    on_goal 3=> apply ih2\n    on_goal 2=> exists ∅, (fun _ _ ↦ True) ; simp [isubst]=> //\n    clear ih1 ih2 ; simp [isubst]\n    -- trm_fun -> val_fun\n    trans\n    on_goal 2=> apply eval_like_val_fun_trm_fun_app_left\n    -- raw eval, then subst reasoning\n    move=> s Q h\n    apply eval.eval_app_fun=> //\n    srw -isubst_insert\n    rw [isubst_perm _ (List.mkAlist_snoc_to_cons xs vs x v hnotin hlen hnodup)] at h\n    apply h", "nesting_depth": 5, "transitive_dep_count": 102, "subset_aristotle": false, "category": "Framework"}
{"id": 403, "thm_name": "eval_frame", "thm_stmt": "lemma eval_frame (h1 h2 : state) t (Q : val -> hProp) :\n  eval h1 t (ofhProp Q) →\n  Finmap.Disjoint h1 h2 →\n  eval (h1 ∪ h2) t (Q ∗ (tohProp (fun h ↦ h = h2)))", "lean_root": "splean", "rel_path": "SPLean/Theories/SepLog.lean", "imports": ["import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Mathlib.Data.Multiset.Nodup", "import SPLean.Theories.XSimp", "import Mathlib.Data.Finset.Basic"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "Finmap.insert", "module": "Mathlib.Data.Finmap"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "Finmap.lookup", "module": "Mathlib.Data.Finmap"}], "used_repo_defs": [{"name": "syntax \"fun\" ident+ \" => \" lang : lang", "content": "syntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "read_state", "content": "def read_state (p : loc) (h : state) :=\n  match Finmap.lookup p h with\n  | some v => v\n  | none   => default"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}], "lib_lemmas": [{"name": "Finmap.disjoint_union_left", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.insert_union", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.union_assoc", "module": "Mathlib.Data.Finmap"}], "repo_lemmas": [{"name": "union_diff_disjoint_r", "content": "lemma union_diff_disjoint_r (h₁ h₂ h₃ : state) :\n  h₂.Disjoint h₃ →\n  (h₁ ∪ h₂) \\ h₃ = (h₁ \\ h₃) ∪ h₂"}, {"name": "lookup_diff", "content": "lemma lookup_diff (h₁ h₂ : state) :\n  p ∉ h₂ →\n  (h₁ \\ h₂).lookup p = h₁.lookup p"}, {"name": "lookup_diff_none", "content": "lemma lookup_diff_none (h₁ h₂ : state) :\n  p ∈ h₂ →\n  (h₁ \\ h₂).lookup p = none"}, {"name": "diff_non_mem", "content": "theorem diff_non_mem (h₁ h₂ : state) :\n  p ∈ h₂ → p ∉ h₁ \\ h₂"}, {"name": "disjoint_update_not_r", "content": "lemma disjoint_update_not_r (h1 h2 : state) (x : loc) (v: val) :\n  Finmap.Disjoint h1 h2 →\n  x ∉ h2 →\n  Finmap.Disjoint (Finmap.insert x v h1) h2"}, {"name": "in_read_union_l", "content": "lemma in_read_union_l (h1 h2 : state) (x : loc) :\n  x ∈ h1 → read_state x (h1 ∪ h2) = read_state x h1"}, {"name": "remove_not_in_r", "content": "lemma remove_not_in_r (h1 h2 : state) (p : loc) :\n  p ∉ h2 →\n  (h1 ∪ h2).erase p = h1.erase p ∪ h2"}, {"name": "disjoint_insert_l", "content": "lemma disjoint_insert_l (h1 h2 : state) (x : loc) (v : val) :\n  Finmap.Disjoint h1 h2 →\n  x ∈ h1 →\n  Finmap.Disjoint (Finmap.insert x v h1) h2"}, {"name": "disjoint_disjoint_diff", "content": "lemma disjoint_disjoint_diff (h₁ h₂ h₃ : state) :\n  h₁.Disjoint h₂ →\n  (h₁ \\ h₃).Disjoint h₂"}, {"name": "erase_disjoint", "content": "lemma erase_disjoint (h1 h2 : state) (p : loc) :\n  h1.Disjoint h2 →\n  (h1.erase p).Disjoint h2"}], "used_local_defs": [{"name": "tohProp", "content": "abbrev tohProp (h : heap -> Prop) : hProp := h"}, {"name": "ofhProp", "content": "abbrev ofhProp (h : val -> hProp) : val -> heap -> Prop := h"}], "used_local_lemmas": [{"name": "eval_conseq", "content": "lemma eval_conseq s t Q1 Q2 :\n  eval s t Q1 →\n  Q1 ===> Q2 →\n  eval s t Q2"}], "local_ctx": "import Mathlib.Data.Finmap\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Multiset.Nodup\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nopen trm val prim\n\nnotation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)\n\nsection evalProp\n\nabbrev tohProp (h : heap -> Prop) : hProp := h\n\nabbrev ofhProp (h : val -> hProp) : val -> heap -> Prop := h", "target_theorem": "lemma eval_frame (h1 h2 : state) t (Q : val -> hProp) :\n  eval h1 t (ofhProp Q) →\n  Finmap.Disjoint h1 h2 →\n  eval (h1 ∪ h2) t (Q ∗ (tohProp (fun h ↦ h = h2))) :=", "ground_truth_proof": ":=\nby\n  unfold ofhProp tohProp; elim=> //\n  { move=> > ?? _ ih' *; apply eval.eval_app_arg1=> //\n    move=> > ![] ?? ? -> ? ->; aesop }\n  { move=> *; apply eval.eval_app_arg2=> //\n    move=> > ![] ?? ? -> ? ->; aesop }\n  { move=> *; apply eval.eval_app_fun=> // }\n  { move=> *; apply eval.eval_app_fix=> // }\n  { move=> *; apply eval.eval_seq=> //\n    move=> > ![] ?? ? -> ? ->; aesop }\n  { move=> *; apply eval.eval_let=> //\n    move=> > ![] ?? ? -> ? ->; aesop }\n  { move=> > ? Pp *; apply eval.eval_unop=> //\n    move=> ? /Pp ?; exists s, h2 }\n  { move=> > ? Pp *; apply eval.eval_binop=> //\n    move=> ? /Pp ?; exists s, h2 }\n  { move=> > ? _ dj ih' ?\n    constructor; apply dj=> //\n    move=> > ![] s1 ? ? -> dj' -> p /== ??\n    rw [@Finmap.insert_union]\n    apply eval_conseq; apply ih'=> //\n    { sby apply disjoint_update_not_r s1 h2 p v1 dj' }\n    move=> v s /= ![] h ? /== ? -> ? ->\n    rw [remove_not_in_r h h2 p]=> //\n    exists (h.erase p), h2=> ⟨|⟩//⟨|⟩//⟨|⟩//\n    sby apply erase_disjoint h h2 p }\n  { move=> > *; apply eval.eval_get\n    simp; aesop; exists s, h2=> ⟨|⟩//\n    sby rw [in_read_union_l s h2 p] }\n  { move=> > *; apply eval.eval_set=> //\n    exists (Finmap.insert p v' s), h2=> ⟨|⟩// ⟨|⟩// ⟨|⟩\n    { apply disjoint_insert_l s h2 p v'=> // }\n    rw [@Finmap.insert_union] }\n  { move=> *; apply eval.eval_alloc_arg=> //\n    move=> > ![] ??? -> ? ->; aesop }\n  { move=> > ? ih ih' dj\n    apply eval.eval_alloc=> // > ?? dj';\n    srw -Finmap.union_assoc; apply eval_conseq; apply ih'=> //\n    { move: dj'; sby rw [@Finmap.disjoint_union_left] }\n    { move: dj'; srw ?Finmap.disjoint_union_left /===> ? ? ⟨|⟩//\n      sby apply (Finmap.Disjoint.symm h2 sb) }\n    move=> > /= ? /= ![] s ? /= ? -> ? ->\n    exists (s \\ sb), h2=> ⟨|⟩//⟨|⟩//⟨|⟩\n    { sby apply disjoint_disjoint_diff s h2 sb }\n    apply union_diff_disjoint_r=> //\n    move: dj'; sby rw [@Finmap.disjoint_union_left] }\n  move=> *; constructor=> // ?? ![] ??? -> ? ->; aesop", "nesting_depth": 6, "transitive_dep_count": 55, "subset_aristotle": false, "category": "Framework"}
{"id": 404, "thm_name": "evalExact_WellAlloc", "thm_stmt": "lemma evalExact_WellAlloc :\n  evalExact s t Q →\n  Q v s' →\n  s'.keys = s.keys", "lean_root": "splean", "rel_path": "SPLean/Theories/SepLog.lean", "imports": ["import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Mathlib.Data.Multiset.Nodup", "import SPLean.Theories.XSimp", "import Mathlib.Data.Finset.Basic"], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "Int.natAbs", "module": "Init.Data.Int.Basic"}, {"name": "Finmap.lookup", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "seq", "module": "Talk.DemoLeanSSR"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "sdone", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "sapply", "module": "Ssreflect.ApplyIn"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Not", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax \"fun\" ident+ \" => \" lang : lang", "content": "syntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" <= \" : bop\n\nsyntax \" >= \" : bop\n\nsyntax \"not\" : uop\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"ref\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"fix\" ident ident+ \" => \" lang : lang\n\nsyntax \"for\" ident \" in \" \"[\" lang \" : \" lang \"]\" \" {\"  (ppLine lang) ( \" }\") : lang\n\nsyntax \"while\" lang  \" {\"  (ppLine lang) ( \" }\") : lang\n\nsyntax \"alloc\" lang \" as \" ident \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang\n\n  syntax \"sdo\" num tactic : tactic"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term| {| $seq |}) => `(withMainContext do evalTactic $ <- `(tacticSeq| $seq))"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "make_list", "content": "def make_list {A} (n : Nat) (v : A) : List A :=\n  match n with\n  | 0      => []\n  | n' + 1 => v :: make_list n' v\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "trm_is_val", "content": "abbrev trm_is_val : trm → Prop\n  | trm_val _ => true\n  | _         => false\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "conseq", "content": "def conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=\n  match vs with\n  | [] => ∅\n  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))"}, {"name": "null", "content": "def null : loc := 0"}, {"name": "evalbinop", "content": "inductive evalbinop : val → val → val → (val->Prop) → Prop where\n  | evalbinop_eq : forall v1 v2,\n      evalbinop val_eq v1 v2 (fun v => v = val_bool (is_true (v1 = v2)))\n  | evalbinop_neq : forall v1 v2,\n      evalbinop val_neq v1 v2 (fun v => v = val_bool (is_true (v1 ≠ v2)))\n  | evalbinop_add : forall n1 n2,\n      evalbinop val_add (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 + n2))\n  | evalbinop_addr : forall r₁ r₂,\n      evalbinop val_add (val_real r₁) (val_real r₂)\n        (fun v => v = val_real (r₁ + r₂))\n  | evalbinop_sub : forall n1 n2,\n      evalbinop val_sub (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 - n2))\n  | evalbinop_subr : forall r1 r2,\n      evalbinop val_sub (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 - r2))\n  | evalbinop_mul : forall n1 n2,\n      evalbinop val_mul (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 * n2))\n  | evalbinop_mulr : forall r1 r2,\n      evalbinop val_mul (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 * r2))\n  | evalbinop_div : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_div (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 / n2))\n  | evalbinop_divr : forall r1 r2,\n      ¬(r2 = 0) →\n      evalbinop val_div (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 / r2))\n  | evalbinop_mod : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_mod (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 % n2))\n  | evalbinop_le : forall n1 n2,\n      evalbinop val_le (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 <= n2))\n  | evalbinop_ler : forall r1 r2,\n      evalbinop val_le (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 <= r2))\n  | evalbinop_lt : forall n1 n2,\n      evalbinop val_lt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 < n2))\n  | evalbinop_ltr : forall r1 r2,\n      evalbinop val_lt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 < r2))\n  | evalbinop_ge : forall n1 n2,\n      evalbinop val_ge (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 >= n2))\n  | evalbinop_ger : forall r1 r2,\n      evalbinop val_ge (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 >= r2))\n  | evalbinop_gt : forall n1 n2,\n      evalbinop val_gt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 > n2))\n  | evalbinop_gtr : forall r1 r2,\n      evalbinop val_gt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 > r2))\n\n  \n  \n  \n  \n  | evalbinop_ptr_add : forall (p1 : loc) (p2 : Int) n,\n      p2 = p1 + n ->\n      evalbinop val_ptr_add (val_loc p1) (val_int n)\n        (fun v => v = val_loc (Int.natAbs p2))"}, {"name": "is_true", "content": "noncomputable def is_true (P : Prop) : Bool :=\n  if P then true else false"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "evalunop", "content": "inductive evalunop : prim → val → (val → Prop) → Prop where\n  | evalunop_neg : forall b1,\n      evalunop val_neg (val_bool b1) (fun v => v = val_bool (¬ b1))\n  | evalunop_opp : forall n1,\n      evalunop val_opp (val_int n1) (fun v => v = val_int (- n1))\n  | evalunop_oppr : forall r1,\n      evalunop val_opp (val_real r1) (fun v => v = val_real (- r1))"}, {"name": "purepost", "content": "def purepost (s : state) (P : val → Prop) : val → state → Prop :=\n  fun v s' => P v ∧ s' = s"}, {"name": "read_state", "content": "def read_state (p : loc) (h : state) :=\n  match Finmap.lookup p h with\n  | some v => v\n  | none   => default"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}], "lib_lemmas": [{"name": "Finmap.mem_keys", "module": "Mathlib.Data.Finmap"}, {"name": "Finset.le_max_of_eq", "module": "Mathlib.Data.Finset.Max"}, {"name": "Finset.max_of_nonempty", "module": "Mathlib.Data.Finset.Max"}, {"name": "Finset.nonempty_iff_ne_empty", "module": "Mathlib.Data.Finset.Empty"}, {"name": "Nat.lt_succ_iff", "module": "Init.Data.Nat.Basic"}, {"name": "Finmap.Disjoint.symm_iff", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.disjoint_union_left", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.union_comm_of_disjoint", "module": "Mathlib.Data.Finmap"}], "repo_lemmas": [{"name": "non_mem_union", "content": "lemma non_mem_union (h1 h2 : state) :\n  a ∉ h1 ∪ h2 ↔ a ∉ h1 ∧ a ∉ h2"}, {"name": "insert_mem_keys", "content": "lemma insert_mem_keys (s : state) :\n  p ∈ s →\n  (s.insert p v).keys = s.keys"}, {"name": "insert_same", "content": "lemma insert_same (h1 h2 : state) :\n  p ∉ h1 → p ∉ h2 →\n  (h1.insert p v).keys = (h2.insert p v').keys →\n  h1.keys = h2.keys"}, {"name": "union_same_keys", "content": "lemma union_same_keys (h₁ h₂ h₃ : state) :\n  h₁.Disjoint h₃ → h₂.Disjoint h₃ →\n  (h₁ ∪ h₃).keys = (h₂ ∪ h₃).keys →\n  h₁.keys = h₂.keys"}, {"name": "insert_delete_id", "content": "lemma insert_delete_id (h : state) (p : loc) :\n  p ∉ h →\n  h = (h.insert p v).erase p"}, {"name": "union_difference_id", "content": "lemma union_difference_id (h₁ h₂ : state) :\n  h₁.Disjoint h₂ →\n  (h₁ ∪ h₂) \\ h₂ = h₁"}, {"name": "insert_disjoint_l", "content": "lemma insert_disjoint_l (h1 h2 : state) (x : loc) (v : val) :\n  h2.Disjoint (h1.insert x v) →\n  x ∉ h2 ∧ h2.Disjoint h1"}, {"name": "remove_not_in_l", "content": "lemma remove_not_in_l (h1 h2 : state) (p : loc) :\n  p ∉ h1 →\n  (h1 ∪ h2).erase p = h1 ∪ h2.erase p"}], "used_local_defs": [], "used_local_lemmas": [{"name": "finite_state", "content": "lemma finite_state (s : state) :\n  ∃ p, p ∉ s"}, {"name": "conseq_ind", "content": "lemma conseq_ind (n : ℕ) (v : val) (p : loc) :\n  x ∈ conseq (make_list n v) p → x ≥ p"}, {"name": "finite_state'", "content": "lemma finite_state' n (s : state) :\n  ∃ p, p ≠ null ∧\n    Finmap.Disjoint s (conseq (make_list n val_uninit) p)"}, {"name": "evalExact_sat", "content": "lemma evalExact_sat :\n  evalExact s t Q → ∃ v s, Q v s"}], "local_ctx": "import Mathlib.Data.Finmap\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Multiset.Nodup\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nopen trm val prim\n\nnotation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)\n\nsection evalProp", "target_theorem": "lemma evalExact_WellAlloc :\n  evalExact s t Q →\n  Q v s' →\n  s'.keys = s.keys :=", "ground_truth_proof": ":= by\n  move=> hev\n  elim: hev s' v\n  { sby move=> > [] }\n  { sby move=> > [] }\n  { sby move=> > [] }\n  { move=> > _ /evalExact_sat ![>] /[dup] hQ1 /[swap] _ /[swap] /[apply] heq\n    move: hQ1=> /[swap] /[apply] /[apply]\n    sby srw heq=> {}heq > /heq }\n  { move=> > _ /evalExact_sat ![>] /[dup] hQ1 /[swap] _ /[swap] /[apply] heq\n    move: hQ1=> /[swap] /[apply] /[apply]\n    sby srw heq=> {}heq > /heq }\n  { sby move=> > _ _ ih > /ih }\n  { sby move=> > _ _ ih > /ih }\n  { move=> > /evalExact_sat ![>] /[dup] hQ1 /[swap] _ /[swap] /[apply] heq\n    move: hQ1=> /[swap] /[apply] /[apply]\n    sby srw heq=> {}heq > /heq }\n  { move=> > /evalExact_sat ![>] /[dup] hQ1 /[swap] _ /[swap] /[apply] heq\n    move: hQ1=> /[swap] /[apply] /[apply]\n    sby srw heq=> {}heq > /heq }\n  { sby move=> > _ ih > /ih }\n  { sby unfold purepost }\n  { sby unfold purepost }\n  { move=> > /evalExact_sat ![>] /[dup] hQ₁ /[swap] /[apply] ? ih1 ih2 >\n    move: hQ₁=> /[dup] hQ₁ /ih1 {}ih1\n    scase: (finite_state (s' ∪ w_1))=> p /non_mem_union []\n    move=> /[dup] /insert_same hins ? /[dup] /hins {}hins\n    move: hQ₁=> /ih2 /[apply] /== {}ih2\n    srw [1](insert_delete_id s' p)=> //\n    sby move=> /ih2 /hins }\n  { sdone }\n  { move=> > _ ? > /= [] _ []\n    sby srw insert_mem_keys }\n  { move=> > _ /evalExact_sat ![>] /[dup] hQ₁ /[swap] _ /[swap] /[apply] ?\n    sby move: hQ₁=> /[swap] /[apply] ih > /ih }\n  { move=> > ? _ ih >\n    scase: (finite_state' n.natAbs (sa ∪ s'))\n    move=> p [] /ih /== {}ih /Finmap.disjoint_union_left /[dup] hdisj [] /ih {}ih\n    move=> /union_difference_id heq\n    srw -[1]heq=> /ih\n    srw [2]Finmap.union_comm_of_disjoint ; rotate_left\n    { sby srw Finmap.Disjoint.symm_iff }\n    sby move: hdisj=> [] /[swap] /union_same_keys /[apply] }\n  { sby move=> > _ ih > /ih }\n  move=> > /evalExact_sat ![>] /[dup] hQ1 /[swap] _ /[swap] /[apply] heq\n  move: hQ1=> /[swap] /[apply] /[apply]\n  sby srw heq=> {}heq > /heq", "nesting_depth": 6, "transitive_dep_count": 62, "subset_aristotle": false, "category": "Framework"}
{"id": 405, "thm_name": "evalExact_post", "thm_stmt": "lemma evalExact_post :\n  eval s t Q → evalExact s t Q' → Q' ===> Q", "lean_root": "splean", "rel_path": "SPLean/Theories/SepLog.lean", "imports": ["import SPLean.Theories.Lang", "import Mathlib.Data.Multiset.Nodup", "import SPLean.Theories.XSimp", "import Mathlib.Data.Finset.Basic", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "Int.natAbs", "module": "Init.Data.Int.Basic"}, {"name": "Finmap.lookup", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "seq", "module": "Talk.DemoLeanSSR"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "sapply", "module": "Ssreflect.ApplyIn"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Not", "module": "Init.Prelude"}, {"name": "sdone", "module": "Ssreflect.Done"}], "used_repo_defs": [{"name": "syntax \"fun\" ident+ \" => \" lang : lang", "content": "syntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" <= \" : bop\n\nsyntax \" >= \" : bop\n\nsyntax \"not\" : uop\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"ref\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"fix\" ident ident+ \" => \" lang : lang\n\nsyntax \"for\" ident \" in \" \"[\" lang \" : \" lang \"]\" \" {\"  (ppLine lang) ( \" }\") : lang\n\nsyntax \"while\" lang  \" {\"  (ppLine lang) ( \" }\") : lang\n\nsyntax \"alloc\" lang \" as \" ident \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "make_list", "content": "def make_list {A} (n : Nat) (v : A) : List A :=\n  match n with\n  | 0      => []\n  | n' + 1 => v :: make_list n' v"}, {"name": "trm_is_val", "content": "abbrev trm_is_val : trm → Prop\n  | trm_val _ => true\n  | _         => false"}, {"name": "conseq", "content": "def conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=\n  match vs with\n  | [] => ∅\n  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))"}, {"name": "null", "content": "def null : loc := 0"}, {"name": "evalbinop", "content": "inductive evalbinop : val → val → val → (val->Prop) → Prop where\n  | evalbinop_eq : forall v1 v2,\n      evalbinop val_eq v1 v2 (fun v => v = val_bool (is_true (v1 = v2)))\n  | evalbinop_neq : forall v1 v2,\n      evalbinop val_neq v1 v2 (fun v => v = val_bool (is_true (v1 ≠ v2)))\n  | evalbinop_add : forall n1 n2,\n      evalbinop val_add (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 + n2))\n  | evalbinop_addr : forall r₁ r₂,\n      evalbinop val_add (val_real r₁) (val_real r₂)\n        (fun v => v = val_real (r₁ + r₂))\n  | evalbinop_sub : forall n1 n2,\n      evalbinop val_sub (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 - n2))\n  | evalbinop_subr : forall r1 r2,\n      evalbinop val_sub (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 - r2))\n  | evalbinop_mul : forall n1 n2,\n      evalbinop val_mul (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 * n2))\n  | evalbinop_mulr : forall r1 r2,\n      evalbinop val_mul (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 * r2))\n  | evalbinop_div : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_div (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 / n2))\n  | evalbinop_divr : forall r1 r2,\n      ¬(r2 = 0) →\n      evalbinop val_div (val_real r1) (val_real r2)\n        (fun v => v = val_real (r1 / r2))\n  | evalbinop_mod : forall n1 n2,\n      ¬(n2 = 0) →\n      evalbinop val_mod (val_int n1) (val_int n2)\n        (fun v => v = val_int (n1 % n2))\n  | evalbinop_le : forall n1 n2,\n      evalbinop val_le (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 <= n2))\n  | evalbinop_ler : forall r1 r2,\n      evalbinop val_le (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 <= r2))\n  | evalbinop_lt : forall n1 n2,\n      evalbinop val_lt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 < n2))\n  | evalbinop_ltr : forall r1 r2,\n      evalbinop val_lt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 < r2))\n  | evalbinop_ge : forall n1 n2,\n      evalbinop val_ge (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 >= n2))\n  | evalbinop_ger : forall r1 r2,\n      evalbinop val_ge (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 >= r2))\n  | evalbinop_gt : forall n1 n2,\n      evalbinop val_gt (val_int n1) (val_int n2)\n        (fun v => v = val_bool (n1 > n2))\n  | evalbinop_gtr : forall r1 r2,\n      evalbinop val_gt (val_real r1) (val_real r2)\n        (fun v => v = val_bool (r1 > r2))\n\n  \n  \n  \n  \n  | evalbinop_ptr_add : forall (p1 : loc) (p2 : Int) n,\n      p2 = p1 + n ->\n      evalbinop val_ptr_add (val_loc p1) (val_int n)\n        (fun v => v = val_loc (Int.natAbs p2))"}, {"name": "is_true", "content": "noncomputable def is_true (P : Prop) : Bool :=\n  if P then true else false"}, {"name": "evalunop", "content": "inductive evalunop : prim → val → (val → Prop) → Prop where\n  | evalunop_neg : forall b1,\n      evalunop val_neg (val_bool b1) (fun v => v = val_bool (¬ b1))\n  | evalunop_opp : forall n1,\n      evalunop val_opp (val_int n1) (fun v => v = val_int (- n1))\n  | evalunop_oppr : forall r1,\n      evalunop val_opp (val_real r1) (fun v => v = val_real (- r1))"}, {"name": "purepost", "content": "def purepost (s : state) (P : val → Prop) : val → state → Prop :=\n  fun v s' => P v ∧ s' = s"}, {"name": "read_state", "content": "def read_state (p : loc) (h : state) :=\n  match Finmap.lookup p h with\n  | some v => v\n  | none   => default"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}], "lib_lemmas": [{"name": "Finmap.mem_keys", "module": "Mathlib.Data.Finmap"}, {"name": "Finset.le_max_of_eq", "module": "Mathlib.Data.Finset.Max"}, {"name": "Finset.max_of_nonempty", "module": "Mathlib.Data.Finset.Max"}, {"name": "Finset.nonempty_iff_ne_empty", "module": "Mathlib.Data.Finset.Empty"}, {"name": "Nat.lt_succ_iff", "module": "Init.Data.Nat.Basic"}, {"name": "Finmap.disjoint_union_left", "module": "Mathlib.Data.Finmap"}], "repo_lemmas": [{"name": "non_mem_union", "content": "lemma non_mem_union (h1 h2 : state) :\n  a ∉ h1 ∪ h2 ↔ a ∉ h1 ∧ a ∉ h2"}, {"name": "evalbinop_unique", "content": "lemma evalbinop_unique :\n  evalbinop op v1 v2 P → evalbinop op v1 v2 P' → P = P'"}, {"name": "insert_delete_id", "content": "lemma insert_delete_id (h : state) (p : loc) :\n  p ∉ h →\n  h = (h.insert p v).erase p"}, {"name": "evalunop_unique", "content": "lemma evalunop_unique :\n  evalunop op v P → evalunop op v P' → P = P'"}, {"name": "union_difference_id", "content": "lemma union_difference_id (h₁ h₂ : state) :\n  h₁.Disjoint h₂ →\n  (h₁ ∪ h₂) \\ h₂ = h₁"}, {"name": "insert_disjoint_l", "content": "lemma insert_disjoint_l (h1 h2 : state) (x : loc) (v : val) :\n  h2.Disjoint (h1.insert x v) →\n  x ∉ h2 ∧ h2.Disjoint h1"}, {"name": "remove_not_in_l", "content": "lemma remove_not_in_l (h1 h2 : state) (p : loc) :\n  p ∉ h1 →\n  (h1 ∪ h2).erase p = h1 ∪ h2.erase p"}], "used_local_defs": [], "used_local_lemmas": [{"name": "finite_state", "content": "lemma finite_state (s : state) :\n  ∃ p, p ∉ s"}, {"name": "conseq_ind", "content": "lemma conseq_ind (n : ℕ) (v : val) (p : loc) :\n  x ∈ conseq (make_list n v) p → x ≥ p"}, {"name": "finite_state'", "content": "lemma finite_state' n (s : state) :\n  ∃ p, p ≠ null ∧\n    Finmap.Disjoint s (conseq (make_list n val_uninit) p)"}, {"name": "evalExact_sat", "content": "lemma evalExact_sat :\n  evalExact s t Q → ∃ v s, Q v s"}], "local_ctx": "import Mathlib.Data.Finmap\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Multiset.Nodup\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nopen trm val prim\n\nnotation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)\n\nsection evalProp", "target_theorem": "lemma evalExact_post :\n  eval s t Q → evalExact s t Q' → Q' ===> Q :=", "ground_truth_proof": ":= by\n  move=> H\n  elim: H Q'=> >\n  -- elim=> >\n  { sby move=> ? > [] v h /== }\n  { sby move=> ? > [] v h /== }\n  { sby move=> ? > [] v h /== }\n  { move=> ??? ih1 ih2 > [] // >\n    { move=> > _ /[dup] h h'\n      apply evalExact_sat in h=> ![] v s' /[dup] hQ1_1 hQ1_1'\n      apply ih1 in h'=> himp hev\n      apply himp in hQ1_1\n      sby apply hev in hQ1_1'=> ? /ih2 }\n    { move=> ?\n      scase: op=> > ? //\n      scase: a=> > ? // [] // }\n    move=> ? [] // }\n  { move=> ? _ _ ih1 ih2 > [] // > _ /[dup] h h'\n    apply evalExact_sat in h=> ![] v s' /[dup] hQ1_1 hQ1_1'\n    apply ih1 in h'=> himp hev\n    apply himp in hQ1_1\n    sby apply hev in hQ1_1'=> ? /ih2 }\n  { sby move=> [] ?? > [] }\n  { sby move=> [] ?? > [] }\n  { move=> _ _ ih1 ih2 > [] > /[dup] h h'\n    apply evalExact_sat in h=> ![] v s' > /[dup] hQ1_1 hQ1_1' hev\n    apply ih1 in h'=> himp\n    apply himp in hQ1_1\n    sby apply hev in hQ1_1'=> ? /ih2 }\n  { move=> _ _ ih1 ih2 > [] > /[dup] h /evalExact_sat ![] v s'\n    move=> /[dup] hQ1_1 hQ1_1' hev\n    apply ih1 in h=> himp\n    apply himp in hQ1_1\n    sby apply hev in hQ1_1'=> ? /ih2 }\n  { sby move=> _ ih > [] }\n  { unfold purepostin=> hOP ? > [] //\n    apply evalunop_unique in hOP=> hP\n    move=> > /hP []\n    sby unfold purepost=> ?? }\n  { unfold purepostin=> hOP ? > [] //\n    { scase: op=> > //\n      scase: a=> > // ? > ? [] // }\n    apply evalbinop_unique in hOP=> hP\n    move=> > /hP []\n    sby unfold purepost=> ?? }\n  { move=> ?? ih1 ih2 > [Q₁'] hevEx hev\n    move=> > h\n    move: hevEx hev\n    move=> /[dup] /ih1 {} ih1 /evalExact_sat ![>] /[dup] /ih1 {}ih1\n    move=> /[swap] /[apply]\n    scase: (finite_state (h ∪ w_1))=> p /non_mem_union [] ? hp\n    move: hp=> /[dup] hp /[swap] /[apply]\n    move: ih1 hp=> /ih2 /[apply] {}ih2\n    specialize (@ih2 fun v (s : state) ↦ Q' v (s.erase p))\n    move: ih2=> /[apply]\n    unfold qimpl himpl=> /== ?\n    sby srw (insert_delete_id h p) }\n  { sby move=> ?? > [] // _ ?? }\n  { move=> [] ?? > [] //\n    { move=> > ? [] // }\n    sby move=> > _ [] ?? }\n  { move=> ? _ _ ih1 ih2 > [] //\n    move=> Q₁' ? /[dup] /ih1 {}ih1 /evalExact_sat ![>] /[dup] /ih1 {} ih1\n    move=> /[swap] /[apply]\n    sby move: ih1=> /ih2 }\n  { move=> ? _ /== ih > hev\n    move=> > h\n    move: hev=> [] // _ /== hev\n    scase: (finite_state' n.natAbs (sa ∪ h))\n    move=> p [] /[dup] /ih {}ih /hev {}hev /Finmap.disjoint_union_left []\n    move=> /[dup] /hev /[swap] /ih {}ih {hev} /ih {}ih /union_difference_id <-\n    sby move=> /ih }\n  { sby move=> ?? > [] }\n  { move=> ?? ih1 ih2 > [] Q₁'\n    move=> /[dup] /ih1 {}ih1 /evalExact_sat ![>] /[dup] /ih1 {}ih1\n    move=> /[swap] /[apply]\n    sby move: ih1=> /ih2 }", "nesting_depth": 6, "transitive_dep_count": 72, "subset_aristotle": false, "category": "Framework"}
{"id": 406, "thm_name": "qwand_equiv", "thm_stmt": "lemma qwand_equiv H A (Q1 Q2 : A → hProp) :\n  H ==> (Q1 -∗ Q2) ↔ (Q1 ∗ H) ===> Q2", "lean_root": "splean", "rel_path": "SPLean/Theories/HProp.lean", "imports": ["import Mathlib.Data.Finmap", "import SPLean.Common.Heap", "import SPLean.Theories.Lang", "import SPLean.Common.Util", "import Mathlib.Algebra.BigOperators.Group.Finset"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "Iff", "module": "Init.Core"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "propext", "module": "Init.Core"}, {"name": "sapply", "module": "Ssreflect.ApplyIn"}, {"name": "And", "module": "Init.Prelude"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "scase!", "module": "Ssreflect.Elim"}], "used_repo_defs": [{"name": "syntax \"sdo\" num tactic : tactic", "content": "syntax \"sdo\" num tactic : tactic"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty\n\nsyntax \"fun\" ident+ \" => \" lang : lang"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term| {| $seq |}) => `(withMainContext do evalTactic $ <- `(tacticSeq| $seq))"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)"}, {"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "Finmap.union_comm_of_disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.disjoint_union_left", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.disjoint_union_right", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.union_assoc", "module": "Mathlib.Data.Finmap"}, {"name": "Iff.mpr", "module": "Init.Core"}, {"name": "Finmap.disjoint_empty", "module": "Mathlib.Data.Finmap"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "hstar", "content": "def hstar (H1 H2 : hProp) : hProp :=\n  fun h => exists h1 h2,\n    H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2"}, {"name": "hexists", "content": "def hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h"}, {"name": "hforall", "content": "def hforall {A} (J : A → hProp) : hProp :=\n  fun h => forall x, J x h"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "_inst_HStar", "content": "@[default_instance]\ninstance : HStar hProp hProp hProp where\n  hStar := hstar"}, {"name": "hpure", "content": "def hpure (P : Prop) : hProp :=\n  hexists (fun (_ : P) => emp)"}, {"name": "hwand", "content": "def hwand (H1 H2 : hProp) : hProp :=\n  hexists (fun (H0 : hProp) => H0 ∗ hpure ((H1 ∗ H0) ==> H2))"}, {"name": "qwand", "content": "def qwand {A} (Q1 Q2 : A → hProp) : hProp :=\n  hforall (fun (x : A) => hwand (Q1 x) (Q2 x))"}, {"name": "qstar", "content": "def qstar {A} (Q : A → hProp) (H : hProp) : A → hProp :=\n  fun x => hstar (Q x) H"}, {"name": "_inst_Type", "content": "instance (A : Type) : HStar (A → hProp) hProp (A → hProp) where\n  hStar := qstar"}, {"name": "HWand", "content": "class HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hWand : α → β → γ"}, {"name": "_inst_HWand", "content": "@[default_instance]\ninstance : HWand hProp hProp hProp where\n  hWand := hwand"}, {"name": "_inst_Type", "content": "instance (α : Type) : HWand (α → hProp) (α → hProp) hProp where\n  hWand := qwand"}], "used_local_lemmas": [{"name": "himpl_trans", "content": "lemma himpl_trans H2 H1 H3 :\n  (H1 ==> H2) → (H2 ==> H3) → (H1 ==> H3)"}, {"name": "himpl_antisym", "content": "lemma himpl_antisym H1 H2:\n  (H1 ==> H2) → (H2 ==> H1) → (H1 = H2)"}, {"name": "hprop_op_comm", "content": "lemma hprop_op_comm (op : hProp → hProp → hProp) :\n  (forall H1 H2, op H1 H2 ==> op H2 H1) →\n  (forall H1 H2, op H1 H2 = op H2 H1)"}, {"name": "hempty_inv", "content": "lemma hempty_inv h :\n  emp h → h = ∅"}, {"name": "hstar_inv", "content": "lemma hstar_inv (H1 H2 : hProp) h:\n  (H1 ∗ H2) h →\n  exists h1 h2, H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2"}, {"name": "hstar_comm", "content": "lemma hstar_comm H1 H2 :\n  H1 ∗ H2 = H2 ∗ H1"}, {"name": "hstar_assoc", "content": "lemma hstar_assoc H1 H2 H3 :\n  (H1 ∗ H2) ∗ H3 = H1 ∗ (H2 ∗ H3)"}, {"name": "hstar_hempty_l", "content": "lemma hstar_hempty_l H :\n  emp ∗ H = H"}, {"name": "hstar_hempty_r", "content": "lemma hstar_hempty_r H :\n  H ∗ emp = H"}, {"name": "hstar_hexists", "content": "lemma hstar_hexists A (J : A → hProp) H :\n  (hexists J) ∗ H = hexists (fun x => (J x) ∗ H)"}, {"name": "hstar_hforall", "content": "lemma hstar_hforall A (J : A → hProp) H :\n  (hforall J) ∗ H ==> hforall (J ∗ H)"}, {"name": "himpl_frame_r", "content": "lemma himpl_frame_r H1 H2 H2' :\n  H2 ==> H2' →\n  (H1 ∗ H2) ==> (H1 ∗ H2')"}, {"name": "himpl_hstar_trans_l", "content": "lemma himpl_hstar_trans_l H1 H2 H3 H4 :\n  H1 ==> H2 →\n  H2 ∗ H3 ==> H4 →\n  H1 ∗ H3 ==> H4"}, {"name": "hstar_hpure_l", "content": "lemma hstar_hpure_l P H h :\n  (⌜P⌝ ∗ H) h = (P ∧ H h)"}, {"name": "himpl_hempty_hpure", "content": "lemma himpl_hempty_hpure P :\n  P → emp ==> ⌜P⌝"}, {"name": "himpl_hstar_hpure_l", "content": "lemma himpl_hstar_hpure_l P H H' :\n  (P → H ==> H') →\n  (⌜P⌝ ∗ H) ==> H'"}, {"name": "himpl_hexists_l", "content": "lemma himpl_hexists_l A H (J : A → hProp) :\n  (forall x, J x ==> H) → (hexists J) ==> H"}, {"name": "himpl_hexists_r", "content": "lemma himpl_hexists_r A (x : A) H (J : A → hProp) :\n  (H ==> J x) →\n  H ==> (hexists J)"}, {"name": "himpl_hforall_r", "content": "lemma himpl_hforall_r A (J : A → hProp) H :\n  (forall x, H ==> J x) →\n  H ==> (hforall J)"}, {"name": "himpl_hforall_l", "content": "lemma himpl_hforall_l A (x : A) (J : A → hProp) H :\n  (J x ==> H) →\n  (hforall J) ==> H"}, {"name": "hwandE", "content": "lemma hwandE :\n  H1 -∗ H2 = hexists (fun H0 => H0 ∗ hpure ((H1 ∗ H0) ==> H2))"}, {"name": "hwand_equiv", "content": "lemma hwand_equiv H0 H1 H2 :\n  (H0 ==> H1 -∗ H2) ↔ (H1 ∗ H0 ==> H2)"}, {"name": "himpl_hwand_r_inv", "content": "lemma himpl_hwand_r_inv H1 H2 H3 :\n  H1 ==> (H2 -∗ H3) →\n  H2 ∗ H1 ==> H3"}, {"name": "hwand_cancel", "content": "lemma hwand_cancel H1 H2 :\n  H1 ∗ (H1 -∗ H2) ==> H2"}, {"name": "qwandE", "content": "lemma qwandE α (Q1 Q2 : α → hProp) :\n  Q1 -∗ Q2 = hforall (fun x => (Q1 x) -∗ (Q2 x))"}, {"name": "qstarE", "content": "lemma qstarE α (Q1 : α → hProp)  (H : hProp):\n  Q1 ∗ H = fun x => Q1 x ∗ H"}], "local_ctx": "import Mathlib.Data.Finmap\n\nimport Mathlib.Algebra.BigOperators.Group.Finset\n\nimport SPLean.Common.Heap\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.Lang\n\nopen Classical\n\ndef hProp := heap -> Prop\n\nabbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h\n\ninfixr:51 \" ==> \" => himpl\n\ndef qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v\n\ninfixr:51 \" ===> \" => qimpl\n\ndef hempty : hProp :=\n  fun h => (h = ∅)\n\ndef hstar (H1 H2 : hProp) : hProp :=\n  fun h => exists h1 h2,\n    H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2\n\ndef hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h\n\ndef hforall {A} (J : A → hProp) : hProp :=\n  fun h => forall x, J x h\n\nnotation:max \"emp\" => hempty\n\ninfixr:60 \" ~~> \" => hsingle\n\nclass HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ\n\ninfixr:55 \" ∗ \" => HStar.hStar\n\n@[default_instance]\ninstance : HStar hProp hProp hProp where\n  hStar := hstar\n\nsection\n\nopen Lean.TSyntax.Compat\n\nend\n\ndef hpure (P : Prop) : hProp :=\n  hexists (fun (_ : P) => emp)\n\ndef hwand (H1 H2 : hProp) : hProp :=\n  hexists (fun (H0 : hProp) => H0 ∗ hpure ((H1 ∗ H0) ==> H2))\n\ndef qwand {A} (Q1 Q2 : A → hProp) : hProp :=\n  hforall (fun (x : A) => hwand (Q1 x) (Q2 x))\n\nnotation:max \"⌜\" P \"⌝\" => hpure P\n\nnotation (priority := high) \"⊤\" => htop\n\ndef qstar {A} (Q : A → hProp) (H : hProp) : A → hProp :=\n  fun x => hstar (Q x) H\n\ninstance (A : Type) : HStar (A → hProp) hProp (A → hProp) where\n  hStar := qstar\n\nclass HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hWand : α → β → γ\n\ninfixr:55 \" -∗ \" => HWand.hWand\n\n@[default_instance]\ninstance : HWand hProp hProp hProp where\n  hWand := hwand\n\ninstance (α : Type) : HWand (α → hProp) (α → hProp) hProp where\n  hWand := qwand", "target_theorem": "lemma qwand_equiv H A (Q1 Q2 : A → hProp) :\n  H ==> (Q1 -∗ Q2) ↔ (Q1 ∗ H) ===> Q2 :=", "ground_truth_proof": ":=\nby\n  srw qwandE ; apply Iff.intro\n  { move=> ? x\n    srw qstarE hstar_comm\n    apply (himpl_hstar_trans_l H (hforall fun x' ↦ Q1 x' -∗ Q2 x'))=>//\n    apply (himpl_trans (hforall fun x0 ↦ ((Q1 x0 -∗ Q2 x0) ∗ Q1 x)))\n    apply hstar_hforall ; apply himpl_hforall_l\n    rw [hstar_comm] ; apply hwand_cancel }\n  srw qimpl qstarE => ?\n  apply himpl_hforall_r => ?\n  sby srw hwand_equiv=> ?", "nesting_depth": 7, "transitive_dep_count": 63, "subset_aristotle": false, "category": "Framework"}
{"id": 407, "thm_name": "Theories.xwp_lemma_funs", "thm_stmt": "lemma xwp_lemma_funs (xs : List _) (vs : List val) :\n  t = trm_apps v0 ts ->\n  v0 = val_funs xs t1 ->\n  trms_to_vals ts = vs ->\n  var_funs xs vs.length ->\n  H ==> wpgen (isubst (xs.mkAlist vs) t1) Q ->\n  triple t H Q", "lean_root": "splean", "rel_path": "SPLean/Theories/WP1.lean", "imports": ["import SPLean.Theories.XChange", "import SPLean.Theories.Lang", "import Mathlib.Data.List.Indexes", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Lean", "import SPLean.Theories.WPUtil"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "AList.erase", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup", "module": "Mathlib.Data.List.AList"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "L", "module": "Archive.Hairer"}, {"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "And", "module": "Init.Prelude"}, {"name": "sdone", "module": "Ssreflect.Done"}, {"name": "AList.entries", "module": "Mathlib.Data.List.AList"}, {"name": "List.NodupKeys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.keys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "List.zip", "module": "Init.Data.List.Basic"}, {"name": "List.zipWith", "module": "Init.Data.List.Basic"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "AList.insert", "module": "Mathlib.Data.List.AList"}, {"name": "List.kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kinsert", "module": "Mathlib.Data.List.Sigma"}, {"name": "AList.keys", "module": "Mathlib.Data.List.AList"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}], "used_repo_defs": [{"name": "syntax \" != \" : bop", "content": "syntax \" != \" : bop\n\nsyntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang\n\nsyntax \" := \" : bop"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang"}, {"name": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hexists xs b"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang"}, {"name": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hforall xs b\n\nsyntax \" ++ \" : bop\n\nsyntax \"not\" : uop"}, {"name": "macro \"xpull\" : tactic =>", "content": "macro \"xpull\" : tactic =>\n  `(tactic| (\n    xpull_start\n    repeat' xsimp_step\n    try rev_pure\n    hsimp\n  ))\n\n  syntax \"sdo\" num tactic : tactic"}, {"name": "macro \"xchange\" l:term : tactic =>", "content": "macro \"xchange\" l:term : tactic =>\n  `(tactic| (xchange_core $l; xsimp))"}, {"name": "macro \"xsimp\" : tactic =>", "content": "macro \"xsimp\" : tactic =>\n  `(tactic| (\n    xsimp_start\n    repeat xsimp_step\n    try rev_pure\n    try hide_mvars\n    try hsimp\n    rotate_left\n\n  ))\n\nsyntax \"while\" lang  \" {\"  (ppLine lang) ( \" }\") : lang\n\nsyntax \"for\" ident \" in \" \"[\" lang \" : \" lang \"]\" \" {\"  (ppLine lang) ( \" }\") : lang"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term| {| $seq |}) => `(withMainContext do evalTactic $ <- `(tacticSeq| $seq))"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)"}, {"name": "var", "content": "abbrev var := String\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "hpure", "content": "def hpure (P : Prop) : hProp :=\n  hexists (fun (_ : P) => emp)"}, {"name": "hexists", "content": "def hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "hsingle", "content": "def hsingle (p : loc) (v : val) : hProp :=\n  fun h => (h = Finmap.singleton p v)"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "HWand", "content": "class HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hWand : α → β → γ"}, {"name": "protect", "content": "@[heapSimp]\ndef protect (x : α) := x"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "make_list", "content": "def make_list {A} (n : Nat) (v : A) : List A :=\n  match n with\n  | 0      => []\n  | n' + 1 => v :: make_list n' v"}, {"name": "hrange", "content": "def hrange (L : List val) (p : loc) : hProp :=\n  match L with\n  | []      => emp\n  | x :: L' => (p ~~> x) ∗ (hrange L' (p + 1))"}, {"name": "null", "content": "def null : loc := 0"}, {"name": "triple", "content": "abbrev triple (t : trm) (H : hProp) (Q : val → hProp) : Prop :=\n  forall s, H s → eval s t Q"}, {"name": "wp", "content": "def wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q"}, {"name": "val_funs", "content": "def val_funs (xs:List var) (t:trm) : val :=\n  match xs with\n  | [] => panic! \"function with zero argumets!\"\n  | x1::xs' => val_fun x1 (trm_funs xs' t)"}, {"name": "trm_funs", "content": "def trm_funs (xs:List var) (t:trm) : trm :=\n  match xs with\n  | [] => t\n  | x1::xs' => trm_fun x1 (trm_funs xs' t)"}, {"name": "trm_apps", "content": "def trm_apps (f:trm) (ts:List trm) : trm :=\n  match ts with\n  | [] => f\n  | ti::ts' => trm_apps (trm_app f ti) ts'"}, {"name": "eval_like", "content": "def eval_like (t1 t2:trm) : Prop :=\n  forall s Q, eval s t1 Q -> eval s t2 Q"}, {"name": "trm_is_val", "content": "abbrev trm_is_val : trm → Prop\n  | trm_val _ => true\n  | _         => false"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "hforall", "content": "def hforall {A} (J : A → hProp) : hProp :=\n  fun h => forall x, J x h"}, {"name": "tohProp", "content": "abbrev tohProp (h : heap -> Prop) : hProp := h"}, {"name": "ofhProp", "content": "abbrev ofhProp (h : val -> hProp) : val -> heap -> Prop := h"}, {"name": "conseq", "content": "def conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=\n  match vs with\n  | [] => ∅\n  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:60 \" ~~> \" => hsingle", "content": "infixr:60 \" ~~> \" => hsingle"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}, {"name": "infixr:55 \" -∗ \" => HWand.hWand", "content": "infixr:55 \" -∗ \" => HWand.hWand"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "AList.keys_nodup", "module": "Mathlib.Data.List.AList"}, {"name": "List.dlookup_dedupKeys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.lookup_ext", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.perm_dlookup", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.perm_nodupKeys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.map_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.map_zipWith", "module": "Init.Data.List.Zip"}, {"name": "List.nodup_cons", "module": "Init.Data.List.Pairwise"}, {"name": "List.perm_append_singleton", "module": "Init.Data.List.Perm"}, {"name": "List.zip_append", "module": "Init.Data.List.Zip"}, {"name": "List.kerase_cons_ne", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kerase_kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "AList.perm_erase", "module": "Mathlib.Data.List.AList"}, {"name": "AList.perm_lookup", "module": "Mathlib.Data.List.AList"}, {"name": "AList.mem_keys", "module": "Mathlib.Data.List.AList"}, {"name": "List.eraseP_of_forall_not", "module": "Init.Data.List.Erase"}, {"name": "AList.erase_erase", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup_erase_ne", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup_insert_ne", "module": "Mathlib.Data.List.AList"}, {"name": "congrArg", "module": "Init.Prelude"}, {"name": "AList.toAList_cons", "module": "Mathlib.Data.List.AList"}, {"name": "List.nodup_middle", "module": "Mathlib.Data.List.Nodup"}, {"name": "Finmap.union_comm_of_disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.Disjoint.symm_iff", "module": "Mathlib.Data.Finmap"}], "repo_lemmas": [{"name": "eval_app_arg1'", "content": "lemma eval_app_arg1' s1 t1 t2 Q1 Q :\n  eval s1 t1 Q1 ->\n  (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n  eval s1 (trm_app t1 t2) Q"}, {"name": "eval_conseq", "content": "lemma eval_conseq s t Q1 Q2 :\n  eval s t Q1 →\n  Q1 ===> Q2 →\n  eval s t Q2"}, {"name": "hwand_inv", "content": "lemma hwand_inv h1 h2 H1 H2 :\n  (H1 -∗ H2) h2 →\n  H1 h1 →\n  Finmap.Disjoint h1 h2 →\n  H2 (h1 ∪ h2)"}, {"name": "hsingl_inv", "content": "lemma hsingl_inv p v h :\n  (p ~~> v) h →\n  h = Finmap.singleton p v"}, {"name": "hforall_inv", "content": "lemma hforall_inv A (J : A → hProp) h :\n  (hforall J) h → forall x, J x h"}, {"name": "union_singleton_eq_insert", "content": "lemma union_singleton_eq_insert (h : state) :\n  Finmap.singleton p v ∪ h = h.insert p v"}, {"name": "insert_delete_id", "content": "lemma insert_delete_id (h : state) (p : loc) :\n  p ∉ h →\n  h = (h.insert p v).erase p"}, {"name": "himpl_trans", "content": "lemma himpl_trans H2 H1 H3 :\n  (H1 ==> H2) → (H2 ==> H3) → (H1 ==> H3)"}, {"name": "himpl_hforall_l", "content": "lemma himpl_hforall_l A (x : A) (J : A → hProp) H :\n  (J x ==> H) →\n  (hforall J) ==> H"}, {"name": "hwand_hpure_l", "content": "lemma hwand_hpure_l P H :\n  P → (⌜P⌝ -∗ H) = H"}, {"name": "himpl_antisym", "content": "lemma himpl_antisym H1 H2:\n  (H1 ==> H2) → (H2 ==> H1) → (H1 = H2)"}, {"name": "himpl_hstar_hpure_l", "content": "lemma himpl_hstar_hpure_l P H H' :\n  (P → H ==> H') →\n  (⌜P⌝ ∗ H) ==> H'"}, {"name": "hstar_hpure_l", "content": "lemma hstar_hpure_l P H h :\n  (⌜P⌝ ∗ H) h = (P ∧ H h)"}, {"name": "hstar_hempty_l", "content": "lemma hstar_hempty_l H :\n  emp ∗ H = H"}, {"name": "hempty_inv", "content": "lemma hempty_inv h :\n  emp h → h = ∅"}, {"name": "hstar_hexists", "content": "lemma hstar_hexists A (J : A → hProp) H :\n  (hexists J) ∗ H = hexists (fun x => (J x) ∗ H)"}, {"name": "himpl_hstar_hpure_r", "content": "lemma himpl_hstar_hpure_r P H H' :\n   P →\n   (H ==> H') →\n   H ==> ⌜P⌝ ∗ H'"}, {"name": "himpl_refl", "content": "lemma himpl_refl H : H ==> H"}, {"name": "hwand_cancel", "content": "lemma hwand_cancel H1 H2 :\n  H1 ∗ (H1 -∗ H2) ==> H2"}, {"name": "himpl_hwand_r_inv", "content": "lemma himpl_hwand_r_inv H1 H2 H3 :\n  H1 ==> (H2 -∗ H3) →\n  H2 ∗ H1 ==> H3"}, {"name": "hwand_equiv", "content": "lemma hwand_equiv H0 H1 H2 :\n  (H0 ==> H1 -∗ H2) ↔ (H1 ∗ H0 ==> H2)"}, {"name": "hstar_assoc", "content": "lemma hstar_assoc H1 H2 H3 :\n  (H1 ∗ H2) ∗ H3 = H1 ∗ (H2 ∗ H3)"}, {"name": "himpl_hstar_trans_l", "content": "lemma himpl_hstar_trans_l H1 H2 H3 H4 :\n  H1 ==> H2 →\n  H2 ∗ H3 ==> H4 →\n  H1 ∗ H3 ==> H4"}, {"name": "hstar_comm", "content": "lemma hstar_comm H1 H2 :\n  H1 ∗ H2 = H2 ∗ H1"}, {"name": "hprop_op_comm", "content": "lemma hprop_op_comm (op : hProp → hProp → hProp) :\n  (forall H1 H2, op H1 H2 ==> op H2 H1) →\n  (forall H1 H2, op H1 H2 = op H2 H1)"}, {"name": "hstar_inv", "content": "lemma hstar_inv (H1 H2 : hProp) h:\n  (H1 ∗ H2) h →\n  exists h1 h2, H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2"}, {"name": "himpl_hempty_hpure", "content": "lemma himpl_hempty_hpure P :\n  P → emp ==> ⌜P⌝"}, {"name": "himpl_hexists_r", "content": "lemma himpl_hexists_r A (x : A) H (J : A → hProp) :\n  (H ==> J x) →\n  H ==> (hexists J)"}, {"name": "hwandE", "content": "lemma hwandE :\n  H1 -∗ H2 = hexists (fun H0 => H0 ∗ hpure ((H1 ∗ H0) ==> H2))"}, {"name": "himpl_frame_r", "content": "lemma himpl_frame_r H1 H2 H2' :\n  H2 ==> H2' →\n  (H1 ∗ H2) ==> (H1 ∗ H2')"}, {"name": "hstar_hempty_r", "content": "lemma hstar_hempty_r H :\n  H ∗ emp = H"}, {"name": "himpl_hexists_l", "content": "lemma himpl_hexists_l A H (J : A → hProp) :\n  (forall x, J x ==> H) → (hexists J) ==> H"}, {"name": "qstarE", "content": "lemma qstarE α (Q1 : α → hProp)  (H : hProp):\n  Q1 ∗ H = fun x => Q1 x ∗ H"}, {"name": "eval_frame", "content": "lemma eval_frame (h1 h2 : state) t (Q : val -> hProp) :\n  eval h1 t (ofhProp Q) →\n  Finmap.Disjoint h1 h2 →\n  eval (h1 ∪ h2) t (Q ∗ (tohProp (fun h ↦ h = h2)))"}, {"name": "qwand_cancel", "content": "lemma qwand_cancel A (Q1 Q2 : A → hProp) :\n  Q1 ∗ (Q1 -∗ Q2) ===> Q2"}, {"name": "qwand_equiv", "content": "lemma qwand_equiv H A (Q1 Q2 : A → hProp) :\n  H ==> (Q1 -∗ Q2) ↔ (Q1 ∗ H) ===> Q2"}, {"name": "hstar_hforall", "content": "lemma hstar_hforall A (J : A → hProp) H :\n  (hforall J) ∗ H ==> hforall (J ∗ H)"}, {"name": "himpl_hforall_r", "content": "lemma himpl_hforall_r A (J : A → hProp) H :\n  (forall x, H ==> J x) →\n  H ==> (hforall J)"}, {"name": "qwandE", "content": "lemma qwandE α (Q1 Q2 : α → hProp) :\n  Q1 -∗ Q2 = hforall (fun x => (Q1 x) -∗ (Q2 x))"}, {"name": "hpure_inv", "content": "lemma hpure_inv P h :\n  ⌜P⌝ h →\n  P ∧ h = ∅"}, {"name": "hrange_eq_conseq", "content": "lemma hrange_eq_conseq (L : List val) (n : ℤ) (p : loc) (s : state) :\n  L.length = n →\n  hrange L p s →\n  s.keys = (conseq (make_list n.natAbs val_uninit) p).keys"}, {"name": "int_eq_sub", "content": "lemma int_eq_sub (l m n : ℤ) :\n  l + m = n → l = n - m"}, {"name": "list_inc_natabs", "content": "lemma list_inc_natabs {α : Type} (L : List α) :\n  ((L.length : ℤ) + 1).natAbs = (L.length : ℤ).natAbs + 1"}, {"name": "hexists_inv", "content": "lemma hexists_inv A (J : A → hProp) h :\n  (hexists J) h → exists x, J x h"}, {"name": "diff_disjoint_eq", "content": "lemma diff_disjoint_eq (s₁ s₂ s₃ : state) :\n  s₁.Disjoint s₂ →\n  s₂.keys = s₃.keys →\n  (s₁ ∪ s₂) \\ s₃  = s₁"}, {"name": "lookup_diff", "content": "lemma lookup_diff (h₁ h₂ : state) :\n  p ∉ h₂ →\n  (h₁ \\ h₂).lookup p = h₁.lookup p"}, {"name": "lookup_diff_none", "content": "lemma lookup_diff_none (h₁ h₂ : state) :\n  p ∈ h₂ →\n  (h₁ \\ h₂).lookup p = none"}, {"name": "diff_non_mem", "content": "theorem diff_non_mem (h₁ h₂ : state) :\n  p ∈ h₂ → p ∉ h₁ \\ h₂"}], "used_local_defs": [{"name": "Theories.wp", "content": "def wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q"}, {"name": "Theories.formula", "content": "abbrev formula := (val → hProp) → hProp"}, {"name": "Theories.mkstruct", "content": "def mkstruct (F : formula) :=\n  fun (Q : val -> hProp) ↦ ∃ʰ Q', F Q' ∗ (Q' -∗ Q)"}, {"name": "Theories.wpgen_val", "content": "def wpgen_val (v : val) : formula :=\n  fun Q ↦ Q v"}, {"name": "Theories.wpgen_fun", "content": "def wpgen_fun (Fof : val → formula) : formula :=\n  fun Q ↦ h∀ vf, ⌜forall vx Q', Fof vx Q' ==>\n    wp (trm_app (trm_val vf) (trm_val vx)) Q'⌝ -∗ Q vf"}, {"name": "Theories.wpgen_fix", "content": "def wpgen_fix (Fof : val → val → formula) : formula :=\n  fun Q ↦ h∀ vf, ⌜forall vx Q', Fof vf vx Q' ==>\n    wp (trm_app vf vx) Q'⌝ -∗ Q vf"}, {"name": "Theories.wpgen_seq", "content": "def wpgen_seq (F1 F2 : formula) : formula :=\n  fun Q ↦ F1 (fun _ ↦ F2 Q)"}, {"name": "Theories.wpgen_let", "content": "def wpgen_let (F1 : formula) (F2of : val → formula) : formula :=\n  fun Q ↦ F1 (fun v ↦ F2of v Q)"}, {"name": "Theories.wpgen_if", "content": "def wpgen_if (t : trm) (F1 F2 : formula) : formula :=\n  fun Q ↦ ∃ʰ (b : Bool),\n    ⌜t = trm_val (val_bool b)⌝ ∗ (if b then F1 Q else F2 Q)"}, {"name": "Theories.wpgen_app", "content": "@[simp]\ndef wpgen_app (t : trm) : formula :=\n  fun Q ↦ ∃ʰ H, H ∗ ⌜triple t H Q⌝"}, {"name": "Theories.wpgen_ref", "content": "def wpgen_ref (x : var) (t1 t2 : trm) : formula :=\n  fun Q ↦ ∃ʰ v, ⌜t1 = trm_val v⌝ ∗\n    h∀ p, (p ~~> v) -∗ protect (wp (subst x p t2) (fun hv ↦ Q hv ∗ ∃ʰ u, p ~~> u))"}, {"name": "Theories.wpgen_alloc", "content": "def wpgen_alloc (x : var) (t1 t2 : trm) : formula :=\n  fun Q ↦ ∃ʰ n : ℤ,\n    ⌜n ≥ 0 ∧ t1 = trm_val n⌝ ∗\n    h∀ p,\n      (hrange (make_list n.natAbs val_uninit) p) -∗\n      protect wp (subst x p t2) (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p)"}, {"name": "Theories.wpgen", "content": "def wpgen (t : trm) : formula :=\n  mkstruct (match t with\n  | trm_val v          => wpgen_val v\n  | trm_fun x t1       => wpgen_fun (fun v ↦ wp $ subst x v t1)\n  | trm_fix f x t1     => wpgen_fix\n      (fun vf v => wp $ subst x v $ subst f vf t1)\n  | trm_if t0 t1 t2    => wpgen_if t0 (wp t1) (wp t2)\n  | trm_seq t1 t2      => wpgen_seq (wp t1) (wp t2)\n  | trm_let x t1 t2    => wpgen_let (wp t1) (fun v ↦ wp $ subst x v t2)\n  | trm_app _ _        => wpgen_app t\n  \n  \n  | trm_ref x t1 t2    => wpgen_ref x t1 t2\n  | trm_alloc x t1 t2  => wpgen_alloc x t1 t2\n  | _ => wp t\n  )"}, {"name": "Theories.formula_sound", "content": "def formula_sound (t : trm) (F : formula) : Prop :=\n  forall Q, F Q ==> wp t Q"}, {"name": "Theories.var_funs", "content": "@[simp]\nabbrev var_funs (xs:List var) (n:Nat) : Prop :=\n     xs.Nodup\n  /\\ xs.length = n\n  /\\ xs != []"}, {"name": "Theories.trms_to_vals", "content": "@[simp]\ndef trms_to_vals (ts:List trm) : Option (List val) := do\n  match ts with\n  | [] => return []\n  | (trm_val v) :: ts' => v :: (<- trms_to_vals ts')\n  | _ => failure"}, {"name": "Theories.ctx", "content": "abbrev ctx := AList (fun _ : var ↦ val)"}, {"name": "Theories.isubst", "content": "def isubst (E : ctx) (t : trm) : trm :=\n  match t with\n  | trm_val v =>\n      v\n  | trm_var x =>\n      match lookup x E with\n      | none   => t\n      | some v => v\n  | trm_fun x t1 =>\n      trm_fun x (isubst (erase x E) t1)\n  | trm_fix f x t1 =>\n      trm_fix f x (isubst (erase x (erase f E)) t1)\n  | trm_if t0 t1 t2 =>\n      trm_if (isubst E t0) (isubst E t1) (isubst E t2)\n  | trm_seq t1 t2 =>\n      trm_seq (isubst E t1) (isubst E t2)\n  | trm_let x t1 t2 =>\n      trm_let x (isubst E t1) (isubst (erase x E) t2)\n  | trm_ref x t1 t2 =>\n      trm_ref x (isubst E t1) (isubst (erase x E) t2)\n  | trm_alloc x t1 t2 =>\n      trm_alloc x (isubst E t1) (isubst (erase x E) t2)\n  | trm_app t1 t2 =>\n      trm_app (isubst E t1) (isubst E t2)\n  | trm_for x n1 n2 t =>\n      trm_for x (isubst E n1) (isubst E n2) (isubst (erase x E) t)\n  | trm_while c t =>\n      trm_while (isubst E c) (isubst E t)"}], "used_local_lemmas": [{"name": "Theories.wp_equiv", "content": "lemma wp_equiv t H Q :\n  (H ==> wp t Q) ↔ triple t H Q"}, {"name": "Theories.wp_conseq", "content": "lemma wp_conseq t Q1 Q2 :\n  Q1 ===> Q2 →\n  wp t Q1 ==> wp t Q2"}, {"name": "Theories.wp_frame", "content": "lemma wp_frame t H Q :\n  (wp t Q) ∗ H ==> wp t (Q ∗ H)"}, {"name": "Theories.wp_ramified", "content": "lemma wp_ramified t (Q1 Q2 : val -> hProp) :\n  (wp t Q1) ∗ (Q1 -∗ Q2) ==> (wp t Q2)"}, {"name": "Theories.wp_eval_like", "content": "lemma wp_eval_like t1 t2 Q :\n  eval_like t1 t2 →\n  wp t1 Q ==> wp t2 Q"}, {"name": "Theories.wp_val", "content": "lemma wp_val v Q :\n  Q v ==> wp (trm_val v) Q"}, {"name": "Theories.wp_fun", "content": "lemma wp_fun x t Q :\n  Q (val_fun x t) ==> wp (trm_fun x t) Q"}, {"name": "Theories.wp_fix", "content": "lemma wp_fix f x t Q :\n  Q (val_fix f x t) ==> wp (trm_fix f x t) Q"}, {"name": "Theories.wp_app_fun", "content": "lemma wp_app_fun x v1 v2 t1 Q :\n  v1 = val_fun x t1 →\n  wp (subst x v2 t1) Q ==> wp (trm_app v1 v2) Q"}, {"name": "Theories.wp_app_fix", "content": "lemma wp_app_fix f x v1 v2 t1 Q :\n  v1 = val_fix f x t1 →\n  wp (subst x v2 (subst f v1 t1)) Q ==> wp (trm_app v1 v2) Q"}, {"name": "Theories.wp_seq", "content": "lemma wp_seq t1 t2 Q :\n  wp t1 (fun _ ↦ wp t2 Q) ==> wp (trm_seq t1 t2) Q"}, {"name": "Theories.wp_let", "content": "lemma wp_let x t1 t2 Q :\n  wp t1 (fun v ↦ wp (subst x v t2) Q) ==> wp (trm_let x t1 t2) Q"}, {"name": "Theories.wp_ref", "content": "lemma wp_ref x v t Q :\n  (h∀ p, (p ~~> v) -∗ wp (subst x p t) (Q ∗ ∃ʰ v', (p ~~> v'))) ==>\n  wp (trm_ref x v t) Q"}, {"name": "Theories.mem_conseq", "content": "lemma mem_conseq :\n  x ∈ conseq L p → p ≤ x"}, {"name": "Theories.hrange_of_conseq", "content": "lemma hrange_of_conseq :\n  (hrange L p) (conseq L p)"}, {"name": "Theories.wp_alloc", "content": "lemma wp_alloc x (n : ℤ) t Q :\n  n ≥ 0 →\n  (h∀ p, (hrange (make_list n.natAbs val_uninit) p) -∗\n    wp (subst x p t) (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p)) ==>\n  wp (trm_alloc x n t) Q"}, {"name": "Theories.mkstruct_erase", "content": "lemma mkstruct_erase Q F :\n  F Q ==> mkstruct F Q"}, {"name": "Theories.mkstruct_monotone", "content": "lemma mkstruct_monotone F1 F2 Q :\n  (forall Q, F1 Q ==> F2 Q) →\n  mkstruct F1 Q ==> mkstruct F2 Q"}, {"name": "Theories.mkstruct_wp", "content": "lemma mkstruct_wp t :\n  mkstruct (wp t) = wp t"}, {"name": "Theories.mkstruct_sound", "content": "lemma mkstruct_sound t F :\n  formula_sound t F →\n  formula_sound t (mkstruct F)"}, {"name": "Theories.wpgen_val_sound", "content": "lemma wpgen_val_sound v :\n  formula_sound (trm_val v) (wpgen_val v)"}, {"name": "Theories.wpgen_fun_sound", "content": "lemma wpgen_fun_sound x t1 Fof :\n  (forall vx, formula_sound (subst x vx t1) (Fof vx)) →\n  formula_sound (trm_fun x t1) (wpgen_fun Fof)"}, {"name": "Theories.wpgen_fix_sound", "content": "lemma wpgen_fix_sound f x t1 Fof :\n  (forall vf vx, formula_sound (subst x vx (subst f vf t1)) (Fof vf vx)) →\n  formula_sound (trm_fix f x t1) (wpgen_fix Fof)"}, {"name": "Theories.wpgen_seq_sound", "content": "lemma wpgen_seq_sound F1 F2 t1 t2 :\n  formula_sound t1 F1 →\n  formula_sound t2 F2 →\n  formula_sound (trm_seq t1 t2) (wpgen_seq F1 F2)"}, {"name": "Theories.wpgen_let_sound", "content": "lemma wpgen_let_sound F1 F2of x t1 t2 :\n  formula_sound t1 F1 →\n  (forall v, formula_sound (subst x v t2) (F2of v)) →\n  formula_sound (trm_let x t1 t2) (wpgen_let F1 F2of)"}, {"name": "Theories.wpgen_if_sound", "content": "lemma wpgen_if_sound F1 F2 t0 t1 t2 :\n  formula_sound t1 F1 →\n  formula_sound t2 F2 →\n  formula_sound (trm_if t0 t1 t2) (wpgen_if t0 F1 F2)"}, {"name": "Theories.wpgen_app_sound", "content": "lemma wpgen_app_sound t :\n  formula_sound t (wpgen_app t)"}, {"name": "Theories.wpgen_ref_sound", "content": "lemma wpgen_ref_sound x t1 t2 :\n  formula_sound (trm_ref x t1 t2) (wpgen_ref x t1 t2)"}, {"name": "Theories.wpgen_alloc_sound", "content": "lemma wpgen_alloc_sound x t1 t2 :\n  formula_sound (trm_alloc x t1 t2) (wpgen_alloc x t1 t2)"}, {"name": "Theories.wpgen_sound", "content": "lemma wpgen_sound t :\n  formula_sound t (wpgen t)"}, {"name": "Theories.himpl_wpgen_wp", "content": "lemma himpl_wpgen_wp t Q :\n  wpgen t Q ==> wp t Q"}, {"name": "Theories.triple_of_wpgen", "content": "lemma triple_of_wpgen t H Q :\n  H ==> wpgen t Q →\n  triple t H Q"}, {"name": "Theories.wp_of_wpgen", "content": "lemma wp_of_wpgen :\n  H ==> wpgen t Q →\n  H ==> wp t Q"}, {"name": "Theories.trms_to_vals_some_equiv", "content": "lemma trms_to_vals_some_equiv ts vs : trms_to_vals ts = some vs → ts = vs.map trm_val"}, {"name": "Theories.List.toAList_perm", "content": "lemma List.toAList_perm {α : Type u} {β : α → Type v} [DecidableEq α]\n  (es es' : List (Sigma β)) (hnodup : es.NodupKeys) (hp : es.Perm es') :\n  es.toAList.entries.Perm es'.toAList.entries"}, {"name": "Theories.List.mkAlist_snoc_to_cons", "content": "lemma List.mkAlist_snoc_to_cons [DecidableEq α] (xs : List α) (vs : List β)\n    (x : α) (v : β) : x ∉ xs → xs.length = vs.length → xs.Nodup →\n    ((xs ++ [x]).mkAlist (vs ++ [v])).entries.Perm (((x :: xs).mkAlist (v :: vs)).entries)"}, {"name": "Theories.AList.erase_insert_cancel", "content": "lemma AList.erase_insert_cancel {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) (l : AList β) :\n    (AList.erase a (AList.insert a b l)).entries.Perm (AList.erase a l).entries"}, {"name": "Theories.AList.erase_insert_swap", "content": "lemma AList.erase_insert_swap {α : Type u} {β : α → Type v} [DecidableEq α] (a a' : α) (b : β a) (l : AList β) :\n  a ≠ a' → (AList.erase a' (AList.insert a b l)).entries.Perm (AList.insert a b (AList.erase a' l)).entries"}, {"name": "Theories.AList.erase_noop", "content": "lemma AList.erase_noop {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : AList β) :\n    a ∉ l → (AList.erase a l).entries.Perm l.entries"}, {"name": "Theories.AList.erase_twice", "content": "lemma AList.erase_twice {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : AList β) :\n    (AList.erase a (AList.erase a l)).entries.Perm (AList.erase a l).entries"}, {"name": "Theories.AList.erase_empty", "content": "lemma AList.erase_empty {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) :\n  AList.erase a (∅ : AList β) = ∅"}, {"name": "Theories.isubst_empty", "content": "lemma isubst_empty t : isubst ∅ t = t"}, {"name": "Theories.isubst_perm", "content": "lemma isubst_perm {al al'} t (hp : al.entries.Perm al'.entries) :\n  isubst al t = isubst al' t"}, {"name": "Theories.isubst_insert", "content": "lemma isubst_insert (al : ctx) x v t :\n  isubst (al.insert x v) t = subst x v (isubst (al.erase x) t)"}, {"name": "Theories.isubst_single", "content": "lemma isubst_single x v t : isubst (List.mkAlist [x] [v]) t = subst x v t"}, {"name": "Theories.trm_apps2", "content": "lemma trm_apps2 :\n  trm_apps (trm_app t1 t2) ts = trm_apps t1 (t2::ts)"}, {"name": "Theories.trm_apps_app", "content": "lemma trm_apps_app :\n  trm_apps t1 (ts ++ ts') = trm_apps (trm_apps t1 ts) ts'"}, {"name": "Theories.trm_funs_app", "content": "lemma trm_funs_app :\n  trm_funs (xs ++ xs') t1 = trm_funs xs (trm_funs xs' t1)"}, {"name": "Theories.eval_like_trm_fun_val_fun", "content": "lemma eval_like_trm_fun_val_fun x t : eval_like (trm_fun x t) (val_fun x t)"}, {"name": "Theories.eval_like_val_fun_trm_fun", "content": "lemma eval_like_val_fun_trm_fun x t : eval_like (val_fun x t) (trm_fun x t)"}, {"name": "Theories.eval_like_trm_app_left", "content": "lemma eval_like_trm_app_left t1 t1' t2 (hsat : ∃ s Q, eval s t1 Q) : eval_like t1 t1' → eval_like (trm_app t1 t2) (trm_app t1' t2)"}, {"name": "Theories.eval_like_trm_fun_val_fun_app_left", "content": "lemma eval_like_trm_fun_val_fun_app_left (x : var) (t1 t2 : trm) :\n  eval_like (trm_app (trm_fun x t1) t2) (trm_app (val_fun x t1) t2)"}, {"name": "Theories.eval_like_val_fun_trm_fun_app_left", "content": "lemma eval_like_val_fun_trm_fun_app_left (x : var) (t1 t2 : trm) :\n  eval_like (trm_app (val_fun x t1) t2) (trm_app (trm_fun x t1) t2)"}, {"name": "Theories.val_funs_snoc", "content": "lemma val_funs_snoc (xs : List var) (x : var) (h : xs ≠ []) (t : trm) :\n  val_funs (xs ++ [x]) t = val_funs xs (trm_fun x t)"}, {"name": "Theories.List.not_nil_snoc", "content": "lemma List.not_nil_snoc {α : Type u} (l : List α) : l ≠ [] → ∃ l' x, l = l' ++ [x]"}, {"name": "Theories.eval_like_trm_apps_funs_pre", "content": "lemma eval_like_trm_apps_funs_pre (heqv0 : v0 = trm_funs xs t1) :\n  eval_like t (trm_apps (val_funs xs t1) ts) ∧  -- NOTE: this part do not require `xs.Nodup`, but anyway\n  eval_like (isubst (xs.mkAlist vs) t1) t"}, {"name": "Theories.eval_like_trm_apps_funs", "content": "lemma eval_like_trm_apps_funs (heqv0 : v0 = trm_funs xs t1) :\n  eval_like (isubst (xs.mkAlist vs) t1) (trm_apps (val_funs xs t1) ts)"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Indexes\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WPUtil\n\nopen trm val prim\n\nnamespace Theories\n\ndef wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q\n\nabbrev formula := (val → hProp) → hProp\n\ndef mkstruct (F : formula) :=\n  fun (Q : val -> hProp) ↦ ∃ʰ Q', F Q' ∗ (Q' -∗ Q)\n\ndef wpgen_val (v : val) : formula :=\n  fun Q ↦ Q v\n\ndef wpgen_fun (Fof : val → formula) : formula :=\n  fun Q ↦ h∀ vf, ⌜forall vx Q', Fof vx Q' ==>\n    wp (trm_app (trm_val vf) (trm_val vx)) Q'⌝ -∗ Q vf\n\ndef wpgen_fix (Fof : val → val → formula) : formula :=\n  fun Q ↦ h∀ vf, ⌜forall vx Q', Fof vf vx Q' ==>\n    wp (trm_app vf vx) Q'⌝ -∗ Q vf\n\ndef wpgen_seq (F1 F2 : formula) : formula :=\n  fun Q ↦ F1 (fun _ ↦ F2 Q)\n\ndef wpgen_let (F1 : formula) (F2of : val → formula) : formula :=\n  fun Q ↦ F1 (fun v ↦ F2of v Q)\n\ndef wpgen_if (t : trm) (F1 F2 : formula) : formula :=\n  fun Q ↦ ∃ʰ (b : Bool),\n    ⌜t = trm_val (val_bool b)⌝ ∗ (if b then F1 Q else F2 Q)\n\n@[simp]\ndef wpgen_app (t : trm) : formula :=\n  fun Q ↦ ∃ʰ H, H ∗ ⌜triple t H Q⌝\n\ndef wpgen_ref (x : var) (t1 t2 : trm) : formula :=\n  fun Q ↦ ∃ʰ v, ⌜t1 = trm_val v⌝ ∗\n    h∀ p, (p ~~> v) -∗ protect (wp (subst x p t2) (fun hv ↦ Q hv ∗ ∃ʰ u, p ~~> u))\n\ndef wpgen_alloc (x : var) (t1 t2 : trm) : formula :=\n  fun Q ↦ ∃ʰ n : ℤ,\n    ⌜n ≥ 0 ∧ t1 = trm_val n⌝ ∗\n    h∀ p,\n      (hrange (make_list n.natAbs val_uninit) p) -∗\n      protect wp (subst x p t2) (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p)\n\ndef wpgen (t : trm) : formula :=\n  mkstruct (match t with\n  | trm_val v          => wpgen_val v\n  | trm_fun x t1       => wpgen_fun (fun v ↦ wp $ subst x v t1)\n  | trm_fix f x t1     => wpgen_fix\n      (fun vf v => wp $ subst x v $ subst f vf t1)\n  | trm_if t0 t1 t2    => wpgen_if t0 (wp t1) (wp t2)\n  | trm_seq t1 t2      => wpgen_seq (wp t1) (wp t2)\n  | trm_let x t1 t2    => wpgen_let (wp t1) (fun v ↦ wp $ subst x v t2)\n  | trm_app _ _        => wpgen_app t\n  \n  \n  | trm_ref x t1 t2    => wpgen_ref x t1 t2\n  | trm_alloc x t1 t2  => wpgen_alloc x t1 t2\n  | _ => wp t\n  )\n\ndef formula_sound (t : trm) (F : formula) : Prop :=\n  forall Q, F Q ==> wp t Q\n\nsection tactics\n\nopen Lean Elab Tactic\n\nsection xapp\n\nend xapp\n\nend tactics\n\n@[simp]\nabbrev var_funs (xs:List var) (n:Nat) : Prop :=\n     xs.Nodup\n  /\\ xs.length = n\n  /\\ xs != []\n\n@[simp]\ndef trms_to_vals (ts:List trm) : Option (List val) := do\n  match ts with\n  | [] => return []\n  | (trm_val v) :: ts' => v :: (<- trms_to_vals ts')\n  | _ => failure\n\nopen AList\n\nabbrev ctx := AList (fun _ : var ↦ val)\n\ndef isubst (E : ctx) (t : trm) : trm :=\n  match t with\n  | trm_val v =>\n      v\n  | trm_var x =>\n      match lookup x E with\n      | none   => t\n      | some v => v\n  | trm_fun x t1 =>\n      trm_fun x (isubst (erase x E) t1)\n  | trm_fix f x t1 =>\n      trm_fix f x (isubst (erase x (erase f E)) t1)\n  | trm_if t0 t1 t2 =>\n      trm_if (isubst E t0) (isubst E t1) (isubst E t2)\n  | trm_seq t1 t2 =>\n      trm_seq (isubst E t1) (isubst E t2)\n  | trm_let x t1 t2 =>\n      trm_let x (isubst E t1) (isubst (erase x E) t2)\n  | trm_ref x t1 t2 =>\n      trm_ref x (isubst E t1) (isubst (erase x E) t2)\n  | trm_alloc x t1 t2 =>\n      trm_alloc x (isubst E t1) (isubst (erase x E) t2)\n  | trm_app t1 t2 =>\n      trm_app (isubst E t1) (isubst E t2)\n  | trm_for x n1 n2 t =>\n      trm_for x (isubst E n1) (isubst E n2) (isubst (erase x E) t)\n  | trm_while c t =>\n      trm_while (isubst E c) (isubst E t)\n\nsection funs_fixs_eval_like\n\nvariable (xs : List var) (vs : List val) (t : trm) (v0 : trm)\n  (heqt : t = trm_apps v0 ts)\n  (hconv : trms_to_vals ts = vs)\n  (hform : var_funs xs vs.length)   -- NOTE: can be relaxed to `vs.length ≤ xs.length`\n\nvariable (f : var) (hf : f ∉ xs)\n\nend funs_fixs_eval_like", "target_theorem": "lemma xwp_lemma_funs (xs : List _) (vs : List val) :\n  t = trm_apps v0 ts ->\n  v0 = val_funs xs t1 ->\n  trms_to_vals ts = vs ->\n  var_funs xs vs.length ->\n  H ==> wpgen (isubst (xs.mkAlist vs) t1) Q ->\n  triple t H Q :=", "ground_truth_proof": ":= by\n  move=> -> -> ?? h\n  srw -wp_equiv ; apply himpl_trans ; apply (wp_of_wpgen h)\n  apply wp_eval_like\n  apply eval_like_trm_apps_funs=> //", "nesting_depth": 11, "transitive_dep_count": 228, "subset_aristotle": false, "category": "Framework"}
{"id": 408, "thm_name": "Theories.xapp_simpl_lemma", "thm_stmt": "lemma xapp_simpl_lemma (F : formula) :\n  H ==> F Q ->\n  H ==> F Q ∗ (Q -∗ protect Q)", "lean_root": "splean", "rel_path": "SPLean/Theories/WP1.lean", "imports": ["import SPLean.Theories.XChange", "import Mathlib.Data.List.Indexes", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Lean", "import SPLean.Theories.WPUtil"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "move", "module": "Ssreflect.Basic"}], "used_repo_defs": [{"name": "macro \"xsimp\" : tactic =>", "content": "macro \"xsimp\" : tactic =>\n  `(tactic| (\n    xsimp_start\n    repeat xsimp_step\n    try rev_pure\n    try hide_mvars\n    try hsimp\n    rotate_left\n\n  ))"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "HWand", "content": "class HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hWand : α → β → γ"}, {"name": "protect", "content": "@[heapSimp]\ndef protect (x : α) := x"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}, {"name": "infixr:55 \" -∗ \" => HWand.hWand", "content": "infixr:55 \" -∗ \" => HWand.hWand"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "himpl_trans", "content": "lemma himpl_trans H2 H1 H3 :\n  (H1 ==> H2) → (H2 ==> H3) → (H1 ==> H3)"}, {"name": "himpl_hempty_hwand_same", "content": "lemma himpl_hempty_hwand_same H :\n  emp ==> (H -∗ H)"}], "used_local_defs": [{"name": "Theories.formula", "content": "abbrev formula := (val → hProp) → hProp"}], "used_local_lemmas": [], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Indexes\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WPUtil\n\nopen trm val prim\n\nnamespace Theories\n\nabbrev formula := (val → hProp) → hProp\n\nsection tactics\n\nopen Lean Elab Tactic", "target_theorem": "lemma xapp_simpl_lemma (F : formula) :\n  H ==> F Q ->\n  H ==> F Q ∗ (Q -∗ protect Q) :=", "ground_truth_proof": ":= by move=> hh; apply himpl_trans ; apply hh ; xsimp", "nesting_depth": 7, "transitive_dep_count": 20, "subset_aristotle": false, "category": "Framework"}
{"id": 409, "thm_name": "hseg_focus_relative", "thm_stmt": "lemma hseg_focus_relative (k : Nat) L p j (v : 0 <= k ∧ k < L.length):\n  hseg L p j ==>\n  hcell L[k]! p (j + k)\n    ∗ (h∀ w, hcell w p (j + k) -∗ hseg (L.set k w) p j)", "lean_root": "splean", "rel_path": "SPLean/Theories/Arrays.lean", "imports": ["import SPLean.Theories.XChange", "import SPLean.Theories.Lang", "import SPLean.Theories.WP1", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "Int.natAbs", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "L", "module": "Archive.Hairer"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "sdone", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "elim", "module": "Ssreflect.Elim"}], "used_repo_defs": [{"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty\n\nsyntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \" >= \" : bop"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \" ++ \" : bop"}, {"name": "macro \"hsimp\" : tactic => `(tactic| (simp only [heapSimp]; t", "content": "macro \"hsimp\" : tactic => `(tactic| (simp only [heapSimp]; try dsimp))\n\nsyntax \" <= \" : bop"}, {"name": "macro \"xsimp\" : tactic =>", "content": "macro \"xsimp\" : tactic =>\n  `(tactic| (\n    xsimp_start\n    repeat xsimp_step\n    try rev_pure\n    try hide_mvars\n    try hsimp\n    rotate_left\n\n  ))"}, {"name": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hforall xs b"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "hseg", "content": "def hseg (L : List val) (p : loc) (j : Int) : hProp :=\n  match L with\n  | []      => emp\n  | x :: L' => (hcell x p j) ∗ (hseg L' p (j + 1))"}, {"name": "hcell", "content": "def hcell (v : val) (p : loc) (i : Int) : hProp :=\n  ((p + 1 + (Int.natAbs i)) ~~> v) ∗ ⌜i >= 0⌝"}, {"name": "hsingle", "content": "def hsingle (p : loc) (v : val) : hProp :=\n  fun h => (h = Finmap.singleton p v)"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "hpure", "content": "def hpure (P : Prop) : hProp :=\n  hexists (fun (_ : P) => emp)"}, {"name": "hexists", "content": "def hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "HWand", "content": "class HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hWand : α → β → γ"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "hforall", "content": "def hforall {A} (J : A → hProp) : hProp :=\n  fun h => forall x, J x h"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:60 \" ~~> \" => hsingle", "content": "infixr:60 \" ~~> \" => hsingle"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}, {"name": "infixr:55 \" -∗ \" => HWand.hWand", "content": "infixr:55 \" -∗ \" => HWand.hWand"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "List.concat_append", "module": "Init.Data.List.Lemmas"}], "repo_lemmas": [{"name": "hstar_comm_assoc", "content": "lemma hstar_comm_assoc (H1 H2 H3 : hProp) :\n  H1 ∗ H2 ∗ H3 = H2 ∗ H1 ∗ H3"}, {"name": "himpl_frame_r", "content": "lemma himpl_frame_r H1 H2 H2' :\n  H2 ==> H2' →\n  (H1 ∗ H2) ==> (H1 ∗ H2')"}, {"name": "himpl_hforall_r", "content": "lemma himpl_hforall_r A (J : A → hProp) H :\n  (forall x, H ==> J x) →\n  H ==> (hforall J)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "hseg_cons", "content": "lemma hseg_cons v p j L :\n  hseg (v :: L) p j = hcell v p j ∗ hseg L p (j + 1)"}, {"name": "hseg_app", "content": "lemma hseg_app L1 L2 p j :\n  hseg (L1 ++ L2) p j =\n  hseg L1 p j ∗ hseg L2 p (j + L1.length)"}, {"name": "list_cons_length", "content": "lemma list_cons_length (A : Type) (x : A) (L : List A) :\n  (x :: L).length = 1 + L.length"}, {"name": "list_middle_inv", "content": "lemma list_middle_inv (A : Type) (n : Nat) (l : List A) :\n  n < l.length →\n  exists (x : A) (l1 l2 : List A),\n    l = l1 ++ x :: l2 ∧ l1.length = n"}, {"name": "nth_app_r", "content": "lemma nth_app_r {A : Type} (_ : Inhabited A) n (l1 l2 : List A) :\n  n ≥ l1.length →\n  (l1 ++ l2)[n]! = l2[n - l1.length]!"}, {"name": "nth_middle", "content": "lemma nth_middle (A : Type) (IA : Inhabited A) (n : Nat)\n  (l1 l2 : List A) (x : A) :\n  n = l1.length → (l1 ++ x :: l2)[n]! = x"}, {"name": "update_middle", "content": "lemma update_middle (A : Type) (_ : Inhabited A) (n : Nat)\n  (l1 l2 : List A) (x v : A) :\n  n = l1.length → (l1 ++ x :: l2).set n v = (l1.concat v) ++ l2"}], "local_ctx": "import SPLean.Common.State\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WP1\n\nimport SPLean.Theories.Lang\n\nopen val trm prim\n\nopen Theories", "target_theorem": "lemma hseg_focus_relative (k : Nat) L p j (v : 0 <= k ∧ k < L.length):\n  hseg L p j ==>\n  hcell L[k]! p (j + k)\n    ∗ (h∀ w, hcell w p (j + k) -∗ hseg (L.set k w) p j) :=", "ground_truth_proof": ":= by\n  move: v=> [] ? /list_middle_inv ![> ?] hlen\n  move: (Eq.symm hlen) => /nth_middle hmid\n  subst L ; srw (hmid _ w_2 w) hseg_app hseg_cons hlen -hstar_comm_assoc\n  apply himpl_frame_r\n  apply himpl_hforall_r => ?\n  move: (Eq.symm hlen) => /(update_middle val _ k w_1 w_2 w) hset\n  srw hset ?List.concat_append ?hseg_app ?hseg_cons ?hlen\n  { sby xsimp }\n  sdone", "nesting_depth": 9, "transitive_dep_count": 49, "subset_aristotle": false, "category": "Framework"}
{"id": 410, "thm_name": "triple_ref", "thm_stmt": "lemma triple_ref (v : val) :\n  (forall (p : loc), triple (subst x p t2) (H ∗ (p ~~> v)) (Q ∗ ∃ʰ v, p ~~> v)) →\n  triple (trm_ref x (trm_val v) t2) H Q", "lean_root": "splean", "rel_path": "SPLean/Theories/SepLog.lean", "imports": ["import Mathlib.Data.Multiset.Nodup", "import SPLean.Theories.XSimp", "import Mathlib.Data.Finset.Basic", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "elim", "module": "Ssreflect.Elim"}], "used_repo_defs": [{"name": "syntax \"if \" lang \"then \" lang \"end \" : lang", "content": "syntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"fun\" ident+ \" => \" lang : lang"}, {"name": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hexists xs b\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "hsingle", "content": "def hsingle (p : loc) (v : val) : hProp :=\n  fun h => (h = Finmap.singleton p v)"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "hexists", "content": "def hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "infixr:60 \" ~~> \" => hsingle", "content": "infixr:60 \" ~~> \" => hsingle"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "disjoint_single", "content": "lemma disjoint_single (h : state) :\n  p ∉ h →\n  h.Disjoint (Finmap.singleton p v)"}, {"name": "hsingl_inv", "content": "lemma hsingl_inv p v h :\n  (p ~~> v) h →\n  h = Finmap.singleton p v"}, {"name": "union_singleton_eq_erase", "content": "lemma union_singleton_eq_erase (h h' : state) :\n  h.Disjoint (Finmap.singleton p v) →\n  h' = h ∪ Finmap.singleton p v →\n  h = h'.erase p"}, {"name": "hexists_inv", "content": "lemma hexists_inv A (J : A → hProp) h :\n  (hexists J) h → exists x, J x h"}, {"name": "hsingle_intro", "content": "lemma hsingle_intro p v :\n  (p ~~> v) (Finmap.singleton p v)"}, {"name": "insert_eq_union_single", "content": "lemma insert_eq_union_single (h : state) :\n  p ∉ h →\n  h.insert p v = h ∪ (Finmap.singleton p v)"}], "used_local_defs": [{"name": "triple", "content": "abbrev triple (t : trm) (H : hProp) (Q : val → hProp) : Prop :=\n  forall s, H s → eval s t Q"}], "used_local_lemmas": [{"name": "eval_conseq", "content": "lemma eval_conseq s t Q1 Q2 :\n  eval s t Q1 →\n  Q1 ===> Q2 →\n  eval s t Q2"}, {"name": "triple_conseq", "content": "lemma triple_conseq t H' Q' H Q :\n  triple t H' Q' →\n  H ==> H'→\n  Q' ===> Q →\n  triple t H Q"}], "local_ctx": "import Mathlib.Data.Finmap\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Multiset.Nodup\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nopen trm val prim\n\nabbrev triple (t : trm) (H : hProp) (Q : val → hProp) : Prop :=\n  forall s, H s → eval s t Q\n\nnotation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)\n\nsection evalProp\n\nend evalProp", "target_theorem": "lemma triple_ref (v : val) :\n  (forall (p : loc), triple (subst x p t2) (H ∗ (p ~~> v)) (Q ∗ ∃ʰ v, p ~~> v)) →\n  triple (trm_ref x (trm_val v) t2) H Q :=", "ground_truth_proof": ":=\nby\n  move=> htriple h ?\n  apply eval.eval_ref\n  { sby apply (eval.eval_val h v (fun v' h' ↦ v' = v ∧ h' = h)) }\n  move=> > [->->] > ?\n  move: (htriple p)=> /triple_conseq {}htriple\n  have eqn:(triple (subst x p t2) (H ∗ p ~~> v) fun v s ↦ Q v (s.erase p)) := by\n    apply htriple=> //\n    move=> > h /= ![>] ? /hexists_inv [v'] /hsingl_inv ->\n    sby move=> /union_singleton_eq_erase /[apply] <-\n  move=> {htriple}\n  apply eqn\n  exists h, Finmap.singleton p v\n  move=> ⟨//|⟩ ⟨|⟩\n  apply hsingle_intro=> ⟨|⟩\n  apply disjoint_single=>//\n  sby apply insert_eq_union_single=> //", "nesting_depth": 5, "transitive_dep_count": 48, "subset_aristotle": false, "category": "Framework"}
{"id": 411, "thm_name": "triple_alloc", "thm_stmt": "lemma triple_alloc (n : Int) :\n  n ≥ 0 →\n  (∀ (p : loc), triple (subst x p t)\n    (H ∗ ⌜p ≠ null⌝ ∗ hrange (make_list n.natAbs val_uninit) p)\n    (Q ∗ ⌜p ≠ null⌝  ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p) ) →\n  triple (trm_alloc x n t) H Q", "lean_root": "splean", "rel_path": "SPLean/Theories/SepLog.lean", "imports": ["import SPLean.Theories.Lang", "import Mathlib.Data.Multiset.Nodup", "import SPLean.Theories.XSimp", "import Mathlib.Data.Finset.Basic", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "List", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "L", "module": "Archive.Hairer"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "sdone", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "elim", "module": "Ssreflect.Elim"}], "used_repo_defs": [{"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}, {"name": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hexists xs b"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "hsingle", "content": "def hsingle (p : loc) (v : val) : hProp :=\n  fun h => (h = Finmap.singleton p v)"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "make_list", "content": "def make_list {A} (n : Nat) (v : A) : List A :=\n  match n with\n  | 0      => []\n  | n' + 1 => v :: make_list n' v"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "null", "content": "def null : loc := 0"}, {"name": "hpure", "content": "def hpure (P : Prop) : hProp :=\n  hexists (fun (_ : P) => emp)"}, {"name": "hexists", "content": "def hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "conseq", "content": "def conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=\n  match vs with\n  | [] => ∅\n  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:60 \" ~~> \" => hsingle", "content": "infixr:60 \" ~~> \" => hsingle"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}], "lib_lemmas": [{"name": "Finmap.mem_keys", "module": "Mathlib.Data.Finmap"}, {"name": "Finset.ext_iff", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finmap.Disjoint.symm_iff", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.union_comm_of_disjoint", "module": "Mathlib.Data.Finmap"}], "repo_lemmas": [{"name": "hempty_inv", "content": "lemma hempty_inv h :\n  emp h → h = ∅"}, {"name": "hsingl_inv", "content": "lemma hsingl_inv p v h :\n  (p ~~> v) h →\n  h = Finmap.singleton p v"}, {"name": "hstar_intro", "content": "lemma hstar_intro (H1 H2 : hProp) h1 h2 :\n  H1 h1 →\n  H2 h2 →\n  Finmap.Disjoint h1 h2 →\n  (H1 ∗ H2) (h1 ∪ h2)"}, {"name": "disjoint_single_conseq", "content": "lemma disjoint_single_conseq B l l' L (v : B) :\n  (l < l') ∨ (l ≥ l' + L.length) →\n  Finmap.Disjoint (Finmap.singleton l v) (conseq L l')"}, {"name": "conseq_cons", "content": "lemma conseq_cons B (l : Nat) (v : B) (vs : List B) :\n  conseq (v :: vs) l = (Finmap.singleton l v) ∪ (conseq vs (l + 1))"}, {"name": "conseq_nil", "content": "lemma conseq_nil B (l : Nat) :\n  conseq ([] : List B) l = ∅"}, {"name": "hexists_inv", "content": "lemma hexists_inv A (J : A → hProp) h :\n  (hexists J) h → exists x, J x h"}, {"name": "hpure_inv", "content": "lemma hpure_inv P h :\n  ⌜P⌝ h →\n  P ∧ h = ∅"}, {"name": "diff_disjoint_eq", "content": "lemma diff_disjoint_eq (s₁ s₂ s₃ : state) :\n  s₁.Disjoint s₂ →\n  s₂.keys = s₃.keys →\n  (s₁ ∪ s₂) \\ s₃  = s₁"}, {"name": "lookup_diff", "content": "lemma lookup_diff (h₁ h₂ : state) :\n  p ∉ h₂ →\n  (h₁ \\ h₂).lookup p = h₁.lookup p"}, {"name": "lookup_diff_none", "content": "lemma lookup_diff_none (h₁ h₂ : state) :\n  p ∈ h₂ →\n  (h₁ \\ h₂).lookup p = none"}, {"name": "diff_non_mem", "content": "theorem diff_non_mem (h₁ h₂ : state) :\n  p ∈ h₂ → p ∉ h₁ \\ h₂"}], "used_local_defs": [{"name": "triple", "content": "abbrev triple (t : trm) (H : hProp) (Q : val → hProp) : Prop :=\n  forall s, H s → eval s t Q"}, {"name": "hrange", "content": "def hrange (L : List val) (p : loc) : hProp :=\n  match L with\n  | []      => emp\n  | x :: L' => (p ~~> x) ∗ (hrange L' (p + 1))"}], "used_local_lemmas": [{"name": "eval_conseq", "content": "lemma eval_conseq s t Q1 Q2 :\n  eval s t Q1 →\n  Q1 ===> Q2 →\n  eval s t Q2"}, {"name": "triple_conseq", "content": "lemma triple_conseq t H' Q' H Q :\n  triple t H' Q' →\n  H ==> H'→\n  Q' ===> Q →\n  triple t H Q"}, {"name": "hrange_intro", "content": "lemma hrange_intro L p :\n  (hrange L p) (conseq L p)"}, {"name": "int_eq_sub", "content": "lemma int_eq_sub (l m n : ℤ) :\n  l + m = n → l = n - m"}, {"name": "list_inc_natabs", "content": "lemma list_inc_natabs {α : Type} (L : List α) :\n  ((L.length : ℤ) + 1).natAbs = (L.length : ℤ).natAbs + 1"}, {"name": "hrange_eq_conseq", "content": "lemma hrange_eq_conseq (L : List val) (n : ℤ) (p : loc) (s : state) :\n  L.length = n →\n  hrange L p s →\n  s.keys = (conseq (make_list n.natAbs val_uninit) p).keys"}], "local_ctx": "import Mathlib.Data.Finmap\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Multiset.Nodup\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nopen trm val prim\n\nabbrev triple (t : trm) (H : hProp) (Q : val → hProp) : Prop :=\n  forall s, H s → eval s t Q\n\nnotation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)\n\nsection evalProp\n\nend evalProp\n\ndef hrange (L : List val) (p : loc) : hProp :=\n  match L with\n  | []      => emp\n  | x :: L' => (p ~~> x) ∗ (hrange L' (p + 1))", "target_theorem": "lemma triple_alloc (n : Int) :\n  n ≥ 0 →\n  (∀ (p : loc), triple (subst x p t)\n    (H ∗ ⌜p ≠ null⌝ ∗ hrange (make_list n.natAbs val_uninit) p)\n    (Q ∗ ⌜p ≠ null⌝  ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p) ) →\n  triple (trm_alloc x n t) H Q :=", "ground_truth_proof": ":= by\n  move=> ? htriple h ?\n  apply eval.eval_alloc=> // > *\n  move: (htriple p)=> /triple_conseq {}htriple\n  specialize (htriple (H ∗ ⌜p ≠ null⌝ ∗ hrange (make_list n.natAbs val_uninit) p))\n  specialize (htriple (fun v s ↦ Q v (s \\ sb)))\n  have eqn:(triple (subst x p t)\n    (H ∗ ⌜p ≠ null⌝ ∗ hrange (make_list n.natAbs val_uninit) p)\n    fun v s ↦ Q v (s \\ sb)) := by\n    { apply htriple=> // {htriple}\n      move=> > s ![>] ? ![>] /hpure_inv [] _ ->\n      move=> /hexists_inv [L] ![>] /hpure_inv [] ? -> ? _\n      move=> /== -> _ -> ? ->\n      srw diff_disjoint_eq=> //\n      subst sb ; sby apply hrange_eq_conseq }\n  move=> {htriple}\n  apply eqn\n  exists h, sb=> ⟨//|⟩ ⟨|⟩\n  { exists ∅, sb => ⟨//|/==⟩ ⟨|⟩\n    subst sb ; apply hrange_intro\n    sdone }\n  constructor=> //\n  sby srw Finmap.union_comm_of_disjoint Finmap.Disjoint.symm_iff", "nesting_depth": 8, "transitive_dep_count": 72, "subset_aristotle": false, "category": "Framework"}
{"id": 412, "thm_name": "Theories.wp_alloc", "thm_stmt": "lemma wp_alloc x (n : ℤ) t Q :\n  n ≥ 0 →\n  (h∀ p, (hrange (make_list n.natAbs val_uninit) p) -∗\n    wp (subst x p t) (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p)) ==>\n  wp (trm_alloc x n t) Q", "lean_root": "splean", "rel_path": "SPLean/Theories/WP1.lean", "imports": ["import SPLean.Theories.XChange", "import Mathlib.Data.List.Indexes", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Lean", "import SPLean.Theories.WPUtil"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "L", "module": "Archive.Hairer"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "elim", "module": "Ssreflect.Elim"}], "used_repo_defs": [{"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}, {"name": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"h∀\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hforall xs b"}, {"name": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hexists xs b"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "make_list", "content": "def make_list {A} (n : Nat) (v : A) : List A :=\n  match n with\n  | 0      => []\n  | n' + 1 => v :: make_list n' v"}, {"name": "hrange", "content": "def hrange (L : List val) (p : loc) : hProp :=\n  match L with\n  | []      => emp\n  | x :: L' => (p ~~> x) ∗ (hrange L' (p + 1))"}, {"name": "wpgen_alloc", "content": "def wpgen_alloc (x : var) (t1 t2 : trm) : formula :=\n  fun Q ↦ ∃ʰ n : ℤ,\n    ⌜n ≥ 0 ∧ t1 = trm_val n⌝ ∗\n    h∀ p,\n      (hrange (make_list n.natAbs val_uninit) p) -∗\n      protect wp (subst x p t2) (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p)"}, {"name": "wpgen", "content": "def wpgen (t : trm) : formula :=\n  mkstruct (match t with\n  | trm_val v          => wpgen_val v\n  | trm_fun x t1       => wpgen_fun (fun v ↦ wp $ subst x v t1)\n  | trm_fix f x t1     => wpgen_fix\n      (fun vf v => wp $ subst x v $ subst f vf t1)\n  | trm_if t0 t1 t2    => wpgen_if t0 (wp t1) (wp t2)\n  | trm_seq t1 t2      => wpgen_seq (wp t1) (wp t2)\n  | trm_let x t1 t2    => wpgen_let (wp t1) (fun v ↦ wp $ subst x v t2)\n  | trm_app _ _        => wpgen_app t\n  \n  \n  | trm_ref x t1 t2    => wpgen_ref x t1 t2\n  | trm_alloc x t1 t2  => wpgen_alloc x t1 t2\n  | _ => wp t\n  )"}, {"name": "hsingle", "content": "def hsingle (p : loc) (v : val) : hProp :=\n  fun h => (h = Finmap.singleton p v)"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "null", "content": "def null : loc := 0"}, {"name": "HWand", "content": "class HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hWand : α → β → γ"}, {"name": "hpure", "content": "def hpure (P : Prop) : hProp :=\n  hexists (fun (_ : P) => emp)"}, {"name": "hexists", "content": "def hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "conseq", "content": "def conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=\n  match vs with\n  | [] => ∅\n  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "hforall", "content": "def hforall {A} (J : A → hProp) : hProp :=\n  fun h => forall x, J x h"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:60 \" ~~> \" => hsingle", "content": "infixr:60 \" ~~> \" => hsingle"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}, {"name": "infixr:55 \" -∗ \" => HWand.hWand", "content": "infixr:55 \" -∗ \" => HWand.hWand"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "Finmap.Disjoint.symm_iff", "module": "Mathlib.Data.Finmap"}], "repo_lemmas": [{"name": "eval_conseq", "content": "lemma eval_conseq s t Q1 Q2 :\n  eval s t Q1 →\n  Q1 ===> Q2 →\n  eval s t Q2"}, {"name": "hpure_inv", "content": "lemma hpure_inv P h :\n  ⌜P⌝ h →\n  P ∧ h = ∅"}, {"name": "hwand_inv", "content": "lemma hwand_inv h1 h2 H1 H2 :\n  (H1 -∗ H2) h2 →\n  H1 h1 →\n  Finmap.Disjoint h1 h2 →\n  H2 (h1 ∪ h2)"}, {"name": "hrange_eq_conseq", "content": "lemma hrange_eq_conseq (L : List val) (n : ℤ) (p : loc) (s : state) :\n  L.length = n →\n  hrange L p s →\n  s.keys = (conseq (make_list n.natAbs val_uninit) p).keys"}, {"name": "hseg_eq_hrange", "content": "lemma hseg_eq_hrange L p (k : Nat) :\n  hseg L p k = hrange L (p + 1 + k)"}, {"name": "hrange_intro", "content": "lemma hrange_intro L p :\n  (hrange L p) (conseq L p)"}, {"name": "int_eq_sub", "content": "lemma int_eq_sub (l m n : ℤ) :\n  l + m = n → l = n - m"}, {"name": "list_inc_natabs", "content": "lemma list_inc_natabs {α : Type} (L : List α) :\n  ((L.length : ℤ) + 1).natAbs = (L.length : ℤ).natAbs + 1"}, {"name": "nonneg_eq_abs", "content": "lemma nonneg_eq_abs (n : Int) :\n  0 ≤ n → n.natAbs = n"}, {"name": "neg_mul_abs", "content": "lemma neg_mul_abs (n : Int) :\n  n < 0 → -1 * n = n.natAbs"}, {"name": "triple_abs", "content": "lemma triple_abs (i : Int) :\n  triple [lang| val_abs i]\n    emp\n    (fun r ↦ ⌜r = val_int i.natAbs⌝)"}, {"name": "add_Int.natAbs", "content": "lemma add_Int.natAbs i j :\n  0 <= i - j → j + Int.natAbs (i - j) = i"}, {"name": "hseg_focus", "content": "lemma hseg_focus (i j : Int) L p (v : 0 <= i - j ∧ i - j < L.length) :\n  0 <= i - j ∧ i - j < L.length →\n  hseg L p j ==>\n  hcell L[(i - j).natAbs]! p i\n    ∗ (h∀ w, hcell w p i -∗ hseg (L.set (i - j).natAbs w) p j)"}, {"name": "harray_focus", "content": "lemma harray_focus i L p (v : 0 <= i ∧ i < L.length) :\n  0 <= i ∧ i < L.length →\n  harray L p ==>\n  hcell L[Int.natAbs i]! p i\n    ∗ (h∀ w, hcell w p i -∗ harray (L.set (Int.natAbs i) w) p)"}, {"name": "natabs_eq", "content": "lemma natabs_eq (a : Nat) :\n  a = (Int.ofNat a).natAbs"}, {"name": "set_nth_same", "content": "lemma set_nth_same (A : Type) (IA : Inhabited A) (n : Nat) (l : List A) :\n  n < l.length → l.set n l[n]! = l"}, {"name": "harray_focus_read", "content": "lemma harray_focus_read i L p :\n  0 <= i ∧ i < L.length →\n  harray L p ==>\n  hcell L[Int.natAbs i]! p i ∗ (hcell L[Int.natAbs i]! p i -∗ harray L p)"}, {"name": "hforall_inv", "content": "lemma hforall_inv A (J : A → hProp) h :\n  (hforall J) h → forall x, J x h"}, {"name": "hexists_inv", "content": "lemma hexists_inv A (J : A → hProp) h :\n  (hexists J) h → exists x, J x h"}, {"name": "diff_disjoint_eq", "content": "lemma diff_disjoint_eq (s₁ s₂ s₃ : state) :\n  s₁.Disjoint s₂ →\n  s₂.keys = s₃.keys →\n  (s₁ ∪ s₂) \\ s₃  = s₁"}, {"name": "lookup_diff", "content": "lemma lookup_diff (h₁ h₂ : state) :\n  p ∉ h₂ →\n  (h₁ \\ h₂).lookup p = h₁.lookup p"}, {"name": "lookup_diff_none", "content": "lemma lookup_diff_none (h₁ h₂ : state) :\n  p ∈ h₂ →\n  (h₁ \\ h₂).lookup p = none"}, {"name": "diff_non_mem", "content": "theorem diff_non_mem (h₁ h₂ : state) :\n  p ∈ h₂ → p ∉ h₁ \\ h₂"}, {"name": "triple", "content": "abbrev triple (t : trm) (H : hProp) (Q : val → hProp) : Prop :=\n  forall s, H s → eval s t Q"}, {"name": "triple_alloc", "content": "lemma triple_alloc (n : Int) :\n  n ≥ 0 →\n  (∀ (p : loc), triple (subst x p t)\n    (H ∗ ⌜p ≠ null⌝ ∗ hrange (make_list n.natAbs val_uninit) p)\n    (Q ∗ ⌜p ≠ null⌝  ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p) ) →\n  triple (trm_alloc x n t) H Q"}, {"name": "hstar_comm", "content": "lemma hstar_comm H1 H2 :\n  H1 ∗ H2 = H2 ∗ H1"}, {"name": "hstar_assoc", "content": "lemma hstar_assoc H1 H2 H3 :\n  (H1 ∗ H2) ∗ H3 = H1 ∗ (H2 ∗ H3)"}, {"name": "hstar_comm_assoc", "content": "lemma hstar_comm_assoc (H1 H2 H3 : hProp) :\n  H1 ∗ H2 ∗ H3 = H2 ∗ H1 ∗ H3"}, {"name": "hwand_cancel", "content": "lemma hwand_cancel H1 H2 :\n  H1 ∗ (H1 -∗ H2) ==> H2"}, {"name": "himpl_trans", "content": "lemma himpl_trans H2 H1 H3 :\n  (H1 ==> H2) → (H2 ==> H3) → (H1 ==> H3)"}, {"name": "himpl_hstar_hpure_l", "content": "lemma himpl_hstar_hpure_l P H H' :\n  (P → H ==> H') →\n  (⌜P⌝ ∗ H) ==> H'"}, {"name": "hforall_specialize", "content": "lemma hforall_specialize A (x : A) (J : A → hProp) :\n  (hforall J) ==> (J x)"}, {"name": "himpl_frame_r", "content": "lemma himpl_frame_r H1 H2 H2' :\n  H2 ==> H2' →\n  (H1 ∗ H2) ==> (H1 ∗ H2')"}, {"name": "finite_state'", "content": "lemma finite_state' n (s : state) :\n  ∃ p, p ≠ null ∧\n    Finmap.Disjoint s (conseq (make_list n val_uninit) p)"}, {"name": "conseq_nil", "content": "lemma conseq_nil B (l : Nat) :\n  conseq ([] : List B) l = ∅"}, {"name": "conseq_cons", "content": "lemma conseq_cons B (l : Nat) (v : B) (vs : List B) :\n  conseq (v :: vs) l = (Finmap.singleton l v) ∪ (conseq vs (l + 1))"}, {"name": "disjoint_single_conseq", "content": "lemma disjoint_single_conseq B l l' L (v : B) :\n  (l < l') ∨ (l ≥ l' + L.length) →\n  Finmap.Disjoint (Finmap.singleton l v) (conseq L l')"}, {"name": "union_difference_id", "content": "lemma union_difference_id (h₁ h₂ : state) :\n  h₁.Disjoint h₂ →\n  (h₁ ∪ h₂) \\ h₂ = h₁"}, {"name": "hstar_hempty_l", "content": "lemma hstar_hempty_l H :\n  emp ∗ H = H"}, {"name": "hstar_hempty_r", "content": "lemma hstar_hempty_r H :\n  H ∗ emp = H"}, {"name": "himpl_refl", "content": "lemma himpl_refl H : H ==> H"}, {"name": "hstar_intro", "content": "lemma hstar_intro (H1 H2 : hProp) h1 h2 :\n  H1 h1 →\n  H2 h2 →\n  Finmap.Disjoint h1 h2 →\n  (H1 ∗ H2) (h1 ∪ h2)"}, {"name": "hstar_inv", "content": "lemma hstar_inv (H1 H2 : hProp) h:\n  (H1 ∗ H2) h →\n  exists h1 h2, H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2"}, {"name": "diff_disjoint_eq", "content": "lemma diff_disjoint_eq (s₁ s₂ s₃ : state) :\n  s₁.Disjoint s₂ →\n  s₂.keys = s₃.keys →\n  (s₁ ∪ s₂) \\ s₃ = s₁"}, {"name": "qstar_simp", "content": "lemma qstar_simp (Q1 : α -> hProp) :\n  (Q1 ∗ H) x = Q1 x ∗ H"}, {"name": "himpl_hexists_l", "content": "lemma himpl_hexists_l A H (J : A → hProp) :\n  (forall x, J x ==> H) → (hexists J) ==> H"}, {"name": "himpl_hexists_r", "content": "lemma himpl_hexists_r A (x : A) H (J : A → hProp) :\n  (H ==> J x) →\n  H ==> (hexists J)"}, {"name": "triple_conseq", "content": "lemma triple_conseq t H' Q' H Q :\n  triple t H' Q' →\n  H ==> H'→\n  Q' ===> Q →\n  triple t H Q"}, {"name": "triple_frame", "content": "lemma triple_frame t H (Q : val -> hProp) H' :\n  triple t H Q →\n  triple t (H ∗ H') (Q ∗ H')"}, {"name": "triple_hpure", "content": "lemma triple_hpure t P H Q :\n  (P → triple t H Q) →\n  triple t (⌜P⌝ ∗ H) Q"}, {"name": "triple_hexists", "content": "lemma triple_hexists t A (J : A → hProp) Q :\n  (forall x, triple t (J x) Q) →\n  triple t (hexists J) Q"}, {"name": "hwand_equiv", "content": "lemma hwand_equiv H0 H1 H2 :\n  (H0 ==> H1 -∗ H2) ↔ (H1 ∗ H0 ==> H2)"}, {"name": "himpl_hwand_r", "content": "lemma himpl_hwand_r H1 H2 H3 :\n  H2 ∗ H1 ==> H3 →\n  H1 ==> (H2 -∗ H3)"}, {"name": "himpl_hwand_r_inv", "content": "lemma himpl_hwand_r_inv H1 H2 H3 :\n  H1 ==> (H2 -∗ H3) →\n  H2 ∗ H1 ==> H3"}, {"name": "hwand_hempty_l", "content": "lemma hwand_hempty_l H :\n  (emp -∗ H) = H"}, {"name": "himpl_hforall_r", "content": "lemma himpl_hforall_r A (J : A → hProp) H :\n  (forall x, H ==> J x) →\n  H ==> (hforall J)"}, {"name": "himpl_hforall_l", "content": "lemma himpl_hforall_l A (x : A) (J : A → hProp) H :\n  (J x ==> H) →\n  (hforall J) ==> H"}], "used_local_defs": [{"name": "Theories.wp", "content": "def wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q"}], "used_local_lemmas": [{"name": "Theories.mem_conseq", "content": "lemma mem_conseq :\n  x ∈ conseq L p → p ≤ x"}, {"name": "Theories.hrange_of_conseq", "content": "lemma hrange_of_conseq :\n  (hrange L p) (conseq L p)"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Indexes\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WPUtil\n\nopen trm val prim\n\nnamespace Theories\n\ndef wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q", "target_theorem": "lemma wp_alloc x (n : ℤ) t Q :\n  n ≥ 0 →\n  (h∀ p, (hrange (make_list n.natAbs val_uninit) p) -∗\n    wp (subst x p t) (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜L.length = n⌝ ∗ hrange L p)) ==>\n  wp (trm_alloc x n t) Q :=", "ground_truth_proof": ":=\nby\n  move=> ? h /hforall_inv hwp\n  apply eval.eval_alloc=> // > *\n  apply (eval_conseq _ _ (Q ∗ ⌜p ≠ null⌝ ∗ ∃ʰ L, ⌜↑L.length = n⌝ ∗ hrange L p))\n  { move: (hwp p)=> /(hwand_inv sb)\n    srw Finmap.Disjoint.symm_iff=> {}hwp\n    apply hwp=> // ; subst sb\n    apply hrange_of_conseq }\n  move=> > s ![>] ? ![>] /hpure_inv [_ ->] /==\n  move=> /hexists_inv [L] ![>] /hpure_inv [? ->] /== ? _ -> _ -> ? ->\n  srw diff_disjoint_eq=> // ; subst sb\n  sby apply hrange_eq_conseq", "nesting_depth": 5, "transitive_dep_count": 65, "subset_aristotle": false, "category": "Framework"}
{"id": 413, "thm_name": "xfor_lemma", "thm_stmt": "lemma xfor_lemma (z n : ℤ) (x : var) (I : ℤ -> hProp) :\n  z <= n ->\n  (H ==> H' ∗ I z) ->\n  (∀ i, z <= i -> i < n -> I i ==> wp (subst x i F1) (fun _ => I (i + 1))) ->\n  ((fun _ => I n ∗ H') ===> Q) ->\n  H ==> wp (trm_for x z n F1) Q", "lean_root": "splean", "rel_path": "SPLean/Theories/WP1.lean", "imports": ["import SPLean.Theories.XChange", "import Mathlib.Data.List.Indexes", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Lean", "import SPLean.Theories.WPUtil"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Id", "module": "Init.Control.Id"}, {"name": "BitVec", "module": "Init.Prelude"}, {"name": "Hashable", "module": "Init.Prelude"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sdone", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}], "used_repo_defs": [{"name": "syntax \"if \" lang \"then \" lang \"end \" : lang", "content": "syntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "macro \"xsimp\" : tactic =>", "content": "macro \"xsimp\" : tactic =>\n  `(tactic| (\n    xsimp_start\n    repeat xsimp_step\n    try rev_pure\n    try hide_mvars\n    try hsimp\n    rotate_left\n\n  ))\n\n  syntax \"sdo\" num tactic : tactic"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P\n\nsyntax \" <= \" : bop"}, {"name": "macro \"xchange\" l:term : tactic =>", "content": "macro \"xchange\" l:term : tactic =>\n  `(tactic| (xchange_core $l; xsimp))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(term| {| $seq |}) => `(withMainContext do evalTactic $ <- `(tacticSeq| $seq))"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "step", "content": "inductive step : state → trm → state → trm → Prop where\n\n  \n  | step_seq_ctx : forall s1 s2 t1 t1' t2,\n      step s1 t1 s2 t1' →\n      step s1 (trm_seq t1 t2) s2 (trm_seq t1' t2)\n  | step_let_ctx : forall s1 s2 x t1 t1' t2,\n      step s1 t1 s2 t1' →\n      step s1 (trm_let x t1 t2) s2 (trm_let x t1' t2)\n  | step_app_arg_1 : forall s1 s2 t1 t1' t2,\n      step s1 t1 s2 t1' →\n      step s1 (trm_app t1 t2) s2 (trm_app t1' t2)\n  | step_app_arg2 : forall s1 s2 v1 t2 t2',\n      step s1 t2 s2 t2' →\n      step s1 (trm_app v1 t2) s2 (trm_app v1 t2')\n\n  \n  | step_fun : forall s x t1,\n      step s (trm_fun x t1) s (val_fun x t1)\n  | step_fix : forall s f x t1,\n      step s (trm_fix f x t1) s (val_fix f x t1)\n  | step_app_fun : forall s v1 v2 x t1,\n      v1 = val_fun x t1 →\n      v2 = trm_val v2' →\n      step s (trm_app v1 v2) s (subst x v2' t1)\n  | step_app_fix : forall s v1 v2 f x t1,\n      v1 = val_fix f x t1 →\n      v2 = trm_val v2' →\n      step s (trm_app v1 v2) s (subst x v2' (subst f v1 t1))\n  | step_if : forall s b t1 t2,\n      step s (trm_if (val_bool b) t1 t2) s (if b then t1 else t2)\n  | step_seq : forall s t2 v1,\n      step s (trm_seq v1 t2) s t2\n  | step_let : forall s x t2 v1,\n      v1 = trm_val v1' →\n      step s (trm_let x v1 t2) s (subst x v1' t2)\n\n  \n  | step_neg : forall s b,\n      step s (trm_app val_neg (val_bool b)) s (val_bool (¬ b))\n  | step_opp : forall s n,\n      step s (trm_app val_opp (val_int n)) s (val_int (- n))\n  \n  \n  \n\n  \n  | step_eq : forall s v1 v2,\n      step s (trm_app (trm_app val_eq v1) v2) s (val_bool (is_true (v1 = v2)))\n  | step_neq : forall s v1 v2,\n      step s (trm_app (trm_app val_neq v1) v2) s (val_bool (is_true (v1 ≠ v2)))\n  | step_add : forall s n1 n2,\n      step s (trm_app (trm_app val_add (val_int n1)) (val_int n2)) s\n        (val_int (n1 + n2))\n  | step_sub : forall s n1 n2,\n      step s (trm_app (trm_app val_sub (val_int n1)) (val_int n2)) s\n        (val_int (n1 - n2))\n  | step_mul : forall s n1 n2,\n      step s (trm_app (trm_app val_mul (val_int n1)) (val_int n2)) s\n        (val_int (n1 * n2))\n  | step_div : forall s n1 n2,\n      n2 ≠ 0 →\n      step s (trm_app (trm_app val_div (val_int n1)) (val_int n2)) s\n        (n1 / n2)\n  | step_mod : forall s n1 n2,\n      n2 ≠ 0 →\n      step s (trm_app (trm_app val_mod (val_int n1)) (val_int n2)) s\n        (n1 % n2)\n  | step_le : forall s n1 n2,\n      step s (trm_app (trm_app val_le (val_int n1)) (val_int n2)) s\n        (val_bool (n1 <= n2))\n  | step_lt : forall s n1 n2,\n      step s (trm_app (trm_app val_lt (val_int n1)) (val_int n2)) s\n        (val_bool (n1 < n2))\n  | step_ge : forall s n1 n2,\n      step s (trm_app (trm_app val_ge (val_int n1)) (val_int n2)) s\n        (val_bool (n1 >= n2))\n  | step_gt : forall s n1 n2,\n      step s (trm_app (trm_app val_gt (val_int n1)) (val_int n2)) s\n        (val_bool (n1 > n2))\n  | step_ptr_add : forall s p1 p2 n,\n      (p2:ℤ) = (p1:loc) + n →\n      step s (trm_app (trm_app val_ptr_add (val_loc p1)) (val_int n)) s\n        (val_loc (Int.toNat p2))\n\n  \n  | step_ref : forall s v p,\n      ¬ p ∈ s →\n      v = trm_val v' →\n      step s (trm_app val_ref v) (Finmap.insert p v' s) (val_loc p)\n  | step_get : forall s p,\n      p ∈ s →\n      step s (trm_app val_get (val_loc p)) s (read_state p s)\n  | step_set : forall s p v,\n      p ∈ s ->\n      v = trm_val v' →\n      step s (trm_app (trm_app val_set (val_loc p)) v)\n        (Finmap.insert p v' s) val_unit\n  | step_free : forall s p,\n      p ∈ s →\n      step s (trm_app val_free (val_loc p)) (Finmap.erase p s) val_unit\n\nsyntax ident : lang"}, {"name": "HWand", "content": "class HWand (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hWand : α → β → γ"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "tohProp", "content": "abbrev tohProp (h : heap -> Prop) : hProp := h"}, {"name": "ofhProp", "content": "abbrev ofhProp (h : val -> hProp) : val -> heap -> Prop := h"}, {"name": "hforall", "content": "def hforall {A} (J : A → hProp) : hProp :=\n  fun h => forall x, J x h"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "hpure", "content": "def hpure (P : Prop) : hProp :=\n  hexists (fun (_ : P) => emp)"}, {"name": "hexists", "content": "def hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}, {"name": "infixr:55 \" -∗ \" => HWand.hWand", "content": "infixr:55 \" -∗ \" => HWand.hWand"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "Int.le_induction_down", "module": "Mathlib.Data.Int.Init"}, {"name": "if_pos", "module": "Init.Core"}], "repo_lemmas": [{"name": "eval_conseq", "content": "lemma eval_conseq s t Q1 Q2 :\n  eval s t Q1 →\n  Q1 ===> Q2 →\n  eval s t Q2"}, {"name": "qstarE", "content": "lemma qstarE α (Q1 : α → hProp)  (H : hProp):\n  Q1 ∗ H = fun x => Q1 x ∗ H"}, {"name": "eval_frame", "content": "lemma eval_frame (h1 h2 : state) t (Q : val -> hProp) :\n  eval h1 t (ofhProp Q) →\n  Finmap.Disjoint h1 h2 →\n  eval (h1 ∪ h2) t (Q ∗ (tohProp (fun h ↦ h = h2)))"}, {"name": "qwand_cancel", "content": "lemma qwand_cancel A (Q1 Q2 : A → hProp) :\n  Q1 ∗ (Q1 -∗ Q2) ===> Q2"}, {"name": "qwand_equiv", "content": "lemma qwand_equiv H A (Q1 Q2 : A → hProp) :\n  H ==> (Q1 -∗ Q2) ↔ (Q1 ∗ H) ===> Q2"}, {"name": "hstar_hforall", "content": "lemma hstar_hforall A (J : A → hProp) H :\n  (hforall J) ∗ H ==> hforall (J ∗ H)"}, {"name": "hstar_comm", "content": "lemma hstar_comm H1 H2 :\n  H1 ∗ H2 = H2 ∗ H1"}, {"name": "hprop_op_comm", "content": "lemma hprop_op_comm (op : hProp → hProp → hProp) :\n  (forall H1 H2, op H1 H2 ==> op H2 H1) →\n  (forall H1 H2, op H1 H2 = op H2 H1)"}, {"name": "himpl_antisym", "content": "lemma himpl_antisym H1 H2:\n  (H1 ==> H2) → (H2 ==> H1) → (H1 = H2)"}, {"name": "hstar_inv", "content": "lemma hstar_inv (H1 H2 : hProp) h:\n  (H1 ∗ H2) h →\n  exists h1 h2, H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2"}, {"name": "himpl_hstar_trans_l", "content": "lemma himpl_hstar_trans_l H1 H2 H3 H4 :\n  H1 ==> H2 →\n  H2 ∗ H3 ==> H4 →\n  H1 ∗ H3 ==> H4"}, {"name": "himpl_hforall_l", "content": "lemma himpl_hforall_l A (x : A) (J : A → hProp) H :\n  (J x ==> H) →\n  (hforall J) ==> H"}, {"name": "himpl_hforall_r", "content": "lemma himpl_hforall_r A (J : A → hProp) H :\n  (forall x, H ==> J x) →\n  H ==> (hforall J)"}, {"name": "hwand_equiv", "content": "lemma hwand_equiv H0 H1 H2 :\n  (H0 ==> H1 -∗ H2) ↔ (H1 ∗ H0 ==> H2)"}, {"name": "hstar_assoc", "content": "lemma hstar_assoc H1 H2 H3 :\n  (H1 ∗ H2) ∗ H3 = H1 ∗ (H2 ∗ H3)"}, {"name": "himpl_hstar_hpure_l", "content": "lemma himpl_hstar_hpure_l P H H' :\n  (P → H ==> H') →\n  (⌜P⌝ ∗ H) ==> H'"}, {"name": "hstar_hpure_l", "content": "lemma hstar_hpure_l P H h :\n  (⌜P⌝ ∗ H) h = (P ∧ H h)"}, {"name": "hstar_hempty_l", "content": "lemma hstar_hempty_l H :\n  emp ∗ H = H"}, {"name": "hempty_inv", "content": "lemma hempty_inv h :\n  emp h → h = ∅"}, {"name": "hstar_hexists", "content": "lemma hstar_hexists A (J : A → hProp) H :\n  (hexists J) ∗ H = hexists (fun x => (J x) ∗ H)"}, {"name": "himpl_hempty_hpure", "content": "lemma himpl_hempty_hpure P :\n  P → emp ==> ⌜P⌝"}, {"name": "himpl_hexists_r", "content": "lemma himpl_hexists_r A (x : A) H (J : A → hProp) :\n  (H ==> J x) →\n  H ==> (hexists J)"}, {"name": "hwandE", "content": "lemma hwandE :\n  H1 -∗ H2 = hexists (fun H0 => H0 ∗ hpure ((H1 ∗ H0) ==> H2))"}, {"name": "himpl_frame_r", "content": "lemma himpl_frame_r H1 H2 H2' :\n  H2 ==> H2' →\n  (H1 ∗ H2) ==> (H1 ∗ H2')"}, {"name": "hstar_hempty_r", "content": "lemma hstar_hempty_r H :\n  H ∗ emp = H"}, {"name": "himpl_hexists_l", "content": "lemma himpl_hexists_l A H (J : A → hProp) :\n  (forall x, J x ==> H) → (hexists J) ==> H"}, {"name": "qwandE", "content": "lemma qwandE α (Q1 Q2 : α → hProp) :\n  Q1 -∗ Q2 = hforall (fun x => (Q1 x) -∗ (Q2 x))"}, {"name": "hwand_cancel", "content": "lemma hwand_cancel H1 H2 :\n  H1 ∗ (H1 -∗ H2) ==> H2"}, {"name": "himpl_hwand_r_inv", "content": "lemma himpl_hwand_r_inv H1 H2 H3 :\n  H1 ==> (H2 -∗ H3) →\n  H2 ∗ H1 ==> H3"}, {"name": "himpl_trans", "content": "lemma himpl_trans H2 H1 H3 :\n  (H1 ==> H2) → (H2 ==> H3) → (H1 ==> H3)"}, {"name": "himpl_trans_r", "content": "lemma himpl_trans_r H2 H1 H3:\n  H2 ==> H3 → H1 ==> H2 → H1 ==> H3"}], "used_local_defs": [{"name": "Theories.wp", "content": "def wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q"}], "used_local_lemmas": [{"name": "Theories.wp_conseq", "content": "lemma wp_conseq t Q1 Q2 :\n  Q1 ===> Q2 →\n  wp t Q1 ==> wp t Q2"}, {"name": "Theories.wp_frame", "content": "lemma wp_frame t H Q :\n  (wp t Q) ∗ H ==> wp t (Q ∗ H)"}, {"name": "Theories.wp_ramified", "content": "lemma wp_ramified t (Q1 Q2 : val -> hProp) :\n  (wp t Q1) ∗ (Q1 -∗ Q2) ==> (wp t Q2)"}, {"name": "Theories.wp_conseq_frame", "content": "lemma wp_conseq_frame t H (Q1 Q2 : val -> hProp) :\n  Q1 ∗ H ===> Q2 →\n  (wp t Q1) ∗ H ==> (wp t Q2)"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Indexes\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WPUtil\n\nopen trm val prim\n\nnamespace Theories\n\ndef wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q\n\nsection tactics\n\nopen Lean Elab Tactic\n\nsection xapp\n\nend xapp\n\nend tactics\n\nopen AList\n\nsection funs_fixs_eval_like\n\nvariable (xs : List var) (vs : List val) (t : trm) (v0 : trm)\n  (heqt : t = trm_apps v0 ts)\n  (hconv : trms_to_vals ts = vs)\n  (hform : var_funs xs vs.length)   -- NOTE: can be relaxed to `vs.length ≤ xs.length`\n\nvariable (f : var) (hf : f ∉ xs)\n\nend funs_fixs_eval_like\n\nend Theories\n\nopen Theories\n\nopen Lean.Elab.Tactic in", "target_theorem": "lemma xfor_lemma (z n : ℤ) (x : var) (I : ℤ -> hProp) :\n  z <= n ->\n  (H ==> H' ∗ I z) ->\n  (∀ i, z <= i -> i < n -> I i ==> wp (subst x i F1) (fun _ => I (i + 1))) ->\n  ((fun _ => I n ∗ H') ===> Q) ->\n  H ==> wp (trm_for x z n F1) Q :=", "ground_truth_proof": ":= by\n  move=> ? hini hstep hfin\n  xchange hini\n  apply himpl_trans_r; apply wp_conseq_frame=> //\n  xsimp\n  move: z hfin {hini}=> z; apply Int.le_induction_down\n  { move=> ?? ??\n    constructor=> /==;constructor; aesop }\n  move=> j ? ih step hfin\n  move=> ??;\n  constructor=> /==; srw if_pos; rotate_left; omega\n  constructor\n  { apply step <;> try omega\n    sdone }\n  move=> _ > ?; apply ih=> // *\n  apply step <;> omega", "nesting_depth": 9, "transitive_dep_count": 90, "subset_aristotle": false, "category": "Framework"}
{"id": 414, "thm_name": "Theories.isubst_insert", "thm_stmt": "lemma isubst_insert (al : ctx) x v t :\n  isubst (al.insert x v) t = subst x v (isubst (al.erase x) t)", "lean_root": "splean", "rel_path": "SPLean/Theories/WP1.lean", "imports": ["import SPLean.Theories.XChange", "import Mathlib.Data.List.Indexes", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Lean", "import SPLean.Theories.WPUtil"], "used_lib_defs": [{"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "AList.erase", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup", "module": "Mathlib.Data.List.AList"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "AList.entries", "module": "Mathlib.Data.List.AList"}, {"name": "AList.insert", "module": "Mathlib.Data.List.AList"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kinsert", "module": "Mathlib.Data.List.Sigma"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "AList.keys", "module": "Mathlib.Data.List.AList"}, {"name": "List.keys", "module": "Mathlib.Data.List.Sigma"}], "used_repo_defs": [{"name": "syntax \"fun\" ident+ \" => \" lang : lang", "content": "syntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"if \" lang \"then \" lang \"end \" : lang\n\nsyntax \" := \" : bop\n\nsyntax \"let\" ident \" := \" lang \" in\" ppDedent(ppLine lang) : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "subst", "content": "def subst (y : var) (v' : val) (t : trm) : trm :=\n  \n  let if_y_eq x t1 t2 := if x = y then t1 else t2\n  match t with\n  | trm_val v          => trm_val v\n  | trm_var x          => if_y_eq x (trm_val v') t\n  | trm_fun x t1       => trm_fun x (if_y_eq x t1 (subst y v' t1))\n  | trm_fix f x t1     => trm_fix f x (if_y_eq f t1 (if_y_eq x t1 (subst y v' t1)))\n  | trm_app t1 t2      => trm_app (subst y v' t1) (subst y v' t2)\n  | trm_seq t1 t2      => trm_seq (subst y v' t1) (subst y v' t2)\n  | trm_let x t1 t2    => trm_let x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_if t0 t1 t2    => trm_if (subst y v' t0) (subst y v' t1) (subst y v' t2)\n  | trm_for x t1 t2 t3 => trm_for x (subst y v' t1) (subst y v' t2) (if_y_eq x t3 (subst y v' t3))\n  | trm_while t1 t2    => trm_while (subst y v' t1) (subst y v' t2)\n  | trm_ref x t1 t2    => trm_ref x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))\n  | trm_alloc x t1 t2  => trm_alloc x (subst y v' t1) (if_y_eq x t2 (subst y v' t2))"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "List.kerase_cons_ne", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kerase_kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "AList.perm_erase", "module": "Mathlib.Data.List.AList"}, {"name": "AList.perm_lookup", "module": "Mathlib.Data.List.AList"}, {"name": "AList.mem_keys", "module": "Mathlib.Data.List.AList"}, {"name": "List.eraseP_of_forall_not", "module": "Init.Data.List.Erase"}, {"name": "AList.erase_erase", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup_erase_ne", "module": "Mathlib.Data.List.AList"}, {"name": "AList.lookup_insert_ne", "module": "Mathlib.Data.List.AList"}, {"name": "congrArg", "module": "Init.Prelude"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Theories.ctx", "content": "abbrev ctx := AList (fun _ : var ↦ val)"}, {"name": "Theories.isubst", "content": "def isubst (E : ctx) (t : trm) : trm :=\n  match t with\n  | trm_val v =>\n      v\n  | trm_var x =>\n      match lookup x E with\n      | none   => t\n      | some v => v\n  | trm_fun x t1 =>\n      trm_fun x (isubst (erase x E) t1)\n  | trm_fix f x t1 =>\n      trm_fix f x (isubst (erase x (erase f E)) t1)\n  | trm_if t0 t1 t2 =>\n      trm_if (isubst E t0) (isubst E t1) (isubst E t2)\n  | trm_seq t1 t2 =>\n      trm_seq (isubst E t1) (isubst E t2)\n  | trm_let x t1 t2 =>\n      trm_let x (isubst E t1) (isubst (erase x E) t2)\n  | trm_ref x t1 t2 =>\n      trm_ref x (isubst E t1) (isubst (erase x E) t2)\n  | trm_alloc x t1 t2 =>\n      trm_alloc x (isubst E t1) (isubst (erase x E) t2)\n  | trm_app t1 t2 =>\n      trm_app (isubst E t1) (isubst E t2)\n  | trm_for x n1 n2 t =>\n      trm_for x (isubst E n1) (isubst E n2) (isubst (erase x E) t)\n  | trm_while c t =>\n      trm_while (isubst E c) (isubst E t)"}], "used_local_lemmas": [{"name": "Theories.AList.erase_insert_cancel", "content": "lemma AList.erase_insert_cancel {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) (l : AList β) :\n    (AList.erase a (AList.insert a b l)).entries.Perm (AList.erase a l).entries"}, {"name": "Theories.AList.erase_insert_swap", "content": "lemma AList.erase_insert_swap {α : Type u} {β : α → Type v} [DecidableEq α] (a a' : α) (b : β a) (l : AList β) :\n  a ≠ a' → (AList.erase a' (AList.insert a b l)).entries.Perm (AList.insert a b (AList.erase a' l)).entries"}, {"name": "Theories.AList.erase_noop", "content": "lemma AList.erase_noop {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : AList β) :\n    a ∉ l → (AList.erase a l).entries.Perm l.entries"}, {"name": "Theories.AList.erase_twice", "content": "lemma AList.erase_twice {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : AList β) :\n    (AList.erase a (AList.erase a l)).entries.Perm (AList.erase a l).entries"}, {"name": "Theories.isubst_perm", "content": "lemma isubst_perm {al al'} t (hp : al.entries.Perm al'.entries) :\n  isubst al t = isubst al' t"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Indexes\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WPUtil\n\nopen trm val prim\n\nnamespace Theories\n\nsection tactics\n\nopen Lean Elab Tactic\n\nsection xapp\n\nend xapp\n\nend tactics\n\nopen AList\n\nabbrev ctx := AList (fun _ : var ↦ val)\n\ndef isubst (E : ctx) (t : trm) : trm :=\n  match t with\n  | trm_val v =>\n      v\n  | trm_var x =>\n      match lookup x E with\n      | none   => t\n      | some v => v\n  | trm_fun x t1 =>\n      trm_fun x (isubst (erase x E) t1)\n  | trm_fix f x t1 =>\n      trm_fix f x (isubst (erase x (erase f E)) t1)\n  | trm_if t0 t1 t2 =>\n      trm_if (isubst E t0) (isubst E t1) (isubst E t2)\n  | trm_seq t1 t2 =>\n      trm_seq (isubst E t1) (isubst E t2)\n  | trm_let x t1 t2 =>\n      trm_let x (isubst E t1) (isubst (erase x E) t2)\n  | trm_ref x t1 t2 =>\n      trm_ref x (isubst E t1) (isubst (erase x E) t2)\n  | trm_alloc x t1 t2 =>\n      trm_alloc x (isubst E t1) (isubst (erase x E) t2)\n  | trm_app t1 t2 =>\n      trm_app (isubst E t1) (isubst E t2)\n  | trm_for x n1 n2 t =>\n      trm_for x (isubst E n1) (isubst E n2) (isubst (erase x E) t)\n  | trm_while c t =>\n      trm_while (isubst E c) (isubst E t)", "target_theorem": "lemma isubst_insert (al : ctx) x v t :\n  isubst (al.insert x v) t = subst x v (isubst (al.erase x) t) :=", "ground_truth_proof": ":= by\n  move: al\n  induction t using (subst.induct x v)=> >\n  all_goals (simp [isubst, subst]=> //)\n  all_goals (split_ands=> //)\n  all_goals ((try split_ifs=> //) <;> (try subst_eqs))\n  all_goals (try srw (fun t => isubst_perm t (AList.erase_twice x al)))\n  all_goals (try srw (fun v t => isubst_perm t (AList.erase_insert_cancel x v al)))\n  all_goals (try solve\n    | srw (fun x' hneq v t => isubst_perm t (AList.erase_insert_swap x x' v al hneq)) <;>\n      (try solve | aesop) ; -- `aesop` does autorewrite here\n      (try simp [*])\n      apply congrArg ; srw AList.erase_erase)\n  all_goals (try apply isubst_perm)\n  { rename_i x' ; by_cases h : x' = x\n    { subst x' ; simp [subst] }\n    { srw AList.lookup_insert_ne // AList.lookup_erase_ne //\n      scase: (lookup x' al)=> //=\n      simp [subst, h] } }\n  { apply AList.perm_erase\n    trans ; apply AList.erase_insert_cancel ; symm ; apply AList.erase_twice }\n  { srw [2]AList.erase_erase [1]AList.erase_erase\n    apply AList.perm_erase\n    trans ; apply AList.erase_insert_cancel ; symm ; apply AList.erase_twice }\n  { rename_i ih ha hb\n    srw [3]AList.erase_erase [2]AList.erase_erase -ih\n    apply isubst_perm\n    trans\n    on_goal 2=> apply AList.erase_insert_swap ; aesop\n    apply AList.perm_erase\n    apply AList.erase_insert_swap ; aesop }", "nesting_depth": 5, "transitive_dep_count": 42, "subset_aristotle": false, "category": "Framework"}
{"id": 415, "thm_name": "Perm.kmerge", "thm_stmt": "theorem Perm.kmerge {l₁ l₂ l₃ l₄ : List (Sigma (fun _ : loc => val))} (nd₁ : l₁.NodupKeys) /- nd₁ is necessary -/ (nd₃ : l₃.NodupKeys)\n    (p₁₂ : l₁.Perm l₂) (p₃₄ : l₃.Perm l₄) : (kmerge l₁ l₃).Perm $ kmerge l₂ l₄", "lean_root": "splean", "rel_path": "SPLean/Common/Heap.lean", "imports": ["import Mathlib.Algebra.BigOperators.Group.Finset", "import Mathlib.Algebra.BigOperators.Intervals", "import Mathlib.Algebra.Group.Basic", "import Ssreflect.Lang", "import Mathlib.Data.Finmap", "import Mathlib.Order.Interval.Finset.Basic", "import Mathlib.Data.Int.Interval", "import Lean", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.NodupKeys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.Perm", "module": "Init.Data.List.Basic"}, {"name": "List.keys", "module": "Mathlib.Data.List.Sigma"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "List.kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "DecidableEq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.NodupKeys.kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kerase_cons_eq", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kerase_cons_ne", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.mem_keys_of_mem", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.NodupKeys.nodup", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.Perm.map", "module": "Init.Data.List.Perm"}, {"name": "List.Perm.mem_iff", "module": "Init.Data.List.Perm"}, {"name": "List.perm_dlookup", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.perm_ext_iff_of_nodup", "module": "Batteries.Data.List.Perm"}, {"name": "List.perm_nodupKeys", "module": "Mathlib.Data.List.Sigma"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "loc", "content": "abbrev loc := Nat"}, {"name": "kmerge1", "content": "private def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v"}, {"name": "kmerge", "content": "@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)"}], "used_local_lemmas": [{"name": "List.kerase_noterased", "content": "lemma List.kerase_noterased {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (Sigma β))\n  (a a' : α) (hneq : a ≠ a') (b : β a) : ⟨a, b⟩ ∈ l ↔ ⟨a, b⟩ ∈ List.kerase a' l"}, {"name": "kmerge_mem2", "content": "lemma kmerge_mem2 (l₁ l₂ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) -- necessary\n    (a : Sigma (fun _ : loc => val)) : a ∈ (kmerge l₁ l₂) ↔\n      if a.1 ∈ l₁.keys\n      then (if a.1 ∈ l₂.keys then Option.merge (· + ·) (l₁.dlookup a.1) (l₂.dlookup a.1) = .some a.2 else a ∈ l₁)\n      else a ∈ l₂"}, {"name": "kmerge_NodupKeys", "content": "lemma kmerge_NodupKeys (l₁ l₂ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) : (kmerge l₁ l₂).NodupKeys"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Algebra.Group.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Data.Int.Interval\n\nimport Mathlib.Order.Interval.Finset.Basic\n\nimport Batteries.Data.List.Perm\n\nimport Ssreflect.Lang\n\nopen Classical\n\nabbrev loc := Nat\n\nsection Option.merge\n\nvariable {α : Type u} (f : α → α → α)\n\nend Option.merge\n\nopen PartialCommMonoid (valid)\n\nsection\n\nvariable {val : Type} [PartialCommMonoid val] -- [Inhabited val]\n\nlocal notation \"heap\" => Heap.heap val\n\nprivate def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v\n\n@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)", "target_theorem": "theorem Perm.kmerge {l₁ l₂ l₃ l₄ : List (Sigma (fun _ : loc => val))} (nd₁ : l₁.NodupKeys) /- nd₁ is necessary -/ (nd₃ : l₃.NodupKeys)\n    (p₁₂ : l₁.Perm l₂) (p₃₄ : l₃.Perm l₄) : (kmerge l₁ l₃).Perm $ kmerge l₂ l₄ :=", "ground_truth_proof": ":= by\n  have nd₂ := nd₁\n  rw [List.perm_nodupKeys p₁₂] at nd₂\n  have nd₄ := nd₃\n  rw [List.perm_nodupKeys p₃₄] at nd₄\n  rw [List.perm_ext_iff_of_nodup] <;> try (apply List.NodupKeys.nodup ; apply kmerge_NodupKeys=> //)\n  move=> [] l v\n  srw !kmerge_mem2 // (List.perm_dlookup _ nd₁ nd₂ p₁₂) // (List.perm_dlookup _ nd₃ nd₄ p₃₄)\n  dsimp [List.keys]\n  srw (List.Perm.mem_iff (List.Perm.map Sigma.fst p₁₂)) (List.Perm.mem_iff (List.Perm.map Sigma.fst p₃₄)) (List.Perm.mem_iff p₁₂) (List.Perm.mem_iff p₃₄)", "nesting_depth": 3, "transitive_dep_count": 30, "subset_aristotle": false, "category": "Framework"}
{"id": 416, "thm_name": "validInter_hop_distr_l", "thm_stmt": "lemma validInter_hop_distr_l (h₁ h₂ h₃ : heap) :\n  (h₁ +ʰ h₂) ⊥ʰ h₃ -> (h₁ ⊥ʰ h₃ ∧ h₂ ⊥ʰ h₃)", "lean_root": "splean", "rel_path": "SPLean/Common/Heap.lean", "imports": ["import Mathlib.Algebra.BigOperators.Group.Finset", "import Mathlib.Algebra.BigOperators.Intervals", "import Mathlib.Algebra.Group.Basic", "import SPLean/Theories/HProp.lean", "import Ssreflect.Lang", "import Mathlib.Data.Finmap", "import Mathlib.Order.Interval.Finset.Basic", "import Mathlib.Data.Int.Interval", "import Lean", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.liftOn₂", "module": "Mathlib.Data.Finmap"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap.lookup", "module": "Mathlib.Data.Finmap"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "HAdd", "module": "Init.Prelude"}, {"name": "HAdd.hAdd", "module": "Init.Prelude"}, {"name": "instHAdd", "module": "Init.Prelude"}, {"name": "sdone", "module": "Ssreflect.Done"}], "used_repo_defs": [{"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "scoped instance : PartialCommMonoid val where", "content": "scoped instance : PartialCommMonoid val where\n  add := add\n  add_assoc := by admit /- proof elided -/"}, {"name": "add", "content": "@[simp]\nabbrev add : val -> val -> val\n  | .val_int i, .val_int j => val.val_int (i + j)\n  | _, _ => val.val_unit"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "valid", "content": "@[simp]\nabbrev valid : val -> Prop\n  | .val_int _ => True\n  | _ => False"}, {"name": "PartialCommMonoid", "content": "class PartialCommMonoid (α : Type) extends AddCommSemigroup α where\n  valid : α -> Prop\n  valid_add : ∀ x, valid (x + y) -> valid x\n  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)"}, {"name": "scoped instance inst : PartialCommMonoidWRT val add valid wh", "content": "scoped instance inst : PartialCommMonoidWRT val add valid where\n  validE := by admit /- proof elided -/"}, {"name": "AddCommMonoidWRT", "content": "class AddCommMonoidWRT (α : Type) (add' : semiOutParam $ α -> α -> α) extends AddCommMonoid α where\n  addE : (· + ·) = add'"}, {"name": "scoped instance : PartialCommMonoid val where", "content": "scoped instance : PartialCommMonoid val where\n  add := add\n  add_assoc := by admit /- proof elided -/"}, {"name": "add", "content": "@[simp]\nabbrev add : val -> val -> val\n  | .val_bool i, .val_bool j => val.val_bool (i || j)\n  | _, _ => val.val_unit"}, {"name": "valid", "content": "@[simp]\nabbrev valid : val -> Prop\n  | .val_bool _ => True\n  | _ => False"}, {"name": "scoped instance : PartialCommMonoid val where", "content": "scoped instance : PartialCommMonoid val where\n  add := add\n  add_assoc := by admit /- proof elided -/"}, {"name": "add", "content": "@[simp]\nabbrev add : val -> val -> val\n  | .val_real i, .val_real j => val.val_real (i + j)\n  | _, _ => val.val_unit"}, {"name": "valid", "content": "@[simp]\nabbrev valid : val -> Prop\n  | .val_real _ => True\n  | _ => False"}], "lib_lemmas": [{"name": "Finmap.mem_iff", "module": "Mathlib.Data.Finmap"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "loc", "content": "abbrev loc := Nat"}, {"name": "var", "content": "abbrev var := String"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "PartialCommMonoid", "content": "class PartialCommMonoid (α : Type) extends AddCommSemigroup α where\n  valid : α -> Prop\n  valid_add : ∀ x, valid (x + y) -> valid x\n  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)"}, {"name": "kmerge1", "content": "private def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v"}, {"name": "kmerge", "content": "@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)"}, {"name": "AList.merge", "content": "noncomputable def AList.merge (h₁ h₂ : AList (fun _ : loc => val)) :  AList (fun _ : loc => val) :=\n  ⟨kmerge h₁.entries h₂.entries, by admit /- proof elided -/\n    ⟩"}, {"name": "Heap.add", "content": "noncomputable def Heap.add (h₁ h₂ : heap) : heap :=\n  Finmap.liftOn₂ h₁ h₂ (fun h₁ h₂ => (h₁.merge h₂).toFinmap) (by admit /- proof elided -/\n    )"}, {"name": "validInter", "content": "def validInter (h₁ h₂ : heap) : Prop :=\n  ∀ l ∈ h₁, l ∈ h₂ -> ((h₁ +ʰ h₂).lookup l).any (valid (α := val))"}], "used_local_lemmas": [], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Algebra.Group.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Data.Int.Interval\n\nimport Mathlib.Order.Interval.Finset.Basic\n\nimport Batteries.Data.List.Perm\n\nimport Ssreflect.Lang\n\nopen Classical\n\nabbrev loc := Nat\n\nabbrev var := String\n\nsection Option.merge\n\nvariable {α : Type u} (f : α → α → α)\n\nend Option.merge\n\nabbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)\n\nclass PartialCommMonoid (α : Type) extends AddCommSemigroup α where\n  valid : α -> Prop\n  valid_add : ∀ x, valid (x + y) -> valid x\n  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)\n\nopen PartialCommMonoid (valid)\n\nsection\n\nvariable {val : Type} [PartialCommMonoid val] -- [Inhabited val]\n\nlocal notation \"heap\" => Heap.heap val\n\nprivate def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v\n\n@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)\n\nnoncomputable def AList.merge (h₁ h₂ : AList (fun _ : loc => val)) :  AList (fun _ : loc => val) :=\n  ⟨kmerge h₁.entries h₂.entries, by admit /- proof elided -/\n    ⟩\n\nnoncomputable def Heap.add (h₁ h₂ : heap) : heap :=\n  Finmap.liftOn₂ h₁ h₂ (fun h₁ h₂ => (h₁.merge h₂).toFinmap) (by admit /- proof elided -/\n    )\n\ninfixr:55 \" +ʰ \" => Heap.add\n\ndef validInter (h₁ h₂ : heap) : Prop :=\n  ∀ l ∈ h₁, l ∈ h₂ -> ((h₁ +ʰ h₂).lookup l).any (valid (α := val))\n\ninfixr:55 \" ⊥ʰ \" => validInter", "target_theorem": "lemma validInter_hop_distr_l (h₁ h₂ h₃ : heap) :\n  (h₁ +ʰ h₂) ⊥ʰ h₃ -> (h₁ ⊥ʰ h₃ ∧ h₂ ⊥ʰ h₃) :=", "ground_truth_proof": ":= by\n  simp [validInter]\n  move=> h ⟨|⟩ l /[tac (specialize h l)]-- | [] h1 h2 l [] /[tac (specialize h1 l; specialize h2 l)] ⟩\n  all_goals (move=> /[dup] hin1 /[swap] /[dup] hin2)\n  all_goals (srw [1]Finmap.mem_iff=> []v3 hv3 ; srw Finmap.mem_iff=> []v hv)\n  all_goals (srw hv hv3 at h ⊢)\n  all_goals (dsimp [Option.merge]; try solve\n    | aesop)\n  { move: h; scase: (Finmap.lookup l h₂)=> > //==\n    all_goals (simp [Option.merge]=> //)\n    srw add_assoc (add_comm _ v3) -add_assoc\n    have hq := fun y => PartialCommMonoid.valid_add (v + v3) (y := y)\n    aesop }\n  { move: h; scase: (Finmap.lookup l h₁)=> > //==\n    all_goals (simp [Option.merge]=> //)\n    srw add_assoc [1]add_comm\n    have hq := fun y => PartialCommMonoid.valid_add (v + v3) (y := y)\n    aesop }", "nesting_depth": 6, "transitive_dep_count": 31, "subset_aristotle": false, "category": "Framework"}
{"id": 417, "thm_name": "xwhile_inv_basic_lemma", "thm_stmt": "lemma xwhile_inv_basic_lemma (I : Bool -> α -> hProp) R\n  -- (F1 F2 : formula)\n  :\n  WellFounded R ->\n  -- structural F1 ->\n  -- structural F2 ->\n  (H ==> H' ∗ ∃ʰ b a, I b a) ->\n  (∀ b X, I b X ==> wp F1 (fun bv => I b X ∗ ⌜bv = b⌝)) ->\n  (∀ X, I true X ==> wp F2 (fun _ => ∃ʰ b X', ⌜R X' X⌝ ∗ I b X')) ->\n  H ==> wp (trm_while F1 F2) (fun _ => H' ∗ ∃ʰ a, I false a)", "lean_root": "splean", "rel_path": "SPLean/Theories/WP1.lean", "imports": ["import SPLean.Theories.XChange", "import Mathlib.Data.List.Indexes", "import SPLean.Theories.XSimp", "import SPLean.Theories.SepLog", "import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Lean", "import SPLean.Theories.WPUtil"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "ite", "module": "Init.Prelude"}, {"name": "Computation", "module": "Mathlib.Data.Seq.Computation"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Max", "module": "Init.Prelude"}, {"name": "Max.max", "module": "Init.Prelude"}, {"name": "WellFounded", "module": "Init.WF"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sapply", "module": "Ssreflect.ApplyIn"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "t", "module": "Ssreflect.IntroPats"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "Finmap.Disjoint", "module": "Mathlib.Data.Finmap"}], "used_repo_defs": [{"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty\n\nsyntax \"fun\" ident+ \" => \" lang : lang\n\nsyntax \"⟨\" term \"⟩\" : lang\n\nsyntax \"⟨\" term \":\" term \"⟩\" : lang"}, {"name": "macro \"xsimp\" : tactic =>", "content": "macro \"xsimp\" : tactic =>\n  `(tactic| (\n    xsimp_start\n    repeat xsimp_step\n    try rev_pure\n    try hide_mvars\n    try hsimp\n    rotate_left\n\n  ))"}, {"name": "macro \"xif\" : tactic => do", "content": "macro \"xif\" : tactic => do\n  `(tactic|\n  (xseq_xlet_if_needed; xstruct_if_needed; apply xif_lemma))"}, {"name": "macro \"xif\" : tactic => `(tactic| (xwp; xif))", "content": "macro \"xif\" : tactic => `(tactic| (xwp; xif))\n\nsyntax \" := \" : bop"}, {"name": "macro \"xstruct\" : tactic => do", "content": "macro \"xstruct\" : tactic => do\n  `(tactic| apply xstruct_lemma)"}, {"name": "macro \"xval\" : tactic => do", "content": "macro \"xval\" : tactic => do\n  `(tactic| (xstruct_if_needed; apply xval_lemma))"}, {"name": "macro \"xval\" : tactic => `(tactic| (xwp; xval))", "content": "macro \"xval\" : tactic => `(tactic| (xwp; xval))"}, {"name": "macro \"xapp\" : tactic =>", "content": "macro \"xapp\" : tactic =>\n  `(tactic|\n    (xapp_nosubst;\n     try xapp_try_subst\n     first\n       | done\n       | all_goals try srw wp_equiv\n         all_goals try subst_vars))"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}, {"name": "macro \"xwp\" : tactic =>", "content": "macro \"xwp\" : tactic =>\n  `(tactic|\n    (intros\n     first | apply xwp_lemma_fix; rfl\n           | apply xwp_lemma_fun; rfl))"}, {"name": "macro \"xwp\" : tactic =>", "content": "macro \"xwp\" : tactic =>\n  `(tactic|\n    (intros\n     try srw trm_apps1\n     srw ?trm_apps2\n     first\n           \n           | (apply xwp_lemma_funs; rfl; rfl; rfl; sdone)=> //\n           | apply wp_of_wpgen\n     all_goals try simp [wpgen, subst, isubst, subst, trm_apps, AList.lookup, List.dlookup]))"}, {"name": "macro \"xseq\" : tactic => do", "content": "macro \"xseq\" : tactic => do\n  `(tactic| (xstruct_if_needed; apply xseq_lemma))"}, {"name": "macro \"xchange\" l:term : tactic =>", "content": "macro \"xchange\" l:term : tactic =>\n  `(tactic| (xchange_core $l; xsimp))"}, {"name": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lea", "content": "macro \"∃ʰ\" xs:Lean.explicitBinders \", \" b:term : term => Lean.expandExplicitBinders ``hexists xs b"}, {"name": "macro \"xpull\" : tactic =>", "content": "macro \"xpull\" : tactic =>\n  `(tactic| (\n    xpull_start\n    repeat' xsimp_step\n    try rev_pure\n    hsimp\n  ))"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules", "content": "macro_rules\n  | `({ $P }[$t:lang]{$v, $Q}) => `(triple [lang| $t] $P (fun $v => $Q))\n  | `({ $P }[$t:lang]{$Q})     => `(triple [lang| $t] $P (fun _ => $Q))\n  | `(WP[$t:lang]{$v, $Q})     => `(wp [lang| $t] (fun $v => $Q))\n  | `(WP[$t:lang]{$Q})         => `(wp [lang| $t] (fun _ => $Q))"}, {"name": "macro_rules", "content": "macro_rules\n  | `(tactic| xstep $(t)? ) => `(tactic| (xwp; xapp $(t)?))"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "eval", "content": "inductive eval : state → trm → (val → state → Prop) -> Prop where\n  | eval_val : forall s v Q,\n      Q v s ->\n      eval s (trm_val v) Q\n  | eval_fun : forall s x t1 Q,\n      Q (val_fun x t1) s ->\n      eval s (trm_fun x t1) Q\n  | eval_fix : forall s f x t1 Q,\n      Q (val_fix f x t1) s ->\n      eval s (trm_fix f x t1) Q\n  | eval_app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      eval s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (trm_app v1 t2) Q) ->\n      eval s1 (trm_app t1 t2) Q\n  | eval_app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      eval s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> eval s2 (trm_app v1 v2) Q) ->\n      eval s1 (trm_app v1 t2) Q\n  | eval_app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      eval s1 (subst x v2 t1) Q ->\n      eval s1 (trm_app v1 v2) Q\n  | eval_app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      eval s (subst x v2 (subst f v1 t1)) Q ->\n      eval s (trm_app v1 v2) Q\n  | eval_seq : forall Q1 s t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 t2 Q) ->\n      eval s (trm_seq t1 t2) Q\n  | eval_let : forall Q1 s x t1 t2 Q,\n      eval s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> eval s2 (subst x v1 t2) Q) ->\n      eval s (trm_let x t1 t2) Q\n  | eval_if : forall s (b : Bool) t1 t2 Q,\n      eval s (if b then t1 else t2) Q ->\n      eval s (trm_if (val_bool b) t1 t2) Q\n  | eval_unop : forall op s v1 P Q,\n      evalunop op v1 P ->\n      purepostin s P Q ->\n      eval s (trm_app op v1) Q\n  | eval_binop : forall op s (v1 v2 : val) P Q,\n      evalbinop op v1 v2 P ->\n      purepostin s P Q ->\n      eval s (trm_app (trm_app op v1) v2) Q\n  | eval_ref : forall s x t1 t2 (Q Q₁ : val → state → Prop),\n    eval s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     eval (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n    eval s (trm_ref x t1 t2) Q\n  | eval_get : forall s p Q,\n      p ∈ s ->\n      Q (read_state p s) s ->\n      eval s (trm_app val_get (val_loc p)) Q\n  | eval_set : forall s p v Q,\n      v = trm_val v' ->\n      p ∈ s ->\n      Q val_unit (Finmap.insert p v' s) ->\n      eval s (trm_app (trm_app val_set (val_loc p)) v) Q\n  | eval_alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    eval s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → eval s' (trm_alloc x v' t2) Q) →\n    eval s (trm_alloc x t1 t2) Q\n  | eval_alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      eval (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    eval sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  \n  \n  | eval_for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    eval s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    eval s (trm_for x n₁ n₂ t₁) Q\n  | eval_while (t₁ t₂ : trm) (Q : val -> state -> Prop) :\n    eval s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> eval s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    eval s (trm_while t₁ t₂) Q"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "hpure", "content": "def hpure (P : Prop) : hProp :=\n  hexists (fun (_ : P) => emp)"}, {"name": "hexists", "content": "def hexists {A} (J : A → hProp) : hProp :=\n  fun h => exists x, J x h"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "himpl", "content": "abbrev himpl (H1 H2 : hProp) : Prop :=\n  forall h, H1 h -> H2 h"}, {"name": "qimpl", "content": "def qimpl {A} (Q1 Q2 : A → hProp) : Prop :=\n  forall (v:A), Q1 v ==> Q2 v"}, {"name": "purepostin", "content": "def purepostin (s : state) (P : val → Prop) (Q : val → state → Prop) : Prop :=\n  \n  forall v, P v → Q v s"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "tohProp", "content": "abbrev tohProp (h : heap -> Prop) : hProp := h"}, {"name": "ofhProp", "content": "abbrev ofhProp (h : val -> hProp) : val -> heap -> Prop := h"}, {"name": "infixr:51 \" ==> \" => himpl", "content": "infixr:51 \" ==> \" => himpl"}, {"name": "infixr:51 \" ===> \" => qimpl", "content": "infixr:51 \" ===> \" => qimpl"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}, {"name": "notation:max \"⌜\" P \"⌝\" => hpure P", "content": "notation:max \"⌜\" P \"⌝\" => hpure P"}, {"name": "fun", "content": "notation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "eval_conseq", "content": "lemma eval_conseq s t Q1 Q2 :\n  eval s t Q1 →\n  Q1 ===> Q2 →\n  eval s t Q2"}, {"name": "qstarE", "content": "lemma qstarE α (Q1 : α → hProp)  (H : hProp):\n  Q1 ∗ H = fun x => Q1 x ∗ H"}, {"name": "eval_frame", "content": "lemma eval_frame (h1 h2 : state) t (Q : val -> hProp) :\n  eval h1 t (ofhProp Q) →\n  Finmap.Disjoint h1 h2 →\n  eval (h1 ∪ h2) t (Q ∗ (tohProp (fun h ↦ h = h2)))"}, {"name": "hstar_comm", "content": "lemma hstar_comm H1 H2 :\n  H1 ∗ H2 = H2 ∗ H1"}, {"name": "hprop_op_comm", "content": "lemma hprop_op_comm (op : hProp → hProp → hProp) :\n  (forall H1 H2, op H1 H2 ==> op H2 H1) →\n  (forall H1 H2, op H1 H2 = op H2 H1)"}, {"name": "himpl_antisym", "content": "lemma himpl_antisym H1 H2:\n  (H1 ==> H2) → (H2 ==> H1) → (H1 = H2)"}, {"name": "hstar_inv", "content": "lemma hstar_inv (H1 H2 : hProp) h:\n  (H1 ∗ H2) h →\n  exists h1 h2, H1 h1 ∧ H2 h2 ∧ Finmap.Disjoint h1 h2 ∧ h = h1 ∪ h2"}, {"name": "himpl_trans", "content": "lemma himpl_trans H2 H1 H3 :\n  (H1 ==> H2) → (H2 ==> H3) → (H1 ==> H3)"}], "used_local_defs": [{"name": "Theories.wp", "content": "def wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q"}], "used_local_lemmas": [{"name": "Theories.wp_conseq", "content": "lemma wp_conseq t Q1 Q2 :\n  Q1 ===> Q2 →\n  wp t Q1 ==> wp t Q2"}, {"name": "Theories.wp_frame", "content": "lemma wp_frame t H Q :\n  (wp t Q) ∗ H ==> wp t (Q ∗ H)"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Data.List.Indexes\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nimport SPLean.Theories.XChange\n\nimport SPLean.Theories.SepLog\n\nimport SPLean.Theories.WPUtil\n\nopen trm val prim\n\nnamespace Theories\n\ndef wp (t : trm) (Q : val → hProp) : hProp :=\n  fun s ↦ eval s t Q\n\nsection tactics\n\nopen Lean Elab Tactic\n\nsection xapp\n\nend xapp\n\nend tactics\n\nopen AList\n\nsection funs_fixs_eval_like\n\nvariable (xs : List var) (vs : List val) (t : trm) (v0 : trm)\n  (heqt : t = trm_apps v0 ts)\n  (hconv : trms_to_vals ts = vs)\n  (hform : var_funs xs vs.length)   -- NOTE: can be relaxed to `vs.length ≤ xs.length`\n\nvariable (f : var) (hf : f ∉ xs)\n\nend funs_fixs_eval_like\n\nend Theories\n\nopen Theories\n\nopen Lean.Elab.Tactic in", "target_theorem": "lemma xwhile_inv_basic_lemma (I : Bool -> α -> hProp) R\n  -- (F1 F2 : formula)\n  :\n  WellFounded R ->\n  -- structural F1 ->\n  -- structural F2 ->\n  (H ==> H' ∗ ∃ʰ b a, I b a) ->\n  (∀ b X, I b X ==> wp F1 (fun bv => I b X ∗ ⌜bv = b⌝)) ->\n  (∀ X, I true X ==> wp F2 (fun _ => ∃ʰ b X', ⌜R X' X⌝ ∗ I b X')) ->\n  H ==> wp (trm_while F1 F2) (fun _ => H' ∗ ∃ʰ a, I false a) :=", "ground_truth_proof": ":= by\n  move=> wf hini hf1 hf2\n  xchange hini=> b sR\n  move: b\n  apply WellFounded.induction wf sR=> X ih []\n  -- apply eval.eval_while\n  -- unfold wpgen_while ; unfold_let ; xstruct ; xsimp=> [] sR hstep; rename_i wfR\n  -- frame H' out, using `structural`?\n  { xchange hf1\n    apply himpl_trans;  rotate_left\n    { srw hstar_comm; apply wp_frame }\n    xsimp\n    apply himpl_trans; rotate_left\n    { move=> ? H\n      apply eval.eval_while; apply H\n      move=> > // }\n    apply wp_conseq=> ? /=; xpull\n    xwp; xif=> // _; xwp; xval; xsimp }\n  -- move: sR\n  -- apply himpl_trans;  rotate_left\n  -- { srw hstar_comm; apply wp_frame }\n  -- xsimp\n  apply himpl_trans; rotate_left\n  { move=> ? H\n    apply eval.eval_while; apply H\n    move=> > // }\n  xchange hf1\n  apply himpl_trans; apply wp_frame\n\n  apply wp_conseq=> ? /==; srw qstarE /=; xpull\n  xwp; xif=> // _\n  xwp; xseq; xapp hf2=> // ?? /ih;\n  srw [2]hstar_comm //; sapply", "nesting_depth": 5, "transitive_dep_count": 52, "subset_aristotle": false, "category": "Framework"}
{"id": 418, "thm_name": "kmerge_assoc_perm", "thm_stmt": "lemma kmerge_assoc_perm (l₁ l₂ l₃ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) (nd₃ : l₃.NodupKeys) :\n  (kmerge (kmerge l₁ l₂) l₃).Perm $ (kmerge l₁ (kmerge l₂ l₃))", "lean_root": "splean", "rel_path": "SPLean/Common/Heap.lean", "imports": ["import Mathlib.Algebra.BigOperators.Group.Finset", "import Mathlib.Algebra.BigOperators.Intervals", "import Mathlib.Algebra.Group.Basic", "import Ssreflect.Lang", "import Mathlib.Data.Finmap", "import Mathlib.Order.Interval.Finset.Basic", "import Mathlib.Data.Int.Interval", "import Lean", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "List.NodupKeys", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "DecidableEq", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.mem_keys_kerase_of_ne", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kerase_cons_eq", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.kerase_cons_ne", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.NodupKeys.kerase", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.mem_keys_of_mem", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.dlookup_eq_none", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.mem_dlookup", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.of_mem_dlookup", "module": "Mathlib.Data.List.Sigma"}, {"name": "List.lookup_ext", "module": "Mathlib.Data.List.Sigma"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "loc", "content": "abbrev loc := Nat"}, {"name": "kmerge1", "content": "private def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v"}, {"name": "kmerge", "content": "@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)"}], "used_local_lemmas": [{"name": "List.kerase_noterased", "content": "lemma List.kerase_noterased {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (Sigma β))\n  (a a' : α) (hneq : a ≠ a') (b : β a) : ⟨a, b⟩ ∈ l ↔ ⟨a, b⟩ ∈ List.kerase a' l"}, {"name": "Option.merge_assoc", "content": "lemma Option.merge_assoc (h : Associative f) (a b c : Option α) :\n  Option.merge f (Option.merge f a b) c = Option.merge f a (Option.merge f b c)"}, {"name": "kmerge_mem", "content": "@[simp]\nlemma kmerge_mem (l₁ : List (Sigma (fun _ : loc => val))) : l ∈ (kmerge l₁ l₂).keys ↔ l ∈ l₁.keys ∨ l ∈ l₂.keys"}, {"name": "kmerge_mem2", "content": "lemma kmerge_mem2 (l₁ l₂ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) -- necessary\n    (a : Sigma (fun _ : loc => val)) : a ∈ (kmerge l₁ l₂) ↔\n      if a.1 ∈ l₁.keys\n      then (if a.1 ∈ l₂.keys then Option.merge (· + ·) (l₁.dlookup a.1) (l₂.dlookup a.1) = .some a.2 else a ∈ l₁)\n      else a ∈ l₂"}, {"name": "kmerge_dlookup", "content": "lemma kmerge_dlookup (l₁ l₂ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys)\n    (a : loc) : (kmerge l₁ l₂).dlookup a = Option.merge (· + ·) (l₁.dlookup a) (l₂.dlookup a)"}, {"name": "kmerge_NodupKeys", "content": "lemma kmerge_NodupKeys (l₁ l₂ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) : (kmerge l₁ l₂).NodupKeys"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Algebra.Group.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Data.Int.Interval\n\nimport Mathlib.Order.Interval.Finset.Basic\n\nimport Batteries.Data.List.Perm\n\nimport Ssreflect.Lang\n\nopen Classical\n\nabbrev loc := Nat\n\nsection Option.merge\n\nvariable {α : Type u} (f : α → α → α)\n\nend Option.merge\n\nopen PartialCommMonoid (valid)\n\nsection\n\nvariable {val : Type} [PartialCommMonoid val] -- [Inhabited val]\n\nlocal notation \"heap\" => Heap.heap val\n\nprivate def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v\n\n@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)", "target_theorem": "lemma kmerge_assoc_perm (l₁ l₂ l₃ : List (Sigma (fun _ : loc => val))) (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys) (nd₃ : l₃.NodupKeys) :\n  (kmerge (kmerge l₁ l₂) l₃).Perm $ (kmerge l₁ (kmerge l₂ l₃)) :=", "ground_truth_proof": ":= by\n  apply List.lookup_ext <;> try (repeat'(apply kmerge_NodupKeys=> //))\n  move=> l v\n  (srw !kmerge_dlookup=> //) <;> try (repeat'(apply kmerge_NodupKeys=> //))\n  rw [Option.merge_assoc]=> // ; apply add_assoc", "nesting_depth": 4, "transitive_dep_count": 31, "subset_aristotle": false, "category": "Framework"}
{"id": 419, "thm_name": "validInter_assoc_r", "thm_stmt": "lemma validInter_assoc_r (h₁ h₂ h₃ : heap) :\n  h₂ ⊥ʰ h₃ -> h₁ ⊥ʰ (h₂ +ʰ h₃) -> (h₁ +ʰ h₂) ⊥ʰ h₃", "lean_root": "splean", "rel_path": "SPLean/Common/Heap.lean", "imports": ["import Mathlib.Algebra.BigOperators.Group.Finset", "import Mathlib.Algebra.BigOperators.Intervals", "import Mathlib.Algebra.Group.Basic", "import SPLean/Theories/HProp.lean", "import Ssreflect.Lang", "import Mathlib.Data.Finmap", "import Mathlib.Order.Interval.Finset.Basic", "import Mathlib.Data.Int.Interval", "import Lean", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.liftOn₂", "module": "Mathlib.Data.Finmap"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap.lookup", "module": "Mathlib.Data.Finmap"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "HAdd", "module": "Init.Prelude"}, {"name": "HAdd.hAdd", "module": "Init.Prelude"}, {"name": "instHAdd", "module": "Init.Prelude"}, {"name": "sdone", "module": "Ssreflect.Done"}], "used_repo_defs": [{"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "scoped instance : PartialCommMonoid val where", "content": "scoped instance : PartialCommMonoid val where\n  add := add\n  add_assoc := by admit /- proof elided -/"}, {"name": "add", "content": "@[simp]\nabbrev add : val -> val -> val\n  | .val_int i, .val_int j => val.val_int (i + j)\n  | _, _ => val.val_unit"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "valid", "content": "@[simp]\nabbrev valid : val -> Prop\n  | .val_int _ => True\n  | _ => False"}, {"name": "PartialCommMonoid", "content": "class PartialCommMonoid (α : Type) extends AddCommSemigroup α where\n  valid : α -> Prop\n  valid_add : ∀ x, valid (x + y) -> valid x\n  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)"}, {"name": "scoped instance inst : PartialCommMonoidWRT val add valid wh", "content": "scoped instance inst : PartialCommMonoidWRT val add valid where\n  validE := by admit /- proof elided -/"}, {"name": "AddCommMonoidWRT", "content": "class AddCommMonoidWRT (α : Type) (add' : semiOutParam $ α -> α -> α) extends AddCommMonoid α where\n  addE : (· + ·) = add'"}, {"name": "scoped instance : PartialCommMonoid val where", "content": "scoped instance : PartialCommMonoid val where\n  add := add\n  add_assoc := by admit /- proof elided -/"}, {"name": "add", "content": "@[simp]\nabbrev add : val -> val -> val\n  | .val_bool i, .val_bool j => val.val_bool (i || j)\n  | _, _ => val.val_unit"}, {"name": "valid", "content": "@[simp]\nabbrev valid : val -> Prop\n  | .val_bool _ => True\n  | _ => False"}, {"name": "scoped instance : PartialCommMonoid val where", "content": "scoped instance : PartialCommMonoid val where\n  add := add\n  add_assoc := by admit /- proof elided -/"}, {"name": "add", "content": "@[simp]\nabbrev add : val -> val -> val\n  | .val_real i, .val_real j => val.val_real (i + j)\n  | _, _ => val.val_unit"}, {"name": "valid", "content": "@[simp]\nabbrev valid : val -> Prop\n  | .val_real _ => True\n  | _ => False"}], "lib_lemmas": [{"name": "Finmap.mem_of_lookup_eq_some", "module": "Mathlib.Data.Finmap"}, {"name": "Or.intro_right", "module": "Init.Prelude"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "loc", "content": "abbrev loc := Nat"}, {"name": "var", "content": "abbrev var := String"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "PartialCommMonoid", "content": "class PartialCommMonoid (α : Type) extends AddCommSemigroup α where\n  valid : α -> Prop\n  valid_add : ∀ x, valid (x + y) -> valid x\n  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)"}, {"name": "kmerge1", "content": "private def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v"}, {"name": "kmerge", "content": "@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)"}, {"name": "AList.merge", "content": "noncomputable def AList.merge (h₁ h₂ : AList (fun _ : loc => val)) :  AList (fun _ : loc => val) :=\n  ⟨kmerge h₁.entries h₂.entries, by admit /- proof elided -/\n    ⟩"}, {"name": "Heap.add", "content": "noncomputable def Heap.add (h₁ h₂ : heap) : heap :=\n  Finmap.liftOn₂ h₁ h₂ (fun h₁ h₂ => (h₁.merge h₂).toFinmap) (by admit /- proof elided -/\n    )"}, {"name": "validInter", "content": "def validInter (h₁ h₂ : heap) : Prop :=\n  ∀ l ∈ h₁, l ∈ h₂ -> ((h₁ +ʰ h₂).lookup l).any (valid (α := val))"}], "used_local_lemmas": [{"name": "Option.merge_none_l", "content": "lemma Option.merge_none_l (a : Option α) : Option.merge f none a = a"}, {"name": "Option.merge_assoc", "content": "lemma Option.merge_assoc (h : Associative f) (a b c : Option α) :\n  Option.merge f (Option.merge f a b) c = Option.merge f a (Option.merge f b c)"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Algebra.Group.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Data.Int.Interval\n\nimport Mathlib.Order.Interval.Finset.Basic\n\nimport Batteries.Data.List.Perm\n\nimport Ssreflect.Lang\n\nopen Classical\n\nabbrev loc := Nat\n\nabbrev var := String\n\nsection Option.merge\n\nvariable {α : Type u} (f : α → α → α)\n\nend Option.merge\n\nabbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)\n\nclass PartialCommMonoid (α : Type) extends AddCommSemigroup α where\n  valid : α -> Prop\n  valid_add : ∀ x, valid (x + y) -> valid x\n  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)\n\nopen PartialCommMonoid (valid)\n\nsection\n\nvariable {val : Type} [PartialCommMonoid val] -- [Inhabited val]\n\nlocal notation \"heap\" => Heap.heap val\n\nprivate def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v\n\n@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)\n\nnoncomputable def AList.merge (h₁ h₂ : AList (fun _ : loc => val)) :  AList (fun _ : loc => val) :=\n  ⟨kmerge h₁.entries h₂.entries, by admit /- proof elided -/\n    ⟩\n\nnoncomputable def Heap.add (h₁ h₂ : heap) : heap :=\n  Finmap.liftOn₂ h₁ h₂ (fun h₁ h₂ => (h₁.merge h₂).toFinmap) (by admit /- proof elided -/\n    )\n\ninfixr:55 \" +ʰ \" => Heap.add\n\ndef validInter (h₁ h₂ : heap) : Prop :=\n  ∀ l ∈ h₁, l ∈ h₂ -> ((h₁ +ʰ h₂).lookup l).any (valid (α := val))\n\ninfixr:55 \" ⊥ʰ \" => validInter", "target_theorem": "lemma validInter_assoc_r (h₁ h₂ h₃ : heap) :\n  h₂ ⊥ʰ h₃ -> h₁ ⊥ʰ (h₂ +ʰ h₃) -> (h₁ +ʰ h₂) ⊥ʰ h₃ :=", "ground_truth_proof": ":= by\n  simp [validInter]\n  move=> h1' h2' l /[swap] hin3 /[tac (have h1 := (fun H => h1' _ H hin3) ; have h2 := (fun H => h2' _ H (Or.intro_right _ hin3)) ; clear h1' h2')] [ hin1 | hin2 ]\n  { rw [Option.merge_assoc, h2]=> //\n    apply add_assoc }\n  { rcases h : Finmap.lookup l h₁\n    { rw [Option.merge_none_l] ; aesop }\n    { srw h at h2 ; rw [Option.merge_assoc, h2] ; apply Finmap.mem_of_lookup_eq_some at h=> //\n      apply add_assoc } }", "nesting_depth": 6, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Framework"}
{"id": 420, "thm_name": "hrange_eq_conseq", "thm_stmt": "lemma hrange_eq_conseq (L : List val) (n : ℤ) (p : loc) (s : state) :\n  L.length = n →\n  hrange L p s →\n  s.keys = (conseq (make_list n.natAbs val_uninit) p).keys", "lean_root": "splean", "rel_path": "SPLean/Theories/SepLog.lean", "imports": ["import Mathlib.Data.Finmap", "import SPLean.Common.State", "import SPLean.Theories.HProp", "import SPLean.Common.Util", "import Mathlib.Data.Multiset.Nodup", "import SPLean.Theories.XSimp", "import Mathlib.Data.Finset.Basic"], "used_lib_defs": [{"name": "String", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.singleton", "module": "Mathlib.Data.Finmap"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "elim", "module": "Ssreflect.Elim"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "sby", "module": "Ssreflect.Done"}, {"name": "srw", "module": "Ssreflect.Rewrite"}], "used_repo_defs": [{"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty\n\nsyntax \"fun\" ident+ \" => \" lang : lang"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| ()])                       => `(trm_val (val_unit))\n  | `([lang| $n:num])                   => `(trm_val (val_int $n))\n  | `([lang| $t1 $t2])                  => `(trm_app [lang| $t1] [lang| $t2])\n  | `([lang| if $t1 then $t2 else $t3]) => `(trm_if [lang| $t1] [lang| $t2] [lang| $t3])\n  | `([lang| if $t1 then $t2 end])      => `(trm_if [lang| $t1] [lang| $t2] (trm_val val_unit))\n  | `([lang| let $x := $t1:lang in $t2:lang])     =>\n    `(trm_let $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| ref $x := $t1:lang in $t2:lang])     =>\n    `(trm_ref $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| alloc $t1:lang as $x in $t2:lang])   =>\n    `(trm_alloc $(%x) [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ; $t2])                => `(trm_seq [lang| $t1] [lang| $t2])\n  | `([lang| fun_ $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(trm_funs [ $xs,* ] [lang| $t])\n  | `([lang| fun $xs* => $t])             => do\n    let xs <- xs.mapM fun x => `(term| $(%x))\n    `(val_funs [ $xs,* ] [lang| $t])\n  | `([lang| fix_ $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(trm_fixs $(%f) [ $xs,* ] [lang| $t])\n  | `([lang| fix $f $xs* => $t])    => do\n      let xs <- xs.mapM fun x => `(term| $(%x))\n      `(val_fixs $(%f) [ $xs,* ] [lang| $t])\n  \n  | `([lang| free $t])                  => `(trm_val (val_prim val_free) [lang| $t])\n  | `([lang| not $t])                   => `(trm_val (val_prim val_not) [lang| $t])\n  \n  | `([lang| !$t])                      => `(trm_val val_get [lang| $t])\n  | `([lang| $t1 := $t2])               => `(trm_val val_set [lang| $t1] [lang| $t2])\n  | `([lang| $t1 + $t2])                => `(trm_val val_add [lang| $t1] [lang| $t2])\n  | `([lang| $t1 * $t2])                => `(trm_val val_mul [lang| $t1] [lang| $t2])\n  | `([lang| $t1 - $t2])                => `(trm_val val_sub [lang| $t1] [lang| $t2])\n  | `([lang| $t1 / $t2])                => `(trm_val val_div [lang| $t1] [lang| $t2])\n  | `([lang| $t1 < $t2])                => `(trm_val val_lt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 > $t2])                => `(trm_val val_gt [lang| $t1] [lang| $t2])\n  | `([lang| $t1 <= $t2])               => `(trm_val val_le [lang| $t1] [lang| $t2])\n  | `([lang| $t1 >= $t2])               => `(trm_val val_ge [lang| $t1] [lang| $t2])\n  | `([lang| -$t])                      => `(trm_val val_opp [lang| $t])\n  | `([lang| $t1 = $t2])                => `(trm_val val_eq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 != $t2])               => `(trm_val val_neq [lang| $t1] [lang| $t2])\n  | `([lang| $t1 mod $t2])              => `(trm_val val_mod [lang| $t1] [lang| $t2])\n  | `([lang| $t1 ++ $t2])               => `(trm_val val_ptr_add [lang| $t1] [lang| $t2])\n  | `([lang| ($t)]) => `([lang| $t])\n  | `([lang| ⟨$t : $tp⟩]) => `(trm_val (($t : $tp)))\n  | `([lang| for $x in [$n1 : $n2] { $t } ]) =>\n    `(trm_for $(%x) [lang| $n1] [lang| $n2] [lang| $t])\n  | `([lang| while $c:lang { $t:lang } ]) =>\n     `(trm_while [lang| $c] [lang| $t] )"}, {"name": "macro_rules", "content": "macro_rules\n  | `([lang| len $p])                => `(trm_val val_array_length [lang| $p])\n  | `([lang| $arr[$i] ])              => `(trm_val val_array_get [lang| $arr] [lang| $i])\n  \n  | `([lang| $arr[$i] := $v])        => `(trm_app val_array_set [lang| $arr] [lang| $i] [lang| $v])\n  | `([lang| mkarr $n:lang $v:lang]) => `(trm_val val_array_make [lang| $n] [lang| $v])"}, {"name": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)", "content": "macro_rules | `($x ∗ $y)   => `(binop% HStar.hStar $x $y)"}, {"name": "HStar", "content": "class HStar (α : Type u) (β : Type v) (γ : outParam (Type w)) where\n   \n  hStar : α → β → γ"}, {"name": "hsingle", "content": "def hsingle (p : loc) (v : val) : hProp :=\n  fun h => (h = Finmap.singleton p v)"}, {"name": "hProp", "content": "def hProp := heap -> Prop"}, {"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "var", "content": "abbrev var := String"}, {"name": "loc", "content": "abbrev loc := Nat"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "hempty", "content": "def hempty : hProp :=\n  fun h => (h = ∅)"}, {"name": "state", "content": "abbrev state := Finmap (λ _ : loc ↦ val)"}, {"name": "make_list", "content": "def make_list {A} (n : Nat) (v : A) : List A :=\n  match n with\n  | 0      => []\n  | n' + 1 => v :: make_list n' v"}, {"name": "conseq", "content": "def conseq {B : Type} (vs : List B) (l : Nat) : Finmap (fun _ : Nat ↦ B) :=\n  match vs with\n  | [] => ∅\n  | v :: vs' => (Finmap.singleton l v) ∪ (conseq vs' (l + 1))"}, {"name": "notation:max \"emp\" => hempty", "content": "notation:max \"emp\" => hempty"}, {"name": "infixr:60 \" ~~> \" => hsingle", "content": "infixr:60 \" ~~> \" => hsingle"}, {"name": "infixr:55 \" ∗ \" => HStar.hStar", "content": "infixr:55 \" ∗ \" => HStar.hStar"}], "lib_lemmas": [{"name": "Finmap.mem_keys", "module": "Mathlib.Data.Finmap"}, {"name": "Finset.ext_iff", "module": "Mathlib.Data.Finset.Defs"}], "repo_lemmas": [{"name": "hempty_inv", "content": "lemma hempty_inv h :\n  emp h → h = ∅"}, {"name": "hsingl_inv", "content": "lemma hsingl_inv p v h :\n  (p ~~> v) h →\n  h = Finmap.singleton p v"}], "used_local_defs": [{"name": "hrange", "content": "def hrange (L : List val) (p : loc) : hProp :=\n  match L with\n  | []      => emp\n  | x :: L' => (p ~~> x) ∗ (hrange L' (p + 1))"}], "used_local_lemmas": [{"name": "int_eq_sub", "content": "lemma int_eq_sub (l m n : ℤ) :\n  l + m = n → l = n - m"}, {"name": "list_inc_natabs", "content": "lemma list_inc_natabs {α : Type} (L : List α) :\n  ((L.length : ℤ) + 1).natAbs = (L.length : ℤ).natAbs + 1"}], "local_ctx": "import Mathlib.Data.Finmap\n\nimport Mathlib.Data.Finset.Basic\n\nimport Mathlib.Data.Multiset.Nodup\n\nimport SPLean.Common.State\n\nimport SPLean.Common.Util\n\nimport SPLean.Theories.HProp\n\nimport SPLean.Theories.XSimp\n\nopen trm val prim\n\nnotation \"funloc\" p \"↦\" H =>\n  fun (r : val) ↦ hexists (fun p ↦ ⌜r = val_loc p⌝ ∗ H)\n\nsection evalProp\n\nend evalProp\n\ndef hrange (L : List val) (p : loc) : hProp :=\n  match L with\n  | []      => emp\n  | x :: L' => (p ~~> x) ∗ (hrange L' (p + 1))", "target_theorem": "lemma hrange_eq_conseq (L : List val) (n : ℤ) (p : loc) (s : state) :\n  L.length = n →\n  hrange L p s →\n  s.keys = (conseq (make_list n.natAbs val_uninit) p).keys :=", "ground_truth_proof": ":= by\n  elim: L n p s=> > ; unfold hrange\n  { sby move=> /= <- /= /hempty_inv -> }\n  move=> ih > /== /[dup] /int_eq_sub /[dup] hn /ih {}ih  <-\n  srw -hn at ih\n  move: ih=> /= ih {hn}\n  unfold hrange=> ![>] /hsingl_inv ? /ih {}ih ? ->\n  unfold conseq make_list\n  srw list_inc_natabs=> /== >\n  move: ih\n  sby srw ?Finset.ext_iff Finmap.mem_keys=> ?", "nesting_depth": 8, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Framework"}
{"id": 421, "thm_name": "validInter_assoc_l", "thm_stmt": "lemma validInter_assoc_l (h₁ h₂ h₃ : heap) :\n  h₁ ⊥ʰ h₂ -> (h₁ +ʰ h₂) ⊥ʰ h₃ -> h₁ ⊥ʰ (h₂ +ʰ h₃)", "lean_root": "splean", "rel_path": "SPLean/Common/Heap.lean", "imports": ["import Mathlib.Algebra.BigOperators.Group.Finset", "import Mathlib.Algebra.BigOperators.Intervals", "import Mathlib.Algebra.Group.Basic", "import SPLean/Theories/HProp.lean", "import Ssreflect.Lang", "import Mathlib.Data.Finmap", "import Mathlib.Order.Interval.Finset.Basic", "import Mathlib.Data.Int.Interval", "import Lean", "import Batteries.Data.List.Perm"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Sigma", "module": "Init.Core"}, {"name": "AList", "module": "Mathlib.Data.List.AList"}, {"name": "scase", "module": "Ssreflect.Elim"}, {"name": "Finmap", "module": "Mathlib.Data.Finmap"}, {"name": "Finmap.liftOn₂", "module": "Mathlib.Data.Finmap"}, {"name": "move", "module": "Ssreflect.Basic"}, {"name": "srw", "module": "Ssreflect.Rewrite"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Finmap.lookup", "module": "Mathlib.Data.Finmap"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Option.merge", "module": "Init.Data.Option.Basic"}, {"name": "HAdd", "module": "Init.Prelude"}, {"name": "HAdd.hAdd", "module": "Init.Prelude"}, {"name": "instHAdd", "module": "Init.Prelude"}, {"name": "sdone", "module": "Ssreflect.Done"}], "used_repo_defs": [{"name": "heap", "content": "abbrev heap := Heap.heap val\n\n  inductive val : Type where\n    | val_unit   : val\n    | val_bool   : Bool → val\n    | val_int    : Int → val\n    | val_real   : ℝ → val\n    | val_loc    : loc → val\n    | val_prim   : prim → val\n    | val_fun    : var -> trm -> val\n    | val_fix    : var -> var -> trm -> val\n    | val_uninit : val\n    | val_error : val"}, {"name": "prim", "content": "inductive prim where\n  \n  | val_get : prim\n  | val_set : prim\n  \n  | val_neg : prim\n  | val_opp : prim\n  | val_eq : prim\n  | val_add : prim\n  | val_neq : prim\n  | val_sub : prim\n  | val_mul : prim\n  | val_div : prim\n  | val_mod : prim\n  \n  | val_le : prim\n  | val_lt : prim\n  | val_ge : prim\n  | val_gt : prim\n  | val_ptr_add : prim\n\n  inductive trm : Type where\n    | trm_val   : val -> trm\n    | trm_var   : var -> trm\n    | trm_fun   : var -> trm -> trm\n    | trm_fix   : var -> var -> trm -> trm\n    | trm_app   : trm -> trm -> trm\n    | trm_seq   : trm -> trm -> trm\n    | trm_let   : var -> trm -> trm -> trm\n    | trm_if    : trm -> trm -> trm -> trm\n    | trm_for   : var -> trm -> trm -> trm -> trm\n    | trm_while : trm -> trm -> trm\n    | trm_ref   : var → trm → trm → trm\n    | trm_alloc : var → trm → trm → trm"}, {"name": "scoped instance : PartialCommMonoid val where", "content": "scoped instance : PartialCommMonoid val where\n  add := add\n  add_assoc := by admit /- proof elided -/"}, {"name": "add", "content": "@[simp]\nabbrev add : val -> val -> val\n  | .val_int i, .val_int j => val.val_int (i + j)\n  | _, _ => val.val_unit"}, {"name": "evalExact", "content": "inductive evalExact : state → trm → (val → state → Prop) -> Prop where\n  | val : forall s v,\n      evalExact s (trm_val v) (fun v' s' ↦ v' = v ∧ s' = s)\n  | fun : forall s x t1,\n      evalExact s (trm_fun x t1) (fun v' s' ↦ v' = val_fun x t1 ∧ s' = s)\n  | fix : forall s f x t1,\n      evalExact s (trm_fix f x t1) (fun v' s' ↦ v' = val_fix f x t1 ∧ s' = s)\n  | app_arg1 : forall s1 t1 t2 Q1 Q,\n      ¬ trm_is_val t1 ->\n      evalExact s1 t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (trm_app v1 t2) Q) ->\n      evalExact s1 (trm_app t1 t2) Q\n  | app_arg2 : forall s1 (v1 : val) t2 Q1 Q,\n      ¬ trm_is_val t2 ->\n      evalExact s1 t2 Q1 ->\n      (forall v2 s2, Q1 v2 s2 -> evalExact s2 (trm_app v1 v2) Q) ->\n      evalExact s1 (trm_app v1 t2) Q\n  | app_fun : forall s1 v1 (v2 :val) x t1 Q,\n      v1 = val_fun x t1 ->\n      evalExact s1 (subst x v2 t1) Q ->\n      evalExact s1 (trm_app v1 v2) Q\n  | app_fix : forall s (v1 v2 : val) f x t1 Q,\n      v1 = val_fix f x t1 ->\n      evalExact s (subst x v2 (subst f v1 t1)) Q ->\n      evalExact s (trm_app v1 v2) Q\n  | seq : forall Q1 s t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 t2 Q) ->\n      evalExact s (trm_seq t1 t2) Q\n  | let : forall Q1 s x t1 t2 Q,\n      evalExact s t1 Q1 ->\n      (forall v1 s2, Q1 v1 s2 -> evalExact s2 (subst x v1 t2) Q) ->\n      evalExact s (trm_let x t1 t2) Q\n  | if : forall s (b : Bool) t1 t2 Q,\n      evalExact s (if b then t1 else t2) Q ->\n      evalExact s (trm_if (val_bool b) t1 t2) Q\n  | unop : forall op s v1 P,\n      evalunop op v1 P ->\n      evalExact s (trm_app op v1) (purepost s P)\n  | binop : forall op s (v1 v2 : val) P,\n      evalbinop op v1 v2 P ->\n      evalExact s (trm_app (trm_app op v1) v2) (purepost s P)\n  | ref : forall s x t1 t2 Q Q₁,\n    evalExact s t1 Q₁ →\n    (∀ v1 s1, Q₁ v1 s1 → ∀ p ∉ s1,\n     evalExact (s1.insert p v1) (subst x p t2) fun v s ↦ Q v (s.erase p)) →\n      evalExact s (trm_ref x t1 t2) Q\n  | get : forall s p,\n      p ∈ s ->\n      evalExact s (trm_app val_get (val_loc p))\n        (fun v' s' ↦ v' = read_state p s ∧ s' = s)\n  | set : forall s p v,\n      v = trm_val v' ->\n      p ∈ s ->\n      evalExact s (trm_app (trm_app val_set (val_loc p)) v)\n        (fun v'' s' ↦ v'' = val_unit ∧ s' = s.insert p v')\n  | alloc_arg : forall s Q₁ Q,\n    ¬ trm_is_val t1 →\n    evalExact s t1 Q₁ →\n    (∀ v' s', Q₁ v' s' → evalExact s' (trm_alloc x v' t2) Q) →\n    evalExact s (trm_alloc x t1 t2) Q\n  | alloc : forall (sa : state) (n : ℤ) Q,\n    n ≥ 0 →\n    (∀ (p : loc) (sb : state),\n      sb = conseq (make_list n.natAbs val_uninit) p →\n      p ≠ null →\n      Finmap.Disjoint sa sb →\n      evalExact (sb ∪ sa) (subst x p t2) fun v s ↦ Q v (s \\ sb)) →\n    evalExact sa (trm_alloc x n t2) Q\n  \n  \n  \n  \n  \n  \n  | for (n₁ n₂ : Int) (Q : val -> state -> Prop) :\n    evalExact s (if (n₁ < n₂) then\n               (trm_seq (subst x n₁ t₁) (trm_for x (val_int (n₁ + 1)) n₂ t₁))\n            else val_unit) Q ->\n    evalExact s (trm_for x n₁ n₂ t₁) Q\n  | while (t₁ t₂ : trm) (Q Q₁ : val -> state -> Prop) :\n    evalExact s t₁ Q₁ ->\n    (∀ s v, Q₁ v s -> evalExact s (trm_if v (trm_seq t₂ (trm_while t₁ t₂)) val_unit) Q) ->\n    evalExact s (trm_while t₁ t₂) Q"}, {"name": "valid", "content": "@[simp]\nabbrev valid : val -> Prop\n  | .val_int _ => True\n  | _ => False"}, {"name": "PartialCommMonoid", "content": "class PartialCommMonoid (α : Type) extends AddCommSemigroup α where\n  valid : α -> Prop\n  valid_add : ∀ x, valid (x + y) -> valid x\n  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)"}, {"name": "scoped instance inst : PartialCommMonoidWRT val add valid wh", "content": "scoped instance inst : PartialCommMonoidWRT val add valid where\n  validE := by admit /- proof elided -/"}, {"name": "AddCommMonoidWRT", "content": "class AddCommMonoidWRT (α : Type) (add' : semiOutParam $ α -> α -> α) extends AddCommMonoid α where\n  addE : (· + ·) = add'"}, {"name": "scoped instance : PartialCommMonoid val where", "content": "scoped instance : PartialCommMonoid val where\n  add := add\n  add_assoc := by admit /- proof elided -/"}, {"name": "add", "content": "@[simp]\nabbrev add : val -> val -> val\n  | .val_bool i, .val_bool j => val.val_bool (i || j)\n  | _, _ => val.val_unit"}, {"name": "valid", "content": "@[simp]\nabbrev valid : val -> Prop\n  | .val_bool _ => True\n  | _ => False"}, {"name": "scoped instance : PartialCommMonoid val where", "content": "scoped instance : PartialCommMonoid val where\n  add := add\n  add_assoc := by admit /- proof elided -/"}, {"name": "add", "content": "@[simp]\nabbrev add : val -> val -> val\n  | .val_real i, .val_real j => val.val_real (i + j)\n  | _, _ => val.val_unit"}, {"name": "valid", "content": "@[simp]\nabbrev valid : val -> Prop\n  | .val_real _ => True\n  | _ => False"}], "lib_lemmas": [{"name": "Finmap.mem_of_lookup_eq_some", "module": "Mathlib.Data.Finmap"}, {"name": "Or.intro_left", "module": "Init.Prelude"}, {"name": "add_assoc", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "loc", "content": "abbrev loc := Nat"}, {"name": "var", "content": "abbrev var := String"}, {"name": "Heap.heap", "content": "abbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)"}, {"name": "PartialCommMonoid", "content": "class PartialCommMonoid (α : Type) extends AddCommSemigroup α where\n  valid : α -> Prop\n  valid_add : ∀ x, valid (x + y) -> valid x\n  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)"}, {"name": "kmerge1", "content": "private def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v"}, {"name": "kmerge", "content": "@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)"}, {"name": "AList.merge", "content": "noncomputable def AList.merge (h₁ h₂ : AList (fun _ : loc => val)) :  AList (fun _ : loc => val) :=\n  ⟨kmerge h₁.entries h₂.entries, by admit /- proof elided -/\n    ⟩"}, {"name": "Heap.add", "content": "noncomputable def Heap.add (h₁ h₂ : heap) : heap :=\n  Finmap.liftOn₂ h₁ h₂ (fun h₁ h₂ => (h₁.merge h₂).toFinmap) (by admit /- proof elided -/\n    )"}, {"name": "validInter", "content": "def validInter (h₁ h₂ : heap) : Prop :=\n  ∀ l ∈ h₁, l ∈ h₂ -> ((h₁ +ʰ h₂).lookup l).any (valid (α := val))"}], "used_local_lemmas": [{"name": "Option.merge_none_r", "content": "lemma Option.merge_none_r (a : Option α) : Option.merge f a none = a"}, {"name": "Option.merge_assoc", "content": "lemma Option.merge_assoc (h : Associative f) (a b c : Option α) :\n  Option.merge f (Option.merge f a b) c = Option.merge f a (Option.merge f b c)"}], "local_ctx": "import Lean\n\nimport Mathlib.Data.Finmap\n\nimport Mathlib.Algebra.Group.Basic\n\nimport Mathlib.Algebra.BigOperators.Group.Finset\n\nimport Mathlib.Algebra.BigOperators.Intervals\n\nimport Mathlib.Data.Int.Interval\n\nimport Mathlib.Order.Interval.Finset.Basic\n\nimport Batteries.Data.List.Perm\n\nimport Ssreflect.Lang\n\nopen Classical\n\nabbrev loc := Nat\n\nabbrev var := String\n\nsection Option.merge\n\nvariable {α : Type u} (f : α → α → α)\n\nend Option.merge\n\nabbrev Heap.heap (val : Type) := Finmap (λ _ : loc ↦ val)\n\nclass PartialCommMonoid (α : Type) extends AddCommSemigroup α where\n  valid : α -> Prop\n  valid_add : ∀ x, valid (x + y) -> valid x\n  add_valid : ∀ x y, valid x -> valid y -> valid (x + y)\n\nopen PartialCommMonoid (valid)\n\nsection\n\nvariable {val : Type} [PartialCommMonoid val] -- [Inhabited val]\n\nlocal notation \"heap\" => Heap.heap val\n\nprivate def kmerge1 (l : loc) (v : val) (l₂ : List (Sigma (fun _ : loc => val))) : val :=\n  match l₂.dlookup l with\n  | .some v' => v + v'\n  | _ => v\n\n@[simp]\ndef kmerge : List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val)) → List (Sigma (fun _ : loc => val))\n  | [], l₂ => l₂\n  | s :: l₁, l₂ =>\n    (if s.1 ∈ l₂.keys then\n      ⟨s.1, kmerge1 s.1 s.2 l₂⟩ :: kmerge l₁ (l₂.kerase s.1)\n    else s :: kmerge l₁ l₂)\n\nnoncomputable def AList.merge (h₁ h₂ : AList (fun _ : loc => val)) :  AList (fun _ : loc => val) :=\n  ⟨kmerge h₁.entries h₂.entries, by admit /- proof elided -/\n    ⟩\n\nnoncomputable def Heap.add (h₁ h₂ : heap) : heap :=\n  Finmap.liftOn₂ h₁ h₂ (fun h₁ h₂ => (h₁.merge h₂).toFinmap) (by admit /- proof elided -/\n    )\n\ninfixr:55 \" +ʰ \" => Heap.add\n\ndef validInter (h₁ h₂ : heap) : Prop :=\n  ∀ l ∈ h₁, l ∈ h₂ -> ((h₁ +ʰ h₂).lookup l).any (valid (α := val))\n\ninfixr:55 \" ⊥ʰ \" => validInter", "target_theorem": "lemma validInter_assoc_l (h₁ h₂ h₃ : heap) :\n  h₁ ⊥ʰ h₂ -> (h₁ +ʰ h₂) ⊥ʰ h₃ -> h₁ ⊥ʰ (h₂ +ʰ h₃) :=", "ground_truth_proof": ":= by\n  simp [validInter]\n  move=> h1 h2 l hin1 /[tac (specialize h1 _ hin1 ; specialize h2 _ (Or.intro_left _ hin1))] [ hin2 | hin3 ]\n  { rcases h : Finmap.lookup l h₃\n    { rw [Option.merge_none_r] ; aesop }\n    { srw h at h2 ; rw [← Option.merge_assoc, h2] ; apply Finmap.mem_of_lookup_eq_some at h=> //\n      apply add_assoc } }\n  { rw [← Option.merge_assoc, h2]=> //\n    apply add_assoc }", "nesting_depth": 6, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Framework"}
{"id": 422, "thm_name": "wtPar", "thm_stmt": "theorem wtPar {Γ} {a b A : Term} (r : a ⇒ b) (h : Γ ⊢ a ∶ A) : Γ ⊢ b ∶ A", "lean_root": "TTBFL", "rel_path": "src/safety.lean", "imports": ["import «src».typing", "import src.syntactics", "import src.reduction", "import src.typing"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.sub", "module": "Init.Prelude"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "In", "content": "inductive In : Nat → Term → Ctxt → Prop where\n  | here {Γ A} : In 0 (rename succ A) (Γ ∷ A)\n  | there {Γ x A B} : In x A Γ → In (succ x) (rename succ A) (Γ ∷ B)"}, {"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "substRename", "content": "def substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )"}, {"name": "renameSubst", "content": "def renameSubst ξ σ : ∀ s, rename ξ (subst σ s) = subst (rename ξ ∘ σ) s :=\n  renameSubst' _ _ (rename ξ ∘ σ) (by admit /- proof elided -/\n  )"}, {"name": "substComp", "content": "def substComp σ τ : ∀ s, (subst σ ∘ subst τ) s = subst (subst σ ∘ τ) s :=\n  substComp' _ _ (subst σ ∘ τ) (by admit /- proof elided -/\n  )"}, {"name": "substId", "content": "def substId : ∀ s, subst var s = s :=\n  substId' var (by admit /- proof elided -/\n  )"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "infix:40 (priority := 1001) \"≈\" => Eqv", "content": "infix:40 (priority := 1001) \"≈\" => Eqv"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}], "lib_lemmas": [{"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "renameLiftRename", "content": "theorem renameLiftRename ξ a : rename succ (rename ξ a) = rename (lift ξ) (rename succ a)"}, {"name": "liftSucc", "content": "omit lc in\ntheorem liftSucc ξ : ∀ x, (lift ξ ∘ succ) x = (succ ∘ ξ) x"}, {"name": "renameComp", "content": "theorem renameComp ξ ζ s : rename ξ (rename ζ s) = rename (ξ ∘ ζ) s"}, {"name": "renameComp'", "content": "theorem renameComp' ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) : ∀ s, (rename ξ ∘ rename ζ) s = rename ς s"}, {"name": "liftComp", "content": "omit lc in\ntheorem liftComp ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) :\n  ∀ x, (lift ξ ∘ lift ζ) x = lift ς x"}, {"name": "renameExt", "content": "theorem renameExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ s, rename ξ s = rename ζ s"}, {"name": "liftExt", "content": "omit lc in\ntheorem liftExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ x, lift ξ x = lift ζ x"}, {"name": "wtfPiInvA", "content": "theorem wtfPiInvA {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j"}, {"name": "wtfPiInvA𝒰", "content": "theorem wtfPiInvA𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j ∧ 𝒰 j ≈ 𝒰'"}, {"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}, {"name": "convRename", "content": "theorem convRename {a b} ξ : a ⇔ b → rename ξ a ⇔ rename ξ b"}, {"name": "parsRename", "content": "theorem parsRename {a b} ξ (r : a ⇒⋆ b) : rename ξ a ⇒⋆ rename ξ b"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "renameDist", "content": "theorem renameDist ξ a s : subst (rename ξ a +: var) (rename (lift ξ) s) = rename ξ (subst (a +: var) s)"}, {"name": "substExt", "content": "theorem substExt σ τ (h : ∀ x, σ x = τ x) : ∀ s, subst σ s = subst τ s"}, {"name": "upExt", "content": "theorem upExt σ τ (h : ∀ x, σ x = τ x) : ∀ x, (⇑ σ) x = (⇑ τ) x"}, {"name": "wRenameLift", "content": "theorem wRenameLift {ξ : ℕ → ℕ} {Γ Δ A}\n  (h : Δ ⊢ ξ ∶ Γ) :\n  Δ ∷ (rename ξ A) ⊢ lift ξ ∶ Γ ∷ A"}, {"name": "inHere", "content": "theorem inHere {Γ A A'} (e : A' = rename succ A) : (Γ ∷ A) ∋ 0 ∶ A'"}, {"name": "inThere", "content": "theorem inThere {Γ x A A' B} (h : Γ ∋ x ∶ A) (e : A' = rename succ A) : Γ ∷ B ∋ succ x ∶ A'"}, {"name": "convEqv", "content": "theorem convEqv {a b} : a ⇔ b → a ≈ b"}, {"name": "parsEqv", "content": "theorem parsEqv {a b} (r : a ⇒⋆ b) : a ≈ b"}, {"name": "parEqv", "content": "theorem parEqv {a b} (r : a ⇒ b) : a ≈ b"}, {"name": "wRenameSucc", "content": "theorem wRenameSucc {Γ A} : Γ ∷ A ⊢ succ ∶ Γ"}, {"name": "renameUpSubst", "content": "theorem renameUpSubst σ a : rename succ (subst σ a) = subst (⇑ σ) (rename succ a)"}, {"name": "upSucc", "content": "theorem upSucc σ : ∀ x, (⇑ σ ∘ succ) x = (rename succ ∘ σ) x"}, {"name": "wtWf", "content": "theorem wtWf {Γ} {a A : Term} (h : Γ ⊢ a ∶ A) : ⊢ Γ"}, {"name": "convSubst", "content": "theorem convSubst {a b} σ : a ⇔ b → subst σ a ⇔ subst σ b"}, {"name": "parsSubst", "content": "theorem parsSubst {a b} σ (r : a ⇒⋆ b) : subst σ a ⇒⋆ subst σ b"}, {"name": "parSubst", "content": "theorem parSubst {a b} σ (r : a ⇒ b) : subst σ a ⇒ subst σ b"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "substDist", "content": "theorem substDist σ a s : subst (subst σ a +: var) (subst (⇑ σ) s) = subst σ (subst (a +: var) s)"}, {"name": "substUnion", "content": "theorem substUnion σ a s : subst (a +: σ) s = subst (a +: var) (subst (⇑ σ) s)"}, {"name": "substDropAll", "content": "theorem substDropAll a b : b = subst (a +: var) (rename succ b)"}, {"name": "wtfLvlInv", "content": "theorem wtfLvlInv {Γ a 𝒰'}\n  (h : Γ ⊢ lvl a ∶ 𝒰') :\n  ∃ b k, Γ ⊢ a ∶ lvl b ∧ 𝒰 k ≈ 𝒰'"}, {"name": "wtfPiInvB", "content": "theorem wtfPiInvB {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ∷ A ⊢ B ∶ 𝒰 j"}, {"name": "convSym", "content": "theorem convSym {a b} : a ⇔ b → b ⇔ a"}, {"name": "parCong", "content": "theorem parCong {a a' b b'} (ra : a ⇒ a') (rb : b ⇒ b') : subst (a +: var) b ⇒ subst (a' +: var) b'"}, {"name": "wtfAbsInv", "content": "theorem wtfAbsInv {Γ A' b C}\n  (h : Γ ⊢ abs A' b ∶ C) :\n  ∃ A B, Γ ∷ A ⊢ b ∶ B ∧ A ≈ A' ∧ pi A B ≈ C"}, {"name": "convCong", "content": "theorem convCong {a a' b b'} : a ⇔ a' → b ⇔ b' → subst (a +: var) b ⇔ subst (a' +: var) b'"}, {"name": "parsCong", "content": "theorem parsCong {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : subst (a +: var) b ⇒⋆ subst (a' +: var) b'"}, {"name": "convPiInv", "content": "theorem convPiInv {a₁ a₂ b₁ b₂} : pi a₁ b₁ ⇔ pi a₂ b₂ → a₁ ⇔ a₂ ∧ b₁ ⇔ b₂"}, {"name": "parsPiInv", "content": "theorem parsPiInv {a b c} (r : pi a b ⇒⋆ c) : ∃ a' b', c = pi a' b' ∧ a ⇒⋆ a' ∧ b ⇒⋆ b'"}, {"name": "parConv", "content": "theorem parConv {a b} (r : a ⇒ b) : a ⇔ b"}, {"name": "parPars", "content": "theorem parPars {a b} (r : a ⇒ b) : a ⇒⋆ b"}, {"name": "parsConv", "content": "theorem parsConv {a b} (r : a ⇒⋆ b) : a ⇔ b"}], "used_local_defs": [], "used_local_lemmas": [{"name": "wtRename", "content": "theorem wtRename {ξ : ℕ → ℕ} {Γ Δ} {a A : Term}\n  (hξ : Δ ⊢ ξ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ rename ξ a ∶ rename ξ A"}, {"name": "wtWeaken", "content": "theorem wtWeaken {Γ k} {a A B : Term}\n  (hΓ : ⊢ Γ) (hB : Γ ⊢ B ∶ 𝒰 k) (h : Γ ⊢ a ∶ A) :\n  Γ ∷ B ⊢ rename succ a ∶ rename succ A"}, {"name": "wSubstUp", "content": "theorem wSubstUp {σ Δ Γ k A}\n  (hA : Δ ⊢ subst σ A ∶ 𝒰 k)\n  (h : Δ ⊢ σ ∶ Γ) :\n  Δ ∷ subst σ A ⊢ ⇑ σ ∶ Γ ∷ A"}, {"name": "wSubstCons", "content": "theorem wSubstCons {Γ} {a A : Term}\n  (h : Γ ⊢ a ∶ A) :\n  Γ ⊢ a +: var ∶ Γ ∷ A"}, {"name": "wtMorph", "content": "theorem wtMorph {σ : ℕ → Term} {Γ Δ} {a A : Term}\n  (hσ : Δ ⊢ σ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ subst σ a ∶ subst σ A"}, {"name": "wtSubst", "content": "theorem wtSubst {Γ} {a A b B : Term}\n  (hb : Γ ⊢ b ∶ B) (h : Γ ∷ B ⊢ a ∶ A) :\n  Γ ⊢ subst (b +: var) a ∶ subst (b +: var) A"}, {"name": "wtReplace", "content": "theorem wtReplace {Γ} {A B c C k : Term}\n  (e : A ≈ B)\n  (hB : Γ ⊢ B ∶ 𝒰 k)\n  (h : Γ ∷ A ⊢ c ∶ C) :\n  Γ ∷ B ⊢ c ∶ C"}, {"name": "wtMem", "content": "theorem wtMem {Γ x A} (mem : Γ ∋ x ∶ A) (h : ⊢ Γ) : ∃ k, Γ ⊢ A ∶ 𝒰 k"}, {"name": "wtRegularity", "content": "theorem wtRegularity {Γ} {a A : Term} (h : Γ ⊢ a ∶ A) : ∃ k, Γ ⊢ A ∶ 𝒰 k"}], "local_ctx": "import «src».typing\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nnotation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "target_theorem": "theorem wtPar {Γ} {a b A : Term} (r : a ⇒ b) (h : Γ ⊢ a ∶ A) : Γ ⊢ b ∶ A :=", "ground_truth_proof": ":= by\n  induction h generalizing b\n  case var => cases r; constructor <;> assumption\n  case 𝒰 ih => cases r with | 𝒰 r' => exact Wt.𝒰 (ih r')\n  case pi ihA ihB =>\n    cases r with | pi ra rb =>\n    let ihA' := ihA ra\n    exact Wt.pi ihA' (wtReplace (parEqv ra) ihA' (ihB rb))\n  case abs B _ _ hPi _ _ ihPi ihA ihb => cases r with | abs rA rb =>\n    let rPi := Par.pi rA (parRefl B)\n    let hb := wtReplace (convEqv (parConv rA)) (ihA rA) (ihb rb)\n    exact Wt.conv\n      (convEqv (convSym (parConv rPi)))\n      (Wt.abs (ihPi rPi) (ihA rA) hb) hPi\n  case app hb ha ihb iha =>\n    cases r\n    case β rb ra =>\n      let ⟨_, hA⟩ := wtRegularity ha\n      let ⟨_, hPi⟩ := wtRegularity hb\n      let ⟨_, hB⟩ := wtfPiInvB hPi\n      let ⟨A', B', hb', _, e⟩ := wtfAbsInv (ihb (Par.abs (parRefl _) rb))\n      let ⟨eA, eB⟩ := convPiInv (eqvConv e)\n      exact Wt.conv\n        (convEqv (convCong (convSym (parConv ra)) eB))\n        (wtSubst (iha ra) (wtReplace (convEqv eA) hA hb'))\n        (wtSubst ha hB)\n    case app rb ra =>\n      let ⟨k, hBa⟩ := wtRegularity (Wt.app hb ha)\n      exact Wt.conv\n        (convEqv (convSym (parConv (parCong ra (parRefl _)))))\n        (Wt.app (ihb rb) (iha ra)) hBa\n  case mty ih => cases r; exact Wt.mty (ih (parRefl _))\n  case exf hA _ ihA ihb => cases r with | exf rA rb =>\n    exact Wt.conv\n      (convEqv (convSym (parConv rA)))\n      (Wt.exf (ihA rA) (ihb rb)) hA\n  case lvl hj iha _ => cases r with | lvl r' => exact Wt.lvl (iha r') hj\n  case lof => cases r; constructor <;> assumption\n  case trans hj ihi _ => exact Wt.trans (ihi r) hj\n  case conv e _ hB iha _ => exact Wt.conv e (iha r) hB\n  case sub hj _ _ ihA => exact Wt.sub hj (ihA r)", "nesting_depth": 10, "transitive_dep_count": 88, "subset_aristotle": false, "category": "Type systems"}
{"id": 423, "thm_name": "wtMorph", "thm_stmt": "theorem wtMorph {σ : ℕ → Term} {Γ Δ} {a A : Term}\n  (hσ : Δ ⊢ σ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ subst σ a ∶ subst σ A", "lean_root": "TTBFL", "rel_path": "src/safety.lean", "imports": ["import «src».typing", "import src.syntactics", "import src.reduction", "import src.typing"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.sub", "module": "Init.Prelude"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "In", "content": "inductive In : Nat → Term → Ctxt → Prop where\n  | here {Γ A} : In 0 (rename succ A) (Γ ∷ A)\n  | there {Γ x A B} : In x A Γ → In (succ x) (rename succ A) (Γ ∷ B)"}, {"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "substRename", "content": "def substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )"}, {"name": "renameSubst", "content": "def renameSubst ξ σ : ∀ s, rename ξ (subst σ s) = subst (rename ξ ∘ σ) s :=\n  renameSubst' _ _ (rename ξ ∘ σ) (by admit /- proof elided -/\n  )"}, {"name": "substComp", "content": "def substComp σ τ : ∀ s, (subst σ ∘ subst τ) s = subst (subst σ ∘ τ) s :=\n  substComp' _ _ (subst σ ∘ τ) (by admit /- proof elided -/\n  )"}, {"name": "substId", "content": "def substId : ∀ s, subst var s = s :=\n  substId' var (by admit /- proof elided -/\n  )"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "infix:40 (priority := 1001) \"≈\" => Eqv", "content": "infix:40 (priority := 1001) \"≈\" => Eqv"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}], "lib_lemmas": [{"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "renameLiftRename", "content": "theorem renameLiftRename ξ a : rename succ (rename ξ a) = rename (lift ξ) (rename succ a)"}, {"name": "liftSucc", "content": "omit lc in\ntheorem liftSucc ξ : ∀ x, (lift ξ ∘ succ) x = (succ ∘ ξ) x"}, {"name": "renameComp", "content": "theorem renameComp ξ ζ s : rename ξ (rename ζ s) = rename (ξ ∘ ζ) s"}, {"name": "renameComp'", "content": "theorem renameComp' ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) : ∀ s, (rename ξ ∘ rename ζ) s = rename ς s"}, {"name": "liftComp", "content": "omit lc in\ntheorem liftComp ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) :\n  ∀ x, (lift ξ ∘ lift ζ) x = lift ς x"}, {"name": "renameExt", "content": "theorem renameExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ s, rename ξ s = rename ζ s"}, {"name": "liftExt", "content": "omit lc in\ntheorem liftExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ x, lift ξ x = lift ζ x"}, {"name": "wtfPiInvA", "content": "theorem wtfPiInvA {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j"}, {"name": "wtfPiInvA𝒰", "content": "theorem wtfPiInvA𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j ∧ 𝒰 j ≈ 𝒰'"}, {"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}, {"name": "convRename", "content": "theorem convRename {a b} ξ : a ⇔ b → rename ξ a ⇔ rename ξ b"}, {"name": "parsRename", "content": "theorem parsRename {a b} ξ (r : a ⇒⋆ b) : rename ξ a ⇒⋆ rename ξ b"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "renameDist", "content": "theorem renameDist ξ a s : subst (rename ξ a +: var) (rename (lift ξ) s) = rename ξ (subst (a +: var) s)"}, {"name": "substExt", "content": "theorem substExt σ τ (h : ∀ x, σ x = τ x) : ∀ s, subst σ s = subst τ s"}, {"name": "upExt", "content": "theorem upExt σ τ (h : ∀ x, σ x = τ x) : ∀ x, (⇑ σ) x = (⇑ τ) x"}, {"name": "wRenameLift", "content": "theorem wRenameLift {ξ : ℕ → ℕ} {Γ Δ A}\n  (h : Δ ⊢ ξ ∶ Γ) :\n  Δ ∷ (rename ξ A) ⊢ lift ξ ∶ Γ ∷ A"}, {"name": "inHere", "content": "theorem inHere {Γ A A'} (e : A' = rename succ A) : (Γ ∷ A) ∋ 0 ∶ A'"}, {"name": "inThere", "content": "theorem inThere {Γ x A A' B} (h : Γ ∋ x ∶ A) (e : A' = rename succ A) : Γ ∷ B ∋ succ x ∶ A'"}, {"name": "convEqv", "content": "theorem convEqv {a b} : a ⇔ b → a ≈ b"}, {"name": "parsEqv", "content": "theorem parsEqv {a b} (r : a ⇒⋆ b) : a ≈ b"}, {"name": "parEqv", "content": "theorem parEqv {a b} (r : a ⇒ b) : a ≈ b"}, {"name": "wRenameSucc", "content": "theorem wRenameSucc {Γ A} : Γ ∷ A ⊢ succ ∶ Γ"}, {"name": "renameUpSubst", "content": "theorem renameUpSubst σ a : rename succ (subst σ a) = subst (⇑ σ) (rename succ a)"}, {"name": "upSucc", "content": "theorem upSucc σ : ∀ x, (⇑ σ ∘ succ) x = (rename succ ∘ σ) x"}, {"name": "wtWf", "content": "theorem wtWf {Γ} {a A : Term} (h : Γ ⊢ a ∶ A) : ⊢ Γ"}, {"name": "convSubst", "content": "theorem convSubst {a b} σ : a ⇔ b → subst σ a ⇔ subst σ b"}, {"name": "parsSubst", "content": "theorem parsSubst {a b} σ (r : a ⇒⋆ b) : subst σ a ⇒⋆ subst σ b"}, {"name": "parSubst", "content": "theorem parSubst {a b} σ (r : a ⇒ b) : subst σ a ⇒ subst σ b"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "substDist", "content": "theorem substDist σ a s : subst (subst σ a +: var) (subst (⇑ σ) s) = subst σ (subst (a +: var) s)"}, {"name": "substUnion", "content": "theorem substUnion σ a s : subst (a +: σ) s = subst (a +: var) (subst (⇑ σ) s)"}, {"name": "substDropAll", "content": "theorem substDropAll a b : b = subst (a +: var) (rename succ b)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "wtRename", "content": "theorem wtRename {ξ : ℕ → ℕ} {Γ Δ} {a A : Term}\n  (hξ : Δ ⊢ ξ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ rename ξ a ∶ rename ξ A"}, {"name": "wtWeaken", "content": "theorem wtWeaken {Γ k} {a A B : Term}\n  (hΓ : ⊢ Γ) (hB : Γ ⊢ B ∶ 𝒰 k) (h : Γ ⊢ a ∶ A) :\n  Γ ∷ B ⊢ rename succ a ∶ rename succ A"}, {"name": "wSubstUp", "content": "theorem wSubstUp {σ Δ Γ k A}\n  (hA : Δ ⊢ subst σ A ∶ 𝒰 k)\n  (h : Δ ⊢ σ ∶ Γ) :\n  Δ ∷ subst σ A ⊢ ⇑ σ ∶ Γ ∷ A"}], "local_ctx": "import «src».typing\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nnotation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "target_theorem": "theorem wtMorph {σ : ℕ → Term} {Γ Δ} {a A : Term}\n  (hσ : Δ ⊢ σ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ subst σ a ∶ subst σ A :=", "ground_truth_proof": ":= by\n  induction h generalizing σ Δ\n  case var mem _ => exact hσ _ _ mem\n  case 𝒰 ih => exact Wt.𝒰 (ih hσ hΔ)\n  case pi ihA ihB =>\n    let ihA' := ihA hσ hΔ\n    refine Wt.pi ihA' ?_\n    rw [renameUpSubst]\n    exact ihB (wSubstUp ihA' hσ) (Wf.cons hΔ ihA')\n  case abs ihPi ihA ihb =>\n    let ihPi' := ihPi hσ hΔ\n    let ⟨k, hA⟩ := wtfPiInvA ihPi'\n    exact Wt.abs ihPi' (ihA hσ hΔ) (ihb (wSubstUp hA hσ) (Wf.cons hΔ hA))\n  case app ihb iha => rw [← substDist]; exact Wt.app (ihb hσ hΔ) (iha hσ hΔ)\n  case mty ih => exact Wt.mty (ih hσ hΔ)\n  case exf ihb ihA => exact Wt.exf (ihb hσ hΔ) (ihA hσ hΔ)\n  case lvl iha ihj => exact Wt.lvl (iha hσ hΔ) (ihj hσ hΔ)\n  case lof => constructor <;> assumption\n  case trans ihi ihj => exact Wt.trans (ihi hσ hΔ) (ihj hσ hΔ)\n  case conv e _ _ iha ihA =>\n    refine Wt.conv (convEqv (convSubst σ (eqvConv e))) (iha hσ hΔ) (ihA hσ hΔ)\n  case sub ihj ihA => exact Wt.sub (ihj hσ hΔ) (ihA hσ hΔ)", "nesting_depth": 8, "transitive_dep_count": 69, "subset_aristotle": false, "category": "Type systems"}
{"id": 424, "thm_name": "antirenaming", "thm_stmt": "theorem antirenaming {ξ a b'} (r : rename ξ a ⇒ b') : ∃ b, b' = rename ξ b ∧ a ⇒ b", "lean_root": "TTBFL", "rel_path": "src/reduction.lean", "imports": ["import src.syntactics", "import «src».syntactics", "import «src».tactics"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "nf", "content": "@[simp]\ndef nf : Term → Prop\n  | 𝒰 a => nf a\n  | pi a b => nf a ∧ nf b\n  | abs a b => nf a ∧ nf b\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | lvl a => nf a\n  | _ => True"}, {"name": "ne", "content": "@[simp]\ndef ne : Term → Prop\n  | var _ => True\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | _ => False"}, {"name": "wnf", "content": "@[simp] def wnf (a : Term) : Prop := ∃ b, nf b ∧ a ⇒⋆ b"}, {"name": "wne", "content": "@[simp] def wne (a : Term) : Prop := ∃ b, ne b ∧ a ⇒⋆ b"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "substRename", "content": "def substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )"}, {"name": "renameSubst", "content": "def renameSubst ξ σ : ∀ s, rename ξ (subst σ s) = subst (rename ξ ∘ σ) s :=\n  renameSubst' _ _ (rename ξ ∘ σ) (by admit /- proof elided -/\n  )"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "liftExt", "content": "theorem liftExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ x, lift ξ x = lift ζ x"}, {"name": "liftId", "content": "theorem liftId ξ (h : ∀ x, ξ x = x) : ∀ x, lift ξ x = x"}, {"name": "liftSucc", "content": "theorem liftSucc ξ : ∀ x, (lift ξ ∘ succ) x = (succ ∘ ξ) x"}, {"name": "renameExt", "content": "theorem renameExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ s, rename ξ s = rename ζ s"}, {"name": "renameId", "content": "theorem renameId s : rename id s = s"}, {"name": "renameComp", "content": "theorem renameComp ξ ζ s : rename ξ (rename ζ s) = rename (ξ ∘ ζ) s"}, {"name": "upExt", "content": "theorem upExt σ τ (h : ∀ x, σ x = τ x) : ∀ x, (⇑ σ) x = (⇑ τ) x"}, {"name": "upSucc", "content": "theorem upSucc σ : ∀ x, (⇑ σ ∘ succ) x = (rename succ ∘ σ) x"}, {"name": "substExt", "content": "theorem substExt σ τ (h : ∀ x, σ x = τ x) : ∀ s, subst σ s = subst τ s"}, {"name": "substId'", "content": "theorem substId' σ (h : ∀ x, σ x = var x) : ∀ s, subst σ s = s"}, {"name": "substRename'", "content": "theorem substRename' ξ σ τ (h : ∀ x, (σ ∘ ξ) x = τ x) : ∀ s, subst σ (rename ξ s) = subst τ s"}, {"name": "renameSubst'", "content": "theorem renameSubst' ξ σ τ (h : ∀ x, (rename ξ ∘ σ) x = τ x) : ∀ s, rename ξ (subst σ s) = subst τ s"}, {"name": "renameDist", "content": "theorem renameDist ξ a s : subst (rename ξ a +: var) (rename (lift ξ) s) = rename ξ (subst (a +: var) s)"}, {"name": "substDrop", "content": "theorem substDrop σ a b : subst (a +: σ) (rename succ b) = subst σ b"}, {"name": "substUnion", "content": "theorem substUnion σ a s : subst (a +: σ) s = subst (a +: var) (subst (⇑ σ) s)"}, {"name": "neNf", "content": "theorem neNf {a} : ne a → nf a"}, {"name": "nfPars", "content": "theorem nfPars {a b} (r : a ⇒⋆ b) : nf a → nf b"}, {"name": "nePars", "content": "theorem nePars {a b} (r : a ⇒⋆ b) : ne a → ne b"}, {"name": "wnfRename", "content": "theorem wnfRename {ξ a} : wnf (rename ξ a) → wnf a"}, {"name": "wneRename", "content": "theorem wneRename {ξ a} : wne (rename ξ a) → wne a"}, {"name": "renameToSubst", "content": "theorem renameToSubst ξ : ∀ s, rename ξ s = subst (var ∘ ξ) s"}, {"name": "renameLiftRename", "content": "theorem renameLiftRename ξ a : rename succ (rename ξ a) = rename (lift ξ) (rename succ a)"}, {"name": "renameUpSubst", "content": "theorem renameUpSubst σ a : rename succ (subst σ a) = subst (⇑ σ) (rename succ a)"}, {"name": "substDropAll", "content": "theorem substDropAll a b : b = subst (a +: var) (rename succ b)"}, {"name": "wnfBwds", "content": "theorem wnfBwds {a b} (r : a ⇒⋆ b) : wnf b → wnf a"}, {"name": "wneBwds", "content": "theorem wneBwds {a b} (r : a ⇒⋆ b) : wne b → wne a"}, {"name": "wnfFwds", "content": "theorem wnfFwds {a b} (r : a ⇒⋆ b) : wnf a → wnf b"}, {"name": "wneFwds", "content": "theorem wneFwds {a b} (r : a ⇒⋆ b) : wne a → wne b"}], "used_local_defs": [{"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}], "used_local_lemmas": [{"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}], "local_ctx": "import «src».tactics\n\nimport «src».syntactics\n\nopen Term\n\nvariable [LevelClass]\n\nsection\n\ninductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k\n\nend\n\ninfix:40 \"⇒\" => Par", "target_theorem": "theorem antirenaming {ξ a b'} (r : rename ξ a ⇒ b') : ∃ b, b' = rename ξ b ∧ a ⇒ b :=", "ground_truth_proof": ":= by\n  generalize e : rename ξ a = a' at r\n  induction r generalizing ξ a\n  all_goals cases a <;> injections; subst_eqs; specialize_rfls\n  case β ihb b _ e _ iha =>\n    cases b <;> injections; subst_eqs; specialize_rfls\n    let ⟨a, ea, ra⟩ := iha\n    let ⟨b, eb, rb⟩ := ihb\n    subst ea; subst eb\n    exact ⟨subst (a +: var) b, renameDist ξ a b, Par.β rb ra⟩\n  case var => exact ⟨var _, rfl, parRefl _⟩\n  case 𝒰 ih =>\n    let ⟨a, e, r⟩ := ih\n    subst e\n    exact ⟨𝒰 a, rfl, Par.𝒰 r⟩\n  case pi iha ihb =>\n    let ⟨a, ea, ra⟩ := iha\n    let ⟨b, eb, rb⟩ := ihb\n    subst ea; subst eb\n    exact ⟨pi a b, rfl, Par.pi ra rb⟩\n  case abs iha ihb =>\n    let ⟨a, ea, ra⟩ := iha\n    let ⟨b, eb, rb⟩ := ihb\n    subst ea; subst eb\n    exact ⟨abs a b, rfl, Par.abs ra rb⟩\n  case app ihb iha =>\n    let ⟨a, ea, ra⟩ := iha\n    let ⟨b, eb, rb⟩ := ihb\n    subst ea; subst eb\n    exact ⟨app b a, rfl, Par.app rb ra⟩\n  case mty => exact ⟨mty, rfl, Par.mty⟩\n  case exf iha ihb =>\n    let ⟨a, ea, ra⟩ := iha\n    let ⟨b, eb, rb⟩ := ihb\n    subst ea; subst eb\n    exact ⟨exf a b, rfl, Par.exf ra rb⟩\n  case lvl ih =>\n    let ⟨a, e, r⟩ := ih\n    subst e\n    exact ⟨lvl a, rfl, Par.lvl r⟩\n  case lof => exact ⟨lof _, rfl, parRefl _⟩", "nesting_depth": 4, "transitive_dep_count": 23, "subset_aristotle": false, "category": "Type systems"}
{"id": 425, "thm_name": "wtRegularity", "thm_stmt": "theorem wtRegularity {Γ} {a A : Term} (h : Γ ⊢ a ∶ A) : ∃ k, Γ ⊢ A ∶ 𝒰 k", "lean_root": "TTBFL", "rel_path": "src/safety.lean", "imports": ["import «src».typing", "import src.syntactics", "import src.reduction", "import src.typing"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.sub", "module": "Init.Prelude"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "In", "content": "inductive In : Nat → Term → Ctxt → Prop where\n  | here {Γ A} : In 0 (rename succ A) (Γ ∷ A)\n  | there {Γ x A B} : In x A Γ → In (succ x) (rename succ A) (Γ ∷ B)"}, {"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "substRename", "content": "def substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )"}, {"name": "renameSubst", "content": "def renameSubst ξ σ : ∀ s, rename ξ (subst σ s) = subst (rename ξ ∘ σ) s :=\n  renameSubst' _ _ (rename ξ ∘ σ) (by admit /- proof elided -/\n  )"}, {"name": "substComp", "content": "def substComp σ τ : ∀ s, (subst σ ∘ subst τ) s = subst (subst σ ∘ τ) s :=\n  substComp' _ _ (subst σ ∘ τ) (by admit /- proof elided -/\n  )"}, {"name": "substId", "content": "def substId : ∀ s, subst var s = s :=\n  substId' var (by admit /- proof elided -/\n  )"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "infix:40 (priority := 1001) \"≈\" => Eqv", "content": "infix:40 (priority := 1001) \"≈\" => Eqv"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}], "lib_lemmas": [{"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "renameLiftRename", "content": "theorem renameLiftRename ξ a : rename succ (rename ξ a) = rename (lift ξ) (rename succ a)"}, {"name": "liftSucc", "content": "omit lc in\ntheorem liftSucc ξ : ∀ x, (lift ξ ∘ succ) x = (succ ∘ ξ) x"}, {"name": "renameComp", "content": "theorem renameComp ξ ζ s : rename ξ (rename ζ s) = rename (ξ ∘ ζ) s"}, {"name": "renameComp'", "content": "theorem renameComp' ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) : ∀ s, (rename ξ ∘ rename ζ) s = rename ς s"}, {"name": "liftComp", "content": "omit lc in\ntheorem liftComp ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) :\n  ∀ x, (lift ξ ∘ lift ζ) x = lift ς x"}, {"name": "renameExt", "content": "theorem renameExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ s, rename ξ s = rename ζ s"}, {"name": "liftExt", "content": "omit lc in\ntheorem liftExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ x, lift ξ x = lift ζ x"}, {"name": "wtfPiInvA", "content": "theorem wtfPiInvA {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j"}, {"name": "wtfPiInvA𝒰", "content": "theorem wtfPiInvA𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j ∧ 𝒰 j ≈ 𝒰'"}, {"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}, {"name": "convRename", "content": "theorem convRename {a b} ξ : a ⇔ b → rename ξ a ⇔ rename ξ b"}, {"name": "parsRename", "content": "theorem parsRename {a b} ξ (r : a ⇒⋆ b) : rename ξ a ⇒⋆ rename ξ b"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "renameDist", "content": "theorem renameDist ξ a s : subst (rename ξ a +: var) (rename (lift ξ) s) = rename ξ (subst (a +: var) s)"}, {"name": "substExt", "content": "theorem substExt σ τ (h : ∀ x, σ x = τ x) : ∀ s, subst σ s = subst τ s"}, {"name": "upExt", "content": "theorem upExt σ τ (h : ∀ x, σ x = τ x) : ∀ x, (⇑ σ) x = (⇑ τ) x"}, {"name": "wRenameLift", "content": "theorem wRenameLift {ξ : ℕ → ℕ} {Γ Δ A}\n  (h : Δ ⊢ ξ ∶ Γ) :\n  Δ ∷ (rename ξ A) ⊢ lift ξ ∶ Γ ∷ A"}, {"name": "inHere", "content": "theorem inHere {Γ A A'} (e : A' = rename succ A) : (Γ ∷ A) ∋ 0 ∶ A'"}, {"name": "inThere", "content": "theorem inThere {Γ x A A' B} (h : Γ ∋ x ∶ A) (e : A' = rename succ A) : Γ ∷ B ∋ succ x ∶ A'"}, {"name": "convEqv", "content": "theorem convEqv {a b} : a ⇔ b → a ≈ b"}, {"name": "parsEqv", "content": "theorem parsEqv {a b} (r : a ⇒⋆ b) : a ≈ b"}, {"name": "parEqv", "content": "theorem parEqv {a b} (r : a ⇒ b) : a ≈ b"}, {"name": "wRenameSucc", "content": "theorem wRenameSucc {Γ A} : Γ ∷ A ⊢ succ ∶ Γ"}, {"name": "renameUpSubst", "content": "theorem renameUpSubst σ a : rename succ (subst σ a) = subst (⇑ σ) (rename succ a)"}, {"name": "upSucc", "content": "theorem upSucc σ : ∀ x, (⇑ σ ∘ succ) x = (rename succ ∘ σ) x"}, {"name": "wtWf", "content": "theorem wtWf {Γ} {a A : Term} (h : Γ ⊢ a ∶ A) : ⊢ Γ"}, {"name": "convSubst", "content": "theorem convSubst {a b} σ : a ⇔ b → subst σ a ⇔ subst σ b"}, {"name": "parsSubst", "content": "theorem parsSubst {a b} σ (r : a ⇒⋆ b) : subst σ a ⇒⋆ subst σ b"}, {"name": "parSubst", "content": "theorem parSubst {a b} σ (r : a ⇒ b) : subst σ a ⇒ subst σ b"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "substDist", "content": "theorem substDist σ a s : subst (subst σ a +: var) (subst (⇑ σ) s) = subst σ (subst (a +: var) s)"}, {"name": "substUnion", "content": "theorem substUnion σ a s : subst (a +: σ) s = subst (a +: var) (subst (⇑ σ) s)"}, {"name": "substDropAll", "content": "theorem substDropAll a b : b = subst (a +: var) (rename succ b)"}, {"name": "wtfLvlInv", "content": "theorem wtfLvlInv {Γ a 𝒰'}\n  (h : Γ ⊢ lvl a ∶ 𝒰') :\n  ∃ b k, Γ ⊢ a ∶ lvl b ∧ 𝒰 k ≈ 𝒰'"}, {"name": "wtfPiInvB", "content": "theorem wtfPiInvB {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ∷ A ⊢ B ∶ 𝒰 j"}], "used_local_defs": [], "used_local_lemmas": [{"name": "wtRename", "content": "theorem wtRename {ξ : ℕ → ℕ} {Γ Δ} {a A : Term}\n  (hξ : Δ ⊢ ξ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ rename ξ a ∶ rename ξ A"}, {"name": "wtWeaken", "content": "theorem wtWeaken {Γ k} {a A B : Term}\n  (hΓ : ⊢ Γ) (hB : Γ ⊢ B ∶ 𝒰 k) (h : Γ ⊢ a ∶ A) :\n  Γ ∷ B ⊢ rename succ a ∶ rename succ A"}, {"name": "wSubstUp", "content": "theorem wSubstUp {σ Δ Γ k A}\n  (hA : Δ ⊢ subst σ A ∶ 𝒰 k)\n  (h : Δ ⊢ σ ∶ Γ) :\n  Δ ∷ subst σ A ⊢ ⇑ σ ∶ Γ ∷ A"}, {"name": "wSubstCons", "content": "theorem wSubstCons {Γ} {a A : Term}\n  (h : Γ ⊢ a ∶ A) :\n  Γ ⊢ a +: var ∶ Γ ∷ A"}, {"name": "wtMorph", "content": "theorem wtMorph {σ : ℕ → Term} {Γ Δ} {a A : Term}\n  (hσ : Δ ⊢ σ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ subst σ a ∶ subst σ A"}, {"name": "wtSubst", "content": "theorem wtSubst {Γ} {a A b B : Term}\n  (hb : Γ ⊢ b ∶ B) (h : Γ ∷ B ⊢ a ∶ A) :\n  Γ ⊢ subst (b +: var) a ∶ subst (b +: var) A"}, {"name": "wtMem", "content": "theorem wtMem {Γ x A} (mem : Γ ∋ x ∶ A) (h : ⊢ Γ) : ∃ k, Γ ⊢ A ∶ 𝒰 k"}], "local_ctx": "import «src».typing\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nnotation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "target_theorem": "theorem wtRegularity {Γ} {a A : Term} (h : Γ ⊢ a ∶ A) : ∃ k, Γ ⊢ A ∶ 𝒰 k :=", "ground_truth_proof": ":= by\n  induction h\n  case var wf mem _ => exact wtMem mem wf\n  case pi ih _ | trans ih => exact ih\n  case abs h _ _ _ _ _ | exf h _ _ _ | conv h _ _ => exact ⟨_, h⟩\n  case 𝒰 ih =>\n    let ⟨_, ihk⟩ := ih\n    let ⟨l, _, hk, _⟩ := wtfLvlInv ihk\n    exact ⟨l, Wt.𝒰 hk⟩\n  case app ha ihb _ =>\n    let ⟨_, hPi⟩ := ihb\n    let ⟨k, hB⟩ := wtfPiInvB hPi\n    exact ⟨subst _ k, wtSubst ha hB⟩\n  case mty hj _ => exact ⟨_, hj⟩\n  case lvl hj _ _ => exact ⟨_, hj⟩\n  case lof k wf _ _ =>\n    let ⟨l, klgt⟩ := exists_gt k\n    let ⟨m, lmgt⟩ := exists_gt l\n    refine ⟨lof l, ?_⟩\n    apply Wt.lvl (Wt.lof wf klgt)\n    apply Wt.𝒰 (Wt.lof wf lmgt)\n  case sub ih _ =>\n    let ⟨_, ihk⟩ := ih\n    let ⟨l, _, hk, _⟩ := wtfLvlInv ihk\n    exact ⟨l, Wt.𝒰 hk⟩", "nesting_depth": 10, "transitive_dep_count": 75, "subset_aristotle": false, "category": "Type systems"}
{"id": 426, "thm_name": "wtProgress", "thm_stmt": "theorem wtProgress {a A : Term} (h : ⬝ ⊢ a ∶ A) : Nonempty (Value a) ∨ ∃ b, a ⇒β b", "lean_root": "TTBFL", "rel_path": "src/safety.lean", "imports": ["import «src».typing", "import src.reduction", "import src.typing"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Or.inr", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}, {"name": "infix:40 (priority := 1001) \"≈\" => Eqv", "content": "infix:40 (priority := 1001) \"≈\" => Eqv"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}], "lib_lemmas": [{"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "wtfLvlInv", "content": "theorem wtfLvlInv {Γ a 𝒰'}\n  (h : Γ ⊢ lvl a ∶ 𝒰') :\n  ∃ b k, Γ ⊢ a ∶ lvl b ∧ 𝒰 k ≈ 𝒰'"}, {"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}, {"name": "wtfMtyInv", "content": "theorem wtfMtyInv {Γ 𝒰'}\n  (h : Γ ⊢ mty ∶ 𝒰') :\n  ∃ k, 𝒰 k ≈ 𝒰'"}, {"name": "wtfPiInv𝒰", "content": "theorem wtfPiInv𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, 𝒰 j ≈ 𝒰'"}, {"name": "wtfPiInvA𝒰", "content": "theorem wtfPiInvA𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j ∧ 𝒰 j ≈ 𝒰'"}, {"name": "wtfAbsInv", "content": "theorem wtfAbsInv {Γ A' b C}\n  (h : Γ ⊢ abs A' b ∶ C) :\n  ∃ A B, Γ ∷ A ⊢ b ∶ B ∧ A ≈ A' ∧ pi A B ≈ C"}, {"name": "wtfLofInv", "content": "theorem wtfLofInv {Γ j 𝒰'}\n  (h : Γ ⊢ lof j ∶ 𝒰') :\n  ∃ k, lvl k ≈ 𝒰'"}, {"name": "wtf𝒰Inv", "content": "theorem wtf𝒰Inv {Γ j 𝒰'}\n  (h : Γ ⊢ 𝒰 j ∶ 𝒰') :\n  ∃ k, 𝒰 k ≈ 𝒰'"}, {"name": "conv𝒰Pi", "content": "theorem conv𝒰Pi {c a b} : ¬ 𝒰 c ⇔ pi a b"}, {"name": "parsPiInv", "content": "theorem parsPiInv {a b c} (r : pi a b ⇒⋆ c) : ∃ a' b', c = pi a' b' ∧ a ⇒⋆ a' ∧ b ⇒⋆ b'"}, {"name": "pars𝒰Inv", "content": "theorem pars𝒰Inv {a b} (r : 𝒰 a ⇒⋆ b) : ∃ a', b = 𝒰 a' ∧ a ⇒⋆ a'"}, {"name": "convLvlPi", "content": "theorem convLvlPi {a b k} : ¬ lvl k ⇔ pi a b"}, {"name": "parsLvlInv", "content": "theorem parsLvlInv {i b} (r : lvl i ⇒⋆ b) : ∃ j, b = lvl j ∧ i ⇒⋆ j"}], "used_local_defs": [{"name": "Value", "content": "inductive Value : Term → Type where\n  | 𝒰 {k} : Value (𝒰 k)\n  | pi {a b} : Value (pi a b)\n  | abs {a b} : Value (abs a b)\n  | mty : Value mty\n  | lvl {k} : Value (lvl k)\n  | lof {k} : Value (lof k)"}, {"name": "CBN", "content": "inductive CBN : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ⇒β subst (a +: var) b\n  | app {b b' a} : b ⇒β  b' → app b a ⇒β app b' a\n  | exf {a b b'} : b ⇒β b' → exf a b ⇒β exf a b'"}, {"name": "valueType", "content": "@[simp] \ndef valueType {a} (A : Term) : Value a → Prop\n  | Value.𝒰 | Value.pi | Value.mty | Value.lvl => ∃ k, 𝒰 k ≈ A\n  | Value.abs => ∃ B C, pi B C ≈ A\n  | Value.lof => ∃ k, lvl k ≈ A"}], "used_local_lemmas": [{"name": "wtValue", "content": "theorem wtValue {Γ} {a A B : Term} (h : Γ ⊢ a ∶ A) (e : A ≈ B) : (v : Value a) → valueType B v\n  | Value.𝒰 => let ⟨_, e𝒰⟩"}, {"name": "wtAbs", "content": "theorem wtAbs {Γ} {b A B : Term} (v : Value b) (h : Γ ⊢ b ∶ pi A B) : ∃ a' b', b = abs a' b'"}], "local_ctx": "import «src».typing\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nnotation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ\n\ninductive Value : Term → Type where\n  | 𝒰 {k} : Value (𝒰 k)\n  | pi {a b} : Value (pi a b)\n  | abs {a b} : Value (abs a b)\n  | mty : Value mty\n  | lvl {k} : Value (lvl k)\n  | lof {k} : Value (lof k)\n\nsection\n\ninductive CBN : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ⇒β subst (a +: var) b\n  | app {b b' a} : b ⇒β  b' → app b a ⇒β app b' a\n  | exf {a b b'} : b ⇒β b' → exf a b ⇒β exf a b'\n\nend\n\ninfix:40 \"⇒β\" => CBN\n\ninfix:40 \"⇒β⋆\" => CBNs\n\n@[simp] \ndef valueType {a} (A : Term) : Value a → Prop\n  | Value.𝒰 | Value.pi | Value.mty | Value.lvl => ∃ k, 𝒰 k ≈ A\n  | Value.abs => ∃ B C, pi B C ≈ A\n  | Value.lof => ∃ k, lvl k ≈ A", "target_theorem": "theorem wtProgress {a A : Term} (h : ⬝ ⊢ a ∶ A) : Nonempty (Value a) ∨ ∃ b, a ⇒β b :=", "ground_truth_proof": ":= by\n  generalize e : (⬝) = Γ at h\n  induction h\n  all_goals subst e; specialize_rfls\n  case var mem => cases mem\n  case 𝒰 | pi | abs | mty | lvl | lof => repeat constructor\n  case trans ih _ | conv ih _ | sub ih => exact ih\n  case app hb _ ihb _ =>\n    cases ihb\n    case inl v =>\n      cases v with | intro v =>\n      let ⟨_, _, e⟩ := wtAbs v hb; subst e\n      exact Or.inr ⟨_, CBN.β⟩\n    case inr r => let ⟨_, r⟩ := r; exact Or.inr ⟨_, CBN.app r⟩\n  case exf _ hb _ ihb =>\n    cases ihb\n    case inl v => cases v with | intro v => cases wtMty v hb\n    case inr r => let ⟨_, r⟩ := r; exact Or.inr ⟨_, CBN.exf r⟩", "nesting_depth": 6, "transitive_dep_count": 40, "subset_aristotle": false, "category": "Type systems"}
{"id": 427, "thm_name": "interpDet'", "thm_stmt": "theorem interpDet' {i I a P Q} (hP : ⟦ a ⟧ i , I ↘ P) (hQ : ⟦ a ⟧ i , I ↘ Q) : P = Q", "lean_root": "TTBFL", "rel_path": "src/candidates.lean", "imports": ["import «src».normal", "import src.reduction", "import src.normal"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "propext", "module": "Init.Core"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Or.inl", "module": "Init.Prelude"}, {"name": "Or.inr", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P", "content": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P"}, {"name": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "content": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "nf", "content": "@[simp]\ndef nf : Term → Prop\n  | 𝒰 a => nf a\n  | pi a b => nf a ∧ nf b\n  | abs a b => nf a ∧ nf b\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | lvl a => nf a\n  | _ => True"}, {"name": "ne", "content": "@[simp]\ndef ne : Term → Prop\n  | var _ => True\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | _ => False"}, {"name": "LevelClass", "content": "class LevelClass where\n  L : Type\n  lc : LevelClasses L"}, {"name": "wne", "content": "@[simp] def wne (a : Term) : Prop := ∃ b, ne b ∧ a ⇒⋆ b"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "wnf", "content": "@[simp] def wnf (a : Term) : Prop := ∃ b, nf b ∧ a ⇒⋆ b"}, {"name": "taka", "content": "@[simp]\ndef taka : Term → Term\n  | 𝒰 a => 𝒰 (taka a)\n  | pi a b => pi (taka a) (taka b)\n  | abs a b => abs (taka a) (taka b)\n  | app b a => match b with\n    | abs _ b => subst (taka a +: var) (taka b)\n    | b => app (taka b) (taka a)\n  | exf a b => exf (taka a) (taka b)\n  | lvl a => lvl (taka a)\n  | t => t"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "parCong", "content": "theorem parCong {a a' b b'} (ra : a ⇒ a') (rb : b ⇒ b') : subst (a +: var) b ⇒ subst (a' +: var) b'"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "nePar", "content": "theorem nePar {a b} (r : a ⇒ b) : ne a → ne b"}, {"name": "nfPar", "content": "theorem nfPar {a b} (r : a ⇒ b) : nf a → nf b"}, {"name": "diacon", "content": "theorem diacon {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diamond", "content": "theorem diamond {a b c} (r₁ : a ⇒ b) (r₂ : a ⇒ c) : ∃ d, b ⇒ d ∧ c ⇒ d"}, {"name": "parTaka", "content": "theorem parTaka {a b} (r : a ⇒ b) : b ⇒ taka a"}, {"name": "parPars", "content": "theorem parPars {a b} (r : a ⇒ b) : a ⇒⋆ b"}, {"name": "parsLofInv", "content": "theorem parsLofInv {j b} (r : lof j ⇒⋆ b) : b = lof j"}, {"name": "wnfBwds", "content": "theorem wnfBwds {a b} (r : a ⇒⋆ b) : wnf b → wnf a"}, {"name": "nfWnf", "content": "theorem nfWnf {a} (nfa : nf a) : wnf a"}, {"name": "nfPars", "content": "theorem nfPars {a b} (r : a ⇒⋆ b) : nf a → nf b"}], "used_local_defs": [{"name": "Interp", "content": "inductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | ne a : ne a → Interp i I a wne\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty wne\n  | lvl b : nf b → Interp i I (lvl b)\n    (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P"}], "used_local_lemmas": [{"name": "interpLvlEq", "content": "theorem interpLvlEq {b c} (r : b ⇒ c) :\n  (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a) =\n  (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ c ⇒⋆ lof k) ∨ wne a)"}, {"name": "interpNeInv", "content": "theorem interpNeInv {i I a P} (h : ⟦ a ⟧ i , I ↘ P) :\n  ne a → P = wne"}, {"name": "interpPiInv", "content": "theorem interpPiInv {i I a b P} (h : ⟦ pi a b ⟧ i , I ↘ P) :\n  ∃ (Pa : Term → Prop) (Pf : Term → (Term → Prop) → Prop),\n    (⟦ a ⟧ i , I ↘ Pa) ∧\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) ∧\n    (∀ x Pb, Pf x Pb → ⟦ subst (x +: var) b ⟧ i, I ↘ Pb) ∧\n    P = (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))"}, {"name": "interp𝒰Inv", "content": "theorem interp𝒰Inv {i I a P} (h : ⟦ 𝒰 a ⟧ i , I ↘ P) :\n  ∃ j lt, a ⇒⋆ lof j ∧ P = I j lt"}, {"name": "interpMtyInv", "content": "theorem interpMtyInv {i I P} (h : ⟦ mty ⟧ i , I ↘ P) : P = wne"}, {"name": "interpLvlInv", "content": "theorem interpLvlInv {i I b P} (h : ⟦ lvl b ⟧ i , I ↘ P) :\n  wnf b ∧ P = (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)"}, {"name": "interpFwd", "content": "theorem interpFwd {i I a b P} (r : a ⇒ b) (h : ⟦ a ⟧ i , I ↘ P) : ⟦ b ⟧ i , I ↘ P"}], "local_ctx": "import «src».normal\n\nopen Term\n\nvariable [lc : LevelClass]\n\ninductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | ne a : ne a → Interp i I a wne\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty wne\n  | lvl b : nf b → Interp i I (lvl b)\n    (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "target_theorem": "theorem interpDet' {i I a P Q} (hP : ⟦ a ⟧ i , I ↘ P) (hQ : ⟦ a ⟧ i , I ↘ Q) : P = Q :=", "ground_truth_proof": ":= by\n  induction hP generalizing Q\n  case ne nea => exact symm (interpNeInv hQ nea)\n  case pi Pa Pf _ hPf _ iha ihb =>\n    let ⟨Pa', Pf', ha', hPf', hb', e⟩ := interpPiInv hQ\n    subst e; apply funext; intro f\n    apply propext; constructor\n    . intro h x Pb' Pax' PfxPb'\n      have Pax : Pa x := by rw [iha ha']; exact Pax'\n      let ⟨Pb, PfxPb⟩ := hPf x Pax\n      rw [← ihb x Pb PfxPb (hb' x Pb' PfxPb')]\n      exact h x Pb Pax PfxPb\n    . intro h x Pb Pax PfxPb\n      have Pax' : Pa' x := by rw [← iha ha']; exact Pax\n      let ⟨Pb', PfxPb'⟩ := hPf' x Pax'\n      rw [ihb x Pb PfxPb (hb' x Pb' PfxPb')]\n      exact h x Pb' Pax' PfxPb'\n  case 𝒰 =>\n    let ⟨j, _, r, e⟩ := interp𝒰Inv hQ\n    injection (parsLofInv r) with ej; subst ej; simp [e]\n  case mty => simp [interpMtyInv hQ]\n  case lvl =>\n    let ⟨_, e⟩ := interpLvlInv hQ; rw [e]\n  case step r _ ih => exact ih (interpFwd r hQ)", "nesting_depth": 8, "transitive_dep_count": 51, "subset_aristotle": false, "category": "Type systems"}
{"id": 428, "thm_name": "wtMty", "thm_stmt": "theorem wtMty {Γ} {b : Term} (v : Value b) (h : Γ ⊢ b ∶ mty) : False", "lean_root": "TTBFL", "rel_path": "src/safety.lean", "imports": ["import «src».typing", "import src.reduction", "import src.typing"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}, {"name": "infix:40 (priority := 1001) \"≈\" => Eqv", "content": "infix:40 (priority := 1001) \"≈\" => Eqv"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "wtfLvlInv", "content": "theorem wtfLvlInv {Γ a 𝒰'}\n  (h : Γ ⊢ lvl a ∶ 𝒰') :\n  ∃ b k, Γ ⊢ a ∶ lvl b ∧ 𝒰 k ≈ 𝒰'"}, {"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}, {"name": "wtfMtyInv", "content": "theorem wtfMtyInv {Γ 𝒰'}\n  (h : Γ ⊢ mty ∶ 𝒰') :\n  ∃ k, 𝒰 k ≈ 𝒰'"}, {"name": "wtfPiInv𝒰", "content": "theorem wtfPiInv𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, 𝒰 j ≈ 𝒰'"}, {"name": "wtfPiInvA𝒰", "content": "theorem wtfPiInvA𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j ∧ 𝒰 j ≈ 𝒰'"}, {"name": "wtfAbsInv", "content": "theorem wtfAbsInv {Γ A' b C}\n  (h : Γ ⊢ abs A' b ∶ C) :\n  ∃ A B, Γ ∷ A ⊢ b ∶ B ∧ A ≈ A' ∧ pi A B ≈ C"}, {"name": "wtfLofInv", "content": "theorem wtfLofInv {Γ j 𝒰'}\n  (h : Γ ⊢ lof j ∶ 𝒰') :\n  ∃ k, lvl k ≈ 𝒰'"}, {"name": "wtf𝒰Inv", "content": "theorem wtf𝒰Inv {Γ j 𝒰'}\n  (h : Γ ⊢ 𝒰 j ∶ 𝒰') :\n  ∃ k, 𝒰 k ≈ 𝒰'"}, {"name": "conv𝒰Mty", "content": "theorem conv𝒰Mty {a} : ¬ 𝒰 a ⇔ mty"}, {"name": "parsMtyInv", "content": "theorem parsMtyInv {b} (r : mty ⇒⋆ b) : b = mty"}, {"name": "pars𝒰Inv", "content": "theorem pars𝒰Inv {a b} (r : 𝒰 a ⇒⋆ b) : ∃ a', b = 𝒰 a' ∧ a ⇒⋆ a'"}, {"name": "convMtyPi", "content": "theorem convMtyPi {a b} : ¬ mty ⇔ pi a b"}, {"name": "parsPiInv", "content": "theorem parsPiInv {a b c} (r : pi a b ⇒⋆ c) : ∃ a' b', c = pi a' b' ∧ a ⇒⋆ a' ∧ b ⇒⋆ b'"}, {"name": "convLvlMty", "content": "theorem convLvlMty {j} : ¬ lvl j ⇔ mty"}, {"name": "parsLvlInv", "content": "theorem parsLvlInv {i b} (r : lvl i ⇒⋆ b) : ∃ j, b = lvl j ∧ i ⇒⋆ j"}], "used_local_defs": [{"name": "Value", "content": "inductive Value : Term → Type where\n  | 𝒰 {k} : Value (𝒰 k)\n  | pi {a b} : Value (pi a b)\n  | abs {a b} : Value (abs a b)\n  | mty : Value mty\n  | lvl {k} : Value (lvl k)\n  | lof {k} : Value (lof k)"}, {"name": "valueType", "content": "@[simp] \ndef valueType {a} (A : Term) : Value a → Prop\n  | Value.𝒰 | Value.pi | Value.mty | Value.lvl => ∃ k, 𝒰 k ≈ A\n  | Value.abs => ∃ B C, pi B C ≈ A\n  | Value.lof => ∃ k, lvl k ≈ A"}], "used_local_lemmas": [{"name": "wtValue", "content": "theorem wtValue {Γ} {a A B : Term} (h : Γ ⊢ a ∶ A) (e : A ≈ B) : (v : Value a) → valueType B v\n  | Value.𝒰 => let ⟨_, e𝒰⟩"}], "local_ctx": "import «src».typing\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nnotation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ\n\ninductive Value : Term → Type where\n  | 𝒰 {k} : Value (𝒰 k)\n  | pi {a b} : Value (pi a b)\n  | abs {a b} : Value (abs a b)\n  | mty : Value mty\n  | lvl {k} : Value (lvl k)\n  | lof {k} : Value (lof k)\n\nsection\n\nend\n\ninfix:40 \"⇒β\" => CBN\n\ninfix:40 \"⇒β⋆\" => CBNs\n\n@[simp] \ndef valueType {a} (A : Term) : Value a → Prop\n  | Value.𝒰 | Value.pi | Value.mty | Value.lvl => ∃ k, 𝒰 k ≈ A\n  | Value.abs => ∃ B C, pi B C ≈ A\n  | Value.lof => ∃ k, lvl k ≈ A", "target_theorem": "theorem wtMty {Γ} {b : Term} (v : Value b) (h : Γ ⊢ b ∶ mty) : False :=", "ground_truth_proof": ":= by\n  generalize e : mty = T at h\n  induction h\n  all_goals try first | subst e | injection e\n  case var | app | exf => contradiction\n  case conv h v emty _ _ =>\n    let _e := wtValue h emty v\n    cases v <;> let ⟨_, e⟩ := _e\n    case 𝒰 | pi | mty | lvl => cases conv𝒰Mty (eqvConv e)\n    case abs => let ⟨_, e⟩ := e; cases convMtyPi (eqvConv (Eqv.sym e))\n    case lof => cases convLvlMty (eqvConv e)", "nesting_depth": 5, "transitive_dep_count": 36, "subset_aristotle": false, "category": "Type systems"}
{"id": 429, "thm_name": "interpDet'", "thm_stmt": "theorem interpDet' {i I a P Q} (hP : ⟦ a ⟧ i , I ↘ P) (hQ : ⟦ a ⟧ i , I ↘ Q) : P = Q", "lean_root": "TTBFL", "rel_path": "src/semantics.lean", "imports": ["import src.reduction", "import «src».reduction"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "propext", "module": "Init.Core"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P", "content": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P"}, {"name": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "content": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "LevelClass", "content": "class LevelClass where\n  L : Type\n  lc : LevelClasses L"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "taka", "content": "@[simp]\ndef taka : Term → Term\n  | 𝒰 a => 𝒰 (taka a)\n  | pi a b => pi (taka a) (taka b)\n  | abs a b => abs (taka a) (taka b)\n  | app b a => match b with\n    | abs _ b => subst (taka a +: var) (taka b)\n    | b => app (taka b) (taka a)\n  | exf a b => exf (taka a) (taka b)\n  | lvl a => lvl (taka a)\n  | t => t"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "parCong", "content": "theorem parCong {a a' b b'} (ra : a ⇒ a') (rb : b ⇒ b') : subst (a +: var) b ⇒ subst (a' +: var) b'"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "diamond", "content": "theorem diamond {a b c} (r₁ : a ⇒ b) (r₂ : a ⇒ c) : ∃ d, b ⇒ d ∧ c ⇒ d"}, {"name": "parTaka", "content": "theorem parTaka {a b} (r : a ⇒ b) : b ⇒ taka a"}, {"name": "parsLofInv", "content": "theorem parsLofInv {j b} (r : lof j ⇒⋆ b) : b = lof j"}], "used_local_defs": [{"name": "Interp", "content": "inductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty (λ _ ↦ False)\n  | lvl k : Interp i I (lvl (lof k)) (λ a ↦ ∃ j, a ⇒⋆ lof j ∧ j < k)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P"}], "used_local_lemmas": [{"name": "interpPiInv", "content": "theorem interpPiInv {i I a b P} (h : ⟦ pi a b ⟧ i , I ↘ P) :\n  ∃ (Pa : Term → Prop) (Pf : Term → (Term → Prop) → Prop),\n    (⟦ a ⟧ i , I ↘ Pa) ∧\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) ∧\n    (∀ x Pb, Pf x Pb → ⟦ subst (x +: var) b ⟧ i, I ↘ Pb) ∧\n    P = (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))"}, {"name": "interp𝒰Inv", "content": "theorem interp𝒰Inv {i I a P} (h : ⟦ 𝒰 a ⟧ i , I ↘ P) :\n  ∃ j lt, a ⇒⋆ lof j ∧ P = I j lt"}, {"name": "interpMtyInv", "content": "theorem interpMtyInv {i I P} (h : ⟦ mty ⟧ i , I ↘ P) : P = (λ _ ↦ False)"}, {"name": "interpLvlInv", "content": "theorem interpLvlInv {i I a P} (h : ⟦ lvl a ⟧ i , I ↘ P) :\n  ∃ k, a ⇒⋆ lof k ∧ P = (λ a ↦ ∃ j, a ⇒⋆ lof j ∧ j < k)"}, {"name": "interpFwd", "content": "theorem interpFwd {i I a b P} (r : a ⇒ b) (h : ⟦ a ⟧ i , I ↘ P) : ⟦ b ⟧ i , I ↘ P"}], "local_ctx": "import «src».reduction\n\nopen Term\n\nvariable [lc : LevelClass]\n\ninductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty (λ _ ↦ False)\n  | lvl k : Interp i I (lvl (lof k)) (λ a ↦ ∃ j, a ⇒⋆ lof j ∧ j < k)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "target_theorem": "theorem interpDet' {i I a P Q} (hP : ⟦ a ⟧ i , I ↘ P) (hQ : ⟦ a ⟧ i , I ↘ Q) : P = Q :=", "ground_truth_proof": ":= by\n  induction hP generalizing Q\n  case pi Pa Pf _ hPf _ iha ihb =>\n    let ⟨Pa', Pf', ha', hPf', hb', e⟩ := interpPiInv hQ\n    subst e; apply funext; intro f\n    apply propext; constructor\n    . intro h x Pb' Pax' PfxPb'\n      have Pax : Pa x := by rw [iha ha']; exact Pax'\n      let ⟨Pb, PfxPb⟩ := hPf x Pax\n      rw [← ihb x Pb PfxPb (hb' x Pb' PfxPb')]\n      exact h x Pb Pax PfxPb\n    . intro h x Pb Pax PfxPb\n      have Pax' : Pa' x := by rw [← iha ha']; exact Pax\n      let ⟨Pb', PfxPb'⟩ := hPf' x Pax'\n      rw [ihb x Pb PfxPb (hb' x Pb' PfxPb')]\n      exact h x Pb' Pax' PfxPb'\n  case 𝒰 =>\n    let ⟨j, _, r, e⟩ := interp𝒰Inv hQ\n    injection (parsLofInv r) with ej; subst ej; simp [e]\n  case mty => simp [interpMtyInv hQ]\n  case lvl =>\n    let ⟨k, r, e⟩ := interpLvlInv hQ\n    injection (parsLofInv r) with ek; subst ek; simp [e]\n  case step r _ ih => exact ih (interpFwd r hQ)", "nesting_depth": 8, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Type systems"}
{"id": 430, "thm_name": "interpFwd", "thm_stmt": "theorem interpFwd {i I a b P} (r : a ⇒ b) (h : ⟦ a ⟧ i , I ↘ P) : ⟦ b ⟧ i , I ↘ P", "lean_root": "TTBFL", "rel_path": "src/candidates.lean", "imports": ["import «src».normal", "import src.reduction", "import src.normal"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Or.inl", "module": "Init.Prelude"}, {"name": "Or.inr", "module": "Init.Prelude"}, {"name": "propext", "module": "Init.Core"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P", "content": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P"}, {"name": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "content": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "nf", "content": "@[simp]\ndef nf : Term → Prop\n  | 𝒰 a => nf a\n  | pi a b => nf a ∧ nf b\n  | abs a b => nf a ∧ nf b\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | lvl a => nf a\n  | _ => True"}, {"name": "ne", "content": "@[simp]\ndef ne : Term → Prop\n  | var _ => True\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | _ => False"}, {"name": "LevelClass", "content": "class LevelClass where\n  L : Type\n  lc : LevelClasses L"}, {"name": "wne", "content": "@[simp] def wne (a : Term) : Prop := ∃ b, ne b ∧ a ⇒⋆ b"}, {"name": "taka", "content": "@[simp]\ndef taka : Term → Term\n  | 𝒰 a => 𝒰 (taka a)\n  | pi a b => pi (taka a) (taka b)\n  | abs a b => abs (taka a) (taka b)\n  | app b a => match b with\n    | abs _ b => subst (taka a +: var) (taka b)\n    | b => app (taka b) (taka a)\n  | exf a b => exf (taka a) (taka b)\n  | lvl a => lvl (taka a)\n  | t => t"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "diacon", "content": "theorem diacon {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diamond", "content": "theorem diamond {a b c} (r₁ : a ⇒ b) (r₂ : a ⇒ c) : ∃ d, b ⇒ d ∧ c ⇒ d"}, {"name": "parTaka", "content": "theorem parTaka {a b} (r : a ⇒ b) : b ⇒ taka a"}, {"name": "parCong", "content": "theorem parCong {a a' b b'} (ra : a ⇒ a') (rb : b ⇒ b') : subst (a +: var) b ⇒ subst (a' +: var) b'"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "parPars", "content": "theorem parPars {a b} (r : a ⇒ b) : a ⇒⋆ b"}, {"name": "parsLofInv", "content": "theorem parsLofInv {j b} (r : lof j ⇒⋆ b) : b = lof j"}, {"name": "nfPars", "content": "theorem nfPars {a b} (r : a ⇒⋆ b) : nf a → nf b"}, {"name": "nfPar", "content": "theorem nfPar {a b} (r : a ⇒ b) : nf a → nf b"}, {"name": "nePar", "content": "theorem nePar {a b} (r : a ⇒ b) : ne a → ne b"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}], "used_local_defs": [{"name": "Interp", "content": "inductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | ne a : ne a → Interp i I a wne\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty wne\n  | lvl b : nf b → Interp i I (lvl b)\n    (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P"}], "used_local_lemmas": [{"name": "interpLvlEq", "content": "theorem interpLvlEq {b c} (r : b ⇒ c) :\n  (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a) =\n  (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ c ⇒⋆ lof k) ∨ wne a)"}], "local_ctx": "import «src».normal\n\nopen Term\n\nvariable [lc : LevelClass]\n\ninductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | ne a : ne a → Interp i I a wne\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty wne\n  | lvl b : nf b → Interp i I (lvl b)\n    (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "target_theorem": "theorem interpFwd {i I a b P} (r : a ⇒ b) (h : ⟦ a ⟧ i , I ↘ P) : ⟦ b ⟧ i , I ↘ P :=", "ground_truth_proof": ":= by\n  induction h generalizing b\n  case pi iha ihb =>\n    cases r; constructor\n    all_goals intros; apply_rules [parCong, parRefl]\n  case ne nea => constructor; exact nePar r nea\n  case 𝒰 => cases r; case 𝒰 r => cases r; constructor\n  case mty => cases r; exact Interp.mty\n  case lvl => cases r; case lvl nfb _ r =>\n    rw [interpLvlEq r]; constructor; exact nfPars (parPars r) nfb\n  case step r' _ ih =>\n    let ⟨c, rc, rc'⟩ := diamond r r'\n    constructor <;> apply_rules", "nesting_depth": 11, "transitive_dep_count": 42, "subset_aristotle": false, "category": "Type systems"}
{"id": 431, "thm_name": "wtReplace", "thm_stmt": "theorem wtReplace {Γ} {A B c C k : Term}\n  (e : A ≈ B)\n  (hB : Γ ⊢ B ∶ 𝒰 k)\n  (h : Γ ∷ A ⊢ c ∶ C) :\n  Γ ∷ B ⊢ c ∶ C", "lean_root": "TTBFL", "rel_path": "src/safety.lean", "imports": ["import «src».typing", "import src.syntactics", "import src.reduction", "import src.typing"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.sub", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "In", "content": "inductive In : Nat → Term → Ctxt → Prop where\n  | here {Γ A} : In 0 (rename succ A) (Γ ∷ A)\n  | there {Γ x A B} : In x A Γ → In (succ x) (rename succ A) (Γ ∷ B)"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}, {"name": "substId", "content": "def substId : ∀ s, subst var s = s :=\n  substId' var (by admit /- proof elided -/\n  )"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "substRename", "content": "def substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )"}, {"name": "renameSubst", "content": "def renameSubst ξ σ : ∀ s, rename ξ (subst σ s) = subst (rename ξ ∘ σ) s :=\n  renameSubst' _ _ (rename ξ ∘ σ) (by admit /- proof elided -/\n  )"}, {"name": "substComp", "content": "def substComp σ τ : ∀ s, (subst σ ∘ subst τ) s = subst (subst σ ∘ τ) s :=\n  substComp' _ _ (subst σ ∘ τ) (by admit /- proof elided -/\n  )"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}, {"name": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ", "content": "notation:40 Γ:41 \"∋\" x:41 \"∶\" A:41 => In x A Γ"}, {"name": "infix:40 (priority := 1001) \"≈\" => Eqv", "content": "infix:40 (priority := 1001) \"≈\" => Eqv"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}], "lib_lemmas": [{"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "renameLiftRename", "content": "theorem renameLiftRename ξ a : rename succ (rename ξ a) = rename (lift ξ) (rename succ a)"}, {"name": "liftSucc", "content": "omit lc in\ntheorem liftSucc ξ : ∀ x, (lift ξ ∘ succ) x = (succ ∘ ξ) x"}, {"name": "renameComp", "content": "theorem renameComp ξ ζ s : rename ξ (rename ζ s) = rename (ξ ∘ ζ) s"}, {"name": "renameComp'", "content": "theorem renameComp' ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) : ∀ s, (rename ξ ∘ rename ζ) s = rename ς s"}, {"name": "liftComp", "content": "omit lc in\ntheorem liftComp ξ ζ ς (h : ∀ x, (ξ ∘ ζ) x = ς x) :\n  ∀ x, (lift ξ ∘ lift ζ) x = lift ς x"}, {"name": "renameExt", "content": "theorem renameExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ s, rename ξ s = rename ζ s"}, {"name": "liftExt", "content": "omit lc in\ntheorem liftExt ξ ζ (h : ∀ x, ξ x = ζ x) : ∀ x, lift ξ x = lift ζ x"}, {"name": "wtfPiInvA", "content": "theorem wtfPiInvA {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j"}, {"name": "wtfPiInvA𝒰", "content": "theorem wtfPiInvA𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j ∧ 𝒰 j ≈ 𝒰'"}, {"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}, {"name": "convRename", "content": "theorem convRename {a b} ξ : a ⇔ b → rename ξ a ⇔ rename ξ b"}, {"name": "parsRename", "content": "theorem parsRename {a b} ξ (r : a ⇒⋆ b) : rename ξ a ⇒⋆ rename ξ b"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "renameDist", "content": "theorem renameDist ξ a s : subst (rename ξ a +: var) (rename (lift ξ) s) = rename ξ (subst (a +: var) s)"}, {"name": "substExt", "content": "theorem substExt σ τ (h : ∀ x, σ x = τ x) : ∀ s, subst σ s = subst τ s"}, {"name": "upExt", "content": "theorem upExt σ τ (h : ∀ x, σ x = τ x) : ∀ x, (⇑ σ) x = (⇑ τ) x"}, {"name": "wRenameLift", "content": "theorem wRenameLift {ξ : ℕ → ℕ} {Γ Δ A}\n  (h : Δ ⊢ ξ ∶ Γ) :\n  Δ ∷ (rename ξ A) ⊢ lift ξ ∶ Γ ∷ A"}, {"name": "inHere", "content": "theorem inHere {Γ A A'} (e : A' = rename succ A) : (Γ ∷ A) ∋ 0 ∶ A'"}, {"name": "inThere", "content": "theorem inThere {Γ x A A' B} (h : Γ ∋ x ∶ A) (e : A' = rename succ A) : Γ ∷ B ∋ succ x ∶ A'"}, {"name": "convEqv", "content": "theorem convEqv {a b} : a ⇔ b → a ≈ b"}, {"name": "parsEqv", "content": "theorem parsEqv {a b} (r : a ⇒⋆ b) : a ≈ b"}, {"name": "parEqv", "content": "theorem parEqv {a b} (r : a ⇒ b) : a ≈ b"}, {"name": "wRenameSucc", "content": "theorem wRenameSucc {Γ A} : Γ ∷ A ⊢ succ ∶ Γ"}, {"name": "renameUpSubst", "content": "theorem renameUpSubst σ a : rename succ (subst σ a) = subst (⇑ σ) (rename succ a)"}, {"name": "upSucc", "content": "theorem upSucc σ : ∀ x, (⇑ σ ∘ succ) x = (rename succ ∘ σ) x"}, {"name": "wtWf", "content": "theorem wtWf {Γ} {a A : Term} (h : Γ ⊢ a ∶ A) : ⊢ Γ"}, {"name": "convSubst", "content": "theorem convSubst {a b} σ : a ⇔ b → subst σ a ⇔ subst σ b"}, {"name": "parsSubst", "content": "theorem parsSubst {a b} σ (r : a ⇒⋆ b) : subst σ a ⇒⋆ subst σ b"}, {"name": "parSubst", "content": "theorem parSubst {a b} σ (r : a ⇒ b) : subst σ a ⇒ subst σ b"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "substDist", "content": "theorem substDist σ a s : subst (subst σ a +: var) (subst (⇑ σ) s) = subst σ (subst (a +: var) s)"}, {"name": "substUnion", "content": "theorem substUnion σ a s : subst (a +: σ) s = subst (a +: var) (subst (⇑ σ) s)"}, {"name": "substDropAll", "content": "theorem substDropAll a b : b = subst (a +: var) (rename succ b)"}, {"name": "convSym", "content": "theorem convSym {a b} : a ⇔ b → b ⇔ a"}], "used_local_defs": [], "used_local_lemmas": [{"name": "wtRename", "content": "theorem wtRename {ξ : ℕ → ℕ} {Γ Δ} {a A : Term}\n  (hξ : Δ ⊢ ξ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ rename ξ a ∶ rename ξ A"}, {"name": "wtWeaken", "content": "theorem wtWeaken {Γ k} {a A B : Term}\n  (hΓ : ⊢ Γ) (hB : Γ ⊢ B ∶ 𝒰 k) (h : Γ ⊢ a ∶ A) :\n  Γ ∷ B ⊢ rename succ a ∶ rename succ A"}, {"name": "wSubstUp", "content": "theorem wSubstUp {σ Δ Γ k A}\n  (hA : Δ ⊢ subst σ A ∶ 𝒰 k)\n  (h : Δ ⊢ σ ∶ Γ) :\n  Δ ∷ subst σ A ⊢ ⇑ σ ∶ Γ ∷ A"}, {"name": "wtMorph", "content": "theorem wtMorph {σ : ℕ → Term} {Γ Δ} {a A : Term}\n  (hσ : Δ ⊢ σ ∶ Γ) (hΔ : ⊢ Δ) (h : Γ ⊢ a ∶ A) :\n  Δ ⊢ subst σ a ∶ subst σ A"}], "local_ctx": "import «src».typing\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nnotation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "target_theorem": "theorem wtReplace {Γ} {A B c C k : Term}\n  (e : A ≈ B)\n  (hB : Γ ⊢ B ∶ 𝒰 k)\n  (h : Γ ∷ A ⊢ c ∶ C) :\n  Γ ∷ B ⊢ c ∶ C :=", "ground_truth_proof": ":= by\n  cases wtWf h with | cons wfΓ hA =>\n  let wfΓB := Wf.cons wfΓ hB\n  rw [← substId c, ← substId C]\n  refine wtMorph ?_ wfΓB h\n  intro x A mem; rw [substId]; cases mem\n  case here =>\n    exact Wt.conv\n      (convEqv (convRename succ (convSym (eqvConv e))))\n      (Wt.var wfΓB In.here)\n      (wtWeaken wfΓ hB hA)\n  case there mem => exact Wt.var wfΓB (In.there mem)", "nesting_depth": 9, "transitive_dep_count": 71, "subset_aristotle": false, "category": "Type systems"}
{"id": 432, "thm_name": "wtAbs", "thm_stmt": "theorem wtAbs {Γ} {b A B : Term} (v : Value b) (h : Γ ⊢ b ∶ pi A B) : ∃ a' b', b = abs a' b'", "lean_root": "TTBFL", "rel_path": "src/safety.lean", "imports": ["import «src».typing", "import src.reduction", "import src.typing"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}, {"name": "infix:40 (priority := 1001) \"≈\" => Eqv", "content": "infix:40 (priority := 1001) \"≈\" => Eqv"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "wtfLvlInv", "content": "theorem wtfLvlInv {Γ a 𝒰'}\n  (h : Γ ⊢ lvl a ∶ 𝒰') :\n  ∃ b k, Γ ⊢ a ∶ lvl b ∧ 𝒰 k ≈ 𝒰'"}, {"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}, {"name": "wtfMtyInv", "content": "theorem wtfMtyInv {Γ 𝒰'}\n  (h : Γ ⊢ mty ∶ 𝒰') :\n  ∃ k, 𝒰 k ≈ 𝒰'"}, {"name": "wtfPiInv𝒰", "content": "theorem wtfPiInv𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, 𝒰 j ≈ 𝒰'"}, {"name": "wtfPiInvA𝒰", "content": "theorem wtfPiInvA𝒰 {Γ A B 𝒰'}\n  (h : Γ ⊢ pi A B ∶ 𝒰') :\n  ∃ j, Γ ⊢ A ∶ 𝒰 j ∧ 𝒰 j ≈ 𝒰'"}, {"name": "wtfAbsInv", "content": "theorem wtfAbsInv {Γ A' b C}\n  (h : Γ ⊢ abs A' b ∶ C) :\n  ∃ A B, Γ ∷ A ⊢ b ∶ B ∧ A ≈ A' ∧ pi A B ≈ C"}, {"name": "wtfLofInv", "content": "theorem wtfLofInv {Γ j 𝒰'}\n  (h : Γ ⊢ lof j ∶ 𝒰') :\n  ∃ k, lvl k ≈ 𝒰'"}, {"name": "wtf𝒰Inv", "content": "theorem wtf𝒰Inv {Γ j 𝒰'}\n  (h : Γ ⊢ 𝒰 j ∶ 𝒰') :\n  ∃ k, 𝒰 k ≈ 𝒰'"}, {"name": "conv𝒰Pi", "content": "theorem conv𝒰Pi {c a b} : ¬ 𝒰 c ⇔ pi a b"}, {"name": "parsPiInv", "content": "theorem parsPiInv {a b c} (r : pi a b ⇒⋆ c) : ∃ a' b', c = pi a' b' ∧ a ⇒⋆ a' ∧ b ⇒⋆ b'"}, {"name": "pars𝒰Inv", "content": "theorem pars𝒰Inv {a b} (r : 𝒰 a ⇒⋆ b) : ∃ a', b = 𝒰 a' ∧ a ⇒⋆ a'"}, {"name": "convLvlPi", "content": "theorem convLvlPi {a b k} : ¬ lvl k ⇔ pi a b"}, {"name": "parsLvlInv", "content": "theorem parsLvlInv {i b} (r : lvl i ⇒⋆ b) : ∃ j, b = lvl j ∧ i ⇒⋆ j"}], "used_local_defs": [{"name": "Value", "content": "inductive Value : Term → Type where\n  | 𝒰 {k} : Value (𝒰 k)\n  | pi {a b} : Value (pi a b)\n  | abs {a b} : Value (abs a b)\n  | mty : Value mty\n  | lvl {k} : Value (lvl k)\n  | lof {k} : Value (lof k)"}, {"name": "valueType", "content": "@[simp] \ndef valueType {a} (A : Term) : Value a → Prop\n  | Value.𝒰 | Value.pi | Value.mty | Value.lvl => ∃ k, 𝒰 k ≈ A\n  | Value.abs => ∃ B C, pi B C ≈ A\n  | Value.lof => ∃ k, lvl k ≈ A"}], "used_local_lemmas": [{"name": "wtValue", "content": "theorem wtValue {Γ} {a A B : Term} (h : Γ ⊢ a ∶ A) (e : A ≈ B) : (v : Value a) → valueType B v\n  | Value.𝒰 => let ⟨_, e𝒰⟩"}], "local_ctx": "import «src».typing\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nnotation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ\n\ninductive Value : Term → Type where\n  | 𝒰 {k} : Value (𝒰 k)\n  | pi {a b} : Value (pi a b)\n  | abs {a b} : Value (abs a b)\n  | mty : Value mty\n  | lvl {k} : Value (lvl k)\n  | lof {k} : Value (lof k)\n\nsection\n\nend\n\ninfix:40 \"⇒β\" => CBN\n\ninfix:40 \"⇒β⋆\" => CBNs\n\n@[simp] \ndef valueType {a} (A : Term) : Value a → Prop\n  | Value.𝒰 | Value.pi | Value.mty | Value.lvl => ∃ k, 𝒰 k ≈ A\n  | Value.abs => ∃ B C, pi B C ≈ A\n  | Value.lof => ∃ k, lvl k ≈ A", "target_theorem": "theorem wtAbs {Γ} {b A B : Term} (v : Value b) (h : Γ ⊢ b ∶ pi A B) : ∃ a' b', b = abs a' b' :=", "ground_truth_proof": ":= by\n  generalize e : pi A B = T at h\n  induction h\n  all_goals try first | subst e | injection e\n  case var | app | exf => contradiction\n  case abs => exact ⟨_, _, rfl⟩\n  case conv h v epi _ _ =>\n    let _e := wtValue h epi v\n    cases v <;> let ⟨_, e⟩ := _e\n    case 𝒰 | pi | mty | lvl => cases conv𝒰Pi (eqvConv e)\n    case abs => exact ⟨_, _, rfl⟩\n    case lof => cases convLvlPi (eqvConv e)", "nesting_depth": 5, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Type systems"}
{"id": 433, "thm_name": "interpsBwdsP", "thm_stmt": "theorem interpsBwdsP {i a x y P} (r : x ⇒⋆ y) (h : ⟦ a ⟧ i ↘ P) : P y → P x", "lean_root": "TTBFL", "rel_path": "src/candidates.lean", "imports": ["import «src».normal", "import src.reduction", "import src.normal"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Or.inl", "module": "Init.Prelude"}, {"name": "Or.inr", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P", "content": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P"}, {"name": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "content": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "nf", "content": "@[simp]\ndef nf : Term → Prop\n  | 𝒰 a => nf a\n  | pi a b => nf a ∧ nf b\n  | abs a b => nf a ∧ nf b\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | lvl a => nf a\n  | _ => True"}, {"name": "ne", "content": "@[simp]\ndef ne : Term → Prop\n  | var _ => True\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | _ => False"}, {"name": "LevelClass", "content": "class LevelClass where\n  L : Type\n  lc : LevelClasses L"}, {"name": "wne", "content": "@[simp] def wne (a : Term) : Prop := ∃ b, ne b ∧ a ⇒⋆ b"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "wneBwds", "content": "theorem wneBwds {a b} (r : a ⇒⋆ b) : wne b → wne a"}, {"name": "parsApp", "content": "theorem parsApp {a a' b b'} (rb : b ⇒⋆ b') (ra : a ⇒⋆ a') : app b a ⇒⋆ app b' a'"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "parsTrans", "content": "theorem parsTrans {a b c} (r₁ : a ⇒⋆ b) (r₂ : b ⇒⋆ c) : a ⇒⋆ c"}], "used_local_defs": [{"name": "Interp", "content": "inductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | ne a : ne a → Interp i I a wne\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty wne\n  | lvl b : nf b → Interp i I (lvl b)\n    (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P"}, {"name": "Interps", "content": "def Interps (i : lc.L) : Term → (Term → Prop) → Prop :=\n  Interp i (λ j _ a ↦ ∃ P, Interps j a P)\ntermination_by i"}], "used_local_lemmas": [{"name": "interpsBwd", "content": "theorem interpsBwd {i a b P} (r : a ⇒ b) (h : ⟦ b ⟧ i ↘ P) : ⟦ a ⟧ i ↘ P"}, {"name": "interpsBwds", "content": "theorem interpsBwds {i a b P} (r : a ⇒⋆ b) (h : ⟦ b ⟧ i ↘ P) : ⟦ a ⟧ i ↘ P"}], "local_ctx": "import «src».normal\n\nopen Term\n\nvariable [lc : LevelClass]\n\ninductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | ne a : ne a → Interp i I a wne\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty wne\n  | lvl b : nf b → Interp i I (lvl b)\n    (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P\n\ndef Interps (i : lc.L) : Term → (Term → Prop) → Prop :=\n  Interp i (λ j _ a ↦ ∃ P, Interps j a P)\ntermination_by i\n\nnotation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "target_theorem": "theorem interpsBwdsP {i a x y P} (r : x ⇒⋆ y) (h : ⟦ a ⟧ i ↘ P) : P y → P x :=", "ground_truth_proof": ":= by\n  unfold Interps at h; induction h generalizing x y\n  case ne => exact wneBwds r\n  case pi ihb =>\n    intro h x Pb Pax PfxPb\n    exact ihb x Pb PfxPb (parsApp r (Pars.refl x)) (h x Pb Pax PfxPb)\n  case 𝒰 => exact λ ⟨P, h⟩ ↦ ⟨P, interpsBwds r h⟩\n  case mty => exact wneBwds r\n  case lvl =>\n    intro Py; rcases Py with ⟨j, k, lt, rj, rk⟩ | wney\n    . exact Or.inl ⟨j, k, lt, parsTrans r rj, rk⟩\n    . exact Or.inr (wneBwds r wney)\n  case step ih => exact ih r", "nesting_depth": 4, "transitive_dep_count": 27, "subset_aristotle": false, "category": "Type systems"}
{"id": 434, "thm_name": "interpsBwdsP", "thm_stmt": "theorem interpsBwdsP {i a x y P} (r : x ⇒⋆ y) (h : ⟦ a ⟧ i ↘ P) : P y → P x", "lean_root": "TTBFL", "rel_path": "src/semantics.lean", "imports": ["import src.reduction", "import «src».reduction"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}], "used_repo_defs": [{"name": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P", "content": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P"}, {"name": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "content": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "LevelClass", "content": "class LevelClass where\n  L : Type\n  lc : LevelClasses L"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "parsApp", "content": "theorem parsApp {a a' b b'} (rb : b ⇒⋆ b') (ra : a ⇒⋆ a') : app b a ⇒⋆ app b' a'"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "parsTrans", "content": "theorem parsTrans {a b c} (r₁ : a ⇒⋆ b) (r₂ : b ⇒⋆ c) : a ⇒⋆ c"}], "used_local_defs": [{"name": "Interp", "content": "inductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty (λ _ ↦ False)\n  | lvl k : Interp i I (lvl (lof k)) (λ a ↦ ∃ j, a ⇒⋆ lof j ∧ j < k)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P"}, {"name": "Interps", "content": "def Interps (i : lc.L) : Term → (Term → Prop) → Prop :=\n  Interp i (λ j _ a ↦ ∃ P, Interps j a P)\ntermination_by i"}], "used_local_lemmas": [{"name": "interpsBwd", "content": "theorem interpsBwd {i a b P} (r : a ⇒ b) (h : ⟦ b ⟧ i ↘ P) : ⟦ a ⟧ i ↘ P"}, {"name": "interpsBwds", "content": "theorem interpsBwds {i a b P} (r : a ⇒⋆ b) (h : ⟦ b ⟧ i ↘ P) : ⟦ a ⟧ i ↘ P"}], "local_ctx": "import «src».reduction\n\nopen Term\n\nvariable [lc : LevelClass]\n\ninductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty (λ _ ↦ False)\n  | lvl k : Interp i I (lvl (lof k)) (λ a ↦ ∃ j, a ⇒⋆ lof j ∧ j < k)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P\n\ndef Interps (i : lc.L) : Term → (Term → Prop) → Prop :=\n  Interp i (λ j _ a ↦ ∃ P, Interps j a P)\ntermination_by i\n\nnotation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "target_theorem": "theorem interpsBwdsP {i a x y P} (r : x ⇒⋆ y) (h : ⟦ a ⟧ i ↘ P) : P y → P x :=", "ground_truth_proof": ":= by\n  unfold Interps at h; induction h generalizing x y\n  case pi ihb =>\n    intro h x Pb Pax PfxPb\n    exact ihb x Pb PfxPb (parsApp r (Pars.refl x)) (h x Pb Pax PfxPb)\n  case 𝒰 => exact λ ⟨P, h⟩ ↦ ⟨P, interpsBwds r h⟩\n  case mty => simp\n  case lvl =>\n    intro ⟨j, r₂, lt⟩\n    exact ⟨j, parsTrans r r₂, lt⟩\n  case step ih => exact ih r", "nesting_depth": 3, "transitive_dep_count": 19, "subset_aristotle": false, "category": "Type systems"}
{"id": 435, "thm_name": "wtfAbsInv", "thm_stmt": "theorem wtfAbsInv {Γ A' b C}\n  (h : Γ ⊢ abs A' b ∶ C) :\n  ∃ A B, Γ ∷ A ⊢ b ∶ B ∧ A ≈ A' ∧ pi A B ≈ C", "lean_root": "TTBFL", "rel_path": "src/typing.lean", "imports": ["import src.reduction", "import «src».reduction"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat.sub", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "taka", "content": "@[simp]\ndef taka : Term → Term\n  | 𝒰 a => 𝒰 (taka a)\n  | pi a b => pi (taka a) (taka b)\n  | abs a b => abs (taka a) (taka b)\n  | app b a => match b with\n    | abs _ b => subst (taka a +: var) (taka b)\n    | b => app (taka b) (taka a)\n  | exf a b => exf (taka a) (taka b)\n  | lvl a => lvl (taka a)\n  | t => t"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}], "lib_lemmas": [{"name": "refl", "module": "Mathlib.Order.Defs.Unbundled"}, {"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "conv𝒰", "content": "theorem conv𝒰 {a a'} : a ⇔ a' → 𝒰 a ⇔ 𝒰 a'"}, {"name": "pars𝒰", "content": "theorem pars𝒰 {a a'} (r : a ⇒⋆ a') : 𝒰 a ⇒⋆ 𝒰 a'"}, {"name": "convAbs", "content": "theorem convAbs {a a' b b'} : a ⇔ a' → b ⇔ b' → abs a b ⇔ abs a' b'"}, {"name": "parsAbs", "content": "theorem parsAbs {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : abs a b ⇒⋆ abs a' b'"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "convPi", "content": "theorem convPi {a a' b b'} : a ⇔ a' → b ⇔ b' → pi a b ⇔ pi a' b'"}, {"name": "parsPi", "content": "theorem parsPi {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : pi a b ⇒⋆ pi a' b'"}, {"name": "convApp", "content": "theorem convApp {b b' a a'} : b ⇔ b' → a ⇔ a' → app b a ⇔ app b' a'"}, {"name": "parsApp", "content": "theorem parsApp {a a' b b'} (rb : b ⇒⋆ b') (ra : a ⇒⋆ a') : app b a ⇒⋆ app b' a'"}, {"name": "convSym", "content": "theorem convSym {a b} : a ⇔ b → b ⇔ a"}, {"name": "convTrans", "content": "theorem convTrans {a b c} : a ⇔ b → b ⇔ c → a ⇔ c"}, {"name": "confluence", "content": "theorem confluence {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒⋆ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diacon", "content": "theorem diacon {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diamond", "content": "theorem diamond {a b c} (r₁ : a ⇒ b) (r₂ : a ⇒ c) : ∃ d, b ⇒ d ∧ c ⇒ d"}, {"name": "parTaka", "content": "theorem parTaka {a b} (r : a ⇒ b) : b ⇒ taka a"}, {"name": "parCong", "content": "theorem parCong {a a' b b'} (ra : a ⇒ a') (rb : b ⇒ b') : subst (a +: var) b ⇒ subst (a' +: var) b'"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "parPars", "content": "theorem parPars {a b} (r : a ⇒ b) : a ⇒⋆ b"}, {"name": "parsTrans", "content": "theorem parsTrans {a b c} (r₁ : a ⇒⋆ b) (r₂ : b ⇒⋆ c) : a ⇒⋆ c"}, {"name": "convExf", "content": "theorem convExf {a a' b b'} : a ⇔ a' → b ⇔ b' → exf a b ⇔ exf a' b'"}, {"name": "parsExf", "content": "theorem parsExf {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : exf a b ⇒⋆ exf a' b'"}, {"name": "convRefl", "content": "theorem convRefl {a} : a ⇔ a"}, {"name": "parConv", "content": "theorem parConv {a b} (r : a ⇒ b) : a ⇔ b"}, {"name": "parsConv", "content": "theorem parsConv {a b} (r : a ⇒⋆ b) : a ⇔ b"}, {"name": "convLvl", "content": "theorem convLvl {a a'} : a ⇔ a' → lvl a ⇔ lvl a'"}, {"name": "parsLvl", "content": "theorem parsLvl {a a'} (r : a ⇒⋆ a') : lvl a ⇒⋆ lvl a'"}, {"name": "conv𝒰Pi", "content": "theorem conv𝒰Pi {c a b} : ¬ 𝒰 c ⇔ pi a b"}, {"name": "parsPiInv", "content": "theorem parsPiInv {a b c} (r : pi a b ⇒⋆ c) : ∃ a' b', c = pi a' b' ∧ a ⇒⋆ a' ∧ b ⇒⋆ b'"}, {"name": "pars𝒰Inv", "content": "theorem pars𝒰Inv {a b} (r : 𝒰 a ⇒⋆ b) : ∃ a', b = 𝒰 a' ∧ a ⇒⋆ a'"}, {"name": "convLvlPi", "content": "theorem convLvlPi {a b k} : ¬ lvl k ⇔ pi a b"}, {"name": "parsLvlInv", "content": "theorem parsLvlInv {i b} (r : lvl i ⇒⋆ b) : ∃ j, b = lvl j ∧ i ⇒⋆ j"}], "used_local_defs": [{"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}], "used_local_lemmas": [{"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}], "local_ctx": "import «src».reduction\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nsection\n\ninductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c\n\nend\n\ninfix:40 (priority := 1001) \"≈\" => Eqv\n\nsection\n\ninductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A\n\ninductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k\n\nend\n\nend\n\nnotation:40 \"⊢\" Γ:40 => Wf Γ\n\nnotation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "target_theorem": "theorem wtfAbsInv {Γ A' b C}\n  (h : Γ ⊢ abs A' b ∶ C) :\n  ∃ A B, Γ ∷ A ⊢ b ∶ B ∧ A ≈ A' ∧ pi A B ≈ C :=", "ground_truth_proof": ":= by\n  generalize e : abs A' b = t at h\n  induction h\n  all_goals injections <;> subst_eqs <;> specialize_rfls\n  case abs hb _ => exact ⟨_, _, hb, Eqv.refl, Eqv.refl⟩\n  case trans ih =>\n    let ⟨_, _, _, _, eC⟩ := ih\n    cases convLvlPi (convSym (eqvConv eC))\n  case conv DC _ _ _ ih =>\n    let ⟨A, B, hb, AA', ABD⟩ := ih\n    exact ⟨A, B, hb, AA', Eqv.trans ABD DC⟩\n  case sub ih =>\n    let ⟨_, _, _, _, e⟩ := ih\n    cases conv𝒰Pi (convSym (eqvConv e))", "nesting_depth": 13, "transitive_dep_count": 62, "subset_aristotle": false, "category": "Type systems"}
{"id": 436, "thm_name": "interpLvlInv", "thm_stmt": "theorem interpLvlInv {i I b P} (h : ⟦ lvl b ⟧ i , I ↘ P) :\n  wnf b ∧ P = (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)", "lean_root": "TTBFL", "rel_path": "src/candidates.lean", "imports": ["import «src».normal", "import src.reduction", "import src.normal"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Or", "module": "Init.Prelude"}, {"name": "Or.inl", "module": "Init.Prelude"}, {"name": "Or.inr", "module": "Init.Prelude"}, {"name": "propext", "module": "Init.Core"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P", "content": "notation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P"}, {"name": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "content": "notation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "nf", "content": "@[simp]\ndef nf : Term → Prop\n  | 𝒰 a => nf a\n  | pi a b => nf a ∧ nf b\n  | abs a b => nf a ∧ nf b\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | lvl a => nf a\n  | _ => True"}, {"name": "ne", "content": "@[simp]\ndef ne : Term → Prop\n  | var _ => True\n  | app b a => ne b ∧ nf a\n  | exf a b => nf a ∧ ne b\n  | _ => False"}, {"name": "LevelClass", "content": "class LevelClass where\n  L : Type\n  lc : LevelClasses L"}, {"name": "wne", "content": "@[simp] def wne (a : Term) : Prop := ∃ b, ne b ∧ a ⇒⋆ b"}, {"name": "wnf", "content": "@[simp] def wnf (a : Term) : Prop := ∃ b, nf b ∧ a ⇒⋆ b"}, {"name": "taka", "content": "@[simp]\ndef taka : Term → Term\n  | 𝒰 a => 𝒰 (taka a)\n  | pi a b => pi (taka a) (taka b)\n  | abs a b => abs (taka a) (taka b)\n  | app b a => match b with\n    | abs _ b => subst (taka a +: var) (taka b)\n    | b => app (taka b) (taka a)\n  | exf a b => exf (taka a) (taka b)\n  | lvl a => lvl (taka a)\n  | t => t"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "diacon", "content": "theorem diacon {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diamond", "content": "theorem diamond {a b c} (r₁ : a ⇒ b) (r₂ : a ⇒ c) : ∃ d, b ⇒ d ∧ c ⇒ d"}, {"name": "parTaka", "content": "theorem parTaka {a b} (r : a ⇒ b) : b ⇒ taka a"}, {"name": "parCong", "content": "theorem parCong {a a' b b'} (ra : a ⇒ a') (rb : b ⇒ b') : subst (a +: var) b ⇒ subst (a' +: var) b'"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "parPars", "content": "theorem parPars {a b} (r : a ⇒ b) : a ⇒⋆ b"}, {"name": "parsLofInv", "content": "theorem parsLofInv {j b} (r : lof j ⇒⋆ b) : b = lof j"}, {"name": "wnfBwds", "content": "theorem wnfBwds {a b} (r : a ⇒⋆ b) : wnf b → wnf a"}, {"name": "nfWnf", "content": "theorem nfWnf {a} (nfa : nf a) : wnf a"}], "used_local_defs": [{"name": "Interp", "content": "inductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | ne a : ne a → Interp i I a wne\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty wne\n  | lvl b : nf b → Interp i I (lvl b)\n    (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P"}], "used_local_lemmas": [{"name": "interpLvlEq", "content": "theorem interpLvlEq {b c} (r : b ⇒ c) :\n  (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a) =\n  (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ c ⇒⋆ lof k) ∨ wne a)"}], "local_ctx": "import «src».normal\n\nopen Term\n\nvariable [lc : LevelClass]\n\ninductive Interp (i : lc.L) (I : ∀ j, j < i → Term → Prop) : Term → (Term → Prop) → Prop where\n  | ne a : ne a → Interp i I a wne\n  | pi a b Pa (Pf : Term → (Term → Prop) → Prop) :\n    Interp i I a Pa →\n    (∀ x, Pa x → ∃ Pb, Pf x Pb) →\n    (∀ x Pb, Pf x Pb → Interp i I (subst (x +: var) b) Pb) →\n    Interp i I (pi a b) (λ f ↦ ∀ x Pb, Pa x → Pf x Pb → Pb (app f x))\n  | 𝒰 j (lt : j < i) : Interp i I (𝒰 (lof j)) (I j lt)\n  | mty : Interp i I mty wne\n  | lvl b : nf b → Interp i I (lvl b)\n    (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a)\n  | step a b P :\n    a ⇒ b →\n    Interp i I b P →\n    Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \",\" I \"↘\" P => Interp i I a P\n\nnotation:40 \"⟦\" a \"⟧\" i \"↘\" P => Interps i a P", "target_theorem": "theorem interpLvlInv {i I b P} (h : ⟦ lvl b ⟧ i , I ↘ P) :\n  wnf b ∧ P = (λ a ↦ (∃ j k, j < k ∧ a ⇒⋆ lof j ∧ b ⇒⋆ lof k) ∨ wne a) :=", "ground_truth_proof": ":= by\n  generalize e : lvl b = c at h\n  induction h generalizing b\n  case ne => subst e; contradiction\n  case lvl nfb => injection e with e; subst e; exact ⟨nfWnf nfb, rfl⟩\n  case step r _ ih =>\n    subst e; let (Par.lvl r₁) := r\n    let ⟨nfc, e⟩ := ih rfl; subst e\n    rw [interpLvlEq r₁]\n    exact ⟨wnfBwds (parPars r₁) nfc, rfl⟩\n  all_goals contradiction", "nesting_depth": 11, "transitive_dep_count": 41, "subset_aristotle": false, "category": "Type systems"}
{"id": 437, "thm_name": "wtfLvlInv", "thm_stmt": "theorem wtfLvlInv {Γ a 𝒰'}\n  (h : Γ ⊢ lvl a ∶ 𝒰') :\n  ∃ b k, Γ ⊢ a ∶ lvl b ∧ 𝒰 k ≈ 𝒰'", "lean_root": "TTBFL", "rel_path": "src/typing.lean", "imports": ["import src.reduction", "import «src».reduction"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat.sub", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "taka", "content": "@[simp]\ndef taka : Term → Term\n  | 𝒰 a => 𝒰 (taka a)\n  | pi a b => pi (taka a) (taka b)\n  | abs a b => abs (taka a) (taka b)\n  | app b a => match b with\n    | abs _ b => subst (taka a +: var) (taka b)\n    | b => app (taka b) (taka a)\n  | exf a b => exf (taka a) (taka b)\n  | lvl a => lvl (taka a)\n  | t => t"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}], "lib_lemmas": [{"name": "refl", "module": "Mathlib.Order.Defs.Unbundled"}, {"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "conv𝒰", "content": "theorem conv𝒰 {a a'} : a ⇔ a' → 𝒰 a ⇔ 𝒰 a'"}, {"name": "pars𝒰", "content": "theorem pars𝒰 {a a'} (r : a ⇒⋆ a') : 𝒰 a ⇒⋆ 𝒰 a'"}, {"name": "convAbs", "content": "theorem convAbs {a a' b b'} : a ⇔ a' → b ⇔ b' → abs a b ⇔ abs a' b'"}, {"name": "parsAbs", "content": "theorem parsAbs {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : abs a b ⇒⋆ abs a' b'"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "convPi", "content": "theorem convPi {a a' b b'} : a ⇔ a' → b ⇔ b' → pi a b ⇔ pi a' b'"}, {"name": "parsPi", "content": "theorem parsPi {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : pi a b ⇒⋆ pi a' b'"}, {"name": "convApp", "content": "theorem convApp {b b' a a'} : b ⇔ b' → a ⇔ a' → app b a ⇔ app b' a'"}, {"name": "parsApp", "content": "theorem parsApp {a a' b b'} (rb : b ⇒⋆ b') (ra : a ⇒⋆ a') : app b a ⇒⋆ app b' a'"}, {"name": "convSym", "content": "theorem convSym {a b} : a ⇔ b → b ⇔ a"}, {"name": "convTrans", "content": "theorem convTrans {a b c} : a ⇔ b → b ⇔ c → a ⇔ c"}, {"name": "confluence", "content": "theorem confluence {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒⋆ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diacon", "content": "theorem diacon {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diamond", "content": "theorem diamond {a b c} (r₁ : a ⇒ b) (r₂ : a ⇒ c) : ∃ d, b ⇒ d ∧ c ⇒ d"}, {"name": "parTaka", "content": "theorem parTaka {a b} (r : a ⇒ b) : b ⇒ taka a"}, {"name": "parCong", "content": "theorem parCong {a a' b b'} (ra : a ⇒ a') (rb : b ⇒ b') : subst (a +: var) b ⇒ subst (a' +: var) b'"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "parPars", "content": "theorem parPars {a b} (r : a ⇒ b) : a ⇒⋆ b"}, {"name": "parsTrans", "content": "theorem parsTrans {a b c} (r₁ : a ⇒⋆ b) (r₂ : b ⇒⋆ c) : a ⇒⋆ c"}, {"name": "convExf", "content": "theorem convExf {a a' b b'} : a ⇔ a' → b ⇔ b' → exf a b ⇔ exf a' b'"}, {"name": "parsExf", "content": "theorem parsExf {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : exf a b ⇒⋆ exf a' b'"}, {"name": "convRefl", "content": "theorem convRefl {a} : a ⇔ a"}, {"name": "parConv", "content": "theorem parConv {a b} (r : a ⇒ b) : a ⇔ b"}, {"name": "parsConv", "content": "theorem parsConv {a b} (r : a ⇒⋆ b) : a ⇔ b"}, {"name": "convLvl", "content": "theorem convLvl {a a'} : a ⇔ a' → lvl a ⇔ lvl a'"}, {"name": "parsLvl", "content": "theorem parsLvl {a a'} (r : a ⇒⋆ a') : lvl a ⇒⋆ lvl a'"}, {"name": "convLvl𝒰", "content": "theorem convLvl𝒰 {j k} : ¬ lvl j ⇔ 𝒰 k"}, {"name": "parsLvlInv", "content": "theorem parsLvlInv {i b} (r : lvl i ⇒⋆ b) : ∃ j, b = lvl j ∧ i ⇒⋆ j"}, {"name": "pars𝒰Inv", "content": "theorem pars𝒰Inv {a b} (r : 𝒰 a ⇒⋆ b) : ∃ a', b = 𝒰 a' ∧ a ⇒⋆ a'"}], "used_local_defs": [{"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}], "used_local_lemmas": [{"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}], "local_ctx": "import «src».reduction\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nsection\n\ninductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c\n\nend\n\ninfix:40 (priority := 1001) \"≈\" => Eqv\n\nsection\n\ninductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A\n\ninductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k\n\nend\n\nend\n\nnotation:40 \"⊢\" Γ:40 => Wf Γ\n\nnotation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "target_theorem": "theorem wtfLvlInv {Γ a 𝒰'}\n  (h : Γ ⊢ lvl a ∶ 𝒰') :\n  ∃ b k, Γ ⊢ a ∶ lvl b ∧ 𝒰 k ≈ 𝒰' :=", "ground_truth_proof": ":= by\n  generalize e : lvl a = t at h\n  induction h\n  all_goals injections <;> subst_eqs <;> specialize_rfls\n  case lvl ha _ => exact ⟨_, _, ha, Eqv.refl⟩\n  case trans ih =>\n    let ⟨_, _, _, e⟩ := ih\n    cases convLvl𝒰 (convSym (eqvConv e))\n  case conv e₁ _ _ _ ih =>\n    let ⟨b, _, ha, e₂⟩ := ih\n    exact ⟨b, _, ha, Eqv.trans e₂ e₁⟩\n  case sub ih =>\n    let ⟨b, _, ha, _⟩ := ih\n    exact ⟨b, _, ha, Eqv.refl⟩", "nesting_depth": 13, "transitive_dep_count": 60, "subset_aristotle": false, "category": "Type systems"}
{"id": 438, "thm_name": "wtfMtyInv", "thm_stmt": "theorem wtfMtyInv {Γ 𝒰'}\n  (h : Γ ⊢ mty ∶ 𝒰') :\n  ∃ k, 𝒰 k ≈ 𝒰'", "lean_root": "TTBFL", "rel_path": "src/typing.lean", "imports": ["import src.reduction", "import «src».reduction"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "taka", "content": "@[simp]\ndef taka : Term → Term\n  | 𝒰 a => 𝒰 (taka a)\n  | pi a b => pi (taka a) (taka b)\n  | abs a b => abs (taka a) (taka b)\n  | app b a => match b with\n    | abs _ b => subst (taka a +: var) (taka b)\n    | b => app (taka b) (taka a)\n  | exf a b => exf (taka a) (taka b)\n  | lvl a => lvl (taka a)\n  | t => t"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}], "lib_lemmas": [{"name": "refl", "module": "Mathlib.Order.Defs.Unbundled"}, {"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "conv𝒰", "content": "theorem conv𝒰 {a a'} : a ⇔ a' → 𝒰 a ⇔ 𝒰 a'"}, {"name": "pars𝒰", "content": "theorem pars𝒰 {a a'} (r : a ⇒⋆ a') : 𝒰 a ⇒⋆ 𝒰 a'"}, {"name": "convAbs", "content": "theorem convAbs {a a' b b'} : a ⇔ a' → b ⇔ b' → abs a b ⇔ abs a' b'"}, {"name": "parsAbs", "content": "theorem parsAbs {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : abs a b ⇒⋆ abs a' b'"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "convPi", "content": "theorem convPi {a a' b b'} : a ⇔ a' → b ⇔ b' → pi a b ⇔ pi a' b'"}, {"name": "parsPi", "content": "theorem parsPi {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : pi a b ⇒⋆ pi a' b'"}, {"name": "convApp", "content": "theorem convApp {b b' a a'} : b ⇔ b' → a ⇔ a' → app b a ⇔ app b' a'"}, {"name": "parsApp", "content": "theorem parsApp {a a' b b'} (rb : b ⇒⋆ b') (ra : a ⇒⋆ a') : app b a ⇒⋆ app b' a'"}, {"name": "convSym", "content": "theorem convSym {a b} : a ⇔ b → b ⇔ a"}, {"name": "convTrans", "content": "theorem convTrans {a b c} : a ⇔ b → b ⇔ c → a ⇔ c"}, {"name": "confluence", "content": "theorem confluence {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒⋆ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diacon", "content": "theorem diacon {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diamond", "content": "theorem diamond {a b c} (r₁ : a ⇒ b) (r₂ : a ⇒ c) : ∃ d, b ⇒ d ∧ c ⇒ d"}, {"name": "parTaka", "content": "theorem parTaka {a b} (r : a ⇒ b) : b ⇒ taka a"}, {"name": "parCong", "content": "theorem parCong {a a' b b'} (ra : a ⇒ a') (rb : b ⇒ b') : subst (a +: var) b ⇒ subst (a' +: var) b'"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "parPars", "content": "theorem parPars {a b} (r : a ⇒ b) : a ⇒⋆ b"}, {"name": "parsTrans", "content": "theorem parsTrans {a b c} (r₁ : a ⇒⋆ b) (r₂ : b ⇒⋆ c) : a ⇒⋆ c"}, {"name": "convExf", "content": "theorem convExf {a a' b b'} : a ⇔ a' → b ⇔ b' → exf a b ⇔ exf a' b'"}, {"name": "parsExf", "content": "theorem parsExf {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : exf a b ⇒⋆ exf a' b'"}, {"name": "convRefl", "content": "theorem convRefl {a} : a ⇔ a"}, {"name": "parConv", "content": "theorem parConv {a b} (r : a ⇒ b) : a ⇔ b"}, {"name": "parsConv", "content": "theorem parsConv {a b} (r : a ⇒⋆ b) : a ⇔ b"}, {"name": "convLvl", "content": "theorem convLvl {a a'} : a ⇔ a' → lvl a ⇔ lvl a'"}, {"name": "parsLvl", "content": "theorem parsLvl {a a'} (r : a ⇒⋆ a') : lvl a ⇒⋆ lvl a'"}, {"name": "convLvl𝒰", "content": "theorem convLvl𝒰 {j k} : ¬ lvl j ⇔ 𝒰 k"}, {"name": "parsLvlInv", "content": "theorem parsLvlInv {i b} (r : lvl i ⇒⋆ b) : ∃ j, b = lvl j ∧ i ⇒⋆ j"}, {"name": "pars𝒰Inv", "content": "theorem pars𝒰Inv {a b} (r : 𝒰 a ⇒⋆ b) : ∃ a', b = 𝒰 a' ∧ a ⇒⋆ a'"}], "used_local_defs": [{"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}], "used_local_lemmas": [{"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}], "local_ctx": "import «src».reduction\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nsection\n\ninductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c\n\nend\n\ninfix:40 (priority := 1001) \"≈\" => Eqv\n\nsection\n\ninductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A\n\ninductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k\n\nend\n\nend\n\nnotation:40 \"⊢\" Γ:40 => Wf Γ\n\nnotation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "target_theorem": "theorem wtfMtyInv {Γ 𝒰'}\n  (h : Γ ⊢ mty ∶ 𝒰') :\n  ∃ k, 𝒰 k ≈ 𝒰' :=", "ground_truth_proof": ":= by\n  generalize e : mty = t at h\n  induction h\n  all_goals injections <;> subst_eqs <;> specialize_rfls\n  case mty | sub => exact ⟨_, Eqv.refl⟩\n  case trans ih =>\n    let ⟨_, e⟩ := ih\n    cases convLvl𝒰 (convSym (eqvConv e))\n  case conv e₁ _ _ _ ih =>\n    let ⟨_, e₂⟩ := ih\n    exact ⟨_, Eqv.trans e₂ e₁⟩", "nesting_depth": 13, "transitive_dep_count": 59, "subset_aristotle": false, "category": "Type systems"}
{"id": 439, "thm_name": "parsAntirenaming", "thm_stmt": "theorem parsAntirenaming {ξ a b'} (r : rename ξ a ⇒⋆ b') : ∃ b, b' = rename ξ b ∧ a ⇒⋆ b", "lean_root": "TTBFL", "rel_path": "src/reduction.lean", "imports": ["import src.syntactics", "import «src».syntactics", "import «src».tactics"], "used_lib_defs": [{"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "substRename", "content": "def substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )"}, {"name": "renameSubst", "content": "def renameSubst ξ σ : ∀ s, rename ξ (subst σ s) = subst (rename ξ ∘ σ) s :=\n  renameSubst' _ _ (rename ξ ∘ σ) (by admit /- proof elided -/\n  )"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "renameDist", "content": "theorem renameDist ξ a s : subst (rename ξ a +: var) (rename (lift ξ) s) = rename ξ (subst (a +: var) s)"}, {"name": "substExt", "content": "theorem substExt σ τ (h : ∀ x, σ x = τ x) : ∀ s, subst σ s = subst τ s"}, {"name": "upExt", "content": "theorem upExt σ τ (h : ∀ x, σ x = τ x) : ∀ x, (⇑ σ) x = (⇑ τ) x"}], "used_local_defs": [{"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}], "used_local_lemmas": [{"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "antirenaming", "content": "theorem antirenaming {ξ a b'} (r : rename ξ a ⇒ b') : ∃ b, b' = rename ξ b ∧ a ⇒ b"}], "local_ctx": "import «src».tactics\n\nimport «src».syntactics\n\nopen Term\n\nvariable [LevelClass]\n\nsection\n\ninductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k\n\nend\n\ninfix:40 \"⇒\" => Par\n\nsection\n\ninductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c\n\nend\n\ninfix:40 \"⇒⋆\" => Pars\n\nopen Pars", "target_theorem": "theorem parsAntirenaming {ξ a b'} (r : rename ξ a ⇒⋆ b') : ∃ b, b' = rename ξ b ∧ a ⇒⋆ b :=", "ground_truth_proof": ":= by\n  generalize e : rename ξ a = a' at r\n  induction r generalizing ξ a <;> subst e\n  case refl => exact ⟨a, rfl, Pars.refl a⟩\n  case trans ih ra =>\n    let ⟨b, eb, rb⟩ := antirenaming ra; subst eb\n    let ⟨c, ec, rc⟩ := ih rfl\n    exact ⟨c, ec, Pars.trans rb rc⟩", "nesting_depth": 5, "transitive_dep_count": 25, "subset_aristotle": false, "category": "Type systems"}
{"id": 440, "thm_name": "substUnion", "thm_stmt": "theorem substUnion σ a s : subst (a +: σ) s = subst (a +: var) (subst (⇑ σ) s)", "lean_root": "TTBFL", "rel_path": "src/syntactics.lean", "imports": ["import «src».level"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "LevelClass", "content": "class LevelClass where\n  L : Type\n  lc : LevelClasses L"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "substId", "content": "def substId : ∀ s, subst var s = s :=\n  substId' var (by admit /- proof elided -/\n  )"}, {"name": "substRename", "content": "def substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )"}, {"name": "substComp", "content": "def substComp σ τ : ∀ s, (subst σ ∘ subst τ) s = subst (subst σ ∘ τ) s :=\n  substComp' _ _ (subst σ ∘ τ) (by admit /- proof elided -/\n  )"}], "used_local_lemmas": [{"name": "upExt", "content": "theorem upExt σ τ (h : ∀ x, σ x = τ x) : ∀ x, (⇑ σ) x = (⇑ τ) x"}, {"name": "substExt", "content": "theorem substExt σ τ (h : ∀ x, σ x = τ x) : ∀ s, subst σ s = subst τ s"}, {"name": "substDropAll", "content": "theorem substDropAll a b : b = subst (a +: var) (rename succ b)"}], "local_ctx": "import «src».level\n\nopen Nat\n\nvariable [lc : LevelClass]\n\n@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n\n\ninfixr:50 \"+:\" => cons\n\ninductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term\n\nopen Term\n\n@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)\n\n@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k\n\n@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)\n\nprefix:95 \"⇑\" => up\n\n@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k\n\ndef substId : ∀ s, subst var s = s :=\n  substId' var (by admit /- proof elided -/\n  )\n\ndef substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )\n\ndef substComp σ τ : ∀ s, (subst σ ∘ subst τ) s = subst (subst σ ∘ τ) s :=\n  substComp' _ _ (subst σ ∘ τ) (by admit /- proof elided -/\n  )", "target_theorem": "theorem substUnion σ a s : subst (a +: σ) s = subst (a +: var) (subst (⇑ σ) s) :=", "ground_truth_proof": ":= by\n  calc\n    subst (a +: σ) s\n      = subst (subst (a +: var) ∘ (var 0 +: (rename succ ∘ σ))) s :=\n        by apply substExt; intro n; cases n <;> simp; rw [← substDropAll]\n    _ = subst (a +: var) (subst (⇑ σ) s) :=\n        by rw [← substComp]; rfl", "nesting_depth": 4, "transitive_dep_count": 16, "subset_aristotle": false, "category": "Type systems"}
{"id": 441, "thm_name": "substDist", "thm_stmt": "theorem substDist σ a s : subst (subst σ a +: var) (subst (⇑ σ) s) = subst σ (subst (a +: var) s)", "lean_root": "TTBFL", "rel_path": "src/syntactics.lean", "imports": ["import «src».level"], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "LevelClass", "content": "class LevelClass where\n  L : Type\n  lc : LevelClasses L"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "substId", "content": "def substId : ∀ s, subst var s = s :=\n  substId' var (by admit /- proof elided -/\n  )"}, {"name": "substRename", "content": "def substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )"}, {"name": "substComp", "content": "def substComp σ τ : ∀ s, (subst σ ∘ subst τ) s = subst (subst σ ∘ τ) s :=\n  substComp' _ _ (subst σ ∘ τ) (by admit /- proof elided -/\n  )"}], "used_local_lemmas": [{"name": "upExt", "content": "theorem upExt σ τ (h : ∀ x, σ x = τ x) : ∀ x, (⇑ σ) x = (⇑ τ) x"}, {"name": "substExt", "content": "theorem substExt σ τ (h : ∀ x, σ x = τ x) : ∀ s, subst σ s = subst τ s"}, {"name": "substDropAll", "content": "theorem substDropAll a b : b = subst (a +: var) (rename succ b)"}, {"name": "substUnion", "content": "theorem substUnion σ a s : subst (a +: σ) s = subst (a +: var) (subst (⇑ σ) s)"}], "local_ctx": "import «src».level\n\nopen Nat\n\nvariable [lc : LevelClass]\n\n@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n\n\ninfixr:50 \"+:\" => cons\n\ninductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term\n\nopen Term\n\n@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)\n\n@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k\n\n@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)\n\nprefix:95 \"⇑\" => up\n\n@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k\n\ndef substId : ∀ s, subst var s = s :=\n  substId' var (by admit /- proof elided -/\n  )\n\ndef substRename ξ σ : ∀ s, subst σ (rename ξ s) = subst (σ ∘ ξ) s :=\n  substRename' _ _ (σ ∘ ξ) (by admit /- proof elided -/\n  )\n\ndef substComp σ τ : ∀ s, (subst σ ∘ subst τ) s = subst (subst σ ∘ τ) s :=\n  substComp' _ _ (subst σ ∘ τ) (by admit /- proof elided -/\n  )", "target_theorem": "theorem substDist σ a s : subst (subst σ a +: var) (subst (⇑ σ) s) = subst σ (subst (a +: var) s) :=", "ground_truth_proof": ":= by\n  calc\n    subst (subst σ a +: var) (subst (⇑ σ) s)\n      = subst (subst σ a +: σ) s       := by rw [← substUnion]\n    _ = subst (subst σ ∘ (a +: var)) s := by apply substExt; intro n; cases n <;> rfl\n    _ = (subst σ ∘ subst (a +: var)) s := by rw [← substComp]", "nesting_depth": 4, "transitive_dep_count": 17, "subset_aristotle": false, "category": "Type systems"}
{"id": 442, "thm_name": "wtfLofInv", "thm_stmt": "theorem wtfLofInv {Γ j 𝒰'}\n  (h : Γ ⊢ lof j ∶ 𝒰') :\n  ∃ k, lvl k ≈ 𝒰'", "lean_root": "TTBFL", "rel_path": "src/typing.lean", "imports": ["import src.reduction", "import «src».reduction"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Lean.ToExpr", "module": "Lean.ToExpr"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Repr", "module": "Init.Data.Repr"}, {"name": "Nat.sub", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "String", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat.succ", "module": "Init.Prelude"}, {"name": "Nat.zero", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" ξ:41 \"∶\" Γ:41 => wRename ξ Γ Δ"}, {"name": "notation:40 \"⊢\" Γ:40 => Wf Γ", "content": "notation:40 \"⊢\" Γ:40 => Wf Γ"}, {"name": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "content": "notation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A"}, {"name": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ", "content": "notation:40 Δ:41 \"⊢\" σ:41 \"∶\" Γ:41 => wSubst σ Γ Δ"}, {"name": "Ctxt", "content": "inductive Ctxt : Type where\n  | nil : Ctxt\n  | cons : Ctxt → Term → Ctxt"}, {"name": "Term", "content": "inductive Term : Type where\n  | var : Nat → Term\n  | 𝒰 : Term → Term\n  | pi : Term → Term → Term\n  | abs : Term → Term → Term\n  | app : Term → Term → Term\n  | mty : Term\n  | exf : Term → Term → Term\n  | lvl : Term → Term\n  | lof : lc.L → Term"}, {"name": "Par", "content": "inductive Par : Term → Term → Prop where\n  | β {b b' a a' c} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app (abs c b) a ⇒ subst (a' +: var) b'\n  | var s : var s ⇒ var s\n  | 𝒰 {a a'} :\n    a ⇒ a' →\n    \n    𝒰 a ⇒ 𝒰 a'\n  | pi {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    pi a b ⇒ pi a' b'\n  | abs {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    abs a b ⇒ abs a' b'\n  | app {b b' a a'} :\n    b ⇒ b' →\n    a ⇒ a' →\n    \n    app b a ⇒ app b' a'\n  | mty : mty ⇒ mty\n  | exf {a a' b b'} :\n    a ⇒ a' →\n    b ⇒ b' →\n    \n    exf a b ⇒ exf a' b'\n  | lvl {a a'} :\n    a ⇒ a' →\n    \n    lvl a ⇒ lvl a'\n  | lof k : lof k ⇒ lof k"}, {"name": "Conv", "content": "def Conv (a : Term) (b : Term) : Prop := ∃ c, a ⇒⋆ c ∧ b ⇒⋆ c"}, {"name": "Pars", "content": "inductive Pars : Term → Term → Prop where\n  | refl a : a ⇒⋆ a\n  | trans {a b c} : a ⇒ b → b ⇒⋆ c → a ⇒⋆ c"}, {"name": "subst", "content": "@[simp]\ndef subst (σ : Nat → Term) : Term → Term\n  | var s => σ s\n  | 𝒰 a => 𝒰 (subst σ a)\n  | pi a b => pi (subst σ a) (subst (⇑ σ) b)\n  | abs a b => abs (subst σ a) (subst (⇑ σ) b)\n  | app b a => app (subst σ b) (subst σ a)\n  | mty => mty\n  | exf a b => exf (subst σ a) (subst σ b)\n  | lvl a => lvl (subst σ a)\n  | lof k => lof k"}, {"name": "taka", "content": "@[simp]\ndef taka : Term → Term\n  | 𝒰 a => 𝒰 (taka a)\n  | pi a b => pi (taka a) (taka b)\n  | abs a b => abs (taka a) (taka b)\n  | app b a => match b with\n    | abs _ b => subst (taka a +: var) (taka b)\n    | b => app (taka b) (taka a)\n  | exf a b => exf (taka a) (taka b)\n  | lvl a => lvl (taka a)\n  | t => t"}, {"name": "up", "content": "@[simp]\ndef up (σ : Nat → Term) : Nat → Term :=\n  var 0 +: (rename succ ∘ σ)"}, {"name": "rename", "content": "@[simp]\ndef rename (ξ : Nat → Nat) : Term → Term\n  | var s => var (ξ s)\n  | 𝒰 a => 𝒰 (rename ξ a)\n  | pi a b => pi (rename ξ a) (rename (lift ξ) b)\n  | abs a b => abs (rename ξ a) (rename (lift ξ) b)\n  | app b a => app (rename ξ b) (rename ξ a)\n  | mty => mty\n  | exf a b => exf (rename ξ a) (rename ξ b)\n  | lvl a => lvl (rename ξ a)\n  | lof k => lof k"}, {"name": "lift", "content": "@[simp]\ndef lift (ξ : Nat → Nat) : Nat → Nat :=\n  zero +: (succ ∘ ξ)"}, {"name": "cons", "content": "@[simp]\ndef cons {A : Type} (x : A) (ξ : Nat → A) : Nat → A\n  | 0 => x\n  | n + 1 => ξ n"}, {"name": "infix:40 \"⇒\" => Par", "content": "infix:40 \"⇒\" => Par"}, {"name": "infix:40 \"⇒⋆\" => Pars", "content": "infix:40 \"⇒⋆\" => Pars"}, {"name": "infix:40 \"⇔\" => Conv", "content": "infix:40 \"⇔\" => Conv"}, {"name": "infixr:50 \"+:\" => cons", "content": "infixr:50 \"+:\" => cons"}, {"name": "prefix:95 \"⇑\" => up", "content": "prefix:95 \"⇑\" => up"}, {"name": "notation:50 \"⬝\" => Ctxt.nil", "content": "notation:50 \"⬝\" => Ctxt.nil"}, {"name": "infixl:50 \"∷\" => Ctxt.cons", "content": "infixl:50 \"∷\" => Ctxt.cons"}], "lib_lemmas": [{"name": "refl", "module": "Mathlib.Order.Defs.Unbundled"}, {"name": "trans", "module": "Mathlib.Order.Defs.Unbundled"}], "repo_lemmas": [{"name": "conv𝒰", "content": "theorem conv𝒰 {a a'} : a ⇔ a' → 𝒰 a ⇔ 𝒰 a'"}, {"name": "pars𝒰", "content": "theorem pars𝒰 {a a'} (r : a ⇒⋆ a') : 𝒰 a ⇒⋆ 𝒰 a'"}, {"name": "convAbs", "content": "theorem convAbs {a a' b b'} : a ⇔ a' → b ⇔ b' → abs a b ⇔ abs a' b'"}, {"name": "parsAbs", "content": "theorem parsAbs {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : abs a b ⇒⋆ abs a' b'"}, {"name": "parRefl", "content": "theorem parRefl a : a ⇒ a"}, {"name": "convPi", "content": "theorem convPi {a a' b b'} : a ⇔ a' → b ⇔ b' → pi a b ⇔ pi a' b'"}, {"name": "parsPi", "content": "theorem parsPi {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : pi a b ⇒⋆ pi a' b'"}, {"name": "convApp", "content": "theorem convApp {b b' a a'} : b ⇔ b' → a ⇔ a' → app b a ⇔ app b' a'"}, {"name": "parsApp", "content": "theorem parsApp {a a' b b'} (rb : b ⇒⋆ b') (ra : a ⇒⋆ a') : app b a ⇒⋆ app b' a'"}, {"name": "convSym", "content": "theorem convSym {a b} : a ⇔ b → b ⇔ a"}, {"name": "convTrans", "content": "theorem convTrans {a b c} : a ⇔ b → b ⇔ c → a ⇔ c"}, {"name": "confluence", "content": "theorem confluence {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒⋆ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diacon", "content": "theorem diacon {a b c} (r₁ : a ⇒⋆ b) (r₂ : a ⇒ c) : ∃ d, b ⇒⋆ d ∧ c ⇒⋆ d"}, {"name": "diamond", "content": "theorem diamond {a b c} (r₁ : a ⇒ b) (r₂ : a ⇒ c) : ∃ d, b ⇒ d ∧ c ⇒ d"}, {"name": "parTaka", "content": "theorem parTaka {a b} (r : a ⇒ b) : b ⇒ taka a"}, {"name": "parCong", "content": "theorem parCong {a a' b b'} (ra : a ⇒ a') (rb : b ⇒ b') : subst (a +: var) b ⇒ subst (a' +: var) b'"}, {"name": "parMorphing", "content": "theorem parMorphing {a b} σ τ (h : ∀ x, σ x ⇒ τ x) (r : a ⇒ b) : subst σ a ⇒ subst τ b"}, {"name": "parLift", "content": "theorem parLift σ τ (h : ∀ x, σ x ⇒ τ x) : ∀ x, (⇑ σ) x ⇒ (⇑ τ) x"}, {"name": "parRename", "content": "theorem parRename {a b} ξ (r : a ⇒ b) : rename ξ a ⇒ rename ξ b"}, {"name": "parPars", "content": "theorem parPars {a b} (r : a ⇒ b) : a ⇒⋆ b"}, {"name": "parsTrans", "content": "theorem parsTrans {a b c} (r₁ : a ⇒⋆ b) (r₂ : b ⇒⋆ c) : a ⇒⋆ c"}, {"name": "convExf", "content": "theorem convExf {a a' b b'} : a ⇔ a' → b ⇔ b' → exf a b ⇔ exf a' b'"}, {"name": "parsExf", "content": "theorem parsExf {a a' b b'} (ra : a ⇒⋆ a') (rb : b ⇒⋆ b') : exf a b ⇒⋆ exf a' b'"}, {"name": "convRefl", "content": "theorem convRefl {a} : a ⇔ a"}, {"name": "parConv", "content": "theorem parConv {a b} (r : a ⇒ b) : a ⇔ b"}, {"name": "parsConv", "content": "theorem parsConv {a b} (r : a ⇒⋆ b) : a ⇔ b"}, {"name": "convLvl", "content": "theorem convLvl {a a'} : a ⇔ a' → lvl a ⇔ lvl a'"}, {"name": "parsLvl", "content": "theorem parsLvl {a a'} (r : a ⇒⋆ a') : lvl a ⇒⋆ lvl a'"}, {"name": "convLvl𝒰", "content": "theorem convLvl𝒰 {j k} : ¬ lvl j ⇔ 𝒰 k"}, {"name": "parsLvlInv", "content": "theorem parsLvlInv {i b} (r : lvl i ⇒⋆ b) : ∃ j, b = lvl j ∧ i ⇒⋆ j"}, {"name": "pars𝒰Inv", "content": "theorem pars𝒰Inv {a b} (r : 𝒰 a ⇒⋆ b) : ∃ a', b = 𝒰 a' ∧ a ⇒⋆ a'"}], "used_local_defs": [{"name": "Eqv", "content": "inductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c"}, {"name": "Wf", "content": "inductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A"}, {"name": "Wt", "content": "inductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k"}], "used_local_lemmas": [{"name": "eqvConv", "content": "theorem eqvConv {a b} (r : a ≈ b) : a ⇔ b"}], "local_ctx": "import «src».reduction\n\nopen Nat\n\nopen Term\n\nvariable [LevelClass]\n\nsection\n\ninductive Eqv : Term → Term → Prop where\n  | β {b a c} : app (abs c b) a ≈ subst (a +: var) b\n  | 𝒰 {a a'} :\n    a ≈ a' →\n    \n    𝒰 a ≈ 𝒰 a'\n  | pi {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    pi a b ≈ pi a' b'\n  | abs {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    abs a b ≈ abs a' b'\n  | app {b b' a a'} :\n    b ≈ b' →\n    a ≈ a' →\n    \n    app b a ≈ app b' a'\n  | exf {a a' b b'} :\n    a ≈ a' →\n    b ≈ b' →\n    \n    exf a b ≈ exf a' b'\n  | lvl {a a'} :\n    a ≈ a' →\n    \n    lvl a ≈ lvl a'\n  | refl {a} : a ≈ a\n  | sym {a b} :\n    a ≈ b →\n    \n    b ≈ a\n  | trans {a b c} :\n    a ≈ b →\n    b ≈ c →\n    \n    a ≈ c\n\nend\n\ninfix:40 (priority := 1001) \"≈\" => Eqv\n\nsection\n\ninductive Wf : Ctxt → Prop where\n  | nil : ⊢ ⬝\n  | cons {Γ A k} :\n    ⊢ Γ →\n    Γ ⊢ A ∶ 𝒰 k →\n    \n    ⊢ Γ ∷ A\n\ninductive Wt : Ctxt → Term → Term → Prop where\n  | var {Γ x A} :\n    ⊢ Γ →\n    Γ ∋ x ∶ A →\n    \n    Γ ⊢ var x ∶ A\n  | 𝒰 {Γ j k} :\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ 𝒰 j ∶ 𝒰 k\n  | pi {Γ A B k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ B ∶ 𝒰 (rename succ k) →\n    \n    Γ ⊢ pi A B ∶ 𝒰 k\n  | abs {Γ A B b k} :\n    Γ ⊢ pi A B ∶ 𝒰 k →\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ∷ A ⊢ b ∶ B →\n    \n    Γ ⊢ abs A b ∶ pi A B\n  | app {Γ A B b a} :\n    Γ ⊢ b ∶ pi A B →\n    Γ ⊢ a ∶ A →\n    \n    Γ ⊢ app b a ∶ subst (a +: var) B\n  | mty {Γ j k} :\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ mty ∶ 𝒰 j\n  | exf {Γ A b k} :\n    Γ ⊢ A ∶ 𝒰 k →\n    Γ ⊢ b ∶ mty →\n    \n    Γ ⊢ exf A b ∶ A\n  | lvl {Γ a b j k} :\n    Γ ⊢ a ∶ lvl b →\n    Γ ⊢ 𝒰 j ∶ 𝒰 k →\n    \n    Γ ⊢ lvl a ∶ 𝒰 j\n  | lof {Γ j k} :\n    ⊢ Γ →\n    j < k →\n    \n    Γ ⊢ lof j ∶ lvl (lof k)\n  | trans {Γ i j k} :\n    Γ ⊢ i ∶ lvl j →\n    Γ ⊢ j ∶ lvl k →\n    \n    Γ ⊢ i ∶ lvl k\n  | conv {Γ A B a k} :\n    A ≈ B →\n    Γ ⊢ a ∶ A →\n    Γ ⊢ B ∶ 𝒰 k →\n    \n    Γ ⊢ a ∶ B\n  | sub {Γ j k A} :\n    Γ ⊢ j ∶ lvl k →\n    Γ ⊢ A ∶ 𝒰 j →\n    \n    Γ ⊢ A ∶ 𝒰 k\n\nend\n\nend\n\nnotation:40 \"⊢\" Γ:40 => Wf Γ\n\nnotation:40 Γ:41 \"⊢\" a:41 \"∶\" A:41 => Wt Γ a A", "target_theorem": "theorem wtfLofInv {Γ j 𝒰'}\n  (h : Γ ⊢ lof j ∶ 𝒰') :\n  ∃ k, lvl k ≈ 𝒰' :=", "ground_truth_proof": ":= by\n  generalize e : lof j = t at h\n  induction h\n  all_goals injections <;> subst_eqs <;> specialize_rfls\n  case lof | trans => exact ⟨_, Eqv.refl⟩\n  case conv e₁ _ _ _ ih =>\n    let ⟨_, e₂⟩ := ih\n    exact ⟨_, Eqv.trans e₂ e₁⟩\n  case sub ih =>\n    let ⟨_, e⟩ := ih\n    cases convLvl𝒰 (eqvConv e)", "nesting_depth": 13, "transitive_dep_count": 60, "subset_aristotle": false, "category": "Type systems"}
{"id": 443, "thm_name": "StateT.set_get", "thm_stmt": "theorem set_get : (do let s ← @StateT.get σ m _; StateT.set s) = pure ⟨⟩", "lean_root": "VCV-io", "rel_path": "ToMathlib/Control/Lawful/MonadState.lean", "imports": [], "used_lib_defs": [{"name": "StateT", "module": "Init.Control.State"}, {"name": "StateT.get", "module": "Init.Control.State"}, {"name": "StateT.bind", "module": "Init.Control.State"}, {"name": "StateT.instMonad", "module": "Init.Control.State"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "namespace LawfulMonadStateOf\n\nvariable {σ : Type u} {m : Type u → Type v}\n\nvariable [Monad m] [LawfulMonad m] [MonadStateOf σ m] [LawfulMonadStateOf σ m]\n\nend LawfulMonadStateOf\n\nnamespace LawfulMonadState\n\nvariable {σ : Type u} {m : Type u → Type v} [Monad m] [LawfulMonad m] [LawfulMonadState σ m]\n\nend LawfulMonadState\n\nnamespace StateT\n\nvariable {σ : Type u} {m : Type u → Type v} [Monad m] [LawfulMonad m]", "target_theorem": "theorem set_get : (do let s ← @StateT.get σ m _; StateT.set s) = pure ⟨⟩ :=", "ground_truth_proof": ":= by\n  unfold StateT.get StateT.instMonad StateT.bind StateT.set StateT.pure\n  simp only [bind_pure_comp, map_pure]", "nesting_depth": 1, "transitive_dep_count": 4, "subset_aristotle": false, "category": "Applied verif."}
{"id": 444, "thm_name": "triple_forIn_deacreasing", "thm_stmt": "theorem triple_forIn_deacreasing {β} {measure : β -> ℕ}\n  {init : β} {f : β → m (ForInStep β)}\n  (inv : β → l)\n  (hstep : ∀ b,\n    measure b <= measure init ->\n    triple\n      (inv b)\n      (f b)\n      (fun | .yield b' => inv b' ⊓ ⌜measure b' < measure b⌝ | .done b' => ⌜ measure b' = 0 ⌝ ⊓ inv b')) :\n  triple (inv init) (forIn [0:measure init] init (fun _ => f)) (fun b => inv b ⊓ ⌜measure b = 0⌝)", "lean_root": "loom", "rel_path": "Loom/MonadAlgebras/WP/Gen.lean", "imports": ["import Loom.MonadAlgebras.WP.Liberal", "import Mathlib.Order.Lattice", "import Mathlib.Order.Basic", "import Loom.MonadAlgebras.WP.DoNames'", "import Mathlib.Order.CompleteBooleanAlgebra", "import Mathlib.Logic.Function.Basic", "import Loom.MonadAlgebras.WP.Basic"], "used_lib_defs": [{"name": "Cont", "module": "Mathlib.Control.Monad.Cont"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "ForInStep", "module": "Init.Core"}, {"name": "Std.Range", "module": "Init.Data.Range.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.range'", "module": "Init.Data.List.Basic"}], "used_repo_defs": [{"name": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)", "content": "macro \"⌜\" p:term \"⌝\" : term => `(LE.pure $p)"}, {"name": "Kind", "content": "inductive Kind where\n  | regular\n  | forIn\n  | forInWithReturn\n  | nestedBC\n  | nestedPR\n  | nestedSBC\n  | nestedPRBC"}, {"name": "triple", "content": "def triple (pre : l) (c : m α) (post : α -> l) : Prop :=\n  pre ≤ wp c post"}, {"name": "wp", "content": "def wp (c : m α) (post : α -> l) : l := liftM (n := Cont l) c post"}, {"name": "LogicLift", "content": "class LogicLift (l : outParam (Type u)) ( k : Type u) [CompleteLattice l] [CompleteLattice k] where\n  [lift : MonadLift (Cont l) (Cont k)]\n  lift_top {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊤) = ⊤\n  lift_bot {α : Type u} :\n    monadLift (m := Cont l) (n := Cont k) (fun (_ : α -> l) => ⊥) = ⊥"}, {"name": "triple", "content": "notation \"{\" P \"}\" c \"{\" v \",\" Q \"}\" => triple P c (fun v => Q)"}], "lib_lemmas": [{"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans'", "module": "Mathlib.Order.Basic"}], "repo_lemmas": [{"name": "wp_cons", "content": "lemma wp_cons (x : m α) (post post' : α -> l) :\n  (∀ y, post y ≤ post' y) ->\n  wp x post ≤ wp x post'"}, {"name": "triple_forIn_range_step1", "content": "theorem triple_forIn_range_step1 {β}\n  {xs : Std.Range} {init : β} {f : ℕ → β → m (ForInStep β)}\n  (inv : ℕ → β → l)\n  (hstep : ∀ i b,\n    triple\n      (inv i b)\n      (f i b)\n      (fun | .yield b' => inv (i + xs.step) b' | .done b' => inv xs.stop b')) :\n  xs.step = 1 ->\n  xs.start <= xs.stop ->\n  triple (inv xs.start init) (forIn xs init f) (inv xs.stop)"}, {"name": "triple_forIn_range", "content": "theorem triple_forIn_range {β}\n  (xs : Std.Range) (init : β) (f : ℕ → β → m (ForInStep β))\n  (inv : ℕ → β → l)\n  (hstep : ∀ i b,\n    triple\n      (inv i b)\n      (f i b)\n      (fun | .yield b' => inv (i + xs.step) b' | .done b' => inv (xs.start + ((xs.stop - xs.start + xs.step - 1) / xs.step) * xs.step) b')) :\n  triple (inv xs.start init) (forIn xs init f) (inv (xs.start + ((xs.stop - xs.start + xs.step - 1) / xs.step) * xs.step))"}, {"name": "triple_forIn_range'", "content": "theorem triple_forIn_range' {β}\n  {xstart : ℕ} {step : ℕ} (n : ℕ) {init : β} {f : ℕ → β → m (ForInStep β)}\n  (inv : ℕ → β → l)\n  (hstep : ∀ i b,\n    triple\n      (inv i b)\n      (f i b)\n      (fun | .yield b' => inv (i + step) b' | .done b' => inv (xstart + n * step) b')) :\n  triple (inv xstart init) (forIn (List.range' xstart n step) init f) (inv (xstart + n * step))"}, {"name": "triple_forIn_range'_aux", "content": "theorem triple_forIn_range'_aux {β}\n  {xstart : ℕ} {step : ℕ} (n : ℕ) (init : β) {f : ℕ → β → m (ForInStep β)}\n  (inv : ℕ → β → l)\n  (hstep : ∀ i b,\n    triple\n      (inv i b)\n      (f i b)\n      (fun | .yield b' => inv (i + step) b' | .done b' => inv (xstart + n * step) b')) :\n  i <= n ->\n  triple (inv (xstart + (n - i) * step) init) (forIn (List.range' (xstart + (n - i) * step) i step) init f) (inv (xstart + n * step))"}, {"name": "triple_bind", "content": "lemma triple_bind {β} (pre : l) (x : m α) (cut : α -> l)\n  (f : α -> m β) (post : β -> l) :\n  triple pre x cut ->\n  (∀ y, triple (cut y) (f y) post) ->\n  triple pre (x >>= f) post"}, {"name": "wp_bind", "content": "lemma wp_bind {β} (x : m α) (f : α -> m β) (post : β -> l) :\n    wp (x >>= f) post = wp x (fun x => wp (f x) post)"}, {"name": "triple_pure", "content": "lemma triple_pure (pre : l) (x : α) (post : α -> l) :\n  triple pre (pure (f := m) x) post <-> pre ≤ (post x)"}], "used_local_defs": [], "used_local_lemmas": [], "local_ctx": "import Mathlib.Logic.Function.Basic\n\nimport Mathlib.Order.CompleteBooleanAlgebra\n\nimport Mathlib.Order.Lattice\n\nimport Mathlib.Order.Basic\n\nimport Loom.MonadAlgebras.WP.Basic\n\nimport Loom.MonadAlgebras.WP.Liberal\n\nimport Loom.MonadAlgebras.WP.DoNames'\n\nopen Lean Meta Elab Command Term\n\nsection\n\nvariable {m : Type u -> Type v} [Monad m] [LawfulMonad m] {α : Type u} {l : Type u} [CompleteLattice l]\n\nvariable [MAlgOrdered m l]", "target_theorem": "theorem triple_forIn_deacreasing {β} {measure : β -> ℕ}\n  {init : β} {f : β → m (ForInStep β)}\n  (inv : β → l)\n  (hstep : ∀ b,\n    measure b <= measure init ->\n    triple\n      (inv b)\n      (f b)\n      (fun | .yield b' => inv b' ⊓ ⌜measure b' < measure b⌝ | .done b' => ⌜ measure b' = 0 ⌝ ⊓ inv b')) :\n  triple (inv init) (forIn [0:measure init] init (fun _ => f)) (fun b => inv b ⊓ ⌜measure b = 0⌝) :=", "ground_truth_proof": ":= by\n  apply le_trans'; apply wp_cons; rotate_left 2; apply le_trans; rotate_left 1\n  apply triple_forIn_range_step1 (inv := fun i b => ⌜ measure b + i <= measure init ⌝ ⊓ inv b) <;>\n    try solve | aesop\n  { simp; intro i b\n    by_cases h : measure b + i ≤ measure init <;> simp [h, triple]\n    apply le_trans; apply hstep; omega\n    apply wp_cons; rintro (b'|b') <;> simp\n    by_cases h: measure b' = 0 <;> simp [h]\n    by_cases h': measure b' < measure b <;> simp [h']\n    have : measure b' + (i + 1) ≤ measure init := by omega\n    simp [this] }\n  { simp; intro b\n    by_cases h : measure b + measure init ≤ measure init <;> simp [h]\n    by_cases h' : measure b = 0 <;> simp [h']\n    omega }\n  simp", "nesting_depth": 6, "transitive_dep_count": 20, "subset_aristotle": true, "category": "Framework"}
{"id": 445, "thm_name": "OracleComp.evalDist_liftComp", "thm_stmt": "@[simp]\nlemma evalDist_liftComp [spec.FiniteRange] [superSpec.FiniteRange]\n    (oa : OracleComp spec α) : evalDist (liftComp oa superSpec) = evalDist oa", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/Coercions/SubSpec.lean", "imports": ["import VCVio.OracleComp.Constructions.UniformSelect", "import VCVio.OracleComp.DistSemantics.EvalDist", "import VCVio.OracleComp.SimSemantics.SimulateQ"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "Bind", "module": "Init.Prelude"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "AlternativeMonad", "module": "Batteries.Control.AlternativeMonad"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.getM", "module": "Init.Data.Option.Basic"}, {"name": "MonadLift", "module": "Init.Prelude"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "StateT", "module": "Init.Control.State"}, {"name": "StateT.run", "module": "Init.Control.State"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "OptionT.lift", "module": "Init.Control.Option"}, {"name": "PMF", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "PMF.uniformOfFintype", "module": "Mathlib.Probability.Distributions.Uniform"}], "used_repo_defs": [{"name": "simulateQ", "content": "def simulateQ [AlternativeMonad m] (so : QueryImpl spec m) (oa : OracleComp spec α) : m α :=\n  do Option.getM (← FreeMonad.mapM oa.run so.impl)"}, {"name": "QueryImpl.Inhabited", "content": "instance QueryImpl.Inhabited [∀ i, Inhabited (spec.range i)] [Pure m] :\n  Inhabited (QueryImpl spec m) := ⟨{impl q := pure q.defaultOutput}⟩"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "mapM", "content": "protected def mapM [Pure m] [Bind m] :\n    (oa : FreeMonad f α) → (s : {α : Type u} → f α → m α) → m α\n  | .pure x, _ => pure x\n  | .roll x r, s => s x >>= λ u ↦ (r u).mapM s"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : OracleComp spec α → Prop}\n    (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}, {"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "FiniteRange", "content": "class FiniteRange (spec : OracleSpec ι) where\n    range_inhabited' (i : ι) : Inhabited (spec.range i)\n    range_fintype' (i : ι) : Fintype (spec.range i)"}, {"name": "cases", "content": "def cases {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive (tl.insertIdx 0 hd)) {m : ℕ} :\n    (v : Vector α m) → motive v := match hm : m with\n  | 0 => fun v => match v with | ⟨⟨[]⟩, rfl⟩ => v_empty\n  | n + 1 => fun v => match hv : v with\n    | ⟨⟨hd :: tl⟩, hSize⟩ => by admit /- proof elided -/"}], "lib_lemmas": [{"name": "Function.comp_apply", "module": "Init.Core"}, {"name": "StateT.run'_eq", "module": "Init.Control.Lawful.Instances"}, {"name": "StateT.run_bind", "module": "Init.Control.Lawful.Instances"}, {"name": "StateT.run_monadLift", "module": "Init.Control.Lawful.Instances"}, {"name": "bind_map_left", "module": "Init.Control.Lawful.Basic"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "map_bind", "module": "Init.Control.Lawful.Basic"}], "repo_lemmas": [{"name": "simulateQ_bind", "content": "@[simp] lemma simulateQ_bind (oa : OracleComp spec α) (ob : α → OracleComp spec β) :\n    simulateQ so (oa >>= ob) = simulateQ so oa >>= (simulateQ so ∘ ob)"}, {"name": "simulateQ_query_bind", "content": "@[simp] lemma simulateQ_query_bind (q : OracleQuery spec α) (ob : α → OracleComp spec β) :\n    simulateQ so (liftM q >>= ob) = so.impl q >>= (simulateQ so ∘ ob)"}, {"name": "evalDist_liftM", "content": "@[simp]\nlemma evalDist_liftM [Nonempty α] [Fintype α] (q : OracleQuery spec α) :\n    evalDist (q : OracleComp spec α) = OptionT.lift (PMF.uniformOfFintype α)"}, {"name": "simulateQ_query", "content": "@[simp] lemma simulateQ_query (q : OracleQuery spec α) : simulateQ so q = so.impl q"}], "used_local_defs": [{"name": "OracleSpec.SubSpec", "content": "class SubSpec (spec : OracleSpec.{u,w} ι) (superSpec : OracleSpec τ)\n  extends MonadLift (OracleQuery spec) (OracleQuery superSpec) where"}, {"name": "OracleComp.liftComp", "content": "def liftComp (oa : OracleComp spec α) (superSpec : OracleSpec τ)\n      [h : MonadLift (OracleQuery spec) (OracleQuery superSpec)] :\n      OracleComp superSpec α := simulateQ ⟨liftM⟩ oa"}], "used_local_lemmas": [{"name": "OracleComp.liftComp_pure", "content": "@[simp]\nlemma liftComp_pure (x : α) : liftComp (pure x : OracleComp spec α) superSpec = pure x"}], "local_ctx": "import VCVio.OracleComp.SimSemantics.SimulateQ\n\nimport VCVio.OracleComp.Constructions.UniformSelect\n\nopen OracleSpec OracleComp BigOperators ENNReal\n\nvariable {ι : Type u} {τ : Type v}\n  {spec : OracleSpec ι} {superSpec : OracleSpec τ} {α β γ : Type w}\n\nnamespace OracleSpec\n\ninfix : 50 \" ⊂ₒ \" => SubSpec\n\nnamespace SubSpec\n\nvariable [h : MonadLift (OracleQuery spec) (OracleQuery superSpec)]\n\nend SubSpec\n\nend OracleSpec\n\nnamespace OracleComp\n\nsection liftComp\n\ndef liftComp (oa : OracleComp spec α) (superSpec : OracleSpec τ)\n      [h : MonadLift (OracleQuery spec) (OracleQuery superSpec)] :\n      OracleComp superSpec α := simulateQ ⟨liftM⟩ oa\n\nvariable (superSpec : OracleSpec τ) [h : MonadLift (OracleQuery spec) (OracleQuery superSpec)]", "target_theorem": "@[simp]\nlemma evalDist_liftComp [spec.FiniteRange] [superSpec.FiniteRange]\n    (oa : OracleComp spec α) : evalDist (liftComp oa superSpec) = evalDist oa :=", "ground_truth_proof": ":= by\n  induction oa using OracleComp.inductionOn with\n  | pure x => simp [liftComp_pure]\n  | query_bind i t oa hoa =>\n      simp only [liftComp, simulateQ_bind, simulateQ_query, StateT.run'_eq, StateT.run_bind,\n        StateT.run_monadLift, SubSpec.liftM_query_eq_liftM_liftM, bind_pure_comp,\n        Function.comp_apply, bind_map_left, map_bind, evalDist_bind, OracleComp.evalDist_liftM]\n      refine congr_arg (_ >>= ·) (funext λ u ↦ ?_)\n      simpa [StateT.run, liftComp] using hoa u\n  | failure => simp", "nesting_depth": 6, "transitive_dep_count": 46, "subset_aristotle": true, "category": "Applied verif."}
{"id": 446, "thm_name": "OracleComp.pure_eq_bind_iff", "thm_stmt": "@[simp]\nlemma pure_eq_bind_iff (oa : OracleComp spec α) (ob : α → OracleComp spec β) (y : β) :\n    pure y = oa >>= ob ↔ ∃ x : α, oa = pure x ∧ ob x = pure y", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/OracleComp.lean", "imports": ["import ToMathlib.Control.AlternativeMonad", "import ToMathlib.Control.Monad.Free", "import VCVio.OracleComp.OracleSpec", "import ToMathlib.Control.WriterT", "import Mathlib.Control.Lawful", "import ToMathlib.Control.OptionT"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Option", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}], "lib_lemmas": [{"name": "eq_comm", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "OracleSpec.OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "OracleComp.induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}], "used_local_lemmas": [{"name": "OracleComp.bind_eq_pure_iff", "content": "@[simp]\nlemma bind_eq_pure_iff (oa : OracleComp spec α) (ob : α → OracleComp spec β) (y : β) :\n    oa >>= ob = pure y ↔ ∃ x : α, oa = pure x ∧ ob x = pure y"}], "local_ctx": "import ToMathlib.Control.Monad.Free\n\nimport ToMathlib.Control.WriterT\n\nimport ToMathlib.Control.AlternativeMonad\n\nimport ToMathlib.Control.OptionT\n\nimport Mathlib.Control.Lawful\n\nimport VCVio.OracleComp.OracleSpec\n\nnamespace OracleSpec\n\ninductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)\n\nnamespace OracleQuery\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β : Type v}\n\nend OracleQuery\n\nend OracleSpec\n\nopen OracleSpec\n\ndef OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β : Type v}\n\nsection lift\n\nend lift\n\nnotation \"$[0..\" n \"]\" => uniformFin n\n\nnotation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m\n\nexample : OracleComp probSpec ℕ := do\n  let x ← $[314⋯31415]; let y ← $[0⋯x]\n  return x + 2 * y\n\n@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa\n\nsection construct\n\nvariable {C : OracleComp spec α → Type w}\n  (h_pure : (a : α) → C (pure a))\n  (h_query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n  (h_failure : C failure) (oa : OracleComp spec α)\n\nend construct\n\nsection noConfusion\n\nvariable (x : α) (y : β) (q : OracleQuery spec β) (oa : β → OracleComp spec α)\n\nend noConfusion\n\nsection mapM\n\nvariable {m : Type v → Type w} [Monad m]\n  (fail : {α : Type v} → m α) (qm : {α : Type v} → OracleQuery spec α → m α)\n\nend mapM\n\nsection inj", "target_theorem": "@[simp]\nlemma pure_eq_bind_iff (oa : OracleComp spec α) (ob : α → OracleComp spec β) (y : β) :\n    pure y = oa >>= ob ↔ ∃ x : α, oa = pure x ∧ ob x = pure y :=", "ground_truth_proof": ":= by\n  apply eq_comm.trans (bind_eq_pure_iff oa ob y)\n\nalias ⟨_, bind_eq_pure⟩ := bind_eq_pure_iff\nalias ⟨_, pure_eq_bind⟩ := pure_eq_bind_iff", "nesting_depth": 4, "transitive_dep_count": 13, "subset_aristotle": true, "category": "Applied verif."}
{"id": 447, "thm_name": "OracleComp.evalDist_uniformSelectFinset", "thm_stmt": "@[simp] lemma evalDist_uniformSelectFinset [DecidableEq α] (s : Finset α) :\n    evalDist ($ s) = if hs : s.Nonempty then\n      OptionT.lift (PMF.uniformOfFinset s hs) else failure", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/Constructions/UniformSelect.lean", "imports": ["import Batteries.Control.OptionT", "import VCVio.OracleComp.DistSemantics.Prod", "import VCVio.OracleComp.DistSemantics.EvalDist", "import ToMathlib.General"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "OptionT.lift", "module": "Init.Control.Option"}, {"name": "PMF", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "PMF.uniformOfFinset", "module": "Mathlib.Probability.Distributions.Uniform"}, {"name": "ENNReal", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "OptionT.mk", "module": "Init.Control.Option"}, {"name": "reduceDIte", "module": "Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core"}, {"name": "PEmpty", "module": "Init.Prelude"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "IsEmpty", "module": "Mathlib.Logic.IsEmpty"}, {"name": "List", "module": "Init.Prelude"}, {"name": "List.count", "module": "Init.Data.List.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Pure", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "emptySpec", "content": "notation \"[]ₒ\" => emptySpec"}, {"name": "uniformFin", "content": "notation \"$[0..\" n \"]\" => uniformFin n"}, {"name": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m", "content": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m\n\nexample : OracleComp probSpec ℕ := do\n  let x ← $[314⋯31415]; let y ← $[0⋯x]\n  return x + 2 * y"}, {"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "DecidableEq", "content": "protected class DecidableEq (spec : OracleSpec ι) where\n  domain_decidableEq' (i : ι) : DecidableEq (spec.domain i)\n  range_decidableEq' (i : ι) : DecidableEq (spec.range i)"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "probFailure", "content": "noncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "probOutput", "content": "noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)"}, {"name": "isEmpty", "content": "instance isEmpty : IsEmpty (OracleQuery []ₒ α) where false | query i t => i.elim"}, {"name": "emptySpec", "content": "@[inline, reducible]\ndef emptySpec : OracleSpec PEmpty := λ _ ↦ (PUnit, PUnit)"}, {"name": "uniformFin", "content": "@[reducible, inline] def uniformFin (n : ℕ) : ProbComp (Fin (n + 1)) :=\n  unifSpec.query n ()"}, {"name": "ProbComp", "content": "abbrev ProbComp : Type z → Type (max z 1) := OracleComp unifSpec"}, {"name": "unifSpec", "content": "@[inline, reducible] def unifSpec : OracleSpec.{0,0} ℕ :=\n  λ n ↦ (Unit, Fin (n + 1))"}, {"name": "uniformFin'", "content": "@[reducible, inline] def uniformFin' (n m : ℕ) : OracleComp probSpec (Fin (m + 1)) :=\n  probSpec.query m n"}, {"name": "probSpec", "content": "def probSpec : OracleSpec.{0,0} ℕ :=\n  fun n => (ℕ, Fin (n + 1))"}, {"name": "induction", "content": "@[elab_as_elim]\ndef induction {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive tl → motive (tl.insertIdx 0 hd))\n    {m : ℕ} : (v : Vector α m) → motive v :="}, {"name": "probOutput", "content": "notation \"Pr[=\" x \"|\" mx \"]\" => probOutput mx x"}, {"name": "probFailure", "content": "notation \"Pr[⊥\" \"|\" mx \"]\" => probFailure mx"}, {"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "uniformFin", "content": "notation \"$[0..\" n \"]\" => uniformFin n"}, {"name": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m", "content": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m\n\nexample : OracleComp probSpec ℕ := do\n  let x ← $[314⋯31415]; let y ← $[0⋯x]\n  return x + 2 * y"}, {"name": "emptySpec", "content": "notation \"[]ₒ\" => emptySpec"}], "lib_lemmas": [{"name": "Finset.nonempty_iff_ne_empty", "module": "Mathlib.Data.Finset.Empty"}, {"name": "div_eq_mul_inv", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "Finset.mem_toList", "module": "Mathlib.Data.Finset.Dedup"}, {"name": "List.isEmpty_eq_false_iff_exists_mem", "module": "Init.Data.List.Lemmas"}, {"name": "ENNReal.inv_eq_zero", "module": "Mathlib.Data.ENNReal.Inv"}, {"name": "ENNReal.natCast_ne_top", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "Finset.not_nonempty_iff_eq_empty", "module": "Mathlib.Data.Finset.Empty"}, {"name": "PMF.bind_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Monad"}, {"name": "PMF.pure_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Monad"}, {"name": "PMF.uniformOfFinset_apply", "module": "Mathlib.Probability.Distributions.Uniform"}, {"name": "imp_false", "module": "Init.Core"}, {"name": "ite_eq_ite", "module": "Init.PropLemmas"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "mul_ite", "module": "Mathlib.Algebra.Notation.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "symm", "module": "Mathlib.Order.Defs.Unbundled"}, {"name": "tsum_eq_single", "module": "Mathlib.Topology.Algebra.InfiniteSum.Basic"}], "repo_lemmas": [{"name": "probOutput_map_eq_sum_fintype_ite", "content": "lemma probOutput_map_eq_sum_fintype_ite [Fintype α] [DecidableEq β] (y : β) :\n    [= y | f <$> oa] = ∑ x : α, if y = f x then [= x | oa] else 0"}, {"name": "probOutput_map_eq_tsum_ite", "content": "lemma probOutput_map_eq_tsum_ite [DecidableEq β] (y : β) :\n    [= y | f <$> oa] = ∑' x : α, if y = f x then [= x | oa] else 0"}, {"name": "probOutput_bind_eq_tsum", "content": "lemma probOutput_bind_eq_tsum (y : β) :\n    [= y | oa >>= ob] = ∑' x : α, [= x | oa] * [= y | ob x]"}, {"name": "probOutput_pure", "content": "@[simp]\nlemma probOutput_pure [DecidableEq α] (y : α) :\n    [= y | (pure x : OracleComp spec α)] = if y = x then 1 else 0"}, {"name": "List.countP_eq_sum_fin_ite", "content": "lemma List.countP_eq_sum_fin_ite {α : Type*} (xs : List α) (p : α → Bool) :\n    (∑ i : Fin xs.length, if p xs[i] then 1 else 0) = xs.countP p"}, {"name": "PMF.monad_pure_eq_pure", "content": "@[simp]\nlemma PMF.monad_pure_eq_pure {α : Type u} (x : α) :\n    (Pure.pure x : PMF α) = PMF.pure x"}, {"name": "PMF.monad_bind_eq_bind", "content": "@[simp]\nlemma PMF.monad_bind_eq_bind {α β : Type u}\n      (p : PMF α) (q : α → PMF β) : p >>= q = p.bind q"}], "used_local_defs": [], "used_local_lemmas": [{"name": "OracleComp.uniformSelectList_cons", "content": "lemma uniformSelectList_cons (x : α) (xs : List α) :\n    ($ x :: xs : ProbComp α) = ((x :: xs)[·]) <$> $[0..xs.length]"}, {"name": "OracleComp.probOutput_uniformSelectList", "content": "@[simp] lemma probOutput_uniformSelectList [DecidableEq α] (xs : List α) (x : α) :\n    [= x | $ xs] = if xs.isEmpty then 0 else (xs.count x : ℝ≥0∞) / xs.length"}, {"name": "OracleComp.uniformSelectFinset_def", "content": "lemma uniformSelectFinset_def {α : Type} [DecidableEq α] (s : Finset α) :\n    ($ s) = ($ s.toList)"}, {"name": "OracleComp.probOutput_uniformSelectFinset", "content": "@[simp] lemma probOutput_uniformSelectFinset [DecidableEq α] (s : Finset α) (x : α) :\n    [= x | $ s] = if x ∈ s then (s.card : ℝ≥0∞)⁻¹ else 0"}, {"name": "OracleComp.probFailure_uniformSelectFinset", "content": "@[simp] lemma probFailure_uniformSelectFinset [DecidableEq α] (s : Finset α) :\n    [⊥ | $ s] = if s.Nonempty then 0 else 1"}], "local_ctx": "import VCVio.OracleComp.DistSemantics.Prod\n\nimport Batteries.Control.OptionT\n\nopen OracleSpec BigOperators ENNReal\n\nnamespace OracleComp\n\nsection uniformSelect\n\nprefix : 50 \"$\" => uniformSelect _\n\nvariable {cont β : Type} [h : HasUniformSelect cont β]\n\nend uniformSelect\n\nsection uniformSelectList\n\nvariable {α : Type}\n\nend uniformSelectList\n\nsection uniformSelectVector\n\nend uniformSelectVector\n\nsection uniformSelectFinset\n\nvariable {α : Type}", "target_theorem": "@[simp] lemma evalDist_uniformSelectFinset [DecidableEq α] (s : Finset α) :\n    evalDist ($ s) = if hs : s.Nonempty then\n      OptionT.lift (PMF.uniformOfFinset s hs) else failure :=", "ground_truth_proof": ":= by\n  refine PMF.ext λ x ↦ ?_\n  by_cases hs : s.Nonempty\n  · cases x with\n    | none =>\n        refine (probFailure_uniformSelectFinset _).trans ?_\n        simp [hs, OptionT.lift, OptionT.mk]\n    | some x =>\n        simp only [hs, ↓reduceDIte]\n        refine (probOutput_uniformSelectFinset _ _).trans ?_\n        simp only [OptionT.lift, OptionT.mk, PMF.monad_pure_eq_pure, PMF.monad_bind_eq_bind,\n          PMF.bind_apply, PMF.uniformOfFinset_apply, PMF.pure_apply, Option.some.injEq, mul_ite,\n          mul_one, mul_zero]\n        refine symm <| (tsum_eq_single x ?_).trans ?_\n        · simp only [ne_eq, @eq_comm _ x, ite_eq_right_iff, ENNReal.inv_eq_zero,\n            natCast_ne_top, imp_false]\n          intros\n          tauto\n        · simp only [↓reduceIte, ite_eq_ite]\n  · simp only [Finset.not_nonempty_iff_eq_empty] at hs\n    simp [hs, uniformSelectFinset_def]", "nesting_depth": 7, "transitive_dep_count": 71, "subset_aristotle": true, "category": "Applied verif."}
{"id": 448, "thm_name": "OracleComp.query_bind_eq_roll", "thm_stmt": "lemma query_bind_eq_roll (q : OracleQuery spec α) (ob : α → OracleComp spec β) :\n    (q : OracleComp spec α) >>= ob = OptionT.mk (FreeMonad.roll q ob) := rfl", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/OracleComp.lean", "imports": ["import ToMathlib.Control.AlternativeMonad", "import ToMathlib.Control.Monad.Free", "import VCVio.OracleComp.OracleSpec", "import ToMathlib.Control.WriterT", "import Mathlib.Control.Lawful", "import ToMathlib.Control.OptionT"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "OptionT.mk", "module": "Init.Control.Option"}], "used_repo_defs": [{"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "OracleSpec.OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}], "used_local_lemmas": [], "local_ctx": "import ToMathlib.Control.Monad.Free\n\nimport ToMathlib.Control.WriterT\n\nimport ToMathlib.Control.AlternativeMonad\n\nimport ToMathlib.Control.OptionT\n\nimport Mathlib.Control.Lawful\n\nimport VCVio.OracleComp.OracleSpec\n\nnamespace OracleSpec\n\ninductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)\n\nnamespace OracleQuery\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β : Type v}\n\nend OracleQuery\n\nend OracleSpec\n\nopen OracleSpec\n\ndef OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β : Type v}\n\nsection lift", "target_theorem": "lemma query_bind_eq_roll (q : OracleQuery spec α) (ob : α → OracleComp spec β) :\n    (q : OracleComp spec α) >>= ob = OptionT.mk (FreeMonad.roll q ob) :=", "ground_truth_proof": ":= rfl", "nesting_depth": 3, "transitive_dep_count": 9, "subset_aristotle": false, "category": "Applied verif."}
{"id": 449, "thm_name": "OracleComp.queryBind_inj", "thm_stmt": "@[simp]\nlemma queryBind_inj (q q' : OracleQuery spec α) (ob ob' : α → OracleComp spec β) :\n    (q : OracleComp spec α) >>= ob = (q' : OracleComp spec α) >>= ob' ↔ q = q' ∧ ob = ob'", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/OracleComp.lean", "imports": ["import ToMathlib.Control.AlternativeMonad", "import ToMathlib.Control.Monad.Free", "import VCVio.OracleComp.OracleSpec", "import ToMathlib.Control.WriterT", "import Mathlib.Control.Lawful", "import ToMathlib.Control.OptionT"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "OptionT.bind", "module": "Init.Control.Option"}, {"name": "OptionT.lift", "module": "Init.Control.Option"}, {"name": "OptionT.mk", "module": "Init.Control.Option"}], "used_repo_defs": [{"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "lift", "content": "@[always_inline, inline]\ndef lift (x : f α) : FreeMonad f α := FreeMonad.roll x FreeMonad.pure"}, {"name": "bind", "content": "@[always_inline, inline]\nprotected def bind : FreeMonad f α → (α → FreeMonad f β) → FreeMonad f β\n  | FreeMonad.pure x, g => g x\n  | FreeMonad.roll x r, g => FreeMonad.roll x (λ u ↦ FreeMonad.bind (r u) g)"}, {"name": "notation:50 m₁ \" >>=ₕ \" m₂ => bind m₁ m₂", "content": "notation:50 m₁ \" >>=ₕ \" m₂ => bind m₁ m₂"}], "lib_lemmas": [{"name": "heq_eq_eq", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "bind_pure", "content": "@[simp]\nlemma bind_pure (x : α) (r : α → FreeMonad f β) :\n    FreeMonad.bind (FreeMonad.pure x) r = r x"}, {"name": "monad_bind_def", "content": "@[simp]\nlemma monad_bind_def (x : FreeMonad f α) (g : α → FreeMonad f β) :\n    x >>= g = FreeMonad.bind x g"}, {"name": "bind_lift", "content": "@[simp]\nlemma bind_lift (x : f α) (r : α → FreeMonad f β) :\n    FreeMonad.bind (FreeMonad.lift x) r = FreeMonad.roll x r"}, {"name": "bind_roll", "content": "@[simp]\nlemma bind_roll (x : f α) (r : α → FreeMonad f β) (g : β → FreeMonad f γ) :\n    FreeMonad.bind (FreeMonad.roll x r) g = FreeMonad.roll x (λ u ↦ FreeMonad.bind (r u) g)"}, {"name": "monad_pure_def", "content": "@[simp]\nlemma monad_pure_def (x : α) : (pure x : FreeMonad f α) = FreeMonad.pure x"}], "used_local_defs": [{"name": "OracleSpec.OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}], "used_local_lemmas": [{"name": "OracleComp.liftM_def", "content": "protected lemma liftM_def (q : OracleQuery spec α) :\n    (q : OracleComp spec α) = OptionT.lift (FreeMonad.lift q)"}, {"name": "OracleComp.bind_def", "content": "protected lemma bind_def (oa : OracleComp spec α) (ob : α → OracleComp spec β) :\n    oa >>= ob = OptionT.bind oa ob"}], "local_ctx": "import ToMathlib.Control.Monad.Free\n\nimport ToMathlib.Control.WriterT\n\nimport ToMathlib.Control.AlternativeMonad\n\nimport ToMathlib.Control.OptionT\n\nimport Mathlib.Control.Lawful\n\nimport VCVio.OracleComp.OracleSpec\n\nnamespace OracleSpec\n\ninductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)\n\nnamespace OracleQuery\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β : Type v}\n\nend OracleQuery\n\nend OracleSpec\n\nopen OracleSpec\n\ndef OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β : Type v}\n\nsection lift\n\nend lift\n\nnotation \"$[0..\" n \"]\" => uniformFin n\n\nnotation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m\n\nexample : OracleComp probSpec ℕ := do\n  let x ← $[314⋯31415]; let y ← $[0⋯x]\n  return x + 2 * y\n\nsection construct\n\nvariable {C : OracleComp spec α → Type w}\n  (h_pure : (a : α) → C (pure a))\n  (h_query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n  (h_failure : C failure) (oa : OracleComp spec α)\n\nend construct\n\nsection noConfusion\n\nvariable (x : α) (y : β) (q : OracleQuery spec β) (oa : β → OracleComp spec α)\n\nend noConfusion\n\nsection mapM\n\nvariable {m : Type v → Type w} [Monad m]\n  (fail : {α : Type v} → m α) (qm : {α : Type v} → OracleQuery spec α → m α)\n\nend mapM\n\nsection inj", "target_theorem": "@[simp]\nlemma queryBind_inj (q q' : OracleQuery spec α) (ob ob' : α → OracleComp spec β) :\n    (q : OracleComp spec α) >>= ob = (q' : OracleComp spec α) >>= ob' ↔ q = q' ∧ ob = ob' :=", "ground_truth_proof": ":= by\n  simp only [OracleComp.liftM_def, OptionT.lift, OptionT.mk, FreeMonad.monad_pure_def,\n    FreeMonad.monad_bind_def, FreeMonad.bind_lift, OracleComp.bind_def, OptionT.bind,\n    FreeMonad.bind_roll, FreeMonad.bind_pure]\n  rw [FreeMonad.roll.injEq]\n  simp only [heq_eq_eq, true_and]", "nesting_depth": 3, "transitive_dep_count": 22, "subset_aristotle": false, "category": "Applied verif."}
{"id": 450, "thm_name": "OracleComp.probFailure_bind_eq_sub_mul", "thm_stmt": "lemma probFailure_bind_eq_sub_mul {oa : OracleComp spec α} {ob : α → OracleComp spec β}\n    (r : ℝ≥0∞) (h : ∀ x, [⊥ | ob x] = r) :\n    [⊥ | oa >>= ob] = 1 - (1 - [⊥ | oa]) * (1 - r)", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/DistSemantics/EvalDist.lean", "imports": ["import Mathlib.Probability.Distributions.Uniform", "import VCVio.OracleComp.SimSemantics.SimulateQ", "import VCVio.OracleComp.Traversal", "import ToMathlib.General"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "AddLECancellable", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "ENNReal", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "Eq", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}, {"name": "AddCommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ContinuousAdd", "module": "Mathlib.Topology.Algebra.Monoid.Defs"}, {"name": "Function.update", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Summable", "module": "Mathlib.Topology.Algebra.InfiniteSum.Defs"}, {"name": "T2Space", "module": "Mathlib.Topology.Separation.Hausdorff"}, {"name": "TopologicalSpace", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "Option.elimM", "module": "Init.Data.Option.Basic"}, {"name": "PMF", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}], "used_repo_defs": [{"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "cases", "content": "def cases {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive (tl.insertIdx 0 hd)) {m : ℕ} :\n    (v : Vector α m) → motive v := match hm : m with\n  | 0 => fun v => match v with | ⟨⟨[]⟩, rfl⟩ => v_empty\n  | n + 1 => fun v => match hv : v with\n    | ⟨⟨hd :: tl⟩, hSize⟩ => by admit /- proof elided -/"}], "lib_lemmas": [{"name": "ENNReal.summable", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "symm", "module": "Mathlib.Order.Defs.Unbundled"}, {"name": "le_add_self", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}, {"name": "le_of_le_of_eq", "module": "Init.Core"}, {"name": "ENNReal.eq_sub_of_add_eq", "module": "Mathlib.Data.ENNReal.Operations"}, {"name": "ENNReal.one_ne_top", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "ne_top_of_le_ne_top", "module": "Mathlib.Order.BoundedOrder.Basic"}, {"name": "PMF.coe_le_one", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "PMF.apply_ne_top", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "AddLECancellable.tsub_tsub_assoc", "module": "Mathlib.Algebra.Order.Sub.Unbundled.Basic"}, {"name": "ENNReal.mul_sub", "module": "Mathlib.Data.ENNReal.Operations"}, {"name": "ENNReal.tsum_add", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "ENNReal.tsum_le_tsum", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "ENNReal.tsum_mul_right", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "le_rfl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "mul_le_mul'", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "tsub_add_cancel_of_le", "module": "Mathlib.Algebra.Order.Sub.Unbundled.Basic"}, {"name": "tsum_congr", "module": "Mathlib.Topology.Algebra.InfiniteSum.Basic"}], "repo_lemmas": [{"name": "tsum_option", "content": "lemma tsum_option {α β : Type*} [AddCommMonoid α] [TopologicalSpace α]\n    [ContinuousAdd α] [T2Space α]\n    (f : Option β → α) (hf : Summable (Function.update f none 0)) :\n    ∑' x : Option β, f x = f none + ∑' x : β, f (some x)"}], "used_local_defs": [{"name": "OracleComp.probOutput", "content": "noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)"}, {"name": "OracleComp.probFailure", "content": "noncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none"}], "used_local_lemmas": [{"name": "OracleComp.probOutput_def", "content": "lemma probOutput_def (oa : OracleComp spec α) (x : α) :\n    [= x | oa] = (evalDist oa).run (some x)"}, {"name": "OracleComp.probFailure_add_tsum_probOutput", "content": "@[simp]\nlemma probFailure_add_tsum_probOutput (oa : OracleComp spec α) :\n    [⊥ | oa] + ∑' x, [= x | oa] = 1"}, {"name": "OracleComp.tsum_probOutput_add_probFailure", "content": "@[simp]\nlemma tsum_probOutput_add_probFailure (oa : OracleComp spec α) :\n    ∑' x, [= x | oa] + [⊥ | oa] = 1"}, {"name": "OracleComp.probFailure_le_one", "content": "@[simp] lemma probFailure_le_one : [⊥ | oa] ≤ 1"}, {"name": "OracleComp.tsum_probOutput_le_one", "content": "@[simp] lemma tsum_probOutput_le_one : ∑' x : α, [= x | oa] ≤ 1"}, {"name": "OracleComp.probOutput_ne_top", "content": "@[simp] lemma probOutput_ne_top : [= x | oa] ≠ ∞"}, {"name": "OracleComp.probFailure_ne_top", "content": "@[simp] lemma probFailure_ne_top : [⊥ | oa] ≠ ∞"}, {"name": "OracleComp.tsum_probOutput_eq_sub", "content": "lemma tsum_probOutput_eq_sub (oa : OracleComp spec α) :\n    ∑' x : α, [= x | oa] = 1 - [⊥ | oa]"}, {"name": "OracleComp.probFailure_eq_sub_tsum", "content": "lemma probFailure_eq_sub_tsum (oa : OracleComp spec α) :\n    [⊥ | oa] = 1 - ∑' x : α, [= x | oa]"}, {"name": "OracleComp.probFailure_bind_eq_tsum", "content": "lemma probFailure_bind_eq_tsum :\n    [⊥ | oa >>= ob] = [⊥ | oa] + ∑' x : α, [= x | oa] * [⊥ | ob x]"}], "local_ctx": "import VCVio.OracleComp.Traversal\n\nimport VCVio.OracleComp.SimSemantics.SimulateQ\n\nimport Mathlib.Probability.Distributions.Uniform\n\nimport ToMathlib.General\n\nopen OracleSpec Option ENNReal BigOperators\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {ι' : Type v} {spec' : OracleSpec ι'}\n  {α β γ : Type w} [spec.FiniteRange] [spec'.FiniteRange]\n\nsection evalDist\n\nend evalDist\n\nnoncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)\n\nnotation \"[=\" x \"|\" oa \"]\" => probOutput oa x\n\nnoncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none\n\nnotation \"[⊥\" \"|\" oa \"]\" => probFailure oa\n\nnotation \"[\" p \"|\" oa \"]\" => probEvent oa p\n\nsection bounds\n\nvariable {oa : OracleComp spec α} {x : α} {p : α → Prop}\n\nend bounds\n\nsection support\n\nvariable (oa : OracleComp spec α) (x : α) (p q : α → Prop)\n\nvariable {oa x p q}\n\nend support\n\nsection sums\n\nvariable (oa : OracleComp spec α) (p : α → Prop)\n\nend sums\n\nsection pure\n\nvariable (x : α)\n\nend pure\n\nsection bind\n\nvariable (oa : OracleComp spec α) (ob : α → OracleComp spec β)", "target_theorem": "lemma probFailure_bind_eq_sub_mul {oa : OracleComp spec α} {ob : α → OracleComp spec β}\n    (r : ℝ≥0∞) (h : ∀ x, [⊥ | ob x] = r) :\n    [⊥ | oa >>= ob] = 1 - (1 - [⊥ | oa]) * (1 - r) :=", "ground_truth_proof": ":= by\n  rw [probFailure_bind_eq_tsum]\n  rw [← tsum_probOutput_eq_sub]\n  rw [← ENNReal.tsum_mul_right]\n  have hl : ∀ x, [=x|oa] * [⊥|ob x] ≤ [=x|oa] :=\n    λ x ↦ le_of_le_of_eq (mul_le_mul' le_rfl probFailure_le_one) (mul_one _)\n  calc [⊥ | oa] + ∑' x, [= x | oa] * [⊥ | ob x]\n    _ = 1 - (∑' x, [= x | oa]) + (∑' x, [= x | oa] * [⊥ | ob x]) := by\n      rw [probFailure_eq_sub_tsum]\n    _ = 1 - (∑' x, [= x | oa] - ∑' x, [= x | oa] * [⊥ | ob x]) := by\n      exact Eq.symm (AddLECancellable.tsub_tsub_assoc\n        (by simp) tsum_probOutput_le_one (ENNReal.tsum_le_tsum hl))\n    _ = 1 - ∑' x, ([= x | oa] - [= x | oa] * [⊥ | ob x]) := by\n      refine congr_arg (1 - ·) (ENNReal.eq_sub_of_add_eq ?_ ?_).symm\n      · refine ne_top_of_le_ne_top one_ne_top ?_\n        refine le_trans ?_ (@tsum_probOutput_le_one _ _ _ _ oa)\n        refine ENNReal.tsum_le_tsum λ x ↦ ?_\n        exact hl x\n      rw [← ENNReal.tsum_add]\n      refine tsum_congr λ x ↦ tsub_add_cancel_of_le (hl x)\n    _ = 1 - ∑' x : α, [= x | oa] * (1 - r) := by\n      refine congr_arg (1 - ·) (tsum_congr λ x ↦ ?_)\n      rw [ENNReal.mul_sub (λ _ _ ↦ probOutput_ne_top), mul_one, ← h x]", "nesting_depth": 5, "transitive_dep_count": 56, "subset_aristotle": true, "category": "Applied verif."}
{"id": 451, "thm_name": "BindEquiv.map_bind_inv", "thm_stmt": "@[simp]\nlemma map_bind_inv (f : BindEquiv m n) {α β : Type u} (x : n α) (y : α → n β) :\n    f.invFun (x >>= y) = f.invFun x >>= (fun a => f.invFun (y a))", "lean_root": "VCV-io", "rel_path": "ToMathlib/Control/Monad/Equiv.lean", "imports": ["import Mathlib.Logic.Function.Defs", "import ToMathlib.Control.Monad.Hom"], "used_lib_defs": [{"name": "Function.LeftInverse", "module": "Init.Data.Function"}, {"name": "Function.RightInverse", "module": "Init.Data.Function"}, {"name": "Bind", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "NatHom", "content": "structure NatHom (m : Type u → Type v) (n : Type u → Type w) where\n  toFun : {α : Type u} → m α → n α"}], "lib_lemmas": [{"name": "Function.LeftInverse.injective", "module": "Init.Data.Function"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "NatEquiv", "content": "structure NatEquiv (m : Type u → Type v) (n : Type u → Type w) where\n  toFun : {α : Type u} → m α → n α\n  invFun : {α : Type u} → n α → m α\n  left_inv : ∀ {α}, Function.LeftInverse (@invFun α) (@toFun α) := by admit /- proof elided -/"}, {"name": "BindEquiv", "content": "structure BindEquiv (m : Type u → Type v) [Bind m] (n : Type u → Type w) [Bind n]\n    extends NatEquiv m n where\n  map_bind' {α β : Type u} (x : m α) (y : α → m β) :\n    toFun (x >>= y) = toFun x >>= (fun a => toFun (y a))"}], "used_local_lemmas": [], "local_ctx": "import ToMathlib.Control.Monad.Hom\n\nimport Mathlib.Logic.Function.Defs\n\nstructure NatEquiv (m : Type u → Type v) (n : Type u → Type w) where\n  toFun : {α : Type u} → m α → n α\n  invFun : {α : Type u} → n α → m α\n  left_inv : ∀ {α}, Function.LeftInverse (@invFun α) (@toFun α) := by admit /- proof elided -/\n\nnamespace NatEquiv\n\nvariable {m : Type u → Type v} {n : Type u → Type w} {p : Type u → Type z}\n\nend NatEquiv\n\nnamespace PureEquiv\n\nvariable {m : Type u → Type v} [Pure m] {n : Type u → Type w} [Pure n]\n\nend PureEquiv\n\nstructure BindEquiv (m : Type u → Type v) [Bind m] (n : Type u → Type w) [Bind n]\n    extends NatEquiv m n where\n  map_bind' {α β : Type u} (x : m α) (y : α → m β) :\n    toFun (x >>= y) = toFun x >>= (fun a => toFun (y a))\n\nnamespace BindEquiv\n\nvariable {m : Type u → Type v} [Bind m] {n : Type u → Type w} [Bind n]", "target_theorem": "@[simp]\nlemma map_bind_inv (f : BindEquiv m n) {α β : Type u} (x : n α) (y : α → n β) :\n    f.invFun (x >>= y) = f.invFun x >>= (fun a => f.invFun (y a)) :=", "ground_truth_proof": ":= by\n  -- We'll show f.toFun applied to both sides gives the same result\n  have h1 : f.toFun (f.invFun (x >>= y)) = x >>= y := f.right_inv (x >>= y)\n  have h2 : f.toFun (f.invFun x >>= (fun a => f.invFun (y a))) =\n            f.toFun (f.invFun x) >>= (fun a => f.toFun (f.invFun (y a))) := f.map_bind' _ _\n  have h3 : f.toFun (f.invFun x) = x := f.right_inv x\n  have h4 : ∀ a, f.toFun (f.invFun (y a)) = y a := fun a => f.right_inv (y a)\n  have h5 : f.toFun (f.invFun x >>= (fun a => f.invFun (y a))) = x >>= y := by\n    rw [h2, h3]\n    congr 1\n    ext a\n    exact h4 a\n  have h6 : f.toFun (f.invFun (x >>= y)) = f.toFun (f.invFun x >>= (fun a => f.invFun (y a))) := by\n    rw [h1, h5]\n  exact Function.LeftInverse.injective f.left_inv h6", "nesting_depth": 2, "transitive_dep_count": 7, "subset_aristotle": true, "category": "Applied verif."}
{"id": 452, "thm_name": "OracleComp.mul_le_probEvent_bind", "thm_stmt": "lemma mul_le_probEvent_bind {oa : OracleComp spec α} {ob : α → OracleComp spec β}\n    {p : α → Prop} {q : β → Prop} {r r' : ℝ≥0∞}\n    (h : r ≤ [p | oa]) (h' : ∀ x ∈ oa.support, p x → r' ≤ [q | ob x]) :\n    r * r' ≤ [q | oa >>= ob]", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/DistSemantics/EvalDist.lean", "imports": ["import Mathlib.Probability.Distributions.Uniform", "import VCVio.OracleComp.SimSemantics.SimulateQ", "import VCVio.OracleComp.Traversal", "import ToMathlib.General"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}, {"name": "ENNReal", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "Option.elimM", "module": "Init.Data.Option.Basic"}, {"name": "PMF", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}, {"name": "AddCommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ContinuousAdd", "module": "Mathlib.Topology.Algebra.Monoid.Defs"}, {"name": "Function.update", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Summable", "module": "Mathlib.Topology.Algebra.InfiniteSum.Defs"}, {"name": "T2Space", "module": "Mathlib.Topology.Separation.Hausdorff"}, {"name": "TopologicalSpace", "module": "Mathlib.Topology.Defs.Basic"}], "used_repo_defs": [{"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "support", "content": "def support (oa : OracleComp spec α) : Set α :=\n  oa.supportWhen fun _ => Set.univ"}, {"name": "supportWhen", "content": "def supportWhen (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Set α) : Set α :="}, {"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : OracleComp spec α → Type*}\n    (pure : (a : α) → C (pure a))\n    (query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.construct (Option.rec failure pure) query_bind oa"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : FreeMonad f α → Type*}\n    (pure : (x : α) → C (pure x))\n    (roll : {β : Type u} → (x : f β) → (r : β → FreeMonad f α) →\n      ((y : β) → C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | .pure x => pure x\n  | .roll x r => roll x _ (λ u ↦ FreeMonad.construct pure roll (r u))"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "cases", "content": "def cases {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive (tl.insertIdx 0 hd)) {m : ℕ} :\n    (v : Vector α m) → motive v := match hm : m with\n  | 0 => fun v => match v with | ⟨⟨[]⟩, rfl⟩ => v_empty\n  | n + 1 => fun v => match hv : v with\n    | ⟨⟨hd :: tl⟩, hSize⟩ => by admit /- proof elided -/"}], "lib_lemmas": [{"name": "ENNReal.summable", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "PMF.toOuterMeasure_bind_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Monad"}, {"name": "PMF.apply_ne_top", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "Option.some_injective", "module": "Mathlib.Data.Option.Basic"}, {"name": "PMF.toOuterMeasure_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "Set.indicator_image", "module": "Mathlib.Algebra.Notation.Indicator"}, {"name": "ENNReal.mul_le_mul_left", "module": "Mathlib.Data.ENNReal.Operations"}, {"name": "ENNReal.tsum_le_tsum", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "ENNReal.tsum_mul_right", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "Set.indicator_apply_le'", "module": "Mathlib.Algebra.Order.Group.Indicator"}, {"name": "Set.indicator_mul_const", "module": "Mathlib.Algebra.GroupWithZero.Indicator"}, {"name": "mul_le_mul_right'", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "zero_le'", "module": "Mathlib.Algebra.Order.GroupWithZero.Canonical"}], "repo_lemmas": [{"name": "tsum_option", "content": "lemma tsum_option {α β : Type*} [AddCommMonoid α] [TopologicalSpace α]\n    [ContinuousAdd α] [T2Space α]\n    (f : Option β → α) (hf : Summable (Function.update f none 0)) :\n    ∑' x : Option β, f x = f none + ∑' x : β, f (some x)"}], "used_local_defs": [{"name": "OracleComp.probOutput", "content": "noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)"}], "used_local_lemmas": [{"name": "OracleComp.evalDist_apply_some", "content": "lemma evalDist_apply_some (oa : OracleComp spec α) (x : α) :\n    (evalDist oa).run (some x) = [= x | oa]"}, {"name": "OracleComp.probEvent_def", "content": "lemma probEvent_def (oa : OracleComp spec α) (p : α → Prop) :\n    [p | oa] = (evalDist oa).run.toOuterMeasure (Option.some '' {x | p x})"}, {"name": "OracleComp.probEvent_eq_tsum_indicator", "content": "lemma probEvent_eq_tsum_indicator (oa : OracleComp spec α) (p : α → Prop) :\n    [p | oa] = ∑' x : α, {x | p x}.indicator ([= · | oa]) x"}, {"name": "OracleComp.probOutput_ne_top", "content": "@[simp] lemma probOutput_ne_top : [= x | oa] ≠ ∞"}, {"name": "OracleComp.probOutput_ne_zero", "content": "lemma probOutput_ne_zero (h : x ∈ oa.support) : [= x | oa] ≠ 0"}, {"name": "OracleComp.probEvent_bind_eq_tsum", "content": "lemma probEvent_bind_eq_tsum (q : β → Prop) :\n    [q | oa >>= ob] = ∑' x : α, [= x | oa] * [q | ob x]"}], "local_ctx": "import VCVio.OracleComp.Traversal\n\nimport VCVio.OracleComp.SimSemantics.SimulateQ\n\nimport Mathlib.Probability.Distributions.Uniform\n\nimport ToMathlib.General\n\nopen OracleSpec Option ENNReal BigOperators\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {ι' : Type v} {spec' : OracleSpec ι'}\n  {α β γ : Type w} [spec.FiniteRange] [spec'.FiniteRange]\n\nsection evalDist\n\nend evalDist\n\nnoncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)\n\nnotation \"[=\" x \"|\" oa \"]\" => probOutput oa x\n\nnotation \"[⊥\" \"|\" oa \"]\" => probFailure oa\n\nnotation \"[\" p \"|\" oa \"]\" => probEvent oa p\n\nsection bounds\n\nvariable {oa : OracleComp spec α} {x : α} {p : α → Prop}\n\nend bounds\n\nsection support\n\nvariable (oa : OracleComp spec α) (x : α) (p q : α → Prop)\n\nvariable {oa x p q}\n\nend support\n\nsection sums\n\nvariable (oa : OracleComp spec α) (p : α → Prop)\n\nend sums\n\nsection pure\n\nvariable (x : α)\n\nend pure\n\nsection bind\n\nvariable (oa : OracleComp spec α) (ob : α → OracleComp spec β)\n\nend bind\n\nsection mul_le_probEvent_bind", "target_theorem": "lemma mul_le_probEvent_bind {oa : OracleComp spec α} {ob : α → OracleComp spec β}\n    {p : α → Prop} {q : β → Prop} {r r' : ℝ≥0∞}\n    (h : r ≤ [p | oa]) (h' : ∀ x ∈ oa.support, p x → r' ≤ [q | ob x]) :\n    r * r' ≤ [q | oa >>= ob] :=", "ground_truth_proof": ":= by\n  rw [probEvent_bind_eq_tsum]\n  refine (mul_le_mul_right' h r').trans ?_\n  rw [probEvent_eq_tsum_indicator, ← ENNReal.tsum_mul_right]\n  refine ENNReal.tsum_le_tsum fun x => ?_\n  rw [← Set.indicator_mul_const]\n  by_cases hx : x ∈ oa.support\n  · refine Set.indicator_apply_le' (fun h => ?_) (fun _ => zero_le')\n    exact (ENNReal.mul_le_mul_left (probOutput_ne_zero _ _ hx) probOutput_ne_top).mpr (h' x hx h)\n  · simp [probOutput_eq_zero _ _ hx]", "nesting_depth": 5, "transitive_dep_count": 48, "subset_aristotle": true, "category": "Applied verif."}
{"id": 453, "thm_name": "OracleComp.probEvent_seq_map_eq_mul", "thm_stmt": "lemma probEvent_seq_map_eq_mul {ι : Type u} {spec : OracleSpec ι}\n    {α β γ : Type v} (f : α → β → γ) [spec.FiniteRange]\n    (oa : OracleComp spec α) (ob : OracleComp spec β)\n    (p : γ → Prop) (q1 : α → Prop) (q2 : β → Prop)\n    (h : ∀ x ∈ oa.support, ∀ y ∈ ob.support, p (f x y) ↔ q1 x ∧ q2 y) :\n    [p | f <$> oa <*> ob] = [q1 | oa] * [q2 | ob]", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/DistSemantics/Seq.lean", "imports": ["import VCVio.OracleComp.DistSemantics.Monad", "import VCVio.OracleComp.DistSemantics.EvalDist", "import VCVio.OracleComp.Support", "import ToMathlib.General"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Classical.decPred", "module": "Mathlib.Logic.Basic"}, {"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "ENNReal", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Function.comp", "module": "Init.Prelude"}, {"name": "Functor", "module": "Init.Prelude"}, {"name": "Set.image2", "module": "Mathlib.Data.Set.Operations"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Function.Injective2", "module": "Mathlib.Logic.Function.Basic"}], "used_repo_defs": [{"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "uniformFin", "content": "notation \"$[0..\" n \"]\" => uniformFin n"}, {"name": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m", "content": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m\n\nexample : OracleComp probSpec ℕ := do\n  let x ← $[314⋯31415]; let y ← $[0⋯x]\n  return x + 2 * y"}, {"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "probOutput", "content": "noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "support", "content": "def support (oa : OracleComp spec α) : Set α :=\n  oa.supportWhen fun _ => Set.univ"}, {"name": "supportWhen", "content": "def supportWhen (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Set α) : Set α :="}, {"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : OracleComp spec α → Type*}\n    (pure : (a : α) → C (pure a))\n    (query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.construct (Option.rec failure pure) query_bind oa"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : FreeMonad f α → Type*}\n    (pure : (x : α) → C (pure x))\n    (roll : {β : Type u} → (x : f β) → (r : β → FreeMonad f α) →\n      ((y : β) → C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | .pure x => pure x\n  | .roll x r => roll x _ (λ u ↦ FreeMonad.construct pure roll (r u))"}, {"name": "FiniteRange", "content": "class FiniteRange (spec : OracleSpec ι) where\n    range_inhabited' (i : ι) : Inhabited (spec.range i)\n    range_fintype' (i : ι) : Fintype (spec.range i)"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : OracleComp spec α → Prop}\n    (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "probEvent", "content": "noncomputable def probEvent (oa : OracleComp spec α) (p : α → Prop) : ℝ≥0∞ :=\n  (evalDist oa).run.toOuterMeasure (Option.some '' {x | p x})"}, {"name": "cases", "content": "def cases {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive (tl.insertIdx 0 hd)) {m : ℕ} :\n    (v : Vector α m) → motive v := match hm : m with\n  | 0 => fun v => match v with | ⟨⟨[]⟩, rfl⟩ => v_empty\n  | n + 1 => fun v => match hv : v with\n    | ⟨⟨hd :: tl⟩, hSize⟩ => by admit /- proof elided -/"}, {"name": "probFailure", "content": "noncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none"}, {"name": "uniformFin", "content": "@[reducible, inline] def uniformFin (n : ℕ) : ProbComp (Fin (n + 1)) :=\n  unifSpec.query n ()"}, {"name": "ProbComp", "content": "abbrev ProbComp : Type z → Type (max z 1) := OracleComp unifSpec"}, {"name": "unifSpec", "content": "@[inline, reducible] def unifSpec : OracleSpec.{0,0} ℕ :=\n  λ n ↦ (Unit, Fin (n + 1))"}, {"name": "uniformFin'", "content": "@[reducible, inline] def uniformFin' (n m : ℕ) : OracleComp probSpec (Fin (m + 1)) :=\n  probSpec.query m n"}, {"name": "probSpec", "content": "def probSpec : OracleSpec.{0,0} ℕ :=\n  fun n => (ℕ, Fin (n + 1))"}, {"name": "probOutput", "content": "notation \"Pr[=\" x \"|\" mx \"]\" => probOutput mx x"}, {"name": "probEvent", "content": "notation \"Pr[\" p \"|\" mx \"]\" => probEvent mx p"}, {"name": "probFailure", "content": "notation \"Pr[⊥\" \"|\" mx \"]\" => probFailure mx"}, {"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "probEvent", "content": "notation \"[\" p \"|\" oa \"]\" => probEvent oa p"}, {"name": "uniformFin", "content": "notation \"$[0..\" n \"]\" => uniformFin n"}, {"name": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m", "content": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m\n\nexample : OracleComp probSpec ℕ := do\n  let x ← $[314⋯31415]; let y ← $[0⋯x]\n  return x + 2 * y"}], "lib_lemmas": [{"name": "ENNReal.tsum_mul_left", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "map_eq_bind_pure_comp", "module": "Mathlib.Control.Monad.Basic"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "seq_eq_bind", "module": "Init.Control.Lawful.Basic"}, {"name": "ENNReal.tsum_prod", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "Prod.eq_iff_fst_eq_snd_eq", "module": "Mathlib.Data.Prod.Basic"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "tsum_eq_single", "module": "Mathlib.Topology.Algebra.InfiniteSum.Basic"}, {"name": "seq_eq_bind_map", "module": "Init.Control.Lawful.Basic"}, {"name": "Set.biUnion_and'", "module": "Mathlib.Data.Set.Lattice"}, {"name": "Set.ext_iff", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.iUnion_exists", "module": "Mathlib.Data.Set.Lattice"}, {"name": "Set.iUnion_iUnion_eq_right", "module": "Mathlib.Data.Set.Lattice"}, {"name": "Set.mem_iUnion", "module": "Mathlib.Order.SetNotation"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_image2", "module": "Mathlib.Data.Set.Operations"}, {"name": "exists_prop", "module": "Init.PropLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "Function.uncurry_apply_pair", "module": "Init.Data.Function"}, {"name": "Functor.map_map", "module": "Init.Control.Lawful.Basic"}, {"name": "Set.image2_mk_eq_prod", "module": "Mathlib.Data.Set.NAry"}, {"name": "Set.image_uncurry_prod", "module": "Mathlib.Data.Set.NAry"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "map_seq", "module": "Mathlib.Control.Basic"}, {"name": "ENNReal.tsum_mul_right", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "ENNReal.tsum_prod'", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "ite_mul", "module": "Mathlib.Algebra.Notation.Defs"}, {"name": "tsum_congr", "module": "Mathlib.Topology.Algebra.InfiniteSum.Basic"}], "repo_lemmas": [{"name": "probOutput_bind_eq_tsum", "content": "lemma probOutput_bind_eq_tsum (y : β) :\n    [= y | oa >>= ob] = ∑' x : α, [= x | oa] * [= y | ob x]"}, {"name": "probEvent_pure", "content": "@[simp]\nlemma probEvent_pure (p : α → Prop) [DecidablePred p] :\n    [p | (pure x : OracleComp spec α)] = if p x then 1 else 0"}, {"name": "probOutput_pure_self", "content": "@[simp]\nlemma probOutput_pure_self (x : α) : [= x | (pure x : OracleComp spec α)] = 1"}, {"name": "support_map", "content": "@[simp] lemma support_map (oa : OracleComp spec α) (f : α → β) :\n    (f <$> oa).support = f '' oa.support"}, {"name": "support_bind", "content": "@[simp] lemma support_bind (oa : OracleComp spec α) (ob : α → OracleComp spec β) :\n    (oa >>= ob).support = ⋃ x ∈ oa.support, (ob x).support"}, {"name": "support_pure", "content": "@[simp] lemma support_pure (x : α) :\n    (pure x : OracleComp spec α).support = {x}"}, {"name": "probEvent_congr'", "content": "lemma probEvent_congr' {p q : α → Prop} {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h1 : ∀ x, x ∈ oa.support → x ∈ oa'.support → (p x ↔ q x))\n    (h2 : evalDist oa = evalDist oa') : [p | oa] = [q | oa']"}, {"name": "mem_support_iff_of_evalDist_eq", "content": "lemma mem_support_iff_of_evalDist_eq {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h : evalDist oa = evalDist oa') (x : α) : x ∈ oa.support ↔ x ∈ oa'.support"}, {"name": "mem_support_iff_probOutput_ne_zero", "content": "lemma mem_support_iff_probOutput_ne_zero : x ∈ oa.support ↔ [= x | oa] ≠ 0"}, {"name": "probOutput_eq_zero_iff", "content": "@[simp low]\nlemma probOutput_eq_zero_iff : [= x | oa] = 0 ↔ x ∉ oa.support"}, {"name": "mem_support_evalDist_iff", "content": "@[simp]\nlemma mem_support_evalDist_iff (oa : OracleComp spec α) (x : α) :\n    some x ∈ (evalDist oa).run.support ↔ x ∈ oa.support"}, {"name": "probOutput_def", "content": "lemma probOutput_def (oa : OracleComp spec α) (x : α) :\n    [= x | oa] = (evalDist oa).run (some x)"}, {"name": "probEvent_eq_tsum_indicator", "content": "lemma probEvent_eq_tsum_indicator (oa : OracleComp spec α) (p : α → Prop) :\n    [p | oa] = ∑' x : α, {x | p x}.indicator ([= · | oa]) x"}, {"name": "probEvent_def", "content": "lemma probEvent_def (oa : OracleComp spec α) (p : α → Prop) :\n    [p | oa] = (evalDist oa).run.toOuterMeasure (Option.some '' {x | p x})"}, {"name": "probOutput_congr", "content": "lemma probOutput_congr {x y : α} {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h1 : x = y) (h2 : evalDist oa = evalDist oa') : [= x | oa] = [= y | oa']"}, {"name": "probEvent_comp", "content": "lemma probEvent_comp (q : β → Prop) : [q ∘ f | oa] = [q | f <$> oa]"}, {"name": "probEvent_map", "content": "@[simp]\nlemma probEvent_map (q : β → Prop) : [q | f <$> oa] = [q ∘ f | oa]"}, {"name": "evalDist_map", "content": "@[simp]\nlemma evalDist_map (oa : OracleComp spec α) (f : α → β) :\n    evalDist (f <$> oa) = f <$> (evalDist oa)"}, {"name": "probEvent_eq_tsum_ite", "content": "lemma probEvent_eq_tsum_ite (oa : OracleComp spec α) (p : α → Prop) [DecidablePred p] :\n    [p | oa] = ∑' x : α, if p x then [= x | oa] else 0"}, {"name": "probEvent_ext", "content": "lemma probEvent_ext (h : ∀ x ∈ oa.support, p x ↔ q x) : [p | oa] = [q | oa]"}, {"name": "probEvent_mono", "content": "lemma probEvent_mono (h : ∀ x ∈ oa.support, p x → q x) : [p | oa] ≤ [q | oa]"}, {"name": "evalDist_apply_eq_zero_iff", "content": "@[simp]\nlemma evalDist_apply_eq_zero_iff (x : Option α) :\n    (evalDist oa).run x = 0 ↔ x.rec ([⊥ | oa] = 0) (· ∉ oa.support)"}, {"name": "probFailure_def", "content": "lemma probFailure_def (oa : OracleComp spec α) :\n    [⊥ | oa] = (evalDist oa).run none"}, {"name": "Prod.mk.injective2", "content": "lemma Prod.mk.injective2 {α β : Type*} :\n    Function.Injective2 (Prod.mk : α → β → α × β)"}, {"name": "probEvent_bind_eq_tsum", "content": "lemma probEvent_bind_eq_tsum (q : β → Prop) :\n    [q | oa >>= ob] = ∑' x : α, [= x | oa] * [q | ob x]"}], "used_local_defs": [], "used_local_lemmas": [{"name": "OracleComp.support_seq", "content": "@[simp low]\nlemma support_seq : (og <*> oa).support = ⋃ g ∈ og.support, g '' oa.support"}, {"name": "OracleComp.support_seq_map_eq_image2", "content": "@[simp low + 1]\nlemma support_seq_map_eq_image2 :\n    (f <$> oa <*> ob).support = Set.image2 f oa.support ob.support"}, {"name": "OracleComp.probOutput_seq_map_eq_tsum", "content": "lemma probOutput_seq_map_eq_tsum [spec.FiniteRange]\n    (z : γ) : [= z | f <$> oa <*> ob] = ∑' (x : α) (y : β),\n      [= x | oa] * [= y | ob] * [= z | (pure (f x y) : OracleComp spec γ)]"}, {"name": "OracleComp.probEvent_seq_map_eq_probEvent_comp_uncurry", "content": "lemma probEvent_seq_map_eq_probEvent_comp_uncurry [spec.FiniteRange]\n    (p : γ → Prop) : [p | f <$> oa <*> ob] =\n      [p ∘ f.uncurry | Prod.mk <$> oa <*> ob]"}, {"name": "OracleComp.probEvent_seq_map_eq_probEvent", "content": "lemma probEvent_seq_map_eq_probEvent [spec.FiniteRange] (p : γ → Prop) :\n    [p | f <$> oa <*> ob] = [λ z ↦ p (f z.1 z.2) | Prod.mk <$> oa <*> ob]"}, {"name": "OracleComp.probOutput_seq_map_eq_mul_of_injective2", "content": "lemma probOutput_seq_map_eq_mul_of_injective2 [spec.FiniteRange]\n    (hf : f.Injective2) (x : α) (y : β) :\n    [= f x y | f <$> oa <*> ob] = [= x | oa] * [= y | ob]"}], "local_ctx": "import VCVio.OracleComp.DistSemantics.Monad\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β γ : Type v}\n\nvariable (oa : OracleComp spec α) (og : OracleComp spec (α → β))\n\nsection seq_map\n\nvariable (oa : OracleComp spec α) (ob : OracleComp spec β) (f : α → β → γ)\n\nsection swap\n\nend swap\n\nsection injective2\n\nend injective2", "target_theorem": "lemma probEvent_seq_map_eq_mul {ι : Type u} {spec : OracleSpec ι}\n    {α β γ : Type v} (f : α → β → γ) [spec.FiniteRange]\n    (oa : OracleComp spec α) (ob : OracleComp spec β)\n    (p : γ → Prop) (q1 : α → Prop) (q2 : β → Prop)\n    (h : ∀ x ∈ oa.support, ∀ y ∈ ob.support, p (f x y) ↔ q1 x ∧ q2 y) :\n    [p | f <$> oa <*> ob] = [q1 | oa] * [q2 | ob] :=", "ground_truth_proof": ":= by\n  have : DecidablePred q1 := Classical.decPred q1\n  have : DecidablePred q2 := Classical.decPred q2\n  rw [probEvent_seq_map_eq_probEvent]\n  calc [λ z : α × β ↦ p (f z.1 z.2) | Prod.mk <$> oa <*> ob] =\n      [λ z ↦ q1 z.1 ∧ q2 z.2 | Prod.mk <$> oa <*> ob] :=\n        probEvent_ext <| by simpa using λ x y hx hy ↦ h x hx y hy\n    _ = ∑' (x : α) (y : β), if q1 x ∧ q2 y then [= (x, y) | Prod.mk <$> oa <*> ob] else 0 := by\n      rw [probEvent_eq_tsum_ite, ENNReal.tsum_prod']\n    _ = ∑' (x : α) (y : β), if q1 x ∧ q2 y then [= x | oa] * [= y | ob] else 0 := by\n      simp_rw [probOutput_seq_map_eq_mul_of_injective2 oa ob Prod.mk Prod.mk.injective2]\n    _ = ∑' x : α, if q1 x then [= x | oa] * (∑' y : β, if q2 y then [= y | ob] else 0) else 0 :=\n      tsum_congr (λ x ↦ by by_cases hx : q1 x <;> simp [hx, ← ENNReal.tsum_mul_left])\n    _ = ∑' x : α, if q1 x then [= x | oa] * [q2 | ob] else 0 := by rw [probEvent_eq_tsum_ite]\n    _ = [q1 | oa] * [q2 | ob] := by\n      simp only [probEvent_eq_tsum_ite oa, ← ENNReal.tsum_mul_right, ite_mul, zero_mul]", "nesting_depth": 8, "transitive_dep_count": 96, "subset_aristotle": true, "category": "Applied verif."}
{"id": 454, "thm_name": "PMF.probOutput_eq", "thm_stmt": "@[simp] lemma probOutput_eq : probOutput p = p", "lean_root": "VCV-io", "rel_path": "VCVio/EvalDist/Basic.lean", "imports": ["import ToMathlib.General", "import Mathlib.Probability.ProbabilityMassFunction.Monad"], "used_lib_defs": [{"name": "OptionT", "module": "Init.Control.Option"}, {"name": "PMF", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "Monad", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "liftM", "module": "Init.Prelude"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "id", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "probEvent", "content": "notation \"Pr[\" p \"|\" mx \"]\" => probEvent mx p"}, {"name": "probOutput", "content": "notation \"Pr[=\" x \"|\" mx \"]\" => probOutput mx x"}, {"name": "macro_rules (kind := probEventBinding1)", "content": "macro_rules (kind := probEventBinding1)\n  | `(Pr[$cond:term | $var:ident ← $src:term]) => `(Pr[fun $var => $cond | $src])"}, {"name": "macro_rules (kind := probEventBinding2)", "content": "macro_rules (kind := probEventBinding2)\n  \n  | `(Pr{{$items*}}[$t]) => `(probOutput (do $items:doSeqItem* return $t:term) True)\n  \n  | `(Pr{$items*}[$t]) => `(probOutput (do $items:doSeqItem* return $t:term) True)"}, {"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "probEvent", "content": "notation \"[\" p \"|\" oa \"]\" => probEvent oa p"}], "lib_lemmas": [{"name": "OptionT.run_mk", "module": "Init.Control.Lawful.Instances"}, {"name": "PMF.map_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Constructions"}, {"name": "PMF.pure_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Monad"}, {"name": "mul_ite", "module": "Mathlib.Algebra.Notation.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "tsum_eq_single", "module": "Mathlib.Topology.Algebra.InfiniteSum.Basic"}], "repo_lemmas": [{"name": "PMF.monad_pure_eq_pure", "content": "@[simp]\nlemma PMF.monad_pure_eq_pure {α : Type u} (x : α) :\n    (Pure.pure x : PMF α) = PMF.pure x"}, {"name": "PMF.monad_bind_eq_bind", "content": "@[simp]\nlemma PMF.monad_bind_eq_bind {α β : Type u}\n      (p : PMF α) (q : α → PMF β) : p >>= q = p.bind q"}], "used_local_defs": [{"name": "SPMF", "content": "@[reducible] def SPMF : Type u → Type u := OptionT PMF"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "probOutput", "content": "def probOutput [HasEvalDist m] (mx : m α) (x : α) : ℝ≥0∞ := evalDist mx x"}, {"name": "probEvent", "content": "noncomputable def probEvent [HasEvalDist m] (mx : m α) (p : α → Prop) : ℝ≥0∞ :=\n  (evalDist mx).run.toOuterMeasure (some '' {x | p x})"}, {"name": "SPMF.hasEvalDist", "content": "instance hasEvalDist : HasEvalDist SPMF where\n  evalDist := id\n  evalDist_pure _ := rfl\n  evalDist_bind _ _ := rfl"}, {"name": "PMF.hasEvalDist", "content": "noncomputable instance hasEvalDist : HasEvalDist PMF where\n  evalDist p := OptionT.mk p\n  evalDist_pure _ := by admit /- proof elided -/"}], "used_local_lemmas": [{"name": "probOutput_def", "content": "lemma probOutput_def (mx : m α) (x : α) : Pr[= x | mx] = (evalDist mx).run (some x)"}, {"name": "PMF.evalDist_eq", "content": "@[simp] lemma evalDist_eq : evalDist p = liftM p"}], "local_ctx": "import Mathlib.Probability.ProbabilityMassFunction.Monad\n\nimport ToMathlib.General\n\nopen ENNReal\n\nvariable {α β γ : Type u} {m : Type u → Type v} [Monad m]\n\n@[reducible] def SPMF : Type u → Type u := OptionT PMF\n\nnamespace SPMF\n\nend SPMF\n\nclass HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)\n\ndef probOutput [HasEvalDist m] (mx : m α) (x : α) : ℝ≥0∞ := evalDist mx x\n\nnoncomputable def probEvent [HasEvalDist m] (mx : m α) (p : α → Prop) : ℝ≥0∞ :=\n  (evalDist mx).run.toOuterMeasure (some '' {x | p x})\n\nnotation \"Pr[=\" x \"|\" mx \"]\" => probOutput mx x\n\nnotation \"Pr[\" p \"|\" mx \"]\" => probEvent mx p\n\nnotation \"Pr[⊥\" \"|\" mx \"]\" => probFailure mx\n\nvariable [HasEvalDist m]\n\nsection sums\n\nend sums\n\nsection bounds\n\nvariable {mx : m α} {mxe : OptionT m α} {x : α} {p : α → Prop}\n\nend bounds\n\nsection bind\n\nvariable (mx : m α) (my : α → m β)\n\nend bind\n\nnamespace SPMF\n\ninstance hasEvalDist : HasEvalDist SPMF where\n  evalDist := id\n  evalDist_pure _ := rfl\n  evalDist_bind _ _ := rfl\n\nvariable (p : SPMF α) (x : α)\n\nend SPMF\n\nnamespace PMF\n\nnoncomputable instance hasEvalDist : HasEvalDist PMF where\n  evalDist p := OptionT.mk p\n  evalDist_pure _ := by admit /- proof elided -/\n\nvariable (p : PMF α) (x : α)", "target_theorem": "@[simp] lemma probOutput_eq : probOutput p = p :=", "ground_truth_proof": ":= by\n  refine funext fun x => ?_\n  simp only [probOutput_def, evalDist_eq, monad_pure_eq_pure, monad_bind_eq_bind, OptionT.run_mk,\n    pure_apply, Option.some.injEq, mul_ite, mul_one, mul_zero]\n  simp\n  refine (PMF.map_apply _ _ _).trans ?_\n  refine (tsum_eq_single x ?_).trans ?_\n  · simp\n    refine fun x h h' => ?_\n    refine (h h'.symm).elim\n  simp only [↓reduceIte]", "nesting_depth": 2, "transitive_dep_count": 20, "subset_aristotle": true, "category": "Applied verif."}
{"id": 455, "thm_name": "OracleComp.probEvent_congr'", "thm_stmt": "lemma probEvent_congr' {p q : α → Prop} {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h1 : ∀ x, x ∈ oa.support → x ∈ oa'.support → (p x ↔ q x))\n    (h2 : evalDist oa = evalDist oa') : [p | oa] = [q | oa']", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/DistSemantics/EvalDist.lean", "imports": ["import Mathlib.Probability.Distributions.Uniform", "import VCVio.OracleComp.SimSemantics.SimulateQ", "import VCVio.OracleComp.Traversal", "import ToMathlib.General"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.indicator", "module": "Mathlib.Algebra.Notation.Indicator"}, {"name": "PMF", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "Option.elimM", "module": "Init.Data.Option.Basic"}, {"name": "OptionT.lift", "module": "Init.Control.Option"}, {"name": "ENNReal", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}, {"name": "AddCommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ContinuousAdd", "module": "Mathlib.Topology.Algebra.Monoid.Defs"}, {"name": "Function.update", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Summable", "module": "Mathlib.Topology.Algebra.InfiniteSum.Defs"}, {"name": "T2Space", "module": "Mathlib.Topology.Separation.Hausdorff"}, {"name": "TopologicalSpace", "module": "Mathlib.Topology.Defs.Basic"}], "used_repo_defs": [{"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "support", "content": "def support (oa : OracleComp spec α) : Set α :=\n  oa.supportWhen fun _ => Set.univ"}, {"name": "supportWhen", "content": "def supportWhen (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Set α) : Set α :="}, {"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : OracleComp spec α → Type*}\n    (pure : (a : α) → C (pure a))\n    (query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.construct (Option.rec failure pure) query_bind oa"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : FreeMonad f α → Type*}\n    (pure : (x : α) → C (pure x))\n    (roll : {β : Type u} → (x : f β) → (r : β → FreeMonad f α) →\n      ((y : β) → C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | .pure x => pure x\n  | .roll x r => roll x _ (λ u ↦ FreeMonad.construct pure roll (r u))"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : OracleComp spec α → Prop}\n    (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "cases", "content": "def cases {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive (tl.insertIdx 0 hd)) {m : ℕ} :\n    (v : Vector α m) → motive v := match hm : m with\n  | 0 => fun v => match v with | ⟨⟨[]⟩, rfl⟩ => v_empty\n  | n + 1 => fun v => match hv : v with\n    | ⟨⟨hd :: tl⟩, hSize⟩ => by admit /- proof elided -/"}], "lib_lemmas": [{"name": "PMF.apply_eq_zero_iff", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "ENNReal.summable", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "Option.some_injective", "module": "Mathlib.Data.Option.Basic"}, {"name": "PMF.toOuterMeasure_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "Set.indicator_image", "module": "Mathlib.Algebra.Notation.Indicator"}, {"name": "if_neg", "module": "Init.Core"}, {"name": "tsum_congr", "module": "Mathlib.Topology.Algebra.InfiniteSum.Basic"}], "repo_lemmas": [{"name": "tsum_option", "content": "lemma tsum_option {α β : Type*} [AddCommMonoid α] [TopologicalSpace α]\n    [ContinuousAdd α] [T2Space α]\n    (f : Option β → α) (hf : Summable (Function.update f none 0)) :\n    ∑' x : Option β, f x = f none + ∑' x : β, f (some x)"}], "used_local_defs": [{"name": "OracleComp.probOutput", "content": "noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)"}], "used_local_lemmas": [{"name": "OracleComp.probOutput_def", "content": "lemma probOutput_def (oa : OracleComp spec α) (x : α) :\n    [= x | oa] = (evalDist oa).run (some x)"}, {"name": "OracleComp.probEvent_def", "content": "lemma probEvent_def (oa : OracleComp spec α) (p : α → Prop) :\n    [p | oa] = (evalDist oa).run.toOuterMeasure (Option.some '' {x | p x})"}, {"name": "OracleComp.probEvent_eq_tsum_indicator", "content": "lemma probEvent_eq_tsum_indicator (oa : OracleComp spec α) (p : α → Prop) :\n    [p | oa] = ∑' x : α, {x | p x}.indicator ([= · | oa]) x"}, {"name": "OracleComp.mem_support_evalDist_iff", "content": "@[simp]\nlemma mem_support_evalDist_iff (oa : OracleComp spec α) (x : α) :\n    some x ∈ (evalDist oa).run.support ↔ x ∈ oa.support"}, {"name": "OracleComp.probOutput_eq_zero_iff", "content": "@[simp low]\nlemma probOutput_eq_zero_iff : [= x | oa] = 0 ↔ x ∉ oa.support"}, {"name": "OracleComp.mem_support_iff_probOutput_ne_zero", "content": "lemma mem_support_iff_probOutput_ne_zero : x ∈ oa.support ↔ [= x | oa] ≠ 0"}, {"name": "OracleComp.mem_support_iff_of_evalDist_eq", "content": "lemma mem_support_iff_of_evalDist_eq {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h : evalDist oa = evalDist oa') (x : α) : x ∈ oa.support ↔ x ∈ oa'.support"}, {"name": "OracleComp.probOutput_congr", "content": "lemma probOutput_congr {x y : α} {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h1 : x = y) (h2 : evalDist oa = evalDist oa') : [= x | oa] = [= y | oa']"}], "local_ctx": "import VCVio.OracleComp.Traversal\n\nimport VCVio.OracleComp.SimSemantics.SimulateQ\n\nimport Mathlib.Probability.Distributions.Uniform\n\nimport ToMathlib.General\n\nopen OracleSpec Option ENNReal BigOperators\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {ι' : Type v} {spec' : OracleSpec ι'}\n  {α β γ : Type w} [spec.FiniteRange] [spec'.FiniteRange]\n\nsection evalDist\n\nend evalDist\n\nnoncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)\n\nnotation \"[=\" x \"|\" oa \"]\" => probOutput oa x\n\nnotation \"[⊥\" \"|\" oa \"]\" => probFailure oa\n\nnotation \"[\" p \"|\" oa \"]\" => probEvent oa p\n\nsection bounds\n\nvariable {oa : OracleComp spec α} {x : α} {p : α → Prop}\n\nend bounds\n\nsection support\n\nvariable (oa : OracleComp spec α) (x : α) (p q : α → Prop)\n\nvariable {oa x p q}\n\nend support\n\nsection sums\n\nvariable (oa : OracleComp spec α) (p : α → Prop)\n\nend sums", "target_theorem": "lemma probEvent_congr' {p q : α → Prop} {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h1 : ∀ x, x ∈ oa.support → x ∈ oa'.support → (p x ↔ q x))\n    (h2 : evalDist oa = evalDist oa') : [p | oa] = [q | oa'] :=", "ground_truth_proof": ":= by\n  have h : ∀ x, x ∈ oa.support ↔ x ∈ oa'.support := mem_support_iff_of_evalDist_eq h2\n  have h' : ∀ x, [= x | oa] = [= x | oa'] := λ x ↦ probOutput_congr rfl h2\n  rw [probEvent_eq_tsum_indicator, probEvent_eq_tsum_indicator]\n  refine tsum_congr λ x ↦ ?_\n  simp [Set.indicator, h']\n  by_cases hp : p x\n  · by_cases hq : q x\n    · simp [hp, hq]\n    · simp [hp, hq, h]\n      refine λ hoa ↦ hq ?_\n      refine (h1 _ ?_ hoa).1 hp\n      refine (h _).2 hoa\n  · by_cases hq : q x\n    · simp [hp, hq]\n      simp [h] at h1\n      intro hoa\n      specialize h1 _ hoa\n      tauto\n    · rw [if_neg hp, if_neg hq]", "nesting_depth": 5, "transitive_dep_count": 48, "subset_aristotle": true, "category": "Applied verif."}
{"id": 456, "thm_name": "PureEquiv.map_pure_inv", "thm_stmt": "@[simp]\nlemma map_pure_inv (f : PureEquiv m n) {α : Type u} (x : α) :\n    f.invFun (pure x) = (pure x : m α)", "lean_root": "VCV-io", "rel_path": "ToMathlib/Control/Monad/Equiv.lean", "imports": ["import Mathlib.Logic.Function.Defs", "import ToMathlib.Control.Monad.Hom"], "used_lib_defs": [{"name": "Function.LeftInverse", "module": "Init.Data.Function"}, {"name": "Function.RightInverse", "module": "Init.Data.Function"}, {"name": "Pure", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "NatHom", "content": "structure NatHom (m : Type u → Type v) (n : Type u → Type w) where\n  toFun : {α : Type u} → m α → n α"}], "lib_lemmas": [{"name": "Function.LeftInverse.injective", "module": "Init.Data.Function"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "NatEquiv", "content": "structure NatEquiv (m : Type u → Type v) (n : Type u → Type w) where\n  toFun : {α : Type u} → m α → n α\n  invFun : {α : Type u} → n α → m α\n  left_inv : ∀ {α}, Function.LeftInverse (@invFun α) (@toFun α) := by admit /- proof elided -/"}, {"name": "PureEquiv", "content": "structure PureEquiv (m : Type u → Type v) [Pure m] (n : Type u → Type w) [Pure n]\n    extends NatEquiv m n where\n  map_pure' {α : Type u} (x : α) : toFun (pure x) = (pure x : n α)"}], "used_local_lemmas": [], "local_ctx": "import ToMathlib.Control.Monad.Hom\n\nimport Mathlib.Logic.Function.Defs\n\nstructure NatEquiv (m : Type u → Type v) (n : Type u → Type w) where\n  toFun : {α : Type u} → m α → n α\n  invFun : {α : Type u} → n α → m α\n  left_inv : ∀ {α}, Function.LeftInverse (@invFun α) (@toFun α) := by admit /- proof elided -/\n\nnamespace NatEquiv\n\nvariable {m : Type u → Type v} {n : Type u → Type w} {p : Type u → Type z}\n\nend NatEquiv\n\nstructure PureEquiv (m : Type u → Type v) [Pure m] (n : Type u → Type w) [Pure n]\n    extends NatEquiv m n where\n  map_pure' {α : Type u} (x : α) : toFun (pure x) = (pure x : n α)\n\nnamespace PureEquiv\n\nvariable {m : Type u → Type v} [Pure m] {n : Type u → Type w} [Pure n]", "target_theorem": "@[simp]\nlemma map_pure_inv (f : PureEquiv m n) {α : Type u} (x : α) :\n    f.invFun (pure x) = (pure x : m α) :=", "ground_truth_proof": ":= by\n  have h1 : f.toFun (f.invFun (pure x)) = pure x := f.right_inv (pure x)\n  have h2 : f.toFun (pure x) = pure x := f.map_pure' x\n  have h3 : f.toFun (f.invFun (pure x)) = f.toFun (pure x) := by rw [h1, h2]\n  exact Function.LeftInverse.injective f.left_inv h3", "nesting_depth": 1, "transitive_dep_count": 7, "subset_aristotle": true, "category": "Applied verif."}
{"id": 457, "thm_name": "OracleComp.isQueryBound_iff_probEvent", "thm_stmt": "lemma isQueryBound_iff_probEvent [spec.FiniteRange] {oa : OracleComp spec α} {qb : ι → ℕ} :\n    IsQueryBound oa qb ↔\n      [(· ≤ qb) | snd <$> (simulateQ countingOracle oa).run <|> return 0] = 1", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/QueryBound.lean", "imports": ["import VCVio.OracleComp.DistSemantics.Alternative", "import VCVio.OracleComp.DistSemantics.EvalDist", "import VCVio.OracleComp.QueryTracking.CountingOracle"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "Bind", "module": "Init.Prelude"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "AlternativeMonad", "module": "Batteries.Control.AlternativeMonad"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.getM", "module": "Init.Data.Option.Basic"}, {"name": "WriterT", "module": "Mathlib.Control.Monad.Writer"}, {"name": "Prod.snd", "module": "Init.Prelude"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Iff", "module": "Init.Core"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}], "used_repo_defs": [{"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "simulateQ", "content": "def simulateQ [AlternativeMonad m] (so : QueryImpl spec m) (oa : OracleComp spec α) : m α :=\n  do Option.getM (← FreeMonad.mapM oa.run so.impl)"}, {"name": "QueryImpl.Inhabited", "content": "instance QueryImpl.Inhabited [∀ i, Inhabited (spec.range i)] [Pure m] :\n  Inhabited (QueryImpl spec m) := ⟨{impl q := pure q.defaultOutput}⟩"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "mapM", "content": "protected def mapM [Pure m] [Bind m] :\n    (oa : FreeMonad f α) → (s : {α : Type u} → f α → m α) → m α\n  | .pure x, _ => pure x\n  | .roll x r, s => s x >>= λ u ↦ (r u).mapM s"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "countingOracle", "content": "def countingOracle : QueryImpl spec (WriterT (QueryCount spec) (OracleComp spec)) :=\n  idOracle.withCounting"}, {"name": "QueryCount", "content": "@[reducible] def QueryCount (_spec : OracleSpec ι) : Type u := ι → ℕ"}, {"name": "FiniteRange", "content": "class FiniteRange (spec : OracleSpec ι) where\n    range_inhabited' (i : ι) : Inhabited (spec.range i)\n    range_fintype' (i : ι) : Fintype (spec.range i)"}, {"name": "probFailure", "content": "noncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "probEvent", "content": "noncomputable def probEvent (oa : OracleComp spec α) (p : α → Prop) : ℝ≥0∞ :=\n  (evalDist oa).run.toOuterMeasure (Option.some '' {x | p x})"}, {"name": "support", "content": "def support (oa : OracleComp spec α) : Set α :=\n  oa.supportWhen fun _ => Set.univ"}, {"name": "supportWhen", "content": "def supportWhen (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Set α) : Set α :="}, {"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : OracleComp spec α → Type*}\n    (pure : (a : α) → C (pure a))\n    (query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.construct (Option.rec failure pure) query_bind oa"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : FreeMonad f α → Type*}\n    (pure : (x : α) → C (pure x))\n    (roll : {β : Type u} → (x : f β) → (r : β → FreeMonad f α) →\n      ((y : β) → C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | .pure x => pure x\n  | .roll x r => roll x _ (λ u ↦ FreeMonad.construct pure roll (r u))"}, {"name": "probEvent", "content": "notation \"Pr[\" p \"|\" mx \"]\" => probEvent mx p"}, {"name": "probFailure", "content": "notation \"Pr[⊥\" \"|\" mx \"]\" => probFailure mx"}, {"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "probEvent", "content": "notation \"[\" p \"|\" oa \"]\" => probEvent oa p"}], "lib_lemmas": [{"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_insert_iff", "module": "Mathlib.Data.Set.Insert"}, {"name": "exists_eq_right", "module": "Init.PropLemmas"}, {"name": "forall_eq_or_imp", "module": "Init.PropLemmas"}, {"name": "forall_exists_index", "module": "Init.PropLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "zero_le", "module": "Mathlib.Algebra.Order.Monoid.Canonical.Defs"}], "repo_lemmas": [{"name": "probEvent_eq_one_iff", "content": "@[simp low]\nlemma probEvent_eq_one_iff : [p | oa] = 1 ↔ [⊥ | oa] = 0 ∧ ∀ x ∈ oa.support, p x"}, {"name": "support_evalDist", "content": "lemma support_evalDist : (evalDist oa).run.support = if [⊥ | oa] = 0 then\n    some '' oa.support else insert none (some '' oa.support)"}, {"name": "probFailure_def", "content": "lemma probFailure_def (oa : OracleComp spec α) :\n    [⊥ | oa] = (evalDist oa).run none"}, {"name": "evalDist_orElse", "content": "@[simp]\nlemma evalDist_orElse [h : spec.FiniteRange] (oa oa' : OracleComp spec α) :\n    evalDist (oa <|> oa') = (evalDist oa <|> evalDist oa')"}, {"name": "probFailure_orElse", "content": "@[simp]\nlemma probFailure_orElse {ι : Type u} {spec : OracleSpec ι} {α : Type u} [h : spec.FiniteRange]\n    (oa oa' : OracleComp spec α) : [⊥ | oa <|> oa'] = [⊥ | oa] * [⊥ | oa']"}, {"name": "support_orElse", "content": "@[simp]\nlemma support_orElse {ι : Type u} {spec : OracleSpec ι} {α : Type u}\n    (oa oa' : OracleComp spec α) [Decidable oa.neverFails] : (oa <|> oa').support =\n      if oa.neverFails then oa.support else oa.support ∪ oa'.support"}, {"name": "probFailure_run_simulateQ", "content": "@[simp]\nlemma probFailure_run_simulateQ [spec.FiniteRange] (oa : OracleComp spec α) :\n    [⊥ | (simulateQ countingOracle oa).run] = [⊥ | oa]"}, {"name": "neverFails_run_simulateQ_iff", "content": "@[simp]\nlemma neverFails_run_simulateQ_iff (oa : OracleComp spec α) :\n    neverFails (simulateQ countingOracle oa).run ↔ neverFails oa"}], "used_local_defs": [{"name": "OracleComp.IsQueryBound", "content": "def IsQueryBound (oa : OracleComp spec α) (qb : ι → ℕ) : Prop :=\n    ∀ qc ∈ (snd <$> (simulateQ countingOracle oa).run).support, qc ≤ qb"}], "used_local_lemmas": [{"name": "OracleComp.isQueryBound_def", "content": "lemma isQueryBound_def (oa : OracleComp spec α) (qb : QueryCount spec) :\n    IsQueryBound oa qb ↔ ∀ qc ∈ (snd <$> (simulateQ countingOracle oa).run).support, qc ≤ qb"}], "local_ctx": "import VCVio.OracleComp.QueryTracking.CountingOracle\n\nimport VCVio.OracleComp.DistSemantics.Alternative\n\nopen OracleSpec Prod\n\nnamespace OracleComp\n\nsection IsQueryBound\n\nvariable {ι : Type u} [DecidableEq ι] {spec : OracleSpec ι} {α β γ : Type u}\n\ndef IsQueryBound (oa : OracleComp spec α) (qb : ι → ℕ) : Prop :=\n    ∀ qc ∈ (snd <$> (simulateQ countingOracle oa).run).support, qc ≤ qb", "target_theorem": "lemma isQueryBound_iff_probEvent [spec.FiniteRange] {oa : OracleComp spec α} {qb : ι → ℕ} :\n    IsQueryBound oa qb ↔\n      [(· ≤ qb) | snd <$> (simulateQ countingOracle oa).run <|> return 0] = 1 :=", "ground_truth_proof": ":= by\n  simp [probEvent_eq_one_iff, isQueryBound_def]\n  apply Iff.intro\n  · intro a x a_1\n    split at a_1\n    next h =>\n      simp_all only [Set.mem_image, Prod.exists, exists_eq_right]\n      obtain ⟨w, h_1⟩ := a_1\n      apply a\n      · exact h_1\n    next h =>\n      simp_all only [Set.mem_insert_iff, Set.mem_image, Prod.exists, exists_eq_right]\n      cases a_1 with\n      | inl h_1 =>\n        subst h_1\n        simp_all only [zero_le]\n      | inr h_2 =>\n        obtain ⟨w, h_1⟩ := h_2\n        apply a\n        · exact h_1\n  · intro a qc x h\n    split at a\n    next h_1 =>\n      simp_all only [Set.mem_image, Prod.exists, exists_eq_right, forall_exists_index]\n      apply a\n      · exact h\n    next\n      h_1 =>\n      simp_all only [Set.mem_insert_iff, Set.mem_image,\n                    Prod.exists, exists_eq_right, forall_eq_or_imp, zero_le,\n        forall_exists_index, true_and]\n      apply a\n      · exact h", "nesting_depth": 6, "transitive_dep_count": 49, "subset_aristotle": true, "category": "Applied verif."}
{"id": 458, "thm_name": "OracleComp.probEvent_uniformFin", "thm_stmt": "@[simp]\nlemma probEvent_uniformFin (p : Fin (n + 1) → Prop) [DecidablePred p] :\n    [p | $[0..n]] = (Finset.univ.filter p).card * (n + 1 : ℝ≥0∞)⁻¹", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/DistSemantics/EvalDist.lean", "imports": ["import Mathlib.Probability.Distributions.Uniform", "import VCVio.OracleComp.SimSemantics.SimulateQ", "import VCVio.OracleComp.Traversal", "import ToMathlib.General", "import VCVio.OracleComp.Support"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "DecidablePred", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.filter", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Finset.univ", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Fintype.card", "module": "Mathlib.Data.Fintype.Card"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "OptionT.lift", "module": "Init.Control.Option"}, {"name": "PMF", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "ENNReal", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.indicator", "module": "Mathlib.Algebra.Notation.Indicator"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}, {"name": "AddCommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ContinuousAdd", "module": "Mathlib.Topology.Algebra.Monoid.Defs"}, {"name": "Function.update", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Summable", "module": "Mathlib.Topology.Algebra.InfiniteSum.Defs"}, {"name": "T2Space", "module": "Mathlib.Topology.Separation.Hausdorff"}, {"name": "TopologicalSpace", "module": "Mathlib.Topology.Defs.Basic"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}, {"name": "Bool", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "uniformFin", "content": "notation \"$[0..\" n \"]\" => uniformFin n"}, {"name": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m", "content": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m\n\nexample : OracleComp probSpec ℕ := do\n  let x ← $[314⋯31415]; let y ← $[0⋯x]\n  return x + 2 * y"}, {"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "uniformFin", "content": "@[reducible, inline] def uniformFin (n : ℕ) : ProbComp (Fin (n + 1)) :=\n  unifSpec.query n ()"}, {"name": "ProbComp", "content": "abbrev ProbComp : Type z → Type (max z 1) := OracleComp unifSpec"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "unifSpec", "content": "@[inline, reducible] def unifSpec : OracleSpec.{0,0} ℕ :=\n  λ n ↦ (Unit, Fin (n + 1))"}, {"name": "uniformFin'", "content": "@[reducible, inline] def uniformFin' (n m : ℕ) : OracleComp probSpec (Fin (m + 1)) :=\n  probSpec.query m n"}, {"name": "probSpec", "content": "def probSpec : OracleSpec.{0,0} ℕ :=\n  fun n => (ℕ, Fin (n + 1))"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "DecidableEq", "content": "protected class DecidableEq (spec : OracleSpec ι) where\n  domain_decidableEq' (i : ι) : DecidableEq (spec.domain i)\n  range_decidableEq' (i : ι) : DecidableEq (spec.range i)"}, {"name": "cases", "content": "def cases {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive (tl.insertIdx 0 hd)) {m : ℕ} :\n    (v : Vector α m) → motive v := match hm : m with\n  | 0 => fun v => match v with | ⟨⟨[]⟩, rfl⟩ => v_empty\n  | n + 1 => fun v => match hv : v with\n    | ⟨⟨hd :: tl⟩, hSize⟩ => by admit /- proof elided -/"}, {"name": "support", "content": "def support (oa : OracleComp spec α) : Set α :=\n  oa.supportWhen fun _ => Set.univ"}, {"name": "supportWhen", "content": "def supportWhen (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Set α) : Set α :="}, {"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : OracleComp spec α → Type*}\n    (pure : (a : α) → C (pure a))\n    (query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.construct (Option.rec failure pure) query_bind oa"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : FreeMonad f α → Type*}\n    (pure : (x : α) → C (pure x))\n    (roll : {β : Type u} → (x : f β) → (r : β → FreeMonad f α) →\n      ((y : β) → C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | .pure x => pure x\n  | .roll x r => roll x _ (λ u ↦ FreeMonad.construct pure roll (r u))"}, {"name": "finSupport", "content": "def finSupport [∀ i, Fintype (spec.range i)] [DecidableEq α] (oa : OracleComp spec α) : Finset α :=\n  oa.finSupportWhen fun | query _ _ => Finset.univ"}, {"name": "finSupportWhen", "content": "def finSupportWhen [DecidableEq α] (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Finset α) : Finset α :="}, {"name": "coin", "content": "@[reducible, inline] def coin : OracleComp coinSpec Bool :=\n  coinSpec.query () ()"}, {"name": "coinSpec", "content": "@[inline, reducible] def coinSpec : OracleSpec.{0,0} Unit := Unit →ₒ Bool"}, {"name": "FiniteRange", "content": "class FiniteRange (spec : OracleSpec ι) where\n    range_inhabited' (i : ι) : Inhabited (spec.range i)\n    range_fintype' (i : ι) : Fintype (spec.range i)"}, {"name": "uniformFin", "content": "notation \"$[0..\" n \"]\" => uniformFin n"}, {"name": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m", "content": "notation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m\n\nexample : OracleComp probSpec ℕ := do\n  let x ← $[314⋯31415]; let y ← $[0⋯x]\n  return x + 2 * y"}], "lib_lemmas": [{"name": "Nat.cast_inj", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "PMF.monad_map_eq_map", "module": "Mathlib.Probability.ProbabilityMassFunction.Constructions"}, {"name": "inv_inj", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "not_imp_not", "module": "Mathlib.Logic.Basic"}, {"name": "tsum_eq_single", "module": "Mathlib.Topology.Algebra.InfiniteSum.Basic"}, {"name": "ENNReal.summable", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "PMF.toOuterMeasure_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "Finset.mem_filter", "module": "Mathlib.Data.Finset.Filter"}, {"name": "Finset.sum_congr", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "if_pos", "module": "Init.Core"}, {"name": "tsum_eq_sum'", "module": "Mathlib.Topology.Algebra.InfiniteSum.Basic"}, {"name": "Finset.sum_const", "module": "Mathlib.Algebra.BigOperators.Group.Finset.Basic"}, {"name": "nsmul_eq_mul", "module": "Mathlib.Algebra.Ring.Defs"}], "repo_lemmas": [{"name": "tsum_option", "content": "lemma tsum_option {α β : Type*} [AddCommMonoid α] [TopologicalSpace α]\n    [ContinuousAdd α] [T2Space α]\n    (f : Option β → α) (hf : Summable (Function.update f none 0)) :\n    ∑' x : Option β, f x = f none + ∑' x : β, f (some x)"}, {"name": "finSupport_uniformFin", "content": "@[simp] lemma finSupport_uniformFin (n : ℕ) : ($[0..n]).finSupport = Finset.univ"}, {"name": "finSupport_query", "content": "lemma finSupport_query [spec.FiniteRange] [DecidableEq α] (q : OracleQuery spec α) :\n    (q : OracleComp spec _).finSupport = match q with"}, {"name": "finSupportWhen_query", "content": "@[simp] lemma finSupportWhen_query [DecidableEq α] (q : OracleQuery spec α) :\n    (q : OracleComp spec α).finSupportWhen fin_poss = fin_poss q"}, {"name": "finSupport_def", "content": "lemma finSupport_def [∀ i, Fintype (spec.range i)] [DecidableEq α] (oa : OracleComp spec α) :\n    oa.finSupport = oa.finSupportWhen fun"}], "used_local_defs": [{"name": "OracleComp.probOutput", "content": "noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)"}, {"name": "OracleComp.probFailure", "content": "noncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none"}], "used_local_lemmas": [{"name": "OracleComp.probOutput_def", "content": "lemma probOutput_def (oa : OracleComp spec α) (x : α) :\n    [= x | oa] = (evalDist oa).run (some x)"}, {"name": "OracleComp.probEvent_def", "content": "lemma probEvent_def (oa : OracleComp spec α) (p : α → Prop) :\n    [p | oa] = (evalDist oa).run.toOuterMeasure (Option.some '' {x | p x})"}, {"name": "OracleComp.probEvent_eq_tsum_ite", "content": "lemma probEvent_eq_tsum_ite (oa : OracleComp spec α) (p : α → Prop) [DecidablePred p] :\n    [p | oa] = ∑' x : α, if p x then [= x | oa] else 0"}, {"name": "OracleComp.probEvent_eq_sum_filter_finSupport", "content": "lemma probEvent_eq_sum_filter_finSupport [spec.DecidableEq] [DecidablePred p] [DecidableEq α] :\n    [p | oa] = ∑ x ∈ oa.finSupport.filter p, [= x | oa]"}, {"name": "OracleComp.probOutput_liftM", "content": "@[simp]\nlemma probOutput_liftM [Fintype α] (q : OracleQuery spec α) (u : α) :\n    [= u | (q : OracleComp spec α)] = (Fintype.card α : ℝ≥0∞)⁻¹"}, {"name": "OracleComp.probOutput_query", "content": "lemma probOutput_query (u : spec.range i) :\n    [= u | (query i t : OracleComp spec _)] = (Fintype.card (spec.range i) : ℝ≥0∞)⁻¹"}, {"name": "OracleComp.probOutput_uniformFin", "content": "@[simp]\nlemma probOutput_uniformFin (x : Fin (n + 1)) : [= x | $[0..n]] = ((n : ℝ≥0∞) + 1)⁻¹"}], "local_ctx": "import VCVio.OracleComp.Traversal\n\nimport VCVio.OracleComp.SimSemantics.SimulateQ\n\nimport Mathlib.Probability.Distributions.Uniform\n\nimport ToMathlib.General\n\nopen OracleSpec Option ENNReal BigOperators\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {ι' : Type v} {spec' : OracleSpec ι'}\n  {α β γ : Type w} [spec.FiniteRange] [spec'.FiniteRange]\n\nsection evalDist\n\nend evalDist\n\nnoncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)\n\nnotation \"[=\" x \"|\" oa \"]\" => probOutput oa x\n\nnoncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none\n\nnotation \"[⊥\" \"|\" oa \"]\" => probFailure oa\n\nnotation \"[\" p \"|\" oa \"]\" => probEvent oa p\n\nsection bounds\n\nvariable {oa : OracleComp spec α} {x : α} {p : α → Prop}\n\nend bounds\n\nsection support\n\nvariable (oa : OracleComp spec α) (x : α) (p q : α → Prop)\n\nvariable {oa x p q}\n\nend support\n\nsection sums\n\nvariable (oa : OracleComp spec α) (p : α → Prop)\n\nend sums\n\nsection pure\n\nvariable (x : α)\n\nend pure\n\nsection bind\n\nvariable (oa : OracleComp spec α) (ob : α → OracleComp spec β)\n\nend bind\n\nsection mul_le_probEvent_bind\n\nend mul_le_probEvent_bind\n\nsection bind_const\n\nvariable (oa : OracleComp spec α) (ob : OracleComp spec β)\n\nend bind_const\n\nsection query\n\nvariable (i : ι) (t : spec.domain i)\n\nend query\n\nsection failure\n\nend failure\n\nsection map\n\nvariable (oa : OracleComp spec α) (f : α → β)\n\nend map\n\nsection neverFails\n\nend neverFails\n\nsection unit\n\nend unit\n\nsection bool\n\nend bool\n\nsection eqRec\n\nvariable (oa : OracleComp spec α) (h : α = β)\n\nend eqRec\n\nsection ite\n\nvariable (p : Prop) [Decidable p] (oa oa' : OracleComp spec α)\n\nend ite\n\nsection coin\n\nend coin\n\nsection uniformFin\n\nvariable (n : ℕ)", "target_theorem": "@[simp]\nlemma probEvent_uniformFin (p : Fin (n + 1) → Prop) [DecidablePred p] :\n    [p | $[0..n]] = (Finset.univ.filter p).card * (n + 1 : ℝ≥0∞)⁻¹ :=", "ground_truth_proof": ":= by\n  simp only [probEvent_eq_sum_filter_finSupport, finSupport_uniformFin, probOutput_uniformFin,\n    Finset.sum_const, nsmul_eq_mul]", "nesting_depth": 6, "transitive_dep_count": 77, "subset_aristotle": true, "category": "Applied verif."}
{"id": 459, "thm_name": "OracleComp.probFailure_bind_of_const", "thm_stmt": "lemma probFailure_bind_of_const [Nonempty α] (r : ℝ≥0∞) (h : ∀ x, [⊥ | ob x] = r) :\n    [⊥ | oa >>= ob] = [⊥ | oa] + r - [⊥ | oa] * r", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/DistSemantics/EvalDist.lean", "imports": ["import Mathlib.Probability.Distributions.Uniform", "import VCVio.OracleComp.SimSemantics.SimulateQ", "import VCVio.OracleComp.Traversal", "import ToMathlib.General"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "AddLECancellable", "module": "Mathlib.Algebra.Order.Monoid.Unbundled.Basic"}, {"name": "Classical.arbitrary", "module": "Mathlib.Logic.Nonempty"}, {"name": "ENNReal", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "PMF", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "Option.elimM", "module": "Init.Data.Option.Basic"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}, {"name": "AddCommMonoid", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ContinuousAdd", "module": "Mathlib.Topology.Algebra.Monoid.Defs"}, {"name": "Function.update", "module": "Mathlib.Logic.Function.Basic"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Summable", "module": "Mathlib.Topology.Algebra.InfiniteSum.Defs"}, {"name": "T2Space", "module": "Mathlib.Topology.Separation.Hausdorff"}, {"name": "TopologicalSpace", "module": "Mathlib.Topology.Defs.Basic"}], "used_repo_defs": [{"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "cases", "content": "def cases {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive (tl.insertIdx 0 hd)) {m : ℕ} :\n    (v : Vector α m) → motive v := match hm : m with\n  | 0 => fun v => match v with | ⟨⟨[]⟩, rfl⟩ => v_empty\n  | n + 1 => fun v => match hv : v with\n    | ⟨⟨hd :: tl⟩, hSize⟩ => by admit /- proof elided -/"}], "lib_lemmas": [{"name": "PMF.coe_le_one", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "ENNReal.summable", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "symm", "module": "Mathlib.Order.Defs.Unbundled"}, {"name": "add_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "PMF.apply_ne_top", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "ENNReal.eq_sub_of_add_eq", "module": "Mathlib.Data.ENNReal.Operations"}, {"name": "AddLECancellable.add_tsub_assoc_of_le", "module": "Mathlib.Algebra.Order.Sub.Unbundled.Basic"}, {"name": "ENNReal.addLECancellable_iff_ne", "module": "Mathlib.Data.ENNReal.Operations"}, {"name": "ENNReal.div_self", "module": "Mathlib.Data.ENNReal.Inv"}, {"name": "ENNReal.mul_le_of_le_div", "module": "Mathlib.Data.ENNReal.Inv"}, {"name": "ENNReal.mul_ne_top", "module": "Mathlib.Data.ENNReal.Operations"}, {"name": "ENNReal.sub_mul", "module": "Mathlib.Data.ENNReal.Operations"}, {"name": "ENNReal.tsum_mul_right", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "le_of_le_of_eq", "module": "Init.Core"}, {"name": "le_refl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "one_mul", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "tsum_option", "content": "lemma tsum_option {α β : Type*} [AddCommMonoid α] [TopologicalSpace α]\n    [ContinuousAdd α] [T2Space α]\n    (f : Option β → α) (hf : Summable (Function.update f none 0)) :\n    ∑' x : Option β, f x = f none + ∑' x : β, f (some x)"}], "used_local_defs": [{"name": "OracleComp.probOutput", "content": "noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)"}, {"name": "OracleComp.probFailure", "content": "noncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none"}], "used_local_lemmas": [{"name": "OracleComp.probOutput_def", "content": "lemma probOutput_def (oa : OracleComp spec α) (x : α) :\n    [= x | oa] = (evalDist oa).run (some x)"}, {"name": "OracleComp.probFailure_add_tsum_probOutput", "content": "@[simp]\nlemma probFailure_add_tsum_probOutput (oa : OracleComp spec α) :\n    [⊥ | oa] + ∑' x, [= x | oa] = 1"}, {"name": "OracleComp.tsum_probOutput_add_probFailure", "content": "@[simp]\nlemma tsum_probOutput_add_probFailure (oa : OracleComp spec α) :\n    ∑' x, [= x | oa] + [⊥ | oa] = 1"}, {"name": "OracleComp.probFailure_le_one", "content": "@[simp] lemma probFailure_le_one : [⊥ | oa] ≤ 1"}, {"name": "OracleComp.probFailure_ne_top", "content": "@[simp] lemma probFailure_ne_top : [⊥ | oa] ≠ ∞"}, {"name": "OracleComp.tsum_probOutput_eq_sub", "content": "lemma tsum_probOutput_eq_sub (oa : OracleComp spec α) :\n    ∑' x : α, [= x | oa] = 1 - [⊥ | oa]"}, {"name": "OracleComp.probFailure_bind_eq_tsum", "content": "lemma probFailure_bind_eq_tsum :\n    [⊥ | oa >>= ob] = [⊥ | oa] + ∑' x : α, [= x | oa] * [⊥ | ob x]"}], "local_ctx": "import VCVio.OracleComp.Traversal\n\nimport VCVio.OracleComp.SimSemantics.SimulateQ\n\nimport Mathlib.Probability.Distributions.Uniform\n\nimport ToMathlib.General\n\nopen OracleSpec Option ENNReal BigOperators\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {ι' : Type v} {spec' : OracleSpec ι'}\n  {α β γ : Type w} [spec.FiniteRange] [spec'.FiniteRange]\n\nsection evalDist\n\nend evalDist\n\nnoncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)\n\nnotation \"[=\" x \"|\" oa \"]\" => probOutput oa x\n\nnoncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none\n\nnotation \"[⊥\" \"|\" oa \"]\" => probFailure oa\n\nnotation \"[\" p \"|\" oa \"]\" => probEvent oa p\n\nsection bounds\n\nvariable {oa : OracleComp spec α} {x : α} {p : α → Prop}\n\nend bounds\n\nsection support\n\nvariable (oa : OracleComp spec α) (x : α) (p q : α → Prop)\n\nvariable {oa x p q}\n\nend support\n\nsection sums\n\nvariable (oa : OracleComp spec α) (p : α → Prop)\n\nend sums\n\nsection pure\n\nvariable (x : α)\n\nend pure\n\nsection bind\n\nvariable (oa : OracleComp spec α) (ob : α → OracleComp spec β)", "target_theorem": "lemma probFailure_bind_of_const [Nonempty α] (r : ℝ≥0∞) (h : ∀ x, [⊥ | ob x] = r) :\n    [⊥ | oa >>= ob] = [⊥ | oa] + r - [⊥ | oa] * r :=", "ground_truth_proof": ":= by\n  have : r ≠ ⊤ := λ hr ↦ probFailure_ne_top ((h (Classical.arbitrary α)).trans hr)\n  simp [probFailure_bind_eq_tsum, h, ENNReal.tsum_mul_right, tsum_probOutput_eq_sub]\n  rw [ENNReal.sub_mul λ _ _ ↦ this, one_mul]\n  refine symm (AddLECancellable.add_tsub_assoc_of_le ?_ ?_ _)\n  · refine ENNReal.addLECancellable_iff_ne.2 (ENNReal.mul_ne_top probFailure_ne_top this)\n  · by_cases hr : r = 0\n    · simp only [hr, mul_zero, le_refl]\n    refine mul_le_of_le_div (le_of_le_of_eq probFailure_le_one ?_)\n    refine symm (ENNReal.div_self hr this)", "nesting_depth": 5, "transitive_dep_count": 48, "subset_aristotle": true, "category": "Applied verif."}
{"id": 460, "thm_name": "OracleComp.probEvent_seq_map_eq_probEvent_comp_uncurry", "thm_stmt": "lemma probEvent_seq_map_eq_probEvent_comp_uncurry [spec.FiniteRange]\n    (p : γ → Prop) : [p | f <$> oa <*> ob] =\n      [p ∘ f.uncurry | Prod.mk <$> oa <*> ob]", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/DistSemantics/Seq.lean", "imports": ["import VCVio.OracleComp.DistSemantics.Monad", "import VCVio.OracleComp.DistSemantics.EvalDist", "import VCVio.OracleComp.Support"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Function.comp", "module": "Init.Prelude"}, {"name": "Functor", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.image2", "module": "Mathlib.Data.Set.Operations"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}], "used_repo_defs": [{"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "FiniteRange", "content": "class FiniteRange (spec : OracleSpec ι) where\n    range_inhabited' (i : ι) : Inhabited (spec.range i)\n    range_fintype' (i : ι) : Fintype (spec.range i)"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "support", "content": "def support (oa : OracleComp spec α) : Set α :=\n  oa.supportWhen fun _ => Set.univ"}, {"name": "supportWhen", "content": "def supportWhen (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Set α) : Set α :="}, {"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : OracleComp spec α → Type*}\n    (pure : (a : α) → C (pure a))\n    (query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.construct (Option.rec failure pure) query_bind oa"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : FreeMonad f α → Type*}\n    (pure : (x : α) → C (pure x))\n    (roll : {β : Type u} → (x : f β) → (r : β → FreeMonad f α) →\n      ((y : β) → C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | .pure x => pure x\n  | .roll x r => roll x _ (λ u ↦ FreeMonad.construct pure roll (r u))"}, {"name": "probOutput", "content": "noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : OracleComp spec α → Prop}\n    (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "probEvent", "content": "noncomputable def probEvent (oa : OracleComp spec α) (p : α → Prop) : ℝ≥0∞ :=\n  (evalDist oa).run.toOuterMeasure (Option.some '' {x | p x})"}, {"name": "cases", "content": "def cases {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive (tl.insertIdx 0 hd)) {m : ℕ} :\n    (v : Vector α m) → motive v := match hm : m with\n  | 0 => fun v => match v with | ⟨⟨[]⟩, rfl⟩ => v_empty\n  | n + 1 => fun v => match hv : v with\n    | ⟨⟨hd :: tl⟩, hSize⟩ => by admit /- proof elided -/"}, {"name": "probOutput", "content": "notation \"Pr[=\" x \"|\" mx \"]\" => probOutput mx x"}, {"name": "probEvent", "content": "notation \"Pr[\" p \"|\" mx \"]\" => probEvent mx p"}, {"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "probEvent", "content": "notation \"[\" p \"|\" oa \"]\" => probEvent oa p"}], "lib_lemmas": [{"name": "seq_eq_bind_map", "module": "Init.Control.Lawful.Basic"}, {"name": "Set.biUnion_and'", "module": "Mathlib.Data.Set.Lattice"}, {"name": "Set.ext_iff", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.iUnion_exists", "module": "Mathlib.Data.Set.Lattice"}, {"name": "Set.iUnion_iUnion_eq_right", "module": "Mathlib.Data.Set.Lattice"}, {"name": "Set.mem_iUnion", "module": "Mathlib.Order.SetNotation"}, {"name": "Set.mem_image", "module": "Mathlib.Data.Set.Operations"}, {"name": "Set.mem_image2", "module": "Mathlib.Data.Set.Operations"}, {"name": "exists_prop", "module": "Init.PropLemmas"}, {"name": "implies_true", "module": "Init.SimpLemmas"}, {"name": "Function.uncurry_apply_pair", "module": "Init.Data.Function"}, {"name": "Functor.map_map", "module": "Init.Control.Lawful.Basic"}, {"name": "Set.image2_mk_eq_prod", "module": "Mathlib.Data.Set.NAry"}, {"name": "Set.image_uncurry_prod", "module": "Mathlib.Data.Set.NAry"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "map_seq", "module": "Mathlib.Control.Basic"}], "repo_lemmas": [{"name": "support_map", "content": "@[simp] lemma support_map (oa : OracleComp spec α) (f : α → β) :\n    (f <$> oa).support = f '' oa.support"}, {"name": "support_bind", "content": "@[simp] lemma support_bind (oa : OracleComp spec α) (ob : α → OracleComp spec β) :\n    (oa >>= ob).support = ⋃ x ∈ oa.support, (ob x).support"}, {"name": "support_pure", "content": "@[simp] lemma support_pure (x : α) :\n    (pure x : OracleComp spec α).support = {x}"}, {"name": "probEvent_congr'", "content": "lemma probEvent_congr' {p q : α → Prop} {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h1 : ∀ x, x ∈ oa.support → x ∈ oa'.support → (p x ↔ q x))\n    (h2 : evalDist oa = evalDist oa') : [p | oa] = [q | oa']"}, {"name": "mem_support_iff_of_evalDist_eq", "content": "lemma mem_support_iff_of_evalDist_eq {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h : evalDist oa = evalDist oa') (x : α) : x ∈ oa.support ↔ x ∈ oa'.support"}, {"name": "mem_support_iff_probOutput_ne_zero", "content": "lemma mem_support_iff_probOutput_ne_zero : x ∈ oa.support ↔ [= x | oa] ≠ 0"}, {"name": "probOutput_eq_zero_iff", "content": "@[simp low]\nlemma probOutput_eq_zero_iff : [= x | oa] = 0 ↔ x ∉ oa.support"}, {"name": "mem_support_evalDist_iff", "content": "@[simp]\nlemma mem_support_evalDist_iff (oa : OracleComp spec α) (x : α) :\n    some x ∈ (evalDist oa).run.support ↔ x ∈ oa.support"}, {"name": "probOutput_def", "content": "lemma probOutput_def (oa : OracleComp spec α) (x : α) :\n    [= x | oa] = (evalDist oa).run (some x)"}, {"name": "probEvent_eq_tsum_indicator", "content": "lemma probEvent_eq_tsum_indicator (oa : OracleComp spec α) (p : α → Prop) :\n    [p | oa] = ∑' x : α, {x | p x}.indicator ([= · | oa]) x"}, {"name": "probEvent_def", "content": "lemma probEvent_def (oa : OracleComp spec α) (p : α → Prop) :\n    [p | oa] = (evalDist oa).run.toOuterMeasure (Option.some '' {x | p x})"}, {"name": "probOutput_congr", "content": "lemma probOutput_congr {x y : α} {oa : OracleComp spec α} {oa' : OracleComp spec' α}\n    (h1 : x = y) (h2 : evalDist oa = evalDist oa') : [= x | oa] = [= y | oa']"}, {"name": "probEvent_comp", "content": "lemma probEvent_comp (q : β → Prop) : [q ∘ f | oa] = [q | f <$> oa]"}, {"name": "probEvent_map", "content": "@[simp]\nlemma probEvent_map (q : β → Prop) : [q | f <$> oa] = [q ∘ f | oa]"}, {"name": "evalDist_map", "content": "@[simp]\nlemma evalDist_map (oa : OracleComp spec α) (f : α → β) :\n    evalDist (f <$> oa) = f <$> (evalDist oa)"}], "used_local_defs": [], "used_local_lemmas": [{"name": "OracleComp.support_seq", "content": "@[simp low]\nlemma support_seq : (og <*> oa).support = ⋃ g ∈ og.support, g '' oa.support"}, {"name": "OracleComp.support_seq_map_eq_image2", "content": "@[simp low + 1]\nlemma support_seq_map_eq_image2 :\n    (f <$> oa <*> ob).support = Set.image2 f oa.support ob.support"}], "local_ctx": "import VCVio.OracleComp.DistSemantics.Monad\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β γ : Type v}\n\nvariable (oa : OracleComp spec α) (og : OracleComp spec (α → β))\n\nsection seq_map\n\nvariable (oa : OracleComp spec α) (ob : OracleComp spec β) (f : α → β → γ)", "target_theorem": "lemma probEvent_seq_map_eq_probEvent_comp_uncurry [spec.FiniteRange]\n    (p : γ → Prop) : [p | f <$> oa <*> ob] =\n      [p ∘ f.uncurry | Prod.mk <$> oa <*> ob] :=", "ground_truth_proof": ":= by\n  rw [probEvent_comp]\n  refine probEvent_congr' ?_ (congr_arg evalDist ?_)\n  · simp only [support_seq_map_eq_image2, Set.mem_image2, support_map, Set.image2_mk_eq_prod,\n      Set.image_uncurry_prod, implies_true]\n  · simp only [map_seq, Function.comp, Functor.map_map, Function.uncurry_apply_pair]\n    rfl", "nesting_depth": 6, "transitive_dep_count": 60, "subset_aristotle": true, "category": "Applied verif."}
{"id": 461, "thm_name": "OracleComp.liftM_inj", "thm_stmt": "@[simp]\nlemma liftM_inj (q q' : OracleQuery spec α) : (q : OracleComp spec α) = q' ↔ q = q'", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/OracleComp.lean", "imports": ["import ToMathlib.Control.AlternativeMonad", "import ToMathlib.Control.Monad.Free", "import VCVio.OracleComp.OracleSpec", "import ToMathlib.Control.WriterT", "import Mathlib.Control.Lawful", "import ToMathlib.Control.OptionT"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "OptionT.lift", "module": "Init.Control.Option"}, {"name": "OptionT.mk", "module": "Init.Control.Option"}], "used_repo_defs": [{"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "lift", "content": "@[always_inline, inline]\ndef lift (x : f α) : FreeMonad f α := FreeMonad.roll x FreeMonad.pure"}, {"name": "bind", "content": "@[always_inline, inline]\nprotected def bind : FreeMonad f α → (α → FreeMonad f β) → FreeMonad f β\n  | FreeMonad.pure x, g => g x\n  | FreeMonad.roll x r, g => FreeMonad.roll x (λ u ↦ FreeMonad.bind (r u) g)"}, {"name": "notation:50 m₁ \" >>=ₕ \" m₂ => bind m₁ m₂", "content": "notation:50 m₁ \" >>=ₕ \" m₂ => bind m₁ m₂"}], "lib_lemmas": [{"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "heq_eq_eq", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "monad_bind_def", "content": "@[simp]\nlemma monad_bind_def (x : FreeMonad f α) (g : α → FreeMonad f β) :\n    x >>= g = FreeMonad.bind x g"}, {"name": "bind_lift", "content": "@[simp]\nlemma bind_lift (x : f α) (r : α → FreeMonad f β) :\n    FreeMonad.bind (FreeMonad.lift x) r = FreeMonad.roll x r"}, {"name": "monad_pure_def", "content": "@[simp]\nlemma monad_pure_def (x : α) : (pure x : FreeMonad f α) = FreeMonad.pure x"}], "used_local_defs": [{"name": "OracleSpec.OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}], "used_local_lemmas": [{"name": "OracleComp.liftM_def", "content": "protected lemma liftM_def (q : OracleQuery spec α) :\n    (q : OracleComp spec α) = OptionT.lift (FreeMonad.lift q)"}], "local_ctx": "import ToMathlib.Control.Monad.Free\n\nimport ToMathlib.Control.WriterT\n\nimport ToMathlib.Control.AlternativeMonad\n\nimport ToMathlib.Control.OptionT\n\nimport Mathlib.Control.Lawful\n\nimport VCVio.OracleComp.OracleSpec\n\nnamespace OracleSpec\n\ninductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)\n\nnamespace OracleQuery\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β : Type v}\n\nend OracleQuery\n\nend OracleSpec\n\nopen OracleSpec\n\ndef OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β : Type v}\n\nsection lift\n\nend lift\n\nnotation \"$[0..\" n \"]\" => uniformFin n\n\nnotation:50 \"$[\" n \"⋯\" m \"]\" => uniformFin' n m\n\nexample : OracleComp probSpec ℕ := do\n  let x ← $[314⋯31415]; let y ← $[0⋯x]\n  return x + 2 * y\n\nsection construct\n\nvariable {C : OracleComp spec α → Type w}\n  (h_pure : (a : α) → C (pure a))\n  (h_query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n  (h_failure : C failure) (oa : OracleComp spec α)\n\nend construct\n\nsection noConfusion\n\nvariable (x : α) (y : β) (q : OracleQuery spec β) (oa : β → OracleComp spec α)\n\nend noConfusion\n\nsection mapM\n\nvariable {m : Type v → Type w} [Monad m]\n  (fail : {α : Type v} → m α) (qm : {α : Type v} → OracleQuery spec α → m α)\n\nend mapM\n\nsection inj", "target_theorem": "@[simp]\nlemma liftM_inj (q q' : OracleQuery spec α) : (q : OracleComp spec α) = q' ↔ q = q' :=", "ground_truth_proof": ":= by\n  simp only [OracleComp.liftM_def, OptionT.lift, OptionT.mk, FreeMonad.monad_pure_def,\n    FreeMonad.monad_bind_def, FreeMonad.bind_lift]\n  rw [FreeMonad.roll.injEq]\n  simp only [heq_eq_eq, and_true, true_and]", "nesting_depth": 3, "transitive_dep_count": 19, "subset_aristotle": false, "category": "Applied verif."}
{"id": 462, "thm_name": "PFunctor.Lens.prodPair_fst_snd", "thm_stmt": "@[simp]\ntheorem prodPair_fst_snd :\n    Lens.prodPair Lens.fst Lens.snd = Lens.id.{max uA₁ uA₂, max uB₁ uB₂} (P * Q)", "lean_root": "VCV-io", "rel_path": "ToMathlib/PFunctor/Lens/Basic.lean", "imports": ["import ToMathlib.PFunctor.Basic"], "used_lib_defs": [{"name": "PFunctor", "module": "Mathlib.Data.PFunctor.Univariate.Basic"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.snd", "module": "Init.Prelude"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "Prod.fst", "module": "Init.Prelude"}, {"name": "Sum.inl", "module": "Init.Core"}], "used_repo_defs": [{"name": "prodPair", "content": "notation \"⟨\" l₁ \",\" l₂ \"⟩ₗ\" => prodPair l₁ l₂"}, {"name": "Lens", "content": "structure Lens (P : PFunctor.{uA₁, uB₁}) (Q : PFunctor.{uA₂, uB₂}) where\n  toFunA : P.A → Q.A\n  toFunB : ∀ a, Q.B (toFunA a) → P.B a"}, {"name": "Chart", "content": "structure Chart (P : PFunctor.{uA₁, uB₁}) (Q : PFunctor.{uA₂, uB₂}) where\n  toFunA : P.A → Q.A\n  toFunB : ∀ a, P.B a → Q.B (toFunA a)"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "PFunctor.Lens.prodPair", "content": "def prodPair {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₂}} {R : PFunctor.{uA₃, uB₃}}\n    (l₁ : Lens P Q) (l₂ : Lens P R) :\n    Lens.{uA₁, uB₁, max uA₂ uA₃, max uB₂ uB₃} P (Q * R) :=\n  (fun p => (l₁.toFunA p, l₂.toFunA p)) ⇆\n    (fun p => Sum.elim (l₁.toFunB p) (l₂.toFunB p))"}], "used_local_lemmas": [{"name": "PFunctor.Lens.ext", "content": "@[ext (iff := false)]\ntheorem ext {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₂}} (l₁ l₂ : Lens P Q)\n    (h₁ : ∀ a, l₁.toFunA a = l₂.toFunA a) (h₂ : ∀ a, l₁.toFunB a = (h₁ a) ▸ l₂.toFunB a) :\n    l₁ = l₂"}], "local_ctx": "import ToMathlib.PFunctor.Basic\n\nsection find_home\n\nvariable {α : Sort u} {β : α → Sort v} {γ : α → Sort v}\n\nend find_home\n\nnamespace PFunctor\n\nnamespace Lens\n\n@[inherit_doc] infixl:75 \" ∘ₗ \" => comp\n\n@[inherit_doc] infix:50 \" ≃ₗ \" => Equiv\n\nnamespace Equiv\n\nend Equiv\n\ndef prodPair {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₂}} {R : PFunctor.{uA₃, uB₃}}\n    (l₁ : Lens P Q) (l₂ : Lens P R) :\n    Lens.{uA₁, uB₁, max uA₂ uA₃, max uB₂ uB₃} P (Q * R) :=\n  (fun p => (l₁.toFunA p, l₂.toFunA p)) ⇆\n    (fun p => Sum.elim (l₁.toFunB p) (l₂.toFunB p))\n\n@[inherit_doc] infixl:75 \" ◃ₗ \" => compMap\n\n@[inherit_doc] infixl:75 \" ×ₗ \" => prodMap\n\n@[inherit_doc] infixl:75 \" ⊎ₗ \" => sumMap\n\n@[inherit_doc] infixl:75 \" ⊗ₗ \" => tensorMap\n\nnotation \"[\" l₁ \",\" l₂ \"]ₗ\" => sumPair l₁ l₂\n\nnotation \"⟨\" l₁ \",\" l₂ \"⟩ₗ\" => prodPair l₁ l₂\n\nsection Coprod\n\nvariable {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₁}}\n  {R : PFunctor.{uA₃, uB₃}} {W : PFunctor.{uA₄, uB₃}} {X : PFunctor.{uA₅, uB₅}}\n\nnamespace Equiv\n\nvariable {P : PFunctor.{uA₁, uB}} {Q : PFunctor.{uA₂, uB}} {R : PFunctor.{uA₃, uB}}\n\nend Equiv\n\nend Coprod\n\nsection Prod\n\nvariable {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₂}} {R : PFunctor.{uA₃, uB₃}}\n  {W : PFunctor.{uA₄, uB₄}} {X : PFunctor.{uA₅, uB₅}}", "target_theorem": "@[simp]\ntheorem prodPair_fst_snd :\n    Lens.prodPair Lens.fst Lens.snd = Lens.id.{max uA₁ uA₂, max uB₁ uB₂} (P * Q) :=", "ground_truth_proof": ":= by\n  ext a x\n  · rfl\n  · cases x <;> rfl", "nesting_depth": 2, "transitive_dep_count": 15, "subset_aristotle": true, "category": "Applied verif."}
{"id": 463, "thm_name": "OracleSpec.QuerySeed.eq_takeAtIndex_length_iff", "thm_stmt": "lemma eq_takeAtIndex_length_iff (seed seed' : QuerySeed spec) (i : ι) :\n    seed = seed'.takeAtIndex i (seed i).length ↔\n      seed' = seed.addValues ((seed' i).drop (seed i).length)", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/QueryTracking/Structures.lean", "imports": ["import VCVio.OracleComp.SimSemantics.SimulateQ"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Function.update", "module": "Mathlib.Logic.Function.Basic"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "DFunLike", "module": "Mathlib.Data.FunLike.Basic"}], "used_repo_defs": [{"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "DecidableEq", "content": "protected class DecidableEq (spec : OracleSpec ι) where\n  domain_decidableEq' (i : ι) : DecidableEq (spec.domain i)\n  range_decidableEq' (i : ι) : DecidableEq (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))\n\n/-- Add a list of values to the query seed.-/\ndef addValues [DecidableEq ι] {i : ι}\n    (us : List (spec.range i)) (seed : QuerySeed spec) : QuerySeed spec :=\n  Function.update seed i (seed i ++ us)\n\n/-- Take only the first `n` values of the seed at index `i`. -/\ndef takeAtIndex [DecidableEq ι] (seed : QuerySeed spec) (i : ι) (n : ℕ) : QuerySeed spec :=\n  Function.update seed i ((seed i).take n)"}], "lib_lemmas": [{"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "congr_fun", "module": "Batteries.Logic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "OracleSpec.QuerySeed", "content": "def QuerySeed (spec : OracleSpec ι) : Type _ :=\n  (i : ι) → List (spec.range i)"}], "used_local_lemmas": [], "local_ctx": "import VCVio.OracleComp.SimSemantics.SimulateQ\n\nopen OracleSpec OracleComp\n\nnamespace OracleSpec\n\nvariable {ι : Type u} {spec : OracleSpec ι}\n\nnamespace QueryCache\n\nvariable [spec.DecidableEq] [DecidableEq ι] (cache : QueryCache spec)\n\nend QueryCache\n\nnamespace QueryCount\n\nend QueryCount\n\nnamespace QueryLog\n\nsection getQ\n\nvariable [DecidableEq ι]\n\nend getQ\n\nsection countQ\n\nvariable [DecidableEq ι]\n\nend countQ\n\nsection prod\n\nvariable {ι₁ ι₂ : Type*} {spec₁ : OracleSpec ι₁} {spec₂ : OracleSpec ι₂}\n\nend prod\n\nend QueryLog\n\ndef QuerySeed (spec : OracleSpec ι) : Type _ :=\n  (i : ι) → List (spec.range i)\n\nnamespace QuerySeed\n\nsection addValues\n\nvariable [DecidableEq ι] {i : ι} (seed : QuerySeed spec) (us : List (spec.range i))\n\nend addValues\n\nsection takeAtIndex\n\nvariable [DecidableEq ι] (seed : QuerySeed spec) (i : ι) (n : ℕ)", "target_theorem": "lemma eq_takeAtIndex_length_iff (seed seed' : QuerySeed spec) (i : ι) :\n    seed = seed'.takeAtIndex i (seed i).length ↔\n      seed' = seed.addValues ((seed' i).drop (seed i).length) :=", "ground_truth_proof": ":= by\n  refine ⟨λ h ↦ QuerySeed.ext _ _ (λ j ↦ ?_), λ h ↦ ?_⟩\n  · by_cases hj : j = i\n    · induction hj\n      rw [h]\n      suffices (seed j).length ≤ (seed' j).length\n      by simp [this]\n      simpa using congr_arg List.length (congr_fun h j)\n    · rw [h]\n      simp [hj]\n  · rw [h]\n    simp", "nesting_depth": 4, "transitive_dep_count": 21, "subset_aristotle": false, "category": "Applied verif."}
{"id": 464, "thm_name": "OracleComp.allWhen_bind_iff", "thm_stmt": "@[simp] lemma allWhen_bind_iff (oa : OracleComp spec α) (ob : α → OracleComp spec β) :\n    (oa >>= ob).allWhen Q F possible_outputs ↔ oa.allWhen Q F possible_outputs ∧\n      ∀ x ∈ oa.supportWhen possible_outputs, (ob x).allWhen Q F possible_outputs", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/Traversal.lean", "imports": ["import VCVio.OracleComp.Support"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Or", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : OracleComp spec α → Type*}\n    (pure : (a : α) → C (pure a))\n    (query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.construct (Option.rec failure pure) query_bind oa"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : FreeMonad f α → Type*}\n    (pure : (x : α) → C (pure x))\n    (roll : {β : Type u} → (x : f β) → (r : β → FreeMonad f α) →\n      ((y : β) → C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | .pure x => pure x\n  | .roll x r => roll x _ (λ u ↦ FreeMonad.construct pure roll (r u))"}, {"name": "supportWhen", "content": "def supportWhen (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Set α) : Set α :="}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : OracleComp spec α → Prop}\n    (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "OracleComp.allWhen", "content": "def allWhen (possible_outputs : {α : Type v} → OracleQuery spec α → Set α)\n    (oa : OracleComp spec α) : Prop :="}], "used_local_lemmas": [{"name": "OracleComp.allWhen_query_bind", "content": "@[simp] lemma allWhen_query_bind (q : OracleQuery spec α) (oa : α → OracleComp spec β) :\n    ((q : OracleComp spec α) >>= oa).allWhen Q F possible_outputs ↔\n      Q q ∧ ∀ x ∈ possible_outputs q, (oa x).allWhen Q F possible_outputs"}], "local_ctx": "import VCVio.OracleComp.Support\n\nopen OracleSpec\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β γ : Type v}\n\nsection When\n\nvariable (Q : {α : Type v} → OracleQuery spec α → Prop)\n    (F : Prop) (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Set α)\n\ndef allWhen (possible_outputs : {α : Type v} → OracleQuery spec α → Set α)\n    (oa : OracleComp spec α) : Prop :=", "target_theorem": "@[simp] lemma allWhen_bind_iff (oa : OracleComp spec α) (ob : α → OracleComp spec β) :\n    (oa >>= ob).allWhen Q F possible_outputs ↔ oa.allWhen Q F possible_outputs ∧\n      ∀ x ∈ oa.supportWhen possible_outputs, (ob x).allWhen Q F possible_outputs :=", "ground_truth_proof": ":= by\n  induction oa using OracleComp.inductionOn with\n  | pure x => {\n    simp [supportWhen]\n  }\n  | failure => {\n    simp [supportWhen]\n  }\n  | query_bind i t oa h => {\n    rw [bind_assoc, allWhen_query_bind]\n    simp [h, supportWhen]\n    grind only [cases Or]\n  }", "nesting_depth": 4, "transitive_dep_count": 19, "subset_aristotle": false, "category": "Applied verif."}
{"id": 465, "thm_name": "OracleComp.probFailure_liftM", "thm_stmt": "@[simp]\nlemma probFailure_liftM (q : OracleQuery spec α) :\n    [⊥ | (q : OracleComp spec _)] = 0", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/DistSemantics/EvalDist.lean", "imports": ["import Mathlib.Probability.Distributions.Uniform", "import VCVio.OracleComp.SimSemantics.SimulateQ", "import VCVio.OracleComp.Traversal", "import ToMathlib.General"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "ENNReal", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "PMF", "module": "Mathlib.Probability.ProbabilityMassFunction.Basic"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Vector.insertIdx", "module": "Init.Data.Vector.Basic"}, {"name": "Nonempty", "module": "Init.Prelude"}, {"name": "OptionT.lift", "module": "Init.Control.Option"}, {"name": "PMF.uniformOfFintype", "module": "Mathlib.Probability.Distributions.Uniform"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Finset.card", "module": "Mathlib.Data.Finset.Card"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "Bind", "module": "Init.Prelude"}, {"name": "Pure", "module": "Init.Prelude"}, {"name": "AlternativeMonad", "module": "Batteries.Control.AlternativeMonad"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Option.getM", "module": "Init.Data.Option.Basic"}], "used_repo_defs": [{"name": "probFailure", "content": "notation \"[⊥\" \"|\" oa \"]\" => probFailure oa"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "cases", "content": "def cases {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])\n  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive (tl.insertIdx 0 hd)) {m : ℕ} :\n    (v : Vector α m) → motive v := match hm : m with\n  | 0 => fun v => match v with | ⟨⟨[]⟩, rfl⟩ => v_empty\n  | n + 1 => fun v => match hv : v with\n    | ⟨⟨hd :: tl⟩, hSize⟩ => by admit /- proof elided -/"}, {"name": "simulateQ", "content": "def simulateQ [AlternativeMonad m] (so : QueryImpl spec m) (oa : OracleComp spec α) : m α :=\n  do Option.getM (← FreeMonad.mapM oa.run so.impl)"}, {"name": "QueryImpl.Inhabited", "content": "instance QueryImpl.Inhabited [∀ i, Inhabited (spec.range i)] [Pure m] :\n  Inhabited (QueryImpl spec m) := ⟨{impl q := pure q.defaultOutput}⟩"}, {"name": "mapM", "content": "protected def mapM [Pure m] [Bind m] :\n    (oa : FreeMonad f α) → (s : {α : Type u} → f α → m α) → m α\n  | .pure x, _ => pure x\n  | .roll x r, s => s x >>= λ u ↦ (r u).mapM s"}], "lib_lemmas": [{"name": "Nat.cast_inj", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "PMF.uniformOfFintype_apply", "module": "Mathlib.Probability.Distributions.Uniform"}, {"name": "congr_arg", "module": "Batteries.Logic"}, {"name": "inv_inj", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "monadLift_self", "module": "Init.Control.Lawful.Basic"}, {"name": "ENNReal.inv_eq_zero", "module": "Mathlib.Data.ENNReal.Inv"}, {"name": "ENNReal.natCast_ne_top", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "ENNReal.tsum_eq_zero", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "PMF.bind_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Monad"}, {"name": "PMF.pure_apply", "module": "Mathlib.Probability.ProbabilityMassFunction.Monad"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "mul_eq_zero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}], "repo_lemmas": [{"name": "simulateQ_query", "content": "@[simp] lemma simulateQ_query (q : OracleQuery spec α) : simulateQ so q = so.impl q"}], "used_local_defs": [{"name": "OracleComp.probFailure", "content": "noncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none"}], "used_local_lemmas": [{"name": "OracleComp.evalDist_liftM", "content": "@[simp]\nlemma evalDist_liftM [Nonempty α] [Fintype α] (q : OracleQuery spec α) :\n    evalDist (q : OracleComp spec α) = OptionT.lift (PMF.uniformOfFintype α)"}], "local_ctx": "import VCVio.OracleComp.Traversal\n\nimport VCVio.OracleComp.SimSemantics.SimulateQ\n\nimport Mathlib.Probability.Distributions.Uniform\n\nimport ToMathlib.General\n\nopen OracleSpec Option ENNReal BigOperators\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {ι' : Type v} {spec' : OracleSpec ι'}\n  {α β γ : Type w} [spec.FiniteRange] [spec'.FiniteRange]\n\nsection evalDist\n\nend evalDist\n\nnotation \"[=\" x \"|\" oa \"]\" => probOutput oa x\n\nnoncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=\n  (evalDist oa).run none\n\nnotation \"[⊥\" \"|\" oa \"]\" => probFailure oa\n\nnotation \"[\" p \"|\" oa \"]\" => probEvent oa p\n\nsection bounds\n\nvariable {oa : OracleComp spec α} {x : α} {p : α → Prop}\n\nend bounds\n\nsection support\n\nvariable (oa : OracleComp spec α) (x : α) (p q : α → Prop)\n\nvariable {oa x p q}\n\nend support\n\nsection sums\n\nvariable (oa : OracleComp spec α) (p : α → Prop)\n\nend sums\n\nsection pure\n\nvariable (x : α)\n\nend pure\n\nsection bind\n\nvariable (oa : OracleComp spec α) (ob : α → OracleComp spec β)\n\nend bind\n\nsection mul_le_probEvent_bind\n\nend mul_le_probEvent_bind\n\nsection bind_const\n\nvariable (oa : OracleComp spec α) (ob : OracleComp spec β)\n\nend bind_const\n\nsection query\n\nvariable (i : ι) (t : spec.domain i)", "target_theorem": "@[simp]\nlemma probFailure_liftM (q : OracleQuery spec α) :\n    [⊥ | (q : OracleComp spec _)] = 0 :=", "ground_truth_proof": ":= by\n  have : Fintype α := q.rangeFintype\n  have : Inhabited α := q.rangeInhabited\n  simp only [probFailure, evalDist_liftM]\n  erw [PMF.bind_apply]\n  simp only [PMF.uniformOfFintype_apply, ENNReal.tsum_eq_zero, mul_eq_zero, ENNReal.inv_eq_zero,\n    natCast_ne_top, false_or]\n  intro i\n  erw [PMF.pure_apply]\n  simp", "nesting_depth": 5, "transitive_dep_count": 43, "subset_aristotle": false, "category": "Applied verif."}
{"id": 466, "thm_name": "OracleComp.probOutput_seq_map_eq_mul", "thm_stmt": "lemma probOutput_seq_map_eq_mul [spec.FiniteRange] (x : α) (y : β) (z : γ)\n    (h : ∀ x' ∈ oa.support, ∀ y' ∈ ob.support, z = f x' y' ↔ x' = x ∧ y' = y) :\n    [= z | f <$> oa <*> ob] = [= x | oa] * [= y | ob]", "lean_root": "VCV-io", "rel_path": "VCVio/OracleComp/DistSemantics/Seq.lean", "imports": ["import VCVio.OracleComp.DistSemantics.Monad", "import VCVio.OracleComp.DistSemantics.EvalDist"], "used_lib_defs": [{"name": "inline", "module": "Init.Core"}, {"name": "OptionT", "module": "Init.Control.Option"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "Set.univ", "module": "Mathlib.Data.Set.Defs"}, {"name": "Fintype", "module": "Mathlib.Data.Fintype.Defs"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "Classical.decEq", "module": "Mathlib.Logic.Basic"}, {"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "ENNReal", "module": "Mathlib.Data.ENNReal.Basic"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Function.comp", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}, {"name": "probOutput", "content": "noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=\n  (evalDist oa).run (some x)"}, {"name": "HasEvalDist", "content": "class HasEvalDist (m : Type u → Type v) [Monad m] where\n  evalDist {α : Type u} (mx : m α) : SPMF α\n  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x\n  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :\n    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)"}, {"name": "OracleComp", "content": "def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :\n    Type w → Type (max u (v + 1) w) :=\n  OptionT (FreeMonad (OracleQuery.{u,v} spec))"}, {"name": "OracleQuery", "content": "inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)\n  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)"}, {"name": "domain", "content": "@[inline, reducible]\nprotected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1"}, {"name": "OracleSpec", "content": "def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=\n  (i : ι) → Type v × Type v"}, {"name": "range", "content": "@[inline, reducible]\nprotected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2"}, {"name": "FreeMonad", "content": "inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)\n  | protected pure (x : α) : FreeMonad f α\n  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α"}, {"name": "support", "content": "def support (oa : OracleComp spec α) : Set α :=\n  oa.supportWhen fun _ => Set.univ"}, {"name": "supportWhen", "content": "def supportWhen (oa : OracleComp spec α)\n    (possible_outputs : {α : Type v} → OracleQuery spec α → Set α) : Set α :="}, {"name": "induction", "content": "@[elab_as_elim]\nprotected def induction {C : OracleComp spec α → Prop}\n    (oa : OracleComp spec α) (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : FreeMonad f α → Prop}\n    (pure : ∀ x, C (pure x))\n    (roll : ∀ {β} (x : f β), (r : β → FreeMonad f α) →\n      (∀ y, C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | FreeMonad.pure x => pure x\n  | FreeMonad.roll x r => roll x _ (λ u ↦\n      FreeMonad.inductionOn pure roll (r u))"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : OracleComp spec α → Type*}\n    (pure : (a : α) → C (pure a))\n    (query_bind : {β : Type v} → (q : OracleQuery spec β) →\n      (oa : β → OracleComp spec α) → ((u : β) → C (oa u)) → C (q >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.construct (Option.rec failure pure) query_bind oa"}, {"name": "construct", "content": "@[elab_as_elim]\nprotected def construct {C : FreeMonad f α → Type*}\n    (pure : (x : α) → C (pure x))\n    (roll : {β : Type u} → (x : f β) → (r : β → FreeMonad f α) →\n      ((y : β) → C (r y)) → C (x >>= r)) :\n    (oa : FreeMonad f α) → C oa\n  | .pure x => pure x\n  | .roll x r => roll x _ (λ u ↦ FreeMonad.construct pure roll (r u))"}, {"name": "FiniteRange", "content": "class FiniteRange (spec : OracleSpec ι) where\n    range_inhabited' (i : ι) : Inhabited (spec.range i)\n    range_fintype' (i : ι) : Fintype (spec.range i)"}, {"name": "DecidableEq", "content": "protected class DecidableEq (spec : OracleSpec ι) where\n  domain_decidableEq' (i : ι) : DecidableEq (spec.domain i)\n  range_decidableEq' (i : ι) : DecidableEq (spec.range i)"}, {"name": "inductionOn", "content": "@[elab_as_elim]\nprotected def inductionOn {C : OracleComp spec α → Prop}\n    (pure : (a : α) → C (pure a))\n    (query_bind : (i : ι) → (t : spec.domain i) →\n      (oa : spec.range i → OracleComp spec α) → (∀ u, C (oa u)) → C (query i t >>= oa))\n    (failure : C failure) (oa : OracleComp spec α) : C oa :=\n  FreeMonad.inductionOn (Option.rec failure pure) (λ (query i t) ↦ query_bind i t) oa"}, {"name": "probOutput", "content": "notation \"Pr[=\" x \"|\" mx \"]\" => probOutput mx x"}, {"name": "probOutput", "content": "notation \"[=\" x \"|\" oa \"]\" => probOutput oa x"}], "lib_lemmas": [{"name": "ENNReal.tsum_mul_left", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "map_eq_bind_pure_comp", "module": "Mathlib.Control.Monad.Basic"}, {"name": "mul_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "seq_eq_bind", "module": "Init.Control.Lawful.Basic"}, {"name": "mul_ite", "module": "Mathlib.Algebra.Notation.Defs"}, {"name": "mul_one", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ENNReal.tsum_prod", "module": "Mathlib.Topology.Instances.ENNReal.Lemmas"}, {"name": "ite_eq_left_iff", "module": "Init.PropLemmas"}, {"name": "ite_eq_right_iff", "module": "Init.PropLemmas"}, {"name": "mul_eq_zero", "module": "Mathlib.Algebra.GroupWithZero.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_and_or", "module": "Mathlib.Logic.Basic"}, {"name": "tsum_eq_single", "module": "Mathlib.Topology.Algebra.InfiniteSum.Basic"}, {"name": "zero_eq_mul", "module": "Mathlib.Algebra.GroupWithZero.Defs"}], "repo_lemmas": [{"name": "probOutput_bind_eq_tsum", "content": "lemma probOutput_bind_eq_tsum (y : β) :\n    [= y | oa >>= ob] = ∑' x : α, [= x | oa] * [= y | ob x]"}, {"name": "probOutput_pure", "content": "@[simp]\nlemma probOutput_pure [DecidableEq α] (y : α) :\n    [= y | (pure x : OracleComp spec α)] = if y = x then 1 else 0"}, {"name": "probOutput_eq_zero_iff", "content": "@[simp low]\nlemma probOutput_eq_zero_iff : [= x | oa] = 0 ↔ x ∉ oa.support"}, {"name": "mem_support_evalDist_iff", "content": "@[simp]\nlemma mem_support_evalDist_iff (oa : OracleComp spec α) (x : α) :\n    some x ∈ (evalDist oa).run.support ↔ x ∈ oa.support"}], "used_local_defs": [], "used_local_lemmas": [{"name": "OracleComp.probOutput_seq_map_eq_tsum", "content": "lemma probOutput_seq_map_eq_tsum [spec.FiniteRange]\n    (z : γ) : [= z | f <$> oa <*> ob] = ∑' (x : α) (y : β),\n      [= x | oa] * [= y | ob] * [= z | (pure (f x y) : OracleComp spec γ)]"}, {"name": "OracleComp.probOutput_seq_map_eq_tsum_ite", "content": "lemma probOutput_seq_map_eq_tsum_ite [spec.FiniteRange] [DecidableEq γ]\n    (z : γ) : [= z | f <$> oa <*> ob] =\n      ∑' (x : α) (y : β), if z = f x y then [= x | oa] * [= y | ob] else 0"}], "local_ctx": "import VCVio.OracleComp.DistSemantics.Monad\n\nnamespace OracleComp\n\nvariable {ι : Type u} {spec : OracleSpec ι} {α β γ : Type v}\n\nvariable (oa : OracleComp spec α) (og : OracleComp spec (α → β))\n\nsection seq_map\n\nvariable (oa : OracleComp spec α) (ob : OracleComp spec β) (f : α → β → γ)\n\nsection swap\n\nend swap\n\nsection injective2\n\nend injective2", "target_theorem": "lemma probOutput_seq_map_eq_mul [spec.FiniteRange] (x : α) (y : β) (z : γ)\n    (h : ∀ x' ∈ oa.support, ∀ y' ∈ ob.support, z = f x' y' ↔ x' = x ∧ y' = y) :\n    [= z | f <$> oa <*> ob] = [= x | oa] * [= y | ob] :=", "ground_truth_proof": ":= by\n  have : DecidableEq γ := Classical.decEq γ\n  rw [probOutput_seq_map_eq_tsum_ite, ← ENNReal.tsum_prod]\n  refine (tsum_eq_single (x, y) (λ (x', y') ↦ ?_)).trans ?_\n  · simp only [ne_eq, Prod.mk.injEq, ite_eq_right_iff, mul_eq_zero,\n      probOutput_eq_zero_iff, ← not_and_or]\n    exact λ h1 h2 h3 ↦ h1 ((h x' h3.1 y' h3.2).1 h2)\n  · simp only [ite_eq_left_iff, zero_eq_mul, probOutput_eq_zero_iff, ← not_and_or]\n    intro h1 h2\n    refine h1 ((h x h2.1 y h2.2).2 ⟨rfl, rfl⟩)", "nesting_depth": 5, "transitive_dep_count": 50, "subset_aristotle": true, "category": "Applied verif."}
{"id": 467, "thm_name": "PFunctor.Lens.sumPair_inl_inr", "thm_stmt": "@[simp]\ntheorem sumPair_inl_inr :\n    Lens.sumPair Lens.inl Lens.inr = Lens.id.{max uA₁ uA₂, uB₁} (P + Q)", "lean_root": "VCV-io", "rel_path": "ToMathlib/PFunctor/Lens/Basic.lean", "imports": ["import ToMathlib.PFunctor.Basic"], "used_lib_defs": [{"name": "PFunctor", "module": "Mathlib.Data.PFunctor.Univariate.Basic"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.elim", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.inr", "module": "Init.Core"}], "used_repo_defs": [{"name": "prodPair", "content": "notation \"⟨\" l₁ \",\" l₂ \"⟩ₗ\" => prodPair l₁ l₂"}, {"name": "Lens", "content": "structure Lens (P : PFunctor.{uA₁, uB₁}) (Q : PFunctor.{uA₂, uB₂}) where\n  toFunA : P.A → Q.A\n  toFunB : ∀ a, Q.B (toFunA a) → P.B a"}, {"name": "Chart", "content": "structure Chart (P : PFunctor.{uA₁, uB₁}) (Q : PFunctor.{uA₂, uB₂}) where\n  toFunA : P.A → Q.A\n  toFunB : ∀ a, P.B a → Q.B (toFunA a)"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "PFunctor.Lens.prodPair", "content": "def prodPair {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₂}} {R : PFunctor.{uA₃, uB₃}}\n    (l₁ : Lens P Q) (l₂ : Lens P R) :\n    Lens.{uA₁, uB₁, max uA₂ uA₃, max uB₂ uB₃} P (Q * R) :=\n  (fun p => (l₁.toFunA p, l₂.toFunA p)) ⇆\n    (fun p => Sum.elim (l₁.toFunB p) (l₂.toFunB p))"}], "used_local_lemmas": [{"name": "PFunctor.Lens.ext", "content": "@[ext (iff := false)]\ntheorem ext {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₂}} (l₁ l₂ : Lens P Q)\n    (h₁ : ∀ a, l₁.toFunA a = l₂.toFunA a) (h₂ : ∀ a, l₁.toFunB a = (h₁ a) ▸ l₂.toFunB a) :\n    l₁ = l₂"}], "local_ctx": "import ToMathlib.PFunctor.Basic\n\nsection find_home\n\nvariable {α : Sort u} {β : α → Sort v} {γ : α → Sort v}\n\nend find_home\n\nnamespace PFunctor\n\nnamespace Lens\n\n@[inherit_doc] infixl:75 \" ∘ₗ \" => comp\n\n@[inherit_doc] infix:50 \" ≃ₗ \" => Equiv\n\nnamespace Equiv\n\nend Equiv\n\ndef prodPair {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₂}} {R : PFunctor.{uA₃, uB₃}}\n    (l₁ : Lens P Q) (l₂ : Lens P R) :\n    Lens.{uA₁, uB₁, max uA₂ uA₃, max uB₂ uB₃} P (Q * R) :=\n  (fun p => (l₁.toFunA p, l₂.toFunA p)) ⇆\n    (fun p => Sum.elim (l₁.toFunB p) (l₂.toFunB p))\n\n@[inherit_doc] infixl:75 \" ◃ₗ \" => compMap\n\n@[inherit_doc] infixl:75 \" ×ₗ \" => prodMap\n\n@[inherit_doc] infixl:75 \" ⊎ₗ \" => sumMap\n\n@[inherit_doc] infixl:75 \" ⊗ₗ \" => tensorMap\n\nnotation \"[\" l₁ \",\" l₂ \"]ₗ\" => sumPair l₁ l₂\n\nnotation \"⟨\" l₁ \",\" l₂ \"⟩ₗ\" => prodPair l₁ l₂\n\nsection Coprod\n\nvariable {P : PFunctor.{uA₁, uB₁}} {Q : PFunctor.{uA₂, uB₁}}\n  {R : PFunctor.{uA₃, uB₃}} {W : PFunctor.{uA₄, uB₃}} {X : PFunctor.{uA₅, uB₅}}", "target_theorem": "@[simp]\ntheorem sumPair_inl_inr :\n    Lens.sumPair Lens.inl Lens.inr = Lens.id.{max uA₁ uA₂, uB₁} (P + Q) :=", "ground_truth_proof": ":= by\n  ext a <;> rcases a <;> rfl", "nesting_depth": 2, "transitive_dep_count": 13, "subset_aristotle": true, "category": "Applied verif."}
{"id": 468, "thm_name": "Veil.bind_terminates", "thm_stmt": "theorem bind_terminates m (act : Wp m σ ρ) (act' : ρ -> Wp m σ ρ') s [LawfulAction act] :\n  pre s ->\n  act.alwaysSuccessfullyTerminates pre →\n  (act.bind act').alwaysSuccessfullyTerminates pre ->\n  act.toBigStep s r' s' ->\n  (act' r').alwaysSuccessfullyTerminates (· = s')", "lean_root": "veil", "rel_path": "Veil/DSL/Action/Theory.lean", "imports": ["import Veil.DSL.Base"], "used_lib_defs": [{"name": "r", "module": "Test.Playground.WHNFExamples"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "p", "module": "Smt.Reconstruct.Certified.ModusPonens"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "t", "module": "Test.Unit.Normalize"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "semiOutParam", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax (name:= assumption) \"assumptionDef\" : attr", "content": "syntax (name:= assumption) \"assumptionDef\" : attr"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Veil.Mode", "content": "inductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq"}, {"name": "Veil.SProp", "content": "@[inline] abbrev SProp := σ -> Prop"}, {"name": "Veil.RProp", "content": "@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ"}, {"name": "Veil.Wp", "content": "abbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop"}, {"name": "Veil.Wlp", "content": "abbrev Wlp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop"}, {"name": "Veil.BigStep", "content": "abbrev BigStep := σ -> ρ -> σ -> Prop"}, {"name": "Veil.Wp.bind", "content": "@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)"}, {"name": "Veil.Wp.get", "content": "@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s"}, {"name": "Veil.Wp.set", "content": "@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'"}, {"name": "Veil.Wp.modifyGet", "content": "@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'"}, {"name": "Veil.IsSubStateOf", "content": "class IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ"}, {"name": "Veil.Wp.lift", "content": "@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))"}, {"name": "Veil.BigStep.bind", "content": "def BigStep.bind (act : BigStep σ ρ) (act' : ρ -> BigStep σ ρ') : BigStep σ ρ' :=\n  fun s r' s' => ∃ r s'', act s r s'' ∧ act' r s'' r' s'"}, {"name": "Veil.Wp.toWlp", "content": "@[actSimp]\nabbrev Wp.toWlp {σ ρ : Type} {m : Mode} (wp : Wp m σ ρ) : Wlp m σ ρ :=\n  \n  fun (s : σ) (post : RProp σ ρ) => ¬ wp s (fun r s' => ¬ post r s')"}, {"name": "Veil.Wp.toBigStep", "content": "@[actSimp]\ndef Wp.toBigStep {σ} (wp : Wp m σ ρ) : BigStep σ ρ :=\n  fun s r' s' =>\n    wp.toWlp s (fun r₀ s₀ => r' = r₀ ∧ s' = s₀)"}, {"name": "Veil.Wp.triple", "content": "abbrev Wp.triple {σ ρ} (req : SProp σ) (act : Wp m σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s, req s -> act s ens"}, {"name": "Veil.Wp.alwaysSuccessfullyTerminates", "content": "abbrev Wp.alwaysSuccessfullyTerminates {σ } (req : SProp σ) (act : Wp m σ ρ)  : Prop :=\n  ∀ s, req s -> act s (fun _ _ => True)"}, {"name": "Veil.BigStep.triple", "content": "abbrev BigStep.triple {σ } (req : SProp σ) (act : BigStep σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s r' s', req s -> act s r' s' -> ens r' s'"}, {"name": "Veil.LawfulAction", "content": "class LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'"}, {"name": "Veil.pure_lawful", "content": "instance pure_lawful : LawfulAction (Wp.pure (σ := σ) (m := m) r) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.bind_lawful", "content": "instance bind_lawful (act : Wp m' σ ρ) (act' : ρ -> Wp m σ ρ') [LawfulAction act] [∀ r, LawfulAction (act' r)] : LawfulAction (Wp.bind (m := m) act act') where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.internal_sound", "content": "instance (priority := low) internal_sound (act : Wp m σ ρ) [inst : LawfulAction (m := .internal) act] : LawfulAction (m := .external) act where\n  inter := inst.inter\n  impl := inst.impl"}, {"name": "Veil.assume_lawful", "content": "instance assume_lawful : LawfulAction (Wp.assume (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.assert_lawful", "content": "instance assert_lawful : LawfulAction (Wp.assert (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.require_lawful", "content": "instance require_lawful : LawfulAction (Wp.require (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.fresh_lawful", "content": "instance fresh_lawful : LawfulAction (Wp.fresh (m := m) (σ := σ) τ) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.spec_lawful", "content": "instance spec_lawful : LawfulAction (Wp.spec (m := m) req ens) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil._inst_σ", "content": "instance (r : σ -> σ -> Prop) : LawfulAction (r.toWp (m := m)) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.get_lawful", "content": "instance get_lawful : LawfulAction (Wp.get (m := m) (σ := σ)) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.set_lawful", "content": "instance set_lawful (s : σ) : LawfulAction (Wp.set s (m := m)) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.modifyGet_lawful", "content": "instance modifyGet_lawful : LawfulAction (Wp.modifyGet f (m := m) (σ := σ) (ρ := ρ)) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.if_lawful", "content": "instance if_lawful [Decidable c] [instT: LawfulAction t] [instS : LawfulAction e] : LawfulAction (ite c t e) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.lift_lawful", "content": "instance lift_lawful (act : Wp m σ ρ) [IsSubStateOf σ σ'] [LawfulAction act] :\n  LawfulAction (act.lift (σ' := σ')) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.GenBigStep", "content": "class GenBigStep (σ ρ : Type) (wp : Wp .external σ ρ) (tr : outParam (BigStep σ ρ)) where\n  lawful : LawfulAction wp\n  equiv pre  :\n    wp.alwaysSuccessfullyTerminates pre -> ∀ s, pre s -> tr s = wp.toBigStep s"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ ρ (Wp.pure r) (BigStep.pure r) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ PUnit (Wp.assume asm) (BigStep.assume asm) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ PUnit (Wp.assert asm) (BigStep.assert asm) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ PUnit (Wp.require rq) (BigStep.assume rq) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ τ (Wp.fresh τ) (BigStep.fresh τ) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ Unit (Wp.set s) (BigStep.set s) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ σ (Wp.get) (BigStep.get) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ ρ (Wp.modifyGet act) (BigStep.modifyGet act) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ ρ (Wp.spec req ens) (BigStep.spec req ens) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance [inst : GenBigStep σ ρ act actTr] : LawfulAction act := inst.lawful"}], "used_local_lemmas": [{"name": "Veil.big_step_sound'", "content": "theorem big_step_sound' [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  act.triple req ens → act.toBigStep.triple req ens"}], "local_ctx": "import Veil.DSL.Base\n\nnamespace Veil\n\nsection Veil\n\nopen Classical\n\nsection Types\n\ninductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq\n\nvariable (m : Mode) (σ ρ : Type)\n\n@[inline] abbrev SProp := σ -> Prop\n\n@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ\n\nabbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop\n\nabbrev Wlp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop\n\nabbrev BigStep := σ -> ρ -> σ -> Prop\n\nend Types\n\nsection Theory\n\nvariable {σ ρ : Type}\n\n@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)\n\n@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s\n\n@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'\n\n@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'\n\nclass IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ\n\n@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))\n\ndef BigStep.bind (act : BigStep σ ρ) (act' : ρ -> BigStep σ ρ') : BigStep σ ρ' :=\n  fun s r' s' => ∃ r s'', act s r s'' ∧ act' r s'' r' s'\n\n@[actSimp]\nabbrev Wp.toWlp {σ ρ : Type} {m : Mode} (wp : Wp m σ ρ) : Wlp m σ ρ :=\n  \n  fun (s : σ) (post : RProp σ ρ) => ¬ wp s (fun r s' => ¬ post r s')\n\n@[actSimp]\ndef Wp.toBigStep {σ} (wp : Wp m σ ρ) : BigStep σ ρ :=\n  fun s r' s' =>\n    wp.toWlp s (fun r₀ s₀ => r' = r₀ ∧ s' = s₀)\n\nabbrev Wp.triple {σ ρ} (req : SProp σ) (act : Wp m σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s, req s -> act s ens\n\nabbrev Wp.alwaysSuccessfullyTerminates {σ } (req : SProp σ) (act : Wp m σ ρ)  : Prop :=\n  ∀ s, req s -> act s (fun _ _ => True)\n\nabbrev BigStep.triple {σ } (req : SProp σ) (act : BigStep σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s r' s', req s -> act s r' s' -> ens r' s'\n\nclass LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'\n\nsection TwoStateSoundness\n\nend TwoStateSoundness\n\nsection BigStepSoundness\n\nend BigStepSoundness\n\nsection LawfulActionInstances\n\ninstance pure_lawful : LawfulAction (Wp.pure (σ := σ) (m := m) r) where\n  inter := by admit /- proof elided -/\n\ninstance bind_lawful (act : Wp m' σ ρ) (act' : ρ -> Wp m σ ρ') [LawfulAction act] [∀ r, LawfulAction (act' r)] : LawfulAction (Wp.bind (m := m) act act') where\n  inter := by admit /- proof elided -/\n\ninstance (priority := low) internal_sound (act : Wp m σ ρ) [inst : LawfulAction (m := .internal) act] : LawfulAction (m := .external) act where\n  inter := inst.inter\n  impl := inst.impl\n\ninstance assume_lawful : LawfulAction (Wp.assume (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/\n\ninstance assert_lawful : LawfulAction (Wp.assert (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/\n\ninstance require_lawful : LawfulAction (Wp.require (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/\n\ninstance fresh_lawful : LawfulAction (Wp.fresh (m := m) (σ := σ) τ) where\n  inter := by admit /- proof elided -/\n\ninstance spec_lawful : LawfulAction (Wp.spec (m := m) req ens) where\n  inter := by admit /- proof elided -/\n\ninstance (r : σ -> σ -> Prop) : LawfulAction (r.toWp (m := m)) where\n  inter := by admit /- proof elided -/\n\ninstance get_lawful : LawfulAction (Wp.get (m := m) (σ := σ)) where\n  inter := by admit /- proof elided -/\n\ninstance set_lawful (s : σ) : LawfulAction (Wp.set s (m := m)) where\n  inter := by admit /- proof elided -/\n\ninstance modifyGet_lawful : LawfulAction (Wp.modifyGet f (m := m) (σ := σ) (ρ := ρ)) where\n  inter := by admit /- proof elided -/\n\ninstance if_lawful [Decidable c] [instT: LawfulAction t] [instS : LawfulAction e] : LawfulAction (ite c t e) where\n  inter := by admit /- proof elided -/\n\ninstance lift_lawful (act : Wp m σ ρ) [IsSubStateOf σ σ'] [LawfulAction act] :\n  LawfulAction (act.lift (σ' := σ')) where\n  inter := by admit /- proof elided -/\n\nend LawfulActionInstances\n\nsection GenBigStepInstances\n\nclass GenBigStep (σ ρ : Type) (wp : Wp .external σ ρ) (tr : outParam (BigStep σ ρ)) where\n  lawful : LawfulAction wp\n  equiv pre  :\n    wp.alwaysSuccessfullyTerminates pre -> ∀ s, pre s -> tr s = wp.toBigStep s\n\ninstance : GenBigStep σ ρ (Wp.pure r) (BigStep.pure r) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ PUnit (Wp.assume asm) (BigStep.assume asm) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ PUnit (Wp.assert asm) (BigStep.assert asm) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ PUnit (Wp.require rq) (BigStep.assume rq) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ τ (Wp.fresh τ) (BigStep.fresh τ) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ Unit (Wp.set s) (BigStep.set s) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ σ (Wp.get) (BigStep.get) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ ρ (Wp.modifyGet act) (BigStep.modifyGet act) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ ρ (Wp.spec req ens) (BigStep.spec req ens) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance [inst : GenBigStep σ ρ act actTr] : LawfulAction act := inst.lawful", "target_theorem": "theorem bind_terminates m (act : Wp m σ ρ) (act' : ρ -> Wp m σ ρ') s [LawfulAction act] :\n  pre s ->\n  act.alwaysSuccessfullyTerminates pre →\n  (act.bind act').alwaysSuccessfullyTerminates pre ->\n  act.toBigStep s r' s' ->\n  (act' r').alwaysSuccessfullyTerminates (· = s') :=", "ground_truth_proof": ":= by\n    unfold Wp.alwaysSuccessfullyTerminates Wp.toBigStep Wp.toWlp Wp.bind\n    intros hpre actT act'T\n    have actT := actT s hpre\n    have act'T := act'T s hpre\n    have act''T := big_step_sound' (act := act) (req := (· = s))\n    unfold Wp.triple BigStep.triple Wp.toBigStep Wp.toWlp at act''T\n    simp at act''T; specialize act''T _ act'T s r' s' rfl\n    simp_all", "nesting_depth": 3, "transitive_dep_count": 21, "subset_aristotle": true, "category": "Framework"}
{"id": 469, "thm_name": "Veil.lift_transition_big_step'", "thm_stmt": "theorem lift_transition_big_step' {σ σ'} [IsSubStateOf σ σ'] (m : Mode) (r : Wp m σ ρ) [LawfulAction r] (st : σ') :\n  r.alwaysSuccessfullyTerminates (· = getFrom st) →\n  (@Wp.lift _  m σ σ' _ r).toBigStep st =\n  fun r' st' =>\n    r.toBigStep (getFrom st) r' (getFrom st') ∧\n    st' = (setIn (@getFrom σ σ' _ st') st)", "lean_root": "veil", "rel_path": "Veil/DSL/Action/Theory.lean", "imports": ["import Veil.DSL.Base"], "used_lib_defs": [{"name": "semiOutParam", "module": "Init.Prelude"}, {"name": "BEq", "module": "Init.Prelude"}, {"name": "p", "module": "Smt.Reconstruct.Certified.ModusPonens"}, {"name": "r", "module": "Test.Playground.WHNFExamples"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "t", "module": "Test.Unit.Normalize"}, {"name": "Decidable", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax (name:= assumption) \"assumptionDef\" : attr", "content": "syntax (name:= assumption) \"assumptionDef\" : attr"}], "lib_lemmas": [{"name": "Classical.not_forall", "module": "Init.Classical"}, {"name": "Decidable.not_not", "module": "Init.PropLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "eq_iff_iff", "module": "Init.Core"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Veil.Mode", "content": "inductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq"}, {"name": "Veil.SProp", "content": "@[inline] abbrev SProp := σ -> Prop"}, {"name": "Veil.RProp", "content": "@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ"}, {"name": "Veil.TwoState", "content": "@[inline] abbrev TwoState := σ -> σ -> Prop"}, {"name": "Veil.Wp", "content": "abbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop"}, {"name": "Veil.Wlp", "content": "abbrev Wlp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop"}, {"name": "Veil.BigStep", "content": "abbrev BigStep := σ -> ρ -> σ -> Prop"}, {"name": "Veil.Wp.bind", "content": "@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)"}, {"name": "Veil.Wp.get", "content": "@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s"}, {"name": "Veil.Wp.set", "content": "@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'"}, {"name": "Veil.Wp.modifyGet", "content": "@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'"}, {"name": "Veil.IsSubStateOf", "content": "class IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ"}, {"name": "Veil.Wp.lift", "content": "@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))"}, {"name": "Veil.BigStep.lift", "content": "def BigStep.lift [IsSubStateOf σ σ'] (act : BigStep σ ρ) : BigStep σ' ρ :=\n  fun st r' st' => act (getFrom st) r' (getFrom st') ∧ st' = (setIn (@getFrom σ σ' _ st') st)"}, {"name": "Veil.Wp.toWlp", "content": "@[actSimp]\nabbrev Wp.toWlp {σ ρ : Type} {m : Mode} (wp : Wp m σ ρ) : Wlp m σ ρ :=\n  \n  fun (s : σ) (post : RProp σ ρ) => ¬ wp s (fun r s' => ¬ post r s')"}, {"name": "Veil.Wp.toBigStep", "content": "@[actSimp]\ndef Wp.toBigStep {σ} (wp : Wp m σ ρ) : BigStep σ ρ :=\n  fun s r' s' =>\n    wp.toWlp s (fun r₀ s₀ => r' = r₀ ∧ s' = s₀)"}, {"name": "Veil.BigStep.toWp", "content": "@[actSimp]\ndef BigStep.toWp {σ} (act : BigStep σ ρ) : Wp .internal σ ρ :=\n  fun s post => ∀ r s', act s r s' -> post r s'"}, {"name": "Veil._root_.Function.toWp", "content": "@[actSimp]\ndef _root_.Function.toWp (m : Mode) (r : TwoState σ) : Wp m σ Unit :=\n  fun s post => ∀ s', r s s' -> post () s'"}, {"name": "Veil.Wp.triple", "content": "abbrev Wp.triple {σ ρ} (req : SProp σ) (act : Wp m σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s, req s -> act s ens"}, {"name": "Veil.Wp.alwaysSuccessfullyTerminates", "content": "abbrev Wp.alwaysSuccessfullyTerminates {σ } (req : SProp σ) (act : Wp m σ ρ)  : Prop :=\n  ∀ s, req s -> act s (fun _ _ => True)"}, {"name": "Veil.LawfulAction", "content": "class LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'"}, {"name": "Veil.pure_lawful", "content": "instance pure_lawful : LawfulAction (Wp.pure (σ := σ) (m := m) r) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.bind_lawful", "content": "instance bind_lawful (act : Wp m' σ ρ) (act' : ρ -> Wp m σ ρ') [LawfulAction act] [∀ r, LawfulAction (act' r)] : LawfulAction (Wp.bind (m := m) act act') where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.internal_sound", "content": "instance (priority := low) internal_sound (act : Wp m σ ρ) [inst : LawfulAction (m := .internal) act] : LawfulAction (m := .external) act where\n  inter := inst.inter\n  impl := inst.impl"}, {"name": "Veil.assume_lawful", "content": "instance assume_lawful : LawfulAction (Wp.assume (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.assert_lawful", "content": "instance assert_lawful : LawfulAction (Wp.assert (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.require_lawful", "content": "instance require_lawful : LawfulAction (Wp.require (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.fresh_lawful", "content": "instance fresh_lawful : LawfulAction (Wp.fresh (m := m) (σ := σ) τ) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.spec_lawful", "content": "instance spec_lawful : LawfulAction (Wp.spec (m := m) req ens) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil._inst_σ", "content": "instance (r : σ -> σ -> Prop) : LawfulAction (r.toWp (m := m)) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.get_lawful", "content": "instance get_lawful : LawfulAction (Wp.get (m := m) (σ := σ)) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.set_lawful", "content": "instance set_lawful (s : σ) : LawfulAction (Wp.set s (m := m)) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.modifyGet_lawful", "content": "instance modifyGet_lawful : LawfulAction (Wp.modifyGet f (m := m) (σ := σ) (ρ := ρ)) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.if_lawful", "content": "instance if_lawful [Decidable c] [instT: LawfulAction t] [instS : LawfulAction e] : LawfulAction (ite c t e) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.lift_lawful", "content": "instance lift_lawful (act : Wp m σ ρ) [IsSubStateOf σ σ'] [LawfulAction act] :\n  LawfulAction (act.lift (σ' := σ')) where\n  inter := by admit /- proof elided -/"}, {"name": "Veil.GenBigStep", "content": "class GenBigStep (σ ρ : Type) (wp : Wp .external σ ρ) (tr : outParam (BigStep σ ρ)) where\n  lawful : LawfulAction wp\n  equiv pre  :\n    wp.alwaysSuccessfullyTerminates pre -> ∀ s, pre s -> tr s = wp.toBigStep s"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ ρ (Wp.pure r) (BigStep.pure r) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ PUnit (Wp.assume asm) (BigStep.assume asm) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ PUnit (Wp.assert asm) (BigStep.assert asm) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ PUnit (Wp.require rq) (BigStep.assume rq) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ τ (Wp.fresh τ) (BigStep.fresh τ) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ Unit (Wp.set s) (BigStep.set s) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ σ (Wp.get) (BigStep.get) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ ρ (Wp.modifyGet act) (BigStep.modifyGet act) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance : GenBigStep σ ρ (Wp.spec req ens) (BigStep.spec req ens) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/"}, {"name": "Veil._inst_GenBigStep", "content": "instance [inst : GenBigStep σ ρ act actTr] : LawfulAction act := inst.lawful"}], "used_local_lemmas": [{"name": "Veil.lift_transition_big_step", "content": "theorem lift_transition_big_step {σ σ'} [IsSubStateOf σ σ'] (m : Mode) (tr : BigStep σ ρ) :\n  (@Wp.lift _  m σ σ' _ tr.toWp).toBigStep =\n  fun st r' st' =>\n    tr (getFrom st) r' (getFrom st') ∧\n    st' = (setIn (@getFrom σ σ' _ st') st)"}, {"name": "Veil.big_step_sound", "content": "theorem big_step_sound [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  (¬ ∀ r s, ens r s) ->\n  act.toBigStep.triple req ens -> act.triple req ens"}, {"name": "Veil.big_step_sound'", "content": "theorem big_step_sound' [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  act.triple req ens → act.toBigStep.triple req ens"}, {"name": "Veil.big_step_always_terminating_sound", "content": "theorem big_step_always_terminating_sound [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  act.alwaysSuccessfullyTerminates req ->\n  act.toBigStep.triple req ens -> act.triple req ens"}, {"name": "Veil.big_step_to_wp", "content": "theorem big_step_to_wp (act : Wp m σ ρ) [LawfulAction act] (req : SProp σ) :\n  act.alwaysSuccessfullyTerminates req ->\n  req s ->\n  act s = act.toBigStep.toWp s"}], "local_ctx": "import Veil.DSL.Base\n\nnamespace Veil\n\nsection Veil\n\nopen Classical\n\nsection Types\n\ninductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq\n\nvariable (m : Mode) (σ ρ : Type)\n\n@[inline] abbrev SProp := σ -> Prop\n\n@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ\n\n@[inline] abbrev TwoState := σ -> σ -> Prop\n\nabbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop\n\nabbrev Wlp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop\n\nabbrev BigStep := σ -> ρ -> σ -> Prop\n\nend Types\n\nsection Theory\n\nvariable {σ ρ : Type}\n\n@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)\n\n@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s\n\n@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'\n\n@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'\n\nclass IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ\n\n@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))\n\ndef BigStep.lift [IsSubStateOf σ σ'] (act : BigStep σ ρ) : BigStep σ' ρ :=\n  fun st r' st' => act (getFrom st) r' (getFrom st') ∧ st' = (setIn (@getFrom σ σ' _ st') st)\n\n@[actSimp]\nabbrev Wp.toWlp {σ ρ : Type} {m : Mode} (wp : Wp m σ ρ) : Wlp m σ ρ :=\n  \n  fun (s : σ) (post : RProp σ ρ) => ¬ wp s (fun r s' => ¬ post r s')\n\n@[actSimp]\ndef Wp.toBigStep {σ} (wp : Wp m σ ρ) : BigStep σ ρ :=\n  fun s r' s' =>\n    wp.toWlp s (fun r₀ s₀ => r' = r₀ ∧ s' = s₀)\n\n@[actSimp]\ndef BigStep.toWp {σ} (act : BigStep σ ρ) : Wp .internal σ ρ :=\n  fun s post => ∀ r s', act s r s' -> post r s'\n\n@[actSimp]\ndef _root_.Function.toWp (m : Mode) (r : TwoState σ) : Wp m σ Unit :=\n  fun s post => ∀ s', r s s' -> post () s'\n\nabbrev Wp.triple {σ ρ} (req : SProp σ) (act : Wp m σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s, req s -> act s ens\n\nabbrev Wp.alwaysSuccessfullyTerminates {σ } (req : SProp σ) (act : Wp m σ ρ)  : Prop :=\n  ∀ s, req s -> act s (fun _ _ => True)\n\nclass LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'\n\nsection TwoStateSoundness\n\nend TwoStateSoundness\n\nsection BigStepSoundness\n\nend BigStepSoundness\n\nsection LawfulActionInstances\n\ninstance pure_lawful : LawfulAction (Wp.pure (σ := σ) (m := m) r) where\n  inter := by admit /- proof elided -/\n\ninstance bind_lawful (act : Wp m' σ ρ) (act' : ρ -> Wp m σ ρ') [LawfulAction act] [∀ r, LawfulAction (act' r)] : LawfulAction (Wp.bind (m := m) act act') where\n  inter := by admit /- proof elided -/\n\ninstance (priority := low) internal_sound (act : Wp m σ ρ) [inst : LawfulAction (m := .internal) act] : LawfulAction (m := .external) act where\n  inter := inst.inter\n  impl := inst.impl\n\ninstance assume_lawful : LawfulAction (Wp.assume (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/\n\ninstance assert_lawful : LawfulAction (Wp.assert (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/\n\ninstance require_lawful : LawfulAction (Wp.require (m := m) (σ := σ) rq) where\n  inter := by admit /- proof elided -/\n\ninstance fresh_lawful : LawfulAction (Wp.fresh (m := m) (σ := σ) τ) where\n  inter := by admit /- proof elided -/\n\ninstance spec_lawful : LawfulAction (Wp.spec (m := m) req ens) where\n  inter := by admit /- proof elided -/\n\ninstance (r : σ -> σ -> Prop) : LawfulAction (r.toWp (m := m)) where\n  inter := by admit /- proof elided -/\n\ninstance get_lawful : LawfulAction (Wp.get (m := m) (σ := σ)) where\n  inter := by admit /- proof elided -/\n\ninstance set_lawful (s : σ) : LawfulAction (Wp.set s (m := m)) where\n  inter := by admit /- proof elided -/\n\ninstance modifyGet_lawful : LawfulAction (Wp.modifyGet f (m := m) (σ := σ) (ρ := ρ)) where\n  inter := by admit /- proof elided -/\n\ninstance if_lawful [Decidable c] [instT: LawfulAction t] [instS : LawfulAction e] : LawfulAction (ite c t e) where\n  inter := by admit /- proof elided -/\n\ninstance lift_lawful (act : Wp m σ ρ) [IsSubStateOf σ σ'] [LawfulAction act] :\n  LawfulAction (act.lift (σ' := σ')) where\n  inter := by admit /- proof elided -/\n\nend LawfulActionInstances\n\nsection GenBigStepInstances\n\nclass GenBigStep (σ ρ : Type) (wp : Wp .external σ ρ) (tr : outParam (BigStep σ ρ)) where\n  lawful : LawfulAction wp\n  equiv pre  :\n    wp.alwaysSuccessfullyTerminates pre -> ∀ s, pre s -> tr s = wp.toBigStep s\n\ninstance : GenBigStep σ ρ (Wp.pure r) (BigStep.pure r) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ PUnit (Wp.assume asm) (BigStep.assume asm) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ PUnit (Wp.assert asm) (BigStep.assert asm) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ PUnit (Wp.require rq) (BigStep.assume rq) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ τ (Wp.fresh τ) (BigStep.fresh τ) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ Unit (Wp.set s) (BigStep.set s) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ σ (Wp.get) (BigStep.get) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ ρ (Wp.modifyGet act) (BigStep.modifyGet act) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance : GenBigStep σ ρ (Wp.spec req ens) (BigStep.spec req ens) where\n  lawful := inferInstance\n  equiv := by admit /- proof elided -/\n\ninstance [inst : GenBigStep σ ρ act actTr] : LawfulAction act := inst.lawful", "target_theorem": "theorem lift_transition_big_step' {σ σ'} [IsSubStateOf σ σ'] (m : Mode) (r : Wp m σ ρ) [LawfulAction r] (st : σ') :\n  r.alwaysSuccessfullyTerminates (· = getFrom st) →\n  (@Wp.lift _  m σ σ' _ r).toBigStep st =\n  fun r' st' =>\n    r.toBigStep (getFrom st) r' (getFrom st') ∧\n    st' = (setIn (@getFrom σ σ' _ st') st) :=", "ground_truth_proof": ":= by\n  intro term\n  have rEq : r.lift.toBigStep st = (r.toBigStep.toWp.lift.toBigStep st) := by {\n    unfold Wp.lift Wp.toBigStep Wp.toWlp; ext; simp\n    rw [big_step_to_wp (act := r) (req := (fun x => x = getFrom st))] <;> try simp [*]\n    unfold Wp.toBigStep Wp.toWlp; simp }\n  rw [rEq, lift_transition_big_step]", "nesting_depth": 5, "transitive_dep_count": 35, "subset_aristotle": true, "category": "Framework"}
{"id": 470, "thm_name": "Veil.big_step_to_wp", "thm_stmt": "theorem big_step_to_wp (act : Wp m σ ρ) [LawfulAction act] (req : SProp σ) :\n  act.alwaysSuccessfullyTerminates req ->\n  req s ->\n  act s = act.toBigStep.toWp s", "lean_root": "veil", "rel_path": "Veil/DSL/Action/Theory.lean", "imports": ["import Veil.DSL.Base"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "r", "module": "Test.Playground.WHNFExamples"}, {"name": "p", "module": "Smt.Reconstruct.Certified.ModusPonens"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "t", "module": "Test.Unit.Normalize"}, {"name": "semiOutParam", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax (name:= assumption) \"assumptionDef\" : attr", "content": "syntax (name:= assumption) \"assumptionDef\" : attr"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Veil.Mode", "content": "inductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq"}, {"name": "Veil.SProp", "content": "@[inline] abbrev SProp := σ -> Prop"}, {"name": "Veil.RProp", "content": "@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ"}, {"name": "Veil.TwoState", "content": "@[inline] abbrev TwoState := σ -> σ -> Prop"}, {"name": "Veil.Wp", "content": "abbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop"}, {"name": "Veil.Wlp", "content": "abbrev Wlp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop"}, {"name": "Veil.BigStep", "content": "abbrev BigStep := σ -> ρ -> σ -> Prop"}, {"name": "Veil.Wp.bind", "content": "@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)"}, {"name": "Veil.Wp.get", "content": "@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s"}, {"name": "Veil.Wp.set", "content": "@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'"}, {"name": "Veil.Wp.modifyGet", "content": "@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'"}, {"name": "Veil.IsSubStateOf", "content": "class IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ"}, {"name": "Veil.Wp.lift", "content": "@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))"}, {"name": "Veil.Wp.toWlp", "content": "@[actSimp]\nabbrev Wp.toWlp {σ ρ : Type} {m : Mode} (wp : Wp m σ ρ) : Wlp m σ ρ :=\n  \n  fun (s : σ) (post : RProp σ ρ) => ¬ wp s (fun r s' => ¬ post r s')"}, {"name": "Veil.Wp.toBigStep", "content": "@[actSimp]\ndef Wp.toBigStep {σ} (wp : Wp m σ ρ) : BigStep σ ρ :=\n  fun s r' s' =>\n    wp.toWlp s (fun r₀ s₀ => r' = r₀ ∧ s' = s₀)"}, {"name": "Veil.BigStep.toWp", "content": "@[actSimp]\ndef BigStep.toWp {σ} (act : BigStep σ ρ) : Wp .internal σ ρ :=\n  fun s post => ∀ r s', act s r s' -> post r s'"}, {"name": "Veil._root_.Function.toWp", "content": "@[actSimp]\ndef _root_.Function.toWp (m : Mode) (r : TwoState σ) : Wp m σ Unit :=\n  fun s post => ∀ s', r s s' -> post () s'"}, {"name": "Veil.Wp.triple", "content": "abbrev Wp.triple {σ ρ} (req : SProp σ) (act : Wp m σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s, req s -> act s ens"}, {"name": "Veil.Wp.alwaysSuccessfullyTerminates", "content": "abbrev Wp.alwaysSuccessfullyTerminates {σ } (req : SProp σ) (act : Wp m σ ρ)  : Prop :=\n  ∀ s, req s -> act s (fun _ _ => True)"}, {"name": "Veil.LawfulAction", "content": "class LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'"}], "used_local_lemmas": [{"name": "Veil.big_step_sound", "content": "theorem big_step_sound [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  (¬ ∀ r s, ens r s) ->\n  act.toBigStep.triple req ens -> act.triple req ens"}, {"name": "Veil.big_step_sound'", "content": "theorem big_step_sound' [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  act.triple req ens → act.toBigStep.triple req ens"}, {"name": "Veil.big_step_always_terminating_sound", "content": "theorem big_step_always_terminating_sound [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  act.alwaysSuccessfullyTerminates req ->\n  act.toBigStep.triple req ens -> act.triple req ens"}], "local_ctx": "import Veil.DSL.Base\n\nnamespace Veil\n\nsection Veil\n\nopen Classical\n\nsection Types\n\ninductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq\n\nvariable (m : Mode) (σ ρ : Type)\n\n@[inline] abbrev SProp := σ -> Prop\n\n@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ\n\n@[inline] abbrev TwoState := σ -> σ -> Prop\n\nabbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop\n\nabbrev Wlp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop\n\nabbrev BigStep := σ -> ρ -> σ -> Prop\n\nend Types\n\nsection Theory\n\nvariable {σ ρ : Type}\n\n@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)\n\n@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s\n\n@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'\n\n@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'\n\nclass IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ\n\n@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))\n\n@[actSimp]\nabbrev Wp.toWlp {σ ρ : Type} {m : Mode} (wp : Wp m σ ρ) : Wlp m σ ρ :=\n  \n  fun (s : σ) (post : RProp σ ρ) => ¬ wp s (fun r s' => ¬ post r s')\n\n@[actSimp]\ndef Wp.toBigStep {σ} (wp : Wp m σ ρ) : BigStep σ ρ :=\n  fun s r' s' =>\n    wp.toWlp s (fun r₀ s₀ => r' = r₀ ∧ s' = s₀)\n\n@[actSimp]\ndef BigStep.toWp {σ} (act : BigStep σ ρ) : Wp .internal σ ρ :=\n  fun s post => ∀ r s', act s r s' -> post r s'\n\n@[actSimp]\ndef _root_.Function.toWp (m : Mode) (r : TwoState σ) : Wp m σ Unit :=\n  fun s post => ∀ s', r s s' -> post () s'\n\nabbrev Wp.triple {σ ρ} (req : SProp σ) (act : Wp m σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s, req s -> act s ens\n\nabbrev Wp.alwaysSuccessfullyTerminates {σ } (req : SProp σ) (act : Wp m σ ρ)  : Prop :=\n  ∀ s, req s -> act s (fun _ _ => True)\n\nclass LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'\n\nsection TwoStateSoundness\n\nend TwoStateSoundness\n\nsection BigStepSoundness", "target_theorem": "theorem big_step_to_wp (act : Wp m σ ρ) [LawfulAction act] (req : SProp σ) :\n  act.alwaysSuccessfullyTerminates req ->\n  req s ->\n  act s = act.toBigStep.toWp s :=", "ground_truth_proof": ":= by\n  intro hterm hreq; ext post; constructor\n  { simp [BigStep.toWp]; intro _ _ _\n    solve_by_elim [big_step_sound'] }\n  simp [BigStep.toWp]\n  intro h; apply big_step_always_terminating_sound (req := (s = ·)) <;> try simp\n  { solve_by_elim }\n  intro; simp_all", "nesting_depth": 3, "transitive_dep_count": 24, "subset_aristotle": false, "category": "Framework"}
{"id": 471, "thm_name": "Veil.big_step_always_terminating_sound", "thm_stmt": "theorem big_step_always_terminating_sound [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  act.alwaysSuccessfullyTerminates req ->\n  act.toBigStep.triple req ens -> act.triple req ens", "lean_root": "veil", "rel_path": "Veil/DSL/Action/Theory.lean", "imports": ["import Veil.DSL.Base"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "r", "module": "Test.Playground.WHNFExamples"}, {"name": "t", "module": "Test.Unit.Normalize"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "semiOutParam", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Veil.Mode", "content": "inductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq"}, {"name": "Veil.SProp", "content": "@[inline] abbrev SProp := σ -> Prop"}, {"name": "Veil.RProp", "content": "@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ"}, {"name": "Veil.Wp", "content": "abbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop"}, {"name": "Veil.Wp.bind", "content": "@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)"}, {"name": "Veil.Wp.get", "content": "@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s"}, {"name": "Veil.Wp.set", "content": "@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'"}, {"name": "Veil.Wp.modifyGet", "content": "@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'"}, {"name": "Veil.IsSubStateOf", "content": "class IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ"}, {"name": "Veil.Wp.lift", "content": "@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))"}, {"name": "Veil.LawfulAction", "content": "class LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'"}], "used_local_lemmas": [{"name": "Veil.big_step_sound", "content": "theorem big_step_sound [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  (¬ ∀ r s, ens r s) ->\n  act.toBigStep.triple req ens -> act.triple req ens"}], "local_ctx": "import Veil.DSL.Base\n\nnamespace Veil\n\nsection Veil\n\nopen Classical\n\nsection Types\n\ninductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq\n\nvariable (m : Mode) (σ ρ : Type)\n\n@[inline] abbrev SProp := σ -> Prop\n\n@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ\n\nabbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop\n\nend Types\n\nsection Theory\n\nvariable {σ ρ : Type}\n\n@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)\n\n@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s\n\n@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'\n\n@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'\n\nclass IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ\n\n@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))\n\nclass LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'\n\nsection TwoStateSoundness\n\nend TwoStateSoundness\n\nsection BigStepSoundness", "target_theorem": "theorem big_step_always_terminating_sound [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  act.alwaysSuccessfullyTerminates req ->\n  act.toBigStep.triple req ens -> act.triple req ens :=", "ground_truth_proof": ":= by\n  intro ensTaut htriple s hreq\n  by_cases h: (¬ ∀ r s, ens r s)\n  { solve_by_elim [big_step_sound] }\n  apply LawfulAction.impl (post := fun _ _ => True) <;> try simp_all", "nesting_depth": 4, "transitive_dep_count": 11, "subset_aristotle": true, "category": "Framework"}
{"id": 472, "thm_name": "Veil.TwoState_sound'_ret_unit", "thm_stmt": "theorem TwoState_sound'_ret_unit [LawfulAction act] (req : SProp σ) (ens : RProp σ PUnit) :\n  act.triple req ens → act.toTwoState.triple req (ens () ·)", "lean_root": "veil", "rel_path": "Veil/DSL/Action/Theory.lean", "imports": ["import Veil.DSL.Base"], "used_lib_defs": [{"name": "BEq", "module": "Init.Prelude"}, {"name": "Inhabited", "module": "Init.Prelude"}, {"name": "impl", "module": "Mathlib.Deprecated.MLList.BestFirst"}, {"name": "r", "module": "Test.Playground.WHNFExamples"}, {"name": "t", "module": "Test.Unit.Normalize"}, {"name": "PUnit", "module": "Init.Prelude"}, {"name": "PUnit.unit", "module": "Init.Prelude"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "semiOutParam", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "syntax (name:= assumption) \"assumptionDef\" : attr", "content": "syntax (name:= assumption) \"assumptionDef\" : attr"}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Veil.Mode", "content": "inductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq"}, {"name": "Veil.SProp", "content": "@[inline] abbrev SProp := σ -> Prop"}, {"name": "Veil.RProp", "content": "@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ"}, {"name": "Veil.Wp", "content": "abbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop"}, {"name": "Veil.Wp.bind", "content": "@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)"}, {"name": "Veil.Wp.get", "content": "@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s"}, {"name": "Veil.Wp.set", "content": "@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'"}, {"name": "Veil.Wp.modifyGet", "content": "@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'"}, {"name": "Veil.IsSubStateOf", "content": "class IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ"}, {"name": "Veil.Wp.lift", "content": "@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))"}, {"name": "Veil.Wp.triple", "content": "abbrev Wp.triple {σ ρ} (req : SProp σ) (act : Wp m σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s, req s -> act s ens"}, {"name": "Veil.LawfulAction", "content": "class LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'"}], "used_local_lemmas": [{"name": "Veil.TwoState_sound'", "content": "theorem TwoState_sound' [LawfulAction act] (req : SProp σ) (ens : RProp σ ρ) :\n  act.triple req ens → act.toTwoState.triple req (∃ r, ens r ·)"}, {"name": "Veil.exists_over_PUnit", "content": "theorem exists_over_PUnit (p : PUnit → Prop) : (∃ (u : PUnit), p u) = p ()"}], "local_ctx": "import Veil.DSL.Base\n\nnamespace Veil\n\nsection Veil\n\nopen Classical\n\nsection Types\n\ninductive Mode where\n  | internal : Mode\n  | external : Mode\nderiving BEq\n\nvariable (m : Mode) (σ ρ : Type)\n\n@[inline] abbrev SProp := σ -> Prop\n\n@[inline] abbrev RProp (ρ : Type u) := ρ → SProp σ\n\nabbrev Wp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop\n\nabbrev Wlp (m : Mode) (σ ρ : Type) := σ -> RProp σ ρ -> Prop\n\nreturn value `r : ρ` and a post-state `s' : σ` -/\n\nabbrev BigStep := σ -> ρ -> σ -> Prop\n\nend Types\n\nsection Theory\n\nvariable {σ ρ : Type}\n\n@[actSimp] def Wp.bind (wp : Wp m σ ρ) (wp_cont : ρ -> Wp m σ ρ') : Wp m σ ρ' :=\n  fun s post => wp s (fun r s' => wp_cont r s' post)\n\n@[actSimp] def Wp.get : Wp m σ σ := fun s post => post s s\n\n@[actSimp] def Wp.set (s' : σ) : Wp m σ Unit := fun _ post => post () s'\n\n@[actSimp] def Wp.modifyGet (act : σ -> ρ × σ) : Wp m σ ρ := fun s post => let (ret, s') := act s ; post ret s'\n\nclass IsSubStateOf (σ : semiOutParam Type) (σ' : Type) where\n   \n  setIn : σ -> σ' -> σ'\n   \n  getFrom : σ' -> σ\n\n  setIn_getFrom_idempotent : ∀ σ', setIn (getFrom σ') σ' = σ'\n  getFrom_setIn_idempotent : ∀ σ σ', getFrom (setIn σ σ') = σ\n\n@[actSimp] def Wp.lift {σ σ'} [IsSubStateOf σ σ'] (act : Wp m σ ρ) : Wp m σ' ρ :=\n  fun s' post => act (getFrom s') (fun r s => post r (setIn s s'))\n\nabbrev Wp.triple {σ ρ} (req : SProp σ) (act : Wp m σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s, req s -> act s ens\n\nclass LawfulAction {σ ρ : Type} (act : Wp m σ ρ) where\n  inter {τ : Type} [Inhabited τ] (post : τ -> RProp σ ρ) :\n    ∀ s : σ, (∀ t : τ, act s (post t)) -> act s (∀ t, post t · ·)\n\n  impl (post post' : RProp σ ρ) : ∀ s,\n    (∀ r s, post r s -> post' r s) -> act s post -> act s post'\n\n@[actSimp]\nabbrev Wp.toWlp {σ ρ : Type} {m : Mode} (wp : Wp m σ ρ) : Wlp m σ ρ :=\n  -- `wlp(P, φ, s) = ¬ wp(P, ¬φ, s)`\n  fun (s : σ) (post : RProp σ ρ) => ¬ wp s (fun r s' => ¬ post r s')\n\n@[actSimp]\ndef Wp.toBigStep {σ} (wp : Wp m σ ρ) : BigStep σ ρ :=\n  fun s r' s' =>\n    wp.toWlp s (fun r₀ s₀ => r' = r₀ ∧ s' = s₀)\n\n@[actSimp]\ndef Wp.toTwoState {σ} (wp : Wp m σ ρ) : TwoState σ :=\n  fun s s' =>\n    wp.toWlp s (fun _ s₀ => (s' = s₀))\n\n@[actSimp]\ndef BigStep.toWp {σ} (act : BigStep σ ρ) : Wp .internal σ ρ :=\n  fun s post => ∀ r s', act s r s' -> post r s'\n\n@[actSimp]\ndef _root_.Function.toWp (m : Mode) (r : TwoState σ) : Wp m σ Unit :=\n  fun s post => ∀ s', r s s' -> post () s'\n\nabbrev refines {σ ρ} (act : Wp m σ ρ) (act' : Wp m σ ρ) : Prop :=\n  ∀ s post, act s post -> act' s post\n\ninstance : LE (Wp m σ ρ) where\n  le := refines\n\nabbrev Wp.triple {σ ρ} (req : SProp σ) (act : Wp m σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s, req s -> act s ens\n\nabbrev Wp.alwaysSuccessfullyTerminates {σ } (req : SProp σ) (act : Wp m σ ρ)  : Prop :=\n  ∀ s, req s -> act s (fun _ _ => True)\n\nabbrev TwoState.triple {σ } (req : SProp σ) (act : TwoState σ) (ens : SProp σ) : Prop :=\n  ∀ s s', req s -> act s s' -> ens s'\n\nabbrev BigStep.triple {σ } (req : SProp σ) (act : BigStep σ ρ) (ens : RProp σ ρ) : Prop :=\n  ∀ s r' s', req s -> act s r' s' -> ens r' s'\nsection TwoStateSoundness", "target_theorem": "theorem TwoState_sound'_ret_unit [LawfulAction act] (req : SProp σ) (ens : RProp σ PUnit) :\n  act.triple req ens → act.toTwoState.triple req (ens () ·) :=", "ground_truth_proof": ":= by\n  have heq : (ens () ·) = (∃ r, ens r ·) := by ext ; rw [exists_over_PUnit]\n  rw [heq] ; apply TwoState_sound'", "nesting_depth": 4, "transitive_dep_count": 15, "subset_aristotle": true, "category": "Framework"}
{"id": 473, "thm_name": "FBA.slice_blocks_ne", "thm_stmt": "theorem slice_blocks_ne : ∀ n S I, intact (inst := inst) I → n ∈ I → blocks_slices S n →\n    S ∩ I ≠ ∅", "lean_root": "veil", "rel_path": "Examples/StellarConsensus/SCPTheory.lean", "imports": ["import Mathlib.Data.Set.Basic"], "used_lib_defs": [{"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "p", "module": "Smt.Reconstruct.Certified.ModusPonens"}], "used_repo_defs": [{"name": "syntax (name:= assumption) \"assumptionDef\" : attr", "content": "syntax (name:= assumption) \"assumptionDef\" : attr"}], "lib_lemmas": [{"name": "Set.nonempty_iff_ne_empty", "module": "Mathlib.Data.Set.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "FBA.project", "content": "def project {α β : Type} (slices : β → Set (Set α)) (S : Set α) : β → Set (Set α) :=\n  fun n => { Sl ∩ S | Sl ∈ slices n }"}, {"name": "FBA.System", "content": "class System (Node : Type) where\n   \n  W : Set Node\n  slices : Node → Set (Set Node)\n   \n  slices_ne : ∀ p ∈ W, slices p ≠ ∅"}, {"name": "FBA.System.project", "content": "def System.project (sys : System Node) (I : Set Node) : System Node :=\n  { W := sys.W\n    slices := FBA.project sys.slices I\n    slices_ne := by admit /- proof elided -/"}, {"name": "FBA.quorum", "content": "def quorum (Q : Set Node) : Prop := ∀ p ∈ Q ∩ W, ∃ Sl ∈ slices p, Sl ⊆ Q"}, {"name": "FBA.blocks_slices", "content": "def blocks_slices (S : Set Node) (p : Node) : Prop :=\n  ∀ Sl ∈ slices p, Sl ∩ S ≠ ∅"}, {"name": "FBA.intertwined", "content": "structure intertwined (S : Set Node) where\n  well_behaved : S ⊆ W\n   \n  q_inter : (∀ Q Q',\n    quorum (inst := inst.project S) Q →\n    quorum (inst := inst.project S) Q' →\n    Q ∩ S ≠ ∅ → Q' ∩ S ≠ ∅ → Q ∩ Q' ∩ S ≠ ∅)"}, {"name": "FBA.intact", "content": "structure intact (I : Set Node) extends intertwined I where\n   \n  q_avail : quorum (inst := inst) I"}], "used_local_lemmas": [{"name": "Set.ne_empty_iff_exists_mem", "content": "theorem Set.ne_empty_iff_exists_mem {α : Type u} {s : Set α} : s ≠ ∅ ↔ ∃ a, a ∈ s"}, {"name": "FBA.intact_implies_intertwined", "content": "theorem intact_implies_intertwined : ∀ I, intact (inst := inst) I → intertwined I"}, {"name": "FBA.intertwined_node_is_well_behaved", "content": "theorem intertwined_node_is_well_behaved : ∀ n S, intertwined (inst := inst) S → n ∈ S → n ∈ W"}, {"name": "FBA.intact_node_is_well_behaved", "content": "theorem intact_node_is_well_behaved : ∀ n I, intact (inst := inst) I → n ∈ I → n ∈ W"}], "local_ctx": "import Mathlib.Data.Set.Basic\n\nnamespace FBA\n\ndef project {α β : Type} (slices : β → Set (Set α)) (S : Set α) : β → Set (Set α) :=\n  fun n => { Sl ∩ S | Sl ∈ slices n }\n\nclass System (Node : Type) where\n   \n  W : Set Node\n  slices : Node → Set (Set Node)\n   \n  slices_ne : ∀ p ∈ W, slices p ≠ ∅\n\nvariable {Node : Type}\n\ndef System.project (sys : System Node) (I : Set Node) : System Node :=\n  { W := sys.W\n    slices := FBA.project sys.slices I\n    slices_ne := by admit /- proof elided -/\n\nvariable [inst : System Node]\n\nopen System\n\ndef quorum (Q : Set Node) : Prop := ∀ p ∈ Q ∩ W, ∃ Sl ∈ slices p, Sl ⊆ Q\n\ndef blocks_slices (S : Set Node) (p : Node) : Prop :=\n  ∀ Sl ∈ slices p, Sl ∩ S ≠ ∅\n\nstructure intertwined (S : Set Node) where\n  well_behaved : S ⊆ W\n   \n  q_inter : (∀ Q Q',\n    quorum (inst := inst.project S) Q →\n    quorum (inst := inst.project S) Q' →\n    Q ∩ S ≠ ∅ → Q' ∩ S ≠ ∅ → Q ∩ Q' ∩ S ≠ ∅)\n\nstructure intact (I : Set Node) extends intertwined I where\n   \n  q_avail : quorum (inst := inst) I", "target_theorem": "theorem slice_blocks_ne : ∀ n S I, intact (inst := inst) I → n ∈ I → blocks_slices S n →\n    S ∩ I ≠ ∅ :=", "ground_truth_proof": ":= by\n  intro n S I hI hin hblock\n  unfold blocks_slices at hblock\n  have h := hI.q_avail ; unfold quorum at h\n  simp at h ; specialize h _ hin (intact_node_is_well_behaved _ _ hI hin)\n  rcases h with ⟨Sl, hSl, h⟩ ; specialize hblock _ hSl\n  rw [Set.ne_empty_iff_exists_mem] at hblock ⊢ ; simp at hblock ⊢\n  aesop", "nesting_depth": 3, "transitive_dep_count": 14, "subset_aristotle": true, "category": "Framework"}
{"id": 474, "thm_name": "Term.lower_ite_hEquiv", "thm_stmt": "theorem lower_ite_hEquiv {cond : Term .bool} {pos neg : Term τ}\n    (hc : cond ≈ lower cond) (hp : pos ≈ lower pos) (hn : neg ≈ lower neg) :\n    ite cond pos neg ≈ lower (ite cond pos neg)", "lean_root": "verified-compiler", "rel_path": "VerifiedCompiler.lean", "imports": [], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "cond", "module": "Init.Prelude"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.append_nil", "module": "Init.Data.List.Basic"}, {"name": "List.append_assoc", "module": "Init.Data.List.Basic"}, {"name": "List.length_append", "module": "Init.Data.List.Basic"}, {"name": "List.length_singleton", "module": "Init.Data.List.Basic"}, {"name": "List.singleton_append", "module": "Init.Data.List.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ty", "content": "inductive Ty\n  | bool\n  | nat"}, {"name": "Value", "content": "inductive Value : Ty → Type\n  | bool : Bool → Value .bool\n  | nat  : Nat  → Value .nat"}, {"name": "Term", "content": "inductive Term : Ty → Type\n  | val (v : Value τ)                          : Term τ\n  | ite (cond : Term .bool) (pos neg : Term τ) : Term τ\n  | and (lhs rhs : Term .bool)                 : Term .bool"}, {"name": "Term.eval", "content": "def eval : Term τ → Value τ\n  | val v => v\n  | ite cond pos neg =>\n    match eval cond with\n    | true  => eval pos\n    | false => eval neg\n  | and lhs rhs =>\n    match eval lhs, eval rhs with\n    | true, true => true\n    | _,    _    => false"}, {"name": "Instruction", "content": "inductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)"}, {"name": "Program", "content": "abbrev Program := List Instruction"}, {"name": "Stack", "content": "abbrev Stack := List Nat"}, {"name": "Nat.conj", "content": "def Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1"}, {"name": "Instruction.Result", "content": "structure Instruction.Result where\n  stack : Stack\n  offset := 0"}, {"name": "Instruction.exec", "content": "def Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none"}, {"name": "Program.goto", "content": "def goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset"}, {"name": "Program.exec", "content": "def exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]"}, {"name": "Value.toNat", "content": "def Value.toNat : Value τ → Nat\n  | bool false => 0\n  | bool true  => 1\n  | nat n      => n"}, {"name": "Term.lower", "content": "def Term.lower : Term τ → Program\n  | val v            => [.const v.toNat]\n  | and lhs rhs      => (lower lhs) ++ (lower rhs) ++ [.and]\n  | ite cond pos neg =>\n    let p := lower pos\n    let n := lower neg\n    lower cond ++ [.jez (p.length + 1)] ++ p ++ [.jmp n.length] ++ n"}, {"name": "HEquiv", "content": "def HEquiv (t : Term τ) (prog : Program) : Prop :=\n  prog.exec [] = some [t.eval.toNat]"}], "used_local_lemmas": [{"name": "Instruction.jmp_def", "content": "@[simp]\ntheorem jmp_def (offset s) : exec (.jmp offset) s = some ⟨s, offset⟩"}, {"name": "Instruction.jez_zero", "content": "@[simp]\ntheorem jez_zero (offset s) : exec (.jez offset) (0 :: s) = some ⟨s, offset⟩"}, {"name": "Instruction.jez_succ", "content": "@[simp]\ntheorem jez_succ (n offset s) : exec (.jez offset) ((n + 1) :: s) = some ⟨s, 0⟩"}, {"name": "Program.goto_zero", "content": "@[simp]\ntheorem goto_zero (prog : Program) : goto prog 0 = some prog"}, {"name": "Program.goto_suffix", "content": "@[simp]\ntheorem goto_suffix (prog₁ prog₂ : Program) : goto (prog₁ ++ prog₂) prog₁.length = some prog₂"}, {"name": "Program.goto_end", "content": "@[simp]\ntheorem goto_end {prog : Program} (h : offset = prog.length) : goto prog offset = some []"}, {"name": "Program.exec_goto", "content": "theorem exec_goto (hi : i.exec s₁ = some ⟨s₂, offset⟩) (hg : goto prog₁ offset = some prog₂) :\n    exec (i :: prog₁) s₁ = exec prog₂ s₂"}, {"name": "Program.goto_mono", "content": "theorem goto_mono (prog₂ : Program) (h : goto prog₁ offset = some prog₁') :\n    goto (prog₁ ++ prog₂) offset = some (prog₁' ++ prog₂)"}, {"name": "Program.exec_prog_mono", "content": "@[simp]\ntheorem exec_prog_mono (prog₂ : Program) (h : exec prog₁ s₁ = some s₂) :\n    exec (prog₁ ++ prog₂) s₁ = exec prog₂ s₂"}, {"name": "Term.eq_false_of_toNat_eq_zero", "content": "theorem eq_false_of_toNat_eq_zero {v : Value .bool} (h : v.toNat = 0) : v = false"}, {"name": "Term.eq_true_of_toNat_eq_succ", "content": "theorem eq_true_of_toNat_eq_succ {v : Value .bool} (h : v.toNat = n + 1) : v = true"}], "local_ctx": "inductive Ty\n  | bool\n  | nat\n\ninductive Value : Ty → Type\n  | bool : Bool → Value .bool\n  | nat  : Nat  → Value .nat\n\ninductive Term : Ty → Type\n  | val (v : Value τ)                          : Term τ\n  | ite (cond : Term .bool) (pos neg : Term τ) : Term τ\n  | and (lhs rhs : Term .bool)                 : Term .bool\n\nnamespace Term\n\ndef eval : Term τ → Value τ\n  | val v => v\n  | ite cond pos neg =>\n    match eval cond with\n    | true  => eval pos\n    | false => eval neg\n  | and lhs rhs =>\n    match eval lhs, eval rhs with\n    | true, true => true\n    | _,    _    => false\n\nnamespace Equiv\n\ninfixl:50 \" ~ \" => Equiv\n\nend Equiv\n\nend Term\n\nopen Term (eval)\n\ninductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)\n\nabbrev Program := List Instruction\n\nabbrev Stack := List Nat\n\ndef Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1\n\nstructure Instruction.Result where\n  stack : Stack\n  offset := 0\n\ndef Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none\n\nnamespace Program\n\ndef goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset\n\ndef exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]\n\nend Program\n\ndef Value.toNat : Value τ → Nat\n  | bool false => 0\n  | bool true  => 1\n  | nat n      => n\n\ndef Term.lower : Term τ → Program\n  | val v            => [.const v.toNat]\n  | and lhs rhs      => (lower lhs) ++ (lower rhs) ++ [.and]\n  | ite cond pos neg =>\n    let p := lower pos\n    let n := lower neg\n    lower cond ++ [.jez (p.length + 1)] ++ p ++ [.jmp n.length] ++ n\n\ndef HEquiv (t : Term τ) (prog : Program) : Prop :=\n  prog.exec [] = some [t.eval.toNat]\n\ninfixl:60 \" ≈ \" => HEquiv\n\nnamespace Instruction\n\nend Instruction\n\nnamespace Program\n\nend Program\n\nnamespace Term\n\nopen Program List\n\nopen Instruction hiding exec", "target_theorem": "theorem lower_ite_hEquiv {cond : Term .bool} {pos neg : Term τ}\n    (hc : cond ≈ lower cond) (hp : pos ≈ lower pos) (hn : neg ≈ lower neg) :\n    ite cond pos neg ≈ lower (ite cond pos neg) :=", "ground_truth_proof": ":= by\n  rw [HEquiv, lower, append_assoc, append_assoc, append_assoc] at *\n  -- Give names to terms needed later.\n  let pl    := pos.lower.length + 1\n  let nl    := neg.lower.length\n  let prog₃ := [.jmp nl] ++ neg.lower\n  let prog₂ := pos.lower ++ prog₃\n  let prog₁ := [.jez pl] ++ prog₂\n  -- Peel off the leading `lower cond`.\n  rw [exec_prog_mono prog₁ hc]\n  -- Continue differently depending on what's on top of the stack.\n  cases h : cond.eval.toNat\n  case zero =>\n    -- Peel off the leading `jez`.\n    have hg := goto_suffix (pos.lower ++ [jmp nl]) neg.lower\n    simp only [length_append, length_singleton, append_assoc] at hg\n    have he := exec_goto (jez_zero pl []) hg\n    simp only [prog₁, singleton_append, he]\n    -- Close the goal by establishing that `eval` yields `neg.eval`.\n    simp_all only [prog₂, prog₃, eval, eq_false_of_toNat_eq_zero h]\n  case succ n =>\n    -- Peel off the leading `jez`.\n    have he := exec_goto (jez_succ n pl []) (goto_zero prog₂)\n    simp only [prog₁, singleton_append, he]\n    -- Peel off the leading `lower pos`.\n    rw [exec_prog_mono prog₃ hp]\n    -- Peel off the leading `jmp`.\n    have he := exec_goto (jmp_def nl [pos.eval.toNat]) (goto_end rfl)\n    simp only [prog₃, singleton_append, he]\n    -- Close the goal by establishing that `eval` yields `pos.eval`.\n    simp only [eval, exec, eq_true_of_toNat_eq_succ h]", "nesting_depth": 3, "transitive_dep_count": 39, "subset_aristotle": false, "category": "Compiler"}
{"id": 475, "thm_name": "Term.lower_and_hEquiv", "thm_stmt": "theorem lower_and_hEquiv (hl : lhs ≈ lower lhs) (hr : rhs ≈ lower rhs) :\n    and lhs rhs ≈ lower (and lhs rhs)", "lean_root": "verified-compiler", "rel_path": "VerifiedCompiler.lean", "imports": [], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "cond", "module": "Init.Prelude"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "List.singleton_append", "module": "Init.Data.List.Lemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ty", "content": "inductive Ty\n  | bool\n  | nat"}, {"name": "Value", "content": "inductive Value : Ty → Type\n  | bool : Bool → Value .bool\n  | nat  : Nat  → Value .nat"}, {"name": "Term", "content": "inductive Term : Ty → Type\n  | val (v : Value τ)                          : Term τ\n  | ite (cond : Term .bool) (pos neg : Term τ) : Term τ\n  | and (lhs rhs : Term .bool)                 : Term .bool"}, {"name": "Term.eval", "content": "def eval : Term τ → Value τ\n  | val v => v\n  | ite cond pos neg =>\n    match eval cond with\n    | true  => eval pos\n    | false => eval neg\n  | and lhs rhs =>\n    match eval lhs, eval rhs with\n    | true, true => true\n    | _,    _    => false"}, {"name": "Instruction", "content": "inductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)"}, {"name": "Program", "content": "abbrev Program := List Instruction"}, {"name": "Stack", "content": "abbrev Stack := List Nat"}, {"name": "Nat.conj", "content": "def Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1"}, {"name": "Instruction.Result", "content": "structure Instruction.Result where\n  stack : Stack\n  offset := 0"}, {"name": "Instruction.exec", "content": "def Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none"}, {"name": "Program.goto", "content": "def goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset"}, {"name": "Program.exec", "content": "def exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]"}, {"name": "Value.toNat", "content": "def Value.toNat : Value τ → Nat\n  | bool false => 0\n  | bool true  => 1\n  | nat n      => n"}, {"name": "Term.lower", "content": "def Term.lower : Term τ → Program\n  | val v            => [.const v.toNat]\n  | and lhs rhs      => (lower lhs) ++ (lower rhs) ++ [.and]\n  | ite cond pos neg =>\n    let p := lower pos\n    let n := lower neg\n    lower cond ++ [.jez (p.length + 1)] ++ p ++ [.jmp n.length] ++ n"}, {"name": "HEquiv", "content": "def HEquiv (t : Term τ) (prog : Program) : Prop :=\n  prog.exec [] = some [t.eval.toNat]"}], "used_local_lemmas": [{"name": "Instruction.exec_stack_mono", "content": "@[simp]\ntheorem exec_stack_mono (s : Stack) (h : exec inst s₁ = some ⟨s₂, o⟩) :\n    exec inst (s₁ ++ s) = some ⟨s₂ ++ s, o⟩"}, {"name": "Program.goto_mono", "content": "theorem goto_mono (prog₂ : Program) (h : goto prog₁ offset = some prog₁') :\n    goto (prog₁ ++ prog₂) offset = some (prog₁' ++ prog₂)"}, {"name": "Program.exec_stack_mono", "content": "@[simp]\ntheorem exec_stack_mono (s : Stack) (h : exec prog s₁ = some s₂) :\n    exec prog (s₁ ++ s) = s₂ ++ s"}, {"name": "Program.exec_prog_mono", "content": "@[simp]\ntheorem exec_prog_mono (prog₂ : Program) (h : exec prog₁ s₁ = some s₂) :\n    exec (prog₁ ++ prog₂) s₁ = exec prog₂ s₂"}], "local_ctx": "inductive Ty\n  | bool\n  | nat\n\ninductive Value : Ty → Type\n  | bool : Bool → Value .bool\n  | nat  : Nat  → Value .nat\n\ninductive Term : Ty → Type\n  | val (v : Value τ)                          : Term τ\n  | ite (cond : Term .bool) (pos neg : Term τ) : Term τ\n  | and (lhs rhs : Term .bool)                 : Term .bool\n\nnamespace Term\n\ndef eval : Term τ → Value τ\n  | val v => v\n  | ite cond pos neg =>\n    match eval cond with\n    | true  => eval pos\n    | false => eval neg\n  | and lhs rhs =>\n    match eval lhs, eval rhs with\n    | true, true => true\n    | _,    _    => false\n\nnamespace Equiv\n\ninfixl:50 \" ~ \" => Equiv\n\nend Equiv\n\nend Term\n\nopen Term (eval)\n\ninductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)\n\nabbrev Program := List Instruction\n\nabbrev Stack := List Nat\n\ndef Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1\n\nstructure Instruction.Result where\n  stack : Stack\n  offset := 0\n\ndef Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none\n\nnamespace Program\n\ndef goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset\n\ndef exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]\n\nend Program\n\ndef Value.toNat : Value τ → Nat\n  | bool false => 0\n  | bool true  => 1\n  | nat n      => n\n\ndef Term.lower : Term τ → Program\n  | val v            => [.const v.toNat]\n  | and lhs rhs      => (lower lhs) ++ (lower rhs) ++ [.and]\n  | ite cond pos neg =>\n    let p := lower pos\n    let n := lower neg\n    lower cond ++ [.jez (p.length + 1)] ++ p ++ [.jmp n.length] ++ n\n\ndef HEquiv (t : Term τ) (prog : Program) : Prop :=\n  prog.exec [] = some [t.eval.toNat]\n\ninfixl:60 \" ≈ \" => HEquiv\n\nnamespace Instruction\n\nend Instruction\n\nnamespace Program\n\nend Program\n\nnamespace Term\n\nopen Program List\n\nopen Instruction hiding exec", "target_theorem": "theorem lower_and_hEquiv (hl : lhs ≈ lower lhs) (hr : rhs ≈ lower rhs) :\n    and lhs rhs ≈ lower (and lhs rhs) :=", "ground_truth_proof": ":= by\n  have h₁ := exec_prog_mono (lower rhs ++ [.and]) hl\n  have h₂ := exec_stack_mono [lhs.eval.toNat] hr\n  simp only [nil_append, singleton_append] at h₂\n  have h₃ := exec_prog_mono [.and] h₂\n  simp [h₁, h₃, HEquiv, lower, exec, goto, Instruction.exec]", "nesting_depth": 4, "transitive_dep_count": 28, "subset_aristotle": false, "category": "Compiler"}
{"id": 476, "thm_name": "Term.lower_hEquiv", "thm_stmt": "theorem lower_hEquiv (t : Term τ) : t ≈ lower t", "lean_root": "verified-compiler", "rel_path": "VerifiedCompiler.lean", "imports": [], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "cond", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.cons_append", "module": "Init.Data.List.Basic"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "List.singleton_append", "module": "Init.Data.List.Lemmas"}, {"name": "List.append_nil", "module": "Init.Data.List.Basic"}, {"name": "List.append_assoc", "module": "Init.Data.List.Basic"}, {"name": "List.length_append", "module": "Init.Data.List.Basic"}, {"name": "List.length_singleton", "module": "Init.Data.List.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ty", "content": "inductive Ty\n  | bool\n  | nat"}, {"name": "Value", "content": "inductive Value : Ty → Type\n  | bool : Bool → Value .bool\n  | nat  : Nat  → Value .nat"}, {"name": "Term", "content": "inductive Term : Ty → Type\n  | val (v : Value τ)                          : Term τ\n  | ite (cond : Term .bool) (pos neg : Term τ) : Term τ\n  | and (lhs rhs : Term .bool)                 : Term .bool"}, {"name": "Term.eval", "content": "def eval : Term τ → Value τ\n  | val v => v\n  | ite cond pos neg =>\n    match eval cond with\n    | true  => eval pos\n    | false => eval neg\n  | and lhs rhs =>\n    match eval lhs, eval rhs with\n    | true, true => true\n    | _,    _    => false"}, {"name": "Instruction", "content": "inductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)"}, {"name": "Program", "content": "abbrev Program := List Instruction"}, {"name": "Stack", "content": "abbrev Stack := List Nat"}, {"name": "Nat.conj", "content": "def Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1"}, {"name": "Instruction.Result", "content": "structure Instruction.Result where\n  stack : Stack\n  offset := 0"}, {"name": "Instruction.exec", "content": "def Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none"}, {"name": "Program.goto", "content": "def goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset"}, {"name": "Program.exec", "content": "def exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]"}, {"name": "Value.toNat", "content": "def Value.toNat : Value τ → Nat\n  | bool false => 0\n  | bool true  => 1\n  | nat n      => n"}, {"name": "Term.lower", "content": "def Term.lower : Term τ → Program\n  | val v            => [.const v.toNat]\n  | and lhs rhs      => (lower lhs) ++ (lower rhs) ++ [.and]\n  | ite cond pos neg =>\n    let p := lower pos\n    let n := lower neg\n    lower cond ++ [.jez (p.length + 1)] ++ p ++ [.jmp n.length] ++ n"}, {"name": "HEquiv", "content": "def HEquiv (t : Term τ) (prog : Program) : Prop :=\n  prog.exec [] = some [t.eval.toNat]"}], "used_local_lemmas": [{"name": "Instruction.jmp_def", "content": "@[simp]\ntheorem jmp_def (offset s) : exec (.jmp offset) s = some ⟨s, offset⟩"}, {"name": "Instruction.jez_zero", "content": "@[simp]\ntheorem jez_zero (offset s) : exec (.jez offset) (0 :: s) = some ⟨s, offset⟩"}, {"name": "Instruction.jez_succ", "content": "@[simp]\ntheorem jez_succ (n offset s) : exec (.jez offset) ((n + 1) :: s) = some ⟨s, 0⟩"}, {"name": "Instruction.exec_stack_mono", "content": "@[simp]\ntheorem exec_stack_mono (s : Stack) (h : exec inst s₁ = some ⟨s₂, o⟩) :\n    exec inst (s₁ ++ s) = some ⟨s₂ ++ s, o⟩"}, {"name": "Program.goto_zero", "content": "@[simp]\ntheorem goto_zero (prog : Program) : goto prog 0 = some prog"}, {"name": "Program.goto_suffix", "content": "@[simp]\ntheorem goto_suffix (prog₁ prog₂ : Program) : goto (prog₁ ++ prog₂) prog₁.length = some prog₂"}, {"name": "Program.goto_end", "content": "@[simp]\ntheorem goto_end {prog : Program} (h : offset = prog.length) : goto prog offset = some []"}, {"name": "Program.exec_goto", "content": "theorem exec_goto (hi : i.exec s₁ = some ⟨s₂, offset⟩) (hg : goto prog₁ offset = some prog₂) :\n    exec (i :: prog₁) s₁ = exec prog₂ s₂"}, {"name": "Program.goto_mono", "content": "theorem goto_mono (prog₂ : Program) (h : goto prog₁ offset = some prog₁') :\n    goto (prog₁ ++ prog₂) offset = some (prog₁' ++ prog₂)"}, {"name": "Program.exec_stack_mono", "content": "@[simp]\ntheorem exec_stack_mono (s : Stack) (h : exec prog s₁ = some s₂) :\n    exec prog (s₁ ++ s) = s₂ ++ s"}, {"name": "Program.exec_prog_mono", "content": "@[simp]\ntheorem exec_prog_mono (prog₂ : Program) (h : exec prog₁ s₁ = some s₂) :\n    exec (prog₁ ++ prog₂) s₁ = exec prog₂ s₂"}, {"name": "Term.eq_false_of_toNat_eq_zero", "content": "theorem eq_false_of_toNat_eq_zero {v : Value .bool} (h : v.toNat = 0) : v = false"}, {"name": "Term.eq_true_of_toNat_eq_succ", "content": "theorem eq_true_of_toNat_eq_succ {v : Value .bool} (h : v.toNat = n + 1) : v = true"}, {"name": "Term.lower_val_hEquiv", "content": "theorem lower_val_hEquiv (v : Value τ) : val v ≈ lower (val v)"}, {"name": "Term.lower_and_hEquiv", "content": "theorem lower_and_hEquiv (hl : lhs ≈ lower lhs) (hr : rhs ≈ lower rhs) :\n    and lhs rhs ≈ lower (and lhs rhs)"}, {"name": "Term.lower_ite_hEquiv", "content": "theorem lower_ite_hEquiv {cond : Term .bool} {pos neg : Term τ}\n    (hc : cond ≈ lower cond) (hp : pos ≈ lower pos) (hn : neg ≈ lower neg) :\n    ite cond pos neg ≈ lower (ite cond pos neg)"}], "local_ctx": "inductive Ty\n  | bool\n  | nat\n\ninductive Value : Ty → Type\n  | bool : Bool → Value .bool\n  | nat  : Nat  → Value .nat\n\ninductive Term : Ty → Type\n  | val (v : Value τ)                          : Term τ\n  | ite (cond : Term .bool) (pos neg : Term τ) : Term τ\n  | and (lhs rhs : Term .bool)                 : Term .bool\n\nnamespace Term\n\ndef eval : Term τ → Value τ\n  | val v => v\n  | ite cond pos neg =>\n    match eval cond with\n    | true  => eval pos\n    | false => eval neg\n  | and lhs rhs =>\n    match eval lhs, eval rhs with\n    | true, true => true\n    | _,    _    => false\n\nnamespace Equiv\n\ninfixl:50 \" ~ \" => Equiv\n\nend Equiv\n\nend Term\n\nopen Term (eval)\n\ninductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)\n\nabbrev Program := List Instruction\n\nabbrev Stack := List Nat\n\ndef Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1\n\nstructure Instruction.Result where\n  stack : Stack\n  offset := 0\n\ndef Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none\n\nnamespace Program\n\ndef goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset\n\ndef exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]\n\nend Program\n\ndef Value.toNat : Value τ → Nat\n  | bool false => 0\n  | bool true  => 1\n  | nat n      => n\n\ndef Term.lower : Term τ → Program\n  | val v            => [.const v.toNat]\n  | and lhs rhs      => (lower lhs) ++ (lower rhs) ++ [.and]\n  | ite cond pos neg =>\n    let p := lower pos\n    let n := lower neg\n    lower cond ++ [.jez (p.length + 1)] ++ p ++ [.jmp n.length] ++ n\n\ndef HEquiv (t : Term τ) (prog : Program) : Prop :=\n  prog.exec [] = some [t.eval.toNat]\n\ninfixl:60 \" ≈ \" => HEquiv\n\nnamespace Instruction\n\nend Instruction\n\nnamespace Program\n\nend Program\n\nnamespace Term\n\nopen Program List\n\nopen Instruction hiding exec", "target_theorem": "theorem lower_hEquiv (t : Term τ) : t ≈ lower t :=", "ground_truth_proof": ":= by\n  induction t\n  case val          => exact lower_val_hEquiv _\n  case and hl hr    => exact lower_and_hEquiv hl hr\n  case ite hc hp hn => exact lower_ite_hEquiv hc hp hn", "nesting_depth": 4, "transitive_dep_count": 44, "subset_aristotle": false, "category": "Compiler"}
{"id": 477, "thm_name": "Term.constFold_equiv", "thm_stmt": "theorem constFold_equiv (t : Term τ) : t ~ constFold t", "lean_root": "verified-compiler", "rel_path": "VerifiedCompiler.lean", "imports": [], "used_lib_defs": [{"name": "Bool", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "bool", "module": "Init.Control.Basic"}, {"name": "cond", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ty", "content": "inductive Ty\n  | bool\n  | nat"}, {"name": "Value", "content": "inductive Value : Ty → Type\n  | bool : Bool → Value .bool\n  | nat  : Nat  → Value .nat"}, {"name": "Term", "content": "inductive Term : Ty → Type\n  | val (v : Value τ)                          : Term τ\n  | ite (cond : Term .bool) (pos neg : Term τ) : Term τ\n  | and (lhs rhs : Term .bool)                 : Term .bool"}, {"name": "Term.eval", "content": "def eval : Term τ → Value τ\n  | val v => v\n  | ite cond pos neg =>\n    match eval cond with\n    | true  => eval pos\n    | false => eval neg\n  | and lhs rhs =>\n    match eval lhs, eval rhs with\n    | true, true => true\n    | _,    _    => false"}, {"name": "Term.constFold", "content": "def constFold : Term τ → Term τ\n  | val v                             => val v\n  | ite cond pos neg                  => ite (constFold cond) (constFold pos) (constFold neg)\n  | and true true                     => true\n  | and (Value.bool _) (Value.bool _) => false\n  | and lhs rhs                       => and (constFold lhs) (constFold rhs)"}, {"name": "Term.Equiv", "content": "def Equiv (t₁ t₂ : Term τ) : Prop :=\n  eval t₁ = eval t₂"}], "used_local_lemmas": [], "local_ctx": "inductive Ty\n  | bool\n  | nat\n\ninductive Value : Ty → Type\n  | bool : Bool → Value .bool\n  | nat  : Nat  → Value .nat\n\ninductive Term : Ty → Type\n  | val (v : Value τ)                          : Term τ\n  | ite (cond : Term .bool) (pos neg : Term τ) : Term τ\n  | and (lhs rhs : Term .bool)                 : Term .bool\n\nnamespace Term\n\ndef eval : Term τ → Value τ\n  | val v => v\n  | ite cond pos neg =>\n    match eval cond with\n    | true  => eval pos\n    | false => eval neg\n  | and lhs rhs =>\n    match eval lhs, eval rhs with\n    | true, true => true\n    | _,    _    => false\n\ndef constFold : Term τ → Term τ\n  | val v                             => val v\n  | ite cond pos neg                  => ite (constFold cond) (constFold pos) (constFold neg)\n  | and true true                     => true\n  | and (Value.bool _) (Value.bool _) => false\n  | and lhs rhs                       => and (constFold lhs) (constFold rhs)\n\ndef Equiv (t₁ t₂ : Term τ) : Prop :=\n  eval t₁ = eval t₂\n\nnamespace Equiv\n\ninfixl:50 \" ~ \" => Equiv\n\nend Equiv", "target_theorem": "theorem constFold_equiv (t : Term τ) : t ~ constFold t :=", "ground_truth_proof": ":= by\n  induction t\n  case val v => rfl\n  case ite hc hp hn => exact Equiv.ite_congr hc hp hn\n  case and lhs rhs hl hr =>\n    unfold constFold\n    cases lhs <;> cases rhs\n    case val.val v₁ v₂ => cases v₁; cases v₂; cases ‹Bool› <;> cases ‹Bool› <;> simp\n    all_goals simp only [Equiv.and_congr hl hr]", "nesting_depth": 3, "transitive_dep_count": 12, "subset_aristotle": false, "category": "Compiler"}
{"id": 478, "thm_name": "Program.exec_prog_mono", "thm_stmt": "@[simp]\ntheorem exec_prog_mono (prog₂ : Program) (h : exec prog₁ s₁ = some s₂) :\n    exec (prog₁ ++ prog₂) s₁ = exec prog₂ s₂", "lean_root": "verified-compiler", "rel_path": "VerifiedCompiler.lean", "imports": [], "used_lib_defs": [{"name": "Nat", "module": "Init.Prelude"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "List.cons_append", "module": "Init.Data.List.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Instruction", "content": "inductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)"}, {"name": "Program", "content": "abbrev Program := List Instruction"}, {"name": "Stack", "content": "abbrev Stack := List Nat"}, {"name": "Nat.conj", "content": "def Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1"}, {"name": "Instruction.Result", "content": "structure Instruction.Result where\n  stack : Stack\n  offset := 0"}, {"name": "Instruction.exec", "content": "def Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none"}, {"name": "Program.goto", "content": "def goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset"}, {"name": "Program.exec", "content": "def exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]"}], "used_local_lemmas": [{"name": "Program.goto_mono", "content": "theorem goto_mono (prog₂ : Program) (h : goto prog₁ offset = some prog₁') :\n    goto (prog₁ ++ prog₂) offset = some (prog₁' ++ prog₂)"}], "local_ctx": "namespace Term\n\nnamespace Equiv\n\ninfixl:50 \" ~ \" => Equiv\n\nend Equiv\n\nend Term\n\nopen Term (eval)\n\ninductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)\n\nabbrev Program := List Instruction\n\nabbrev Stack := List Nat\n\ndef Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1\n\nstructure Instruction.Result where\n  stack : Stack\n  offset := 0\n\ndef Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none\n\nnamespace Program\n\ndef goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset\n\ndef exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]\n\nend Program\n\ninfixl:60 \" ≈ \" => HEquiv\n\nnamespace Instruction\n\nend Instruction\n\nnamespace Program", "target_theorem": "@[simp]\ntheorem exec_prog_mono (prog₂ : Program) (h : exec prog₁ s₁ = some s₂) :\n    exec (prog₁ ++ prog₂) s₁ = exec prog₂ s₂ :=", "ground_truth_proof": ":= by\n  induction prog₁, s₁ using exec.induct <;> try (simp_all [exec]; done)\n  next hh hg =>\n    simp only [exec, hh] at h\n    rw [hg] at h\n    contradiction\n  next hh _ hg hi =>\n    simp only [exec, hh, List.cons_append] at *\n    rw [hg] at h\n    rw [Program.goto_mono _ hg, ←(hi h)]", "nesting_depth": 3, "transitive_dep_count": 14, "subset_aristotle": false, "category": "Compiler"}
{"id": 479, "thm_name": "Program.exec_stack_mono", "thm_stmt": "@[simp]\ntheorem exec_stack_mono (s : Stack) (h : exec prog s₁ = some s₂) :\n    exec prog (s₁ ++ s) = s₂ ++ s", "lean_root": "verified-compiler", "rel_path": "VerifiedCompiler.lean", "imports": [], "used_lib_defs": [{"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "List.next", "module": "Mathlib.Data.List.Cycle"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "...", "module": ""}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Instruction", "content": "inductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)"}, {"name": "Program", "content": "abbrev Program := List Instruction"}, {"name": "Stack", "content": "abbrev Stack := List Nat"}, {"name": "Nat.conj", "content": "def Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1"}, {"name": "Instruction.Result", "content": "structure Instruction.Result where\n  stack : Stack\n  offset := 0"}, {"name": "Instruction.exec", "content": "def Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none"}, {"name": "Program.goto", "content": "def goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset"}, {"name": "Program.exec", "content": "def exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]"}], "used_local_lemmas": [{"name": "Instruction.exec_stack_mono", "content": "@[simp]\ntheorem exec_stack_mono (s : Stack) (h : exec inst s₁ = some ⟨s₂, o⟩) :\n    exec inst (s₁ ++ s) = some ⟨s₂ ++ s, o⟩"}], "local_ctx": "namespace Term\n\nnamespace Equiv\n\ninfixl:50 \" ~ \" => Equiv\n\nend Equiv\n\nend Term\n\nopen Term (eval)\n\ninductive Instruction\n  | const (n : Nat)\n  | and\n  | jmp (offset : Nat)\n  | jez (offset : Nat)\n\nabbrev Program := List Instruction\n\nabbrev Stack := List Nat\n\ndef Nat.conj : Nat → Nat → Nat\n  | 0, _ | _, 0 => 0\n  | _, _        => 1\n\nstructure Instruction.Result where\n  stack : Stack\n  offset := 0\n\ndef Instruction.exec : Instruction → Stack → Option Instruction.Result\n  | .const n,    stack        => some { stack := n :: stack }\n  | .and,        r :: l :: tl => some { stack := (l.conj r) :: tl }\n  | .and,        _            => none\n  | .jmp offset, stack        => some { stack, offset }\n  | .jez offset, 0 :: tl      => some { stack := tl, offset }\n  | .jez _,      _ :: tl      => some { stack := tl }\n  | .jez _,      _            => none\n\nnamespace Program\n\ndef goto : Program → Nat → Option Program\n  | prog,    0          => prog\n  | [],      _ + 1      => none\n  | _ :: tl, offset + 1 => goto tl offset\n\ndef exec : Program → Stack → Option Stack\n  | [],       stack => stack\n  | hd :: tl, stack =>\n    match Instruction.exec hd stack with\n    | none => none\n    | some { stack, offset } =>\n      match h : goto tl offset with\n      | none   => none\n      | some p => exec p stack\ntermination_by p => p.length\ndecreasing_by simp +arith [goto_decreasing h]\n\nend Program\n\ninfixl:60 \" ≈ \" => HEquiv\n\nnamespace Instruction\n\nend Instruction\n\nnamespace Program", "target_theorem": "@[simp]\ntheorem exec_stack_mono (s : Stack) (h : exec prog s₁ = some s₂) :\n    exec prog (s₁ ++ s) = s₂ ++ s :=", "ground_truth_proof": ":= by\n  induction prog, s₁ using exec.induct <;> try (simp_all [exec]; done)\n  next hh hg =>\n    simp only [exec, hh] at h\n    rw [hg] at h\n    contradiction\n  next hh _ hg hi =>\n    simp only [exec, hh, Instruction.exec_stack_mono _ hh] at *\n    rw [hg] at h ⊢\n    exact hi h", "nesting_depth": 5, "transitive_dep_count": 13, "subset_aristotle": false, "category": "Compiler"}
{"id": 480, "thm_name": "Intmax.lemma4", "thm_stmt": "lemma lemma4 (h : π₁ ≤ π₂) : Bal π₁ bs ≤ Bal π₂ bs", "lean_root": "FVIntmax", "rel_path": "FVIntmax/Lemma4.lean", "imports": ["import FVIntmax.Wheels.Dictionary", "import FVIntmax.Wheels", "import FVIntmax.Balance"], "used_lib_defs": [{"name": "DecidableEq", "module": "Init.Prelude"}, {"name": "Preorder", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Zero", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Set", "module": "Mathlib.Data.Set.Defs"}, {"name": "IsGLB", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "And", "module": "Init.Prelude"}, {"name": "IsGreatest", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "lowerBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "upperBounds", "module": "Mathlib.Order.Bounds.Defs"}, {"name": "InfSet", "module": "Mathlib.Order.SetNotation"}, {"name": "iInf", "module": "Mathlib.Order.SetNotation"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Vector", "module": "Init.Data.Vector.Basic"}, {"name": "Finite", "module": "Mathlib.Data.Finite.Defs"}, {"name": "Fin", "module": "Init.Prelude"}, {"name": "Finset", "module": "Mathlib.Data.Finset.Defs"}, {"name": "Classical.arbitrary", "module": "Mathlib.Logic.Nonempty"}, {"name": "Monotone", "module": "Mathlib.Order.Monotone.Defs"}, {"name": "Finset.sort", "module": "Mathlib.Data.Finset.Sort"}, {"name": "LE", "module": "Init.Prelude"}, {"name": "List.length", "module": "Init.Prelude"}, {"name": "List.map", "module": "Init.Prelude"}, {"name": "Subtype", "module": "Init.Prelude"}, {"name": "flip", "module": "Init.Core"}, {"name": "LE.le", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$", "content": "local macro:max (priority := high) \"↪\" b:term : term => `(⟨$b, by admit /- proof elided -/\n⟩)"}, {"name": "S", "content": "abbrev S (K₁ K₂ V : Type) [PreWithZero V] := { s : S' K₁ K₂ V // s.isValid }"}, {"name": "S'", "content": "abbrev S' (K₁ K₂ V : Type) := Kbar K₁ K₂ → V"}, {"name": "Kbar", "content": "inductive Kbar (K₁ K₂ : Type) where\n  | key (k : Key K₁ K₂)\n  | Source\nderiving DecidableEq"}, {"name": "Key", "content": "abbrev Key (K₁ K₂ : Type) := K₁ ⊕ K₂"}, {"name": "abbrev", "content": "class abbrev PreWithZero (α : Type) := Preorder α, Zero α"}, {"name": "fStar", "content": "def fStar (Ts : List (Τ K₁ K₂ V)) (s₀ : S K₁ K₂ V) : S K₁ K₂ V :=\n  Ts.foldl f s₀"}, {"name": "f", "content": "def f (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨\n    λ k ↦\n      have : InfSet V := infV b T k\n      ⨅ x : boundedBelow b T, fc x.1 k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "infV", "content": "def infV (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) :\n  InfSet V where\n    sInf := λ s ↦ if s = V' b T k\n                  then (exists_inf b T).1 k\n                  else 0"}, {"name": "exists_inf", "content": "def exists_inf (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : { s : S K₁ K₂ V // ∀ k, IsGLB (V' b T k) (s k) } :=\n  ⟨\n    f' b T,\n    λ k ↦\n      have f'_codomain : (f' b T) k ∈ V' b T k := by admit /- proof elided -/\n  ⟩"}, {"name": "fc", "content": "def fc (τcXb : Τc K₁ K₂ V × S K₁ K₂ V) : S K₁ K₂ V :=\n  ⟨λ k : Kbar K₁ K₂ ↦\n    match τcXb with\n    | ⟨⟨⟨⟨s, r, v⟩, _⟩, hτ⟩, b⟩ =>\n      let v' := v' (v.get hτ) b s\n      b k + (e r - e s) k • v',\n  by admit /- proof elided -/\n  ⟩"}, {"name": "e", "content": "def e (i : Kbar K₁ K₂) : Kbar K₁ K₂ → ℤ := λ j ↦ if i = j then 1 else 0"}, {"name": "Τc", "content": "abbrev Τc (K₁ K₂ V : Type) [PreWithZero V] : Type := { τ : Τ K₁ K₂ V // τ.isComplete }"}, {"name": "Τ", "content": "abbrev Τ (K₁ K₂ V : Type) [PreWithZero V] := { τ : Τ' K₁ K₂ V // τ.isValid }"}, {"name": "Τ'", "content": "abbrev Τ' (K₁ K₂ V : Type) [PreWithZero V] := Kbar K₁ K₂ × Kbar K₁ K₂ × Option V₊"}, {"name": "NonNeg", "content": "def NonNeg (α : Type) [PreWithZero α] := { a : α // 0 ≤ a }"}, {"name": "boundedBelow", "content": "abbrev boundedBelow (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) :=\n  { a : Τc K₁ K₂ V × S K₁ K₂ V | (T, b) ≤ (↑a.1, a.2) }"}, {"name": "f'", "content": "def f' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) : S K₁ K₂ V := \n  ⟨\n    λ k ↦\n      match h : T with\n      | ⟨(_, _, .some _), hT⟩ => fc (⟨T, by admit /- proof elided -/\n      ⟩, b) k\n      | ⟨(s, _, .none), _⟩ => if k = s then 0 else b k,\n    by admit /- proof elided -/\n  ⟩"}, {"name": "Injective", "content": "class Injective {α ω : Type} (f : α → ω) where\n  h : ComputationallyInfeasible (¬ Function.Injective f)"}, {"name": "V'", "content": "def V' (b : S K₁ K₂ V) (T : Τ K₁ K₂ V) (k : Kbar K₁ K₂) : Set V :=\n  { v : V | v ∈ (fc · k) '' boundedBelow b T }"}, {"name": "isComplete", "content": "def isComplete (τ : Τ K₁ K₂ V) :=\n  match τ with | ⟨(_, _, v), _⟩ => v.isSome"}, {"name": "BalanceProof", "content": "abbrev BalanceProof (K₁ K₂ : Type) [Finite K₁] [Finite K₂]\n                    (C Pi V : Type) [PreWithZero V] : Type :=\n  Dict (C × K₂) ((Pi × ExtraDataT) × TransactionBatch K₁ K₂ V) "}, {"name": "TransactionBatch", "content": "abbrev TransactionBatch (K₁ : Type) [Finite K₁]\n                        (K₂ : Type) [Finite K₂]\n                        (V : Type) [PreWithZero V] :=\n  Key K₁ K₂ → V₊"}, {"name": "ExtraDataT", "content": "abbrev ExtraDataT : Type := ℕ"}, {"name": "Dict", "content": "abbrev Dict (α ω : Type) : Type := α → Option ω"}, {"name": "TransactionsInBlocks", "content": "def TransactionsInBlocks\n  (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) : List (Τ K₁ K₂ V) :=\n  (bs.map (TransactionsInBlock π)).flatten"}, {"name": "TransactionsInBlock", "content": "def TransactionsInBlock (π : BalanceProof K₁ K₂ C Pi V) (b : Block K₁ K₂ C Sigma V) : List (Τ K₁ K₂ V) := \n  match h : b with\n  | .deposit ..    => TransactionsInBlock_deposit ↪b\n  | .transfer ..   => TransactionsInBlock_transfer π ↪b\n  | .withdrawal .. => TransactionsInBlock_withdrawal ↪b"}, {"name": "TransactionsInBlock_withdrawal", "content": "def TransactionsInBlock_withdrawal \n  (b : { b : Block K₁ K₂ C Sigma V // b.isWithdrawalBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .withdrawal withdrawals =>\n     \n    let k₁InOrder := { s | s : K₁ }.toFinset.sort (·≤·)\n    k₁InOrder.attach.map λ s : K₁ ↦ ⟨(s, .Source, withdrawals s), by admit /- proof elided -/\n    ⟩\n  | .deposit r v | .transfer .. => by admit /- proof elided -/"}, {"name": "Block", "content": "inductive Block (K₁ K₂ : Type) (C Sigma : Type) (V : Type) [PreWithZero V] where\n   \n  | deposit (recipient : K₂) (amount : V₊)\n   \n  | transfer (aggregator : K₁) (extradata : ExtraDataT) (commitment : C) (senders : List K₂) (sigma : Sigma)\n   \n  | withdrawal (withdrawals : K₁ → V₊)"}, {"name": "isWithdrawalBlock", "content": "abbrev isWithdrawalBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.withdrawal _)"}, {"name": "attach", "content": "noncomputable def attach (α : Type) [Finite α] : UniquelyIndexed α :=\n  have := Finite.exists_equiv_fin α\n  this.choose_spec.some.toEmbedding"}, {"name": "UniquelyIndexed", "content": "abbrev UniquelyIndexed (α : Type) [Finite α] : Type := α ↪ !α"}, {"name": "UniqueTokenT", "content": "abbrev UniqueTokenT (α : Type) [Finite α] : Type := Fin (Finite.exists_equiv_fin α |>.choose)"}, {"name": "TransactionsInBlock_transfer", "content": "def TransactionsInBlock_transfer \n  (π : BalanceProof K₁ K₂ C Pi V) (b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .transfer _ _ commitment S _ =>\n     \n    let senderRecipient : Finset (K₂ × Key K₁ K₂) := { (k₂, k) | (k₂ : K₂) (k : Key K₁ K₂) (_h : k₂ ≠ₖ k) }\n    let sorted : List (K₂ × Key K₁ K₂) := senderRecipient.sort Key.lexLe \n     \n    let v (s : K₂) (r : Key K₁ K₂) : Option V₊ :=\n      if s ∉ S\n      then .some 0\n      else \n        if h : (commitment, s) ∈ π.keys\n        then let (_, t) := π[(commitment, s)]\n             t r\n        else .none\n    sorted.attach.map λ ⟨(s, r), h⟩ ↦ ⟨(s, r, v s r), by admit /- proof elided -/\n    ⟩\n  | .deposit .. | .withdrawal .. => by admit /- proof elided -/"}, {"name": "lexLe", "content": "def lexLe (a b : K₂ × Key K₁ K₂) : Prop :=\n  a.1 < b.1 ∨ (a.1 = b.1 ∧ a.2 ≤ b.2)"}, {"name": "isTransferBlock", "content": "abbrev isTransferBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.transfer _ _ _ _ _)"}, {"name": "keys", "content": "def keys (m : Dict α ω) : Set α := { x | Dict.is_mem m x }"}, {"name": "Dict.is_mem", "content": "def Dict.is_mem (m : Dict α ω) (x : α) : Prop := (m x).isSome"}, {"name": "keys", "content": "abbrev keys (ct : CommitT C K Pi) := ct.dict.keys"}, {"name": "CommitT", "content": "structure CommitT (C K Pi : Type) where\n  commitment : C\n  dict       : Dict K Pi"}, {"name": "keysUneq", "content": "abbrev keysUneq (k₂ : K₂) (k : Key K₁ K₂) : Prop :=\n  match k with\n  | .inl _   => True\n  | .inr k₂' => k₂ ≠ k₂'"}, {"name": "TransactionsInBlock_deposit", "content": "def TransactionsInBlock_deposit\n  (b : { b : Block K₁ K₂ C Sigma V // b.isDepositBlock }) : List (Τ K₁ K₂ V) :=\n  match h : b.1 with\n  | .deposit r v => [⟨(.Source, r, v), by admit /- proof elided -/\n  ⟩]\n  | .withdrawal .. | .transfer .. => by admit /- proof elided -/"}, {"name": "isDepositBlock", "content": "abbrev isDepositBlock (b : Block K₁ K₂ C Sigma V) := b matches (Block.deposit _ _) "}, {"name": "initial", "content": "def initial (K₁ K₂ V : Type) [PreWithZero V] : S K₁ K₂ V :=\n  ⟨S'.initial K₁ K₂ V, S'.isValid_initial⟩"}, {"name": "Bal", "content": "def Bal (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) : S K₁ K₂ V :=\n  fStar (TransactionsInBlocks π bs) (.initial K₁ K₂ V)"}, {"name": "initial", "content": "def initial : Scontract K₁ K₂ V C Sigma := []"}, {"name": "Scontract", "content": "abbrev Scontract (K₁ K₂ V : Type) [PreWithZero V] (C Sigma : Type) :=\n  List (Block K₁ K₂ C Sigma V)"}, {"name": "le", "content": "def le (v₁ v₂ : Vector α n) :=\n  ∀ x ∈ (v₁.1.zip v₂.1), x.1 ≤ x.2"}, {"name": "v'", "content": "def v' (v : V₊) (b : S K₁ K₂ V) (s : Kbar K₁ K₂) : V₊ :=\n  match h : s with\n  | .Source => v\n  | .key _  => ⟨v ⊓ b s, by admit /- proof elided -/\n  ⟩"}, {"name": "(priority", "content": "instance (priority := high) vectorPreorder : Preorder (Vector α n) where\n  le_refl := λ _ ↦ Vec.le_refl\n  le_trans := λ _ _ _ ↦ Vec.le_trans"}, {"name": "infix:50 \" ≠ₖ \" => Key.keysUneq ", "content": "infix:50 \" ≠ₖ \" => Key.keysUneq "}, {"name": "prefix:max \"!\" => UniqueTokenT", "content": "prefix:max \"!\" => UniqueTokenT"}, {"name": "postfix:max \"₊\" => NonNeg", "content": "postfix:max \"₊\" => NonNeg"}], "lib_lemmas": [{"name": "not_and_or", "module": "Mathlib.Logic.Basic"}, {"name": "congr_fun", "module": "Batteries.Logic"}, {"name": "List.ext_get_iff", "module": "Mathlib.Data.List.Basic"}, {"name": "le_isGLB_iff", "module": "Mathlib.Order.Bounds.Basic"}, {"name": "mem_lowerBounds", "module": "Mathlib.Order.Bounds.Basic"}, {"name": "le_refl", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Monotone.comp", "module": "Mathlib.Order.Monotone.Defs"}], "repo_lemmas": [{"name": "le_of_ext_le", "content": "lemma le_of_ext_le {α : Type} [Preorder α] {v₁ v₂ : Vector α n}\n  (h : ∀ i : Fin n, v₁.1[i] ≤ v₂.1[i]) : v₁ ≤ v₂"}, {"name": "mem_dict_iff_mem_keys", "content": "lemma mem_dict_iff_mem_keys {dict : Dict α ω} : k ∈ dict ↔ k ∈ dict.keys"}, {"name": "eq_of_BalanceProof_le", "content": "lemma eq_of_BalanceProof_le (h : π ≤ π') (h₁ : k ∈ π) (h₂ : k ∈ π') :\n  ((π k).get h₁).2 = ((π' k).get h₂).2"}, {"name": "mem_of_BalanceProof_le", "content": "lemma mem_of_BalanceProof_le (h : π ≤ π') (h₁ : k ∈ π) : k ∈ π'"}, {"name": "notin_of_BalanceProof_le_notin", "content": "lemma notin_of_BalanceProof_le_notin (h : π ≤ π') (h₁ : k ∉ π') : k ∉ π"}, {"name": "length_TransactionsInBlock_transfer", "content": "lemma length_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }} :\n  ∀ (π₁ π₂ : BalanceProof K₁ K₂ C Pi V),\n    (TransactionsInBlock_transfer π₁ b).length =\n    (TransactionsInBlock_transfer π₂ b).length"}, {"name": "length_transactionsInBlock", "content": "lemma length_transactionsInBlock :\n  (TransactionsInBlock π₁ b).length = (TransactionsInBlock π₂ b).length"}, {"name": "map_join_unnecessarily_specific", "content": "lemma map_join_unnecessarily_specific\n  {α β γ δ Pi : Type}\n  [LE δ]\n  [LE Pi]\n  {l : List α}\n  {P : (β × γ × δ) → Prop}\n  {π π' : Pi}\n  {f : Pi → α → List (Subtype P)}\n  {i : ℕ}\n  (h₀ : (List.length ∘ f π) = (List.length ∘ f π'))\n  (h₁ : ∀ (a : α)\n          (i : ℕ) (h : i < (f π a).length),\n          (f π a)[i].1.2.2 ≤ ((f π' a)[i]'(by apply congr_fun at h₀; aesop)).1.2.2)\n  (h) :\n  ((List.map (f π) l).flatten[i]'h).1.2.2 ≤\n  ((List.map (f π') l).flatten[i]'(by aesop)).1.2.2"}, {"name": "map_eq_project_triple", "content": "lemma map_eq_project_triple {β γ δ : Type}\n                            {s : β} {r : γ} {v : δ}\n                            {i : ℕ}\n                            {P : (β × γ × δ) → Prop}\n                            {l : List (Subtype P)}\n                            {h₀}\n                            {h : i < l.length} : \n  l[i]'h = ⟨(s, r, v), h₀⟩ → (l[i]'h).1.2.2 = v"}, {"name": "receiver_transactionsInBlocks", "content": "lemma receiver_transactionsInBlocks {bs : List (Block K₁ K₂ C Sigma V)}\n                                    {π₁ π₂ : BalanceProof K₁ K₂ C Pi V} :\n  (TransactionsInBlocks π₁ bs).map (λ s ↦ s.1.2.1) =\n  (TransactionsInBlocks π₂ bs).map (λ s ↦ s.1.2.1)"}, {"name": "receiver_transactionsInBlock", "content": "lemma receiver_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.2.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.2.1)"}, {"name": "length_transactionsInBlocks", "content": "lemma length_transactionsInBlocks {bs : List (Block K₁ K₂ C Sigma V)}\n                                  {π₁ π₂ : BalanceProof K₁ K₂ C Pi V} :\n  (TransactionsInBlocks π₁ bs).length = (TransactionsInBlocks π₂ bs).length"}, {"name": "sender_transactionsInBlocks", "content": "lemma sender_transactionsInBlocks {bs : List (Block K₁ K₂ C Sigma V)}\n                                  {π₁ π₂ : BalanceProof K₁ K₂ C Pi V} :\n  (TransactionsInBlocks π₁ bs).map (λ s ↦ s.1.1) =\n  (TransactionsInBlocks π₂ bs).map (λ s ↦ s.1.1)"}, {"name": "sender_transactionsInBlock", "content": "lemma sender_transactionsInBlock :\n  (TransactionsInBlock π₁ b).map (λ s ↦ s.1.1) =\n  (TransactionsInBlock π₂ b).map (λ s ↦ s.1.1)"}, {"name": "V'_sset_V'_of_le", "content": "lemma V'_sset_V'_of_le {b₁ b₂ : S K₁ K₂ V} {T₁ T₂ : Τ K₁ K₂ V} {k : Kbar K₁ K₂}\n  (h : b₁ ≤ b₂) (h₁ : T₁ ≤ T₂) : \n  V' b₂ T₂ k ⊆ V' b₁ T₁ k"}, {"name": "boundedBelow_sset_boundedBelow_of_le", "content": "lemma boundedBelow_sset_boundedBelow_of_le {b₁ b₂ : S K₁ K₂ V} {T₁ T₂ : Τ K₁ K₂ V}\n  (h : b₁ ≤ b₂) (h₁ : T₁ ≤ T₂) : boundedBelow b₂ T₂ ⊆ boundedBelow b₁ T₁"}, {"name": "f_IsGLB_of_V'", "content": "lemma f_IsGLB_of_V' {b : S K₁ K₂ V} {T : Τ K₁ K₂ V} {k : Kbar K₁ K₂} :\n  IsGLB (V' b T k) (f b T k)"}, {"name": "le_cons", "content": "lemma le_cons\n  (eq₁ : v₁ = ⟨hd₁ :: tl₁, len₁⟩) (eq₂ : v₂ = ⟨hd₂ :: tl₂, len₂⟩)\n  (h : v₁ ≤ v₂) : hd₁ ≤ hd₂ ∧\n  @LE.le (Vector α n) vectorPreorder.toLE ⟨tl₁, by simp at len₁; assumption⟩ ⟨tl₂, by simp at len₂; assumption⟩"}, {"name": "le_cons_aux", "content": "private lemma le_cons_aux \n  (eq₁ : v₁ = ⟨hd₁ :: tl₁, len₁⟩) (eq₂ : v₂ = ⟨hd₂ :: tl₂, len₂⟩)\n  (h : v₁ ≤ v₂) : hd₁ ≤ hd₂ ∧\n  le ⟨tl₁, by simp at len₁; assumption⟩ ⟨tl₂, by simp at len₂; assumption⟩"}], "used_local_defs": [{"name": "Intmax.length_of_TransactionsInBlocks", "content": "private abbrev length_of_TransactionsInBlocks (bs : List (Block K₁ K₂ C Sigma V)) :\n  { n : ℕ // n = (TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length } :=\n  ⟨(TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length, rfl⟩"}, {"name": "Intmax.TransactionsInBlocksFixed", "content": "private def TransactionsInBlocksFixed (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) :\n  Vector (Τ K₁ K₂ V) (length_of_TransactionsInBlocks (Pi := Pi) bs).1 :=\n  ⟨TransactionsInBlocks π bs, by admit /- proof elided -/\n  ⟩"}, {"name": "Intmax.Bal'", "content": "private def Bal' (bs : List (Block K₁ K₂ C Sigma V)) : BalanceProof K₁ K₂ C Pi V → S K₁ K₂ V :=\n  fStar (s₀ := S.initial K₁ K₂ V) ∘ TransactionsInBlocks (bs := bs)"}, {"name": "Intmax.fStarFixed", "content": "private def fStarFixed {n : ℕ}\n  (Ts : Vector (Τ K₁ K₂ V) n) (s₀ : S K₁ K₂ V) : S K₁ K₂ V :=\n  fStar Ts.1 s₀"}, {"name": "Intmax.BalFixed", "content": "private def BalFixed (bs : List (Block K₁ K₂ C Sigma V)) : BalanceProof K₁ K₂ C Pi V → S K₁ K₂ V :=\n  λ π ↦ fStarFixed (s₀ := S.initial K₁ K₂ V) (TransactionsInBlocksFixed π bs)"}, {"name": "Intmax.BalFixed'", "content": "private def BalFixed' (bs : List (Block K₁ K₂ C Sigma V)) : BalanceProof K₁ K₂ C Pi V → S K₁ K₂ V :=\n  fStarFixed (n := (length_of_TransactionsInBlocks (Pi := Pi) bs).1)\n             (s₀ := S.initial K₁ K₂ V) ∘ TransactionsInBlocksFixed (Pi := Pi) (bs := bs)"}], "used_local_lemmas": [{"name": "Intmax.Bal'_eq_Bal", "content": "private lemma Bal'_eq_Bal : Bal' bs π = Bal π bs"}, {"name": "Intmax.BalFixed_eq_BalFixed'", "content": "private lemma BalFixed_eq_BalFixed' : BalFixed bs π = BalFixed' bs π"}, {"name": "Intmax.TransactionsInBlocksFixed_le_of_TransactionsInBlocks", "content": "lemma TransactionsInBlocksFixed_le_of_TransactionsInBlocks\n  (h : ∀ i : Fin (length_of_TransactionsInBlocks bs).1,\n    (TransactionsInBlocks π bs)[i]'(by blast with π i) ≤\n    (TransactionsInBlocks π' bs)[i]'(by blast with π' i)) :\n  TransactionsInBlocksFixed π bs ≤ TransactionsInBlocksFixed π' bs"}, {"name": "Intmax.senderReceiver_transactionsInBlocks", "content": "lemma senderReceiver_transactionsInBlocks {s r v} {s' r' v'} {eq₁ eq₂} {i}\n  (h₀ : i < (TransactionsInBlocks π bs).length)\n  (h₁ : (TransactionsInBlocks π bs)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' bs)[i]'(by blast with π) = ⟨(s', r', v'), eq₂⟩) :\n  s = s' ∧ r = r'"}, {"name": "Intmax.delta_TransactionsInBlock_transfer", "content": "private lemma delta_TransactionsInBlock_transfer\n  {b : { b : Block K₁ K₂ C Sigma V // b.isTransferBlock }}\n  (h : π ≤ π') : \n  ∀ i : ℕ, (hlen : i < (TransactionsInBlock_transfer π b).length) →\n    (TransactionsInBlock_transfer π b)[i]'hlen =\n    (TransactionsInBlock_transfer π' b)[i]'(by rwa [length_TransactionsInBlock_transfer _ π]) ∨\n    ((TransactionsInBlock_transfer π b)[i]'hlen).1.2.2.isNone"}, {"name": "Intmax.v_transactionsInBlocks", "content": "lemma v_transactionsInBlocks {s r v v'} {eq₁ eq₂} {i}\n  (h : π ≤ π')\n  (h₀ : i < (TransactionsInBlocks π Bstar).length)\n  (h₁ : (TransactionsInBlocks π Bstar)[i] = ⟨(s, r, v), eq₁⟩)\n  (h₂ : (TransactionsInBlocks π' Bstar)[i]'(by blast with π) = ⟨(s, r, v'), eq₂⟩) :\n  v ≤ v'"}, {"name": "Intmax.monotone_TransactionsInBlocksFixed", "content": "lemma monotone_TransactionsInBlocksFixed :\n  Monotone λ (π : BalanceProof K₁ K₂ C Pi V) ↦ TransactionsInBlocksFixed π Bstar"}, {"name": "Intmax.monotone_f", "content": "lemma monotone_f (h₁ : b₁ ≤ b₂) (h₂ : T₁ ≤ T₂) : f b₁ T₁ k ≤ f b₂ T₂ k"}, {"name": "Intmax.monotone_fStarFixed_aux", "content": "private theorem monotone_fStarFixed_aux (h : v₁ ≤ v₂) (h₂ : b₁ ≤ b₂) :\n                                        v₁.1.foldl f b₁ ≤ v₂.1.foldl f b₂"}, {"name": "Intmax.monotone_fStarFixed", "content": "lemma monotone_fStarFixed :\n  Monotone λ (Ts : Vector (Τ K₁ K₂ V) n) ↦ Intmax.fStarFixed Ts (S.initial K₁ K₂ V)"}], "local_ctx": "import FVIntmax.Balance\n\nnamespace Intmax\n\nopen Mathlib\n\nnoncomputable section Lemma4\n\nsection HicSuntDracones\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type} [AddCommGroup V] [Lattice V]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nsection HelperFunctionsToAppeaseLean\n\nopen Mathlib\n\nprivate abbrev length_of_TransactionsInBlocks (bs : List (Block K₁ K₂ C Sigma V)) :\n  { n : ℕ // n = (TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length } :=\n  ⟨(TransactionsInBlocks (Classical.arbitrary _ : BalanceProof K₁ K₂ C Pi V) bs).length, rfl⟩\n\nopen Lean.Elab.Tactic in\n\nprivate def TransactionsInBlocksFixed (π : BalanceProof K₁ K₂ C Pi V) (bs : List (Block K₁ K₂ C Sigma V)) :\n  Vector (Τ K₁ K₂ V) (length_of_TransactionsInBlocks (Pi := Pi) bs).1 :=\n  ⟨TransactionsInBlocks π bs, by admit /- proof elided -/\n  ⟩\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs : List (Block K₁ K₂ C Sigma V)}\n\nprivate def Bal' (bs : List (Block K₁ K₂ C Sigma V)) : BalanceProof K₁ K₂ C Pi V → S K₁ K₂ V :=\n  fStar (s₀ := S.initial K₁ K₂ V) ∘ TransactionsInBlocks (bs := bs)\n\nprivate def fStarFixed {n : ℕ}\n  (Ts : Vector (Τ K₁ K₂ V) n) (s₀ : S K₁ K₂ V) : S K₁ K₂ V :=\n  fStar Ts.1 s₀\n\nprivate def BalFixed (bs : List (Block K₁ K₂ C Sigma V)) : BalanceProof K₁ K₂ C Pi V → S K₁ K₂ V :=\n  λ π ↦ fStarFixed (s₀ := S.initial K₁ K₂ V) (TransactionsInBlocksFixed π bs)\n\nprivate def BalFixed' (bs : List (Block K₁ K₂ C Sigma V)) : BalanceProof K₁ K₂ C Pi V → S K₁ K₂ V :=\n  fStarFixed (n := (length_of_TransactionsInBlocks (Pi := Pi) bs).1)\n             (s₀ := S.initial K₁ K₂ V) ∘ TransactionsInBlocksFixed (Pi := Pi) (bs := bs)\n\nend\n\nend HelperFunctionsToAppeaseLean\n\nend\n\nsection\n\nvariable {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs Bstar : List (Block K₁ K₂ C Sigma V)}\n\nend\n\nend HicSuntDracones\n\nsection\n\nvariable {n : ℕ}\n         {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁]\n         {K₂ : Type} [Finite K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs Bstar : List (Block K₁ K₂ C Sigma V)}\n\nsection Monotone\n\nvariable {b₁ b₂ : S K₁ K₂ V}\n         {T₁ T₂ : Τ K₁ K₂ V}\n         {k : Kbar K₁ K₂}\n         {v₁ v₂ : Vector (Τ K₁ K₂ V) n}\n\nend Monotone\n\nend\n\nvariable {n : ℕ}\n         {Pi C Sigma : Type}\n         {K₁ : Type} [Finite K₁] [LinearOrder K₁]\n         {K₂ : Type} [Finite K₂] [LinearOrder K₂]\n         \n         {V : Type}\n         [Lattice V] [AddCommGroup V]\n         [CovariantClass V V (· + ·) (· ≤ ·)]\n         [CovariantClass V V (Function.swap (· + ·)) (· ≤ ·)]\n         {π π' : BalanceProof K₁ K₂ C Pi V} {bs Bstar : List (Block K₁ K₂ C Sigma V)}", "target_theorem": "lemma lemma4 (h : π₁ ≤ π₂) : Bal π₁ bs ≤ Bal π₂ bs :=", "ground_truth_proof": ":= by\n  simp only [←Bal'_eq_Bal]\n  suffices BalFixed bs π₁ ≤ BalFixed bs π₂ by aesop\n  rw [BalFixed_eq_BalFixed', BalFixed_eq_BalFixed']\n  exact Monotone.comp monotone_fStarFixed monotone_TransactionsInBlocksFixed h", "nesting_depth": 11, "transitive_dep_count": 117, "subset_aristotle": false, "category": "Applied verif."}
{"id": 481, "thm_name": "add_aux1", "thm_stmt": "theorem add_aux1 (x : UInt128) : 4 * (x.cast : F).valMinAbs.natAbs < PRIME", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/WadraySigned.lean", "imports": ["import WadrayVerification.Aux", "import WadrayVerification.Wadray"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Bool.toSierraBool", "module": "Aegis.Aux.Bool"}, {"name": "Bool.xor", "module": "Init.Data.Bool"}, {"name": "SierraBool.toBool", "module": "Aegis.Aux.Bool"}, {"name": "Sierra.F", "module": "Aegis.Types"}, {"name": "Sierra.PRIME", "module": "Aegis.Types"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Option", "module": "Init.Prelude"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Sierra.aegis_prove", "module": "Aegis.Commands"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Inhabited", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "RAY_SCALE", "content": "def RAY_SCALE : ℕ := 1000000000000000000000000000"}, {"name": "div", "content": "protected def div : SignedRay :=\n⟨Ray.div (w₁.1 : Ray) (w₂.1 : Ray), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "SignedRay", "content": "def SignedRay := UInt128 × (Unit ⊕ Unit)"}, {"name": "div", "content": "protected def div : Ray := (r.toZMod.val * RAY_SCALE / r'.toZMod.val : UInt128)"}, {"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}, {"name": "WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "toRat", "content": "def toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1"}, {"name": "toRat", "content": "protected def toRat : ℚ := r.toZMod.val / RAY_SCALE"}, {"name": "div", "content": "protected def div : Wad := (w.toZMod.val * WAD_SCALE / w'.toZMod.val : UInt128)"}, {"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "mul", "content": "protected def mul : Wad := (w.toZMod.val * w'.toZMod.val / WAD_SCALE : UInt128)"}, {"name": "toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "toRat", "content": "def toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1"}, {"name": "mul", "content": "protected def mul : Ray := (r.toZMod.val * r'.toZMod.val / RAY_SCALE  : UInt128)"}, {"name": "add", "content": "protected def add : Ray := r.toZMod + r'.toZMod"}, {"name": "toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "DIFF", "content": "def DIFF : ℕ := 1000000000"}, {"name": "div", "content": "protected def div : SignedWad :=\n⟨Wad.div (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "SignedWad", "content": "def SignedWad := UInt128 × (Unit ⊕ Unit)"}, {"name": "mul", "content": "protected def mul : SignedWad :=\n⟨Wad.mul (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "sub", "content": "protected def sub : Ray := r.toZMod - r'.toZMod"}, {"name": "mul", "content": "protected def mul : SignedRay :=\n⟨Ray.mul w₁.1 w₂.1, Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}], "lib_lemmas": [{"name": "le_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Int.cast_abs", "module": "Mathlib.Algebra.Order.Ring.Cast"}, {"name": "Int.cast_add", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Int.cast_natAbs", "module": "Mathlib.Algebra.Order.Ring.Cast"}, {"name": "Int.cast_neg", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Int.natAbs_neg", "module": "Init.Data.Int.Order"}, {"name": "Nat.abs_cast", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.cast_add", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.cast_le", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.mul_lt_mul_of_pos_left", "module": "Init.Data.Nat.Basic"}, {"name": "SierraBool_toBool_inl", "module": "Aegis.Aux.Bool"}, {"name": "SierraBool_toBool_inr", "module": "Aegis.Aux.Bool"}, {"name": "Sum.getLeft?_inl", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.getLeft?_inr", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.isLeft_inl", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.isLeft_inr", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.isRight_inl", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.isRight_inr", "module": "Init.Data.Sum.Basic"}, {"name": "ZMod.intCast_cast", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.natCast_val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.valMinAbs_neg_of_ne_half", "module": "Mathlib.Data.ZMod.ValMinAbs"}, {"name": "ZMod.val_lt", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "abs_div", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "abs_neg", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "add_div", "module": "Mathlib.Algebra.Field.Basic"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "div_lt_div_of_pos_right", "module": "Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_prop_iff_or", "module": "Mathlib.Logic.Basic"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "lt_of_lt_of_le", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "lt_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "ne_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "neg_add", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "neg_div", "module": "Mathlib.Algebra.Ring.Basic"}, {"name": "neg_neg", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "not_lt", "module": "Mathlib.Order.Defs.LinearOrder"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "sub_eq_add_neg", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_neg_eq_add", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "two_pos", "module": "Mathlib.Algebra.Order.Monoid.NatCast"}], "repo_lemmas": [{"name": "mul_def", "content": "protected theorem mul_def :\n    r * r' = (r.toZMod.val * r'.toZMod.val / RAY_SCALE  : UInt128)"}, {"name": "mul_def", "content": "theorem mul_def :\n    w₁ * w₂ = ⟨Ray.mul w₁.1 w₂.1, Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "ZMod.valMinAbs_add_of_four_lt", "content": "theorem ZMod.valMinAbs_add_of_four_lt [NeZero m] {a b : ZMod m}\n    (ha : 4 * a.valMinAbs.natAbs < m) (hb : 4 * b.valMinAbs.natAbs < m) :\n    (a + b).valMinAbs = a.valMinAbs + b.valMinAbs"}, {"name": "ZMod.valMinAbs_add_of_two_lt", "content": "theorem ZMod.valMinAbs_add_of_two_lt [NeZero m] {a b : ZMod m}\n    (h : 2 * (a.valMinAbs.natAbs + b.valMinAbs.natAbs) < m) :\n    (a + b).valMinAbs = a.valMinAbs + b.valMinAbs"}, {"name": "ZMod.valMinAbs_cast_of_lt_half", "content": "theorem ZMod.valMinAbs_cast_of_lt_half [NeZero m] (hm : 2 * m < n) (a : ZMod m) :\n  (a.cast : ZMod n).valMinAbs = a.val"}, {"name": "mul_def", "content": "protected theorem mul_def :\n    w * w' = (w.toZMod.val * w'.toZMod.val / WAD_SCALE  : UInt128)"}, {"name": "div_def", "content": "protected theorem div_def :\n    r / r' = (r.toZMod.val * RAY_SCALE / r'.toZMod.val : UInt128)"}, {"name": "mul_def", "content": "theorem mul_def :\n    w₁ * w₂ = ⟨Wad.mul (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "ZMod.val_cast_lt", "content": "theorem ZMod.val_cast_lt [NeZero m] (n : ℕ) [NeZero n] (a : ZMod m) : (a.cast : ZMod n).val < m"}, {"name": "div_def", "content": "protected theorem div_def :\n    w / w' = (w.toZMod.val * WAD_SCALE / w'.toZMod.val : UInt128)"}, {"name": "div_def", "content": "theorem div_def :\n    w₁ / w₂ = ⟨Ray.div (w₁.1 : Ray) (w₂.1 : Ray), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "Option.get!_none", "content": "@[simp]\ntheorem Option.get!_none [Inhabited α] : (.none : Option α).get! = default"}, {"name": "Option.get!_some", "content": "@[simp]\ntheorem Option.get!_some [Inhabited α] (a : α) : (Option.some a).get! = a"}, {"name": "div_def", "content": "theorem div_def :\n    w₁ / w₂ = ⟨Wad.div (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}], "used_local_defs": [], "used_local_lemmas": [{"name": "two_U128_MOD_lt_PRIME", "content": "theorem two_U128_MOD_lt_PRIME : 2 * U128_MOD < PRIME"}, {"name": "four_U128_MOD_lt_PRIME", "content": "theorem four_U128_MOD_lt_PRIME : 4 * U128_MOD < PRIME"}, {"name": "four_U128_MOD_le_PRIME", "content": "theorem four_U128_MOD_le_PRIME : 4 * U128_MOD ≤ PRIME"}], "local_ctx": "import WadrayVerification.Aux\n\nimport WadrayVerification.Wadray\n\nopen Sierra\n\naegis_spec \"wadray::wadray_signed::SignedWadZeroable::zero\" :=\n  fun _ (ρ : SignedWad) =>\n  ρ = 0\n\naegis_prove \"wadray::wadray_signed::SignedWadZeroable::zero\" :=\n  fun _ (ρ : SignedWad) => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::SignedRayZeroable::zero\" :=\n  fun _ (ρ : SignedRay) =>\n  ρ = 0\n\naegis_prove \"wadray::wadray_signed::SignedRayZeroable::zero\" :=\n  fun _ (ρ : SignedRay) => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::SignedWadOneable::one\" :=\n  fun _ (ρ : SignedWad) =>\n  ρ = 1\n\naegis_prove \"wadray::wadray_signed::SignedWadOneable::one\" :=\n  fun _ (ρ : SignedWad) => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::SignedRayOneable::one\" :=\n  fun _ (ρ : SignedRay) =>\n  ρ = 1\n\naegis_prove \"wadray::wadray_signed::SignedRayOneable::one\" :=\n  fun _ (ρ : SignedRay) => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::SignedWadPartialEq::eq\" :=\n  fun _ (a b : SignedWad) ρ =>\n  ρ = Bool.toSierraBool (a.toRat = b.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedWadPartialEq::eq\" :=\n  fun _ (a b : SignedWad) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadPartialEq::eq»\n  rename_i x\n  intro h_auto\n  aesop (add safe forward [SignedWad.val_eq_of_toRat_eq, SignedWad.val_eq_zero_of_toRat_neg,\n    SignedWad.val_eq_zero_of_toRat_neg'])\n\naegis_spec \"wadray::wadray_signed::SignedRayPartialEq::eq\" :=\n  fun _ (a b : SignedRay) ρ =>\n  ρ = Bool.toSierraBool (a.toRat = b.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedRayPartialEq::eq\" :=\n  fun _ (a b : SignedRay) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayPartialEq::eq»\n  rename_i x\n  intro h_auto\n  aesop (add safe forward [SignedRay.val_eq_of_toRat_eq, SignedRay.val_eq_zero_of_toRat_neg,\n    SignedRay.val_eq_zero_of_toRat_neg'])\n\naegis_spec \"wadray::wadray_signed::SignedWadPartialEq::ne\" :=\n  fun _ (a b : SignedWad) ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≠ b.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedWadPartialEq::ne\" :=\n  fun _ (a b : SignedWad) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadPartialEq::ne»\n  rintro rfl\n  simp\n\naegis_spec \"wadray::wadray_signed::SignedRayPartialEq::ne\" :=\n  fun _ (a b : SignedRay) ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≠ b.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedRayPartialEq::ne\" :=\n  fun _ (a b : SignedRay) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayPartialEq::ne»\n  rintro rfl\n  simp\n\naegis_spec \"wadray::wadray_signed::SignedWadSigned::is_positive\" :=\n  fun _ _ (a : SignedWad) _ ρ =>\n  ρ = Bool.toSierraBool (a.toRat > 0)\n\naegis_prove \"wadray::wadray_signed::SignedWadSigned::is_positive\" :=\n  fun _ _ (a : SignedWad) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadSigned::is_positive»\n  rintro ⟨w,s,rfl,(⟨h,rfl⟩|⟨h,rfl⟩)⟩\n  · aesop (add simp [SignedWad.toRat, Wad.toRat])\n  · rcases s with (s|s)\n    · simp_all only [Int.ofNat_eq_coe, CharP.cast_eq_zero, Int.cast_zero, ZMod.val_zero,\n        SierraBool_toBool_inl, Bool.not_false, Bool.toSierraBool_true, SignedWad.toRat, Wad.toRat,\n        Wad.toZMod, ZMod.natCast_val, ite_false, gt_iff_lt, Bool.toSierraBool_decide_inr']\n      rw [lt_div_iff (by norm_num [Wad.WAD_SCALE]), zero_mul]\n      apply Ne.lt_of_le _ (ZMod.cast_rat_nonneg _)\n      intro he\n      rw [eq_comm, ZMod.cast_rat_eq_zero_iff] at he; subst he\n      simp at h\n    · simp [SignedWad.toRat, Wad.toRat]\n      rw [le_div_iff (by norm_num [Wad.WAD_SCALE]), zero_mul]\n      apply ZMod.cast_rat_nonneg\n\naegis_spec \"wadray::wadray_signed::SignedRaySigned::is_positive\" :=\n  fun _ _ (a : SignedRay) _ ρ =>\n  ρ = Bool.toSierraBool (a.toRat > 0)\n\naegis_prove \"wadray::wadray_signed::SignedRaySigned::is_positive\" :=\n  fun _ _ (a : SignedWad) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRaySigned::is_positive»\n  rintro ⟨w,s,rfl,(⟨h,rfl⟩|⟨h,rfl⟩)⟩\n  · aesop (add simp [SignedRay.toRat, Ray.toRat])\n  · rcases s with (s|s)\n    · simp_all only [Int.ofNat_eq_coe, CharP.cast_eq_zero, Int.cast_zero, ZMod.val_zero,\n        SierraBool_toBool_inl, Bool.not_false, Bool.toSierraBool_true, SignedRay.toRat, Ray.toRat,\n        Ray.toZMod, ZMod.natCast_val, ite_false, gt_iff_lt, Bool.toSierraBool_decide_inr']\n      rw [lt_div_iff (by norm_num [Ray.RAY_SCALE]), zero_mul]\n      apply Ne.lt_of_le _ (ZMod.cast_rat_nonneg _)\n      intro he\n      rw [eq_comm, ZMod.cast_rat_eq_zero_iff] at he; subst he\n      simp at h\n    · simp [SignedRay.toRat, Ray.toRat]\n      rw [le_div_iff (by norm_num [Ray.RAY_SCALE]), zero_mul]\n      apply ZMod.cast_rat_nonneg\n\naegis_spec \"wadray::wadray_signed::SignedWadSigned::is_negative\" :=\n  fun _ _ (a : SignedWad) _ ρ =>\n  ρ = Bool.toSierraBool (a.toRat < 0)\n\naegis_prove \"wadray::wadray_signed::SignedWadSigned::is_negative\" :=\n  fun _ _ (a : SignedWad) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadSigned::is_negative»\n  rintro ⟨w,s,rfl,(⟨h,rfl⟩|⟨h,rfl⟩)⟩\n  · aesop (add simp [SignedWad.toRat, Wad.toRat])\n  · have : 0 < (Wad.WAD_SCALE : ℚ) := by norm_num [Wad.WAD_SCALE]\n    rcases s with (s|s)\n    <;> aesop (add simp [SignedWad.toRat, Wad.toRat, lt_div_iff, le_div_iff,\n      Wad.toZMod, ZMod.cast_rat_nonneg], safe apply [Ne.lt_of_le])\n\naegis_spec \"wadray::wadray_signed::SignedRaySigned::is_negative\" :=\n  fun _ _ (a : SignedRay) _ ρ =>\n  ρ = Bool.toSierraBool (a.toRat < 0)\n\naegis_prove \"wadray::wadray_signed::SignedRaySigned::is_negative\" :=\n  fun _ _ (a : SignedRay) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRaySigned::is_negative»\n  rintro ⟨w,s,rfl,(⟨h,rfl⟩|⟨h,rfl⟩)⟩\n  · aesop (add simp [SignedRay.toRat, Ray.toRat])\n  · have : 0 < (Ray.RAY_SCALE : ℚ) := by norm_num [Ray.RAY_SCALE]\n    rcases s with (s|s)\n    <;> aesop (add simp [SignedRay.toRat, Ray.toRat, lt_div_iff, le_div_iff,\n      Ray.toZMod, ZMod.cast_rat_nonneg], safe apply [Ne.lt_of_le])\n\naegis_spec \"wadray::wadray_signed::SignedWadZeroable::is_zero\" :=\n  fun _ (a : SignedWad) ρ =>\n  ρ = Bool.toSierraBool (a.toRat = 0)\n\naegis_prove \"wadray::wadray_signed::SignedWadZeroable::is_zero\" :=\n  fun _ (a : SignedWad) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadZeroable::is_zero»\n  have : ¬ (Wad.WAD_SCALE = 0) := by\n    norm_num [Wad.WAD_SCALE]\n  aesop (add simp [SignedWad.toRat, Wad.toRat])\n\naegis_spec \"wadray::wadray_signed::SignedRayZeroable::is_zero\" :=\n  fun _ (a : SignedRay) ρ =>\n  ρ = Bool.toSierraBool (a.toRat = 0)\n\naegis_prove \"wadray::wadray_signed::SignedRayZeroable::is_zero\" :=\n  fun _ (a : SignedRay) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayZeroable::is_zero»\n  have : ¬ (Ray.RAY_SCALE = 0) := by\n    norm_num [Ray.RAY_SCALE]\n  aesop (add simp [SignedRay.toRat, Ray.toRat])\n\naegis_spec \"wadray::wadray_signed::SignedWadZeroable::is_non_zero\" :=\n  fun _ (a : SignedWad) ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≠ 0)\n\naegis_prove \"wadray::wadray_signed::SignedWadZeroable::is_non_zero\" :=\n  fun _ (a : SignedWad) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadZeroable::is_non_zero»\n  aesop (add simp [SignedWad.toRat, Wad.toRat, Wad.toZMod])\n\naegis_spec \"wadray::wadray_signed::SignedRayZeroable::is_non_zero\" :=\n  fun _ (a : SignedRay) ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≠ 0)\n\naegis_prove \"wadray::wadray_signed::SignedRayZeroable::is_non_zero\" :=\n  fun _ (a : SignedRay) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayZeroable::is_non_zero»\n  aesop (add simp [SignedRay.toRat, Ray.toRat, Ray.toZMod])\n\naegis_spec \"wadray::wadray_signed::SignedWadOneable::is_one\" :=\n  fun _ (a : SignedWad) ρ =>\n  ρ = Bool.toSierraBool (a.toRat = 1)\n\naegis_prove \"wadray::wadray_signed::SignedWadOneable::is_one\" :=\n  fun _ (a : SignedWad) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadOneable::is_one»\n  rintro ⟨w,s,h⟩\n  have : 1000000000000000000 = ((1000000000000000000 : Sierra.UInt128).cast : ℚ) := by\n    rw [ZMod.cast_eq_val]; aesop\n  have : (1000000000000000000 : ℚ) ≠ 0 := by\n    norm_num\n  have : ¬ (-(w.cast : ℚ) = 1000000000000000000) := by\n    apply ne_of_lt (lt_of_le_of_lt (b := 0) _ (by norm_num))\n    simp only [Left.neg_nonpos_iff, ZMod.cast_rat_nonneg]\n  aesop (add simp [SignedWad.toRat, Wad.toRat, Wad.toZMod, Wad.WAD_SCALE,\n    div_eq_iff, neg_div'], safe forward [ZMod.cast_rat_injective])\n\naegis_spec \"wadray::wadray_signed::SignedRayOneable::is_one\" :=\n  fun _ (a : SignedRay) ρ =>\n  ρ = Bool.toSierraBool (a.toRat = 1)\n\naegis_prove \"wadray::wadray_signed::SignedRayOneable::is_one\" :=\n  fun _ (a : SignedWad) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayOneable::is_one»\n  rintro ⟨w,s,h⟩\n  have : 1000000000000000000000000000 = ((1000000000000000000000000000 : Sierra.UInt128).cast : ℚ) := by\n    rw [ZMod.cast_eq_val]; aesop\n  have : (1000000000000000000000000000 : ℚ) ≠ 0 := by\n    norm_num\n  have : ¬ (-(w.cast : ℚ) = 1000000000000000000000000000) := by\n    apply ne_of_lt (lt_of_le_of_lt (b := 0) _ (by norm_num))\n    simp only [Left.neg_nonpos_iff, ZMod.cast_rat_nonneg]\n  aesop (add simp [SignedRay.toRat, Ray.toRat, Ray.toZMod, Ray.RAY_SCALE,\n    div_eq_iff, neg_div'], safe forward [ZMod.cast_rat_injective])\n\naegis_spec \"wadray::wadray_signed::SignedWadOneable::is_non_one\" :=\n  fun _ (a : SignedWad) ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≠ 1)\n\naegis_prove \"wadray::wadray_signed::SignedWadOneable::is_non_one\" :=\n  fun _ (a : SignedWad) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadOneable::is_non_one»\n  rintro ⟨w,s,rfl,h⟩\n  have : ¬ (-(w.cast : ℚ) = 1000000000000000000) := by\n    apply ne_of_lt (lt_of_le_of_lt (b := 0) _ (by norm_num))\n    simp only [Left.neg_nonpos_iff, ZMod.cast_rat_nonneg]\n  have : ¬w = 1000000000000000000 → ¬(w.cast : ℚ) = 1000000000000000000 := by\n    have hn : 1000000000000000000 = ((1000000000000000000 : Sierra.UInt128).cast : ℚ) := by\n      rw [ZMod.cast_eq_val]; aesop\n    intros he hee; rw [hn] at hee; have := ZMod.cast_rat_injective hee; contradiction\n  aesop (add simp [SignedWad.toRat, Wad.toRat, Wad.toZMod, Wad.WAD_SCALE,\n    div_eq_iff, neg_div'], safe forward [ZMod.cast_rat_injective])\n\naegis_spec \"wadray::wadray_signed::SignedRayOneable::is_non_one\" :=\n  fun _ (a : SignedRay) ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≠ 1)\n\naegis_prove \"wadray::wadray_signed::SignedRayOneable::is_non_one\" :=\n  fun _ (a : SignedRay) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayOneable::is_non_one»\n  rintro ⟨w,s,rfl,h⟩\n  have : ¬ (-(w.cast : ℚ) = 1000000000000000000000000000) := by\n    apply ne_of_lt (lt_of_le_of_lt (b := 0) _ (by norm_num))\n    simp only [Left.neg_nonpos_iff, ZMod.cast_rat_nonneg]\n  have : ¬w = 1000000000000000000000000000 → ¬(w.cast : ℚ) = 1000000000000000000000000000 := by\n    have hn : 1000000000000000000000000000 = ((1000000000000000000000000000 : Sierra.UInt128).cast : ℚ) := by\n      rw [ZMod.cast_eq_val]; aesop\n    intros he hee; rw [hn] at hee; have := ZMod.cast_rat_injective hee; contradiction\n  aesop (add simp [SignedRay.toRat, Ray.toRat, Ray.toZMod, Ray.RAY_SCALE,\n    div_eq_iff, neg_div'], safe forward [ZMod.cast_rat_injective])\n\naegis_spec \"wadray::wadray_signed::BoundedSignedWad::max\" :=\n  fun _ ρ =>\n  ρ = ((U128_MOD - 1 : UInt128), .inl ())\n\naegis_prove \"wadray::wadray_signed::BoundedSignedWad::max\" :=\n  fun _ ρ => by\n  rintro rfl\n  rfl\naegis_spec \"wadray::wadray_signed::BoundedSignedRay::max\" :=\n  fun _ ρ =>\n  ρ = ((U128_MOD - 1 : UInt128), .inl ())\n\naegis_prove \"wadray::wadray_signed::BoundedSignedRay::max\" :=\n  fun _ ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::BoundedSignedWad::min\" :=\n  fun _ ρ =>\n  ρ = ((U128_MOD - 1 : UInt128), .inr ())\n\naegis_prove \"wadray::wadray_signed::BoundedSignedWad::min\" :=\n  fun _ ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::BoundedSignedRay::min\" :=\n  fun _ ρ =>\n  ρ = ((U128_MOD - 1 : UInt128), .inr ())\n\naegis_prove \"wadray::wadray_signed::BoundedSignedRay::min\" :=\n  fun _ ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::SignedWadPartialOrd::gt\" :=\n  fun _ _ (a b : SignedWad) _ ρ =>\n  ρ = Bool.toSierraBool (b.toRat < a.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedWadPartialOrd::gt\" :=\n  fun _ _ (a b : SignedWad) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadPartialOrd::gt»\n  have : 0 < (Wad.WAD_SCALE : ℚ) := by\n    norm_num [Wad.WAD_SCALE]\n  rintro ⟨w₁,s₁,w₂,s₂,u₁,u₂,u₃,u₄,u₅,u₆,rfl,rfl,(h|h)⟩\n    <;> (cases u₁; cases u₂; cases u₃; cases u₄; cases u₅; cases u₆)\n  · rcases h with ⟨rfl,(h|h)⟩\n    · rcases h with ⟨rfl,(⟨hne,h⟩|⟨rfl,rfl,rfl⟩)⟩\n      · rcases h with (⟨hle,rfl⟩|⟨hlt,rfl⟩)\n        · simp [SignedWad.toRat_le_toRat_of_val_le_val_inl hle]\n        · simp only [SierraBool_toBool_inr, SierraBool_toBool_inl, Bool.xor_false,\n            Bool.toSierraBool_true, Bool.toSierraBool_decide_inr']\n          apply lt_of_le_of_ne\n          · apply SignedWad.toRat_le_toRat_of_val_le_val_inl\n            simp [Wad.toZMod, le_of_lt hlt]\n          · intro he\n            exact hne (SignedWad.val_eq_of_toRat_eq _ _ he).symm\n      · simp only [lt_self_iff_false, decide_False, Bool.toSierraBool_false]\n    · rcases h with ⟨rfl,(⟨hne,rfl,rfl⟩|h)⟩\n      · simp only [SierraBool_toBool_inl, Bool.not_false, Bool.toSierraBool_true,\n          Bool.toSierraBool_decide_inr']\n        apply lt_of_le_of_ne\n        · simp [SignedWad.toRat_inr_le_toRat_inl]\n        · intro he\n          exact hne (SignedWad.val_eq_of_toRat_eq _ _ he).symm\n      · rcases h with ⟨rfl,(⟨h,rfl,rfl⟩|⟨h,rfl,rfl⟩)⟩\n        · simp only [Nat.cast_pos, Bool.toSierraBool_decide_inl', SierraBool_toBool_inl,\n            Bool.not_false, Bool.toSierraBool_true, Bool.toSierraBool_decide_inr'] at *\n          apply lt_of_lt_of_le (b := 0)\n          · apply lt_of_le_of_ne\n            · simp [SignedWad.toRat, Wad.toRat_nonneg]\n            · simpa [SignedWad.toRat]\n          · simp [SignedWad.toRat, Wad.toRat_nonneg]\n        · simp only [Nat.cast_pos, SignedWad.toRat, SierraBool_toBool_inl, ite_false,\n            Bool.toSierraBool_decide_inr', SierraBool_toBool_inr, ite_true, neg_lt_self_iff,\n            Bool.toSierraBool_decide_inl', not_lt] at *\n          apply le_of_eq h\n  · rcases h with ⟨rfl,(h|h)⟩\n    · rcases h with ⟨h',(h|⟨rfl,rfl,rfl⟩)⟩ <;> rcases s₂ with (_|s₂); · simp at h'\n      · simp at h'\n        rcases h with ⟨hne,(⟨hle,rfl⟩|⟨h,rfl⟩)⟩\n        · simp at *\n          · cases s₂\n            apply lt_of_le_of_ne\n            · apply SignedWad.toRat_le_toRat_of_val_ge_val_inr hle\n            · intro he\n              exact hne (SignedWad.val_eq_of_toRat_eq _ _ he).symm\n        · simp at *\n          apply SignedWad.toRat_le_toRat_of_val_ge_val_inr (le_of_lt h)\n      · simp at *\n      · simp at *\n    · rcases h with ⟨h',h⟩\n      rcases s₂ with (s₂|s₂) <;> cases s₂\n      · rcases h with (⟨_,rfl,rfl⟩|⟨rfl,(⟨_,rfl,rfl⟩|⟨_,rfl,rfl⟩)⟩)\n          <;> simp [SignedWad.toRat_inr_le_toRat_inl]\n      · simp at h'\n\naegis_spec \"wadray::wadray_signed::SignedWadPartialOrd::lt\" :=\n  fun _ _ (a b : SignedWad) _ ρ =>\n  ρ = Bool.toSierraBool (a.toRat < b.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedWadPartialOrd::lt\" :=\n  fun _ _ (a b : SignedWad) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadPartialOrd::lt»\n  have : 0 < (Wad.WAD_SCALE : ℚ) := by\n    norm_num [Wad.WAD_SCALE]\n  rintro ⟨w₁,s₁,w₂,s₂,u₁,u₂,u₃,u₄,u₅,u₆,rfl,rfl,(h|h)⟩\n    <;> (cases u₁; cases u₂; cases u₃; cases u₄; cases u₅; cases u₆)\n  · rcases h with ⟨rfl,(h|h)⟩\n    · rcases h with ⟨rfl,(⟨hne,h⟩|⟨rfl,rfl,rfl⟩)⟩\n      · rcases h with (⟨hle,rfl⟩|⟨hlt,rfl⟩)\n        · simp [SignedWad.toRat_le_toRat_of_val_le_val_inl hle]\n        · simp only [SierraBool_toBool_inr, SierraBool_toBool_inl, Bool.xor_false,\n            Bool.toSierraBool_true, Bool.toSierraBool_decide_inr']\n          apply lt_of_le_of_ne\n          · apply SignedWad.toRat_le_toRat_of_val_le_val_inl\n            simp [Wad.toZMod, le_of_lt hlt]\n          · intro he\n            exact hne (SignedWad.val_eq_of_toRat_eq _ _ he)\n      · simp only [lt_self_iff_false, decide_False, Bool.toSierraBool_false]\n    · rcases h with ⟨rfl,(⟨_,rfl,rfl⟩|⟨rfl,(⟨_,rfl,rfl⟩|⟨_,rfl,rfl⟩)⟩)⟩\n        <;> simp [SignedWad.toRat_inr_le_toRat_inl]\n  · rcases h with ⟨rfl,(h|h)⟩\n    · rcases h with ⟨h',(h|⟨rfl,rfl,rfl⟩)⟩ <;> rcases s₂ with (_|s₂); · simp at h'\n      · simp at h'\n        rcases h with ⟨hne,(⟨hle,rfl⟩|⟨h,rfl⟩)⟩\n        · simp at *\n          · cases s₂\n            apply lt_of_le_of_ne\n            · apply SignedWad.toRat_le_toRat_of_val_ge_val_inr hle\n            · intro he\n              exact hne (SignedWad.val_eq_of_toRat_eq _ _ he)\n        · simp at *\n          apply SignedWad.toRat_le_toRat_of_val_ge_val_inr (le_of_lt h)\n      · simp at *\n      · simp at *\n    · rcases h with ⟨h',h⟩\n      rcases s₂ with (s₂|s₂) <;> cases s₂\n      · rcases h with (⟨hne,rfl,rfl⟩|⟨rfl,h⟩)\n        · simp\n          apply lt_of_le_of_ne\n          · apply SignedWad.toRat_inr_le_toRat_inl\n          · intro he\n            exact hne (SignedWad.val_eq_of_toRat_eq _ _ he)\n        · rcases h with (⟨h,rfl,rfl⟩|⟨h,rfl,rfl⟩)\n          · simp at *\n            apply lt_of_lt_of_le (b := 0)\n            · apply lt_of_le_of_ne\n              · simp [SignedWad.toRat, Wad.toRat_nonneg]\n              · exact h\n            · simp [SignedWad.toRat, Wad.toRat_nonneg]\n          · simp only [Nat.cast_pos, SierraBool_toBool_inl, Bool.not_false, Bool.toSierraBool_true,\n              SignedWad.toRat, SierraBool_toBool_inr, ite_true, neg_eq_zero,\n              Bool.toSierraBool_decide_inr', ite_false, neg_lt_self_iff,\n              Bool.toSierraBool_decide_inl', not_lt] at *\n            simp [h]\n      · simp at h'\n\naegis_spec \"wadray::wadray_signed::SignedWadPartialOrd::ge\" :=\n  fun _ _ (a b : SignedWad) _ ρ =>\n  ρ = Bool.toSierraBool (b.toRat ≤ a.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedWadPartialOrd::ge\" :=\n  fun _ _ (a b : SignedWad) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadPartialOrd::ge»\n  rintro rfl\n  simp only [← not_lt]\n  rw [decide_not]\n  simp only [Bool.coe_toSierraBool, Bool.toSierraBool_not]\n\naegis_spec \"wadray::wadray_signed::SignedWadPartialOrd::le\" :=\n  fun _ _ (a b : SignedWad) _ ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≤ b.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedWadPartialOrd::le\" :=\n  fun _ _ (a b : SignedWad) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadPartialOrd::le»\n  rintro rfl\n  simp only [← not_lt]\n  rw [decide_not]\n  simp only [Bool.coe_toSierraBool, Bool.toSierraBool_not]\n\naegis_spec \"wadray::wadray_signed::SignedRayPartialOrd::gt\" :=\n  fun _ _ (a b : SignedRay) _ ρ =>\n  ρ = Bool.toSierraBool (b.toRat < a.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedRayPartialOrd::gt\" :=\n  fun _ _ (a b : SignedRay) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayPartialOrd::gt»\n  have : 0 < (Ray.RAY_SCALE : ℚ) := by\n    norm_num [Ray.RAY_SCALE]\n  rintro ⟨w₁,s₁,w₂,s₂,u₁,u₂,u₃,u₄,u₅,u₆,rfl,rfl,(h|h)⟩\n    <;> (cases u₁; cases u₂; cases u₃; cases u₄; cases u₅; cases u₆)\n  · rcases h with ⟨rfl,(h|h)⟩\n    · rcases h with ⟨rfl,(⟨hne,h⟩|⟨rfl,rfl,rfl⟩)⟩\n      · rcases h with (⟨hle,rfl⟩|⟨hlt,rfl⟩)\n        · simp [SignedRay.toRat_le_toRat_of_val_le_val_inl hle]\n        · simp only [SierraBool_toBool_inr, SierraBool_toBool_inl, Bool.xor_false,\n            Bool.toSierraBool_true, Bool.toSierraBool_decide_inr']\n          apply lt_of_le_of_ne\n          · apply SignedRay.toRat_le_toRat_of_val_le_val_inl\n            simp [Ray.toZMod, le_of_lt hlt]\n          · intro he\n            exact hne (SignedRay.val_eq_of_toRat_eq _ _ he).symm\n      · simp only [lt_self_iff_false, decide_False, Bool.toSierraBool_false]\n    · rcases h with ⟨rfl,(⟨hne,rfl,rfl⟩|h)⟩\n      · simp only [SierraBool_toBool_inl, Bool.not_false, Bool.toSierraBool_true,\n          Bool.toSierraBool_decide_inr']\n        apply lt_of_le_of_ne\n        · simp [SignedRay.toRat_inr_le_toRat_inl]\n        · intro he\n          exact hne (SignedRay.val_eq_of_toRat_eq _ _ he).symm\n      · rcases h with ⟨rfl,(⟨h,rfl,rfl⟩|⟨h,rfl,rfl⟩)⟩\n        · simp only [Nat.cast_pos, Bool.toSierraBool_decide_inl', SierraBool_toBool_inl,\n            Bool.not_false, Bool.toSierraBool_true, Bool.toSierraBool_decide_inr'] at *\n          apply lt_of_lt_of_le (b := 0)\n          · apply lt_of_le_of_ne\n            · simp [SignedRay.toRat, Ray.toRat_nonneg]\n            · simpa [SignedRay.toRat]\n          · simp [SignedRay.toRat, Ray.toRat_nonneg]\n        · simp only [Nat.cast_pos, SignedRay.toRat, SierraBool_toBool_inl, ite_false,\n            Bool.toSierraBool_decide_inr', SierraBool_toBool_inr, ite_true, neg_lt_self_iff,\n            Bool.toSierraBool_decide_inl', not_lt] at *\n          apply le_of_eq h\n  · rcases h with ⟨rfl,(h|h)⟩\n    · rcases h with ⟨h',(h|⟨rfl,rfl,rfl⟩)⟩ <;> rcases s₂ with (_|s₂); · simp at h'\n      · simp at h'\n        rcases h with ⟨hne,(⟨hle,rfl⟩|⟨h,rfl⟩)⟩\n        · simp at *\n          · cases s₂\n            apply lt_of_le_of_ne\n            · apply SignedRay.toRat_le_toRat_of_val_ge_val_inr hle\n            · intro he\n              exact hne (SignedRay.val_eq_of_toRat_eq _ _ he).symm\n        · simp at *\n          apply SignedRay.toRat_le_toRat_of_val_ge_val_inr (le_of_lt h)\n      · simp at *\n      · simp at *\n    · rcases h with ⟨h',h⟩\n      rcases s₂ with (s₂|s₂) <;> cases s₂\n      · rcases h with (⟨_,rfl,rfl⟩|⟨rfl,(⟨_,rfl,rfl⟩|⟨_,rfl,rfl⟩)⟩)\n          <;> simp [SignedRay.toRat_inr_le_toRat_inl]\n      · simp at h'\n\naegis_spec \"wadray::wadray_signed::SignedRayPartialOrd::lt\" :=\n  fun _ _ (a b : SignedRay) _ ρ =>\n  ρ = Bool.toSierraBool (a.toRat < b.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedRayPartialOrd::lt\" :=\n  fun _ _ (a b : SignedRay) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayPartialOrd::lt»\n  have : 0 < (Ray.RAY_SCALE : ℚ) := by\n    norm_num [Ray.RAY_SCALE]\n  rintro ⟨w₁,s₁,w₂,s₂,u₁,u₂,u₃,u₄,u₅,u₆,rfl,rfl,(h|h)⟩\n    <;> (cases u₁; cases u₂; cases u₃; cases u₄; cases u₅; cases u₆)\n  · rcases h with ⟨rfl,(h|h)⟩\n    · rcases h with ⟨rfl,(⟨hne,h⟩|⟨rfl,rfl,rfl⟩)⟩\n      · rcases h with (⟨hle,rfl⟩|⟨hlt,rfl⟩)\n        · simp [SignedRay.toRat_le_toRat_of_val_le_val_inl hle]\n        · simp only [SierraBool_toBool_inr, SierraBool_toBool_inl, Bool.xor_false,\n            Bool.toSierraBool_true, Bool.toSierraBool_decide_inr']\n          apply lt_of_le_of_ne\n          · apply SignedRay.toRat_le_toRat_of_val_le_val_inl\n            simp [Ray.toZMod, le_of_lt hlt]\n          · intro he\n            exact hne (SignedRay.val_eq_of_toRat_eq _ _ he)\n      · simp only [lt_self_iff_false, decide_False, Bool.toSierraBool_false]\n    · rcases h with ⟨rfl,(⟨_,rfl,rfl⟩|⟨rfl,(⟨_,rfl,rfl⟩|⟨_,rfl,rfl⟩)⟩)⟩\n        <;> simp [SignedRay.toRat_inr_le_toRat_inl]\n  · rcases h with ⟨rfl,(h|h)⟩\n    · rcases h with ⟨h',(h|⟨rfl,rfl,rfl⟩)⟩ <;> rcases s₂ with (_|s₂); · simp at h'\n      · simp at h'\n        rcases h with ⟨hne,(⟨hle,rfl⟩|⟨h,rfl⟩)⟩\n        · simp at *\n          · cases s₂\n            apply lt_of_le_of_ne\n            · apply SignedRay.toRat_le_toRat_of_val_ge_val_inr hle\n            · intro he\n              exact hne (SignedRay.val_eq_of_toRat_eq _ _ he)\n        · simp at *\n          apply SignedRay.toRat_le_toRat_of_val_ge_val_inr (le_of_lt h)\n      · simp at *\n      · simp at *\n    · rcases h with ⟨h',h⟩\n      rcases s₂ with (s₂|s₂) <;> cases s₂\n      · rcases h with (⟨hne,rfl,rfl⟩|⟨rfl,h⟩)\n        · simp\n          apply lt_of_le_of_ne\n          · apply SignedRay.toRat_inr_le_toRat_inl\n          · intro he\n            exact hne (SignedRay.val_eq_of_toRat_eq _ _ he)\n        · rcases h with (⟨h,rfl,rfl⟩|⟨h,rfl,rfl⟩)\n          · simp at *\n            apply lt_of_lt_of_le (b := 0)\n            · apply lt_of_le_of_ne\n              · simp [SignedRay.toRat, Ray.toRat_nonneg]\n              · exact h\n            · simp [SignedRay.toRat, Ray.toRat_nonneg]\n          · simp only [Nat.cast_pos, SierraBool_toBool_inl, Bool.not_false, Bool.toSierraBool_true,\n              SignedRay.toRat, SierraBool_toBool_inr, ite_true, neg_eq_zero,\n              Bool.toSierraBool_decide_inr', ite_false, neg_lt_self_iff,\n              Bool.toSierraBool_decide_inl', not_lt] at *\n            simp [h]\n      · simp at h'\n\naegis_spec \"wadray::wadray_signed::SignedRayPartialOrd::ge\" :=\n  fun _ _ (a b : SignedRay) _ ρ =>\n  ρ = Bool.toSierraBool (b.toRat ≤ a.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedRayPartialOrd::ge\" :=\n  fun _ _ (a b : SignedRay) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayPartialOrd::ge»\n  rintro rfl\n  simp only [← not_lt]\n  rw [decide_not]\n  simp only [Bool.coe_toSierraBool, Bool.toSierraBool_not]\n\naegis_spec \"wadray::wadray_signed::SignedRayPartialOrd::le\" :=\n  fun _ _ (a b : SignedRay) _ ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≤ b.toRat)\n\naegis_prove \"wadray::wadray_signed::SignedRayPartialOrd::le\" :=\n  fun _ _ (a b : SignedRay) _ ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayPartialOrd::le»\n  rintro rfl\n  simp only [← not_lt]\n  rw [decide_not]\n  simp only [Bool.coe_toSierraBool, Bool.toSierraBool_not]\n\naegis_spec \"wadray::wadray_signed::SignedWadTryIntoWad::try_into\" :=\n  fun _ (a : SignedWad) (ρ : Wad ⊕ _) =>\n  ρ = if SierraBool.toBool a.2 then .inr () else .inl a.1\n\naegis_prove \"wadray::wadray_signed::SignedWadTryIntoWad::try_into\" :=\n  fun _ (a : SignedWad) (ρ : Wad ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedWadTryIntoWad::try_into»\n  aesop\n\naegis_spec \"wadray::wadray_signed::SignedRayTryIntoRay::try_into\" :=\n  fun _ (a : SignedRay) (ρ : Ray ⊕ _) =>\n  ρ = if SierraBool.toBool a.2 then .inr () else .inl a.1\n\naegis_prove \"wadray::wadray_signed::SignedRayTryIntoRay::try_into\" :=\n  fun _ (a : SignedRay) (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedRayTryIntoRay::try_into»\n  aesop\n\naegis_spec \"wadray::wadray_signed::_felt_sign\" :=\n  fun _ _ a _ ρ =>\n  ρ = Bool.toSierraBool (a.valMinAbs < 0)\n\naegis_prove \"wadray::wadray_signed::_felt_sign\" :=\n  fun _ _ a _ ρ => by\n  unfold «spec_wadray::wadray_signed::_felt_sign»\n  rintro ⟨x : UInt256, y : UInt256, h₁, h₂, rfl⟩\n  have : (1809251394333065606848661391547535052811553607665798349986546028067936010240 : F).val\n          = PRIME / 2 := rfl\n  simp only [Int.ofNat_eq_coe, Nat.cast_ofNat, Int.cast_ofNat] at h₂\n  simp only [UInt256.val, h₂, h₁, ← not_le (b := a.valMinAbs) (a := 0), ZMod.valMinAbs_nonneg_iff]\n  congr; apply propext\n  rw [not_le, ← this, ← h₂, UInt256.val, ← h₁, UInt256.val]\n\naegis_spec \"wadray::wadray_signed::_felt_abs\" :=\n  fun _ _ a _ ρ =>\n  ρ = a.valMinAbs.natAbs\n\naegis_prove \"wadray::wadray_signed::_felt_abs\" :=\n  fun _ _ a _ ρ => by\n  unfold «spec_wadray::wadray_signed::_felt_abs»\n  sierra_simp'\n  rw [← not_le, ZMod.valMinAbs_nonneg_iff, not_le, not_lt, ZMod.natCast_natAbs_valMinAbs a]\n  aesop\n\naegis_spec \"wadray::wadray_signed::sign_from_mul\" :=\n  fun _ a b ρ =>\n  ρ = Bool.toSierraBool (xor (SierraBool.toBool a) (SierraBool.toBool b))\n\naegis_prove \"wadray::wadray_signed::sign_from_mul\" :=\n  fun _ a b ρ => by\n  unfold «spec_wadray::wadray_signed::sign_from_mul»\n  aesop\n\naegis_spec \"wadray::wadray_signed::signed_wad_from_felt\" :=\n  fun _ _ a _ (ρ : SignedWad ⊕ _) =>\n  if a.valMinAbs.natAbs < U128_MOD\n  then ρ.isLeft ∧ ρ.getLeft?.get!.toRat = a.valMinAbs / Wad.WAD_SCALE\n  else ρ.isRight\n\naegis_prove \"wadray::wadray_signed::signed_wad_from_felt\" :=\n  fun _ _ a _ (ρ : SignedWad ⊕ _) => by\n  have hlt : a.valMinAbs.natAbs < PRIME := by\n    apply lt_of_le_of_lt (ZMod.natAbs_valMinAbs_le a)\n    norm_num [PRIME]\n  unfold «spec_wadray::wadray_signed::signed_wad_from_felt»\n  rintro ⟨_,_,_,(⟨h₁,rfl⟩|⟨h₁,rfl⟩),(⟨h₂,rfl⟩|⟨h₂,rfl⟩)⟩\n  · cases h₂\n    rw [ZMod.val_natCast_of_lt hlt] at h₁\n    simp only [SignedWad.toRat]\n    split_ifs with h₃\n    · simp_all only [Option.get!, ZMod.cast_nat_cast_of_lt hlt, Sum.getLeft?_inl,\n        Bool.coe_toSierraBool, decide_eq_true_eq, decide_True, Bool.toSierraBool_true, Sum.isLeft_inl,\n        Wad.toRat, Wad.toZMod, ZMod.val_natCast, Nat.mod_eq_of_lt h₁, neg_div', true_and]\n      congr\n      rw [Nat.cast_natAbs, Int.cast_abs, neg_eq_iff_eq_neg]\n      simp only [abs_eq_neg_self, Int.cast_nonpos]\n      exact le_of_lt h₃\n    · simp_all only [Option.get!, Sum.getLeft?_inl, Bool.coe_toSierraBool, decide_eq_true_eq,\n        not_lt, Sum.isLeft_inl, Wad.toRat, Wad.toZMod, ZMod.natCast_val, true_and,\n        ZMod.cast_nat_cast_of_lt hlt, ZMod.cast_nat_cast_of_lt h₁]\n      congr\n      rw [Nat.cast_natAbs]\n      aesop\n  · simp at h₂\n  · simp at h₂\n  · cases h₂\n    rw [ZMod.val_natCast_of_lt hlt, ← not_lt] at h₁\n    simp [h₁]\n\naegis_spec \"wadray::wadray_signed::signed_ray_from_felt\" :=\n  fun _ _ a _ (ρ : SignedRay ⊕ _) =>\n  if a.valMinAbs.natAbs < U128_MOD\n  then ρ.isLeft ∧ ρ.getLeft?.get!.toRat = a.valMinAbs / Ray.RAY_SCALE\n  else ρ.isRight\n\naegis_prove \"wadray::wadray_signed::signed_ray_from_felt\" :=\n  fun _ _ a _ (ρ : SignedRay ⊕ _) => by\n  have hlt : a.valMinAbs.natAbs < PRIME := by\n    apply lt_of_le_of_lt (ZMod.natAbs_valMinAbs_le a)\n    norm_num [PRIME]\n  unfold «spec_wadray::wadray_signed::signed_ray_from_felt»\n  rintro ⟨_,_,_,(⟨h₁,rfl⟩|⟨h₁,rfl⟩),(⟨h₂,rfl⟩|⟨h₂,rfl⟩)⟩\n  · cases h₂\n    rw [ZMod.val_natCast_of_lt hlt] at h₁\n    simp only [SignedRay.toRat]\n    split_ifs with h₃\n    · simp_all only [Option.get!, ZMod.cast_nat_cast_of_lt hlt, Sum.getLeft?_inl,\n        Bool.coe_toSierraBool, decide_eq_true_eq, decide_True, Bool.toSierraBool_true, Sum.isLeft_inl,\n        Ray.toRat, Ray.toZMod, ZMod.val_natCast, Nat.mod_eq_of_lt h₁, neg_div', true_and]\n      congr\n      rw [Nat.cast_natAbs, Int.cast_abs, neg_eq_iff_eq_neg]\n      simp only [abs_eq_neg_self, Int.cast_nonpos]\n      exact le_of_lt h₃\n    · simp_all only [Option.get!, Sum.getLeft?_inl, Bool.coe_toSierraBool, decide_eq_true_eq,\n        not_lt, Sum.isLeft_inl, Ray.toRat, Ray.toZMod, ZMod.natCast_val, true_and,\n        ZMod.cast_nat_cast_of_lt hlt, ZMod.cast_nat_cast_of_lt h₁]\n      congr\n      rw [Nat.cast_natAbs]\n      aesop\n  · simp at h₂\n  · simp at h₂\n  · cases h₂\n    rw [ZMod.val_natCast_of_lt hlt, ← not_lt] at h₁\n    simp [h₁]\n\naegis_spec \"wadray::wadray_signed::SignedWadIntoFelt252::into\" :=\n  fun _ (a : SignedWad) ρ =>\n  ρ = if SierraBool.toBool a.2 then -a.1.cast else a.1.cast\n\naegis_prove \"wadray::wadray_signed::SignedWadIntoFelt252::into\" :=\n  fun _ (a : SignedWad) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedWadIntoFelt252::into»\n  aesop\n\naegis_spec \"wadray::wadray_signed::SignedRayIntoFelt252::into\" :=\n  fun _ (a : SignedRay) ρ =>\n  ρ = if SierraBool.toBool a.2 then -a.1.cast else a.1.cast\n\naegis_prove \"wadray::wadray_signed::SignedRayIntoFelt252::into\" :=\n  fun _ (a : SignedRay) ρ => by\n  unfold «spec_wadray::wadray_signed::SignedRayIntoFelt252::into»\n  aesop\n\naegis_spec \"wadray::wadray_signed::SignedWadAdd::add\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad ⊕ _) =>\n  if |a.toRat + b.toRat| < U128_MOD / Wad.WAD_SCALE\n  then ρ.isLeft ∧ ρ.getLeft?.get!.toRat = a.toRat + b.toRat\n  else ρ.isRight", "target_theorem": "theorem add_aux1 (x : UInt128) : 4 * (x.cast : F).valMinAbs.natAbs < PRIME :=", "ground_truth_proof": ":= by\n  rw [ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME, Int.natAbs_ofNat]\n  apply lt_of_lt_of_le _ four_U128_MOD_le_PRIME\n  apply Nat.mul_lt_mul_of_pos_left (ZMod.val_lt _) (by norm_num)\n\naegis_prove \"wadray::wadray_signed::SignedWadAdd::add\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedWadAdd::add»\n  rcases a with ⟨va,sa⟩\n  rcases b with ⟨vb,sb⟩\n  have hS : 0 < (Wad.WAD_SCALE : ℚ) := by\n    norm_num [Wad.WAD_SCALE]\n  have hane : 2 * (va.cast : ZMod PRIME).val ≠ PRIME := by\n    apply ne_of_lt (lt_trans (Nat.mul_lt_mul_of_pos_left (ZMod.val_cast_lt PRIME va) two_pos)\n      two_U128_MOD_lt_PRIME)\n  have hbne : 2 * (vb.cast : ZMod PRIME).val ≠ PRIME := by\n    apply ne_of_lt (lt_trans (Nat.mul_lt_mul_of_pos_left (ZMod.val_cast_lt PRIME vb) two_pos)\n      two_U128_MOD_lt_PRIME)\n  have ha : 4 * (va.cast : F).valMinAbs.natAbs < PRIME := add_aux1 va\n  have hb : 4 * (vb.cast : F).valMinAbs.natAbs < PRIME := add_aux1 vb\n  have ha' : 4 * (- (va.cast : F)).valMinAbs.natAbs < PRIME := by\n    rwa [ZMod.valMinAbs_neg_of_ne_half hane, Int.natAbs_neg]\n  have hb' : 4 * (- (vb.cast : F)).valMinAbs.natAbs < PRIME := by\n    rwa [ZMod.valMinAbs_neg_of_ne_half hbne, Int.natAbs_neg]\n  rintro ⟨x,y,z,h₁,(⟨rfl,rfl⟩|⟨rfl,rfl⟩)⟩ <;> dsimp only at h₁\n  · simp only [Sum.isLeft_inl, Sum.getLeft?_inl, Option.get!_some, true_and, Sum.isRight_inl,\n      ite_prop_iff_or, not_lt, and_false, or_false] at h₁ ⊢\n    rcases sa with (sa|sa) <;> cases sa <;> rcases sb with (sb|sb) <;> cases sb\n      <;> simp only [SierraBool_toBool_inl, SierraBool_toBool_inr, ite_false, ite_true] at h₁\n      <;> rcases h₁ with ⟨h₁,h₂⟩\n    · simp only [ZMod.valMinAbs_add_of_four_lt ha hb, Int.natAbs_ofNat,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME, ← Nat.cast_add] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedWad.toRat, SierraBool_toBool_inl, Wad.toRat, Wad.toZMod,\n        ite_false, and_true, add_div]\n      rw [← add_div, ← Nat.cast_add, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      rwa [Nat.cast_lt]\n    · simp only [ZMod.valMinAbs_add_of_four_lt ha hb', ZMod.valMinAbs_neg_of_ne_half hbne,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedWad.toRat, SierraBool_toBool_inl, Wad.toRat, Wad.toZMod,\n        ite_false, SierraBool_toBool_inr, ite_true, add_div, neg_div, and_true]\n      rw [← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      simp only [← Nat.cast_lt (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁ ⊢\n      assumption\n    · simp only [ZMod.valMinAbs_add_of_four_lt ha' hb, ZMod.valMinAbs_neg_of_ne_half hane,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedWad.toRat, SierraBool_toBool_inl, Wad.toRat, Wad.toZMod,\n        ite_false, SierraBool_toBool_inr, ite_true, add_div, neg_div, and_true]\n      rw [← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      simp only [← Nat.cast_lt (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁ ⊢\n      assumption\n    · simp only [ZMod.valMinAbs_add_of_four_lt ha' hb',\n         ZMod.valMinAbs_neg_of_ne_half hbne,  ZMod.valMinAbs_neg_of_ne_half hane,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME, neg_add] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedWad.toRat, SierraBool_toBool_inr, Wad.toRat, Wad.toZMod,\n        ite_true, add_div, neg_div, and_true]\n      rw [← neg_div, ← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      rw [← neg_add, abs_neg, ← Nat.cast_add, Nat.abs_cast, Nat.cast_lt]\n      rwa [← neg_add, ← Nat.cast_add, Int.natAbs_neg, Int.natAbs_ofNat] at h₁\n  · simp only [Sum.isLeft_inr, Sum.getLeft?_inr, Option.get!_none, false_and, Sum.isRight_inr,\n      ite_prop_iff_or, and_false, not_lt, and_true, false_or] at h₁ ⊢\n    rcases sa with (sa|sa) <;> cases sa <;> rcases sb with (sb|sb) <;> cases sb\n      <;> simp only [SierraBool_toBool_inl, SierraBool_toBool_inr, ite_false, ite_true,\n        ZMod.valMinAbs_add_of_four_lt ha hb, ZMod.valMinAbs_add_of_four_lt ha' hb,\n        ZMod.valMinAbs_add_of_four_lt ha hb', ZMod.valMinAbs_add_of_four_lt ha' hb',\n        ZMod.valMinAbs_neg_of_ne_half hane, ZMod.valMinAbs_neg_of_ne_half hbne,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME,\n        ← Nat.cast_le (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁\n      <;> simpa [SignedWad.toRat, Wad.toRat, Wad.toZMod, ← neg_div, ← add_div, abs_div,\n        div_le_div_right hS]\n\naegis_spec \"wadray::wadray_signed::SignedWadAddEq::add_eq\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad × Unit  ⊕ _) =>\n  if |a.toRat + b.toRat| < U128_MOD / Wad.WAD_SCALE\n  then ρ.isLeft ∧ ρ.getLeft?.get!.1.toRat = a.toRat + b.toRat\n  else ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedWadAddEq::add_eq\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad × Unit  ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedWadAddEq::add_eq»\n  aesop\n\naegis_spec \"wadray::wadray_signed::SignedRayAdd::add\" :=\n  fun _ _ (a b : SignedRay) _ (ρ : SignedRay ⊕ _) =>\n  if |a.toRat + b.toRat| < U128_MOD / Ray.RAY_SCALE\n  then ρ.isLeft ∧ ρ.getLeft?.get!.toRat = a.toRat + b.toRat\n  else ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedRayAdd::add\" :=\n  fun _ _ (a b : SignedRay) _ (ρ : SignedRay ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedRayAdd::add»\n  rcases a with ⟨va,sa⟩\n  rcases b with ⟨vb,sb⟩\n  have hS : 0 < (Ray.RAY_SCALE : ℚ) := by\n    norm_num [Ray.RAY_SCALE]\n  have hane : 2 * (va.cast : ZMod PRIME).val ≠ PRIME := by\n    apply ne_of_lt (lt_trans (Nat.mul_lt_mul_of_pos_left (ZMod.val_cast_lt PRIME va) two_pos)\n      two_U128_MOD_lt_PRIME)\n  have hbne : 2 * (vb.cast : ZMod PRIME).val ≠ PRIME := by\n    apply ne_of_lt (lt_trans (Nat.mul_lt_mul_of_pos_left (ZMod.val_cast_lt PRIME vb) two_pos)\n      two_U128_MOD_lt_PRIME)\n  have ha : 4 * (va.cast : F).valMinAbs.natAbs < PRIME := add_aux1 va\n  have hb : 4 * (vb.cast : F).valMinAbs.natAbs < PRIME := add_aux1 vb\n  have ha' : 4 * (-(va.cast : F)).valMinAbs.natAbs < PRIME := by\n    rwa [ZMod.valMinAbs_neg_of_ne_half hane, Int.natAbs_neg]\n  have hb' : 4 * (-(vb.cast : F)).valMinAbs.natAbs < PRIME := by\n    rwa [ZMod.valMinAbs_neg_of_ne_half hbne, Int.natAbs_neg]\n  rintro ⟨x,y,z,h₁,(⟨rfl,rfl⟩|⟨rfl,rfl⟩)⟩ <;> dsimp only at h₁\n  · simp only [Sum.isLeft_inl, Sum.getLeft?_inl, Option.get!_some, true_and, Sum.isRight_inl,\n      ite_prop_iff_or, not_lt, and_false, or_false] at h₁ ⊢\n    rcases sa with (sa|sa) <;> cases sa <;> rcases sb with (sb|sb) <;> cases sb\n      <;> simp only [SierraBool_toBool_inl, SierraBool_toBool_inr, ite_false, ite_true] at h₁\n      <;> rcases h₁ with ⟨h₁,h₂⟩\n    · simp only [ZMod.valMinAbs_add_of_four_lt ha hb, Int.natAbs_ofNat,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME, ← Nat.cast_add] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedRay.toRat, SierraBool_toBool_inl, Ray.toRat, Ray.toZMod,\n        ite_false, and_true, add_div]\n      rw [← add_div, ← Nat.cast_add, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      rwa [Nat.cast_lt]\n    · simp only [ZMod.valMinAbs_add_of_four_lt ha hb', ZMod.valMinAbs_neg_of_ne_half hbne,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedRay.toRat, SierraBool_toBool_inl, Ray.toRat, Ray.toZMod,\n        ite_false, SierraBool_toBool_inr, ite_true, add_div, neg_div, and_true]\n      rw [← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      simp only [← Nat.cast_lt (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁ ⊢\n      assumption\n    · simp only [ZMod.valMinAbs_add_of_four_lt ha' hb, ZMod.valMinAbs_neg_of_ne_half hane,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedRay.toRat, SierraBool_toBool_inl, Ray.toRat, Ray.toZMod,\n        ite_false, SierraBool_toBool_inr, ite_true, add_div, neg_div, and_true]\n      rw [← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      simp only [← Nat.cast_lt (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁ ⊢\n      assumption\n    · simp only [ZMod.valMinAbs_add_of_four_lt ha' hb',\n         ZMod.valMinAbs_neg_of_ne_half hbne,  ZMod.valMinAbs_neg_of_ne_half hane,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME, neg_add] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedRay.toRat, SierraBool_toBool_inr, Ray.toRat, Ray.toZMod,\n        ite_true, add_div, neg_div, and_true]\n      rw [← neg_div, ← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      rw [← neg_add, abs_neg, ← Nat.cast_add, Nat.abs_cast, Nat.cast_lt]\n      rwa [← neg_add, ← Nat.cast_add, Int.natAbs_neg, Int.natAbs_ofNat] at h₁\n  · simp only [Sum.isLeft_inr, Sum.getLeft?_inr, Option.get!_none, false_and, Sum.isRight_inr,\n      ite_prop_iff_or, and_false, not_lt, and_true, false_or] at h₁ ⊢\n    rcases sa with (sa|sa) <;> cases sa <;> rcases sb with (sb|sb) <;> cases sb\n      <;> simp only [SierraBool_toBool_inl, SierraBool_toBool_inr, ite_false, ite_true,\n        ZMod.valMinAbs_add_of_four_lt ha hb, ZMod.valMinAbs_add_of_four_lt ha' hb,\n        ZMod.valMinAbs_add_of_four_lt ha hb', ZMod.valMinAbs_add_of_four_lt ha' hb',\n        ZMod.valMinAbs_neg_of_ne_half hane, ZMod.valMinAbs_neg_of_ne_half hbne,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME,\n        ← Nat.cast_le (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁\n      <;> simpa [SignedRay.toRat, Ray.toRat, Ray.toZMod, ← neg_div, ← add_div, abs_div,\n        div_le_div_right hS]\n\naegis_spec \"wadray::wadray_signed::SignedWadSub::sub\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad ⊕ _) =>\n  if |a.toRat - b.toRat| < U128_MOD / Wad.WAD_SCALE\n  then ρ.isLeft ∧ ρ.getLeft?.get!.toRat = a.toRat - b.toRat\n  else ρ.isRight\n\n\naegis_prove \"wadray::wadray_signed::SignedWadSub::sub\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedWadSub::sub»\n  rcases a with ⟨va,sa⟩\n  rcases b with ⟨vb,sb⟩\n  have hS : 0 < (Wad.WAD_SCALE : ℚ) := by norm_num [Wad.WAD_SCALE]\n  have hane : 2 * (va.cast : ZMod PRIME).val ≠ PRIME := by\n    apply ne_of_lt (lt_trans (Nat.mul_lt_mul_of_pos_left (ZMod.val_cast_lt PRIME va) two_pos)\n      two_U128_MOD_lt_PRIME)\n  have hbne : 2 * (vb.cast : ZMod PRIME).val ≠ PRIME := by\n    apply ne_of_lt (lt_trans (Nat.mul_lt_mul_of_pos_left (ZMod.val_cast_lt PRIME vb) two_pos)\n      two_U128_MOD_lt_PRIME)\n  have ha : 4 * (va.cast : F).valMinAbs.natAbs < PRIME := add_aux1 va\n  have hb : 4 * (vb.cast : F).valMinAbs.natAbs < PRIME := add_aux1 vb\n  have ha' : 4 * (- (va.cast : F)).valMinAbs.natAbs < PRIME := by\n    rwa [ZMod.valMinAbs_neg_of_ne_half hane, Int.natAbs_neg]\n  have hb' : 4 * (- (vb.cast : F)).valMinAbs.natAbs < PRIME := by\n    rwa [ZMod.valMinAbs_neg_of_ne_half hbne, Int.natAbs_neg]\n  rintro ⟨x,y,z,h₁,(⟨rfl,rfl⟩|⟨rfl,rfl⟩)⟩ <;> dsimp only at h₁\n  · simp only [Sum.isLeft_inl, Sum.getLeft?_inl, Option.get!_some, true_and, Sum.isRight_inl,\n      ite_prop_iff_or, not_lt, and_false, or_false] at h₁ ⊢\n    rcases sa with (sa|sa) <;> cases sa <;> rcases sb with (sb|sb) <;> cases sb\n      <;> simp only [SierraBool_toBool_inl, SierraBool_toBool_inr, ite_false, ite_true] at h₁\n      <;> rcases h₁ with ⟨h₁,h₂⟩\n    · rw [sub_eq_add_neg] at h₁ h₂ ⊢\n      simp only [ZMod.valMinAbs_add_of_four_lt ha hb', ZMod.valMinAbs_neg_of_ne_half hbne,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedWad.toRat, SierraBool_toBool_inl, Wad.toRat, Wad.toZMod,\n        ite_false, SierraBool_toBool_inr, ite_true, add_div, neg_div, and_true]\n      rw [← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      simp only [← Nat.cast_lt (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁ ⊢\n      assumption\n    · rw [sub_neg_eq_add] at h₁ h₂; rw [sub_eq_add_neg]\n      simp only [ZMod.valMinAbs_add_of_four_lt ha hb, Int.natAbs_ofNat,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME, ← Nat.cast_add] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedWad.toRat, SierraBool_toBool_inl, Wad.toRat, Wad.toZMod,\n        ite_false, SierraBool_toBool_inr, ite_true, neg_neg, add_div, and_true]\n      rw [← add_div, ← Nat.cast_add, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      rwa [Nat.cast_lt]\n    · rw [sub_eq_add_neg] at h₁ h₂ ⊢\n      simp only [ZMod.valMinAbs_add_of_four_lt ha' hb',\n         ZMod.valMinAbs_neg_of_ne_half hbne,  ZMod.valMinAbs_neg_of_ne_half hane,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME, neg_add] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedWad.toRat, SierraBool_toBool_inr, Wad.toRat, Wad.toZMod,\n        ite_true, SierraBool_toBool_inl, ite_false, add_div, neg_div, and_true]\n      rw [← neg_div, ← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      rw [← neg_add, abs_neg, ← Nat.cast_add, Nat.abs_cast, Nat.cast_lt]\n      rwa [← neg_add, ← Nat.cast_add, Int.natAbs_neg, Int.natAbs_ofNat] at h₁\n    · rw [sub_neg_eq_add] at h₁ h₂; rw [sub_eq_add_neg]\n      simp only [ZMod.valMinAbs_add_of_four_lt ha' hb, ZMod.valMinAbs_neg_of_ne_half hane,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedWad.toRat, SierraBool_toBool_inr, Wad.toRat, Wad.toZMod,\n        ite_true, neg_neg, add_div, neg_div, and_true]\n      rw [← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      simp only [← Nat.cast_lt (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁ ⊢\n      assumption\n  · simp only [Sum.isLeft_inr, Sum.getLeft?_inr, Option.get!_none, false_and, Sum.isRight_inr,\n      ite_prop_iff_or, and_false, not_lt, and_true, false_or] at h₁ ⊢\n    rw [sub_eq_add_neg] at h₁ ⊢\n    rcases sa with (sa|sa) <;> cases sa <;> rcases sb with (sb|sb) <;> cases sb\n      <;> simp only [SierraBool_toBool_inl, SierraBool_toBool_inr, ite_false, ite_true,\n        ZMod.valMinAbs_add_of_four_lt ha hb, ZMod.valMinAbs_add_of_four_lt ha' hb,\n        ZMod.valMinAbs_add_of_four_lt ha hb', ZMod.valMinAbs_add_of_four_lt ha' hb',\n        ZMod.valMinAbs_neg_of_ne_half hane, ZMod.valMinAbs_neg_of_ne_half hbne,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME,\n        ← Nat.cast_le (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg, neg_neg] at h₁\n      <;> simpa [SignedWad.toRat, Wad.toRat, Wad.toZMod, ← neg_div, ← add_div, abs_div,\n        div_le_div_right hS]\n\naegis_spec \"wadray::wadray_signed::SignedWadSubEq::sub_eq\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad × Unit ⊕ _) =>\n  if |a.toRat - b.toRat| < U128_MOD / Wad.WAD_SCALE\n  then ρ.isLeft ∧ ρ.getLeft?.get!.1.toRat = a.toRat - b.toRat\n  else ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedWadSubEq::sub_eq\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad × Unit ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedWadSubEq::sub_eq»\n  aesop\n\naegis_spec \"wadray::wadray_signed::SignedRaySub::sub\" :=\n  fun _ _ (a b : SignedRay) _ (ρ : SignedRay ⊕ _) =>\n  if |a.toRat - b.toRat| < U128_MOD / Ray.RAY_SCALE\n  then ρ.isLeft ∧ ρ.getLeft?.get!.toRat = a.toRat - b.toRat\n  else ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedRaySub::sub\" :=\n  fun _ _ (a b : SignedRay) _ (ρ : SignedRay ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedRaySub::sub»\n  rcases a with ⟨va,sa⟩\n  rcases b with ⟨vb,sb⟩\n  have hS : 0 < (Ray.RAY_SCALE : ℚ) := by\n    norm_num [Ray.RAY_SCALE]\n  have hane : 2 * (va.cast : ZMod PRIME).val ≠ PRIME := by\n    apply ne_of_lt (lt_trans (Nat.mul_lt_mul_of_pos_left (ZMod.val_cast_lt PRIME va) two_pos)\n      two_U128_MOD_lt_PRIME)\n  have hbne : 2 * (vb.cast : ZMod PRIME).val ≠ PRIME := by\n    apply ne_of_lt (lt_trans (Nat.mul_lt_mul_of_pos_left (ZMod.val_cast_lt PRIME vb) two_pos)\n      two_U128_MOD_lt_PRIME)\n  have ha : 4 * (va.cast : F).valMinAbs.natAbs < PRIME := add_aux1 va\n  have hb : 4 * (vb.cast : F).valMinAbs.natAbs < PRIME := add_aux1 vb\n  have ha' : 4 * (- (va.cast : F)).valMinAbs.natAbs < PRIME := by\n    rwa [ZMod.valMinAbs_neg_of_ne_half hane, Int.natAbs_neg]\n  have hb' : 4 * (- (vb.cast : F)).valMinAbs.natAbs < PRIME := by\n    rwa [ZMod.valMinAbs_neg_of_ne_half hbne, Int.natAbs_neg]\n  rintro ⟨x,y,z,h₁,(⟨rfl,rfl⟩|⟨rfl,rfl⟩)⟩ <;> dsimp only at h₁\n  · simp only [Sum.isLeft_inl, Sum.getLeft?_inl, Option.get!_some, true_and, Sum.isRight_inl,\n      ite_prop_iff_or, not_lt, and_false, or_false] at h₁ ⊢\n    rcases sa with (sa|sa) <;> cases sa <;> rcases sb with (sb|sb) <;> cases sb\n      <;> simp only [SierraBool_toBool_inl, SierraBool_toBool_inr, ite_false, ite_true] at h₁\n      <;> rcases h₁ with ⟨h₁,h₂⟩\n    · rw [sub_eq_add_neg] at h₁ h₂ ⊢\n      simp only [ZMod.valMinAbs_add_of_four_lt ha hb', ZMod.valMinAbs_neg_of_ne_half hbne,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedRay.toRat, SierraBool_toBool_inl, Ray.toRat, Ray.toZMod,\n        ite_false, SierraBool_toBool_inr, ite_true, add_div, neg_div, and_true]\n      rw [← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      simp only [← Nat.cast_lt (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁ ⊢\n      assumption\n    · rw [sub_neg_eq_add] at h₁ h₂; rw [sub_eq_add_neg]\n      simp only [ZMod.valMinAbs_add_of_four_lt ha hb, Int.natAbs_ofNat,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME, ← Nat.cast_add] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedRay.toRat, SierraBool_toBool_inl, Ray.toRat, Ray.toZMod,\n        ite_false, SierraBool_toBool_inr, ite_true, neg_neg, add_div, and_true]\n      rw [← add_div, ← Nat.cast_add, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      rwa [Nat.cast_lt]\n    · rw [sub_eq_add_neg] at h₁ h₂ ⊢\n      simp only [ZMod.valMinAbs_add_of_four_lt ha' hb',\n         ZMod.valMinAbs_neg_of_ne_half hbne,  ZMod.valMinAbs_neg_of_ne_half hane,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME, neg_add] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedRay.toRat, SierraBool_toBool_inr, Ray.toRat, Ray.toZMod,\n        ite_true, SierraBool_toBool_inl, ite_false, add_div, neg_div, and_true]\n      rw [← neg_div, ← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      rw [← neg_add, abs_neg, ← Nat.cast_add, Nat.abs_cast, Nat.cast_lt]\n      rwa [← neg_add, ← Nat.cast_add, Int.natAbs_neg, Int.natAbs_ofNat] at h₁\n    · rw [sub_neg_eq_add] at h₁ h₂; rw [sub_eq_add_neg]\n      simp only [ZMod.valMinAbs_add_of_four_lt ha' hb, ZMod.valMinAbs_neg_of_ne_half hane,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME] at h₁ h₂\n      push_cast at h₂; rw [h₂]\n      simp only [SignedRay.toRat, SierraBool_toBool_inr, Ray.toRat, Ray.toZMod,\n        ite_true, neg_neg, add_div, neg_div, and_true]\n      rw [← neg_div, ← add_div, abs_div, Nat.abs_cast]\n      apply div_lt_div_of_pos_right _ hS\n      simp only [← Nat.cast_lt (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg] at h₁ ⊢\n      assumption\n  · simp only [Sum.isLeft_inr, Sum.getLeft?_inr, Option.get!_none, false_and, Sum.isRight_inr,\n      ite_prop_iff_or, and_false, not_lt, and_true, false_or] at h₁ ⊢\n    rw [sub_eq_add_neg] at h₁ ⊢\n    rcases sa with (sa|sa) <;> cases sa <;> rcases sb with (sb|sb) <;> cases sb\n      <;> simp only [SierraBool_toBool_inl, SierraBool_toBool_inr, ite_false, ite_true,\n        ZMod.valMinAbs_add_of_four_lt ha hb, ZMod.valMinAbs_add_of_four_lt ha' hb,\n        ZMod.valMinAbs_add_of_four_lt ha hb', ZMod.valMinAbs_add_of_four_lt ha' hb',\n        ZMod.valMinAbs_neg_of_ne_half hane, ZMod.valMinAbs_neg_of_ne_half hbne,\n        ZMod.valMinAbs_cast_of_lt_half two_U128_MOD_lt_PRIME,\n        ← Nat.cast_le (α := ℚ), Int.cast_natAbs, Int.cast_abs, Int.cast_add,\n        ZMod.natCast_val, ZMod.intCast_cast, Int.cast_neg, neg_neg] at h₁\n      <;> simpa [SignedRay.toRat, Ray.toRat, Ray.toZMod, ← neg_div, ← add_div, abs_div,\n        div_le_div_right hS]\n\naegis_spec \"wadray::wadray_signed::SignedWadMul::mul\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad ⊕ _) =>\n  a.1.val * b.1.val / Wad.WAD_SCALE < U128_MOD ∧ ρ = .inl (a * b)\n  ∨ U128_MOD ≤ a.1.val * b.1.val / Wad.WAD_SCALE ∧ ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedWadMul::mul\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedWadMul::mul»\n  rintro ⟨va,sa,vb,sb,_,_,_,rfl,rfl,h₁,h₂⟩\n  rcases h₁ with (⟨h₁,rfl⟩|⟨h₁,h₁'⟩)\n  · simp only [Sum.inl.injEq, false_and, or_false] at h₂\n    rcases h₂ with ⟨rfl,rfl⟩\n    simp [h₁, SignedWad.mul_def, Wad.mul, Wad.toZMod]\n  · rcases h₂ with (⟨rfl,rfl⟩|⟨rfl,rfl⟩)\n    · simp at h₁'\n    · simp [h₁]\n\naegis_spec \"wadray::wadray_signed::SignedRayMul::mul\" :=\n  fun _ _ (a b : SignedRay) _ (ρ : SignedRay ⊕ _) =>\n  a.1.val * b.1.val / Ray.RAY_SCALE < U128_MOD ∧ ρ = .inl (a * b)\n  ∨ U128_MOD ≤ a.1.val * b.1.val / Ray.RAY_SCALE ∧ ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedRayMul::mul\" :=\n  fun _ _ (a b : SignedRay) _ (ρ : SignedRay ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedRayMul::mul»\n  rintro ⟨va,sa,vb,sb,_,_,_,rfl,rfl,h₁,h₂⟩\n  rcases h₁ with (⟨h₁,rfl⟩|⟨h₁,h₁'⟩)\n  · simp only [Sum.inl.injEq, false_and, or_false] at h₂\n    rcases h₂ with ⟨rfl,rfl⟩\n    simp [h₁, SignedRay.mul_def, Ray.mul, Ray.toZMod]\n  · rcases h₂ with (⟨rfl,rfl⟩|⟨rfl,rfl⟩)\n    · simp at h₁'\n    · simp [h₁]\n\naegis_spec \"wadray::wadray_signed::SignedWadMulEq::mul_eq\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad × Unit ⊕ _) =>\n  a.1.val * b.1.val / Wad.WAD_SCALE < U128_MOD ∧ ρ = .inl (a * b, ())\n  ∨ U128_MOD ≤ a.1.val * b.1.val / Wad.WAD_SCALE ∧ ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedWadMulEq::mul_eq\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad × Unit ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedWadMulEq::mul_eq»\n  rename_i x x_1 x_2\n  intro h_auto\n  aesop\n\naegis_spec \"wadray::wadray_signed::SignedWadDiv::div\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad ⊕ _) =>\n  a.1.val * Wad.WAD_SCALE / b.1.val < U128_MOD ∧ b.1.val ≠ 0\n    ∧ ρ = .inl (a / b)\n  ∨ (U128_MOD ≤ a.1.val * Wad.WAD_SCALE / b.1.val ∨ b.1.val = 0)\n    ∧ ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedWadDiv::div\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedWadDiv::div»\n  rintro ⟨va,sa,vb,sb,_,_,_,rfl,rfl,h₁,h₂⟩\n  rcases h₁ with (⟨h₁,hne,rfl⟩|⟨h₁,h₁'⟩)\n  · simp only [Sum.inl.injEq, false_and, or_false] at h₂\n    rcases h₂ with ⟨rfl,rfl⟩\n    simp_all [h₁, Wad.toZMod, SignedWad.div_def, Wad.div, Wad.div_def]\n  · rcases h₂ with (⟨rfl,rfl⟩|⟨rfl,rfl⟩)\n    · simp at h₁'\n    · simp_all [Wad.toZMod]\n\naegis_spec \"wadray::wadray_signed::SignedRayDiv::div\" :=\n  fun _ _ (a b : SignedRay) _ (ρ : SignedRay ⊕ _) =>\n  a.1.val * Ray.RAY_SCALE / b.1.val < U128_MOD ∧ b.1.val ≠ 0\n    ∧ ρ = .inl (a / b)\n  ∨ (U128_MOD ≤ a.1.val * Ray.RAY_SCALE / b.1.val ∨ b.1.val = 0)\n    ∧ ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedRayDiv::div\" :=\n  fun _ _ (a b : SignedRay) _ (ρ : SignedRay ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedRayDiv::div»\n  rintro ⟨va,sa,vb,sb,_,_,_,rfl,rfl,h₁,h₂⟩\n  rcases h₁ with (⟨h₁,hne,rfl⟩|⟨h₁,h₁'⟩)\n  · simp only [Sum.inl.injEq, false_and, or_false] at h₂\n    rcases h₂ with ⟨rfl,rfl⟩\n    simp_all [h₁, Ray.toZMod, SignedRay.div_def, Ray.div, Ray.div_def]\n  · rcases h₂ with (⟨rfl,rfl⟩|⟨rfl,rfl⟩)\n    · simp at h₁'\n    · simp_all [Ray.toZMod]\n\naegis_spec \"wadray::wadray_signed::SignedWadDivEq::div_eq\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad × Unit ⊕ _) =>\n  a.1.val * Wad.WAD_SCALE / b.1.val < U128_MOD ∧ b.1.val ≠ 0\n    ∧ ρ = .inl (a / b, ())\n  ∨ (U128_MOD ≤ a.1.val * Wad.WAD_SCALE / b.1.val ∨ b.1.val = 0)\n    ∧ ρ.isRight\n\naegis_prove \"wadray::wadray_signed::SignedWadDivEq::div_eq\" :=\n  fun _ _ (a b : SignedWad) _ (ρ : SignedWad × Unit ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::SignedWadDivEq::div_eq»\n  rename_i x x_1 x_2\n  intro h_auto\n  aesop\n\naegis_spec \"wadray::wadray_signed::U128IntoSignedWad::into\" :=\n  fun _ a (ρ : SignedWad) =>\n  ρ = ⟨a, Bool.toSierraBool .false⟩\n\naegis_prove \"wadray::wadray_signed::U128IntoSignedWad::into\" :=\n  fun _ a (ρ : SignedWad) => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::U128IntoSignedRay::into\" :=\n  fun _ a (ρ : SignedRay) =>\n  ρ = ⟨a, Bool.toSierraBool .false⟩\n\naegis_prove \"wadray::wadray_signed::U128IntoSignedRay::into\" :=\n  fun _ a (ρ : SignedRay) => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::WadIntoSignedWad::into\" :=\n  fun _ (a : Wad) (ρ : SignedWad) =>\n  ρ = ⟨a, Bool.toSierraBool .false⟩\n\naegis_prove \"wadray::wadray_signed::WadIntoSignedWad::into\" :=\n  fun _ (a : Wad) (ρ : SignedWad) => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::RayIntoSignedRay::into\" :=\n  fun _ (a : Ray) (ρ : SignedRay) =>\n  ρ = ⟨a, Bool.toSierraBool .false⟩\n\naegis_prove \"wadray::wadray_signed::RayIntoSignedRay::into\" :=\n  fun _ (a : Ray) (ρ : SignedRay) => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray_signed::WadIntoSignedRay::into\" :=\n  fun _ _ (a : Wad) _ (ρ : SignedRay ⊕ _) =>\n  if a.toZMod.val * Ray.DIFF < U128_MOD\n  then ρ = .inl ⟨a.toZMod * Ray.DIFF, Bool.toSierraBool .false⟩\n  else ρ.isRight\n\naegis_prove \"wadray::wadray_signed::WadIntoSignedRay::into\" :=\n  fun _ _ (a : Wad) _ (ρ : SignedRay ⊕ _) => by\n  unfold «spec_wadray::wadray_signed::WadIntoSignedRay::into»\n  rintro ⟨_,_,_,h₁,h₂⟩\n  aesop", "nesting_depth": 3, "transitive_dep_count": 104, "subset_aristotle": false, "category": "Applied verif."}
{"id": 482, "thm_name": "RAY_SCALE_val", "thm_stmt": "theorem RAY_SCALE_val :\n    (1000000000000000000000000000 : UInt128).val = 1000000000000000000000000000 := rfl\n\naegis_spec \"wadray::wadray::u128_rmul\"", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Wadray.lean", "imports": ["import WadrayVerification.Load", "import WadrayVerification.Aux", "import CorelibVerification"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "SierraBool.toBool", "module": "Aegis.Aux.Bool"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Bool.toSierraBool", "module": "Aegis.Aux.Bool"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "List", "module": "Init.Prelude"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Prod", "module": "Init.Prelude"}, {"name": "Prod.mk", "module": "Init.Prelude"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Sierra.U256_MOD", "module": "CorelibVerification.Aux.UInt256"}, {"name": "Sierra.aegis_prove", "module": "Aegis.Commands"}, {"name": "Sum", "module": "Init.Core"}, {"name": "Sum.inl", "module": "Init.Core"}, {"name": "Sum.inr", "module": "Init.Core"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "ZMod.ndiv", "module": "Aegis.Aux.ZMod.DivMod"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "sierra_simp'", "module": "CorelibVerification.Corelib.Integer"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Bool.xor", "module": "Init.Data.Bool"}], "used_repo_defs": [{"name": "RAY_SCALE", "content": "def RAY_SCALE : ℕ := 1000000000000000000000000000"}, {"name": "add", "content": "protected def add : Ray := r.toZMod + r'.toZMod"}, {"name": "toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}, {"name": "WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "toRat", "content": "protected def toRat : ℚ := r.toZMod.val / RAY_SCALE"}, {"name": "Wad.toRay", "content": "def Wad.toRay (w : Wad) : Ray := w.toZMod * (Ray.DIFF : UInt128)"}, {"name": "DIFF", "content": "def DIFF : ℕ := 1000000000"}, {"name": "toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "toRat", "content": "def toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1"}, {"name": "sub", "content": "protected def sub : Ray := r.toZMod - r'.toZMod"}, {"name": "ofZMod", "content": "protected def ofZMod (a : UInt128) : Ray := a"}, {"name": "Wad.MAX_CONVERTIBLE_WAD", "content": "def Wad.MAX_CONVERTIBLE_WAD : ℕ := 340282366920938463463374607431"}, {"name": "ofZMod", "content": "protected def ofZMod (a : UInt128) : Wad := a"}, {"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "toWad", "content": "def toWad : Wad := r.toZMod.ndiv DIFF"}, {"name": "toRat", "content": "def toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1"}, {"name": "SignedWad", "content": "def SignedWad := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedRay", "content": "def SignedRay := UInt128 × (Unit ⊕ Unit)"}, {"name": "mul", "content": "protected def mul : Ray := (r.toZMod.val * r'.toZMod.val / RAY_SCALE  : UInt128)"}, {"name": "mul", "content": "protected def mul : Wad := (w.toZMod.val * w'.toZMod.val / WAD_SCALE : UInt128)"}, {"name": "div", "content": "protected def div : Ray := (r.toZMod.val * RAY_SCALE / r'.toZMod.val : UInt128)"}, {"name": "div", "content": "protected def div : Wad := (w.toZMod.val * WAD_SCALE / w'.toZMod.val : UInt128)"}], "lib_lemmas": [{"name": "Aesop.BuiltinRules.not_intro", "module": "Aesop.BuiltinRules"}, {"name": "Bool.false_eq_true", "module": "Init.Data.Bool"}, {"name": "Int.cast_ofNat", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Int.ofNat_eq_coe", "module": "Init.Data.Int.Basic"}, {"name": "List.nil_append", "module": "Init.Data.List.Basic"}, {"name": "Nat.cast_ofNat", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.div_le_div_left", "module": "Init.Data.Nat.Div.Lemmas"}, {"name": "Nat.div_le_div_right", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.eq_zero_of_mul_lt_right", "module": "CorelibVerification.Aux.ZMod"}, {"name": "Nat.lt_of_add_lt", "module": "CorelibVerification.Aux.ZMod"}, {"name": "Nat.mul_le_mul_right", "module": "Init.Data.Nat.Basic"}, {"name": "Nat.pos_of_ne_zero", "module": "Init.Data.Nat.Basic"}, {"name": "Sierra.U256_MOD_div", "module": "CorelibVerification.Aux.UInt256"}, {"name": "Sum.isRight_inl", "module": "Init.Data.Sum.Basic"}, {"name": "Sum.isRight_inr", "module": "Init.Data.Sum.Basic"}, {"name": "ZMod.cast_rat_eq_zero_iff", "module": "CorelibVerification.Aux.ZMod"}, {"name": "ZMod.intCast_zmod_eq_zero_iff_dvd", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.natCast_zmod_val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_eq_zero", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_injective", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_mul_val_eq_hmul", "module": "CorelibVerification.Aux.ZMod"}, {"name": "ZMod.val_zero", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "add_zero", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "and_false", "module": "Init.SimpLemmas"}, {"name": "and_true", "module": "Init.SimpLemmas"}, {"name": "exists_and_left", "module": "Init.PropLemmas"}, {"name": "exists_and_right", "module": "Init.PropLemmas"}, {"name": "exists_const", "module": "Init.PropLemmas"}, {"name": "exists_eq_left", "module": "Init.PropLemmas"}, {"name": "exists_eq_left'", "module": "Init.PropLemmas"}, {"name": "exists_false", "module": "Init.PropLemmas"}, {"name": "false_and", "module": "Init.SimpLemmas"}, {"name": "false_or", "module": "Init.SimpLemmas"}, {"name": "le_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "le_trans", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "mul_comm", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "ne_eq", "module": "Init.SimpLemmas"}, {"name": "not_true", "module": "Init.Core"}, {"name": "or_false", "module": "Init.SimpLemmas"}, {"name": "true_and", "module": "Init.SimpLemmas"}, {"name": "zero_add", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "toRat_lt_toRat_of_val_lt_val", "content": "theorem toRat_lt_toRat_of_val_lt_val (h : @ZMod.val U128_MOD r < @ZMod.val U128_MOD r') :\n    r.toRat < r'.toRat"}, {"name": "RAY_SCALE_rat_pos", "content": "theorem RAY_SCALE_rat_pos : 0 < (RAY_SCALE : ℚ)"}, {"name": "toRat_lt_toRat_of_val_lt_val", "content": "theorem toRat_lt_toRat_of_val_lt_val (h : @ZMod.val U128_MOD w < @ZMod.val U128_MOD w') :\n    w.toRat < w'.toRat"}, {"name": "WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}, {"name": "toRat_le_toRat_of_val_le_val", "content": "theorem toRat_le_toRat_of_val_le_val (h : @ZMod.val U128_MOD r ≤ @ZMod.val U128_MOD r') :\n    r.toRat ≤ r'.toRat"}, {"name": "toRat_le_toRat_of_val_le_val", "content": "theorem toRat_le_toRat_of_val_le_val (h : @ZMod.val U128_MOD w ≤ @ZMod.val U128_MOD w') :\n    w.toRat ≤ w'.toRat"}, {"name": "zero_def", "content": "theorem zero_def : (0 : SignedWad) = (0, false.toSierraBool)"}, {"name": "zero_def", "content": "theorem zero_def : (0 : SignedRay) = (0, false.toSierraBool)"}, {"name": "mul_def", "content": "protected theorem mul_def :\n    r * r' = (r.toZMod.val * r'.toZMod.val / RAY_SCALE  : UInt128)"}, {"name": "mul_def", "content": "theorem mul_def :\n    w₁ * w₂ = ⟨Ray.mul w₁.1 w₂.1, Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "mul_def", "content": "protected theorem mul_def :\n    w * w' = (w.toZMod.val * w'.toZMod.val / WAD_SCALE  : UInt128)"}, {"name": "div_def", "content": "protected theorem div_def :\n    r / r' = (r.toZMod.val * RAY_SCALE / r'.toZMod.val : UInt128)"}, {"name": "mul_def", "content": "theorem mul_def :\n    w₁ * w₂ = ⟨Wad.mul (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "div_def", "content": "protected theorem div_def :\n    w / w' = (w.toZMod.val * WAD_SCALE / w'.toZMod.val : UInt128)"}, {"name": "div_def", "content": "theorem div_def :\n    w₁ / w₂ = ⟨Ray.div (w₁.1 : Ray) (w₂.1 : Ray), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "RAY_SCALE_pos", "content": "theorem RAY_SCALE_pos : 0 < RAY_SCALE"}, {"name": "div_def", "content": "theorem div_def :\n    w₁ / w₂ = ⟨Wad.div (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}], "used_local_defs": [], "used_local_lemmas": [{"name": "Bool.toSierraBool_def", "content": "theorem Bool.toSierraBool_def (b : Bool) : b.toSierraBool = if b then .inr () else .inl ()"}], "local_ctx": "import CorelibVerification\n\nimport WadrayVerification.Aux\n\nimport WadrayVerification.Load\n\nopen Sierra", "target_theorem": "theorem RAY_SCALE_val :\n    (1000000000000000000000000000 : UInt128).val = 1000000000000000000000000000 :=", "ground_truth_proof": ":= rfl\n\naegis_spec \"wadray::wadray::u128_rmul\" :=\n  fun _ _ a b _ ρ =>\n  (a.val * b.val / Ray.RAY_SCALE < U128_MOD ∧ ρ = .inl (a.val * b.val / Ray.RAY_SCALE))\n  ∨ (U128_MOD ≤ a.val * b.val / Ray.RAY_SCALE ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::u128_rmul\" :=\n  fun _ _ a b _ ρ => by\n  unfold «spec_wadray::wadray::u128_rmul»\n  sierra_simp'\n  rintro ⟨_,(⟨h₁,rfl⟩|⟨h₁,h₁'⟩),h₂⟩\n  · simp only [UInt256.val, ZMod.val_zero, mul_zero, zero_add] at h₁\n    rcases h₂ with (⟨_,_,h₂,h₃,h₄⟩|h₂)\n    · injection h₂ with h₂; subst h₂\n      rcases h₃ with (⟨_,_,rfl,h₅⟩|⟨h₃,_⟩)\n      · rcases h₄ with (⟨⟨wₗ,wₕ⟩,v,h₄,h₆,h₇⟩|h₄)\n        · injection h₄ with h₄; subst h₄\n          simp [UInt256.val, UInt256.mul_def, ← ZMod.val_mul_val_eq_hmul] at h₅  -- TODO\n          rw [UInt256.val_lt_U128_MOD_iff, UInt256.U128_MOD_le_val_iff] at h₆\n          rcases h₆ with (⟨rfl,rfl⟩|⟨h₆,rfl⟩)\n          · rw [ZMod.val_zero, mul_zero, zero_add] at h₅\n            simp only [Sum.inl.injEq, exists_eq_left, Nat.cast_ofNat, Int.cast_ofNat,\n              List.nil_append, false_and, exists_const, or_false] at h₇; cases h₇\n            erw [← h₅]\n            simp only [wₗ.val_lt, Sum.inl.injEq, true_and, Sum.isRight_inl, and_false, or_false]\n            rw [@ZMod.natCast_zmod_val]\n          · simp only [false_and, exists_const, Nat.cast_ofNat, Int.cast_ofNat, List.nil_append,\n              true_and, false_or] at h₇; cases h₇\n            rw [← UInt256.U128_MOD_le_val_iff] at h₆\n            erw [← h₅]\n            refine .inr ⟨h₆, ?_⟩\n            simp\n        · simp only [false_and, exists_false] at h₄\n      · injection h₃ with h₃\n        rw [ZMod.intCast_zmod_eq_zero_iff_dvd] at h₃\n        norm_num[U128_MOD] at h₃\n    · simp at h₂\n  · simp only [UInt256.val, ZMod.val_zero, mul_zero, zero_add] at h₁\n    right\n    constructor\n    · rw [Nat.le_div_iff_mul_le' Ray.RAY_SCALE_pos]\n      apply le_trans _ h₁\n      norm_num [U128_MOD, Ray.RAY_SCALE, U256_MOD]\n    · aesop\n\naegis_spec \"wadray::wadray::rmul\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) =>\n  (a.toZMod.val * b.toZMod.val / Ray.RAY_SCALE < U128_MOD ∧ ρ = .inl (a * b))\n  ∨ (U128_MOD ≤ a.toZMod.val * b.toZMod.val / Ray.RAY_SCALE ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::rmul\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray::rmul»\n  sierra_simp'\n  aesop\n\naegis_spec \"wadray::wadray::WadSub::sub\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad ⊕ _) =>\n  (b.toRat ≤ a.toRat ∧ ρ = .inl (a - b))\n  ∨ (a.toRat < b.toRat ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::WadSub::sub\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad ⊕ _) => by\n  unfold «spec_wadray::wadray::WadSub::sub»\n  aesop (add safe forward [Wad.toRat_le_toRat_of_val_le_val, Wad.toRat_lt_toRat_of_val_lt_val])\n\naegis_spec \"wadray::wadray::WadAdd::add\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad ⊕ _) =>\n  (a.toZMod.val + b.toZMod.val < U128_MOD ∧ ρ = .inl (a + b))\n  ∨ (U128_MOD ≤ a.toZMod.val + b.toZMod.val ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::WadAdd::add\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad ⊕ _) => by\n  unfold «spec_wadray::wadray::WadAdd::add»\n  aesop\n\naegis_spec \"wadray::wadray::RaySub::sub\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) =>\n  (b.toRat ≤ a.toRat ∧ ρ = .inl (a - b))\n  ∨ (a.toRat < b.toRat ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::RaySub::sub\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray::RaySub::sub»\n  aesop (add safe forward [Ray.toRat_le_toRat_of_val_le_val, Ray.toRat_lt_toRat_of_val_lt_val])\n\naegis_spec \"wadray::wadray::RayAdd::add\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) =>\n  (a.toZMod.val + b.toZMod.val < U128_MOD ∧ ρ = .inl (a + b))\n  ∨ (U128_MOD ≤ a.toZMod.val + b.toZMod.val ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::RayAdd::add\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray::RayAdd::add»\n  aesop\n\naegis_spec \"wadray::wadray::WadZeroable::is_non_zero\" :=\n  fun _ (a : Wad) ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≠ 0)\n\naegis_prove \"wadray::wadray::WadZeroable::is_non_zero\" :=\n  fun _ (a : Wad) ρ => by\n  unfold «spec_wadray::wadray::WadZeroable::is_non_zero»\n  aesop (add simp norm [Wad.toRat, Wad.toZMod, ZMod.cast_rat_eq_zero_iff])\n\naegis_spec \"wadray::wadray::WadZeroable::is_zero\" :=\n  fun _ (a : Wad) ρ =>\n  ρ = Bool.toSierraBool (a.toRat = 0)\n\naegis_prove \"wadray::wadray::WadZeroable::is_zero\" :=\n  fun _ (a : Wad) ρ => by\n  unfold «spec_wadray::wadray::WadZeroable::is_zero»\n  aesop (add simp norm [Bool.toSierraBool_def, Wad.toRat, ZMod.cast_rat_eq_zero_iff, Wad.toZMod])\n\naegis_spec \"wadray::wadray::RayZeroable::is_zero\" :=\n  fun _ (a : Ray) ρ =>\n  ρ = Bool.toSierraBool (a.toRat = 0)\n\naegis_prove \"wadray::wadray::RayZeroable::is_zero\" :=\n  fun _ (a : Ray) ρ => by\n  unfold «spec_wadray::wadray::RayZeroable::is_zero»\n  aesop (add simp norm [Bool.toSierraBool_def, Ray.toRat, ZMod.cast_rat_eq_zero_iff, Ray.toZMod])\n\naegis_spec \"wadray::wadray::RayZeroable::is_non_zero\" :=\n  fun _ (a : Ray) ρ =>\n  ρ = Bool.toSierraBool (a.toRat ≠ 0)\n\naegis_prove \"wadray::wadray::RayZeroable::is_non_zero\" :=\n  fun _ (a : Ray) ρ => by\n  unfold «spec_wadray::wadray::RayZeroable::is_non_zero»\n  aesop (add simp norm [Ray.toRat, Ray.toZMod, ZMod.cast_rat_eq_zero_iff])\n\naegis_spec \"wadray::wadray::BoundedWad::min\" :=\n  fun _ ρ =>\n  ρ = 0\n\naegis_prove \"wadray::wadray::BoundedWad::min\" :=\n  fun _ ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::BoundedRay::min\" :=\n  fun _ ρ =>\n  ρ = 0\n\naegis_prove \"wadray::wadray::BoundedRay::min\" :=\n  fun _ ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::BoundedWad::max\" :=\n  fun _ ρ =>\n  ρ = U128_MOD - 1\n\naegis_prove \"wadray::wadray::BoundedWad::max\" :=\n  fun _ ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::BoundedRay::max\" :=\n  fun _ ρ =>\n  ρ = U128_MOD - 1\n\naegis_prove \"wadray::wadray::BoundedRay::max\" :=\n  fun _ ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::WadAddEq::add_eq\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad × _ ⊕ _) =>\n  (a.toZMod.val + b.toZMod.val < U128_MOD ∧ ρ = .inl (a + b, ()))\n  ∨ (U128_MOD ≤ a.toZMod.val + b.toZMod.val ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::WadAddEq::add_eq\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad × _ ⊕ _) => by\n  unfold «spec_wadray::wadray::WadAddEq::add_eq»\n  aesop\n\naegis_spec \"wadray::wadray::WadSubEq::sub_eq\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad × _ ⊕ _) =>\n  (b.toRat ≤ a.toRat ∧ ρ = .inl (a - b, ()))\n  ∨ (a.toRat < b.toRat ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::WadSubEq::sub_eq\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad × _ ⊕ _) => by\n  unfold «spec_wadray::wadray::WadSubEq::sub_eq»\n  aesop\n\naegis_spec \"wadray::wadray::WadMulEq::mul_eq\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad × _ ⊕ _) =>\n  (a.toZMod.val * b.toZMod.val / Wad.WAD_SCALE < U128_MOD ∧ ρ = .inl (a * b, ()))\n  ∨ (U128_MOD ≤ a.toZMod.val * b.toZMod.val / Wad.WAD_SCALE ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::WadMulEq::mul_eq\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad × _ ⊕ _) => by\n  unfold «spec_wadray::wadray::WadMulEq::mul_eq»\n  aesop\n\naegis_spec \"wadray::wadray::TIntoWad<core::integer::u128, core::traits::TIntoT<core::integer::u128>>::into\" :=\n  fun _ a (ρ : Wad) =>\n  ρ = a\n\naegis_prove \"wadray::wadray::TIntoWad<core::integer::u128, core::traits::TIntoT<core::integer::u128>>::into\" :=\n  fun _ a (ρ : Wad) => by\n  unfold «spec_wadray::wadray::TIntoWad<core::integer::u128, core::traits::TIntoT<core::integer::u128>>::into»\n  aesop\n\naegis_spec \"wadray::wadray::TIntoRay<core::integer::u128, core::traits::TIntoT<core::integer::u128>>::into\" :=\n  fun _ a (ρ : Ray) =>\n  ρ = a\n\naegis_prove \"wadray::wadray::TIntoRay<core::integer::u128, core::traits::TIntoT<core::integer::u128>>::into\" :=\n  fun _ a (ρ : Ray) => by\n  unfold «spec_wadray::wadray::TIntoRay<core::integer::u128, core::traits::TIntoT<core::integer::u128>>::into»\n  aesop\n\naegis_spec \"wadray::wadray::u128_wdiv\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad ⊕ _) =>\n  a.toZMod.val * Wad.WAD_SCALE / b.toZMod.val < U128_MOD\n    ∧ b.toZMod.val ≠ 0 ∧ ρ = .inl (a / b)\n  ∨ (U128_MOD ≤ a.toZMod.val * Wad.WAD_SCALE / b.toZMod.val ∨ b.toZMod.val = 0) ∧ ρ.isRight\n\naegis_prove \"wadray::wadray::u128_wdiv\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad ⊕ _) => by\n  sierra_simp'\n  unfold «spec_wadray::wadray::u128_wdiv»\n  rintro ⟨(x|x),h₁,h₂⟩\n  · simp_all only [Nat.cast_ofNat, Int.cast_ofNat, Sum.inl.injEq, Sum.isRight_inl,\n      Bool.false_eq_true, and_false, or_false, ne_eq, exists_and_left, List.nil_append,\n      exists_and_right, Sum.exists, exists_eq_left, exists_const, false_and, and_true, false_or,\n      true_and, Prod.exists, exists_eq_left', Prod.mk.injEq, Sum.isRight_inr, Sum.inr.injEq,\n      UInt256.val_mul_of_low, ZMod.val_eq_zero]\n    rcases h₁ with ⟨h₁,rfl⟩\n    simp only [UInt256.val, ZMod.val_zero, mul_zero, zero_add] at h₁\n    rcases h₂ with (⟨_,_,⟨h₂,h₄⟩,_,_,⟨rfl,rfl⟩,h₃⟩|⟨h₂,_,_,rfl⟩)\n    · simp [h₄, UInt256.val] at h₃\n      simp only [UInt256.val, ZMod.val_zero, mul_zero, zero_add,\n        UInt256.mul_def, mul_zero, add_zero, zero_mul] at h₄\n      rcases ρ with (ρ|ρ)\n      · simp only [Sum.inl.injEq, Sum.isRight_inl, and_false, or_false] at h₃ ⊢\n        rcases h₃ with ⟨_,⟨h₃,rfl⟩,rfl⟩\n        refine ⟨?_,?_,?_⟩\n        · rwa [h₄] at h₃\n        · simp only [UInt256.zero_def] at h₂\n          simp only [Wad.toZMod]\n          apply Aesop.BuiltinRules.not_intro\n          intro h\n          subst h\n          simp_all only [not_true]\n        · simp only [Wad.div_def, Wad.WAD_SCALE]\n          erw [← h₄]; clear h₄\n          simp [Nat.eq_zero_of_mul_lt_right (Nat.lt_of_add_lt h₃)]\n      · simp only [and_false, exists_false, Sum.inr.injEq, false_or] at h₃\n        simp only [and_false, Sum.isRight_inr, and_true, false_or]\n        left\n        rcases h₃ with ⟨h₃,rfl⟩\n        rwa [h₄] at h₃\n    · simp only [and_false, Sum.isRight_inr, and_true, false_or]\n      right\n      injection h₂\n  · simp_all only [Nat.cast_ofNat, Int.cast_ofNat, and_false, Sum.isRight_inr, and_true, false_or,\n      ne_eq, exists_and_left, List.nil_append, exists_and_right, Sum.exists, Sum.inl.injEq, or_false,\n      exists_eq_left, exists_const, false_and, true_and, Prod.exists, Sum.inr.injEq, ZMod.val_eq_zero]\n    subst h₂\n    simp only [and_false, Sum.isRight_inr, and_true, false_or]\n    by_cases h₃ : b.toZMod.val = 0\n    · right\n      rwa [ZMod.val_eq_zero] at h₃\n    · left\n      conv => lhs; rw [← U256_MOD_div]\n      trans; apply Nat.div_le_div_left (le_of_lt b.toZMod.val_lt) (Nat.pos_of_ne_zero h₃)\n      simp only [UInt256.val, ZMod.val_zero, mul_zero, zero_add] at h₁\n      apply Nat.div_le_div_right h₁\n\naegis_spec \"wadray::wadray::u128_rdiv\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) =>\n  (a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val < U128_MOD ∧ b.toZMod.val ≠ 0 ∧ ρ = .inl (a / b))\n  ∨ ((U128_MOD ≤ a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val ∨ b.toZMod.val = 0) ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::u128_rdiv\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) => by\n  sierra_simp'\n  unfold «spec_wadray::wadray::u128_rdiv»\n  rintro ⟨(x|x),h₁,h₂⟩\n  · simp_all only [Nat.cast_ofNat, Int.cast_ofNat, Sum.inl.injEq, Sum.isRight_inl,\n      Bool.false_eq_true, and_false, or_false, ne_eq, exists_and_left, List.nil_append,\n      exists_and_right, Sum.exists, exists_eq_left, exists_const, false_and, and_true, false_or,\n      true_and, Prod.exists, exists_eq_left', Prod.mk.injEq, Sum.isRight_inr, Sum.inr.injEq,\n      UInt256.val_mul_of_low, ZMod.val_eq_zero]\n    rcases h₁ with ⟨h₁,rfl⟩\n    simp only [UInt256.val, ZMod.val_zero, mul_zero, zero_add] at h₁\n    rcases h₂ with (⟨_,_,⟨h₂,h₄⟩,_,_,⟨rfl,rfl⟩,h₃⟩|⟨h₂,_,_,rfl⟩)\n    · simp [h₄, UInt256.val] at h₃\n      simp only [UInt256.val, ZMod.val_zero, mul_zero, zero_add,\n        UInt256.mul_def, mul_zero, add_zero, zero_mul] at h₄\n      rcases ρ with (ρ|ρ)\n      · simp only [Sum.inl.injEq, Sum.isRight_inl, and_false, or_false] at h₃ ⊢\n        rcases h₃ with ⟨_,⟨h₃,rfl⟩,rfl⟩\n        refine ⟨?_,?_,?_⟩\n        · rwa [h₄] at h₃\n        · simp only [UInt256.zero_def] at h₂\n          simp only [Ray.toZMod]\n          apply Aesop.BuiltinRules.not_intro\n          intro h\n          subst h\n          simp_all only [not_true]\n        · simp only [Ray.div_def, Ray.RAY_SCALE]\n          erw [← h₄]; clear h₄\n          simp [Nat.eq_zero_of_mul_lt_right (Nat.lt_of_add_lt h₃)]\n      · simp only [and_false, exists_false, Sum.inr.injEq, false_or] at h₃\n        simp only [and_false, Sum.isRight_inr, and_true, false_or]\n        left\n        rcases h₃ with ⟨h₃,rfl⟩\n        rwa [h₄] at h₃\n    · simp only [and_false, Sum.isRight_inr, and_true, false_or]\n      right\n      injection h₂\n  · simp_all only [Nat.cast_ofNat, Int.cast_ofNat, and_false, Sum.isRight_inr, and_true, false_or,\n      ne_eq, exists_and_left, List.nil_append, exists_and_right, Sum.exists, Sum.inl.injEq, or_false,\n      exists_eq_left, exists_const, false_and, true_and, Prod.exists, Sum.inr.injEq, ZMod.val_eq_zero]\n    subst h₂\n    simp only [and_false, Sum.isRight_inr, and_true, false_or]\n    by_cases h₃ : b.toZMod.val = 0\n    · right\n      rwa [ZMod.val_eq_zero] at h₃\n    · left\n      conv => lhs; rw [← U256_MOD_div]\n      trans; apply Nat.div_le_div_left (le_of_lt b.toZMod.val_lt) (Nat.pos_of_ne_zero h₃)\n      simp only [UInt256.val, ZMod.val_zero, mul_zero, zero_add] at h₁\n      apply Nat.div_le_div_right h₁\n\naegis_spec \"wadray::wadray::wdiv\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad ⊕ _) =>\n  (a.toZMod.val * Wad.WAD_SCALE / b.toZMod.val < U128_MOD ∧ b.toZMod.val ≠ 0 ∧ ρ = .inl (a / b))\n  ∨ ((U128_MOD ≤ a.toZMod.val * Wad.WAD_SCALE / b.toZMod.val ∨ b.toZMod.val = 0) ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::wdiv\" :=\n  fun _ _ (a b : Wad) _ (ρ : Wad ⊕ _) => by\n  unfold «spec_wadray::wadray::wdiv»\n  aesop\n\naegis_spec \"wadray::wadray::WadDivEq::div_eq\" :=\n  fun _ _ (a b : Wad) _ (ρ : (Wad × _) ⊕ _) =>\n  (a.toZMod.val * Wad.WAD_SCALE / b.toZMod.val < U128_MOD ∧ b.toZMod.val ≠ 0 ∧ ρ = .inl (a / b, ()))\n  ∨ ((U128_MOD ≤ a.toZMod.val * Wad.WAD_SCALE / b.toZMod.val ∨ b.toZMod.val = 0) ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::WadDivEq::div_eq\" :=\n  fun _ _ (a b : Wad) _ (ρ : (Wad × _) ⊕ _) => by\n  unfold «spec_wadray::wadray::WadDivEq::div_eq»\n  aesop\n\naegis_spec \"wadray::wadray::rdiv\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) =>\n  (a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val < U128_MOD ∧ b.toZMod.val ≠ 0 ∧ ρ = .inl (a / b))\n  ∨ ((U128_MOD ≤ a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val ∨ b.toZMod.val = 0) ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::rdiv\" :=\n  fun _ _ (a b : Ray) _ (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray::rdiv»\n  aesop\n\naegis_spec \"wadray::wadray::RayIntoWad::into\" :=\n  fun _ _ (a : Ray) _ ρ =>\n  ρ = .inl a.toWad\n\naegis_prove \"wadray::wadray::RayIntoWad::into\" :=\n  fun _ _ (a : Ray) _ ρ => by\n  unfold «spec_wadray::wadray::RayIntoWad::into»\n  rintro ⟨_,_,_,(h|h),h'⟩\n  · rcases h' with (⟨rfl,rfl⟩|⟨rfl,rfl⟩)\n    · rcases h with ⟨_,ρ',h₁,h₂⟩\n      cases h₁\n      congr\n      simp only [Ray.toWad, ZMod.ndiv, U128_MOD]\n      apply ZMod.val_injective\n      exact h₂\n    · simp at h\n  · rcases h with ⟨h,-⟩\n    rw [ZMod.intCast_zmod_eq_zero_iff_dvd] at h\n    norm_num [U128_MOD] at h\n\naegis_spec \"wadray::wadray::WadIntoU256::into\" :=\n  fun _ a ρ =>\n  ρ = (a, 0)\n\naegis_prove \"wadray::wadray::WadIntoU256::into\" :=\n  fun _ a ρ => by\n  unfold «spec_wadray::wadray::WadIntoU256::into»\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::U256TryIntoWad::try_into\" :=\n  fun _ (a : UInt256) ρ =>\n  (a.val < U128_MOD ∧ ρ = .inl a.1)\n  ∨ (U128_MOD ≤ a.val ∧ ρ = .inr ())\n\naegis_prove \"wadray::wadray::U256TryIntoWad::try_into\" :=\n  fun _ (a : UInt256) ρ => by\n  unfold «spec_wadray::wadray::U256TryIntoWad::try_into»\n  aesop\n\naegis_spec \"wadray::wadray::TIntoRay<core::integer::u64, core::integer::UpcastableInto<core::integer::u64, core::integer::u128, core::integer::UpcastableU64U128>>::into\" :=\n  fun _ a ρ =>\n  ρ = a.cast\n\naegis_prove \"wadray::wadray::TIntoRay<core::integer::u64, core::integer::UpcastableInto<core::integer::u64, core::integer::u128, core::integer::UpcastableU64U128>>::into\" :=\n  fun _ a ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::TIntoRay<core::integer::u32, core::integer::UpcastableInto<core::integer::u32, core::integer::u128, core::integer::UpcastableU32U128>>::into\" :=\n  fun _ a ρ =>\n  ρ = a.cast\n\naegis_prove \"wadray::wadray::TIntoRay<core::integer::u32, core::integer::UpcastableInto<core::integer::u32, core::integer::u128, core::integer::UpcastableU32U128>>::into\" :=\n  fun _ a ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::TIntoRay<core::integer::u16, core::integer::UpcastableInto<core::integer::u16, core::integer::u128, core::integer::UpcastableU16U128>>::into\" :=\n  fun _ a ρ =>\n  ρ = a.cast\n\naegis_prove \"wadray::wadray::TIntoRay<core::integer::u16, core::integer::UpcastableInto<core::integer::u16, core::integer::u128, core::integer::UpcastableU16U128>>::into\" :=\n  fun _ a ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::TIntoRay<core::integer::u8, core::integer::UpcastableInto<core::integer::u8, core::integer::u128, core::integer::UpcastableU8U128>>::into\" :=\n  fun _ a ρ =>\n  ρ = a.cast\n\naegis_prove \"wadray::wadray::TIntoRay<core::integer::u8, core::integer::UpcastableInto<core::integer::u8, core::integer::u128, core::integer::UpcastableU8U128>>::into\" :=\n  fun _ a ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::TIntoWad<core::integer::u64, core::integer::UpcastableInto<core::integer::u64, core::integer::u128, core::integer::UpcastableU64U128>>::into\" :=\n  fun _ a ρ =>\n  ρ = a.cast\n\naegis_prove \"wadray::wadray::TIntoWad<core::integer::u64, core::integer::UpcastableInto<core::integer::u64, core::integer::u128, core::integer::UpcastableU64U128>>::into\" :=\n  fun _ a ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::TIntoWad<core::integer::u32, core::integer::UpcastableInto<core::integer::u32, core::integer::u128, core::integer::UpcastableU32U128>>::into\" :=\n  fun _ a ρ =>\n  ρ = a.cast\n\naegis_prove \"wadray::wadray::TIntoWad<core::integer::u32, core::integer::UpcastableInto<core::integer::u32, core::integer::u128, core::integer::UpcastableU32U128>>::into\" :=\n  fun _ a ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::TIntoWad<core::integer::u16, core::integer::UpcastableInto<core::integer::u16, core::integer::u128, core::integer::UpcastableU16U128>>::into\" :=\n  fun _ a ρ =>\n  ρ = a.cast\n\naegis_prove \"wadray::wadray::TIntoWad<core::integer::u16, core::integer::UpcastableInto<core::integer::u16, core::integer::u128, core::integer::UpcastableU16U128>>::into\" :=\n  fun _ a ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::TIntoWad<core::integer::u8, core::integer::UpcastableInto<core::integer::u8, core::integer::u128, core::integer::UpcastableU8U128>>::into\" :=\n  fun _ a ρ =>\n  ρ = a.cast\n\naegis_prove \"wadray::wadray::TIntoWad<core::integer::u8, core::integer::UpcastableInto<core::integer::u8, core::integer::u128, core::integer::UpcastableU8U128>>::into\" :=\n  fun _ a ρ => by\n  rintro rfl\n  rfl\n\naegis_spec \"wadray::wadray::WadTryIntoRay::try_into\" :=\n  fun _ _ (a : Wad) _ (ρ : (Ray ⊕ _) ⊕ _) =>\n  ρ = if (a.toZMod.val ≤ Wad.MAX_CONVERTIBLE_WAD) then .inl (.inl a.toRay)\n      else .inl (.inr ())\n\naegis_prove \"wadray::wadray::WadTryIntoRay::try_into\" :=\n  fun _ _ (a : Wad) _ (ρ : (Ray ⊕ _) ⊕ _) => by\n  unfold «spec_wadray::wadray::WadTryIntoRay::try_into»\n  have h₁ : ZMod.val (340282366920938463463374607431 : UInt128) = Wad.MAX_CONVERTIBLE_WAD := rfl\n  have h₂ : ZMod.val (1000000000 : UInt128) = 1000000000 := rfl\n  rintro ⟨_,_,_,(h|h)⟩\n  · rcases h with ⟨h₃,(h₄|h₄),h₅⟩\n    · aesop\n    · rcases h₅ with (h₅|⟨rfl,rfl⟩)\n      · aesop\n      · simp only [Int.ofNat_eq_coe, Nat.cast_ofNat, Int.cast_ofNat, Sum.isRight_inr, and_true] at h₃ h₄\n        rw [h₂] at h₄\n        rw [h₁] at h₃\n        replace h₄ := h₄.trans (Nat.mul_le_mul_right 1000000000 h₃)\n        norm_num [U128_MOD, Wad.MAX_CONVERTIBLE_WAD] at h₄\n  · aesop\n\naegis_spec \"wadray::wadray::scale_u128_by_ray\" :=\n  fun _ _ a (b : Ray) _ ρ =>\n  (a.val * b.toZMod.val / Ray.RAY_SCALE < U128_MOD ∧ ρ = .inl (a.val * b.toZMod.val / Ray.RAY_SCALE))\n  ∨ (U128_MOD ≤ a.val * b.toZMod.val / Ray.RAY_SCALE ∧ ρ.isRight)\n\naegis_prove \"wadray::wadray::scale_u128_by_ray\" :=\n  fun _ _ a (b : Ray) _ ρ => by\n  unfold «spec_wadray::wadray::scale_u128_by_ray»\n  aesop\n\naegis_spec \"wadray::wadray::div_u128_by_ray\" :=\n  fun _ _ a (b : Ray) _ ρ =>\n  a.val * Ray.RAY_SCALE / b.toZMod.val < U128_MOD ∧ b.toZMod.val ≠ 0\n    ∧ ρ = .inl (a.val * Ray.RAY_SCALE / b.toZMod.val)\n  ∨ (U128_MOD ≤ a.val * Ray.RAY_SCALE / b.toZMod.val ∨ b.toZMod.val = 0) ∧ ρ.isRight\n\naegis_prove \"wadray::wadray::div_u128_by_ray\" :=\n  fun _ _ a (b : Ray) _ ρ => by\n  unfold «spec_wadray::wadray::div_u128_by_ray»\n  aesop\n\naegis_spec \"wadray::wadray::wmul_wr\" :=\n  fun _ _ (a : Wad) (b : Ray) _ (ρ : Ray ⊕ _) =>\n  a.toZMod.val * b.toZMod.val / Wad.WAD_SCALE < U128_MOD\n    ∧ ρ = .inl (Ray.ofZMod (a.toZMod.val * b.toZMod.val / Wad.WAD_SCALE))\n  ∨ U128_MOD ≤ a.toZMod.val * b.toZMod.val / Wad.WAD_SCALE ∧ ρ.isRight\n\naegis_prove \"wadray::wadray::wmul_wr\" :=\n  fun _ _ (a : Wad) (b : Ray) _ (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray::wmul_wr»\n  aesop\n\naegis_spec \"wadray::wadray::wmul_rw\" :=\n  fun _ _ (a : Ray) (b : Wad) _ (ρ : Ray ⊕ _) =>\n  a.toZMod.val * b.toZMod.val / Wad.WAD_SCALE < U128_MOD\n    ∧ ρ = .inl (Ray.ofZMod (a.toZMod.val * b.toZMod.val / Wad.WAD_SCALE))\n  ∨ U128_MOD ≤ a.toZMod.val * b.toZMod.val / Wad.WAD_SCALE ∧ ρ.isRight\n\naegis_prove \"wadray::wadray::wmul_rw\" :=\n  fun _ _ (a : Ray) (b : Wad) _ (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray::wmul_rw»\n  aesop (add simp [mul_comm])\n\naegis_spec \"wadray::wadray::rmul_rw\" :=\n  fun _ _ (a : Ray) (b : Wad) _ (ρ : Wad ⊕ _) =>\n  a.toZMod.val * b.toZMod.val / Ray.RAY_SCALE < U128_MOD\n    ∧ ρ = .inl (Ray.ofZMod (a.toZMod.val * b.toZMod.val / Ray.RAY_SCALE))\n  ∨ U128_MOD ≤ a.toZMod.val * b.toZMod.val / Ray.RAY_SCALE ∧ ρ.isRight\n\naegis_prove \"wadray::wadray::rmul_rw\" :=\n  fun _ _ (a : Ray) (b : Wad) _ (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray::rmul_rw»\n  aesop (add simp [mul_comm])\n\naegis_spec \"wadray::wadray::rmul_wr\" :=\n  fun _ _ (a : Wad) (b : Ray) _ (ρ : Wad ⊕ _) =>\n  a.toZMod.val * b.toZMod.val / Ray.RAY_SCALE < U128_MOD\n    ∧ ρ = .inl (Ray.ofZMod (a.toZMod.val * b.toZMod.val / Ray.RAY_SCALE))\n  ∨ U128_MOD ≤ a.toZMod.val * b.toZMod.val / Ray.RAY_SCALE ∧ ρ.isRight\n\naegis_prove \"wadray::wadray::rmul_wr\" :=\n  fun _ _ (a : Ray) (b : Wad) _ (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray::rmul_wr»\n  aesop (add simp [mul_comm])\n\naegis_spec \"wadray::wadray::rdiv_wr\" :=\n  fun _ _ (a : Wad) (b : Ray) _ (ρ : Wad ⊕ _) =>\n  a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val < U128_MOD\n    ∧ b.toZMod.val ≠ 0 ∧ ρ = .inl (Wad.ofZMod (a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val))\n  ∨ (U128_MOD ≤ a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val ∨ b.toZMod.val = 0) ∧ ρ.isRight\n\naegis_prove \"wadray::wadray::rdiv_wr\" :=\n  fun _ _ (a : Wad) (b : Ray) _ (ρ : Wad ⊕ _) => by\n  unfold «spec_wadray::wadray::rdiv_wr»\n  aesop\n\naegis_spec \"wadray::wadray::wdiv_rw\" :=\n  fun _ _ (a : Ray) (b : Wad) _ (ρ : Ray ⊕ _) =>\n  a.toZMod.val * Wad.WAD_SCALE / b.toZMod.val < U128_MOD\n    ∧ b.toZMod.val ≠ 0 ∧ ρ = .inl (Ray.ofZMod (a.toZMod.val * Wad.WAD_SCALE / b.toZMod.val))\n  ∨ (U128_MOD ≤ a.toZMod.val * Wad.WAD_SCALE / b.toZMod.val ∨ b.toZMod.val = 0) ∧ ρ.isRight\n\naegis_prove \"wadray::wadray::wdiv_rw\" :=\n  fun _ _ (a : Ray) (b : Wad) _ (ρ : Wad ⊕ _) => by\n  unfold «spec_wadray::wadray::wdiv_rw»\n  aesop\n\naegis_spec \"wadray::wadray::rdiv_ww\" :=\n  fun _ _ (a b : Wad) _ (ρ : Ray ⊕ _) =>\n  a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val < U128_MOD\n    ∧ b.toZMod.val ≠ 0 ∧ ρ = .inl (Wad.ofZMod (a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val))\n  ∨ (U128_MOD ≤ a.toZMod.val * Ray.RAY_SCALE / b.toZMod.val ∨ b.toZMod.val = 0) ∧ ρ.isRight\n\naegis_prove \"wadray::wadray::rdiv_ww\" :=\n  fun _ _ (a b : Wad) _ (ρ : Ray ⊕ _) => by\n  unfold «spec_wadray::wadray::rdiv_ww»\n  aesop", "nesting_depth": 3, "transitive_dep_count": 104, "subset_aristotle": false, "category": "Applied verif."}
{"id": 483, "thm_name": "Wad.toRat_div", "thm_stmt": "theorem toRat_div (h : w.toZMod.val * WAD_SCALE / w'.toZMod.val < U128_MOD)\n    (h' : w'.toZMod.val ≠ 0) :\n    |(w / w').toRat - w.toRat / w'.toRat| < 1 / WAD_SCALE", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}], "used_repo_defs": [{"name": "Wad.div", "content": "protected def Wad.div : Wad := (w.toZMod.val * WAD_SCALE / w'.toZMod.val : UInt128)"}, {"name": "", "content": "instance : Div Wad := ⟨Wad.div⟩"}], "lib_lemmas": [{"name": "Nat.cast_div", "module": "Mathlib.Data.Nat.Cast.Field"}, {"name": "Nat.cast_ne_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_sub", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "sub_div", "module": "Mathlib.Algebra.Field.Basic"}, {"name": "Nat.abs_cast", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_mul", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Nat.cast_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.pos_of_ne_zero", "module": "Init.Data.Nat.Basic"}, {"name": "ZMod.val_natCast", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "abs_div", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "abs_neg", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "div_div", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_div_div_cancel_right", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_lt_one", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "mul_div_mul_right", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "sub_sub_cancel_left", "module": "Mathlib.Algebra.Group.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}], "used_local_lemmas": [{"name": "Rat.nat_cast_div_eq", "content": "theorem Rat.nat_cast_div_eq {a b : ℕ} :\n    ↑(a / b) = (a : ℚ) / (b : ℚ) - ↑(a % b) / (b : ℚ)"}, {"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}, {"name": "Wad.WAD_SCALE_rat_ne_zero", "content": "theorem WAD_SCALE_rat_ne_zero : (WAD_SCALE : ℚ) ≠ 0"}, {"name": "Wad.div_def", "content": "protected theorem div_def :\n    w / w' = (w.toZMod.val * WAD_SCALE / w'.toZMod.val : UInt128)"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE", "target_theorem": "theorem toRat_div (h : w.toZMod.val * WAD_SCALE / w'.toZMod.val < U128_MOD)\n    (h' : w'.toZMod.val ≠ 0) :\n    |(w / w').toRat - w.toRat / w'.toRat| < 1 / WAD_SCALE :=", "ground_truth_proof": ":= by\n  have h'' : 0 < w'.toZMod.val := Nat.pos_of_ne_zero h'\n  have h''' : (0 : ℚ) < w'.toZMod.val := Nat.cast_pos.mpr h''\n  simp only [Wad.toRat, Wad.toZMod, Wad.div_def, ZMod.val_natCast] at *\n  rw [Nat.mod_eq_of_lt h, Rat.nat_cast_div_eq, sub_div, Nat.cast_mul,\n    div_div, mul_div_mul_right _ _ WAD_SCALE_rat_ne_zero,\n    div_div_div_cancel_right _ WAD_SCALE_rat_ne_zero, sub_sub_cancel_left,\n    abs_neg, abs_div, Nat.abs_cast, div_lt_div_right WAD_SCALE_rat_pos,\n    abs_div, Nat.abs_cast, Nat.abs_cast, div_lt_one h''', Nat.cast_lt]\n  apply Nat.mod_lt _ h''", "nesting_depth": 2, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Applied verif."}
{"id": 484, "thm_name": "SignedWad.toRat_mul", "thm_stmt": "theorem toRat_mul (h₁ : w₁.1.val * w₂.1.val / Wad.WAD_SCALE < U128_MOD ):\n    |SignedWad.toRat (w₁ * w₂) - SignedWad.toRat w₁ * SignedWad.toRat w₂| < 1 / Wad.WAD_SCALE", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Bool.toSierraBool", "module": "Aegis.Aux.Bool"}, {"name": "Bool.xor", "module": "Init.Data.Bool"}, {"name": "SierraBool.toBool", "module": "Aegis.Aux.Bool"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_div", "module": "Mathlib.Data.Nat.Cast.Field"}, {"name": "Nat.cast_ne_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_sub", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "sub_div", "module": "Mathlib.Algebra.Field.Basic"}, {"name": "Nat.abs_cast", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_mul", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "ZMod.natCast_val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "abs_div", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "abs_neg", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "div_lt_one", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "div_mul_eq_mul_div", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_div_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "neg_sub", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_add_cancel", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_right_comm", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_self", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}, {"name": "Wad.mul", "content": "protected def mul : Wad := (w.toZMod.val * w'.toZMod.val / WAD_SCALE : UInt128)"}, {"name": "SignedWad", "content": "def SignedWad := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedWad.toRat", "content": "def toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1"}], "used_local_lemmas": [{"name": "Rat.nat_cast_div_eq", "content": "theorem Rat.nat_cast_div_eq {a b : ℕ} :\n    ↑(a / b) = (a : ℚ) / (b : ℚ) - ↑(a % b) / (b : ℚ)"}, {"name": "Wad.WAD_SCALE_pos", "content": "theorem WAD_SCALE_pos : 0 < WAD_SCALE"}, {"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}, {"name": "SignedWad.mul_def", "content": "theorem mul_def :\n    w₁ * w₂ = ⟨Wad.mul (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE\n\nprotected def mul : Wad := (w.toZMod.val * w'.toZMod.val / WAD_SCALE : UInt128)\n\nend Wad\n\nnamespace Ray\n\nvariable (r r' : Ray)\n\nend Ray\n\ndef SignedWad := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedWad\n\ninstance : Inhabited SignedWad := ⟨default, default⟩\n\nvariable (w w₁ w₂ : SignedWad)\n\ndef sign : ℤ := if SierraBool.toBool w.2 then -1 else 1\n\ninstance : Zero SignedWad := ⟨(0, false.toSierraBool)⟩\n\ndef toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1\n\nprotected def mul : SignedWad :=\n⟨Wad.mul (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩\n\ninstance : Mul SignedWad := ⟨SignedWad.mul⟩", "target_theorem": "theorem toRat_mul (h₁ : w₁.1.val * w₂.1.val / Wad.WAD_SCALE < U128_MOD ):\n    |SignedWad.toRat (w₁ * w₂) - SignedWad.toRat w₁ * SignedWad.toRat w₂| < 1 / Wad.WAD_SCALE :=", "ground_truth_proof": ":= by\n  rcases w₁ with ⟨w₁, s₁⟩\n  rcases w₂ with ⟨w₂, s₂⟩\n  rcases s₁ with (⟨⟨⟩⟩|⟨⟨⟩⟩) <;> rcases s₂ with (⟨⟨⟩⟩|⟨⟨⟩⟩)\n    <;> dsimp only at h₁\n    <;> simp [mul_def, toRat, Wad.mul, Wad.toRat, Wad.toZMod, Nat.mod_eq_of_lt h₁, -one_div]\n    <;> rw [Rat.nat_cast_div_eq, Nat.cast_mul, ZMod.natCast_val, ZMod.natCast_val, mul_div_assoc,\n      sub_div, div_mul_eq_mul_div]\n    <;> [rw [sub_right_comm, sub_self, zero_sub, abs_neg];\n       rw [neg_sub, sub_add_cancel];\n       rw [neg_sub, sub_add_cancel];\n       rw [sub_right_comm, sub_self, zero_sub, abs_neg]]\n    <;> rw [abs_div,\n      Nat.abs_cast, div_lt_div_right Wad.WAD_SCALE_rat_pos, abs_div, Nat.abs_cast,\n      Nat.abs_cast, div_lt_one Wad.WAD_SCALE_rat_pos, Nat.cast_lt]\n    <;> apply Nat.mod_lt _ Wad.WAD_SCALE_pos", "nesting_depth": 2, "transitive_dep_count": 43, "subset_aristotle": false, "category": "Applied verif."}
{"id": 485, "thm_name": "SignedRay.toRat_mul", "thm_stmt": "theorem toRat_mul (h₁ : w₁.1.val * w₂.1.val / Ray.RAY_SCALE < U128_MOD ):\n    |SignedRay.toRat (w₁ * w₂) - SignedRay.toRat w₁ * SignedRay.toRat w₂| < 1 / Ray.RAY_SCALE", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "SierraBool.toBool", "module": "Aegis.Aux.Bool"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Bool.toSierraBool", "module": "Aegis.Aux.Bool"}, {"name": "Bool.xor", "module": "Init.Data.Bool"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_div", "module": "Mathlib.Data.Nat.Cast.Field"}, {"name": "Nat.cast_ne_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_sub", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "sub_div", "module": "Mathlib.Algebra.Field.Basic"}, {"name": "Nat.abs_cast", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_mul", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "ZMod.natCast_val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "abs_div", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "abs_neg", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "div_lt_one", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "div_mul_eq_mul_div", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "mul_div_assoc", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "neg_sub", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_add_cancel", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_right_comm", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_self", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.RAY_SCALE", "content": "def RAY_SCALE : ℕ := 1000000000000000000000000000"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "Ray.toRat", "content": "protected def toRat : ℚ := r.toZMod.val / RAY_SCALE"}, {"name": "Ray.mul", "content": "protected def mul : Ray := (r.toZMod.val * r'.toZMod.val / RAY_SCALE  : UInt128)"}, {"name": "SignedRay", "content": "def SignedRay := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedRay.toRat", "content": "def toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1"}], "used_local_lemmas": [{"name": "Rat.nat_cast_div_eq", "content": "theorem Rat.nat_cast_div_eq {a b : ℕ} :\n    ↑(a / b) = (a : ℚ) / (b : ℚ) - ↑(a % b) / (b : ℚ)"}, {"name": "Ray.RAY_SCALE_pos", "content": "theorem RAY_SCALE_pos : 0 < RAY_SCALE"}, {"name": "Ray.RAY_SCALE_rat_pos", "content": "theorem RAY_SCALE_rat_pos : 0 < (RAY_SCALE : ℚ)"}, {"name": "SignedRay.mul_def", "content": "theorem mul_def :\n    w₁ * w₂ = ⟨Ray.mul w₁.1 w₂.1, Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\nnamespace Wad\n\nvariable (w w' : Wad)\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\ndef RAY_SCALE : ℕ := 1000000000000000000000000000\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nprotected def toRat : ℚ := r.toZMod.val / RAY_SCALE\n\nprotected def mul : Ray := (r.toZMod.val * r'.toZMod.val / RAY_SCALE  : UInt128)\n\nend Ray\n\nnamespace SignedWad\n\nvariable (w w₁ w₂ : SignedWad)\n\nend SignedWad\n\ndef SignedRay := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedRay\n\ninstance : Inhabited SignedRay := ⟨default, default⟩\n\nvariable (w w₁ w₂ : SignedRay)\n\ndef sign : ℤ := if SierraBool.toBool w.2 then -1 else 1\n\ninstance : Zero SignedRay := ⟨(0, false.toSierraBool)⟩\n\ndef toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1\n\nprotected def mul : SignedRay :=\n⟨Ray.mul w₁.1 w₂.1, Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩\n\ninstance : Mul SignedRay := ⟨SignedRay.mul⟩", "target_theorem": "theorem toRat_mul (h₁ : w₁.1.val * w₂.1.val / Ray.RAY_SCALE < U128_MOD ):\n    |SignedRay.toRat (w₁ * w₂) - SignedRay.toRat w₁ * SignedRay.toRat w₂| < 1 / Ray.RAY_SCALE :=", "ground_truth_proof": ":= by\n  rcases w₁ with ⟨w₁, s₁⟩\n  rcases w₂ with ⟨w₂, s₂⟩\n  rcases s₁ with (⟨⟨⟩⟩|⟨⟨⟩⟩) <;> rcases s₂ with (⟨⟨⟩⟩|⟨⟨⟩⟩)\n    <;> dsimp only at h₁\n    <;> simp [mul_def, toRat, Ray.mul, Ray.toRat, Ray.toZMod, Nat.mod_eq_of_lt h₁, -one_div]\n    <;> rw [Rat.nat_cast_div_eq, Nat.cast_mul, ZMod.natCast_val, ZMod.natCast_val, mul_div_assoc,\n      sub_div, div_mul_eq_mul_div]\n    <;> [rw [sub_right_comm, sub_self, zero_sub, abs_neg];\n       rw [neg_sub, sub_add_cancel];\n       rw [neg_sub, sub_add_cancel];\n       rw [sub_right_comm, sub_self, zero_sub, abs_neg]]\n    <;> rw [abs_div,\n      Nat.abs_cast, div_lt_div_right Ray.RAY_SCALE_rat_pos, abs_div, Nat.abs_cast,\n      Nat.abs_cast, div_lt_one Ray.RAY_SCALE_rat_pos, Nat.cast_lt]\n    <;> apply Nat.mod_lt _ Ray.RAY_SCALE_pos", "nesting_depth": 2, "transitive_dep_count": 43, "subset_aristotle": false, "category": "Applied verif."}
{"id": 486, "thm_name": "Wad.toRat_mul", "thm_stmt": "theorem toRat_mul (h : w.toZMod.val * w'.toZMod.val / WAD_SCALE < U128_MOD) :\n    |(w * w').toRat - w.toRat * w'.toRat| < 1 / WAD_SCALE", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Int", "module": "Init.Data.Int.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}], "used_repo_defs": [{"name": "Wad.mul", "content": "protected def Wad.mul : Wad := (w.toZMod.val * w'.toZMod.val / WAD_SCALE : UInt128)"}, {"name": "", "content": "instance : Mul Wad := ⟨Wad.mul⟩"}], "lib_lemmas": [{"name": "Nat.cast_div", "module": "Mathlib.Data.Nat.Cast.Field"}, {"name": "Nat.cast_ne_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_sub", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "sub_div", "module": "Mathlib.Algebra.Field.Basic"}, {"name": "Int.natCast_natAbs", "module": "Mathlib.Algebra.Order.Group.Unbundled.Int"}, {"name": "Nat.abs_cast", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_mul", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "ZMod.natCast_val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val_natCast", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "abs_div", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "abs_neg", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "div_div", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_lt_one", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "div_mul_div_comm", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_sub_cancel_left", "module": "Mathlib.Algebra.Group.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}], "used_local_lemmas": [{"name": "Rat.nat_cast_div_eq", "content": "theorem Rat.nat_cast_div_eq {a b : ℕ} :\n    ↑(a / b) = (a : ℚ) / (b : ℚ) - ↑(a % b) / (b : ℚ)"}, {"name": "Wad.WAD_SCALE_pos", "content": "theorem WAD_SCALE_pos : 0 < WAD_SCALE"}, {"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}, {"name": "Wad.mul_def", "content": "protected theorem mul_def :\n    w * w' = (w.toZMod.val * w'.toZMod.val / WAD_SCALE  : UInt128)"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE", "target_theorem": "theorem toRat_mul (h : w.toZMod.val * w'.toZMod.val / WAD_SCALE < U128_MOD) :\n    |(w * w').toRat - w.toRat * w'.toRat| < 1 / WAD_SCALE :=", "ground_truth_proof": ":= by\n  simp only [Wad.toRat, Wad.toZMod, Wad.mul_def, Int.natCast_natAbs] at *\n  simp only [ZMod.val_natCast, ZMod.natCast_val] at *\n  rw [Nat.mod_eq_of_lt h, div_mul_div_comm, ← div_div, ← sub_div, abs_div,\n    Nat.abs_cast, div_lt_div_right WAD_SCALE_rat_pos, Rat.nat_cast_div_eq]\n  simp only [Nat.cast_mul, ZMod.natCast_val, sub_sub_cancel_left, abs_neg]\n  rw [abs_div, Nat.abs_cast, Nat.abs_cast, div_lt_one WAD_SCALE_rat_pos,\n    Nat.cast_lt]\n  apply Nat.mod_lt _ WAD_SCALE_pos", "nesting_depth": 2, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Applied verif."}
{"id": 487, "thm_name": "Ray.toRat_mul", "thm_stmt": "theorem toRat_mul (h : r.toZMod.val * r'.toZMod.val / RAY_SCALE < U128_MOD) :\n    |(r * r').toRat - r.toRat * r'.toRat| < 1 / RAY_SCALE", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}], "used_repo_defs": [{"name": "Ray.mul", "content": "protected def Ray.mul : Ray := (r.toZMod.val * r'.toZMod.val / RAY_SCALE : UInt128)"}, {"name": "", "content": "instance : Mul Ray := ⟨Ray.mul⟩"}], "lib_lemmas": [{"name": "Nat.cast_div", "module": "Mathlib.Data.Nat.Cast.Field"}, {"name": "Nat.cast_ne_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_sub", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "sub_div", "module": "Mathlib.Algebra.Field.Basic"}, {"name": "Nat.abs_cast", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_mul", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "ZMod.val_natCast", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "abs_div", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "abs_neg", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "div_div", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_lt_one", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "div_mul_div_comm", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_sub_cancel_left", "module": "Mathlib.Algebra.Group.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.RAY_SCALE", "content": "def RAY_SCALE : ℕ := 1000000000000000000000000000"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "Ray.toRat", "content": "protected def toRat : ℚ := r.toZMod.val / RAY_SCALE"}], "used_local_lemmas": [{"name": "Rat.nat_cast_div_eq", "content": "theorem Rat.nat_cast_div_eq {a b : ℕ} :\n    ↑(a / b) = (a : ℚ) / (b : ℚ) - ↑(a % b) / (b : ℚ)"}, {"name": "Ray.RAY_SCALE_pos", "content": "theorem RAY_SCALE_pos : 0 < RAY_SCALE"}, {"name": "Ray.RAY_SCALE_rat_pos", "content": "theorem RAY_SCALE_rat_pos : 0 < (RAY_SCALE : ℚ)"}, {"name": "Ray.mul_def", "content": "protected theorem mul_def :\n    r * r' = (r.toZMod.val * r'.toZMod.val / RAY_SCALE  : UInt128)"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\nnamespace Wad\n\nvariable (w w' : Wad)\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\ndef RAY_SCALE : ℕ := 1000000000000000000000000000\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nprotected def toRat : ℚ := r.toZMod.val / RAY_SCALE", "target_theorem": "theorem toRat_mul (h : r.toZMod.val * r'.toZMod.val / RAY_SCALE < U128_MOD) :\n    |(r * r').toRat - r.toRat * r'.toRat| < 1 / RAY_SCALE :=", "ground_truth_proof": ":= by\n  simp only [Ray.toRat, Ray.toZMod, Ray.mul_def] at *\n  simp only [ZMod.val_natCast]\n  rw [Nat.mod_eq_of_lt h, div_mul_div_comm, ← div_div, ← sub_div, abs_div,\n    Nat.abs_cast, div_lt_div_right RAY_SCALE_rat_pos, Rat.nat_cast_div_eq]\n  simp only [Nat.cast_mul, ZMod.val_natCast, sub_sub_cancel_left, abs_neg]\n  rw [abs_div, Nat.abs_cast, Nat.abs_cast, div_lt_one RAY_SCALE_rat_pos,\n    Nat.cast_lt]\n  apply Nat.mod_lt _ RAY_SCALE_pos", "nesting_depth": 2, "transitive_dep_count": 31, "subset_aristotle": false, "category": "Applied verif."}
{"id": 488, "thm_name": "SignedWad.toRat_div", "thm_stmt": "theorem toRat_div (h₁ : w₁.1.val * Wad.WAD_SCALE / w₂.1.val < U128_MOD)\n    (h₂ : w₂.1.val ≠ 0):\n    |SignedWad.toRat (w₁ / w₂) - SignedWad.toRat w₁ / SignedWad.toRat w₂| < 1 / Wad.WAD_SCALE", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Bool.toSierraBool", "module": "Aegis.Aux.Bool"}, {"name": "Bool.xor", "module": "Init.Data.Bool"}, {"name": "SierraBool.toBool", "module": "Aegis.Aux.Bool"}], "used_repo_defs": [{"name": "SignedWad.div", "content": "protected def SignedWad.div : SignedWad :=\n⟨Wad.div (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "", "content": "instance : Div SignedWad := ⟨SignedWad.div⟩"}], "lib_lemmas": [{"name": "Nat.cast_div", "module": "Mathlib.Data.Nat.Cast.Field"}, {"name": "Nat.cast_ne_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_sub", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "sub_div", "module": "Mathlib.Algebra.Field.Basic"}, {"name": "Nat.abs_cast", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_mul", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Nat.cast_zero", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.pos_of_ne_zero", "module": "Init.Data.Nat.Basic"}, {"name": "ZMod.natCast_val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "abs_div", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "abs_neg", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "div_div", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_div_div_cancel_right", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_lt_one", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "div_neg", "module": "Mathlib.Algebra.Ring.Basic"}, {"name": "mul_div_mul_right", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "neg_div", "module": "Mathlib.Algebra.Ring.Basic"}, {"name": "neg_sub", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_add_cancel", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_neg_eq_add", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_right_comm", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_self", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}, {"name": "Wad.div", "content": "protected def div : Wad := (w.toZMod.val * WAD_SCALE / w'.toZMod.val : UInt128)"}, {"name": "SignedWad", "content": "def SignedWad := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedWad.toRat", "content": "def toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1"}], "used_local_lemmas": [{"name": "Rat.nat_cast_div_eq", "content": "theorem Rat.nat_cast_div_eq {a b : ℕ} :\n    ↑(a / b) = (a : ℚ) / (b : ℚ) - ↑(a % b) / (b : ℚ)"}, {"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}, {"name": "Wad.WAD_SCALE_rat_ne_zero", "content": "theorem WAD_SCALE_rat_ne_zero : (WAD_SCALE : ℚ) ≠ 0"}, {"name": "SignedWad.div_def", "content": "theorem div_def :\n    w₁ / w₂ = ⟨Wad.div (w₁.1 : Wad) (w₂.1 : Wad), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE\n\nprotected def div : Wad := (w.toZMod.val * WAD_SCALE / w'.toZMod.val : UInt128)\n\nend Wad\n\nnamespace Ray\n\nvariable (r r' : Ray)\n\nend Ray\n\ndef SignedWad := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedWad\n\nvariable (w w₁ w₂ : SignedWad)\n\ndef toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1", "target_theorem": "theorem toRat_div (h₁ : w₁.1.val * Wad.WAD_SCALE / w₂.1.val < U128_MOD)\n    (h₂ : w₂.1.val ≠ 0):\n    |SignedWad.toRat (w₁ / w₂) - SignedWad.toRat w₁ / SignedWad.toRat w₂| < 1 / Wad.WAD_SCALE :=", "ground_truth_proof": ":= by\n  rcases w₁ with ⟨w₁, s₁⟩\n  rcases w₂ with ⟨w₂, s₂⟩\n  rcases s₁ with (⟨⟨⟩⟩|⟨⟨⟩⟩) <;> rcases s₂ with (⟨⟨⟩⟩|⟨⟨⟩⟩)\n    <;> dsimp only at h₁ h₂\n    <;> simp [div_def, toRat, Wad.div, Wad.toRat, Wad.toZMod, Nat.mod_eq_of_lt h₁, -one_div]\n    <;> rw [Rat.nat_cast_div_eq, Nat.cast_mul, ZMod.natCast_val, ZMod.natCast_val, sub_div]\n    <;> [rw [div_div_div_cancel_right _ Wad.WAD_SCALE_rat_ne_zero, div_div,\n         mul_div_mul_right _ _ Wad.WAD_SCALE_rat_ne_zero,\n         sub_right_comm, sub_self, zero_sub, abs_neg];\n       rw [neg_sub, div_neg, sub_neg_eq_add, div_div_div_cancel_right _ Wad.WAD_SCALE_rat_ne_zero,\n         div_div (w₁.cast * _), mul_div_mul_right _ _ Wad.WAD_SCALE_rat_ne_zero, sub_add_cancel];\n       rw [neg_sub, neg_div, sub_neg_eq_add, div_div_div_cancel_right _ Wad.WAD_SCALE_rat_ne_zero,\n         div_div (w₁.cast * _), mul_div_mul_right _ _ Wad.WAD_SCALE_rat_ne_zero, sub_add_cancel];\n       rw [div_div_div_cancel_right _ Wad.WAD_SCALE_rat_ne_zero, div_div,\n         mul_div_mul_right _ _ Wad.WAD_SCALE_rat_ne_zero,\n         sub_right_comm, sub_self, zero_sub, abs_neg]]\n    <;> rw [abs_div,\n      Nat.abs_cast, div_lt_div_right Wad.WAD_SCALE_rat_pos, abs_div, ← ZMod.natCast_val, Nat.abs_cast,\n      Nat.abs_cast, div_lt_one (by rw [← Nat.cast_zero, Nat.cast_lt]; apply Nat.pos_of_ne_zero h₂), Nat.cast_lt]\n    <;> apply Nat.mod_lt _ (Nat.pos_of_ne_zero h₂)", "nesting_depth": 2, "transitive_dep_count": 49, "subset_aristotle": false, "category": "Applied verif."}
{"id": 489, "thm_name": "SignedRay.toRat_div", "thm_stmt": "theorem toRat_div (h₁ : w₁.1.val * Ray.RAY_SCALE / w₂.1.val < U128_MOD)\n    (h₂ : w₂.1.val ≠ 0):\n    |SignedRay.toRat (w₁ / w₂) - SignedRay.toRat w₁ / SignedRay.toRat w₂| < 1 / Ray.RAY_SCALE", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "SierraBool.toBool", "module": "Aegis.Aux.Bool"}, {"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Bool", "module": "Init.Prelude"}, {"name": "Bool.toSierraBool", "module": "Aegis.Aux.Bool"}, {"name": "Bool.xor", "module": "Init.Data.Bool"}], "used_repo_defs": [{"name": "SignedRay.div", "content": "protected def SignedRay.div : SignedRay :=\n⟨Ray.div (w₁.1 : Ray) (w₂.1 : Ray), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}, {"name": "", "content": "instance : Div SignedRay := ⟨SignedRay.div⟩"}], "lib_lemmas": [{"name": "Nat.cast_div", "module": "Mathlib.Data.Nat.Cast.Field"}, {"name": "Nat.cast_ne_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_sub", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "sub_div", "module": "Mathlib.Algebra.Field.Basic"}, {"name": "Nat.abs_cast", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_mul", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Nat.cast_zero", "module": "Mathlib.Data.Nat.Cast.Defs"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.pos_of_ne_zero", "module": "Init.Data.Nat.Basic"}, {"name": "ZMod.natCast_val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "abs_div", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "abs_neg", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "div_div", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_div_div_cancel_right", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_lt_one", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "div_neg", "module": "Mathlib.Algebra.Ring.Basic"}, {"name": "mul_div_mul_right", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "neg_div", "module": "Mathlib.Algebra.Ring.Basic"}, {"name": "neg_sub", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "one_div", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_add_cancel", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "sub_neg_eq_add", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_right_comm", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "sub_self", "module": "Mathlib.Algebra.Group.Defs"}, {"name": "zero_sub", "module": "Mathlib.Algebra.Group.Defs"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.RAY_SCALE", "content": "def RAY_SCALE : ℕ := 1000000000000000000000000000"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "Ray.toRat", "content": "protected def toRat : ℚ := r.toZMod.val / RAY_SCALE"}, {"name": "Ray.div", "content": "protected def div : Ray := (r.toZMod.val * RAY_SCALE / r'.toZMod.val : UInt128)"}, {"name": "SignedRay", "content": "def SignedRay := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedRay.toRat", "content": "def toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1"}], "used_local_lemmas": [{"name": "Rat.nat_cast_div_eq", "content": "theorem Rat.nat_cast_div_eq {a b : ℕ} :\n    ↑(a / b) = (a : ℚ) / (b : ℚ) - ↑(a % b) / (b : ℚ)"}, {"name": "Ray.RAY_SCALE_rat_pos", "content": "theorem RAY_SCALE_rat_pos : 0 < (RAY_SCALE : ℚ)"}, {"name": "Ray.RAY_SCALE_rat_ne_zero", "content": "theorem RAY_SCALE_rat_ne_zero : (RAY_SCALE : ℚ) ≠ 0"}, {"name": "SignedRay.div_def", "content": "theorem div_def :\n    w₁ / w₂ = ⟨Ray.div (w₁.1 : Ray) (w₂.1 : Ray), Bool.toSierraBool (Bool.xor (SierraBool.toBool w₁.2) (SierraBool.toBool w₂.2))⟩"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\nnamespace Wad\n\nvariable (w w' : Wad)\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\ndef RAY_SCALE : ℕ := 1000000000000000000000000000\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nprotected def toRat : ℚ := r.toZMod.val / RAY_SCALE\n\nprotected def div : Ray := (r.toZMod.val * RAY_SCALE / r'.toZMod.val : UInt128)\n\nend Ray\n\nnamespace SignedWad\n\nvariable (w w₁ w₂ : SignedWad)\n\nend SignedWad\n\ndef SignedRay := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedRay\n\nvariable (w w₁ w₂ : SignedRay)\n\ndef toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1", "target_theorem": "theorem toRat_div (h₁ : w₁.1.val * Ray.RAY_SCALE / w₂.1.val < U128_MOD)\n    (h₂ : w₂.1.val ≠ 0):\n    |SignedRay.toRat (w₁ / w₂) - SignedRay.toRat w₁ / SignedRay.toRat w₂| < 1 / Ray.RAY_SCALE :=", "ground_truth_proof": ":= by\n  rcases w₁ with ⟨w₁, s₁⟩\n  rcases w₂ with ⟨w₂, s₂⟩\n  rcases s₁ with (⟨⟨⟩⟩|⟨⟨⟩⟩) <;> rcases s₂ with (⟨⟨⟩⟩|⟨⟨⟩⟩)\n    <;> dsimp only at h₁ h₂\n    <;> simp [div_def, toRat, Ray.div, Ray.toRat, Ray.toZMod, Nat.mod_eq_of_lt h₁, -one_div]\n    <;> rw [Rat.nat_cast_div_eq, Nat.cast_mul, ZMod.natCast_val, ZMod.natCast_val, sub_div]\n    <;> [rw [div_div_div_cancel_right _ Ray.RAY_SCALE_rat_ne_zero, div_div,\n         mul_div_mul_right _ _ Ray.RAY_SCALE_rat_ne_zero,\n         sub_right_comm, sub_self, zero_sub, abs_neg];\n       rw [neg_sub, div_neg, sub_neg_eq_add, div_div_div_cancel_right _ Ray.RAY_SCALE_rat_ne_zero,\n         div_div (w₁.cast * _), mul_div_mul_right _ _ Ray.RAY_SCALE_rat_ne_zero, sub_add_cancel];\n       rw [neg_sub, neg_div, sub_neg_eq_add, div_div_div_cancel_right _ Ray.RAY_SCALE_rat_ne_zero,\n         div_div (w₁.cast * _), mul_div_mul_right _ _ Ray.RAY_SCALE_rat_ne_zero, sub_add_cancel];\n       rw [div_div_div_cancel_right _ Ray.RAY_SCALE_rat_ne_zero, div_div,\n         mul_div_mul_right _ _ Ray.RAY_SCALE_rat_ne_zero,\n         sub_right_comm, sub_self, zero_sub, abs_neg]]\n    <;> rw [abs_div,\n      Nat.abs_cast, div_lt_div_right Ray.RAY_SCALE_rat_pos, abs_div, ← ZMod.natCast_val, Nat.abs_cast,\n      Nat.abs_cast, div_lt_one (by rw [← Nat.cast_zero, Nat.cast_lt]; apply Nat.pos_of_ne_zero h₂), Nat.cast_lt]\n    <;> apply Nat.mod_lt _ (Nat.pos_of_ne_zero h₂)", "nesting_depth": 2, "transitive_dep_count": 49, "subset_aristotle": false, "category": "Applied verif."}
{"id": 490, "thm_name": "Wad.toRat_nonneg", "thm_stmt": "theorem toRat_nonneg : 0 ≤ w.toRat", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "ZMod.cast", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_eq_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_nonneg", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "NeZero.ne", "module": "Init.Data.NeZero"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}], "used_local_lemmas": [{"name": "ZMod.cast_rat_nonneg", "content": "theorem ZMod.cast_rat_nonneg [NeZero n] (a : ZMod n) : 0 ≤ (a.cast : ℚ)"}, {"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE", "target_theorem": "theorem toRat_nonneg : 0 ≤ w.toRat :=", "ground_truth_proof": ":= by\n  simp [Wad.toRat]\n  rw [le_div_iff WAD_SCALE_rat_pos, zero_mul]\n  exact ZMod.cast_rat_nonneg (Wad.toZMod w)", "nesting_depth": 2, "transitive_dep_count": 14, "subset_aristotle": false, "category": "Applied verif."}
{"id": 491, "thm_name": "SignedWad.val_eq_of_toRat_eq", "thm_stmt": "theorem val_eq_of_toRat_eq : w₁.toRat = w₂.toRat → w₁.1 = w₂.1", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "ZMod.cast", "module": "Mathlib.Data.ZMod.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_nonneg", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "le_of_eq", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "ZMod.val_injective", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ne_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Nat.cast_eq_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "NeZero.ne", "module": "Init.Data.NeZero"}, {"name": "Left.neg_nonpos_iff", "module": "Mathlib.Algebra.Order.Group.Unbundled.Basic"}, {"name": "SierraBool_toBool_inl", "module": "Aegis.Aux.Bool"}, {"name": "SierraBool_toBool_inr", "module": "Aegis.Aux.Bool"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "le_antisymm", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "neg_inj", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_eq_neg", "module": "Mathlib.Algebra.Group.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}, {"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "SignedWad", "content": "def SignedWad := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedWad.toRat", "content": "def toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1"}], "used_local_lemmas": [{"name": "ZMod.cast_rat_nonneg", "content": "theorem ZMod.cast_rat_nonneg [NeZero n] (a : ZMod n) : 0 ≤ (a.cast : ℚ)"}, {"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}, {"name": "Wad.toRat_lt_toRat_of_val_lt_val", "content": "theorem toRat_lt_toRat_of_val_lt_val (h : @ZMod.val U128_MOD w < @ZMod.val U128_MOD w') :\n    w.toRat < w'.toRat"}, {"name": "Wad.toRat_injective", "content": "theorem toRat_injective : Function.Injective Wad.toRat"}, {"name": "Wad.toRat_nonneg", "content": "theorem toRat_nonneg : 0 ≤ w.toRat"}, {"name": "Wad.toRat_zero", "content": "@[simp]\ntheorem toRat_zero : (0 : Wad).toRat = 0"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nend Ray\n\ndef SignedWad := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedWad\n\nvariable (w w₁ w₂ : SignedWad)\n\ndef toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1", "target_theorem": "theorem val_eq_of_toRat_eq : w₁.toRat = w₂.toRat → w₁.1 = w₂.1 :=", "ground_truth_proof": ":= by\n  rcases w₁ with ⟨w₁, s₁⟩\n  rcases w₂ with ⟨w₂, s₂⟩\n  intro h\n  cases s₁ <;> cases s₂\n  · have := Wad.toRat_injective h\n    cases this\n    rfl\n  · simp only [toRat, SierraBool_toBool_inl, ite_false, SierraBool_toBool_inr, ite_true] at *\n    have h' : Wad.toRat w₁ = 0 := by\n      apply le_antisymm _ _\n      · rw [h, Left.neg_nonpos_iff]\n        apply Wad.toRat_nonneg\n      · apply Wad.toRat_nonneg\n    rw [h', zero_eq_neg, ← Wad.toRat_zero] at h\n    have := Wad.toRat_injective h; cases this\n    rw [← Wad.toRat_zero] at h'\n    have := Wad.toRat_injective h'; cases this\n    rfl\n  · simp only [toRat, SierraBool_toBool_inr, ite_true, SierraBool_toBool_inl, ite_false] at *\n    have h' : Wad.toRat w₂ = 0 := by\n      apply le_antisymm _ _\n      · rw [← h, Left.neg_nonpos_iff]\n        apply Wad.toRat_nonneg\n      · apply Wad.toRat_nonneg\n    rw [h', eq_comm, zero_eq_neg, ← Wad.toRat_zero] at h\n    have := Wad.toRat_injective h; cases this\n    rw [← Wad.toRat_zero] at h'\n    have := Wad.toRat_injective h'; cases this\n    rfl\n  · simp only [toRat, SierraBool_toBool_inr, ite_true, neg_inj] at h\n    have := Wad.toRat_injective h\n    cases this\n    rfl", "nesting_depth": 3, "transitive_dep_count": 36, "subset_aristotle": false, "category": "Applied verif."}
{"id": 492, "thm_name": "SignedRay.val_eq_of_toRat_eq", "thm_stmt": "theorem val_eq_of_toRat_eq : w₁.toRat = w₂.toRat → w₁.1 = w₂.1", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "SierraBool.toBool", "module": "Aegis.Aux.Bool"}, {"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "ZMod.cast", "module": "Mathlib.Data.ZMod.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_nonneg", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "le_of_eq", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "ZMod.val_injective", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ne_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "Nat.cast_eq_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "NeZero.ne", "module": "Init.Data.NeZero"}, {"name": "Left.neg_nonpos_iff", "module": "Mathlib.Algebra.Order.Group.Unbundled.Basic"}, {"name": "SierraBool_toBool_inl", "module": "Aegis.Aux.Bool"}, {"name": "SierraBool_toBool_inr", "module": "Aegis.Aux.Bool"}, {"name": "eq_comm", "module": "Init.Core"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}, {"name": "le_antisymm", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "neg_inj", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "zero_eq_neg", "module": "Mathlib.Algebra.Group.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.RAY_SCALE", "content": "def RAY_SCALE : ℕ := 1000000000000000000000000000"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "Ray.toRat", "content": "protected def toRat : ℚ := r.toZMod.val / RAY_SCALE"}, {"name": "SignedRay", "content": "def SignedRay := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedRay.toRat", "content": "def toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1"}], "used_local_lemmas": [{"name": "ZMod.cast_rat_nonneg", "content": "theorem ZMod.cast_rat_nonneg [NeZero n] (a : ZMod n) : 0 ≤ (a.cast : ℚ)"}, {"name": "Ray.RAY_SCALE_rat_pos", "content": "theorem RAY_SCALE_rat_pos : 0 < (RAY_SCALE : ℚ)"}, {"name": "Ray.toRat_lt_toRat_of_val_lt_val", "content": "theorem toRat_lt_toRat_of_val_lt_val (h : @ZMod.val U128_MOD r < @ZMod.val U128_MOD r') :\n    r.toRat < r'.toRat"}, {"name": "Ray.toRat_injective", "content": "theorem toRat_injective : Function.Injective Ray.toRat"}, {"name": "Ray.toRat_nonneg", "content": "theorem toRat_nonneg : 0 ≤ r.toRat"}, {"name": "Ray.toRat_zero", "content": " @[simp]\ntheorem toRat_zero : (0 : Ray).toRat = 0"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\ndef RAY_SCALE : ℕ := 1000000000000000000000000000\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nprotected def toRat : ℚ := r.toZMod.val / RAY_SCALE\n\nend Ray\n\nnamespace SignedWad\n\nvariable (w w₁ w₂ : SignedWad)\n\nend SignedWad\n\ndef SignedRay := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedRay\n\nvariable (w w₁ w₂ : SignedRay)\n\ndef toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1", "target_theorem": "theorem val_eq_of_toRat_eq : w₁.toRat = w₂.toRat → w₁.1 = w₂.1 :=", "ground_truth_proof": ":= by\n  rcases w₁ with ⟨w₁, s₁⟩\n  rcases w₂ with ⟨w₂, s₂⟩\n  intro h\n  cases s₁ <;> cases s₂\n  · have := Ray.toRat_injective h\n    cases this\n    rfl\n  · simp only [toRat, SierraBool_toBool_inl, ite_false, SierraBool_toBool_inr, ite_true] at *\n    have h' : Ray.toRat w₁ = 0 := by\n      apply le_antisymm _ _\n      · rw [h, Left.neg_nonpos_iff]\n        apply Ray.toRat_nonneg\n      · apply Ray.toRat_nonneg\n    rw [h', zero_eq_neg, ← Ray.toRat_zero] at h\n    have := Ray.toRat_injective h; cases this\n    rw [← Ray.toRat_zero] at h'\n    have := Ray.toRat_injective h'; cases this\n    rfl\n  · simp only [toRat, SierraBool_toBool_inr, ite_true, SierraBool_toBool_inl, ite_false] at *\n    have h' : Ray.toRat w₂ = 0 := by\n      apply le_antisymm _ _\n      · rw [← h, Left.neg_nonpos_iff]\n        apply Ray.toRat_nonneg\n      · apply Ray.toRat_nonneg\n    rw [h', eq_comm, zero_eq_neg, ← Ray.toRat_zero] at h\n    have := Ray.toRat_injective h; cases this\n    rw [← Ray.toRat_zero] at h'\n    have := Ray.toRat_injective h'; cases this\n    rfl\n  · simp only [toRat, SierraBool_toBool_inr, ite_true, neg_inj] at h\n    have := Ray.toRat_injective h\n    cases this\n    rfl", "nesting_depth": 3, "transitive_dep_count": 37, "subset_aristotle": false, "category": "Applied verif."}
{"id": 493, "thm_name": "Ray.toRat_nonneg", "thm_stmt": "theorem toRat_nonneg : 0 ≤ r.toRat", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "NeZero", "module": "Init.Data.NeZero"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "ZMod.cast", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_eq_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_nonneg", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "NeZero.ne", "module": "Init.Data.NeZero"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.RAY_SCALE", "content": "def RAY_SCALE : ℕ := 1000000000000000000000000000"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "Ray.toRat", "content": "protected def toRat : ℚ := r.toZMod.val / RAY_SCALE"}], "used_local_lemmas": [{"name": "ZMod.cast_rat_nonneg", "content": "theorem ZMod.cast_rat_nonneg [NeZero n] (a : ZMod n) : 0 ≤ (a.cast : ℚ)"}, {"name": "Ray.RAY_SCALE_rat_pos", "content": "theorem RAY_SCALE_rat_pos : 0 < (RAY_SCALE : ℚ)"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\nnamespace Wad\n\nvariable (w w' : Wad)\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\ndef RAY_SCALE : ℕ := 1000000000000000000000000000\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nprotected def toRat : ℚ := r.toZMod.val / RAY_SCALE", "target_theorem": "theorem toRat_nonneg : 0 ≤ r.toRat :=", "ground_truth_proof": ":= by\n  simp [Ray.toRat]\n  rw [le_div_iff RAY_SCALE_rat_pos, zero_mul]\n  exact ZMod.cast_rat_nonneg (Ray.toZMod r)", "nesting_depth": 2, "transitive_dep_count": 14, "subset_aristotle": false, "category": "Applied verif."}
{"id": 494, "thm_name": "Ray.toRat_div", "thm_stmt": "theorem toRat_div (h : r.toZMod.val * RAY_SCALE / r'.toZMod.val < U128_MOD)\n    (h' : r'.toZMod.val ≠ 0) :\n    |(r / r').toRat - r.toRat / r'.toRat| < 1 / RAY_SCALE", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "Nat", "module": "Init.Prelude"}, {"name": "Rat", "module": "Init.Data.Rat.Basic"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}], "used_repo_defs": [{"name": "Ray.div", "content": "protected def Ray.div : Ray := (r.toZMod.val * RAY_SCALE / r'.toZMod.val : UInt128)"}, {"name": "", "content": "instance : Div Ray := ⟨Ray.div⟩"}], "lib_lemmas": [{"name": "Nat.cast_div", "module": "Mathlib.Data.Nat.Cast.Field"}, {"name": "Nat.cast_ne_zero", "module": "Mathlib.Algebra.CharZero.Defs"}, {"name": "Nat.cast_sub", "module": "Mathlib.Data.Int.Cast.Basic"}, {"name": "Nat.div_eq_sub_mod_div", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.dvd_sub_mod", "module": "Init.Data.Nat.Lemmas"}, {"name": "Nat.mod_le", "module": "Init.Data.Nat.Div.Basic"}, {"name": "sub_div", "module": "Mathlib.Algebra.Field.Basic"}, {"name": "Nat.abs_cast", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_mul", "module": "Mathlib.Data.Nat.Cast.Basic"}, {"name": "Nat.cast_pos", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "Nat.mod_eq_of_lt", "module": "Init.Data.Nat.Div.Basic"}, {"name": "Nat.mod_lt", "module": "Init.Prelude"}, {"name": "Nat.pos_of_ne_zero", "module": "Init.Data.Nat.Basic"}, {"name": "ZMod.val_natCast", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "abs_div", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "abs_neg", "module": "Mathlib.Algebra.Order.Group.Unbundled.Abs"}, {"name": "div_div", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_div_div_cancel_right", "module": "Mathlib.Algebra.Group.Basic"}, {"name": "div_lt_one", "module": "Mathlib.Algebra.Order.Field.Basic"}, {"name": "mul_div_mul_right", "module": "Mathlib.Algebra.GroupWithZero.Units.Basic"}, {"name": "sub_sub_cancel_left", "module": "Mathlib.Algebra.Group.Basic"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.RAY_SCALE", "content": "def RAY_SCALE : ℕ := 1000000000000000000000000000"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "Ray.toRat", "content": "protected def toRat : ℚ := r.toZMod.val / RAY_SCALE"}], "used_local_lemmas": [{"name": "Rat.nat_cast_div_eq", "content": "theorem Rat.nat_cast_div_eq {a b : ℕ} :\n    ↑(a / b) = (a : ℚ) / (b : ℚ) - ↑(a % b) / (b : ℚ)"}, {"name": "Ray.RAY_SCALE_rat_pos", "content": "theorem RAY_SCALE_rat_pos : 0 < (RAY_SCALE : ℚ)"}, {"name": "Ray.RAY_SCALE_rat_ne_zero", "content": "theorem RAY_SCALE_rat_ne_zero : (RAY_SCALE : ℚ) ≠ 0"}, {"name": "Ray.div_def", "content": "protected theorem div_def :\n    r / r' = (r.toZMod.val * RAY_SCALE / r'.toZMod.val : UInt128)"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\nnamespace Wad\n\nvariable (w w' : Wad)\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\ndef RAY_SCALE : ℕ := 1000000000000000000000000000\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nprotected def toRat : ℚ := r.toZMod.val / RAY_SCALE", "target_theorem": "theorem toRat_div (h : r.toZMod.val * RAY_SCALE / r'.toZMod.val < U128_MOD)\n    (h' : r'.toZMod.val ≠ 0) :\n    |(r / r').toRat - r.toRat / r'.toRat| < 1 / RAY_SCALE :=", "ground_truth_proof": ":= by\n  have h'' : 0 < r'.toZMod.val := Nat.pos_of_ne_zero h'\n  have h''' : (0 : ℚ) < r'.toZMod.val := Nat.cast_pos.mpr h''\n  simp only [Ray.toRat, Ray.toZMod, Ray.div_def, ZMod.val_natCast] at *\n  rw [Nat.mod_eq_of_lt h, Rat.nat_cast_div_eq, sub_div, Nat.cast_mul,\n    div_div, mul_div_mul_right _ _ RAY_SCALE_rat_ne_zero,\n    div_div_div_cancel_right _ RAY_SCALE_rat_ne_zero, sub_sub_cancel_left,\n    abs_neg, abs_div, Nat.abs_cast, div_lt_div_right RAY_SCALE_rat_pos,\n    abs_div, Nat.abs_cast, Nat.abs_cast, div_lt_one h''', Nat.cast_lt]\n  apply Nat.mod_lt _ h''", "nesting_depth": 2, "transitive_dep_count": 34, "subset_aristotle": false, "category": "Applied verif."}
{"id": 495, "thm_name": "SignedWad.toRat_eq_zero_iff", "thm_stmt": "theorem toRat_eq_zero_iff : w.toRat = 0 ↔ w.1 = 0", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "ZMod.cast_rat_eq_zero_iff", "module": "CorelibVerification.Aux.ZMod"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}, {"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "Ray.add", "content": "protected def add : Ray := r.toZMod + r'.toZMod"}, {"name": "SignedWad", "content": "def SignedWad := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedWad.toRat", "content": "def toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1"}], "used_local_lemmas": [{"name": "Wad.WAD_SCALE_pos", "content": "theorem WAD_SCALE_pos : 0 < WAD_SCALE"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nprotected def add : Ray := r.toZMod + r'.toZMod\n\nend Ray\n\ndef SignedWad := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedWad\n\nvariable (w w₁ w₂ : SignedWad)\n\ndef toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1", "target_theorem": "theorem toRat_eq_zero_iff : w.toRat = 0 ↔ w.1 = 0 :=", "ground_truth_proof": ":= by\n  have := Wad.WAD_SCALE_pos\n  rcases w with ⟨w, (s|s)⟩ <;> cases s\n  <;> simp only [SignedWad.toRat, Wad.toRat, Wad.toZMod]\n  <;> aesop (add simp ZMod.cast_rat_eq_zero_iff)", "nesting_depth": 2, "transitive_dep_count": 11, "subset_aristotle": false, "category": "Applied verif."}
{"id": 496, "thm_name": "SignedRay.toRat_eq_zero_iff", "thm_stmt": "theorem toRat_eq_zero_iff : w.toRat = 0 ↔ w.1 = 0", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "SierraBool.toBool", "module": "Aegis.Aux.Bool"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "ZMod.cast_rat_eq_zero_iff", "module": "CorelibVerification.Aux.ZMod"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.RAY_SCALE", "content": "def RAY_SCALE : ℕ := 1000000000000000000000000000"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "Ray.toRat", "content": "protected def toRat : ℚ := r.toZMod.val / RAY_SCALE"}, {"name": "Ray.add", "content": "protected def add : Ray := r.toZMod + r'.toZMod"}, {"name": "SignedRay", "content": "def SignedRay := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedRay.toRat", "content": "def toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1"}], "used_local_lemmas": [{"name": "Ray.RAY_SCALE_pos", "content": "theorem RAY_SCALE_pos : 0 < RAY_SCALE"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\nnamespace Wad\n\nvariable (w w' : Wad)\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\ndef RAY_SCALE : ℕ := 1000000000000000000000000000\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nprotected def toRat : ℚ := r.toZMod.val / RAY_SCALE\n\nprotected def add : Ray := r.toZMod + r'.toZMod\n\nend Ray\n\nnamespace SignedWad\n\nvariable (w w₁ w₂ : SignedWad)\n\nend SignedWad\n\ndef SignedRay := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedRay\n\nvariable (w w₁ w₂ : SignedRay)\n\ndef toRat : ℚ := if SierraBool.toBool w.2 then -(Ray.toRat w.1) else Ray.toRat w.1", "target_theorem": "theorem toRat_eq_zero_iff : w.toRat = 0 ↔ w.1 = 0 :=", "ground_truth_proof": ":= by\n  have := Ray.RAY_SCALE_pos\n  rcases w with ⟨w, (s|s)⟩ <;> cases s\n  <;> simp only [SignedRay.toRat, Ray.toRat, Ray.toZMod]\n  <;> aesop (add simp ZMod.cast_rat_eq_zero_iff)", "nesting_depth": 3, "transitive_dep_count": 11, "subset_aristotle": false, "category": "Applied verif."}
{"id": 497, "thm_name": "SignedWad.val_eq_zero_of_toRat_neg", "thm_stmt": "theorem val_eq_zero_of_toRat_neg (x : Wad) (p q : Unit)\n  (h : SignedWad.toRat ((x, .inl p) : SignedWad) = SignedWad.toRat ((x, .inr q) : SignedWad)) :\n    x = 0", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_nonneg", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "le_of_eq", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "ZMod.val_injective", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ne_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "SierraBool_toBool_inl", "module": "Aegis.Aux.Bool"}, {"name": "SierraBool_toBool_inr", "module": "Aegis.Aux.Bool"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}, {"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "SignedWad", "content": "def SignedWad := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedWad.toRat", "content": "def toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1"}], "used_local_lemmas": [{"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}, {"name": "Wad.toRat_lt_toRat_of_val_lt_val", "content": "theorem toRat_lt_toRat_of_val_lt_val (h : @ZMod.val U128_MOD w < @ZMod.val U128_MOD w') :\n    w.toRat < w'.toRat"}, {"name": "Wad.toRat_injective", "content": "theorem toRat_injective : Function.Injective Wad.toRat"}, {"name": "Wad.toRat_zero", "content": "@[simp]\ntheorem toRat_zero : (0 : Wad).toRat = 0"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nend Ray\n\ndef SignedWad := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedWad\n\nvariable (w w₁ w₂ : SignedWad)\n\ndef toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1", "target_theorem": "theorem val_eq_zero_of_toRat_neg (x : Wad) (p q : Unit)\n  (h : SignedWad.toRat ((x, .inl p) : SignedWad) = SignedWad.toRat ((x, .inr q) : SignedWad)) :\n    x = 0 :=", "ground_truth_proof": ":= by\n  simp only [toRat, SierraBool_toBool_inl, ite_false, SierraBool_toBool_inr, ite_true,\n    eq_neg_self_iff] at h\n  rw [← Wad.toRat_zero] at h\n  exact Wad.toRat_injective h", "nesting_depth": 3, "transitive_dep_count": 26, "subset_aristotle": false, "category": "Applied verif."}
{"id": 498, "thm_name": "SignedWad.val_eq_zero_of_toRat_neg'", "thm_stmt": "theorem val_eq_zero_of_toRat_neg' (x : Wad) (p q : Unit)\n  (h : SignedWad.toRat ((x, .inr p) : SignedWad) = SignedWad.toRat ((x, .inl q) : SignedWad)) :\n    x = 0", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Unit", "module": "Init.Prelude"}, {"name": "Function.Injective", "module": "Init.Data.Function"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_nonneg", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "le_of_eq", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "ZMod.val_injective", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "ne_of_lt", "module": "Mathlib.Order.Defs.PartialOrder"}, {"name": "SierraBool_toBool_inl", "module": "Aegis.Aux.Bool"}, {"name": "SierraBool_toBool_inr", "module": "Aegis.Aux.Bool"}, {"name": "ite_false", "module": "Init.SimpLemmas"}, {"name": "ite_true", "module": "Init.SimpLemmas"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}, {"name": "Ray", "content": "def Ray : Type := UInt128"}, {"name": "Ray.toZMod", "content": "protected def toZMod : UInt128 := r"}, {"name": "SignedWad", "content": "def SignedWad := UInt128 × (Unit ⊕ Unit)"}, {"name": "SignedWad.toRat", "content": "def toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1"}], "used_local_lemmas": [{"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}, {"name": "Wad.toRat_lt_toRat_of_val_lt_val", "content": "theorem toRat_lt_toRat_of_val_lt_val (h : @ZMod.val U128_MOD w < @ZMod.val U128_MOD w') :\n    w.toRat < w'.toRat"}, {"name": "Wad.toRat_injective", "content": "theorem toRat_injective : Function.Injective Wad.toRat"}, {"name": "Wad.toRat_zero", "content": "@[simp]\ntheorem toRat_zero : (0 : Wad).toRat = 0"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE\n\nend Wad\n\ndef Ray : Type := UInt128\n\nnamespace Ray\n\nvariable (r r' : Ray)\n\nprotected def toZMod : UInt128 := r\n\nend Ray\n\ndef SignedWad := UInt128 × (Unit ⊕ Unit)\n\nnamespace SignedWad\n\nvariable (w w₁ w₂ : SignedWad)\n\ndef toRat : ℚ := if w.2 then -(Wad.toRat w.1) else Wad.toRat w.1", "target_theorem": "theorem val_eq_zero_of_toRat_neg' (x : Wad) (p q : Unit)\n  (h : SignedWad.toRat ((x, .inr p) : SignedWad) = SignedWad.toRat ((x, .inl q) : SignedWad)) :\n    x = 0 :=", "ground_truth_proof": ":= by\n  simp only [toRat, SierraBool_toBool_inr, ite_true, SierraBool_toBool_inl, ite_false,\n    neg_eq_self_iff] at h\n  rw [← Wad.toRat_zero] at h\n  exact Wad.toRat_injective h", "nesting_depth": 3, "transitive_dep_count": 26, "subset_aristotle": false, "category": "Applied verif."}
{"id": 499, "thm_name": "Wad.toRat_le_toRat_of_val_le_val", "thm_stmt": "theorem toRat_le_toRat_of_val_le_val (h : @ZMod.val U128_MOD w ≤ @ZMod.val U128_MOD w') :\n    w.toRat ≤ w'.toRat", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_le", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_nonneg", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "le_of_eq", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}], "used_local_lemmas": [{"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE", "target_theorem": "theorem toRat_le_toRat_of_val_le_val (h : @ZMod.val U128_MOD w ≤ @ZMod.val U128_MOD w') :\n    w.toRat ≤ w'.toRat :=", "ground_truth_proof": ":= by\n  simp only [Wad.toRat]\n  apply div_le_div\n  · exact Nat.cast_nonneg (ZMod.val (Wad.toZMod w'))\n  · rwa [Nat.cast_le]\n  · exact WAD_SCALE_rat_pos\n  · apply le_of_eq; rfl", "nesting_depth": 2, "transitive_dep_count": 12, "subset_aristotle": false, "category": "Applied verif."}
{"id": 500, "thm_name": "Wad.toRat_lt_toRat_of_val_lt_val", "thm_stmt": "theorem toRat_lt_toRat_of_val_lt_val (h : @ZMod.val U128_MOD w < @ZMod.val U128_MOD w') :\n    w.toRat < w'.toRat", "lean_root": "wadray_verification", "rel_path": "WadrayVerification/Aux.lean", "imports": ["import Aegis.Aux.Bool", "import CorelibVerification.Aux.ZMod", "import Aegis.Aux.ZMod.DivMod"], "used_lib_defs": [{"name": "Sierra.UInt128", "module": "Aegis.Types"}, {"name": "Sierra.U128_MOD", "module": "Aegis.Types"}, {"name": "ZMod", "module": "Mathlib.Data.ZMod.Defs"}, {"name": "ZMod.val", "module": "Mathlib.Data.ZMod.Basic"}, {"name": "Nat", "module": "Init.Prelude"}], "used_repo_defs": [{"name": "...", "content": "..."}], "lib_lemmas": [{"name": "Nat.cast_lt", "module": "Mathlib.Data.Nat.Cast.Order.Basic"}, {"name": "Nat.cast_nonneg", "module": "Mathlib.Data.Nat.Cast.Order.Ring"}, {"name": "le_of_eq", "module": "Mathlib.Order.Defs.PartialOrder"}], "repo_lemmas": [{"name": "List.getElem_append_left{α", "content": "theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\\n(l₁ ++ l₂)[i] = l₁[i]"}], "used_local_defs": [{"name": "Wad", "content": "def Wad : Type := UInt128"}, {"name": "Wad.WAD_SCALE", "content": "def WAD_SCALE : ℕ := 1000000000000000000"}, {"name": "Wad.toZMod", "content": "protected def toZMod : UInt128 := w"}, {"name": "Wad.toRat", "content": "protected def toRat : ℚ := w.toZMod.val / WAD_SCALE"}], "used_local_lemmas": [{"name": "Wad.WAD_SCALE_rat_pos", "content": "theorem WAD_SCALE_rat_pos : 0 < (WAD_SCALE : ℚ)"}], "local_ctx": "import CorelibVerification.Aux.ZMod\n\nimport Aegis.Aux.Bool\n\nimport Aegis.Aux.ZMod.DivMod\n\nopen Sierra\n\ndef Wad : Type := UInt128\n\nnamespace Wad\n\ndef WAD_SCALE : ℕ := 1000000000000000000\n\nvariable (w w' : Wad)\n\nprotected def toZMod : UInt128 := w\n\nprotected def toRat : ℚ := w.toZMod.val / WAD_SCALE", "target_theorem": "theorem toRat_lt_toRat_of_val_lt_val (h : @ZMod.val U128_MOD w < @ZMod.val U128_MOD w') :\n    w.toRat < w'.toRat :=", "ground_truth_proof": ":= by\n  simp only [Wad.toRat]\n  apply div_lt_div\n  · rwa [Nat.cast_lt, Wad.toZMod, Wad.toZMod]\n  · apply le_of_eq; rfl\n  · apply Nat.cast_nonneg\n  · exact WAD_SCALE_rat_pos", "nesting_depth": 2, "transitive_dep_count": 12, "subset_aristotle": false, "category": "Applied verif."}